Some Classes of Function Spaces, Their Properties, and Applications

Journal of Function Spaces and Applications

Guest Editors: Józef Banaś, Janusz Matkowski, Nelson Merentes, Manuel Pinto, and Jose Luis Sanchez Some Classes of Function Spaces, Their Properties, and Applications Journal of Function Spaces and Applications

Some Classes of Function Spaces, Their Properties, and Applications

This is a special issue published in “Journal of Function Spaces andlications.” App All articles are open access articles distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Editorial Board

John R. Akeroyd, USA Norimichi Hirano, Japan L.E. Persson, Sweden Gerassimos Barbatis, Greece Henryk Hudzik, Poland Adrian Petrusel, Romania Ismat Beg, Pakistan Pankaj Jain, India Konrad Podczeck, Austria Bjorn Birnir, USA Krzysztof Jarosz, USA JoseRodr´ ´ıguez, Spain Messaoud Bounkhel, Saudi Arabia Anna Kaminfhska, USA Natasha Samko, Portugal Huy Qui Bui, New Zealand H. Turgay Kaptanoglu, Turkey Carlo Sbordone, Italy Victor I. Burenkov, Italy Valentin Keyantuo, Puerto Rico Simone Secchi, Italy Jiecheng Chen, China Vakhtang M. Kokilashvili, Georgia Naseer Shahzad, Saudi Arabia Jaeyoung Chung, Korea Alois Kufner, Czech Republic Mitsuru Sugimoto, Japan Sompong Dhompongsa, Thailand David R. Larson, USA Rodolfo H. Torres, USA Luisa Di Piazza, Italy Yuri Latushkin, USA Wilfredo Urbina, USA Lars Diening, Germany Young Joo Lee, Republic of Korea Nikolai L. Vasilevski, Mexico Dragan Djordjevic, Serbia Hugo Leiva, Venezuela P. Veeramani, India Miroslav Engliˇs, Czech Republic Guozhen Lu, USA Igor E. Verbitsky, USA Jose A. Ezquerro, Spain Dag Lukkassen, Norway Dragan Vukotic, Spain Dashan Fan, USA Qiaozhen Ma, China Bruce A. Watson, South Africa Xiang Fang, USA Mark A. McKibben, USA Anthony Weston, USA Hans G. Feichtinger, Austria Mihail Megan, Romania Quanhua Xu, France Alberto Fiorenza, Italy Alfonso Montes-Rodriguez, Spain Gen-Qi Xu, China Ajda Foˇsner, Slovenia Dumitru Motreanu, France Dachun Yang, China Eva A. Gallardo Gutierrez,´ Spain Sivaram K. Narayan, USA Kari Ylinen, Finland Aurelian Gheondea, Turkey Renxing Ni, China Chengbo Zhai, China AntonioS.Granero,Spain Kasso A. Okoudjou, USA Ruhan Zhao, USA Yongsheng S. Han, USA Gestur Olafsson,´ USA Kehe Zhu, USA Seppo Hassi, Finland Josip E. Pecariˇ c,´ Croatia William P. Ziemer, USA Stanislav Hencl, Czech Republic Jose´ Afh´ Pelaez,´ Spain Contents

Some Classes of Function Spaces, Their Properties, and Applications,Jozef´ Bana´s, Janusz Matkowski, Nelson Merentes, Manuel Pinto, and Jose Luis Sanchez Volume 2013, Article ID 360980, 3 pages

Sensitivity Analysis for Nonlinear Set-Valued Variational Equations in Banach Framework, A. Farajzadeh and Salahuddin Volume 2013, Article ID 258543, 6 pages

Nonlinear Kato Class and Unique Continuation of Eigenfunctions for 𝑝-Laplacian Operator, ReneErl´ ´ın Castillo and Julio C. Ramos Fernandez´ Volume 2013, Article ID 512050, 7 pages

Generalized Lorentz Spaces and Applications, Hatem Mejjaoli Volume 2013, Article ID 302941, 14 pages

Density in Spaces of Interpolation by Hankel Translates of a Basis Function, Cristian Arteaga and Isabel Marrero Volume 2013, Article ID 813502, 9 pages

2 On a New Space 𝑚 (𝑀,𝐴,𝜙,𝑝)of Double Sequences,CenapDuyarandOguz˘ Ogur˘ Volume 2013, Article ID 509613, 8 pages

Schauder-Tychonoff Fixed-Point Theorem in Theory of Superconductivity, Mariusz Gil and Stanisław We¸drychowicz Volume 2013, Article ID 692879, 12 pages

𝑠 Frequency-Uniform Decomposition, Function Spaces 𝑋𝑝,𝑞, and Applications to Nonlinear Evolution Equations, Shaolei Ru and Jiecheng Chen Volume 2013, Article ID 176596, 12 pages The Use of an Isometric Isomorphism on the Completion of the Space of Henstock-Kurzweil Integrable Functions,LuisAngel´ Gutierrez´ Mendez,´ Juan Alberto Escamilla Reyna, Francisco Javier Mendoza Torres, and Mar´ıa Guadalupe Morales Mac´ıas Volume 2013, Article ID 715789, 5 pages

Generalized Virtually Stable Maps and Their Associated Sequences,P.Chaoha,S.Iampiboonvatana, and J. Intrakul Volume 2013, Article ID 237858, 8 pages Asymptotics of the Eigenvalues of a Self-Adjoint Fourth Order Boundary Value Problem with Four Eigenvalue Parameter Dependent Boundary Conditions,ManfredMoller¨ and Bertin Zinsou Volume 2013, Article ID 280970, 8 pages

The Uniqueness of Strong Solutions for the Camassa-Holm Equation,MengWuandChongLai Volume 2013, Article ID 409760, 7 pages

On the Hermite-Hadamard Inequality and Other Integral Inequalities Involving Several Functions, Banyat Sroysang Volume 2013, Article ID 921828, 6 pages Positive Solutions for Some Competitive Fractional Systems in Bounded Domains,ImedBachar, Habib Maagli,ˆ and Noureddine Zeddini Volume 2013, Article ID 140130, 6 pages

Commutators of Higher Order Riesz Transform Associated with Schrodinger¨ Operators,YuLiu, Lijuan Wang, and Jianfeng Dong Volume 2013, Article ID 842375, 15 pages

On the Space of Functions with Growths Tempered by a Modulus of Continuity and Its Applications, Jozef´ Bana´sandRafałNalepa Volume 2013, Article ID 820437, 13 pages

Boundary Value Problems for a Class of Sequential Integrodifferential Equations of Fractional Order, Bashir Ahmad and Juan J. Nieto Volume 2013, Article ID 149659, 8 pages

Concerning Asymptotic Behavior for Extremal Polynomials Associated to Nondiagonal Sobolev Norms, Ana Portilla, Yamilet Quintana, JoseM.Rodr´ ´ıguez, and Eva Tour´ıs Volume 2013, Article ID 628031, 11 pages

Functions of Bounded 𝜅𝜑-Variation in the Sense of Riesz-Korenblum, Mariela Castillo, Sergio Rivas, Mar´ıa Sanoja, and Ivan´ Zea Volume 2013, Article ID 718507, 12 pages

Homogeneous Triebel-Lizorkin Spaces on Stratified Lie Groups,GuorongHu Volume 2013, Article ID 475103, 16 pages

A Note on Weighted Besov-Type and Triebel-Lizorkin-Type Spaces, Canqin Tang Volume 2013, Article ID 865835, 12 pages

The Space of Continuous Periodic Functions Is a Set of First Category in 𝐴𝑃(𝑋), Zhe-Ming Zheng, Hui-Sheng Ding, and Gaston M. N’Guer´ ekata´ Volume 2013, Article ID 275702, 3 pages Hindawi Publishing Corporation Journal of Function Spaces and Applications Volume 2013, Article ID 360980, 3 pages http://dx.doi.org/10.1155/2013/360980

Editorial Some Classes of Function Spaces, Their Properties, and Applications

Józef BanaV,1 Janusz Matkowski,2 Nelson Merentes,3 Manuel Pinto,4 and Jose Luis Sanchez3

1 Department of Mathematics, Rzeszow´ University of Technology, al. Powstanc´ ow´ Warszawy 8, 35-959 Rzeszow,´ Poland 2 Division of Functional Equations, Zielona GoraUniversity,ul.Prof.Z.Szafrana4a,65-516ZielonaG´ ora,´ Poland 3 Department of Mathematics, Central University of Venezuela, Paseo Los Ilustres, Urb. Valle Abajo, Apartado Postal 20513, Caracas 1020-A, Venezuela 4 Department of Mathematics, University of Chile, Casilla Postal 653, Santiago, Chile

Correspondence should be addressed to Jozef´ Bana´s; [email protected]

Received 10 October 2013; Accepted 10 October 2013

Copyright © 2013 Jozef´ Bana´s et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Function spaces create the basis of almost all investigations integral, and so on). We describe below the results contained in several branches of mathematics such as functional anal- in the papers published in this special issue. ysis, nonlinear analysis, operator theory, and the theories At the beginning we present eight papers which are of differential and integral equations, among others. Those mainly devoted to describe some function spaces and their spaces are frequently an object of the intensive study, in which various properties. properties of the mentioned function spaces are considered In the paper of C. Tang a weighted Besov-type space and described. First of all, such an approach is presented andweightedTriebel-Lizorkin-typespaceareintroduced. in functional analysis. But some topics of nonlinear analysis Moreover, the author obtained some characterization of those and operator theory are also closely related to the study of spaces expressed in terms of the so-called 𝜑-transforms. function spaces and their properties. The paper of G. Hu is dedicated to some topics of homoge- The second direction of investigations connected with neous Triebel-Lizorkin spaces with full range of parameters. the theory of function spaces depends on the application ThesespacesareintroducedonstratifiedLiegroupsinterms of that theory in the study of equations of various type, of Littlewood-Paley-type decomposition. The main result of such as ordinary and partial differential equations, integral the paper asserts that the scale of the considered Triebel- equations, functional differential, functional integral and Lizorkin spaces is independent of the choice of Littlewood- functional equations, operator equations, and so forth. The Paley-type decomposition and the sub-Laplacian used for the study of the mentioned equations requires to place considera- construction of the decomposition. tions in some function space. Consequently, all investigations The next paper written by H. Mejjaoli discusses the associated with a considered equation are closely linked with Lorentz spaces associated with the Dunkl operators on the a function space in which that equation is treated. space 𝑅4. Estimates of the Strichartz type for the Dunkl- Thisspecialissueisdevotedtodiscussalotoffacts Schrodinger¨ equations are obtained. The author considers connected with the theory of function spaces. We study some also Sobolev inequalities between the homogeneous Dunkl- properties of those spaces and we consider those aspects Besov spaces and Lorentz spaces. C. Duyar and O. Ogur˘ of functions spaces which are essential in miscellaneous introduced in their paper a new space of double sequences applications. The papers published in this issue describe related to 𝑝-absolutely convergent double sequence space, numerous spaces and indicate the applicability of those where 1¬𝑝 <. 1 The main tools used in the construction spaces in the study of several operator equations (differential, of that space are an Orlicz function and an infinite double 2 Journal of Function Spaces and Applications matrix. Some properties of the introduced space are estab- ThepaperofJ.BanasandR.Nalepadiscussesthefunction lished. For example, it is proved that the mentioned space is space consisting of functions having increments tempered by a paranormed space with a suitably defined paranorm. agivenmodulusofcontinuity.Suchaspaceisageneralization The paper of M. Castillo et al. is concerning the space of of the classical spaces of Lipschitz or Holder¨ continuous functions with bounded 𝜅𝜑-variation in the sense of Riesz- functions. A criterion for relative compactness in that space Korenblum. It is shown that the mentioned space creates a is established. To show the applicability of that criterion, the Banach space with an appropriate norm. The main result author proved the solvability of certain quadratic integral asserts that any uniformly bounded composition operator on equation of Fredholm type in the space of Holder¨ continuous that space satisfies the so-called Matkowski weak condition. functions. ThepaperofL.A.G.Mendez´ et al. contains a few results Now, we describe the papers contained in this spacial concerning the completion of the space of functions which issue which contain mainly applications of function spaces are Henstock-Kurzweil integrable. The authors consider that in the study of various operator equations. The first paper in space with the so-called Alexiewicz norm. Moreover, it is this direction, written by M. Moller¨ and B. Zinsou, contains shown that this space has several other properties expressed considerations connected with a boundary value problem for in terms of Radon-Pettis property, Radon-Riesz property, and a fourth order ordinary differential equation depending on isometrical isomorphism, among others. an eigenvalue parameter. Moreover, the formulated boundary P.Chaoha,S.Iampiboonvatana,andJ.Intrakuldiscussin conditions depend linearly upon that parameter. The local- their paper the concept of the virtual stability of continuous ization of eigenvalues is investigated and the first four terms self-mappings. That concept is generalized to arbitrary self- oftheasymptoticexpansionoftheeigenvaluesarefound.The mappings. The structure of sequences connected with uni- considerationsofthepaperareplacedinsomeSobolevspace. formly virtually stable self-mappings is studied. A necessary In the paper of A. Farajzadeh and Salahuddin the study and sufficient condition for a uniform virtual stability of the of the sensitivity analysis for nonlinear set valued variational space of the mentioned sequences is also obtained. The last equations is conducted. The resolvent operator technique is paper belonging to the above announced group is due to Z. the main tool applied in that paper. The authors obtained a M. Zheng et al. In that paper the author showed that the space lot of results in the paper in question. of continuous periodic functions is the set of first category in S. Ru and J. Chen described in their paper a new the space of almost periodic functions. Moreover, it is shown class of function spaces with the help of frequency-uniform that the space of almost periodic functions is the set of first decomposition of a Lebesgue space. The authors considered category in the space of almost automorphic functions. also the Cauchy problem for Ginzburg-Landau equation in Next part of the papers which we would like to describe is the mentioned function spaces. composed of six papers. Those papers discuss various aspects Using classical Schauder-Tychonoff fixed point princi- of function spaces which are connected with some other pleM.GilandS.Wędrychowiczstudysolutionsofthe topics expressed via operators, polynomials, interpolation, time-dependent Ginzburg-Landau equation. Moreover, they and so forth. Moreover, some applications of the discussed characterized those solutions in terms of local and global function spaces are also indicated. We start with the paper attractivity. Investigations of the paper are conducted in some of A. Portilla et al. which describes asymptotic behaviour Frechet´ and Sobolev spaces. of extremal polynomials (with complex coefficients and The uniqueness of strong solutions of the Cauchy problem equipped with a nondiagonal Sobolev norm) and the loca- for the Camassa-Holm equation is studied in the paper of tion of zeros of those polynomials under some hypotheses M. Wu and C. Lai. That equation is considered in a suitable imposed on the diagonal matrix which is obtained via the Sobolev space. The authors gave a new proof of the mentioned unitary factorization. uniqueness result with the help of several auxiliary lemmas. Basic properties of nonlinear Kato class are established In the paper of I. Bachar et al. an existence theorem for a inthepaperofR.E.CastilloandJ.C.RamosFernandez.´ competitive fractional system is proved. The main tool used The authors showed the strong continuation property of the in the proof is the Schauder fixed point theorem. Apart of eigenfunctions for the 𝑝-Laplacian operator defined on some the existence of solutions the authors characterized those Kato class. solutions in terms of global behaviour. The considerations are Schrodinger¨ operators are the main topic discussed in the located in the classical space 𝐶1,1(𝐷),where𝐷 is a bounded paperofY.Liuetal.Itisshown,amongothers,thatcom- subset of 𝑅𝑛. mutators of higher order associated with some Schrodinger¨ A boundary value problem for a class of sequential operators are bounded if those operators act between a Hardy integrodifferential equations of fractional order is considered space and a Lebesgue space with the weak topology. by B. Ahmad and J. J. Nieto. The authors proved the exis- ThepaperofB.Sroysangconcernswithsomeinequalities tence result concerning the mentioned equations using the of Hermite-Hadamard type. The author proved also some standard tools of the fixed point theory. The considerations integral inequalities involving several functions. C. Arteaga are conducted in the classical Banach space consisting of real and I. Marrero discuss a function space arising in the theory functions defined and continuous on a given segment [𝑎,] 𝑏 of interpolation by Hankel translates of a basis function. The and endowed with supremum norm. main result ensures that some Zemanian spaces are dense in The authors of the papers included in this special issue the mentioned function space. Several other nontrivial results represent fourteen countries: China, Colombia, India, Iran, are obtained in this paper. Japan, Mexico, Poland, Saudi Arabia, South Africa, Spain, Journal of Function Spaces and Applications 3

Thailand, Turkey, USA, and Venezuela. Those countries are placed on five continents: Africa, North and South America, Asia, and Europe. This shows that this special issue includes the outcomes of the research work conducted over the whole world.

Acknowledgments The guest editors of this special issue would like to express their immense gratitude to the authors of all papers submitted for considerations. We hope that the results included in this issue will be an inspiration for researchers working in the theory of function spaces and nonlinear analysis. Jozef´ Bana´s Janusz Matkowski Nelson Merentes Manuel Pinto Jose Luis Sanchez Hindawi Publishing Corporation Journal of Function Spaces and Applications Volume 2013, Article ID 258543, 6 pages http://dx.doi.org/10.1155/2013/258543

Research Article Sensitivity Analysis for Nonlinear Set-Valued Variational Equations in Banach Framework

A. Farajzadeh1 and Salahuddin2

1 Department of Mathematics, Razi University, Kermanshah 67149, Iran 2 Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India

Correspondence should be addressed to Salahuddin; dr [email protected]

Received 2 May 2013; Accepted 30 August 2013

Academic Editor: Nelson Merentes

Copyright © 2013 A. Farajzadeh and Salahuddin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The main purpose of this paper is to study the sensitivity analysis for nonlinear set-valued variational equations basedon (𝐴, 𝜂)- resolvent operator technique. The obtained results encompass a broad model of results.

1. Introduction of all the nonempty subset of 𝑋 and CB(𝑋)thefamilyof all nonempty closed bounded subsets of 𝑋. The generalized The set-valued inclusions problem, which was initiated by 𝑋∗ duality mapping 𝐽𝑞 :𝑋 → 2 is defined by Di Bella [1]andHuangetal.[2, 3], is a useful extension of the mathematical analysis and variational equation and 𝐽𝑞 (𝑥) is important context in the set-valued equations. Verma [4], ∗ ∗ ∗ 𝑞 󵄩 ∗󵄩 𝑞−1 Huang [5], Fang and Huang [6], Lan et al. [7], Khan and ={𝑓 ∈𝑋 :⟨𝑢,𝑓 ⟩=‖𝑢‖ , 󵄩𝑓 󵄩 = ‖𝑢‖ }, ∀𝑢∈𝑋, Salahuddin [8, 9], Li et al. [10], and Yen and Lee [11]intro- (1) duced the concepts of 𝐴-monotone, 𝐻-monotone, (𝐴, 𝜂)- 𝑞>1 monotone operator, (𝐴, 𝜂)-accretive mappings, and resolvent where is constant. 𝑋 𝜌 : operator associated with them, respectively. Dafermos [12] The modulus of smoothness of is the function 𝑋 [0, ∞) → [0, ∞) studied the sensitivity property of solutions of particular defined by kinds of variational inequality on parameter which takes 1 𝐾 𝜌 (𝑡) = { (‖𝑢+V‖ + ‖𝑢−V‖) −1:‖𝑢‖ ≤1,‖V‖ <𝑡} . values on an open subset of Euclidean space 𝑅 .Tobin 𝑋 sup 2 [13], Verma [14, 15], Lee and Salahuddin [3], Kyparisis [16], (2) Moudafi [17], Noor [18], Robinson [19], Yen and Lee [11], and ABanachspace𝑋 is called uniformly smooth if Hussain et al. [20] studied the sensitivity analysis of various types of variational inequalities. 𝜌𝑋 (𝑡) lim =0. (3) In this paper, we present the sensitivity analysis for (𝐴, 𝜂)- 𝑡→0 𝑡 (𝐴, 𝜂) accretive variational equations based on the -resolvent 𝑋 is called 𝑞-uniformly smooth if there exists a constant 𝑐>0 operator technique. The obtained results generalize a wide such that range of results on the sensitivity analysis for nonlinear set- 𝑞 valued variational equations in Banach spaces. 𝜌𝑋 (𝑡) ≤𝑐𝑡 (𝑞 > 1). (4) Let 𝜂:𝑋×𝑋→be 𝑋 a single-valued mapping. The map 2. Preliminaries 𝜂 is called 𝜏-Lipschitz continuous if there exists a constant 𝜏>0 ∗ such that Let 𝑋 be a real Banach space with dual space 𝑋 , and let ⟨⋅, ⋅⟩ 󵄩 󵄩 ∗ 𝑋 󵄩𝜂 (𝑢, V)󵄩 ≤𝜏‖𝑢−V‖ ,∀𝑢,V ∈𝑋. be the dual pair between 𝑋 and 𝑋 , 2 denote the family 󵄩 󵄩 (5) 2 Journal of Function Spaces and Applications

Definition 1. Let 𝜂 : 𝑋×𝑋 →𝑋be a single-valued mapping, Definition 3. A single-valued mapping 𝑁 : 𝑋×𝑋 →𝑋is 𝑋 and let 𝑀:𝑋 →2 be a mapping on 𝑋.Aset-valued said to be mapping 𝑀 is said to be (i) (𝜇, 𝜐)-Lipschitz continuous if there exist constants (i) accretive if 𝜇, 𝜐 >0 such that ∗ ∗ 󵄩 󵄩 ⟨𝑢 − V ,𝐽𝑞 (𝑢−V)⟩≥0, 󵄩𝑁(𝑥1,𝑦1)−𝑁(𝑥2,𝑦2)󵄩 (6) ∗ ∗ 󵄩 󵄩 󵄩 󵄩 ∀𝑢, V ∈𝑋, 𝑢 ∈𝑀(𝑢) , V ∈𝑀(V) ; ≤𝜇󵄩𝑥1 −𝑥2󵄩 +𝜐󵄩𝑦1 −𝑦2󵄩 ,∀𝑥𝑖,𝑦𝑖 ∈𝑋,𝑖=1,2; (16) (ii) 𝜂-accretive if (𝜑, 𝜅) −𝑄 𝐴𝑄 ∗ ∗ (ii) -relaxed cocoercive with respect to in ⟨𝑢 − V ,𝐽𝑞 (𝜂 (𝑢, V))⟩≥0, the second argument if there exists constant 𝜑, 𝜅 >0 (7) and for all 𝑥𝑖 ∈𝑋, 𝑦𝑖 ∈𝑄(𝑢𝑖), (𝑖 = 1, 2) such that ∀𝑢, V ∈𝑋,∗ 𝑢 ∈𝑀(𝑢) , V∗ ∈𝑀(V) ;

⟨𝑁 1(𝑦 ,⋅)−𝑁(𝑦2,⋅),𝐽𝑞 (𝐴 1(𝑦 )−𝐴(𝑦2))⟩ (iii) (𝑟, 𝜂)-strongly accretive if (17) 󵄩 󵄩𝑞 󵄩 󵄩𝑞 ≥𝜑󵄩𝑁(𝑦 ,⋅)−𝑁(𝑦 ,⋅)󵄩 +𝜅󵄩𝑥 −𝑥 󵄩 , ∗ ∗ 𝑞 󵄩 1 2 󵄩 󵄩 1 2󵄩 ⟨𝑢 − V ,𝐽𝑞 (𝜂 (𝑢, V))⟩≥𝑟‖𝑢−V‖ , (8) 𝐴, 𝑄 : 𝑋 →𝑋 ∗ ∗ where are single-valued mappings. ∀(𝑢,𝑢 ),(V, V )∈Graph (𝑀) ; 𝑋 Definition 4. Amapping𝑀:𝑋 →2 is said to be maximal (iv) 𝜂-pseudomonotone if (𝑚, 𝜂)-relaxed accretive if ∗ 𝑀 (𝑀, 𝜂) ⟨V ,𝐽𝑞 (𝜂 (𝑢, V))⟩≥0, (9) (i) is -accretive; ∗ (ii) for (𝑢, 𝑢 )∈𝑋×𝑋and implying ∗ ∗ ∗ ∗ ∗ ⟨𝑢 − V ,𝐽𝑞 (𝜂 (𝑢, V))⟩ ⟨𝑢 ,𝐽𝑞 (𝜂 (𝑢, V))⟩≥0, ∀(𝑢,𝑢),(V, V )∈Graph (𝑀) ; (18) 𝑞 ∗ (10) ≥ (−𝑚) ‖𝑢−V‖ ,∀(V, V )∈Graph (𝑀) ,

∗ (v) (𝑚, 𝜂)-relaxed accretive if there exists a positive con- one has 𝑢 ∈ 𝑀(𝑢). stant 𝑚 such that ∗ ∗ 𝑞 Definition 5. Let 𝐴:𝑋and →𝑋 𝜂:𝑋×𝑋→be 𝑋 two ⟨𝑢 − V ,𝐽 (𝜂 (𝑢, V))⟩≥(−𝑚) ‖𝑢−V‖ , 𝑋 𝑞 single-valued mappings; the mapping 𝑀:𝑋 →2 is said to (11) ∗ ∗ be (𝐴, 𝜂)-accretive if ∀(𝑢,𝑢 ),(V, V )∈Graph (𝑀) . (i) 𝑀 is (𝑀, 𝜂)-relaxed accretive; 𝐴:𝑋 →𝑋 Definition 2. A single-valued mapping is said to (ii) 𝑅(𝐴 + 𝜌𝑀) =𝑋 for 𝜌>0. be 𝑋 Note that alternatively, the mapping 𝑀:𝑋 →2 is said to (i) accretive if be (𝐴, 𝜂)-accretive if ⟨𝐴 (𝑢) −𝐴(V) ,𝐽𝑞 (𝑢−V)⟩ ≥ 0, ∀𝑢, V ∈𝑋; (12) (i) 𝑀 is (𝑀, 𝜂)-relaxed accretive; 𝐴+𝜌𝑀 𝜂 𝜌>0 (ii) strictly accretive if 𝐴 is accretive and (ii) is -pseudoaccretive for . The following propositions will be needed in the sequel. ⟨𝐴 (𝑢) −𝐴(V) ,𝐽 (𝑢−V)⟩=0, 𝑢=V,∀𝑢,V ∈𝑋; 𝑞 iff For more details one can refer to [7, 10, 15]. (13) Proposition 6. Let 𝐴:𝑋be →𝑋 an (𝑟, 𝜂)-strongly accretive 𝑋 (iii) (𝑟, 𝜂)-strongly accretive if there exists a constant 𝑟>0 single-valued mapping, and let 𝑀:𝑋 →2 be an (𝐴, 𝜂)- such that accretive mapping. Let 𝜂:𝑋×𝑋→be 𝑋 𝜏-Lipschitz continuous single-valued mapping. Then, 𝑀 is maximal (𝑚, 𝜂)- ⟨𝐴 (𝑢) −𝐴(V) ,𝐽 (𝜂 (𝑢, V))⟩≥𝑟𝑢−V 𝑞,∀𝑢,V ∈𝑋; 𝑞 ‖ ‖ relaxed accretive, and (𝐴+𝜌𝑀)𝑋=𝑋for 0 < 𝜌 < 𝑟/𝑚. (14) Proposition 7. Let 𝐴:𝑋be →𝑋 an (𝑟, 𝜂)-strongly accretive 𝑋 (iv) 𝛼-Lipschitz continuous if there exists a constant 𝛼>0 single-valued mapping, and let 𝑀:𝑋 →2 be an (𝐴, 𝜂)- such that accretive mapping. In addition, let 𝜂 : 𝑋×𝑋 →𝑋be 𝜏- Lipschitz continuous. Then (𝐴 + 𝜌𝑀) is maximal 𝜂-accretive ‖𝐴 (𝑢) −𝐴(V)‖ ≥𝛼‖𝑢−V‖ ,∀𝑢,V ∈𝑋. (15) for 0 < 𝜌 < 𝑟/𝑚. Journal of Function Spaces and Applications 3

𝑞−1 Proposition 8. Let 𝐴:𝑋be →𝑋 an (𝑟, 𝜂)-strongly accretive is (𝜏 /(𝑟 − 𝜌𝑚))-Lipschitz continuous, where 𝜌 ∈ (0, 𝑟/𝑚); 𝑋 mapping, and let 𝑀:𝑋 →2 be an (𝐴, 𝜂)-accretive that is, mapping. If, in addition, 𝜂:𝑋×𝑋→is 𝑋 𝜏-Lipschitz 󵄩 󵄩 −1 󵄩𝑅𝜂,𝜌,𝐴 (𝑢) −𝑅𝜂,𝜌,𝐴 (V)󵄩 continuous, then, the operator (𝐴 + 𝜌𝑀) is single-valued for 󵄩 𝑀(⋅,𝑤,𝜆) 𝑀(⋅,𝑤,𝜆) 󵄩 0 < 𝜌 < 𝑟/𝑚. 𝜏𝑞−1 (24) ≤ ‖𝑢−V‖ ,∀𝑢,V,𝑤∈𝑋. Definition 9. Let 𝜂 : 𝑋×𝑋 →𝑋be a single-valued mapping. 𝑟−𝜌𝑚 Let 𝐴:𝑋be →𝑋 an (𝑟, 𝜂)-strongly accretive mapping, and 𝑋 let 𝑀:𝑋 →2 be an (𝐴, 𝜂)-accretive mapping. Then, the Lemma 12 (see [7, 10, 15]). Let 𝑋 be a real uniformly smooth 𝑅𝜂,𝜌,𝐴 :𝑋 → 𝑋 Banach spaces, let 𝐴:𝑋be →𝑋 the (𝑟, 𝜂)-strongly accretive generalized resolvent operator 𝑀 is defined by 𝑋 mapping, and let 𝑀:𝑋×𝑋×𝐿 →2 be an (𝐴, 𝜂)-accretive 𝜂,𝜌,𝐴 −1 𝜂 : 𝑋×𝑋 →𝑋 𝜏 𝑅 (𝑢) =(𝐴+𝜌𝑀) (𝑢) , (19) inthefirstvariable.Let be a -Lipschitz 𝑀 continuous nonlinear mapping; then the following statements where 𝜌>0is a constant. are mutually equivalent: (i) an element 𝑢∈𝑋, 𝑥∈𝑄(𝑢,𝜆), 𝑦∈𝑇(𝑢,𝜆), 𝑤∈ Definition 10. The mapping 𝑓:𝑋×𝑋×𝐿 →𝑋is said to be 𝑉(𝑢, 𝜆) is a solution to (21); 𝛽-strongly accretive with respect to 𝐴 in the first argument if there exists a positive constant 𝛽 such that (ii) the mapping 𝐺:𝑋×𝐿defined →𝑋 by 𝐺 (𝑢,) 𝜆 ⟨𝑓 (𝑥,) 𝜆 −𝑓(𝑦,𝜆),𝐽𝑞 (𝑥 − 𝑦)⟩ (20) 󵄩 󵄩𝑞 =𝑅𝜂,𝜌,𝐴 (𝐴 (𝑢) −𝜌𝑁(𝑥,𝑔(𝑢,) 𝜆 ,𝜆)+𝜌𝑢−𝜌𝑓(𝑦,𝜆)) ≥𝛽󵄩𝑥−𝑦󵄩 , ∀(𝑥,𝑦,𝑢,𝜆)∈𝑋×𝑋×𝑋×𝐿. 𝑀(⋅,𝑤,𝜆) (25)

3. Sensitivity Analysis has a fixed point. 𝑋 Let 𝑄,let𝑇,andlet𝑉:𝑋×𝐿 →2 be three set-valued Lemma 13 𝑋 𝑁:𝑋×𝑋×𝐿 →𝑋 (see [21]). Let be a real uniformly smooth Banach mappings, and let a nonlinear mapping spaces. Then, 𝑋 is 𝑞-uniformly smooth if and only if there exists be an 𝐴𝑄-accretive mapping with respect to first argument, 𝑋 aconstant𝑐𝑞 >0such that for all 𝑢, V ∈𝑋 and let 𝑀:𝑋×𝑋×𝐿→ 2 bethemapping,where𝐿 is 𝑋 𝑓, 𝑔 : 𝑋 × 𝐿→𝑋 anonemptyopensubsetof .Let be ‖𝑢+V‖𝑞 ≤ ‖𝑢‖𝑞 +𝑞⟨V,𝐽 (𝑢)⟩+𝑐‖V‖𝑞. the single-valued mappings. Furthermore, let 𝜂:𝑋×𝑋 → 𝑞 𝑞 (26) 𝑋 be a nonlinear mapping. Then, the problem of finding an 𝑢∈𝑋𝑥∈𝑄(𝑢,𝜆)𝑦∈𝑇(𝑢,𝜆)𝑤∈𝑉(𝑢,𝜆) Theorem 14. Let 𝑋 be a 𝑞-uniformly smooth Banach spaces, element , , , such 𝜂:𝑋×𝑋 →𝑋 𝜏 that and let be a -Lipschitz continuous nonlinear mapping. Let 𝐴:𝑋be →𝑋 the (𝑟, 𝜂)-strongly accretive, and 𝑢∈𝑓(𝑦,) 𝜆 +𝑁(𝑥, 𝑔 (𝑢,) 𝜆 ,𝜆) +𝑀(𝑢, 𝑤,) 𝜆 , let 𝑠-Lipschitz continuous mapping and 𝑀 : 𝑋×𝑋×𝐿 → (21) 𝑋 2 be the (𝐴, 𝜂)-accretiveinthefirstvariablewith𝑚.Let 𝑇,𝑄,𝑉: 𝑋×𝐿𝑋 →2 H where 𝐴∈𝐿is the set-valued parameter is called a be the set-valued mappings and - 𝛾 𝜎 𝜁 generalized set-valued variational inclusions. Lipschitz continuous with , ,and constants, respectively. 𝑔, 𝑓 : 𝑋 × 𝐿→𝑋 𝜋 𝜒 The solvability of problem (21) depends on the equiva- Let be the -Lipschitz continuous and - 𝑁:𝑋×𝑋×𝐿→ 𝑋 lence between (21) and the problem of finding the fixed point Lipschitz continuous mappings. Let be 𝜇 𝜐 of the associated resolvent operator. the -Lipschitz continuous with first variable and -Lipschitz 𝑁 (𝜑, 𝜅) −𝑄 Note that if 𝑀 is (𝐴, 𝜂)-accretive, then the corresponding continuous in the second variable. Let be the - 𝐴𝑄 resolvent operator defined by relaxed cocoercive with respect to in the first argument. Let 𝑓 be the strongly accretive mapping. If the following condition 𝜂,𝜌,𝐴 −1 holds: 𝑅𝑀(⋅,V,𝜆) (𝑢) = (𝐴 + 𝜌𝑀(⋅, V, 𝜆)) (𝑢) ,∀𝑢∈𝑋, (22) 𝑞−1 󵄩 𝜂,𝜌,𝐴 𝜂,𝜌,𝐴 󵄩 𝜏 󵄩𝑅 (𝑢) −𝑅 (V)󵄩 ≤ ‖𝑢−V‖ , where 𝜌>0and 𝐴 is an (𝑟, 𝜂)-strongly accretive mapping. 󵄩 𝑀(⋅,𝑤,𝜆) 𝑀(⋅,𝑤,𝜆) 󵄩 𝑟−𝜌𝑚 (27) 󵄩 󵄩 Lemma 11 (see [7, 10, 15]). Let 𝑋 be a real Banach spaces, and 󵄩 𝜂,𝜌,𝐴 𝜂,𝜌,𝐴 󵄩 󵄩 󵄩 󵄩𝑅𝑀(⋅,𝑤 ,𝜆) (𝑢) −𝑅𝑀(⋅,𝑤 ,𝜆) (𝑢)󵄩 ≤𝛿󵄩𝑤1 −𝑤2󵄩 , (28) let 𝜂:𝑋×𝑋→be 𝑋 a 𝜏-Lipschitz continuous nonlinear 󵄩 1 2 󵄩 𝐴:𝑋 →𝑋 (𝑟, 𝜂) mapping. Let be -strongly accretive mapping, 𝑞 𝑞 𝑞 𝑞 𝑞 𝑞 𝑞 1/𝑞 𝑋 𝜏 [(𝑠 +𝑐𝜌 𝜇 𝜎 +𝑞𝜌𝜓𝜇 𝜎 −𝑞𝜌𝜅) and let 𝑀:𝑋×𝑋×𝐿 →2 be (𝐴, 𝜂)-accretive in the first 𝑞 variable with 𝑚. Then, the resolvent operator associated with 𝑀(⋅, V,𝜆) V ∈𝑋 𝑞 𝑞 1/𝑞 (29) for a fixed ,definedby +𝜌 (𝜐𝜋 +𝑞 (1+𝑐 𝜒 𝛾 −𝑞𝛽) )]

𝜂,𝜌,𝐴 −1 𝑅𝑀(⋅,V,𝜆) (𝑢) = (𝐴 + 𝜌𝑀(⋅, V, 𝜆)) (𝑢) , ∀𝑢∈𝑋, (23) <(𝑟−𝜌𝑚)(1−𝛿𝜁) , 4 Journal of Function Spaces and Applications

where 𝑐𝑞 >0is the same as in Lemma 13 and 𝜌 ∈ (0, 𝑟/𝑚). Since 𝑇, 𝑉,and𝑄 are H-Lipschitz continuous and 𝑁 Consequently, for each 𝜆∈𝐿,themapping𝐺(𝑢, 𝜆) in the light is Lipschitz continuous with respect to first and second (28) has a unique fixed point 𝑧(𝜆).Hence,inlightofLemma 12, arguments, 𝑓, 𝑔 are Lipschitz continuous, we have 𝑧(𝜆) is a unique solution to (21).Thus,onehas 󵄩 󵄩 󵄩𝑁(𝑥 ,𝑔(𝑢,) 𝜆 ,𝜆)−𝑁(𝑥 ,𝑔(𝑢,) 𝜆 ,𝜆)󵄩 𝐺 (𝑧 (𝜆) ,𝜆) =𝑧(𝜆) . (30) 󵄩 1 2 󵄩 󵄩 󵄩 ≤𝜇󵄩𝑥 −𝑥 󵄩 Proof. For any elements (𝑢, V,𝜆) ∈ 𝑋×𝑋×𝐿and 𝑥1 ∈ 𝑄(𝑢,, 𝜆) 󵄩 1 2󵄩 𝑥 ∈𝑄(V,𝜆) 𝑦 ∈ 𝑇(𝑢, 𝜆) 𝑦 ∈𝑇(V,𝜆) 𝑤 ∈ 𝑉(𝑢, 𝜆) 2 , 1 , 2 , 1 ,and ≤𝜇H (𝑄 (𝑢,) 𝜆 ,𝑄(V,𝜆)) 𝑤2 ∈𝑉(V,𝜆),wehave ≤𝜇𝜎‖𝑢−V‖ , ‖𝐺 (𝑢,) 𝜆 −𝐺(V,𝜆)‖ 󵄩 = 󵄩𝑅𝜂,𝜌,𝐴 (𝐴 𝑢 −𝜌𝑁(𝑥 ,𝑔 𝑢, 𝜆 ,𝜆) 󵄩 󵄩 󵄩 ( ) 1 ( ) 󵄩𝑁(𝑥2,𝑔(𝑢,) 𝜆 ,𝜆)−𝑁(𝑥2,𝑔(V,𝜆) ,𝜆)󵄩 󵄩 𝑀(⋅,𝑤1,𝜆) 󵄩 󵄩 ≤𝜐󵄩𝑔 (𝑢,) 𝜆 −𝑔(V,𝜆)󵄩 +𝜌𝑢−𝜌𝑓(𝑦1,𝜆)) 󵄩 󵄩 ≤𝜐𝜋‖𝑢−V‖ , −𝑅𝜂,𝜌,𝐴 (𝐴 (V) −𝜌𝑁(𝑥 ,𝑔(V,𝜆) ,𝜆) 𝑀 ⋅,𝑤 ,𝜆 2 ( 2 ) 󵄩 󵄩 󵄩 󵄩 󵄩𝑓(𝑦 ,𝜆)−𝑓(𝑦 ,𝜆)󵄩 ≤𝜒󵄩𝑦 −𝑦󵄩 󵄩 󵄩 1 2 󵄩 󵄩 1 2󵄩 󵄩 +𝜌V −𝜌𝑓(𝑦,𝜆))󵄩 2 󵄩 ≤𝜒H (𝑇 (𝑢,) 𝜆 ,𝑇(V,𝜆)) 󵄩 󵄩 𝜂,𝜌,𝐴 ≤𝜒𝛾‖𝑢−V‖ , ≤ 󵄩𝑅 [𝐴 (𝑢) −𝜌𝑁(𝑥1,𝑔(𝑢,) 𝜆 ,𝜆) 󵄩 𝑀(⋅,𝑤1,𝜆) 󵄩 󵄩 󵄩𝑤1 −𝑤2󵄩 ≤ H (𝑉 (𝑢,) 𝜆 ,𝑉(V,𝜆)) ≤𝜁‖𝑢−V‖ . +𝜌𝑢−𝜌𝑓(𝑦1,𝜆)] (32) −𝑅𝜂,𝜌,𝐴 [𝐴 (V) −𝜌𝑁(𝑥 ,𝑔(V,𝜆) ,𝜆) 𝑀(⋅,𝑤 ,𝜆) 2 1 Since 𝑁(⋅, ⋅) is (𝜑, 𝜅)-relaxed −𝑄 cocoercive with respect 󵄩 󵄩 to 𝐴𝑄 in the first argument and 𝑄 is H-Lipschitz continuous +𝜌V −𝜌𝑓(𝑦,𝜆)]󵄩 2 󵄩 mapping, so we obtain 󵄩 󵄩 𝜂,𝜌,𝐴 󵄩 + 󵄩𝑅 󵄩𝐴 (𝑢) −𝐴(V) 󵄩 𝑀(⋅,𝑤1,𝜆) 󵄩 󵄩𝑞 −𝜌 (𝑁 (𝑥 ,𝑔(𝑢,) 𝜆 ,𝜆)−𝑁(𝑥 ,𝑔(𝑢,) 𝜆 ,𝜆))󵄩 ×[𝐴(V) −𝜌𝑁(𝑥2,𝑔(V,𝜆) ,𝜆)+𝜌V −𝜌𝑓(𝑦2,𝜆)] 1 2 󵄩 𝑞 −𝑅𝜂,𝜌,𝐴 ≤ ‖𝐴 (𝑢) −𝐴(V)‖ 𝑀(⋅,𝑤2,𝜆) 󵄩 −𝑞𝜌⟨𝑁(𝑥 ,𝑔(𝑢,) 𝜆 ,𝜆) 󵄩 1 (33) ×[𝐴(V) −𝜌𝑁(𝑥 ,𝑔(V,𝜆) ,𝜆)+𝜌V −𝜌𝑓(𝑦,𝜆)]󵄩 2 2 󵄩 −𝑁 2(𝑥 ,𝑔(𝑢,) 𝜆 ,𝜆),𝐽𝑞 (𝐴 (𝑢) −𝐴(V))⟩ 𝑞−1 󵄩 𝜏 󵄩 ≤ [󵄩𝐴 (𝑢) −𝐴(V) 𝑞󵄩 󵄩𝑞 𝑟−𝜌𝑚 󵄩 +𝑐𝑞𝜌 󵄩𝑁(𝑥1,𝑔(𝑢,) 𝜆 ,𝜆)−𝑁(𝑥2,𝑔(𝑢,) 𝜆 ,𝜆)󵄩 𝑞 𝑞 𝑞 𝑞 𝑞 𝑞 𝑞 − 𝜌 (𝑁1 (𝑥 ,𝑔(𝑢,) 𝜆 ,𝜆)−𝑁(𝑥2,𝑔(V,𝜆) ,𝜆)) ≤[𝑠 −𝑞𝜌𝜅+𝑞𝜌𝜓𝜇 𝜎 +𝑐𝑞𝜌 𝜇 𝜎 ] ‖𝑢−V‖ . 󵄩 󵄩 +𝜌(𝑢−V −(𝑓(𝑦,𝜆)−𝑓(𝑦 ,𝜆)))󵄩] 𝑓 𝛽 𝑇 H 1 2 󵄩 Since is -strongly accretive and is the -Lipschitz continuous, we have 󵄩 󵄩 +𝛿󵄩𝑤1 −𝑤2󵄩 󵄩 󵄩𝑞 󵄩𝑢−V −(𝑓(𝑦,𝜆)−𝑓(𝑦 ,𝜆))󵄩 𝑞−1 󵄩 1 2 󵄩 𝜏 󵄩 ≤ [󵄩𝐴 (𝑢) −𝐴(V) 𝑞 𝑟−𝜌𝑚 ≤ ‖𝑢−V‖ −𝑞⟨𝑓(𝑦1,𝜆)−𝑓(𝑦2,𝜆),𝐽𝑞 (𝑢−V)⟩ 󵄩 (34) − 𝜌 (𝑁 (𝑥 ,𝑔(𝑢,) 𝜆 ,𝜆)−𝑁(𝑥 ,𝑔(𝑢,) 𝜆 ,𝜆))󵄩 󵄩 󵄩𝑞 1 2 󵄩 +𝑐𝑞󵄩𝑓(𝑦1,𝜆)−𝑓(𝑦2,𝜆)󵄩 󵄩 󵄩 +𝜌󵄩𝑁(𝑥2,𝑔(𝑢,) 𝜆 ,𝜆)−𝑁(𝑥2,𝑔(V,𝜆) ,𝜆)󵄩 𝑞 𝑞 𝑞 󵄩 󵄩 ≤[1+𝑐𝑞𝜒 𝛾 −𝑞𝛽]‖𝑢−V‖ . 󵄩 󵄩 +𝜌󵄩𝑢−V −(𝑓(𝑦1,𝜆)−𝑓(𝑦2,𝜆)󵄩] Combining (31)–(34), we can get 󵄩 󵄩 +𝛿󵄩𝑤1 −𝑤2󵄩 . (31) ‖𝐺 (𝑢,) 𝜆 −𝐺(V,𝜆)‖ ≤𝜃‖𝑢−V‖ , (35) Journal of Function Spaces and Applications 5 where It follows that 󵄩 ∗ ∗ ∗ 󵄩 𝑞−1 󵄩𝐺(𝑧(𝜆 ),𝜆)−𝐺(𝑧(𝜆 ),𝜆 )󵄩 𝜏 1/𝑞 󵄩 󵄩 𝜃= [(𝑠𝑞 −𝑞𝜌𝜅+𝑐𝜌𝑞𝜇𝑞𝜎𝑞 +𝑞𝜌𝜓𝜇𝑞𝜎𝑞) 𝑞 󵄩 (𝑟 − 𝜌𝑚) 󵄩 𝜂,𝜌,𝐴 = 󵄩𝑅 󵄩 𝑀(⋅,𝑤(𝑧(𝜆∗),𝜆),𝜆) 𝑞 𝑞 1/𝑞 +𝜌𝜐𝜋 + 𝜌(1𝑞 +𝑐 𝜒 𝛾 −𝑞𝛽) ]+𝛿𝜁. ×[𝐴(𝑧(𝜆∗)) − 𝜌𝑁 (𝑥 (𝑧∗ (𝜆 ),𝜆)), (36) 𝑔 ((𝑧 (𝜆∗),𝜆),𝜆)+𝜌𝑧(𝜆∗) − 𝜌𝑓 (𝑦 (𝑧 (𝜆∗),𝜆),𝜆)] It follows from (29)suchthat𝜃<1;itconcludestheproof. −𝑅𝜂,𝜌,𝐴 𝑀(⋅,𝑤(𝑧(𝜆∗),𝜆∗),𝜆∗)

∗ ∗ ∗ ∗ ∗ ∗ Theorem 15. Let 𝑋 be a real 𝑞-uniformly smooth Banach ×[𝐴(𝑧(𝜆 )) − 𝜌𝑁 (𝑥 (𝑧 (𝜆 ),𝜆 ),𝑔(𝑧(𝜆 ),𝜆 ),𝜆 ) 𝜂 : 𝑋×𝑋 →𝑋 𝜏 spaces, and let be a -Lipschitz continuous 󵄩 ∗ ∗ ∗ ∗ 󵄩 nonlinear mapping. Let 𝐴:𝑋be →𝑋 a (𝑟, 𝜂)-strongly +𝜌𝑧 (𝜆 ) − 𝜌𝑓 (𝑦 (𝑧 (𝜆 ),𝜆 ),𝜆 )] 󵄩 󵄩 accretive and 𝑠-Lipschitz continuous mapping, and let 𝑀: 𝑋 󵄩 󵄩 𝜂,𝜌,𝐴 𝑋×𝑋×𝐿 →2 be (𝐴, 𝜂)-accretive in the first variable ≤ 󵄩𝑅 𝑋 󵄩 𝑀(⋅,𝑤(𝑧(𝜆∗),𝜆),𝜆) with 𝑚.Let𝑇, 𝑄,and𝑉: 𝑋×𝐿 →2 be the set-valued mappings and H-Lipschitz continuous with respect to 𝛾, 𝜎, ×[𝐴(𝑧(𝜆∗)) − 𝜌𝑁 (𝑥 (𝑧∗ (𝜆 ),𝜆),𝑔(𝑧(𝜆∗),𝜆),𝜆) and 𝜁 constants, respectively. Let 𝑔,𝑓: 𝑋×𝐿 →𝑋be the 𝜋-Lipschitz continuous and 𝜒-Lipschitz continuous mappings, +𝜌𝑧(𝜆∗) − 𝜌𝑓 (𝑦 (𝑧 (𝜆∗),𝜆),𝜆)] respectively. Let 𝑁 : 𝑋×𝑋×𝐿 →𝑋be the 𝜇-Lipschitz 𝜐 −𝑅𝜂,𝜌,𝐴 continuous with first variable and -Lipschitz continuous with 𝑀(⋅,𝑤(𝑧(𝜆∗),𝜆),𝜆) second variable. Let 𝑁 be the (𝜑, 𝜅)-relaxed −𝑄 cocoercive with respect to 𝐴𝑄 in the first variable. Let 𝑓 be the 𝛽-strongly ×[𝐴(𝑧(𝜆∗)) − 𝜌𝑁 (𝑥 (𝑧∗ (𝜆 ),𝜆∗),𝑔(𝑧(𝜆∗),𝜆∗),𝜆∗) accretive mapping if the following condition holds: 󵄩 ∗ ∗ ∗ ∗ 󵄩 +𝜌𝑧 (𝜆 ) − 𝜌𝑓 (𝑦 (𝑧 (𝜆 ),𝜆 ),𝜆 )] 󵄩 𝑞−1 󵄩 󵄩 𝜂,𝜌,𝐴 𝜂,𝜌,𝐴 󵄩 𝜏 󵄩𝑅 (𝑢) −𝑅 (V)󵄩 ≤ ‖𝑢−V‖ , 󵄩 󵄩 𝑀(⋅,𝑤,𝜆) 𝑀(⋅,𝑤,𝜆) 󵄩 󵄩 𝜂,𝜌,𝐴 𝑟−𝜌𝑚 + 󵄩𝑅 󵄩 𝑀(⋅,𝑤(𝑧(𝜆∗),𝜆),𝜆) 󵄩 𝜂,𝜌,𝐴 𝜂,𝜌,𝐴 󵄩 󵄩 󵄩 󵄩 󵄩 ∗ ∗ ∗ ∗ ∗ ∗ 󵄩𝑅 (𝑢) −𝑅 (𝑢)󵄩 ≤𝛿󵄩𝑤1 −𝑤2󵄩 , ×[𝐴(𝑧(𝜆 )) − 𝜌𝑁 (𝑥 (𝑧 (𝜆 ),𝜆 ),𝑔(𝑧(𝜆 ),𝜆 ),𝜆 ) 󵄩 𝑀(⋅,𝑤1,𝜆) 𝑀(⋅,𝑤2,𝜆) 󵄩 󵄩 󵄩

𝑞 𝑞 𝑞 𝑞 𝑞 𝑞 𝑞 1/𝑞 (37) +𝜌𝑧 (𝜆∗) − 𝜌𝑓 (𝑦 ∗(𝑧 (𝜆 ),𝜆∗),𝜆∗)] 𝜏 [(𝑠 +𝑐𝑞𝜌 𝜇 𝜎 +𝑞𝜌𝜓𝜇 𝜎 −𝑞𝜌𝜅)

𝜂,𝜌,𝐴 1/𝑞 −𝑅 𝑞 𝑞 𝑀(⋅,𝑤(𝑧(𝜆∗),𝜆∗),𝜆∗) +𝜌(𝜐𝜋+(1+𝑐𝑞𝜒 𝛾 −𝑞𝛽) )] ×[𝐴(𝑧(𝜆∗)) − 𝜌𝑁 (𝑥 (𝑧∗ (𝜆 ),𝜆∗),𝑔(𝑧(𝜆∗),𝜆∗),𝜆∗) <(𝑟−𝜌𝑚)(1−𝛿𝜁) , 󵄩 ∗ ∗ ∗ ∗ 󵄩 +𝜌𝑧 (𝜆 ) − 𝜌𝑓 (𝑦 (𝑧 (𝜆 ),𝜆 ),𝜆 )] 󵄩 󵄩 where 𝑐𝑞 >0is the same as in Lemma 13 and 𝜌 ∈ (0, 𝑟/𝑚). If the mapping 𝜆→𝑁(𝑥1, 𝑔(𝑢, 𝜆),, 𝜆) 𝜆→𝑓(𝑦,𝜆),and 𝑞−1 𝜌𝜏 󵄩 ∗ ∗ 𝜂,𝜌,𝐴 ≤ [󵄩𝑁(𝑥(𝑧(𝜆 ) , 𝜆) , 𝑔 (𝑧 (𝜆 ),𝜆),𝜆) 𝜆→𝑅𝑀(⋅,𝑤,𝜆), 𝜆→𝑇(𝑢,𝜆), 𝜆→𝑉(𝑢,𝜆), 𝜆→𝑔(𝑢,𝜆), 𝑟−𝜌𝑚 𝜆→𝑄(𝑢,𝜆) and are continuous (or Lipschitz continuous) −𝑁 (𝑥 (𝑧 (𝜆∗),𝜆∗),𝑔(𝑧(𝜆∗),𝜆∗),𝜆∗) from 𝐿 to 𝑋, then the solution 𝑧(𝜆) of (21) is continuous (or 𝐿 𝑋 ∗ ∗ ∗ ∗ 󵄩 Lipschitz continuous) from to . −(𝑓(𝑦(𝑧(𝜆 ),𝜆),𝜆)−𝑓(𝑦(𝑧(𝜆 ),𝜆 ),𝜆 ))󵄩] 𝜆,∗ 𝜆 ∈𝐿 󵄩 Proof. From the hypothesis of theorem, for any ,we 󵄩 𝜂,𝜌,𝐴 + 󵄩𝑅 have 󵄩 𝑀(⋅,𝑤(𝑧(𝜆∗),𝜆),𝜆) 󵄩 ∗ 󵄩 ×[𝐴(𝑧(𝜆∗)) − 𝜌𝑁 (𝑥 (𝑧∗ (𝜆 ),𝜆∗),𝑔(𝑧(𝜆∗),𝜆∗),𝜆∗) 󵄩𝑧 (𝜆) −𝑧(𝜆 )󵄩 󵄩 ∗ ∗ 󵄩 ∗ ∗ ∗ ∗ = 󵄩𝐺 (𝑧 (𝜆) ,𝜆) − 𝐺 (𝑧 (𝜆 ),𝜆 )󵄩 +𝜌𝑧 (𝜆 ) − 𝜌𝑓 (𝑦 (𝑧 (𝜆 ),𝜆 ),𝜆 )]

󵄩 ∗ 󵄩 𝜂,𝜌,𝐴 ≤ 󵄩𝐺 (𝑧 (𝜆) ,𝜆) − 𝐺 (𝑧 (𝜆 ),𝜆)󵄩 −𝑅 󵄩 󵄩 𝑀(⋅,𝑤(𝑧(𝜆∗),𝜆∗),𝜆∗) (38) 󵄩 ∗ ∗ ∗ 󵄩 + 󵄩𝐺(𝑧(𝜆 ),𝜆)−𝐺(𝑧(𝜆 ),𝜆 )󵄩 ×[𝐴(𝑧(𝜆∗)) − 𝜌𝑁 (𝑥 (𝑧∗ (𝜆 ),𝜆∗),𝑔(𝑧(𝜆∗),𝜆∗),𝜆∗) 󵄩 ∗ 󵄩 ≤𝜃󵄩𝑧 (𝜆) −𝑧(𝜆 )󵄩 󵄩 󵄩 󵄩 ∗ ∗ ∗ ∗ 󵄩 +𝜌𝑧(𝜆 ) − 𝜌𝑓 (𝑦 (𝑧 (𝜆 ),𝜆 ),𝜆 )] 󵄩 . 󵄩 ∗ ∗ ∗ 󵄩 󵄩 + 󵄩𝐺(𝑧(𝜆 ),𝜆)−𝐺(𝑧(𝜆 ),𝜆 )󵄩 . (39) 6 Journal of Function Spaces and Applications

Hence, we get [8] M. F. Khan and Salahuddin, “Generalized co-complementarity problems in 𝑝-uniformly smooth Banach spaces,” JIPAM: Jour- 󵄩 ∗ 󵄩 󵄩𝑧 (𝜆) −𝑧(𝜆 )󵄩 nal of Inequalities in Pure and Applied Mathematics,vol.7,no.2, article 66, 11 pages, 2006. 𝜌𝜏𝑞−1 [9] M. F. Khan and Salahuddin, “Generalized multivalued non- ≤ (𝑟 − 𝜌𝑚) (1−𝜃) linear co-variational inequalities in Banach spaces,” Functional 󵄩 Differential Equations, vol. 14, no. 2–4, pp. 299–313, 2007. ×[󵄩𝑁(𝑥(𝑧(𝜆∗),𝜆),𝑔(𝑧(𝜆∗),𝜆),𝜆) 󵄩 [10]H.G.Li,A.J.Xu,andM.M.Jin,“AnIshikawa-hybridproximal point algorithm for nonlinear set-valued inclusions problem −𝑁 (𝑥 (𝑧 (𝜆∗),𝜆∗),𝑔(𝑧(𝜆∗),𝜆∗),𝜆∗) based on (𝐴, 𝜂)-accretive framework,” Fixed Point Theory and Applications, vol. 2010, Article ID 501293, 12 pages, 2010. −(𝑓(𝑦(𝑧(𝜆∗),𝜆),𝜆)) [11] N. D. Yen and G. M. Lee, “Solution sensitivity of a class of 󵄩 variational inequalities,” Journal of Mathematical Analysis and − 𝑓 (𝑦 (𝑧 (𝜆∗),𝜆∗),𝜆∗)󵄩] 󵄩 Applications,vol.215,no.1,pp.48–55,1997. 󵄩 [12] S. Dafermos, “Sensitivity analysis in variational inequalities,” 1 󵄩 𝜂,𝜌,𝐴 + 󵄩𝑅 1−𝜃󵄩 𝑀(⋅,𝑤(𝑧(𝜆∗),𝜆),𝜆) Mathematics of Operations Research,vol.13,no.3,pp.421–434, 1988. ∗ ×[𝐴(𝑧(𝜆 )) [13] R. L. Tobin, “Sensitivity analysis for variational inequalities,” Journal of Optimization Theory and Applications,vol.48,no.1, ∗ ∗ ∗ ∗ ∗ ∗ −𝜌𝑁 (𝑥 (𝑧 (𝜆 ),𝜆 ),𝑔(𝑧(𝜆 ),𝜆 ),𝜆 )+𝜌𝑧(𝜆 ) pp.191–209,1986.

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Research Article Nonlinear Kato Class and Unique Continuation of Eigenfunctions for 𝑝-Laplacian Operator

René Erlín Castillo1 and Julio C. Ramos Fernández2

1 Departamento de Matematicas,´ Universidad Nacional de Colombia, Apartado, 360354 Bogota,´ Colombia 2 Departamento de Matematicas,´ Universidad de Oriente, Cumana6101EstadoSucre,Venezuela´

Correspondence should be addressed to ReneErl´ ´ın Castillo; [email protected]

Received 25 April 2013; Revised 27 August 2013; Accepted 28 August 2013

Academic Editor: Janusz Matkowski

Copyright © 2013 R. E. Castillo and J. C. Ramos Fernandez.´ This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

𝑛 ̃ 𝑛 We study some basic properties of nonlinear Kato class 𝑀𝑝(R ) and 𝑀𝑝(R ), respectively, for 1<𝑝<𝑛.Also, we study the 𝑝−2 𝑝−2 𝑛 problem −div(|∇𝑢| ∇𝑢) + 𝑉|𝑢| 𝑢=0in Ω, where Ω is a bounded domain in R and the weight function 𝑉 isassumedtobe ̃ not equivalent to zero and lies in 𝑀𝑝(Ω),inthecasewhere𝑝<𝑛. Finally, we establish the strong unique continuation property of ̃ the eigenfunction for the 𝑝-Laplacian operator in the case where 𝑉∈𝑀𝑝(Ω).

̃ 1. Introduction and the class 𝑀𝑝 of functions 𝑓 such that 𝑓∈𝑀𝑝 and

The Kato class 𝐾𝑛 was introduced and studied by Aizenman and Simon (see [1]). For 𝑛≥3,itconsistsoflocallyintegrable 𝑓 R𝑛 functions on ,suchthat { 1 ∫ lim sup { 󵄨 󵄨𝑛−1 𝑟→0𝑥∈R𝑛 𝐵(𝑥,𝑟) 󵄨𝑥−𝑦󵄨 { 󵄨 󵄨 󵄨 󵄨 󵄨𝑓(𝑦)󵄨 󵄨 󵄨 1/(𝑝−1) 𝑝−1 ∫ 󵄨 󵄨 𝑑𝑦 = 0. 󵄨𝑓(𝑧)󵄨 } lim sup 󵄨 󵄨𝑛−2 (1) 󵄨 󵄨 𝑟→0 𝑛 󵄨 󵄨 ×(∫ 𝑑𝑧) 𝑑𝑦 =0. 𝑥∈R 𝐵(𝑥,𝑟) 󵄨𝑥−𝑦󵄨 󵄨 󵄨𝑛−1 } 󵄨 󵄨 𝐵(𝑥,𝑟) 󵄨𝑦−𝑧󵄨 󵄨 󵄨 } (3)

For 1<𝑝<𝑛, the following classes were defined by Zamboni ̃ (see [2]): the class 𝑀𝑝 of functions 𝑓,suchthat Section 2 of the present paper is devoted to the study of 𝑛 somebasicpropertiesofthenonlinearKatoclass𝑀𝑝(R ) and ̃ 𝑛 𝑀𝑝(R ),respectively,for1<𝑝<𝑛. 󵄨 󵄨 1/(𝑝−1) 𝑝−1 Among other things, we show that the Lorentz space { 1 󵄨𝑓(𝑧)󵄨 } 𝐿(𝑛/2, 1) 𝑀 (R𝑛) ∫ (∫ 󵄨 󵄨 𝑑𝑧) 𝑑𝑦 is embedded into 𝑝 (see Lemma 10)aswell sup{ 󵄨 󵄨𝑛−1 󵄨 󵄨𝑛−1 } 𝑛 𝑥∈R𝑛 𝐵(𝑥,𝑟) 󵄨𝑥−𝑦󵄨 𝐵(𝑥,𝑟) 󵄨𝑦−𝑧󵄨 𝑀̃ (R ) { 󵄨 󵄨 󵄨 󵄨 } as that 𝑝 is a complete topological vector space (see Remark 11 and Lemma 13). <∞, The 𝑝-Laplacian operator is a generalization of the (2) Laplace operator, where 𝑝 is allowed to range over 1< 𝑝 <; ∞ 2 Journal of Function Spaces and Applications in our case where 1<𝑝<𝑛,itiswrittenas In [6], de Figueiredo and Gossez prove (9)inthecase 𝑟,𝑛−2𝑟 𝑛 where 𝑝=2, assuming that 𝑉∈𝐿 (R ) with 1<𝑟≤𝑛/2. (|∇𝑢|𝑝−2∇𝑢) Later in [7],JerisonandKeningshowedthesameresulttaking div 𝑛 𝑉 in the Stummel-Kato class 𝑆(R ).Wepointoutthatitis 𝑟,𝑛−2𝑟 𝑛 =∇⋅(|∇𝑢|𝑝−2∇𝑢) not possible to compare the assumptions 𝑉∈𝐿 (R ) and 𝑛 𝑓∈𝑆(R ). Chiarenza and Frasca [5] generalized Fefferman’s 𝑟,𝑛−2𝑟 𝑛 𝑛 𝜕𝑢 𝜕𝑢 𝜕2𝑢 result proving (9) under the assumption that 𝑉∈𝐿 (R ) = |∇𝑢|𝑝−4 (|∇𝑢|2Δ𝑢 + (𝑝 − 2) ∑ ), 𝑟 ∈ (1, 𝑛/𝑝) 𝑝 ∈ (1, 𝑛) 𝜕𝑥 𝜕𝑥 𝜕𝑥 𝜕𝑥 with and .In[3], Schechter gave a new 𝑖,𝑗=1 𝑖 𝑗 𝑖 𝑗 ̃ 𝑛 proof of (9) assuming that 𝑉∈𝑀𝑝(R ). (4) where 2. Definitions and Notation

1/2 𝜕𝑢 𝜕𝑢 𝑛 𝜕𝑢 2 In this section, we gather definitions and notations that will ∇𝑢 = ( ,..., ), |∇𝑢| =(∑( ) ) , (5) be used throughout the paper. We also include several simple 𝜕𝑥1 𝜕𝑥𝑛 𝜕𝑥𝑖 1 𝑛 𝑖=1 lemmas. By 𝐿loc(R ),wewilldenotethespaceoffunctions 𝑛 1 which are locally integrable on R ,andby𝐿loc,𝑢 the space of 𝑢∈𝐶∞(Ω) Ω R𝑛 and 0 ,with bounded domain in . functions 𝑓,suchthat We are concerned with the following problem:

𝑝−2 𝑝−2 󵄨 󵄨 − div (|∇𝑢| ∇𝑢) +| 𝑉 𝑢| 𝑢=0, in Ω, (6) sup ∫ 󵄨𝑓(𝑦)󵄨 𝑑𝑦 < ∞. (10) 𝑥∈R𝑛 𝐵(𝑥,1) and the weight function 𝑉 isassumedtobenotequivalentto 𝑀̃ (R𝑛) 𝑝<𝑛 1 𝑛 zero and lies in 𝑝 in the case . Definition 3. Let 𝑓∈𝐿loc(R ).Forany1<𝑝<𝑛and 𝑟>0, Specifically,weareinterestedinstudyingafamilyoffunc- we set tions which enjoys the strong unique continuation property, that is, functions besides the possible zero functions which Φ (𝑟) have zero of infinite order. 𝑝 󵄨 󵄨 1/(𝑝−1) 𝑝−1 Definition 1. We say that a function 𝑢∈𝐿 (Ω) vanishes of 1 󵄨𝑓 (𝑧)󵄨 𝑑𝑧 loc = (∫ (∫ 󵄨 󵄨 ) 𝑑𝑦) , infinite order at point 𝑥0 if for any natural number 𝑁 there sup 󵄨 󵄨𝑛−1 󵄨 󵄨𝑛−1 𝑥∈R𝑛 𝐵(𝑥,𝑟) 󵄨𝑥−𝑦󵄨 𝐵(𝑥,𝑟) 󵄨𝑧−𝑦󵄨 exists a constant 𝐶𝑁,suchthat 󵄨 󵄨 󵄨 󵄨 (11) 𝑝 𝑁 ∫ |𝑢 (𝑥)| 𝑑𝑥≤𝐶𝑁𝑟 , (7) 𝐵(𝑥 ,𝑟) 0 where 𝐵(𝑥, 𝑟) = {𝑦 : |𝑥 .Wesaythat− 𝑦|<𝑟} 𝑓 belongs to the ̃ 𝑛 space 𝑀𝑝(R ) if Φ(𝑟) < ∞ for all 𝑟>0. for all 𝑁∈N and for small positive number 𝑟.Here, 𝑓∈𝑀 (R𝑛) 𝑛 󵄨 󵄨 Definition 4. We say that a function 𝑝 if 𝐵 (𝑥0,𝑟) = {𝑦∈R : 󵄨𝑦−𝑥0󵄨 <𝑟} . (8)

Definition 2. We say that (6)hasastronguniquecontinuation Φ (𝑟) =0. 𝑟→0lim (12) property if and only if any solution 𝑢 of (6)inΩ is identically zero in Ω provided that 𝑢 vanishes of infinite order at a point in Ω. We are now ready to formulate some simple properties of ̃ the classes 𝑀𝑝 and 𝑀𝑝. There is an extensive literature on unique continuation. We refer to the work of Zamboni on unique continuation Lemma 5 (see [3], page 152). For 1<𝑝<𝑛,onehas for nonnegative solutions of quasilinear elliptic equation [3], also the work of Jerison-Kenig on the unique continuation 𝑛 ̃ 𝑛 for Schrodinger¨ operators [4]. The same work is done by (i) 𝑀𝑝(R )⊂𝑀𝑝(R ), ChiarenzaandFrasca,butforlinearellipticoperatorinthe 𝑉∈𝐿𝑛/2 𝑛>2 𝑛 case where when [5]. (ii) 𝑀2(R )=𝐾𝑛. LetusrecallsomeknownresultsconcerningFefferman’s 𝑛 ̃ 𝑛 inequality as follows: From Lemma 5 we conclude that both 𝑀𝑝(R ) and 𝑀𝑝(R ) are generalizations of 𝐾𝑛. 𝑝 𝑝 ∞ 𝑛 ∫ |𝑢 (𝑥)| |𝑉 (𝑥)| 𝑑𝑥≤∫ 𝐶 |∇𝑢 (𝑥)| 𝑑𝑥 ∀𝑢0 ∈𝐶 (R ). R𝑛 R𝑛 Remark 6. The following example shows that 𝐾𝑛 is properly 𝑛 (9) contained in 𝑀𝑝(R ) for 𝑝>2. It is known that the function Journal of Function Spaces and Applications 3

−2 𝑛 𝑓(𝑥) = |𝑥| is not in the Kato class 𝐾𝑛.However,𝑓∈𝑀𝑝. Lemma 10. Consider 𝐿(𝑛/2,1) ⊂𝑀𝑝(R ). Indeed, Proof. Let 𝑓∈𝐿(𝑛/2,1);then { 1 ∞ ∫ 2/𝑛−1 ∗ lim sup { 󵄨 󵄨𝑛−2 ∫ 𝑡 𝑓 (𝑡) 𝑑𝑡 < ∞. 𝑟→0 𝑥 𝐵(𝑥,𝑟) 󵄨𝑥−𝑦󵄨 (20) { 󵄨 󵄨 0

1/(𝑝−1) 𝑝−1 Since |𝑓|𝜒𝐵(𝑥,𝑟) ≤|𝑓|,wehave 𝑑𝑧 } ×(∫ ) 𝑑𝑦 =0. 󵄨 󵄨 ∗ 󵄨 󵄨𝑛−1 } 󵄨 󵄨 ∗ 𝐵(𝑥,𝑟) |𝑧|2󵄨𝑧−𝑦󵄨 (󵄨𝑓󵄨 𝜒𝐵(𝑥,𝑟)) (𝑡) ≤𝑓 (𝑡) ; (21) 󵄨 󵄨 } (13) then, ∞ ∞ 2/𝑛−1 󵄨 󵄨 ∗ 2/𝑛−1 ∗ This can be shown by splitting the domain of integra- ∫ 𝑡 (󵄨𝑓󵄨 𝜒𝐵(𝑥,𝑟)) (𝑡) 𝑑𝑡 ≤ ∫ 𝑡 𝑓 (𝑡) 𝑑𝑡 < ∞. tion in the interior integral into the following three parts: 0 0 𝐵(𝑥, 𝑟) ⋂{|𝑧| < (1/2)|𝑦|}, 𝐵(𝑥, 𝑟) ⋂{(1/2)|𝑦| ≤|𝑧| (22) (3/2)|𝑦|},and𝐵(𝑥, 𝑟) ⋂{|𝑧| > (3/2)|𝑦|}. Thus, |𝑓|𝜒𝐵(𝑥,𝑟) ∈𝐿(𝑛/2,1). After routine calculations, we can see that −(1−𝑛) On the other hand, let 𝑔(𝑥) = |𝑥| ;then 𝑑𝑧 ∫ 󵄨 󵄨 −(1−𝑛) 2󵄨 󵄨𝑛−1 (14) 𝑚 ({𝑥 : 󵄨𝑔 (𝑥)󵄨 > 𝜆}) = 𝑚 ({𝑥 : |𝑥| >𝜆}), 𝐵(𝑥,𝑟) |𝑧| 󵄨𝑧−𝑦󵄨 1 1/(𝑛−1) 1 𝑛/(𝑛−1) (23) −1 𝑚({𝑥:|𝑥| <( ) }) = 𝐶 ( ) , is majorized by 𝐶|𝑦| .Finally,wehave 𝜆 𝑛 𝜆

𝑝−1 𝑑𝑦 where 𝐶𝑛 = 𝑚(𝐵(0, .1)) 𝐶 sup{∫ } 󳨀→ 0 as 𝑟󳨀→0. 𝑡=𝐶(1/𝜆)𝑛/(𝑛−1) 𝜆=𝐶𝑡1/𝑛−1 𝑥 󵄨 󵄨1/(𝑝−1)󵄨 󵄨𝑛−1 Next, we set 𝑛 ,andthen 𝑛 . 𝐵(𝑥,𝑟) 󵄨𝑦󵄨 󵄨𝑥−𝑦󵄨 ∗ 1/𝑛−1 Thus, 𝑔 (𝑡) =𝑛 𝐶 𝑡 .Fromthis,weobtain (15) 󵄩 󵄩 󵄩𝑔󵄩(𝑛/(𝑛−2),∞) This shows that (13)holds.Thus,𝑓∈⋂𝑝>2 𝑀𝑝. 󵄩 󵄩 󵄩 1 󵄩 = 󵄩 󵄩 󵄩 𝑛−1 󵄩 Definition 7. The distribution function 𝐷𝑓 of a measurable 󵄩|⋅| 󵄩 (𝑛/(𝑛−2),∞) (24) function 𝑓 is given by 1−2/𝑛 1/𝑛−1 −(1/𝑛) = sup 𝐶𝑛𝑡 𝑡 = sup 𝐶𝑛𝑡 𝑛 󵄨 󵄨 𝑡>1 𝑡>1 𝐷𝑓 (𝜆) =𝑚({𝑥∈R : 󵄨𝑓 (𝑥)󵄨 > 𝜆}) , (16) ≤𝐶𝑛 <∞, 𝑛 where 𝑚 denotes the Lebesgue measure on R .Thedistribu- which means that 𝑔∈𝐿(𝑛/(𝑛−2),∞). Finally, by Fubini’s tion function 𝐷𝑓 provides information about the size of 𝑓 but not about the behavior of 𝑓 itself near any given point. For theorem and Holder’s¨ inequality, we have 𝑛 instance, a function on R and each of its translates have the 𝜙 (𝑟) same distribution function. It follows from Definition 7 that 𝐷𝑓 is a decreasing function of 𝜆 (not strictly necessary). 1 = ( ∫ sup 󵄨 󵄨𝑛−1 𝑛 𝑥∈R𝑛 𝐵(𝑥,𝑟) 󵄨𝑥−𝑦󵄨 Definition 8. Let 𝑓 be a measurable function in R .The 󵄨 󵄨 𝑓 𝑓 1/(𝑝−1) 𝑝−1 decreasing rearrangement of is the function defined on |𝑓(𝑧)| [0, ∞) by ×(∫ 𝑑𝑧) 𝑑𝑦) 󵄨 󵄨𝑛−1 𝐵(𝑥,𝑟) 󵄨𝑦−𝑧󵄨 ∗ 𝑓 (𝑡) = inf {𝜆 :𝑓 𝐷 (𝜆) ≤𝑡} (𝑡≥0) . (17) 1 ≤ ( ∫ sup 󵄨 󵄨𝑛−1 0=∞ 𝑛 󵄨 󵄨 We use here the convention that inf . 𝑥∈R 𝐵(𝑥,𝑟) 󵄨𝑥−𝑦󵄨 󵄨 󵄨 1/(𝑝−1) 𝑝−1 Definition 9 (Lorentz space). Let 𝑓 be a measurable function; 󵄨𝑓 (𝑧)󵄨 𝑓 𝐿 ×(∫ 󵄨 󵄨 𝑑𝑧) 𝑑𝑦) we say that belongs to (𝑛/2,1) if 󵄨 󵄨𝑛−1 𝐵(𝑥,2𝑟) 󵄨𝑦−𝑧󵄨 ∞ 󵄩 󵄩 2/𝑛−1 ∗ 󵄩𝑓󵄩(𝑛/2,1) = ∫ 𝑡 𝑓 (𝑡) 𝑑𝑡 < ∞. (18) 1 0 = ( ∫ sup 𝑛−1 𝑛 󵄨 󵄨 𝑥∈R 𝐵(𝑥,𝑟) 󵄨𝑥−𝑦󵄨 And it belongs to 𝐿(𝑛/(𝑛 − 2), ∞) if 1/(𝑝−1) 𝑝−1 |𝑓(𝑧)|𝜒𝐵(0,2𝑟)(𝑦 − 𝑧)𝑑𝑧 󵄩 󵄩 1−2/𝑛 ∗ ×(∫ ) 𝑑𝑦) 󵄩𝑓󵄩 = 𝑡 𝑓 (𝑡) <∞. 󵄨 󵄨𝑛−1 󵄩 󵄩(𝑛/(𝑛−2),∞) sup (19) R𝑛 󵄨 󵄨 𝑡>1 󵄨𝑦−𝑧󵄨 4 Journal of Function Spaces and Applications

𝑝−1 𝑑𝑦 where ≤ (∫ ) sup 󵄨 󵄨𝑛−1 𝑛−1 𝑝−1 𝑥∈R𝑛 𝐵(𝑥,𝑟) 󵄨𝑥−𝑦󵄨 󵄨 󵄨 𝐵=(2𝑟0) (𝑟0𝑚 (𝐵 (0, 1))) . (31) 󵄩 󵄩 󵄩 󵄩 󵄩 1 󵄩 𝑛 × 󵄩𝑓𝜒 󵄩 󵄩 󵄩 𝐵(𝑥, 1) ⊆ ⋃ 𝐵(𝑥 ,𝑟 ) 󵄩 𝐵(0,2𝑟)󵄩(𝑛/2,1)󵄩 𝑛−1 󵄩 Finally, let 𝑘=1 𝑘 0 ;then 󵄩|⋅| 󵄩(𝑛/(𝑛−2),∞) 𝑛 𝑝−1󵄩 󵄩 󵄨 󵄨 󵄨 󵄨 =𝐶𝑛𝑟 󵄩𝑓𝜒 𝐵(0,2𝑟)󵄩 󳨀→ 0 as 𝑟󳨀→0, sup ∫ 󵄨𝑓 (𝑧)󵄨 𝑑𝑧 ≤ ∑ sup ∫ 󵄨𝑓 (𝑧)󵄨 𝑑𝑧, (32) (𝑛/2,1) 𝑛 𝑛 𝑥∈R 𝐵(𝑥,1) 𝑘=1𝑥∈R 𝐵(𝑥𝑘,𝑟0) (25) so 𝑛 which means that 𝑓∈𝑀𝑝(R ) and the proof is complete. 󵄨 󵄨 sup ∫ 󵄨𝑓 (𝑧)󵄨 𝑑𝑧 < ∞. (33) 𝑥∈R𝑛 𝐵(𝑥,1) Remark 11. (i) For 0<𝑟<1, it is not hard to check that for 1<𝑝≤2, the expression Therefore, ̃ 𝑛 1 𝑛 󵄩 󵄩 𝑀𝑝 (R )⊂𝐿loc,𝑢 (R ). (34) 󵄩𝑓󵄩̃ 𝑛 󵄩 󵄩𝑀𝑝(R )

1 = ( ∫ Lemma 13. 1<𝑝<𝑛 𝑀̃ (R𝑛) sup 𝑛−1 For , 𝑝 is a complete space. 𝑛 󵄨 󵄨 𝑥∈R 𝐵(𝑥,1) 󵄨𝑥−𝑦󵄨 (26) Proof. Let {𝑓𝑛}𝑛∈N be a Cauchy sequence in 󵄨 󵄨 1/(𝑝−1) 𝑝−1 󵄨𝑓(𝑧)󵄨 󵄨 󵄨 ̃ 𝑛 󵄩 󵄩 ×(∫ 𝑑𝑧) 𝑑𝑦) 𝐵 (0, 𝑟) ={𝑓∈𝑀𝑝 (R ):󵄩𝑓󵄩̃ 𝑛 ≤𝑟}. (35) 󵄨 󵄨𝑛−1 𝑀𝑝(R ) 𝐵(𝑥,1) 󵄨𝑧−𝑦󵄨 1 𝑛 By Lemma 10, {𝑓𝑛}𝑛∈N is a Cauchy sequence in 𝐿loc,𝑢(R ). ̃ 𝑛 defines a norm on 𝑀𝑝(R ). Since this space is complete, there exists a function 𝑓∈ 𝑝>2 1 𝑛 1 𝑛 (ii) For , the expression (26) satisfies the following 𝐿loc,𝑢(R ) such that 𝑓𝑛 →𝑓in 𝐿loc,𝑢(R ).ByFatous’s inequality: lemma, we have 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩𝑓󵄩̃ 𝑛 ≤ 󵄩𝑓𝑛󵄩̃ 𝑛 ≤𝑟. 󵄩 󵄩 𝑝−2 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩𝑀𝑝(R ) lim inf 󵄩 󵄩𝑀𝑝(R ) (36) 󵄩𝑓+𝑔󵄩̃ 𝑛 ≤2 (󵄩𝑓󵄩̃ 𝑛 + 󵄩𝑔󵄩̃ 𝑛 ) 󵄩 󵄩𝑀𝑝(R ) 󵄩 󵄩𝑀𝑝(R ) 󵄩 󵄩𝑀𝑝(R ) (27) Thus, 𝑓∈𝐵(0, 𝑟),whichmeansthat𝐵(0, 𝑟) is complete 𝑓 𝑔 𝑀̃ (R𝑛) 1 𝑛 for all and in 𝑝 . with respect to the topology generated by 𝐿loc,𝑢(R )-norm. If 𝑈 is a neighborhood of 0 from (27), we have By Corollary 2 of Proposition 9 in [8, Chapter III, Section 3, no. 5] we obtain the assertion. 𝑝−1 𝑝−1 2 𝑈+2 𝑈⊂𝑈; (28) 𝑛 ̃ 𝑛 Lemma 14. If 1<𝑝<𝑛,then𝑀𝑝(R ) is closed in 𝑀𝑝(R ). 𝑀̃ (R𝑛) then, 𝑝 is a topological vector space. ̃ 𝑛 Proof. Let us define the map 𝜑:𝑀𝑝(R )→[0,∞)by ̃ 𝑛 1 𝑛 𝜑(𝑓) = lim𝑟→0𝜙𝑓(𝑟) (see Definition). 3 Lemma 12. Consider 𝑀𝑝(R )⊂𝐿loc,𝑢(R ) for 1<𝑝<𝑛. It is not hard to prove that the family {𝜑𝑟}𝑟>0 where ̃ 𝑛 𝜑𝑟(𝑓) =𝑓 𝜙 (𝑟) is equicontinuous and 𝜑𝑟 →𝜑pointwise as Proof. Let 𝑓∈𝑀𝑝(R ),andfix𝑟0 >0. Then, there exists a 𝑟→0 𝑀 (R𝑛)=𝜑−1(0) positive constant 𝐶 such that Φ(𝑟0)≤𝐶.Itfollowsthat .Since 𝑝 ,weobtaintheresult. For more details on nonlinear Kato class, we refer the 󵄨 󵄨 1/(𝑝−1) 𝑝−1 1 󵄨𝑓 (𝑧)󵄨 readers to [9]. (∫ (∫ 󵄨 󵄨 𝑑𝑧) 𝑑𝑦) sup 󵄨 󵄨𝑛−1 󵄨 󵄨𝑛−1 𝑛 𝐵(𝑥,𝑟 ) 󵄨 󵄨 𝐵(𝑥,𝑟 ) 󵄨 󵄨 𝑥∈R 0 󵄨𝑥−𝑦󵄨 0 󵄨𝑧−𝑦󵄨 3. Some Useful Inequalities 󵄨 󵄨 1/(𝑝−1) 𝑝−1 𝑑𝑦 󵄨𝑓(𝑧)󵄨 ≥ (∫ (∫ 󵄨 󵄨 𝑑𝑧) ) For the sake of completeness and convenience of the reader, sup 𝑛−1 𝑛−1 𝑛 𝑟 𝑥∈R 𝐵(𝑥,𝑟0) 0 𝐵(𝑥,𝑟0) (2𝑟0) we include the proof of the next result which is due to Schechter [3]. 𝑛−1 𝑝−1 1 𝑚(𝐵(𝑥,𝑟0)) 󵄨 󵄨 ≥ ( ) ( ) ∫ 󵄨𝑓 (𝑧)󵄨 𝑑𝑧. 𝑛 sup 𝑛−1 󵄨 󵄨 Theorem 15. 𝑉∈𝑀̃ (R ) 𝑟>0 𝑛 2𝑟 𝑟 𝐵(𝑥,𝑟 ) Assume that 𝑝 .Then,forany 𝑥∈R 0 0 0 𝐶(𝑛, 𝑝) (29) there exists a positive constant ,suchthat ∫ |𝑉 (𝑥)||𝑢 (𝑥)|𝑝 𝑑𝑥 ≤𝐶 (𝑛,) 𝑝 Φ (2𝑟) ∫ |∇𝑢 (𝑥)|𝑝 𝑑𝑥, Therefore, R𝑛 R𝑛 (37) 󵄨 󵄨 sup ∫ 󵄨𝑓 (𝑧)󵄨 𝑑𝑧 < 𝐵𝐶, ∞ 𝑛 𝑛 (30) 𝑢∈𝐶 (R ) 𝐵(𝑥 ,𝑟) 𝑥∈R 𝐵(𝑥,𝑟0) for any 0 supported in 0 . Journal of Function Spaces and Applications 5

∞ 𝑛 Proof. For any 𝑢∈𝐶0 (R ) supported in 𝐵(𝑥0,𝑟),usingthe By (39)and(40), we obtain well-known inequality 𝑝 󵄨 󵄨 ∫ |𝑉 (𝑥)||𝑢 (𝑥)| 𝑑𝑥 󵄨∇𝑢 (𝑦)󵄨 𝐵(𝑥 ,𝑟) |𝑢 (𝑥)| ≤𝐶(𝑛,𝑝) ∫ 𝑑𝑦, 0 󵄨 󵄨𝑛−1 󵄨𝑥−𝑦󵄨 (38) 𝐵(𝑥 ,𝑟) 󵄨 󵄨 1/𝑝 0 1/(𝑝−1) 󵄨 󵄨𝑝 ≤𝐶(𝑛,𝑝)[Φ (2𝑟)] (∫ 󵄨∇𝑢 (𝑦)󵄨 𝑑𝑦) (41) Fubini’s theorem, and Holder’s¨ inequality, we obtain 𝐵(𝑥0,𝑟)

𝑝 1−1/𝑝 ∫ |𝑉 (𝑥)||𝑢 (𝑥)| 𝑑𝑥 𝑝 R𝑛 ×(∫ |𝑉 (𝑥)||𝑢 (𝑥)| 𝑑𝑥) . 𝐵(𝑥0,𝑟) 𝑝 = ∫ |𝑉 (𝑥)||𝑢 (𝑥)| 𝑑𝑥 Hence 𝐵(𝑥0,𝑟) ∫ |𝑉 (𝑥)||𝑢 (𝑥)|𝑝𝑑𝑥 𝑝−1 𝐵(𝑥 ,𝑟) ≤𝐶(𝑛,𝑝)∫ |𝑉 (𝑥)||𝑢 (𝑥)| 0 𝐵(𝑥0,𝑟) (42) 𝑞/𝑝 𝑝 󵄨 󵄨 ≤𝐶(𝑛,𝑝)[Φ (2𝑟)] ∫ |∇𝑢 (𝑥)| 𝑑𝑥. 󵄨∇𝑢 (𝑦)󵄨 𝐵(𝑥 ,𝑟) ×(∫ 󵄨 󵄨 𝑑𝑦) 𝑑𝑥 0 󵄨 󵄨𝑛−1 𝐵(𝑥 ,𝑟) 󵄨 󵄨 0 󵄨𝑥−𝑦󵄨 󵄨 󵄨 ≤𝐶(𝑛,𝑝)∫ 󵄨∇𝑢 (𝑦)󵄨 The next corollary is an easy consequence of the previous 󵄨 󵄨 theorem. It can be obtained via a standard partition of unity. 𝐵(𝑥0,𝑟) Corollary 16. 𝑉∈𝑀̃ (R𝑛) Ω 𝑝−1 1 Let 𝑝 and let be a bounded subset ×(∫ |𝑉 (𝑥)||𝑢 (𝑥)| 𝑑𝑥) 𝑑𝑦 𝑛 󵄨 󵄨𝑛−1 R 𝑉⊆Ω 𝜎>0 𝐵(𝑥 ,𝑟) 󵄨 󵄨 of , supp .Then,forany there exists a positive 0 󵄨𝑥−𝑦󵄨 constant 𝐾 depending on 𝜎,suchthat 1/𝑝 󵄨 󵄨𝑝 𝑝 ≤𝐶(𝑛,𝑝)(∫ 󵄨∇𝑢 (𝑦)󵄨 𝑑𝑦) ∫ |𝑉 (𝑥)||𝑢 (𝑥)| 𝑑𝑥 𝐵(𝑥0,𝑟) Ω (43) ≤𝜎∫ |∇𝑢 (𝑥)|𝑝 𝑑𝑥 +𝐾 (𝜎) ∫ |𝑢 (𝑥)|𝑝𝑑𝑥, [ × [ ∫ ( ∫ |𝑉 (𝑥)||𝑢 (𝑥)|𝑝−1 Ω Ω 𝐵(𝑥 ,𝑟) 𝐵(𝑥 ,𝑟) ∞ 0 0 𝑢∈𝐶 (Ω) [ for all 0 . 𝑝/(𝑝−1) Proof. Let 𝜎>0.Let𝑟 be a positive number that will be 1 ] {𝛼𝑝} 𝑘 = 1,2,...,𝑁(𝑟) × 𝑑𝑥) 𝑑𝑦] . chosen later. Let 𝑘 , , be a finite partition 󵄨 󵄨𝑛−1 󵄨𝑥−𝑦󵄨 Ω 𝑉⊆𝐵(𝑥,𝑟) 𝑥 ∈ Ω ] of the unity of ,suchthatsupp 𝑘 with 𝑘 . We apply Theorem 15 to the functions 𝛼𝑘 and we get (39) 𝑝 On the other hand, using Holder’s¨ inequality one more ∫ |𝑉 (𝑥)||𝑢 (𝑥)| 𝑑𝑥 Ω time, we have 𝑝/(𝑝−1) 𝑁(𝑟) 𝑝−1 1 𝑝 𝑝 ∫ (∫ |𝑉 (𝑥)||𝑢 (𝑥)| 𝑑𝑥) 𝑑𝑦 = ∫ |𝑉 (𝑥)||𝑢 (𝑥)| ∑ 𝛼𝑘 (𝑥) 𝑑𝑥 |𝑥 − 𝑦|𝑛−1 Ω 𝐵(𝑥0,𝑟) 𝐵(𝑥0,𝑟) 𝑘=1

1/(𝑝−1) 𝑁(𝑟) |𝑉 (𝑧)| 󵄨 󵄨𝑝 ≤ ∫ (∫ 𝑑𝑧) = ∑ ∫ |𝑉 (𝑥)| 󵄨𝑢 (𝑥) 𝛼𝑝 (𝑥)󵄨 𝑑𝑥 󵄨 󵄨𝑛−1 󵄨 𝑘 󵄨 𝐵(𝑥 ,𝑟) 𝐵(𝑥 ,𝑟) 󵄨 󵄨 Ω 0 0 󵄨𝑧−𝑦󵄨 𝑘=1 |𝑉 (𝑥)||𝑢 (𝑥)|𝑝 𝑁(𝑟) × ∫ 𝑑𝑥 𝑑𝑦 𝑝 𝑝 󵄨 󵄨𝑛−1 ≤ ∑ 𝐶Φ𝑉 (2𝑟) (∫ |∇𝑢 (𝑥)| 𝛼 (𝑥) 𝑑𝑥 𝐵(𝑥 ,𝑟) 󵄨𝑥−𝑦󵄨 𝑛 0 󵄨 󵄨 𝑘=1 Ω

𝑝 = ∫ |𝑉 (𝑥)||𝑢 (𝑥)| + ∫ |∇𝑢 (𝑥)|𝑝|𝑢 (𝑥)|𝑝𝑑𝑥) 𝐵(𝑥0,𝑟) Ω 1/(𝑝−1) 𝑁 (𝑟) 1 |𝑉 (𝑧)| ≤𝐶Φ (2𝑟) (∫ |∇𝑢 (𝑥)|𝑝 𝑑𝑥 + ∫ |𝑢 (𝑥)|𝑝𝑑𝑥) . × ∫ (∫ 𝑑𝑧) 𝑑𝑦 𝑑𝑥 𝑉 𝑝 󵄨 󵄨𝑛−1 󵄨 󵄨𝑛−1 Ω 𝑟 Ω 𝐵(𝑥 ,𝑟) 󵄨𝑥−𝑦󵄨 𝐵(𝑥 ,𝑟) 󵄨𝑧−𝑦󵄨 0 󵄨 󵄨 0 󵄨 󵄨 (44)

≤ [Φ (2𝑟)]1/(𝑝−1) ∫ |𝑉 (𝑥)||𝑢 (𝑥)|𝑝𝑑𝑥. Finally, to obtain the result, it is sufficient to choose 𝑟 such 𝐶Φ (2𝑟) = 𝜎 𝑁(𝑟)−𝑛 ≈𝑟 𝐵(𝑥0,𝑟) that 𝑉 .Afterthat,wenotethat and (40) the corollary follows. 6 Journal of Function Spaces and Applications

Lemma 17. Let 𝐵𝑟 and 𝐵2𝑟 be two concentric balls contained 4. Strong Unique Continuation in Ω.Then, In this section, we proceed to establish the strong unique con- 𝐶 𝑝 ∫ |∇𝑢(𝑥)|𝑝𝑑𝑥 ≤ ∫ |𝑢|𝑝, tinuation property of the eigenfunction for the -Laplacian 𝑟𝑝 (45) 𝑉∈𝑀̃ (Ω) 𝐵𝑟 𝐵2𝑟 operator in the case 𝑝 .

∞ ∞ where the constant 𝐶 does not depend on 𝑟 and 𝑢∈𝐶0 (𝐵𝑟). Theorem 19. Let 𝑢∈𝐶0 (Ω) be a solution of (6).If𝑢=0on 𝐸 𝑢 ∞ aset of positive measures, then haszeroofinfiniteorderin Proof. Take 𝜑∈𝐶0 (Ω),withsupp𝜑⊂𝐵2𝑟, 𝜑(𝑥) =1 for 𝑝 𝑝 -mean. 𝑥∈𝐵𝑟 and |∇𝜑| ≤ 𝐶/𝑟 using 𝜑 as a test function in (6); we get Proof. We know that almost every point of 𝐸 is a point of density of 𝐸.Let𝑥0 ∈𝐸be such point. This means that 𝑝−2 𝑝 𝑝−2 𝑝 ∫ − div (|∇𝑢| ∇𝑢) 𝜑 𝑢+∫ 𝑉|𝑢| 𝑢𝜑 𝑢=0. (46) 𝑚(𝐸∩𝐵 ) 𝐵 𝐵 𝑟 2𝑟 2𝑟 =1, 𝑟→0lim (52) 𝑚(𝐵𝑟) Thus, where 𝐵𝑟 denotes the ball of radius 𝑟 centered at 𝑥0,andthus, 𝑝 𝑝 𝜀>0 𝑟 =𝑟(𝜀) ∫ |∇𝑢| 𝜑 given that ,thereisan 0 0 ,suchthat 𝐵 2𝑟 𝑚(𝐸𝑐 ∩𝐵) 𝑚(𝐸∩𝐵 ) 𝑟 <𝜀, 𝑟 >1−𝜀 𝑟≤𝑟, 𝑝−2 𝑝−2 󵄨 󵄨𝑝 𝑚(𝐵 ) 𝑚(𝐵 ) for 0 (53) =−𝑝∫ |∇𝑢| 𝜑 ∇𝑢 ⋅ ∇𝜑 (𝜑𝑢) − ∫ 𝑉󵄨𝜑𝑢󵄨 . 𝑟 𝑟 𝐵 𝐵 2𝑟 2𝑟 𝑐 (47) where 𝐸 denotes the complement of the set 𝐸.Taking𝑟0, 𝐵 ⊂Ω 𝑢=0 smallerifnecessary,wecanassumethat 𝑟0 .Since Using Young’s inequalities for (𝑝 − 1)/𝑝 + 1/𝑝,wecan =1 on 𝐸,byLemmas18 and 11, we have estimate the first integral in the right-hand side of (47)by 𝑛 𝑝 𝑝 𝑟 𝑐 1/𝑛 ∫ |𝑢| = ∫ |𝑢| ≤𝛽 [𝑚 (𝐸 ∩𝐵𝑟)] 𝑐 𝑚(𝐸∩𝐵 ) 𝑝/(𝑝−1) 𝑝 𝑝 −𝑝 󵄨 󵄨𝑝 𝑝 𝐵𝑟 𝐵𝑟∩𝐸 𝑟 (𝑝−1) 𝜀 ∫ |∇𝑢| 𝜑 +𝜀 ∫ 󵄨∇𝜑󵄨 |𝑢| . (48) 𝐵2𝑟 𝐵2𝑟 × ∫ |∇ (𝑢)|𝑝, 𝐵 (54) Also by result of Corollary 16,wecanestimatethesecond 𝑟 integral in the right-hand side of (47)by 𝑟𝑛𝜀1/𝑛 𝑝𝛽 ∫ |𝑢|𝑝−1 |∇ (𝑢)| . 1−1/𝑛 𝑝 𝑝 𝐵 󵄨 󵄨 󵄨 󵄨 [𝑚 𝑟(𝐵 )] (1−𝜀) 𝑟 𝜀 ∫ 󵄨∇ (𝜑𝑢)󵄨 +𝐶𝜀 ∫ 󵄨𝜑𝑢󵄨 . (49) 𝐵2𝑟 𝐵2𝑟 By Holder¨ inequality

Using these estimates in (47), we have 1/𝑛 1/𝑝 (𝑝−1)/𝑝 𝑝 𝜀 𝑝 𝑝 ∫ |𝑢| ≤𝐶 (∫ |∇ (𝑢)| ) (∫ |𝑢| ) , (55) 𝑝 𝑝 𝐵 1−𝜀 𝐵 𝐵 ∫ |∇𝑢| 𝜑 𝑟 𝑟 𝑟 𝐵 2𝑟 andbyusingtheYounginequality,weget ≤((𝑝−1) 𝜀𝑝/(𝑝−1) +𝜀)∫ ∇𝑢 𝑝𝜑𝑝 1/𝑛 | | (50) 𝑝 𝜀 𝑝−1 𝑝 𝑝−1 󵄨 𝑝󵄨 𝐵 ∫ |𝑢| ≤𝐶 𝑟(𝑟 ∫ |∇ (𝑢)| + ∫ 󵄨𝑢 󵄨). 2𝑟 1−𝜀 𝑟 󵄨 󵄨 𝐵𝑟 𝐵𝑟 𝐵𝑟 𝑝 𝑝 𝑝 𝑝󵄨 󵄨𝑝 (56) +(𝜀 +𝜀)∫ |𝑢| |∇𝑢| +𝐶𝜀 ∫ |∇𝑢| 󵄨𝜑󵄨 . 𝐵2𝑟 𝐵2𝑟 Finally, by Lemma 17,wehave Using the fact that |∇𝜑| ≤ 𝐶/𝑟, |𝜑| ≤ 𝐶/𝑟,and𝜑=1in 1/𝑛 𝑝 𝜀 󵄨 𝑝󵄨 𝐵𝑟, we immediately have inequality (45). ∫ |𝑢| ≤𝐶 ∫ 󵄨𝑢 󵄨 , 1−𝜀 󵄨 󵄨 (57) 𝐵𝑟 𝐵2𝑟 ∞ Lemma 18. Let 𝑢∈𝐶0 (𝐵𝑟) where 𝐵𝑟 is the ball of radius 𝑟 𝑛 where 𝐶 is independent of 𝜀 and of 𝑟,as𝑟→0.Now,letus in R and let 𝐸={𝑥∈𝐵𝑟 :𝑢(𝑥)=0}. Then, there exists a constant 𝛽 depending only on 𝑛,suchthat introduce the following function: 𝑛 𝑓 (𝑟) = ∫ |𝑢|𝑝. 𝑟 1/𝑛 (58) ∫ |𝑢| ≤𝛽 [𝑚 (𝐴)] ∫ |∇𝑢| , 𝐵 𝑚 (𝐸) (51) 2𝑟 𝐴 𝐵2𝑟 1/𝑛 And let us fix 𝑛∈N and choose 𝜀>0such that (𝐶𝜀 )(1− 𝐵 𝐴⊂𝐵 −𝑛 for all ball 𝑟, u as above, and all measurable sets 𝑟. 𝜀) ≤ 2 . Observe that consequently 𝑟0 depends on 𝑛.Then, (57)canbewrittenas To prove this lemma see [5]. Note that 𝑚(𝐴) and 𝑚(𝐸) −𝑛 denote the Lebesgue measure of the sets 𝐴 and 𝐸. 𝑓 (𝑟) ≤2 𝑓 (2𝑟) , for 𝑟≤𝑟0. (59) Journal of Function Spaces and Applications 7

Iterating (59), we get −𝑘𝑛 𝑘 𝑘−1 𝑓(𝜌)≤2 𝑓(2𝜌) , if 2 𝜌≤𝑟0. (60)

Now, given that 0<𝑟<𝑟0(𝑛),choose𝑘∈N,suchthat −𝑘 −𝑘+1 2 𝑟0 ≤𝑟≤2 𝑟0. (61) From (60), we obtain −𝑘𝑛 𝑘 −𝑘𝑛 𝑓 (𝑟) ≤2 𝑓(2𝑟) ≤ 2 𝑓 (2𝑟) . (62)

−𝑘 Since 2 ≤ 𝑟/𝑟0,wefinallyobtain 𝑟 𝑛 𝑓 (𝑟) ≤ ( ) 𝑓 (2𝑟0) . (63) 𝑟0 And thus, we have 𝑛 󵄨 󵄨𝑝 𝑟 ∫ 󵄨𝑢 (𝑦)󵄨 𝑑𝑦 ≤ ( ) 𝑓 (2𝑟 ) , 󵄨 󵄨 𝑟 0 (64) 𝐵𝑟(𝑥0) 0 and this shows that (7)holds,whichmeansthat𝑢 has a zero of infinite order in 𝑝-mean at 𝑥0.

Corollary 20. Equation (6) has a strong unique continuation property.

Acknowledgments The authors would like to thank the referee for the useful comments and suggestions which improved the presentation of this paper. The authors were supported by the Banco Central de Venezuela.

References

[1] M. Aizenman and B. Simon, “Brownian motion and harnack inequality for Schrodinger¨ operators,” Communications on Pure and Applied Mathematics,vol.35,no.2,pp.209–273,1982. [2] P. Zamboni, “Unique continuation for non-negative solutions of quasilinear elliptic equations,” Bulletin of the Australian Mathematical Society, vol. 64, no. 1, pp. 149–156, 2001. [3] M. Schechter, Spectra of Partial Differential Operators,vol. 14, North-Holland, Amsterdam, The Netherlands, 2nd edition, 1986. [4] C. L. Fefferman, “The uncertainty principle,” Bulletin of the American Mathematical Society,vol.9,no.2,pp.129–206,1983. [5] F. Chiarenza and M. Frasca, “A remark on a paper by C. Fefferman,” Proceedings of the American Mathematical Society, vol. 108, no. 2, pp. 407–409, 1990. [6] D. G. de Figueiredo and J.-P. Gossez, “Strict monotonicity of eigenvalues and unique continuation,” Communications in Partial Differential Equations,vol.17,no.1-2,pp.339–346,1992. [7] D. Jerison and C. E. Kening, “Unique continuation and absence of positive eigenvalues for Shrodinger¨ operators: with an appendix by E. M. Stein,” Annals of Mathematics,vol.121,no. 3, pp. 463–494, 1985. [8] N. Bourbaki, “Elements of Mathematics,” in General Topology, Part 1, Addison Wesley. [9]R.E.Castillo,“NonlinearBesselpotentialsandgeneralizations of the Kato class,” Proyecciones,vol.30,no.3,pp.285–294,2011. Hindawi Publishing Corporation Journal of Function Spaces and Applications Volume 2013, Article ID 302941, 14 pages http://dx.doi.org/10.1155/2013/302941

Research Article Generalized Lorentz Spaces and Applications

Hatem Mejjaoli

Department of Mathematics, College of Sciences, Taibah University, P.O. B 30002, Al Madinah Al Munawarah, Saudi Arabia

Correspondence should be addressed to Hatem Mejjaoli; [email protected]

Received 21 March 2013; Accepted 6 June 2013

Academic Editor: Jose Luis Sanchez

Copyright © 2013 Hatem Mejjaoli. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

𝑑 We define and study the Lorentz spaces associated with the Dunkl operators on R . Furthermore, we obtain the Strichartz estimates for the Dunkl-Schrodinger¨ equations under the generalized Lorentz norms. The Sobolev inequalities between the homogeneous Dunkl-Besov spaces and generalized Lorentz spaces are also considered.

“Dedicated to Khalifa Trimeche”`

𝑑 2 1. Introduction where △𝑘 =∑𝑗=1 𝑇𝑗 istheDunklLaplaceoperator.Westudy the previous equation focusing on the following problems: Dunkl operators 𝑇𝑗 (𝑗 = 1,...,𝑑)introducedbyDunkl in [1] are parameterized differential-difference operators on 𝑑 (1) Establish the Strichartz estimate under the general- R that are related to finite reflection groups. Over the ized Lorentz norms. last years, much attention has been paid to these operators in various mathematical (and even physical) directions. (2) Illustrate applications to well posedness. In this prospect, Dunkl operators are naturally connected with certain Schrodinger¨ operators for Calogero-Sutherland- type quantum many-body systems [2–4]. Moreover, Dunkl The contents of the paper are as follows. In Section 2, operators allow generalizations of several analytic structures, we recall some basic results about the harmonic analysis such as Laplace operator, Fourier transform, heat semigroup, associated with the Dunkl operators. In Section 3,weintro- wave equations, and Schrodinger¨ equations [5–11]. duce the homogeneous Dunkl-Besov spaces, the homoge- In the present paper, we intend to continue our study neous Dunkl-Triebel-Lizorkin spaces, and the homogeneous of generalized spaces of type Sobolev associated with Dunkl Dunkl-Riesz potential spaces and we prove new embedding operators started in [12, 13]. In this paper, we study the Sobolev theorem. In Section 4,werecallsomefactsabouta generalized Lorentz spaces, and we establish Sobolev inequal- real interpolation method. Next, we define the generalized ities between the homogeneous Dunkl-Besov spaces and Lorentz spaces and will pay special attention to the interpo- many spaces as the homogeneous Dunkl-Riesz spaces and lation definition of these spaces. Section 5 is devoted to give generalized Lorentz spaces. a complete picture of the Sobolev type inequalities for the As an application, we consider the Dunkl-Schrodinger¨ fractional Dunkl-Laplace operators. In Section 6,Strichartz equation estimatesforthesolutionoftheDunkl-Schrodinger¨ evolution equation are considered on a mixed normed space with generalized Lorentz norm with respect to the time variable. 𝑑 Finally, we establish Sobolev inequalities between the homo- 𝑖𝜕𝑡𝑢+△𝑘𝑢=𝑓(𝑡, 𝑥) , (𝑡, 𝑥) ∈ (0, ∞) × R (1) geneous Dunkl-Besov spaces and generalized Lorentz spaces, 𝑢|𝑡=0 =𝑔, andwegivemanyapplications. 2 Journal of Function Spaces and Applications

2. Preliminaries where 𝛼=(𝛼1,𝛼2,...,𝛼𝑑). Similarly, as ordinary derivatives, 1 𝑑 each 𝑇𝑗 satisfies for all 𝑓, 𝑔 in 𝐶 (R ) and at least one of them Inordertoconfirmthebasicandstandardnotations,we is 𝐺-invariant, briefly overview the theory of Dunkl operators and related

harmonic analysis. Main references are [1, 5, 6, 11, 14–17]. 𝑇𝑗 (𝑓𝑔) =𝑗 (𝑇 𝑓)𝑔+𝑓(𝑇𝑗𝑔) , (6)

2.1. Root System, Reflection Group and Multiplicity Function. 𝑓 𝐶1(R𝑑) 𝑔 S(R𝑑) 𝑑 and for all in 𝑏 and in , Let R betheEuclideanspaceequippedwithascalarproduct 𝑑 ⟨, ⟩,andlet||𝑥|| = √⟨𝑥, 𝑥⟩.For𝛼 in R \{0}, 𝜎𝛼 denotes the ∫ 𝑇𝑗𝑓 (𝑥) 𝑔 (𝑥) 𝜔𝑘 (𝑥) 𝑑𝑥=−∫ 𝑓 (𝑥) 𝑇𝑗𝑔 (𝑥) 𝜔𝑘 (𝑥) 𝑑𝑥. 𝑑 𝑑 𝑑 reflection in the hyperplane 𝐻𝛼 ⊂ R perpendicular to 𝛼, R R 𝑑 −2 (7) that is, for 𝑥∈R , 𝜎𝛼(𝑥) = 𝑥 − 2‖𝛼‖ ⟨𝛼, 𝑥⟩𝛼.Afiniteset 𝑑 𝑅⊂R \{0}is called a root system if 𝑅∩R𝛼 = {±𝛼} and Furthermore, according to [1, 14], the Dunkl operators 𝜎𝛼𝑅=𝑅for all 𝛼∈𝑅.Wenormalizeeach𝛼∈𝑅as ⟨𝛼, 𝛼⟩. =2 𝑑 𝑇𝑗,1≤𝑗≤𝑑commute, and there exists the so-called Dunkl’s We fix a 𝛽∈R \∪𝛼∈𝑅𝐻𝛼 and define a positive root system 𝑅+ intertwining operator 𝑉𝑘 such that 𝑇𝑗𝑉𝑘 =𝑉𝑘(𝜕/𝜕𝑥𝑗) for of 𝑅 as 𝑅+ ={𝛼∈𝑅|⟨𝛼,𝛽⟩>0}.Thereflections𝜎𝛼,𝛼 ∈, 𝑅 1≤𝑗≤𝑑and 𝑉𝑘(1) = 1. We define the Dunkl-Laplace generate a finite group 𝐺⊂𝑂(𝑑), called the reflection group. 𝑑 operator △𝑘 on R by Afunction𝑘:𝑅 → C on 𝑅 is called a multiplicity function if it is invariant under the action of 𝐺.Weintroducetheindex 𝑑 𝛾 2 as △𝑘𝑓 (𝑥) := ∑𝑇𝑗 𝑓 (𝑥) 𝛾=𝛾(𝑘) = ∑ 𝑘 (𝛼) . 𝑗=1 𝛼∈𝑅 (2) + ⟨∇𝑓 (𝑥) ,𝛼⟩ =△𝑓(𝑥) +2∑ 𝑘 (𝛼) ( 𝑘(𝛼) ≥0 Throughout this paper, we will assume that for all + ⟨𝛼, 𝑥⟩ 𝑑 𝛼∈𝑅 𝛼∈𝑅. We denote by 𝜔𝑘 the weight function on R given by 𝑓 (𝑥) −𝑓(𝜎𝛼 (𝑥)) 2𝑘(𝛼) − ), 𝜔 (𝑥) = ∏ |⟨𝛼, 𝑥⟩| , 2 𝑘 (3) ⟨𝛼, 𝑥⟩ 𝛼∈𝑅 + (8) which is invariant and homogeneous of degree 2𝛾.Inthecase 𝑑 △ ∇ that the reflection group 𝐺 is the group Z2 of sign changes, where and are the usual Euclidean Laplacian and nabla R𝑑 the weight function 𝜔𝑘 is a product function of the form operators on , respectively. Since the Dunkl operators 𝑑 𝑘𝑗 commute, their joint eigenvalue problem is significant, and ∏ |𝑥𝑗| , 𝑘𝑗 ≥0. We denote by 𝑐𝑘 the Mehta-type constant 𝑗=1 𝑦∈R𝑑 defined by for each ,thesystem

2 −||𝑥|| /2 𝑇𝑗𝑢(𝑥,𝑦)=𝑦𝑗𝑢(𝑥,𝑦), 𝑗=1,...,𝑑,𝑢(0,𝑦)=1 (9) 𝑐𝑘 = ∫ 𝑒 𝜔𝑘 (𝑥) 𝑑𝑥. (4) R𝑑 𝑑 We note that Etingof (cf. [18]) has given a derivation of the admits a unique analytic solution 𝐾(𝑥, 𝑦), 𝑥∈R ,calledthe 𝑑 𝑑 Mehta-type constant valid for all finite reflection groups. Dunkl kernel, which has a holomorphic extension to C ×C . 𝑑 Inthefollowing,wedenoteby For 𝑥, 𝑦 ∈ C , the kernel satisfies 𝐶(R𝑑) R𝑑 : the space of continuous functions on . (a) 𝐾(𝑥, 𝑦) = 𝐾(𝑦,, 𝑥) 𝐶𝑝(R𝑑) 𝐶𝑝 R𝑑 : the space of functions of class on . (b) 𝐾(𝜆𝑥, 𝑦) = 𝐾(𝑥, 𝜆𝑦) for 𝜆∈C, 𝐶𝑝(R𝑑): 𝐶𝑝 𝑏 thespaceofboundedfunctionsofclass . (c) 𝐾(𝑔𝑥, 𝑔𝑦) = 𝐾(𝑥, 𝑦) for 𝑔∈𝐺. 𝑑 ∞ 𝑑 E(R ): the space of 𝐶 functions on R . 𝑑 S(R𝑑) 2.3. The Dunkl Transform. For functions 𝑓 on R , we define : the Schwartz space of rapidly decreasing functions on 𝑝 𝑑 𝐿 𝑓 𝜔 (𝑥)𝑑𝑥 R . -norms of with respect to 𝑘 as 𝑑 ∞ 𝑑 𝐷(R ) 𝐶 R 1/𝑝 : the space of functions on which are of compact 󵄩 󵄩 󵄨 󵄨𝑝 󵄩𝑓󵄩 𝑝 = (∫ 󵄨𝑓 (𝑥)󵄨 𝜔 (𝑥) 𝑑𝑥) , support. 󵄩 󵄩𝐿 (R𝑑) 󵄨 󵄨 𝑘 (10) 𝑘 R𝑑 󸀠 𝑑 𝑑 S (R ): the space of temperate distributions on R . if 1≤𝑝<∞and ‖𝑓‖𝐿∞(R𝑑) = ess sup𝑥∈R𝑑 |𝑓(𝑥)|. We denote 𝑘 𝑘:𝑅 → C 𝑝 𝑑 𝑑 2.2. The Dunkl Operators. Let be a multiplicity by 𝐿𝑘(R ) the space of all measurable functions 𝑓 on R with 𝑅 𝑅 𝑅 𝑝 function on and + afixedpositiverootsystemof .Then, finite 𝐿𝑘-norm. 1 𝑑 1 𝑑 𝑇𝑗,1≤𝑗≤𝑑 𝐶 (R ) the Dunkl operators , are defined on by The Dunkl transform F𝐷 on 𝐿𝑘(R ) is given by 𝜕 𝑓 (𝑥) −𝑓(𝜎𝛼 (𝑥)) 1 𝑇𝑗𝑓 (𝑥) = 𝑓 (𝑥) + ∑ 𝑘 (𝛼) 𝛼𝑗 , (5) F𝐷 (𝑓) (𝑦) = ∫ 𝑓 (𝑥) 𝐾 (𝑥, −𝑖𝑦)𝑘 𝜔 (𝑥) 𝑑𝑥. 𝜕𝑥𝑗 ⟨𝛼, 𝑥⟩ 𝑑 (11) 𝛼∈𝑅+ 𝑐𝑘 R Journal of Function Spaces and Applications 3

Some basic properties are the following (cf. [5, 6]). For all Clearly, 𝜏𝑦𝑓(𝑥)𝑥 =𝜏 𝑓(𝑦),andbyusingtheDunkl’sinter- 1 𝑑 𝑉 𝜏 𝑓 𝑓∈𝐿𝑘(R ), twining operator 𝑘, 𝑦 is related to the usual translation −1 −1 as 𝜏𝑦𝑓(𝑥) =𝑘 (𝑉 )𝑥(𝑉𝑘)𝑦((𝑉𝑘) (𝑓)(𝑥+𝑦)) (cf. [11, 17]). Hence, (a) ‖ F𝐷(𝑓)‖𝐿∞(R𝑑) ≤𝑐𝑘 ‖𝑓‖𝐿1 (R𝑑), 𝑘 𝑘 𝜏 𝑓∈E(R𝑑) 2𝛾+𝑑 𝑦 canalsobedefinedfor . We define the Dunkl (b) F𝐷(𝑓(⋅/𝜆))(𝑦) =𝜆 F𝐷(𝑓)(𝜆𝑦) for 𝜆>0, 𝑑 convolution product 𝑓∗𝐷 𝑔 of functions 𝑓, 𝑔 ∈ S(R ) as 1 𝑑 (c) if F𝐷(𝑓) belongs to 𝐿𝑘(R ),then follows: 1 𝑓(𝑦)= ∫ F𝐷 (𝑓) (𝑥) 𝐾 (𝑖𝑥,𝑘 𝑦)𝜔 (𝑥) 𝑑𝑥, a.e., (12) 𝑓∗𝐷 𝑔 (𝑥) = ∫ 𝜏𝑥𝑓 (−𝑦) 𝑔 (𝑦) 𝜔𝑘 (𝑦) 𝑑𝑦. (19) 𝑐𝑘 R𝑑 R𝑑 𝑑 and moreover, for all 𝑓∈S(R ), This convolution is commutative and associative (cf. [17]). F (𝜏 𝑓)(𝑥) = 𝐾(𝑖𝑥,F 𝑦) (𝑓)(𝑥) (d) F𝐷(𝑇𝑗𝑓)(𝑦) =𝑗 𝑖𝑦 F𝐷(𝑓)(𝑦), Since 𝐷 𝑦 𝐷 by the previous definition of 𝜏𝑦𝑓,itfollowsthat (e) if we define F𝐷(𝑓)(𝑦) = F𝐷(𝑓)(−𝑦),then 𝑑 𝑑 (a) for all 𝑓, 𝑔 ∈ 𝐷(R ) (resp., S(R )), 𝑓∗𝐷 𝑔 belongs to F𝐷F𝐷 = F𝐷F𝐷 =𝐼𝑑. (13) 𝑑 𝑑 𝐷(R ) (resp. S(R ))and Proposition 1. The Dunkl transform F𝐷 is a topological 𝑑 𝑑 F (𝑓∗ 𝑔) (𝑦) = F (𝑓) (𝑦) F (𝑔) (𝑦) . isomorphism from S(R ) onto itself, and for all f in S(R ), 𝐷 𝐷 𝐷 𝐷 (20) 󵄨 󵄨2 󵄨 󵄨2 ∫ 󵄨𝑓 (𝑥)󵄨 𝜔𝑘 (𝑥) 𝑑𝑥 = ∫ 󵄨F𝐷 (𝑓) (𝜉)󵄨 𝜔𝑘 (𝜉) 𝑑𝜉. (14) Moreover, as pointed in [16]andSections4 and 7,the R𝑑 R𝑑 𝑝 𝑑 operator 𝑓→𝑓∗𝐷 𝑔 is bounded on 𝐿𝑘(R ), 1≤𝑝≤∞, 1 𝑑 In particular, the Dunkl transform 𝑓→F𝐷(𝑓) can be 𝑔 𝐿 (R ) 2 𝑑 provided that is a radial function in 𝑘 or an arbitrary uniquely extended to an isometric isomorphism on 𝐿 (R ). 1 𝑑 𝑑 𝑘 function in 𝐿𝑘(R ) for 𝐺=Z2 .Hence,thestandard argument yields the following Young’s inequality. We define the tempered distribution T𝑓 associated with 𝑝 𝑑 1≤𝑝,𝑞,𝑟≤∞ 1/𝑝 + 1/𝑞 − 1/𝑟 =1 𝑓∈𝐿𝑘(R ) by (b) Let such that . 𝑝 𝑑 𝑞 𝑑 Assume that 𝑓∈𝐿𝑘(R ) and 𝑔∈𝐿𝑘(R ).If ⟨T ,𝜙⟩ = ∫ 𝑓 (𝑥) 𝜙 (𝑥) 𝜔 (𝑥) 𝑑𝑥, 𝑑 𝑓 𝑘 (15) ‖𝜏𝑥𝑔‖𝐿𝑞 (R𝑑) ≤ 𝐶‖𝑔‖𝐿𝑞 (R𝑑) for all 𝑥∈R ,then𝑓∗𝐷 𝑔∈ R𝑑 𝑘 𝑘 𝑟 𝑑 𝑑 𝐿𝑘(R ) and for 𝜙∈S(R ) and denote by ⟨𝑓,𝜙⟩𝑘 the integral in the right hand side. 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩𝑓∗𝐷 𝑔󵄩𝐿𝑟 (R𝑑) ≤𝐶󵄩𝑓󵄩𝐿𝑝 (R𝑑)󵄩𝑔󵄩𝐿𝑞 (R𝑑). 𝑘 𝑘 𝑘 (21) Definition 2. The Dunkl transform F𝐷(𝜏) of a distribution 󸀠 𝑑 𝜏∈S (R ) is defined by Definition 4. The Dunkl convolution product of a distribu- 󸀠 𝑑 𝑑 tion 𝑆 in S (R ) and a function 𝜙 in S(R ) is the function ⟨F𝐷 (𝜏) ,𝜙⟩ = ⟨𝜏, F𝐷 (𝜙)⟩ , (16) 𝑆∗𝐷 𝜙 defined by 𝑑 for 𝜙∈S(R ). 𝑆∗𝐷 𝜙 (𝑥) =⟨𝑆𝑦,𝜏−𝑦𝜙 (𝑥)⟩. (22) 𝑝 𝑑 In particular, for 𝑓∈𝐿𝑘(R ),itfollowsthatfor 𝑑 Proposition 5. 𝑓 𝐿𝑝(R𝑑) 1≤𝑝≤∞ 𝜙 𝜙∈S(R ), Let be in 𝑘 , ,and in 𝑑 S(R ).Then,thedistributionT𝑓∗𝐷 𝜙 is given by the function ⟨F (𝑓) , 𝜙⟩ =⟨F (T ),𝜙⟩=⟨T , F (𝜙)⟩ 𝐷 𝐷 𝑓 𝑓 𝐷 𝑓∗𝐷 𝜙.Ifoneassumesthat𝜙 is arbitrary for 𝑑=1and radial (17) 𝑑≥2 T ∗ 𝜙 𝐿𝑝(R𝑑) =⟨𝑓,F (𝜙)⟩ . for ,then 𝑓 𝐷 belongs to 𝑘 .Moreover,forall 𝐷 𝑘 𝑑 𝜓∈S(R ), Proposition 3. The Dunkl transform F𝐷 is a topological 󸀠 𝑑 isomorphism from S (R ) onto itself. ⟨T ∗ 𝜙, 𝜓⟩ = ⟨𝑓,̌𝜙∗ 𝜓̌⟩ , 𝑓 𝐷 𝐷 𝑘 (23)

2.4. The Dunkl Convolution. ByusingtheDunklkernelin where 𝜓(𝑥)̌ = 𝜓(−𝑥),and Section 2.2, we introduce a generalized translation and a convolution structure in our Dunkl setting. For a function F𝐷 (T𝑓∗𝐷 𝜙) = F𝐷 (T𝑓) F𝐷 (𝜙) . 𝑑 𝑑 (24) 𝑓∈S(R ) and 𝑦∈R , the Dunkl translation 𝜏𝑦𝑓 is defined 󸀠 𝑑 by For each 𝑢∈S (R ), we define the distributions 1 𝑇𝑗𝑢, 1≤𝑗≤𝑑,by 𝜏𝑦𝑓 (𝑥) = ∫ F𝐷 (𝑓) (𝑧) 𝐾 (𝑖𝑥,) 𝑧 𝐾(𝑖𝑦,𝑧)𝜔𝑘 (𝑧) 𝑑𝑧. 𝑐𝑘 R𝑑 (18) ⟨𝑇𝑗𝑢, 𝜓⟩ = −𝑗 ⟨𝑢,𝑇 𝜓⟩ , (25) 4 Journal of Function Spaces and Applications

𝑑 for all 𝜓∈S(R ).Then,⟨△𝑘𝑢, 𝜓⟩ =𝑘 ⟨𝑢,△ 𝜓⟩,andthese We put distributions satisfy the following properties (see Section 2.3 𝜙=̃ F−1 (𝜑)̃ , 𝜒=̃ F−1 (𝜓)̃ . (d)): 𝐷 𝐷 (30) 󸀠 𝑑 Definition 8. Let one denote by S ℎ,𝑘(R ) the space of F𝐷 (𝑇𝑗𝑢) = 𝑖𝑦𝑗F𝐷 (𝑢) , (26) tempered distribution such that 󵄩 󵄩2 F𝐷 (△𝑘𝑢) =󵄩 − 𝑦󵄩 F𝐷 (𝑢) . 󸀠 𝑑 lim 𝑆𝑗𝑢=0 in S (R ). 𝑗→−∞ (31) In the following, we denote T𝑓 given by (15)by𝑓 for simplicity. On the follow, we define analogues of the homogeneous Besov, Triebel-Lizorkin, and Riesz potential spaces associated R𝑑 Ḃ𝑠,𝑘 Ḟ𝑠,𝑘 (R𝑑) Ḣ𝑠 with the Dunkl operators on and obtain their basic 3. 𝑝,𝑞, 𝑝,𝑞 ,and 𝑝,𝑘 Spaces and properties. Basic Properties From now, we make the convention that for all non- 𝑟 1/𝑟 negative sequence {𝑎𝑞}𝑞∈Z,thenotation(∑𝑞 𝑎𝑞) stands for Oneofthemaintoolsinthispaperisthehomogeneous sup𝑞𝑎𝑞 in the case 𝑟=∞. Littlewood-Paley decompositions of distributions associated with the Dunkl operators into dyadic blocs of frequencies. Definition 9. Let 𝑠∈R and 𝑝, 𝑞 ∈ [1, ∞].Thehomogeneous ̇𝑠,𝑘 𝑑 Dunkl-Besov spaces B (R ) are the space of distribution in Lemma 6. C 0 𝑝,𝑞 Let one define by the ring of center ,ofsmall S󸀠 (R𝑑) radius 1/2,andgreatradius2. There exist two radial functions ℎ,𝑘 such that 𝜓 and 𝜑 thevaluesofwhichareintheinterval[0, 1] belonging 1/𝑞 𝑑 󵄩 󵄩 𝑞 to 𝐷(R ) such that 󵄩 󵄩 𝑠𝑗 󵄩 󵄩 󵄩𝑓󵄩 ̇𝑠,𝑘 𝑑 := (∑(2 󵄩Δ 𝑗𝑓󵄩 𝑝 𝑑 ) ) <∞. (32) 󵄩 󵄩B𝑝,𝑞(R ) 󵄩 󵄩𝐿 (R ) 𝑗∈Z 𝑘 supp 𝜓⊂𝐵(0, 1) , supp 𝜑⊂C 𝑠∈R 1≤𝑝,𝑞≤∞ ∀𝜉 ∈ R𝑑,𝜓(𝜉) + ∑𝜑(2−𝑗𝜉) = 1 Definition 10. Let and ,the Ḟ𝑠,𝑘 (R𝑑) 𝑗≥0 homogeneous Dunkl-Triebel-Lizorkin space 𝑝,𝑞 is the 󸀠 𝑑 −𝑗 space of distribution in S ℎ,𝑘(R ) such that ∀𝜉 ∈ C, ∑𝜑(2 𝜉) = 1 (27) 𝑗∈Z 󵄩 󵄩 󵄩 1/𝑞󵄩 󵄩 󵄨 󵄨𝑞 󵄩 |𝑛−𝑚| ≥2󳨐⇒ 𝜑(2−𝑛⋅) ∩ 𝜑(2−𝑚⋅) = 0 󵄩 󵄩 󵄩 𝑠𝑗𝑞 󵄨 󵄨 󵄩 supp supp 󵄩𝑓󵄩 ̇𝑠,𝑘 𝑑 := 󵄩(∑2 󵄨Δ 𝑗𝑓󵄨 ) 󵄩 <∞. 󵄩 󵄩F𝑝,𝑞(R ) 󵄩 󵄨 󵄨 󵄩 (33) 󵄩 𝑗∈Z 󵄩 −𝑗 󵄩 󵄩𝐿𝑝 (R𝑑) 𝑗≥1󳨐⇒supp 𝜓∩supp 𝜑(2 ⋅) = 0. 𝑘 𝑠/2 𝑠/2 Letusrecallthattheoperators(−△𝑘) and (𝐼−△𝑘) Notations. We denote by have been defined, respectively, by (cf. [19]) 𝑠/2 −1 𝜉 −1 𝑠 Δ 𝑓=F (𝜑 ( ) F (𝑓)) , 𝑆 𝑓= ∑ Δ 𝑓, (−△𝑘) 𝑓=F𝐷 (‖⋅‖ F𝐷𝑓) , 𝑗 𝐷 2𝑗 𝐷 𝑗 𝑛 𝑛≤𝑗−1 (34) 𝑠/2 −1 2 𝑠/2 (𝐼−△𝑘) 𝑓=F𝐷 ((1 + ‖⋅‖ ) F𝐷𝑓) . ∀𝑗 ∈ Z. (𝐼−△ )−𝑠/2 𝑠>0 (28) The operators 𝑘 ,for , are called Dunkl-Bessel potential operators, and they are given by Dunkl convolution The distribution Δ 𝑗𝑓 is called the jth dyadic block of the with the Dunkl-Bessel potential 𝑓 homogeneous Littlewood-Paley decomposition of associ- −𝑠/2 ated with the Dunkl operators. (𝐼−△𝑘) 𝑓=𝑓∗𝐷 𝐵𝑘,𝑠, (35) −1 where Throughout this paper, we define 𝜙 and 𝜒 by 𝜙=F𝐷 (𝜑) ∞ −1 1 2 𝑑𝑡 and 𝜒=F𝐷 (𝜓). −𝑡 −‖𝑦‖ /4𝑡 (𝑠−𝑑−2𝛾)/2 𝐵𝑘,𝑠 (𝑦) = ∫ 𝑒 𝑒 𝑡 . (36) When dealing with the Littlewood-Paley decomposition, Γ (𝑠/2) 0 𝑡 it is convenient to introduce the functions 𝜓̃ and 𝜑̃ belonging 𝑑 1 𝑑 𝑑 We note that 𝐵𝑘,𝑠(𝑦) ≥ 0 for all 𝑦∈R , 𝐵𝑘,𝑠 ∈𝐿 (R ),and to 𝐷(R ) such that 𝜓≡1̃ on supp 𝜓 and 𝜑≡1̃ on supp 𝜑. 𝑘 󵄩 󵄩𝑠−𝑑−2𝛾 2 󵄩 󵄩 𝐵 (𝑦) ≤󵄩 𝐶 𝑦󵄩 𝑒−‖𝑦‖ /2, 󵄩𝑦󵄩 >0. (37) Remark 7. We remark that 𝑘,𝑠 󵄩 󵄩 󵄩 󵄩 𝜉 Definition 11. For 𝑠∈R and 1≤𝑝≤∞,theDunkl- F (𝑆 𝑓) (𝜉) = 𝜓(̃ ) F (𝑆 𝑓) (𝜉) , 𝐻𝑠 (R𝑑) 𝐷 𝑗 2𝑗 𝐷 𝑗 Bessel potential space 𝑝,𝑘 is defined as the space 𝑠/2 𝑝 𝑑 (29) (𝐼−△𝑘) (𝐿𝑘(R )), equipped with the norm ‖𝑓‖𝐻𝑠 (R𝑑) = 𝜉 𝑝,𝑘 𝑠/2 F𝐷 (Δ 𝑗𝑓) (𝜉) = 𝜑(̃ ) F𝐷 (Δ 𝑗𝑓) (𝜉) . 𝑗 ‖(𝐼 −𝑘 △ ) 𝑓‖𝐿𝑝 (R𝑑). 2 𝑘 Journal of Function Spaces and Applications 5

𝑠 𝑑 𝑠 𝑑 Furthermore, 𝑝=2, 𝐻2,𝑘(R ):=𝐻𝑘(R ). We define 𝑁(𝑥) as the largest index such that

−𝑠/2 𝑗𝑏 −𝑗𝑏 󵄨 󵄨 −𝑎𝑗 𝑎𝑗 󵄨 󵄨 Definition 12. The operators (−△𝑘) , 0<𝑠<𝑑+2𝛾,are 2 (2 󵄨Δ 𝑓 (𝑥)󵄨)≤2 (2 󵄨Δ 𝑓 (𝑥)󵄨), 𝑘 sup 󵄨 𝑗 󵄨 sup 󵄨 𝑗 󵄨 (45) called Dunkl-Riesz potentials operators, and one has 𝑗∈Z 𝑗∈Z (−△ )−𝑠/2𝑓=𝑅 ∗ 𝑓, 𝑘 𝑘,𝑠 𝐷 (38) and we write where 𝑅𝑘,𝑠 is the Dunkl-Riesz potential given by 󵄨 󵄨 󵄨𝑓 (𝑥)󵄨 󵄩 󵄩𝑠−𝑑−2𝛾 󵄨 󵄨 𝑅𝑘,𝑠 (𝑦) := 𝐶 (𝑘, 𝑠,) 𝑑 󵄩𝑦󵄩 , 𝑗𝑏 −𝑗𝑏 󵄨 󵄨 ≤ ∑ 2 sup (2 󵄨Δ 𝑗𝑓 (𝑥)󵄨) 𝑗∈Z Γ((𝑑+2𝛾−𝑠)/2) (39) 𝑗≤𝑁(𝑥) where 𝐶 (𝑘, 𝑠,) 𝑑 = . 󵄨 󵄨 2(𝑑+2𝛾−𝑠)/2Γ (𝑠/2) −𝑎𝑗 𝑎𝑗 󵄨 󵄨 + ∑ 2 sup (2 󵄨Δ 𝑗𝑓 (𝑥)󵄨) 𝑗>𝑁(𝑥) 𝑗∈Z Definition 13. For 𝑠∈R and 1≤𝑝≤∞, the homogeneous ̇𝑠 𝑑 Dunkl-Riesz potential space H (R ) is defined as the space 𝑏/(𝑎+𝑏) 𝑎/(𝑎+𝑏) 𝑝,𝑘 𝑎𝑗 󵄨 󵄨 −𝑏𝑗 󵄨 󵄨 𝑠/2 𝑝 𝑑 ≤𝐶(sup 2 󵄨Δ 𝑗𝑓 (𝑥)󵄨) (sup 2 󵄨Δ 𝑗𝑓 (𝑥)󵄨) . (−△𝑘) (𝐿 (R )), equipped with the norm ‖𝑓‖Ḣ𝑠 (R𝑑) = 󵄨 󵄨 󵄨 󵄨 𝑘 𝑝,𝑘 𝑗∈Z 𝑗∈Z 𝑠/2 ‖(−△𝑘) 𝑓‖𝐿𝑝 (R𝑑). (46) 𝑘

Proposition 14. Let 𝑞 ∈ (1, ∞),andlet𝑠∈R such that Thus, (42) is proved. In order to obtain (43), it is enough to 0 < 𝑠 < (𝑑 + 2𝛾)/𝑞, then one has apply Holder’s¨ inequality in the expression previous since we have 𝜃 = 𝑎/(𝑎 + 𝑏) ∈ (0,1) and 1/𝑝 = (1 − 𝜃)/𝑞1 +𝜃/𝑞2. ̇𝑠,𝑘 𝑑 B𝑞,𝑞 (R ) Corollary 16. Let 𝑞 ∈ (1, ∞),andlet𝑠∈R such that ̇𝑠,𝑘 𝑑 ̇𝑠,𝑘 𝑑 ̇𝑠−(𝑑+2𝛾)/𝑞,𝑘 𝑑 = F𝑞,𝑞 (R )󳨅→F𝑞,∞ (R )󳨅→F∞,∞ (R ), 0 < 𝑠 < (𝑑 + 2𝛾)/𝑞, then one has (40) 󵄩 󵄩 󵄩 󵄩1−𝑞/𝑝 󵄩 󵄩𝑞/𝑝 󵄩𝑓󵄩 𝑝 ≤𝐶󵄩𝑓󵄩 󵄩𝑓󵄩 , 𝑠 𝑑 󵄩 󵄩𝐿 (R𝑑) 󵄩 󵄩 ̇−((2𝛾+𝑑)/𝑞−𝑠),𝑘 𝑑 󵄩 󵄩 ̇𝑠,𝑘 𝑑 (47) ̇ 𝑘 B∞,∞ (R ) B𝑞,𝑞(R ) H𝑞,𝑘 (R ) 󵄩 󵄩 󵄩 󵄩1−𝑞/𝑝 󵄩 󵄩𝑞/𝑝 𝑠,𝑘 𝑑 𝑠,𝑘 𝑑 𝑠−(𝑑+2𝛾)/𝑞,𝑘 𝑑 󵄩𝑓󵄩 𝑝 ≤𝐶󵄩𝑓󵄩 󵄩𝑓󵄩 , = Ḟ(R )󳨅→Ḟ(R )󳨅→Ḟ (R ). 󵄩 󵄩𝐿 (R𝑑) 󵄩 󵄩 ̇−((2𝛾+𝑑)/𝑞−𝑠),𝑘,𝑘 𝑑 󵄩 󵄩Ḣ𝑠 R𝑑 (48) 𝑞,2 𝑞,∞ ∞,∞ 𝑘 B∞,∞ (R ) 𝑞,𝑘( ) (41) where 𝑝 = 𝑞(2𝛾+ 𝑑)/(2𝛾+ 𝑑−𝑞𝑠) . Proof. We obtain these results by similar ideas used in the nonhomogeneous case. (cf. [12]). Proof. We take 𝑎=𝑠>0, −𝑏 = 𝑠 − (𝑑 + 2𝛾)/𝑞 <0, 𝑞1 =𝑞, and 𝑞2 =∞, and we deduce the inequality (47)fromthe Theorem 15. 𝑎, 𝑏 >0 𝑞 ,𝑞 ∈[1,∞] Let ,andlet 1 2 .Let relations (43)and(40).Inthesameway,wededuce(48)from 𝜃 = 𝑎/(𝑎 + 𝑏) ∈ (0,1) 1/𝑝 = (1 − 𝜃)/𝑞 + ,andlet 1 the relations (43)and(41). 𝜃/𝑞2. Then, there exists a constant 𝐶 such that for every 𝑓∈Ḟ𝑎,𝑘 (R𝑑)∩Ḟ−𝑏,𝑘 (R𝑑) Theorem 17 (1) 𝑠>0 𝑝, 𝑟 ∈ [1, ∞] 𝑞1,∞ 𝑞2,∞ , then one has (see [13]). Let and .Then, ̇𝑠,𝑘 𝑑 ∞ 𝑑 B𝑝,𝑟(R )∩𝐿𝑘 (R ) is an algebra, and there exists a positive 󵄨 󵄨 1−𝜃 󵄨 󵄨 𝜃 󵄨 󵄨 𝑎𝑗 󵄨 󵄨 −𝑏𝑗 󵄨 󵄨 constant 𝐶 such that 󵄨𝑓 (𝑥)󵄨 ≤𝐶(sup 2 󵄨Δ 𝑗𝑓 (𝑥)󵄨) (sup 2 󵄨Δ 𝑗𝑓 (𝑥)󵄨) . 𝑗∈Z 𝑗∈Z ‖𝑢V‖ ̇𝑠,𝑘 𝑑 (42) B𝑝,𝑟(R )

In particular, one gets ≤𝐶[‖𝑢‖𝐿∞(R𝑑)‖V‖ ̇𝑠,𝑘 𝑑 + ‖V‖𝐿∞(R𝑑)‖𝑢‖ ̇𝑠,𝑘 𝑑 ]. 𝑘 B𝑝,𝑟(R ) 𝑘 B𝑝,𝑟(R ) 󵄩 󵄩 󵄩 󵄩1−𝜃 󵄩 󵄩𝜃 (49) 󵄩𝑓󵄩𝐿𝑝 (R𝑑) ≤𝐶󵄩𝑓󵄩Ḟ𝑎,𝑘 R𝑑 󵄩𝑓󵄩Ḟ−𝑏,𝑘 R𝑑 . (43) 𝑘 𝑞1,∞( ) 𝑞2,∞( )

Proof. Let 𝑓 beaSchwartzclass,wehave (2) Moreover, for any (𝑠1,𝑠2), any 𝑝2 and any 𝑟2 such that 𝑠 +𝑠 > (𝑑 + 2𝛾)/𝑝 𝑠 < (𝑑 + 2𝛾)/𝑝 󵄨 󵄨 1 2 1 and 1 1,onehas 󵄨 󵄨 󵄨 󵄨 󵄨𝑓 (𝑥)󵄨 ≤ ∑ 󵄨Δ 𝑗𝑓 (𝑥)󵄨 𝑗∈Z ‖𝑢V‖Ḃ𝑠,𝑘 (R𝑑) 𝑝2,𝑟2 󵄨 󵄨 ≤ ∑ (2−𝑎𝑗 (2𝑎𝑗 󵄨Δ 𝑓 (𝑥)󵄨), min sup 󵄨 𝑗 󵄨 ≤𝐶[‖𝑢‖ 𝑠 ,𝑘 ‖V‖ 𝑠 ,𝑘 + ‖𝑢‖ 𝑠 ,𝑘 ‖V‖ 𝑠 ,𝑘 ], 󵄨 󵄨 (44) Ḃ1 (R𝑑) Ḃ2 (R𝑑) Ḃ2 (R𝑑) Ḃ1 (R𝑑) 𝑗∈Z 𝑗∈Z 𝑝1,∞ 𝑝2,𝑟2 𝑝2,𝑟2 𝑝1,∞ (50) 󵄨 󵄨 𝑗𝑏 −𝑗𝑏 󵄨 󵄨 2 sup (2 󵄨Δ 𝑗𝑓 (𝑥)󵄨)) . 𝑗∈Z where 𝑠=𝑠1 +𝑠2 − (𝑑 + 2𝛾)/𝑝1. 6 Journal of Function Spaces and Applications

(3) Moreover, for any (𝑠1,𝑠2), any 𝑝2 and any (𝑟1,𝑟2) such Definition 19 (K-method of interpolation). For 0<𝜃<1 that 𝑠1 +𝑠2 > (𝑑 + 2𝛾)/𝑝1,𝑠1 < (𝑑 + 2𝛾)/𝑝1,1/𝑟1 +1/𝑟2 =1, and 1≤𝑞≤∞, the space [𝐴0,𝐴1]𝜃,𝑞,𝐾 is defined by one has 𝑎∈[𝐴0,𝐴1]𝜃,𝑞,𝐾 if and only if 𝑎∈𝐴0 +𝐴1,and (2−𝑗𝜃𝐾(2𝑗,𝑎;𝐴 ,𝐴 )) ∈𝑙𝑞(Z) ‖𝑢V‖ ̇𝑠,𝑘 𝑑 0 1 𝑗∈Z . B𝑝,∞(R ) The norm of [𝐴0,𝐴1]𝜃,𝑞,𝐾 is defined as follows:

≤𝐶[‖𝑢‖ 𝑠 ,𝑘 ‖V‖ 𝑠 ,𝑘 + ‖𝑢‖ 𝑠 ,𝑘 ‖V‖ 𝑠 ,𝑘 ]. Ḃ1 (R𝑑) Ḃ2 (R𝑑) Ḃ2 (R𝑑) Ḃ1 (R𝑑) 1/𝑞 𝑝1,𝑟1 𝑝2,𝑟2 𝑝2,𝑟2 𝑝1,𝑟1 𝑞 (51) ‖𝑎‖ =(∑2−𝑗𝜃𝑞𝐾(2𝑗,𝑎;𝐴 ,𝐴 ) ) . (56) [𝐴0,𝐴1]𝜃,𝑞,𝐾 0 1 𝑗∈Z (4) Moreover, for any (𝑠1,𝑠2), any (𝑝1,𝑝2,𝑝) and any (𝑟 ,𝑟 ) 𝑠 < (𝑑 + 2𝛾)/𝑝 ,𝑠+𝑠 > (𝑑 + 2𝛾)(1/𝑝 + 1 2 such that 𝑗 𝑗 1 2 1 Proposition 20 (Equivalence theorem). For 0<𝜃<1and 1/𝑝2 −1/𝑝)and 𝑝≥max(𝑝1,𝑝2),onehas 1≤𝑞≤∞,onehas[𝐴0,𝐴1]𝜃,𝑞,𝐾 =[𝐴0,𝐴1]𝜃,𝑞,𝐽.

‖𝑢V‖ 𝑠 ,𝑘 ≤𝐶‖𝑢‖ 𝑠 ,𝑘 ‖V‖ 𝑠 ,𝑘 . Ḃ1,2 (R𝑑) Ḃ1 (R𝑑) Ḃ2 (R𝑑) (52) 𝑝,𝑟 𝑝1,𝑟1 𝑝2,𝑟2 Remark 21. Inthefollowing,wewilldenotethisspaceby [𝐴0,𝐴1]𝜃,𝑞. with 𝑠1,2 =𝑠1 +𝑠2 − (𝑑 + 2𝛾)(1/𝑝1 +1/𝑝2 −1/𝑝)and 𝑟=max(𝑟1,𝑟2). Lemma 22. For 𝑎=∑𝑗∈Z 𝑎𝑗 and 󰜚>0,with󰜚 =1̸,onehas

4. A Primer to Real Interpolation Theory and (1−𝜃)/𝑞 −𝑗𝜃𝑞󵄩 󵄩𝑞 Generalized Lorentz Spaces ‖𝑎‖ ≤𝐶(𝑞,𝜃,󰜚)(∑󰜚 󵄩𝑎 󵄩 ) [𝐴0,𝐴1]𝜃,𝑞 󵄩 𝑗󵄩𝐴 𝑗∈Z 0 𝑙𝑞(Z) (𝑎 ) From now, we denote by the set of sequence 𝑗 𝑗∈Z such (57) that 𝜃/𝑞 󵄩 󵄩𝑞 1/𝑞 𝑗(1−𝜃)𝑞󵄩 󵄩 ×(∑󰜚 󵄩𝑎𝑗󵄩 ) . 󵄨 󵄨𝑞 𝐴1 󵄨 󵄨 𝑗∈Z (∑󵄨𝑎𝑗󵄨 ) <∞ (53) 𝑗∈Z Proposition 23. (i) For 𝜃0 =𝜃̸1,onehas stands for sup𝑗|𝑎𝑗| in the case 𝑞=∞. [[𝐴0,𝐴1] ,[𝐴0,𝐴1] ] =[𝐴0,𝐴1] . The theory of interpolation spaces was introduced in the 𝜃0,𝑞0 𝜃1,𝑞1 𝜃,𝑞 (1−𝜃)𝜃0+𝜃𝜃1,𝑞 (58) early sixties by J. Lions and J. Peetre for the real method and by Calderon´ for the complex method (cf. [20]). (ii) For 𝜃0 =𝜃1, (58) is still valid if 1/𝑞 = (1 − 𝜃)/𝑞0 +𝜃/𝑞1. In this section, we present the real method. There are many equivalent ways to define the method; we will present Proposition 24 (Duality theorem for the real method). One 󸀠 󸀠 󸀠 the discrete J-method and the K-method which are the considers the dual spaces 𝐴0,𝐴1 and [𝐴0,𝐴1]𝜃,𝑞 for 0<𝜃<1 simplest ones. and 1≤𝑞<∞of the spaces 𝐴0,𝐴1 and [𝐴0,𝐴1]𝜃,𝑞.If We consider two Banach spaces 𝐴0 and 𝐴1 which are 󸀠 𝐴0 ⋂𝐴1 is dense in 𝐴0 and in 𝐴1,onehas[𝐴0,𝐴1] = continuously imbedded into a common topological vector 𝜃,𝑞 [𝐴󸀠 ,𝐴󸀠 ] 𝑞󸀠 𝑞 space 𝑉 and 𝑡>0. 0 1 𝜃,𝑞󸀠 ,where is the conjugate component of . The J-method and the K-method consist to consider the 𝑑 J-functional and the K-functional defined on 𝐴0 ⋂ 𝐴1 by For any measurable function 𝑓 on R , we define its distribution and rearrangement functions 𝐽(𝑡,𝑎;𝐴 ,𝐴 )= (‖𝑎‖ ,𝑡‖𝑎‖ ), 0 1 max 𝐴0 𝐴1 󵄩 󵄩 󵄩 󵄩 𝐾(𝑡,𝑎;𝐴 ,𝐴 )= (󵄩𝑎 󵄩 +𝑡󵄩𝑎 󵄩 :𝑎=𝑎 +𝑎). 𝑑𝑓,𝑘 (𝜆) := ∫ 𝜔𝑘 (𝑥) 𝑑𝑥, 0 1 min 󵄩 0󵄩𝐴 󵄩 1󵄩𝐴 0 1 {𝑥∈R𝑑: |𝑓(𝑥)|≥𝜆} 0 1 (59) (54) ∗ 𝑓𝑘 (𝑠) := inf {𝜆 :𝑓,𝑘 𝑑 (𝜆) ≤𝑠}. Definition 18 (J-method of interpolation). For 0<𝜃<1and 1≤𝑞≤∞ [𝐴 ,𝐴 ] , the interpolation space 0 1 𝜃,𝑞,𝐽 is defined as For 1≤𝑝≤∞and 1≤𝑞≤∞, define follows: 𝑎∈[𝐴0,𝐴1]𝜃,𝑞,𝐽 if and only if 𝑎 can be written as a 𝑎=∑ 𝑎 𝐴 +𝐴 󵄩 󵄩 sum 𝑗∈Z 𝑗,wheretheseriesconvergein 0 1, each 󵄩𝑓󵄩𝐿𝑝,𝑞(R𝑑) −𝑗𝜃 𝑗 𝑞 𝑘 𝑎𝑗 belongs to 𝐴0 ⋂ 𝐴1 and (2 𝐽(2 ,𝑎𝑗;𝐴0,𝐴1))𝑗∈Z ∈𝑙(Z). ∞ 1/𝑞 The norm of [𝐴0,𝐴1]𝜃,𝑞,𝐽 is defined by { 1/𝑝 ∗ 𝑞 𝑑𝑠 {(∫ (𝑠 𝑓𝑘 (𝑠)) ) if 1≤𝑝,𝑞<∞ = 0 𝑠 ‖𝑎‖ { 1/𝑝 ∗ [𝐴0,𝐴1] { 𝜃,𝑞,𝐽 sup 𝑠 𝑓𝑘 (𝑠) if 1≤𝑝≤∞,𝑞=∞. { 𝑠>0 1/𝑞 1/𝑞 (60) −𝑗𝜃𝑞󵄩 󵄩𝑞 𝑗(1−𝜃)𝑞󵄩 󵄩𝑞 = inf (∑2 󵄩𝑎𝑗󵄩 ) +(∑2 󵄩𝑎𝑗󵄩 ) . 𝑎=∑ 𝑎 󵄩 󵄩𝐴0 󵄩 󵄩𝐴1 𝑗∈Z 𝑗 𝑗∈Z 𝑗∈Z 𝑝,𝑞 𝑑 The generalized Lorentz spaces 𝐿𝑘 (R ) is defined as the set (55) of all measurable functions 𝑓 such that ||𝑓||𝐿𝑝,𝑞(R𝑑) <∞. 𝑘 Journal of Function Spaces and Applications 7

Proposition 25. (i) For 1<𝑝<∞, 1≤𝑞≤∞ Proof. We obtain this result by similar ideas used for the Dunkl-Riesz potential (cf. [21]). 𝐿𝑝,𝑞 (R𝑑) = [𝐿1 (R𝑑) ,𝐿∞ (R𝑑)] , 𝑘 𝑘 𝑘 𝜃,𝑞 (61) Proposition 28. Let 𝑠 < (𝑑 + 2𝛾)/2 and 𝑞 = (2𝑑 + 4𝛾)/(𝑑 + with 1/𝑝 = 1 − 𝜃. 2𝛾− 2𝑠) .Then, (ii) For 𝑝0 =𝑝̸1,onehas 󵄩 󵄩 󵄩 󵄩 󵄩 𝑠/2 󵄩 𝑠 𝑑 󵄩𝑓󵄩𝐿𝑞 (R𝑑) ≤𝐶󵄩(𝐼−△𝑘) 𝑓󵄩 ,𝑓∈𝐻𝑘 (R ). 𝑝 ,𝑞 𝑝 ,𝑞 𝑝 𝑝 𝑘 󵄩 󵄩𝐿2 (R𝑑) [𝐿 0 0 (R𝑑),𝐿 1 1 (R𝑑)] =[𝐿 0 (R𝑑),𝐿 1 (R𝑑)] 𝑘 𝑘 𝑘 𝜃,𝑞 𝑘 𝑘 𝜃,𝑞 (69) 𝑝,𝑞 𝑑 =𝐿 (R ), 𝑑 𝑘 Proof. Let us first observe that since 𝐷(R ) is dense in 𝑠 𝑑 𝑑 (62) 𝐻𝑘(R ),itisenoughtoprove(69)for𝑓∈𝐷(R ).Let 𝑑 𝑓, 𝑔 ∈ 𝐷(R ).Then,wehave with 1/𝑝 = (1 − 𝜃)/𝑝0 +𝜃/𝑝1. (iii) In the case 𝑝0 =𝑝1 =𝑝,onehas ⟨𝑓,𝑔⟩𝐿2 (R𝑑) 𝑘 𝑝,𝑞 𝑝,𝑞 𝑝,𝑞 [𝐿 0 (R𝑑) ,𝐿 1 (R𝑑)] =𝐿 (R𝑑) , 𝑘 𝑘 𝜃,𝑞 𝑘 (63) =⟨F𝐷(𝑓), F𝐷 (𝑔)⟩𝐿2 (R𝑑) 𝑘 1/𝑞 = (1 − 𝜃)/𝑞 +𝜃/𝑞 with 0 1. 󵄩 󵄩2 𝑠/2 󵄩 󵄩2 −𝑠/2 = ∫ (1 + 󵄩𝜉󵄩 ) F (𝑓) (𝜉) F (𝑔) (𝜉)(1 + 󵄩𝜉󵄩 ) (iv) If 1≤𝑝≤∞and 1≤𝑞1 <𝑞2 ≤∞,then 󵄩 󵄩 𝐷 𝐷 󵄩 󵄩 R𝑑

𝑝,𝑞1 𝑑 𝑝,𝑞2 𝑑 𝐿𝑘 (R )󳨅→𝐿𝑘 (R ). (64) ×𝜔𝑘 (𝜉) 𝑑𝜉

𝑠/2 −𝑠/2 Proof . We obtain these results by similar ideas used in the =⟨(𝐼−△𝑘) 𝑓, (𝐼 −△ 𝑘) 𝑔⟩ . 𝐿2 (R𝑑) Euclidean case. 𝑘 (70) Proposition 26. (i) Let 1<𝑝<∞, 1≤𝑞≤∞.Then, 𝑝,𝑞 𝑑 Hence, there exists a constant 𝐶 such that every 𝑓∈𝐿𝑘 (R ) can be 󵄨 󵄨 󵄩 󵄩 󵄩 󵄩 decomposed as 𝑓=∑𝑗∈Z 𝑓𝑗,where 󵄨 󵄨 󵄩 𝑠/2 󵄩 󵄩 −𝑠/2 󵄩 󵄨⟨𝑓,𝑔⟩𝐿2 (R𝑑)󵄨 ≤ 󵄩(𝐼 − △𝑘) 𝑓󵄩 󵄩(𝐼−△𝑘) 𝑔󵄩 . 󵄨 𝑘 󵄨 󵄩 󵄩𝐿2 (R𝑑)󵄩 󵄩𝐿2 (R𝑑) 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 𝑘 𝑘 󵄩(2−𝑗(𝑝−1)/𝑝󵄩𝑓 󵄩 )󵄩 + 󵄩(2𝑗/𝑝󵄩𝑓 󵄩 )󵄩 (71) 󵄩 󵄩 𝑗󵄩𝐿1 (R𝑑) 󵄩 󵄩 󵄩 𝑗󵄩𝐿∞(R𝑑) 󵄩 󵄩 𝑘 󵄩𝑙𝑟 󵄩 𝑘 󵄩𝑙𝑟 (65) 󵄩 󵄩 Now, by the previous lemma we obtain ≤𝐶󵄩𝑓󵄩𝐿𝑝,𝑞(R𝑑), 𝑘 󵄨 󵄨 󵄩 󵄩 󵄨 󵄨 󵄩 𝑠/2 󵄩 󵄩 󵄩 󵄨⟨𝑓,𝑔⟩𝐿2 (R𝑑)󵄨 ≤𝐶󵄩(𝐼 − △𝑘) 𝑓󵄩 󵄩𝑔󵄩𝐿𝑝 (R𝑑), 𝑓 𝑗 =𝑛̸𝑓 𝑓 =0 󵄨 𝑘 󵄨 󵄩 󵄩𝐿2 (R𝑑) 𝑘 (72) the 𝑗 have disjoint supports if , 𝑗 𝑛 . 𝑘 (ii) Let 1<𝑝<∞, 1≤𝑞≤∞. Then, there exists 𝑝,𝑞 𝑑 𝑝 = (2𝑑 + 4𝛾)/(𝑑 +2𝛾2𝑠) 𝑔=𝑓𝑞−1 aconstant𝐶 such that every 𝑓∈𝐿𝑘 (R ) and every 𝑔∈ where .Now,letustake , 𝑝/(𝑝−1),𝑞/(𝑞−1) 𝑑 1 𝑑 with 1/𝑝 + 1/𝑞,thatis, =1 𝑞 = (2𝑑 + 4𝛾)/(𝑑 +2𝛾−2𝑠).Then, 𝐿 (R ),onehas𝑓𝑔 ∈𝐿 (R ) and 𝑘 𝑘 the relation (72)givesthat 󵄨 󵄨 󵄩 󵄩 󵄩 󵄩 󵄨 󵄨 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩𝑞 󵄩 𝑠/2 󵄩 󵄩 󵄩𝑞−1 󵄨∫ 𝑓 (𝑥) 𝑔 (𝑥) 𝜔𝑘 (𝑥) 𝑑𝑥󵄨 ≤𝐶󵄩𝑓󵄩 𝑝,𝑞 𝑑 󵄩𝑔󵄩 𝑝/(𝑝−1),𝑞/(𝑞−1) 𝑑 . 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄨 𝑑 󵄨 𝐿 (R ) 𝐿 (R ) 󵄩𝑓󵄩 𝑞 𝑑 ≤𝐶󵄩(𝐼 − △𝑘) 𝑓󵄩 󵄩𝑓󵄩 𝑞 𝑑 . 󵄨 R 󵄨 𝑘 𝑘 󵄩 󵄩𝐿 (R ) 󵄩 󵄩𝐿2 (R𝑑)󵄩 󵄩𝐿 (R ) (73) 𝑘 𝑘 𝑘 (66) Thus, we obtain (69). Proof. We obtain these results by similar ideas used in the Euclidean case. Proposition 29. Let 1≤𝑝,𝑝2 <∞, 0<𝜃<𝑝<∞, 0<𝑠<𝑑+2𝛾,and1<𝑝1 < (𝑑 + 2𝛾)/𝑠.Then,onehas 5. Inequalities for the Fractional the inequality

Dunkl-Laplace Operators 󵄩 󵄩 󵄩 𝑠/2 󵄩𝜃/𝑝 󵄩 󵄩(𝑝−𝜃)/𝑝 󵄩𝑓󵄩 𝑝 ≤𝐶󵄩(𝐼 − △ ) 𝑓󵄩 󵄩𝑓󵄩 , 󵄩 󵄩𝐿 (R𝑑) 󵄩 𝑘 󵄩 𝑝 󵄩 󵄩 𝑝2 𝑑 (74) 𝑘 󵄩 󵄩𝐿 1 (R𝑑) 𝐿 (R ) Lemma 27. Let 𝑠 be a real number such that 0<𝑠<𝑑+2𝛾, 𝑘 𝑘 and let 1<𝑝<𝑞<∞satisfy with 1 1 𝑠 − = . 1 𝑠 𝑝−𝜃 𝑝 𝑞 2𝛾+ 𝑑 (67) 𝜃( − )+ =1. (75) 𝑝1 𝑑+2𝛾 𝑝2 𝑓∈𝐿𝑝(R𝑑) For 𝑘 ,onehas Proof. Holder’s¨ inequality yields 󵄩 󵄩 󵄩 −𝑠/2 󵄩 󵄩 󵄩 󵄩 󵄩𝑝 󵄩 󵄩𝜃 󵄩 󵄩𝑝−𝜃 󵄩(𝐼−△ ) 𝑓󵄩 ≤𝐶󵄩𝑓󵄩 𝑝 . 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 𝑘 󵄩 𝑞 󵄩 󵄩𝐿 (R𝑑) 󵄩𝑓󵄩 𝑝 ≤𝐶󵄩𝑓󵄩 𝑝0 𝑑 󵄩𝑓󵄩 𝑝 , 󵄩 󵄩𝐿 (R𝑑) 𝑘 (68) 󵄩 󵄩𝐿 R𝑑 󵄩 󵄩𝐿 (R )󵄩 󵄩𝐿 2 R𝑑 (76) 𝑘 𝑘( ) 𝑘 𝑘 ( ) 8 Journal of Function Spaces and Applications where Corollary 31. Let 0<𝑠<𝑑+2𝛾and 1 < 𝑞 < (𝑑 + ,2𝛾)/𝑠 𝑠 𝑑 𝑓∈𝐻𝑞,𝑘(R ) such that ‖𝑓‖𝐿𝑞 (R𝑑) =1,onehas 1 1 1 1 1 𝑘 = (1 − ), = . 𝑝 𝜃 𝑝 𝑝 (𝑝 − 𝜃) 𝑝 (77) 0 2 𝑠 󵄨 󵄨𝑞 󵄨 󵄨𝑞 exp ( ∫ 󵄨𝑓 (𝑥)󵄨 ln (󵄨𝑓 (𝑥)󵄨 )𝜔𝑘 (𝑥) 𝑑𝑥) 𝑑+2𝛾 R𝑑 Applying Lemma 27,with𝑝1 = ((𝑑 + 2𝛾)𝑝0)/(𝑑 + 2𝛾+ 𝑠𝑝 0), 󵄩 󵄩 (82) we obtain the result. 󵄩 𝑠/2 󵄩 ≤𝐶󵄩(𝐼−△𝑘) 𝑓󵄩 𝑞 . 󵄩 󵄩𝐿 (R𝑑) 𝑘 Theorem 30. Let 1<𝑞<∞, 0<𝑠<𝑑+2𝛾and 𝑞=𝑝 1<𝑝1 < (𝑑 + 2𝛾)/𝑠.Then,theinequality Proof. It suffices to apply the previous theorem for 1.

1 𝑠 1 𝑑 (( + − ) Lemma 32 (see [23]). One assumes that 𝐺=Z2 .If exp 𝑝 ,𝑞 𝑝 ,𝑞 𝑞 𝑑+2𝛾 𝑝1 1 1 𝑑 2 2 𝑑 𝑓∈𝐿𝑘 (R ), 𝑔∈𝐿𝑘 (R ),and1/𝑝1 +1/𝑝2 >1,then 𝑝3,𝑞3 𝑑 𝑓∗𝐷𝑔∈𝐿 (R ) where 1/𝑝3 =1/𝑝1 +1/𝑝2 −1and 𝑞3 ≥1 󵄨 󵄨𝑞 󵄨 󵄨𝑞 𝑘 󵄨𝑓 (𝑥)󵄨 󵄨𝑓 (𝑥)󵄨 is any number such that 1/𝑞3 ≤1/𝑞1 +1/𝑞2.Moreover, × ∫ 󵄩 󵄩𝑞 ln (󵄩 󵄩𝑞 )𝜔𝑘 (𝑥) 𝑑𝑥) 𝑑 󵄩 󵄩 󵄩 󵄩 (78) R 󵄩𝑓󵄩 𝑞 𝑑 󵄩𝑓󵄩 𝑞 𝑑 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩𝐿 (R ) 󵄩 󵄩𝐿 (R ) 󵄩𝑓∗ 𝑔󵄩 𝑝 ,𝑞 ≤𝐶󵄩𝑓󵄩 𝑝 ,𝑞 󵄩𝑔󵄩 𝑝 ,𝑞 . 𝑘 𝑘 󵄩 𝐷 󵄩𝐿 3 3 (R𝑑) 󵄩 󵄩𝐿 1 1 (R𝑑)󵄩 󵄩𝐿 2 2 (R𝑑) 𝑘 𝑘 𝑘 (83) 󵄩 󵄩 󵄩 𝑠/2 󵄩 󵄩(𝐼 −𝑘 △ ) 𝑓󵄩 𝑝 󵄩 󵄩𝐿 1 (R𝑑) Remark 33. Theanaloguesofthislemmaforthegeneral 𝑘 ≤𝐶 󵄩 󵄩 reflection group 𝐺, together with other additional results, will 󵄩𝑓󵄩𝐿𝑞 (R𝑑) 𝑘 appear in a forthcoming paper.

𝑑 holds for Theorem 34. One assumes that 𝐺=Z2 .Let1≤𝑝<∞, 1≤𝑝2,𝑞,𝑞1,𝑞2 <∞, 0<𝜃<𝑞, 0<𝑠<𝑑+2𝛾,and 1 𝑠 1 1<𝑝 < (𝑑 + 2𝛾)/𝑠 + − >0. (79) 1 .Then,theinequality 𝑞 𝑑+2𝛾 𝑝1 󵄩 󵄩 󵄩 𝑠/2 󵄩𝜃/𝑞 󵄩 󵄩(𝑞−𝜃)/𝑞 󵄩𝑓󵄩 𝑝,𝑞 ≤𝐶󵄩(𝐼 − △ ) 𝑓󵄩 󵄩𝑓󵄩 󵄩 󵄩𝐿 (R𝑑) 󵄩 𝑘 󵄩 𝑝 ,𝑞 󵄩 󵄩 𝑝2,𝑞2 𝑑 (84) 𝑔(ℎ) = 𝑘 󵄩 󵄩𝐿 1 1 R𝑑 𝐿 (R ) Proof. Using the convexity of the function 𝑘 ( ) 𝑘 ℎ (∫ |𝑓(𝑥)|1/ℎ𝜔 (𝑥)𝑑𝑥) ln R𝑑 𝑘 ,andthelogarithmicHolder’s¨ inequality proved by Merker [22], we obtain holds for 𝜃 𝑞−𝜃 󵄨 󵄨𝑞 󵄨 󵄨𝑞 + =1, 󵄨𝑓 (𝑥)󵄨 󵄨𝑓 (𝑥)󵄨 𝑞 𝑞 ∫ 󵄩 󵄩𝑞 ln (󵄩 󵄩𝑞 )𝜔𝑘 (𝑥) 𝑑𝑥 1 2 𝑑 󵄩 󵄩 󵄩 󵄩 R 󵄩𝑓󵄩 𝑞 󵄩𝑓󵄩 𝑞 (85) 󵄩 󵄩𝐿 R𝑑 󵄩 󵄩𝐿 R𝑑 𝑘( ) 𝑘( ) 1 𝑠 𝑞−𝜃 𝑞 𝜃( − )+ = . 󵄩 󵄩𝑞 (80) 󵄩𝑓󵄩 𝑝1 𝑑+2𝛾 𝑝2 𝑝 𝑝 󵄩 󵄩𝐿𝑞 R𝑑 𝑘( ) ≤ ln (󵄩 󵄩𝑞 ), 𝑝−𝑞 󵄩𝑓󵄩 Proof. Applying the Holder¨ inequality and simple computa- 󵄩 󵄩𝐿𝑞 (R𝑑) 𝑘 tion yields for 0<𝑞<𝑝≤∞.Wecanchoose𝑝 = ((𝑑 + 2𝛾)𝑞)/(𝑑 + 󵄩 󵄩𝑞 󵄩 󵄩𝜃 󵄩 󵄩𝑞−𝜃 󵄩𝑓󵄩 ≤𝐶󵄩𝑓󵄩 𝑝 ,𝑞 󵄩𝑓󵄩 , 󵄩 󵄩𝐿𝑝,𝑞 R𝑑 󵄩 󵄩𝐿 3 1 (R𝑑)󵄩 󵄩𝐿𝑝2,𝑞2 R𝑑 (86) 2𝛾− 𝑞𝑠) ∈ (𝑞, ∞) for 𝑝2 =𝑞and 𝜃 satisfying the condition of 𝑘 ( ) 𝑘 𝑘 ( ) Proposition 29,andweget where 󵄨 󵄨𝑞 󵄨 󵄨𝑞 1 1 1 1 󵄨𝑓 (𝑥)󵄨 󵄨𝑓 (𝑥)󵄨 = − + , ∫ 󵄩 󵄩𝑞 ln (󵄩 󵄩𝑞 )𝜔𝑘 (𝑥) 𝑑𝑥 𝑝 𝑝 𝑞 𝑞 R𝑑 󵄩𝑓󵄩 󵄩𝑓󵄩 3 1 󵄩 󵄩𝐿𝑞 R𝑑 󵄩 󵄩𝐿𝑞 R𝑑 𝑘( ) 𝑘( ) (87) 1 1 1 1 𝑞 = − + . 󵄩 𝑠/2 󵄩𝜃/𝑝 󵄩 󵄩(𝑝−𝜃)/𝑝 󵄩 󵄩 󵄩 󵄩 𝑝2 𝑝 𝑞 𝑞2 (𝐶󵄩(𝐼−△𝑘) 𝑓󵄩 𝑝 󵄩𝑓󵄩 𝑝2 𝑑 ) 󵄩 󵄩𝐿 1 R𝑑 󵄩 󵄩𝐿 (R ) 𝑝 𝑘 ( ) 𝑘 ≤ ln ( 󵄩 󵄩𝑞 ) 𝑝−𝑞 󵄩𝑓󵄩 Note that 󵄩 󵄩𝐿𝑞 (R𝑑) 𝑘 𝑠/2 𝑓 (𝑥) =(𝐼−△) 𝑓∗ 𝐵 (𝑥) , (88) 󵄩 󵄩 𝑘 𝐷 𝑘,𝑠 󵄩 𝑠/2 󵄩 𝐶󵄩(𝐼−△𝑘) 𝑓󵄩 𝑝 𝑞𝜃 󵄩 󵄩𝐿 1 (R𝑑) 𝐵 𝑘 where 𝑘,𝑠 is the Dunkl-Bessel kernel defined by rela- ≤ ln ( 󵄩 󵄩𝑞 ). 𝐵 ∈ 𝑝−𝑞 󵄩𝑓󵄩 tion (36). From the relation (37), we see that 𝑘,𝑠 󵄩 󵄩𝐿𝑞 R𝑑 𝑘( ) (𝑑+2𝛾)/(𝑑+2𝛾−𝑠),∞ 𝑑 𝐿𝑘 (R ).UsingnowLemma 32,wededucethat (81) 󵄩 󵄩 󵄩 𝑠/2 󵄩 󵄩𝑓󵄩 𝑝 ,𝑞 ≤𝐶󵄩(𝐼−△ ) 𝑓󵄩 , 󵄩 󵄩𝐿 3 1 (R𝑑) 󵄩 𝑘 󵄩 𝑝 ,𝑞 𝑘 󵄩 󵄩𝐿 1 1 (R𝑑) (89) Byasimplecalculation,weobtaintheresult. 𝑘 Journal of Function Spaces and Applications 9 for holds for 1 1 𝑠 𝑑+2𝛾 𝜃 𝑞−𝜃 = − ,0<𝑠< . + =1, (90) 𝑞 𝑞 𝑝3 𝑝1 𝑑+2𝛾 𝑝1 1 2 (98) 1 𝑠 𝑞−𝜃 𝑞 The result then follows. 𝜃( − )+ = . 𝑝1 𝑑+2𝛾 𝑝2 𝑝 Now, we state the results for the Dunkl-Riesz potential 𝑑 operators. The proofs are essentially as for the Dunkl-Bessel Remark 40. (i) We assume that G = Z2 . It follows from the potential operators. We will not repeat them. special case 𝑝1 =𝑞1 and 𝑝2 =𝑞2 of (97)thattheinequality 󵄩 󵄩 󵄩 𝑠/2 󵄩𝜃/𝑞 󵄩 󵄩(𝑞−𝜃)/𝑞 Proposition 35. 𝑠 < (𝑑 + 2𝛾)/2 𝑞 = (2𝑑 + 4𝛾)/(𝑑 + 󵄩𝑓󵄩 𝑝,𝑞 ≤𝐶󵄩(−△ ) 𝑓󵄩 󵄩𝑓󵄩 Let and 󵄩 󵄩𝐿 (R𝑑) 󵄩 𝑘 󵄩 𝑝 󵄩 󵄩 𝑝2 𝑑 (99) 𝑘 󵄩 󵄩𝐿 1 (R𝑑) 𝐿 (R ) 2𝛾− 2𝑠) .Then, 𝑘 𝑘 󵄩 󵄩 with 𝑞=𝑝(1−𝜃𝑠/(𝑑+2𝛾)).Equation(99)canbethoughtof 󵄩 󵄩 󵄩 𝑠/2 󵄩 𝑠 𝑑 󵄩𝑓󵄩𝐿𝑞 (R𝑑) ≤𝐶󵄩(−△𝑘) 𝑓󵄩 ,𝑓∈𝐻𝑘 (R ). a refinement of (92)from(64). 𝑘 󵄩 󵄩𝐿2 (R𝑑) (91) 𝑑 𝑘 (ii) We assume that 𝐺=Z2 . It follows from the special case 𝑝1 =𝑞=𝜃that (99)becomes Proposition 36. Let 1≤𝑝,𝑝2 <∞, 0<𝜃<𝑝<∞, 󵄩 𝑠/2 󵄩 0<𝑠<𝑑+2𝛾,and1<𝑝1 < (𝑑 + 2𝛾)/𝑠. Then, the inequality 󵄩 󵄩 󵄩 󵄩 󵄩𝑓󵄩 𝑞(𝑑+2𝛾)/(𝑑+2𝛾−𝑞𝑠),𝑞 𝑑 ≤𝐶󵄩(−△𝑘) 𝑓󵄩 𝑞 , 󵄩 󵄩𝐿 (R ) 󵄩 󵄩𝐿 (R𝑑) (100) 𝑘 𝑘 󵄩 󵄩 󵄩 𝑠/2 󵄩𝜃/𝑝 󵄩 󵄩(𝑝−𝜃)/𝑝 󵄩𝑓󵄩 𝑝 ≤𝐶󵄩(−△ ) 𝑓󵄩 󵄩𝑓󵄩 󵄩 󵄩𝐿 (R𝑑) 󵄩 𝑘 󵄩 𝑝 󵄩 󵄩 𝑝2 𝑑 (92) which can also be thought of as a refinement of the Hardy- 𝑘 󵄩 󵄩𝐿 1 (R𝑑) 𝐿 (R ) 𝑘 𝑘 Littlewood-Sobolev fractional integration theorem in Dunkl with setting (cf. [21]): 󵄩 −𝑠/2 󵄩 󵄩 󵄩 1 𝑠 𝑝−𝜃 󵄩(−△ ) 𝑓󵄩 ≤𝐶󵄩𝑓󵄩 𝑝 . 󵄩 𝑘 󵄩 𝑞 󵄩 󵄩𝐿 (R𝑑) 𝜃( − )+ =1. 󵄩 󵄩𝐿 (R𝑑) 𝑘 (101) (93) 𝑘 𝑝1 𝑑+2𝛾 𝑝2 (iii) We note that the results of Dunkl-Riesz potential of Theorem 37. Let 1<𝑞<∞, 0<𝑠<𝑑+2𝛾,and this section are in sprit of the classical case (cf. [24]). 1<𝑝1 < (𝑑 + 2𝛾)/𝑠.Then,theinequality 𝑑 Theorem 41. One assumes that 𝐺=Z2 .Let1<𝑝<∞, 1 𝑠 1 0 < 𝑠 < (𝑑 + ,and2𝛾)/𝑝 1≤𝑞≤∞. There exists a positive (( + − ) exp 𝑞 𝑑+2𝛾 𝑝 constant 𝐶 such that one has 1 󵄩 󵄩 󵄩𝑓 (𝑥)󵄩 󵄩 𝑠/2 󵄩 󵄩 󵄩 ≤𝐶󵄩(−△ ) 𝑓󵄩 . 󵄩 𝑠 󵄩 󵄩 𝑘 󵄩 𝑝,𝑞 𝑑 (102) 󵄩 ‖𝑥‖ 󵄩 𝑝,𝑞 𝑑 󵄩 󵄩𝐿 (R ) 󵄨 󵄨𝑞 󵄨 󵄨𝑞 󵄩 󵄩𝐿 (R ) 𝑘 󵄨𝑓 (𝑥)󵄨 󵄨𝑓 (𝑥)󵄨 𝑘 × ∫ 󵄩 󵄩𝑞 ln (󵄩 󵄩𝑞 )𝜔𝑘 (𝑥) 𝑑𝑥) (94) R𝑑 󵄩𝑓󵄩 󵄩𝑓󵄩 For proof of this result, we need the following lemma 󵄩 󵄩𝐿𝑞 (R𝑑) 󵄩 󵄩𝐿𝑞 (R𝑑) 𝑘 𝑘 which we prove as the Euclidean case. 󵄩 𝑠/2 󵄩 󵄩(−△ ) 𝑓󵄩 𝑝1,𝑞1 𝑑 󵄩 𝑘 󵄩 𝑝 󵄩 󵄩𝐿 1 (R𝑑) Lemma 42. Let 1≤𝑝1,𝑝2,𝑞1,𝑞2 ≤∞.If𝑓∈𝐿𝑘 (R ) and 𝑘 𝑝 ,𝑞 ≤𝐶 󵄩 󵄩 𝑔∈𝐿 2 2 (R𝑑) 󵄩 󵄩 𝑞 ,then 󵄩𝑓󵄩𝐿 (R𝑑) 𝑘 𝑘 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩𝑓𝑔 󵄩 𝑝,𝑞 ≤𝐶󵄩𝑓󵄩 𝑝 ,𝑞 󵄩𝑔󵄩 𝑝 ,𝑞 , 󵄩 󵄩𝐿 (R𝑑) 󵄩 󵄩𝐿 1 1 (R𝑑)󵄩 󵄩𝐿 2 2 (R𝑑) (103) holds for 𝑘 𝑘 𝑘 1/𝑝 = 1/𝑝 +1/𝑝 1/𝑞 = 1/𝑞 +1/𝑞 1 𝑠 1 where 1 2 and 1 2. + − >0. (95) 𝑞 𝑑+2𝛾 𝑝1 Proof of Theorem 41. Let 1<𝑝<∞and 𝑠 ∈ (0, (𝑑 + 2𝛾)/𝑝). 𝑠 We take 𝑔(𝑥) = 1/‖𝑥‖ and apply (103)inthespecificform Corollary 38. 0<𝑠<𝑑+2𝛾 1 < 𝑞 < (𝑑 + 2𝛾)/𝑠 Let and , 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 𝑠 𝑑 󵄩𝑓𝑔 󵄩 𝑝,𝑞 ≤𝐶󵄩𝑓󵄩 𝑝 ,𝑞 󵄩𝑔󵄩 𝑟,∞ , ̇ 𝑞 󵄩 󵄩𝐿 (R𝑑) 󵄩 󵄩𝐿 1 (R𝑑)󵄩 󵄩𝐿 (R𝑑) 𝑓∈H𝑞,𝑘(R ) such that ‖𝑓‖𝐿 (R𝑑) =1,onehas 𝑘 𝑘 𝑘 (104) 𝑘 𝑟 = (𝑑 + 2𝛾)/𝑠 𝑝 = (𝑞(𝑑 + 2𝛾))/(𝑑 + 2𝛾−𝑞𝑠) 𝑠 where and 1 .As 󵄨 󵄨𝑞 󵄨 󵄨𝑞 𝑔∈𝐿𝑟,∞(R𝑑) exp ( ∫ 󵄨𝑓 (𝑥)󵄨 ln (󵄨𝑓 (𝑥)󵄨 )𝜔𝑘 (𝑥) 𝑑𝑥) 𝑘 ,wehave 𝑑+2𝛾 R𝑑 󵄩 󵄩 (96) 󵄩𝑓 (𝑥)󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 ≤𝐶󵄩𝑓󵄩 , 󵄩 𝑠/2 󵄩 󵄩 𝑠 󵄩 󵄩 󵄩𝐿((𝑑+2𝛾)𝑝)/(𝑑+2𝛾−𝑝𝑠),𝑞(R𝑑) 󵄩 󵄩 󵄩 󵄩 𝑝,𝑞 𝑘 (105) ≤𝐶󵄩(−△𝑘) 𝑓󵄩 𝑞 . 󵄩 ‖𝑥‖ 󵄩𝐿 (R𝑑) 󵄩 󵄩𝐿 (R𝑑) 𝑘 𝑘 with 1≤𝑞≤∞. On the other hand, from [23], Theorem 1.2, Theorem 39. 𝐺=Z𝑑 1≤𝑝<∞ One assumes that 2 .Let , we have 1≤𝑝,𝑞,𝑞 ,𝑞 <∞0<𝜃<𝑞0<𝑠<𝑑+2𝛾 󵄩 󵄩 2 1 2 , , ,and 󵄩 󵄩 󵄩 𝑠/𝑠 󵄩 󵄩𝑓󵄩 ((𝑑+2𝛾)𝑝)/(𝑑+2𝛾−𝑝𝑠),𝑞 𝑑 ≤𝐶󵄩(−△𝑘) 𝑓󵄩 𝑝,𝑞 , 1<𝑝 < (𝑑 + 2𝛾)/𝑠 󵄩 󵄩𝐿 (R ) 󵄩 󵄩𝐿 (R𝑑) (106) 1 .Then,theinequality 𝑘 𝑘 󵄩 󵄩𝜃/𝑞 𝑝,𝑞 𝑑 󵄩 󵄩 󵄩 𝑠/2 󵄩 󵄩 󵄩(𝑞−𝜃)/𝑞 for any 𝑓∈𝐿 (R ) with 1≤𝑞≤∞, 1<𝑝<∞,and 󵄩𝑓󵄩 𝑝,𝑞 ≤𝐶󵄩(−△ ) 𝑓󵄩 󵄩𝑓󵄩 𝑘 󵄩 󵄩𝐿 (R𝑑) 󵄩 𝑘 󵄩 𝑝 ,𝑞 󵄩 󵄩 𝑝2,𝑞2 𝑑 (97) 𝑘 󵄩 󵄩𝐿 1 1 R𝑑 𝐿 (R ) 0 < 𝑠 < (𝑑 + 2𝛾)/𝑝 𝑘 ( ) 𝑘 .Thus,weobtain(102). 10 Journal of Function Spaces and Applications

6. Dispersion Phenomena Proof. From the dispersion of I𝑘(𝑡) such that 󵄩 󵄩 −(𝑑+2𝛾)(1/2−1/𝑟)󵄩 󵄩 I (𝑡) 󵄩I (𝑡) 𝑔󵄩 ≤𝐶𝑡 󵄩𝑔󵄩 󸀠 , Notations. We denote by 𝑘 the Dunkl-Schrodinger¨ semi- 󵄩 𝑘 󵄩𝐿𝑟 (R𝑑) 󵄩 󵄩𝐿𝑟 (R𝑑) (116) 2 𝑑 𝑘 𝑘 group on 𝐿𝑘(R ) defined by for any 𝑟∈[2,∞],(cf.[8]), and the fact that 1 2 I (𝑡) V := 𝑒−𝑖(𝑑+2𝛾)(𝜋/4) sgn 𝑡𝑒𝑖(‖⋅‖ /4𝑡) 𝑘 𝛾+𝑑/2 𝑡−(𝑑+2𝛾)(1/2−1/𝑟) ∈𝐿2𝑟/(𝑑+2𝛾)(𝑟−2),∞, 𝑐𝑘|𝑡| (107) 𝑖(‖⋅‖2/4𝑡) ⋅ 2(𝑑+2𝛾) (117) ×[F𝐷 (𝑒 V)] ( ). 𝑟 ∈ [2, ], 2𝑡 for any 𝑑+2𝛾−2 1,𝑟 𝑑 𝑊𝑘 (R ) (1≤𝑟≤∞) Banach space of (classes of) 𝑑 𝜇 𝑟 𝑑 onecaneasilyprovetheresult. measurable functions 𝑢:R → C such that 𝑇 𝑢∈𝐿𝑘(𝑅 ) in the sense of distributions, for every multi-index 𝜇 with Theorem 46. Suppose that 𝑑≥1, (𝑞, 𝑟) and (𝑞1,𝑟1) are 1,𝑟 𝑑 (𝑑 + 2𝛾)/2 2<𝑎≤𝑞 𝑢 |𝜇| ≤. 1 𝑊𝑘 (R ) is equipped with the norm -admissible pairs and .If is a solution to the problem 󵄩 𝜇 󵄩 ‖𝑢‖𝑊1,𝑟(R𝑑) = ∑ 󵄩𝑇 𝑢󵄩𝐿𝑟 (R𝑑). 𝑘 𝑘 (108) 𝑑 |𝜇|≤1 𝑖𝜕𝑡𝑢 (𝑡, 𝑥) +△𝑘𝑢 (𝑡, 𝑥) =𝑓(𝑡, 𝑥) , (𝑡, 𝑥) ∈ R × R , (118) 1,𝑟 𝑑 1,𝑟 𝑑 𝑢 =𝑢, 𝑊𝑘,𝐺(R ) (1≤𝑟≤∞)thesubspaceof𝑊𝑘 (R ),whichthese |𝑡=0 0 elements are 𝐺-invariant. for some data, 𝑢0, 𝑓,then (𝑞, 𝑟) Definition 43. One says that the exponent pair is ‖𝑢‖𝐿𝑞,𝑎(R;𝐿𝑟 (R𝑑)) + ‖𝑢‖𝐶(R;𝐿2 (R𝑑)) (𝑑 + 2𝛾)/2-admissible if 𝑞, 𝑟, ≥2 (𝑞, 𝑟, (𝑑 + 2𝛾)/2) ≠ (2, ∞, 1), 𝑘 𝑘 and 󵄩 󵄩 ≤𝐶(󵄩𝑢0󵄩𝐿2 (R𝑑) 1 𝑑+2𝛾 𝑑+2𝛾 𝑘 (119) + ≤ . 𝑞 2𝑟 4 (109) 󵄩 󵄩 +󵄩𝑓󵄩 󸀠 ). 󵄩 󵄩 󸀠 𝑟 (2𝑑+4𝛾)/(𝑑+2𝛾+2),2 (𝑞, 𝑟) (𝑑+2𝛾)/2 𝐿𝑞1,2(R;𝐿 1 (R𝑑)) ⋂ 𝐿2(R;𝐿 (R𝑑)) If equality holds in (109)onesaysthat is sharp - 𝑘 𝑘 (𝑞, 𝑟) (𝑑 + admissible, otherwise, one says that is nonsharp 𝑢 𝑢 2𝛾)/2-admissible. Note in particular that when 𝑑+2𝛾>2, Proof. Let be a solution of (118). We write as the endpoint 𝑡 𝑢 (𝑡, 𝑥) = I (𝑡) 𝑢 (𝑥) + ∫ I (𝑡−𝜏) 𝑓 (𝜏,) 𝑥 𝑑𝜏, 2𝑑 + 4𝛾 𝑘 0 𝑘 𝑃=(2, ) 0 (120) 𝑑+2𝛾−2 (110) (𝑡, 𝑥) ∈ R × R𝑑. (𝑑 + 2𝛾)/2 is sharp -admissible. 󸀠 𝑟1 𝑑 (2𝑑+4𝛾)/(𝑑+2𝛾+2),2 𝑑 Let 𝑘(𝑡, 𝜏) = I𝑘(𝑡 − 𝜏), 𝐸=𝐿𝑘 (R ) or 𝐿𝑘 (R ), ∞ Lemma 44 (see [25]). Let 𝐸 and 𝐹 be Banach spaces, and let 𝐹=𝐿𝑟 (R𝑑) L𝑓(𝑡) =∫ 𝑘(𝑡, 𝜏)𝑓(𝜏)𝑑𝜏 L 𝐿𝑝,𝑟(0,∞;𝐸)→𝐿𝑞,𝑠(0, ∞; 𝐹) 𝑘 ,and 0 .Then,since : be an integral operator for 𝑞󸀠 ≤2<𝑠≤𝑞 some 𝑝, 𝑟,,withakernel 𝑞,𝑠 𝑘(𝑡, 𝜏) such that 1 ,inviewofLemma 44,weonlyhavetoshow that ∞ 󵄩 ∞ 󵄩 L𝑓 (𝑡) = ∫ 𝑘 (𝑡, 𝜏) 𝑓 (𝜏) 𝑑𝜏. (111) 󵄩 󵄩 0 󵄩∫ 𝑘(𝑡, 𝜏)𝑓(𝜏)𝑑𝜏󵄩 󵄩 0 󵄩𝐿𝑞,𝑠(0,∞;𝐿𝑟 (R𝑑)) 𝑘 If 1≤𝑝≤𝑟<𝑠≤𝑞<∞, then one has (121) 󵄩 󵄩 ≤𝐶󵄩𝑓󵄩 󸀠 󸀠 . 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 𝑞 ,2 𝑟1 𝑑 2 (2𝑑+4𝛾)/(𝑑+2𝛾+2),2 𝑑 󵄩L̃𝑓󵄩 ≤𝐶󵄩𝑓󵄩 , 𝐿 1 (0,∞;𝐿 (R ))∩𝐿 (0,∞;𝐿 (R )) 󵄩 󵄩𝐿𝑞,𝑠(0,∞;𝐹) 󵄩 󵄩𝐿𝑝,𝑟(0,∞;𝐸) (112) 𝑘 𝑘 𝑞,𝑠 𝑞,2 ̃ To show this, observe from (114)and𝐿 ⊂𝐿 for all 𝑠≥2 where L is the low diagonal operator defined by that 𝑡 ̃ 󵄩 ∞ 󵄩2 L𝑓 (𝑡) = ∫ 𝑘 (𝑡, 𝜏) 𝑓 (𝜏) 𝑑𝜏. (113) 󵄩 󵄩 0 󵄩∫ 𝑘(𝑡, 𝜏)𝑓(𝜏)𝑑𝜏󵄩 󵄩 0 󵄩𝐿𝑞,𝑠 0,∞;𝐿𝑟 R𝑑 ( 𝑘( )) Lemma 45. For any (𝑑 + 2𝛾)/2-admissible pair (𝑞, 𝑟) with (122) 𝑞>2 ∞ ≤𝐶∫∫ ⟨I𝑘 (−𝜏) 𝑓 (𝜏) , I𝑘 (−𝑦) 𝑓 (𝑦)⟩ 𝑑𝜏𝑑𝑦. 󵄩 󵄩 󵄩 󵄩 0 󵄩I𝑘 (𝑡) 𝑓󵄩𝐿𝑞,2(0,∞;𝐿𝑟 (R𝑑)) ≤𝐶󵄩𝑓󵄩𝐿2 (R𝑑), 𝑘 𝑘 (114) Then,fromtheendpointresultofKeelandTao[26], the right- 󵄩 󵄩 󵄩 𝑡 󵄩 ‖𝑓‖2 󵄩 󵄩 hand side of (122)isboundedby 𝐿2(0,∞;𝐿(2𝑑+4𝛾)/(𝑑+2𝛾+2),2(R𝑑)). 󵄩∫ I𝑘 (𝑡−𝜏) 𝑔 (𝜏) 𝑑𝜏󵄩 𝑘 󵄩 0 󵄩𝐿𝑞,2(0,∞;𝐿𝑟 (R𝑑))∩𝐿∞(0,∞;𝐿2 (R𝑑)) 𝑘 𝑘 The remaining part of theorem can be obtained by the duality (115) 󸀠 󸀠 󵄩 󵄩 (𝐿𝑞,𝑠)󸀠 =𝐿𝑞 ,𝑠 ≤𝐶󵄩𝑔󵄩 󸀠 󸀠 . of Lorentz space and the second part of (115). 󵄩 󵄩𝐿𝑞 ,2(0,∞;𝐿𝑟 (R𝑑)) 𝑘 Journal of Function Spaces and Applications 11

As an application of the previous theorem, we can derive Finally, since we can choose (𝑞1,𝑟1) arbitrarily to be (𝑑+2𝛾)/2 Strichartz estimates of the solution to the following nonlinear -admissible, for any (𝑑 + 2𝛾)/2-admissible pair (𝑞, 𝑟) and 𝑠 problem: with 𝑞>2and 2<𝑠≤𝑞,wehave

‖𝑇𝑢‖𝐿𝑞,𝑠(R;𝐿𝑟 (R𝑑)) 𝑖𝜕𝑡𝑢 (𝑡, 𝑥) +△𝑘𝑢 (𝑡, 𝑥) 𝑘 4/(𝑑+2𝛾−2) 𝑑 󵄩 󵄩 (𝑑+2𝛾+2)/(𝑑+2𝛾−2) =−|𝑢 (𝑡, 𝑥)| 𝑢 (𝑡, 𝑥) , (𝑡, 𝑥) ∈ R × R , ≤𝐶(󵄩𝑇𝑢 󵄩 2 𝑑 + ‖𝑇𝑢‖ 𝑟 ) 󵄩 0󵄩𝐿 (R ) 𝐿𝑞0,𝑠0 R;𝐿 0 R𝑑 𝑘 ( 𝑘 ( )) (129) 1 𝑑 𝑑 𝑢 =𝑢 ∈𝐻 (R ) R . 󵄩 󵄩 󵄩 󵄩(𝑑+2𝛾+2)/(𝑑+2𝛾−2) |𝑡=0 0 𝑘 in ≤𝐶(󵄩𝑇𝑢 󵄩 + 󵄩𝑇𝑢 󵄩 ). 󵄩 0󵄩𝐿2 (R𝑑) 󵄩 0󵄩𝐿2 (R𝑑) (123) 𝑘 𝑘 In a similar way, we can also derive from the smallness of Theorem 47. If the initial data is sufficiently small and ||𝑢0||𝐻1(R𝑑) 𝑘 , 𝐺-invariant, then there exists a unique solution 𝑢∈ 󵄩 󵄩 𝑞,𝑠 1,𝑟 𝑑 2 1,(2𝑑+4𝛾)/(𝑑+2𝛾−2) 𝑑 𝑢 ≤𝐶󵄩𝑢 󵄩 . 𝐿 (0, ∞; 𝑊 (R )) ∩ 𝐿 (0, ∞; 𝑊 (R )) ∩ ‖ ‖𝐿𝑞,𝑠(R;𝐿𝑟 (R𝑑)) 󵄩 0󵄩𝐿2 (R𝑑) 𝑘,𝐺 𝑘,𝐺 𝑘 𝑘 (130) 1 𝑑 𝐶([0, ∞);𝑘,𝐺 𝐻 (R ) for every sharp (𝑑 + 2𝛾)/2-admissible pair (𝑞, 𝑟) with 𝑞>2and 2<𝑠≤𝑞.

1 𝑑 7. Embedding Sobolev Theorems Proof. The existence of a unique 𝐻𝑘,𝐺(R )-solution is proved 𝑞,𝑠 1,𝑟 𝑑 and Applications in [9], it suffices to prove that 𝑢∈𝐿 (0, ∞; 𝑊𝑘,𝐺(R )).From Duhamel’s principle, we deduce that Theorem 48. Let 𝑠, 𝑡, >0 𝑞1,𝑞2 ∈[1,∞]with 𝑞1 =𝑞̸2.Let 𝜃 = 𝑠/(𝑠 + 𝑡) ∈ (0,1), 1/𝑝 = (1 − 𝜃)/𝑞1 +𝜃/𝑞2,and𝑟∈[1,∞]. ̇𝑠,𝑘 𝑑 ̇−𝑡,𝑘 𝑑 𝑝,𝑟 𝑑 𝑢 (𝑡, 𝑥) = I𝑘 (𝑡) 𝑢0 (𝑥) 𝑓∈B (R )∩B (R ) 𝑓∈𝐿 (R ) If 𝑞1,𝑟 𝑞2,𝑟 ,then 𝑘 ,andonehas 𝑡 󵄩 󵄩 󵄩 󵄩1−𝜃 󵄩 󵄩𝜃 4/(𝑑+2𝛾−2) 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩𝑓󵄩𝐿𝑝,𝑟(R𝑑) ≤𝐶󵄩𝑓󵄩Ḃ𝑠,𝑘 R𝑑 󵄩𝑓󵄩Ḃ−𝑡,𝑘(R𝑑). (131) + ∫ I𝑘 (𝑡−𝜏) (|𝑢 (𝜏,) 𝑥 | 𝑢 (𝜏,) 𝑥 )𝑑𝜏. 𝑘 𝑞1,𝑟( ) 𝑞2,𝑟 0 Proof. We start picking 𝑝1,𝑝2 such that 1≤𝑞1 <𝑝1 < (124) 𝑝<𝑝2 <𝑞2 ≤∞with 2/𝑝 = 1/𝑝1 +1/𝑝2.Wehavethen 1/𝑝𝑖 =(1−𝑎𝑖)/𝑞1 +𝑎𝑖/𝑞2 with 𝑎𝑖 ∈ (0, 1) and 𝑖=1,2.We Using (114)and(119), we have write 󵄩 󵄩 󵄩 󵄩1−𝑎 󵄩 󵄩𝑎 󵄩Δ 𝑓󵄩 ≤ 󵄩Δ 𝑓󵄩 𝑖 󵄩Δ 𝑓󵄩 𝑖 . ‖𝑇𝑢‖𝐿𝑞,𝑠(R;𝐿𝑟 (R𝑑)) 󵄩 𝑗 󵄩𝐿𝑝𝑖 (R𝑑) 󵄩 𝑗 󵄩𝐿𝑞1 R𝑑 󵄩 𝑗 󵄩𝐿𝑞2 (R𝑑) (132) 𝑘 𝑘 𝑘 ( ) 𝑘 󵄩 󵄩 󵄩 󵄩 󵄩 4/(𝑑+2𝛾−2) 󵄩 Using Holder’s¨ inequality and by simple calculations, we ≤𝐶(󵄩𝑇𝑢 󵄩 2 𝑑 + 󵄩𝑇(|𝑢| 𝑢)󵄩 𝑞󸀠 ,2 󸀠 ). 󵄩 0󵄩𝐿 (R ) 󵄩 󵄩𝐿 1 (R;𝐿𝑟 1(R𝑑)) 𝑘 𝑘 obtain 󵄩 󵄩𝑟 󵄩 󵄩(1−𝑎 )𝑟 󵄩 󵄩𝑟𝑎 (125) ∑󰜚−𝑗𝑟/2󵄩Δ 𝑓󵄩 ≤ 󵄩𝑓󵄩 1 󵄩𝑓󵄩 1 , 󵄩 𝑗 󵄩𝐿𝑝1 (R𝑑) 󵄩 󵄩Ḃ𝑠,𝑘 (R𝑑)󵄩 󵄩Ḃ−𝑡,𝑘(R𝑑) 𝑗∈Z 𝑘 𝑞1,𝑟 𝑞2,𝑟 (𝑞 ,𝑟 ) 𝑟 <𝑑+2𝛾 Wecanalwaysfindanadmissiblepair 0 0 with 0 (133) 󵄩 󵄩𝑟 󵄩 󵄩(1−𝑎 )𝑟 󵄩 󵄩𝑟𝑎 and 2<𝑠0 <𝑞0 and (𝑞1,𝑟1) and 1<𝑠1 <2such that ∑󰜚𝑗𝑟/2󵄩Δ 𝑓󵄩 ≤ 󵄩𝑓󵄩 2 󵄩𝑓󵄩 2 , 󵄩 𝑗 󵄩𝐿𝑝2 (R𝑑) 󵄩 󵄩Ḃ𝑠,𝑘 (R𝑑)󵄩 󵄩Ḃ−𝑡,𝑘(R𝑑) 𝑗∈Z 𝑘 𝑞1,𝑟 𝑞2,𝑟 1 4 1 1 4 1 = + , = + , −2(𝑠(1−𝑎 )−𝑡𝑎 ) 󰜚=2 1 1 >0 𝑞1 (𝑑 + 2𝛾− 2) 𝑞 0 𝑞0 𝑟1 (𝑑 + 2𝛾− 2) 𝑟 1 𝑟0 where . From this, and applying 𝑓∈Ḃ𝑠,𝑘 (R𝑑)∩Ḃ−𝑡,𝑘(R𝑑) Proposition 25,wededucethatif 𝑞1,𝑟 𝑞2,𝑟 , 1 4 1 𝑝 𝑝 𝑝,𝑟 = + , 𝑓∈[𝐿1 (R𝑑), 𝐿 2 (R𝑑)] =𝐿 (R𝑑) 𝑠 (𝑑 + 2𝛾− 2) 𝑠 𝑠 then 𝑘 𝑘 1/2,𝑟 𝑘 .Furthermore, 1 0 0 using (57), we finally have: (126) 󵄩 󵄩 󵄩 󵄩1−𝜃 󵄩 󵄩𝜃 󵄩𝑓󵄩𝐿𝑝,𝑟(R𝑑) ≤𝐶󵄩𝑓󵄩Ḃ𝑠,𝑘 R𝑑 󵄩𝑓󵄩Ḃ−𝑡,𝑘(R𝑑). (134) ∗ 𝑘 𝑞1,𝑟( ) 𝑞2,𝑟 where 𝑟 = ((𝑑 + 2𝛾)𝑟0)/(𝑑 + 2𝛾− 𝑟 0).Thus,fromthe Leibnitz rule, Holder’s¨ inequality on Lorentz space, and Sobolevembedding,wededucethat Corollary 49. Let 𝑠 be a real number in the interval (0, (𝑑 + 2𝛾)/𝑞) 𝑞 [1, ∞] 𝑟 ,andlet be a real number in .There ‖𝑇𝑢‖𝐿𝑞0,𝑠0 (R;𝐿 0 (R𝑑)) 𝑘 ̇𝑠,𝑘 𝑑 is a constant 𝐶 such that, for any function 𝑓∈B𝑞,𝑞(R ),the (127) 󵄩 󵄩 (𝑑+2𝛾+2)/(𝑑+2𝛾−2) following inequality holds: ≤𝐶(󵄩𝑇𝑢 󵄩 2 𝑑 + ‖𝑇𝑢‖ 𝑟 ). 󵄩 0󵄩𝐿 (R ) 𝐿𝑞0,𝑠0 (R;𝐿 0 (R𝑑)) 𝑘 𝑘 󵄨 󵄨𝑞 1/𝑞 󵄨𝑓 (𝑥)󵄨 󵄩 󵄩𝜃 󵄩 󵄩1−𝜃 (∫ 𝜔 (𝑥) 𝑑𝑥) ≤𝐶󵄩𝑓󵄩 󵄩𝑓󵄩 𝑠−(𝑑+2𝛾)/𝑞,𝑘 , 𝑠𝑞 𝑘 󵄩 󵄩Ḃ𝑠,𝑘 R𝑑 󵄩 󵄩Ḃ R𝑑 ||𝑢0||𝐻1(R𝑑) 𝑑 ‖𝑥‖ 𝑞,𝑞( ) ∞,𝑞 ( ) Since 𝑘 is small, we have R 󵄩 󵄩 (135) ‖𝑇𝑢‖ 𝑟 ≤𝐶󵄩𝑇𝑢 󵄩 . 𝐿𝑞0,𝑠0 (R;𝐿 0 (R𝑑)) 󵄩 0󵄩𝐿2 (R𝑑) (128) 𝑘 𝑘 where 𝜃=1−𝑞𝑠/(𝑑+2𝛾). 12 Journal of Function Spaces and Applications

Proof. Let 𝑝∈(1,∞)and 𝑠 ∈ (0, (𝑑 + 2𝛾)/𝑞) with Theorem 52. Let 𝑠, 𝑡, >0 𝜃 = 𝑠/(𝑠 +𝑡) and let 𝑞1,𝑞2,𝑟1,𝑟2 ∈ 𝑠 1/𝑝 = 1/𝑞 − 𝑠/(𝑑 +2𝛾).Wetake𝑔(𝑥) = 1/||𝑥|| and apply [1, ∞], 𝑝,0 𝑟 ∈[1,∞)with 1/𝑝 = (1 − 𝜃)/𝑞1 +𝜃/𝑞2, (103)inthespecificform 1/𝑟0 =(1−𝜃)/𝑟1 +𝜃/𝑟2. 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 ̇𝑠,𝑘 𝑑 ̇−𝑡,𝑘 𝑑 󵄩𝑓𝑔 󵄩𝐿𝑞,𝑞(R𝑑) ≤𝐶󵄩𝑓󵄩𝐿𝑝,𝑞(R𝑑)󵄩𝑔󵄩𝐿𝑟,∞(R𝑑), (i) For every 𝑓∈B𝑞 ,𝑟 (R )∩B𝑞 ,𝑟 (R ),andif𝑟>𝑟0, 𝑘 𝑘 𝑘 (136) 1 1 2 2 𝑝,𝑟 𝑑 one has 𝑓∈𝐿𝑘 (R ) and where 𝑟 = (𝑑 + 2𝛾)/𝑠 and 𝑝 = (𝑞(𝑑 + 2𝛾))/(𝑑 +2𝛾−𝑞𝑠).As 𝑔∈𝐿𝑟,∞(R𝑑) 󵄩 󵄩 󵄩 󵄩1−𝜃 󵄩 󵄩𝜃 𝑘 ,wehave 󵄩 󵄩 𝑝,𝑟 󵄩 󵄩 󵄩 󵄩 󵄩𝑓󵄩𝐿 (R𝑑) ≤𝐶󵄩𝑓󵄩Ḃ𝑠,𝑘 R𝑑 󵄩𝑓󵄩Ḃ−𝑡,𝑘 R𝑑 . (143) 𝑘 𝑞1,𝑟1 ( ) 𝑞2,𝑟2 ( ) 󵄨 󵄨𝑞 1/𝑞 󵄨𝑓 (𝑥)󵄨 󵄩 󵄩 (∫ 𝑠𝑞 𝜔𝑘 (𝑥) 𝑑𝑥) ≤𝐶󵄩𝑓󵄩𝐿𝑝,𝑞(R𝑑). (137) (ii) Moreover,thisinequalityisvalidfor𝑟=𝑟0 in the 𝑑 ‖𝑥‖ 𝑘 R following cases: Combining this with (131), we obtain (135). (a) 𝑟=𝑟1 =𝑟2, Theorem 50. Let 0 < 𝑠 < (𝑑 + 2𝛾)/2 be given. There exists (b) 𝑟1 =𝑞1 and 𝑟2 =𝑞2, ̇𝑠 𝑑 apositiveconstant𝐶 such that for all function 𝑢∈H2,𝑘(R ), (c) 1<𝑝≤2and 𝑟0 =𝑝. one has (iii) Finally, the condition 𝑟≥𝑟0 is sharp. |𝑢 (𝑥)|2 ∫ 𝜔 (𝑥) 𝑑𝑥≤𝐶‖𝑢‖2 . 2𝑠 𝑘 Ḣ𝑠 (R𝑑) (138) R𝑑 ‖𝑥‖ 2,𝑘 Proof. (i) Case 𝑟>𝑟0. With no loss of generality, we may assume that 𝑞1 <𝑞2,andwefix𝜀>0such that For proof of this theorem, we need the following lemma, 1 1 1 1 1 1 1 1 whichweobtainbysimplecalculations. < −𝜀( − )= < +𝜀( − ) 𝑞2 𝑝 𝑞1 𝑞2 𝑝2 𝑝 𝑞1 𝑞2 Lemma 51. Let 𝑠 be a real number in the interval (0, 𝛾+ 𝑑/2) . (144) −2𝑠 1 1 Then, the function 𝑥 󳨃→ ||𝑥|| belongs to the Dunkl-Besov = < . ̇𝑑+2𝛾−2𝑠,𝑘 𝑑 𝑝 𝑞 space B1,∞ (R ). 1 1

Proof of Theorem 50. Let us define The proof follows essentially the same ideas used in the 𝑀 =2𝑗𝑠‖Δ 𝑓‖ previous theorem. Indeed, we have for 𝑗 𝑗 𝐿𝑞1 (R𝑑) 2 𝑘 |𝑢 (𝑥)| −2𝑠 2 −𝑗𝑡 𝐼 (𝑢) := ∫ 𝜔 (𝑥) 𝑑𝑥 =⟨ ⋅ ,𝑢 ⟩. 𝑁 =2 ‖Δ 𝑓‖ 𝑞 𝜀 =1 𝜀 =−1 𝑠,𝑘 2𝑠 𝑘 ‖ ‖ (139) and 𝑗 𝑗 𝐿 2 (R𝑑) and for 0 and 1 , R𝑑 ‖𝑥‖ 𝑘 󵄩 󵄩 󵄩 󵄩1−𝜃+𝜀𝜀 󵄩 󵄩𝜃−𝜀𝜀 󵄩Δ 𝑓󵄩 ≤ 󵄩Δ 𝑓󵄩 𝑖 󵄩Δ 𝑓󵄩 𝑖 Using homogeneous Littlewood-Paley decomposition and 󵄩 𝑗 󵄩𝐿𝑝𝑖 (R𝑑) 󵄩 𝑗 󵄩𝐿𝑞1 R𝑑 󵄩 𝑗 󵄩𝐿𝑞2 R𝑑 2 󸀠 𝑑 𝑘 𝑘 ( ) 𝑘 ( ) the fact that 𝑢 belongs to S ℎ,𝑘(R ),wecanwrite (145) 1−𝜃+𝜀𝜀 𝜃−𝜀𝜀 −𝑗𝜀𝜀 (𝑠+𝑡) =𝑀 𝑖 𝑁 𝑖 2 𝑖 . −2𝑠 2 𝑗 𝑗 𝐼𝑠,𝑘 (𝑢) = ∑ ⟨Δ 𝑛 (‖⋅‖ ),Δ𝑚 (𝑢 )⟩ |𝑛−𝑚|≤2 1−𝜃+𝜀𝜀𝑖 𝜃−𝜀𝜀𝑖 󰜚𝑖 As 𝑟1 =𝑟̸2,wecanonlysaythat(𝑀𝑗 𝑁𝑗 )𝑗∈Z ∈𝑙, 𝑛((𝑑+2𝛾)/2−2𝑠) ≤𝐶 ∑ ⟨2 where 1/󰜚𝑖 =(1−𝜃+𝜀𝜀𝑖)/𝑟1 +(𝜃−𝜀𝜀𝑖)/𝑟2.Wemayuse(57), but 𝑝,󰜚 𝑑 𝑝 𝑑 𝑝 𝑑 |𝑛−𝑚|≤2 1 2 we get only that 𝑓∈𝐿𝑘 (R )=[𝐿𝑘 (R ), 𝐿𝑘 (R )]1/2,󰜚 with 󰜚=max(󰜚1,󰜚2) and that satisfies (143)with𝑟=󰜚.However, ×Δ (‖⋅‖−2𝑠),2−𝑚((𝑑+2𝛾)/2−2𝑠)Δ (𝑢2)⟩ . 𝑛 𝑚 we may choose 𝜀 as small as we want and thus, 󰜚 as close to 𝑟0 (140) as we want; thus 𝑓 satisfies (143)forevery𝑟>𝑟0.

−2𝑠 (𝑑+2𝛾)/2−2𝑠,𝑘 𝑑 ̇ 𝑟=𝑟0 Lemma 51 claims that ‖⋅‖ belongs to B2,∞ (R ). (ii) Case . Theorem 17 yields (a) If 𝑟=𝑟1 =𝑟2: this case was treated in Theorem 48. 󵄩 󵄩 𝑟 =𝑞 𝑟 =𝑞 󵄩𝑢2󵄩 ≤𝐶‖𝑢‖2 . (b) If 1 1 and 2 2:thisisadirectconsequenceof 󵄩 󵄩Ḃ2𝑠−(𝑑+2𝛾)/2,𝑘(R𝑑) Ḣ𝑠 (R𝑑) (141) 2,1 2,𝑘 (43), since we have 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 Thus, 󵄩𝑓󵄩Ḃ𝑠,𝑘 (R𝑑) = 󵄩𝑓󵄩Ḟ𝑠,𝑘 (R𝑑) ≤𝐶󵄩𝑓󵄩Ḟ𝑠,𝑘 (R𝑑), 𝑞𝑖,𝑞𝑖 𝑞𝑖,𝑞𝑖 𝑞𝑖,∞ 2 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 (146) 𝐼 (𝑢) ≤𝐶‖𝑢‖ 𝑠 𝑑 . 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 𝑠,𝑘 Ḣ(R ) (142) 𝑓 −𝑡,𝑘 = 𝑓 −𝑡,𝑘 ≤𝐶 𝑓 −𝑡,𝑘 , 2,𝑘 󵄩 󵄩Ḃ (R𝑑) 󵄩 󵄩Ḟ (R𝑑) 󵄩 󵄩Ḟ (R𝑑) 𝑞𝑖,𝑞𝑖 𝑞𝑖,𝑞𝑖 𝑞𝑖,∞

we obtain The following results of this section are in sprit of the 󵄩 󵄩 󵄩 󵄩1−𝜃 󵄩 󵄩𝜃 󵄩𝑓󵄩 𝑝 ≤𝐶󵄩𝑓󵄩 󵄩𝑓󵄩 . 󵄩 󵄩𝐿 (R𝑑) 󵄩 󵄩Ḃ𝑠,𝑘 (R𝑑)󵄩 󵄩Ḃ−𝑡,𝑘 (R𝑑) (147) classical case (cf. [27]). 𝑘 𝑞1,𝑞1 𝑞2,𝑞2 Journal of Function Spaces and Applications 13

1/𝑞 −𝑗𝑞2𝑡 𝑞2 2 (c) Case 1<𝑝≤2and 𝑟0 =𝑝. 𝑀 =(∑ 2 ‖Δ 𝑓‖ 𝑞 ) 𝑁 = Let us note that 𝑛 𝑗∈𝑍𝑛 𝑗 𝐿 2 (R𝑑) , 𝑛 𝑘 𝑞 1/𝑞 𝑞 1/𝑞 2−𝑎𝑛(∑ 𝜆 1 ) 1 ,𝐿 =2(1−𝑎)𝑛(∑ 𝜆 2 ) 2 𝑓 = We just write 𝑗∈𝑍𝑛 𝑗 𝑛 𝑗∈𝑍𝑛 𝑗 ,and 𝑛 ∑ Δ 𝑓 𝑗∈𝑍𝑛 𝑗 . We apply now (147)andTheorem 48 to obtain 󵄩 󵄩 󵄩 󵄩1−𝜃 󵄩 󵄩𝜃 󵄩Δ 𝑓󵄩 ≤ 󵄩Δ 𝑓󵄩 󵄩Δ 𝑓󵄩 󵄩 𝑗 󵄩𝐿𝑝 (R𝑑) 󵄩 𝑗 󵄩𝐿𝑞1 (R𝑑)󵄩 𝑗 󵄩𝐿𝑞2 (R𝑑) 󵄩 󵄩 󵄩 󵄩1−𝜃 󵄩 󵄩𝜃 𝑘 𝑘 𝑘 󵄩𝑓𝑛󵄩𝐿𝑝 (R𝑑) ≤𝐶󵄩𝑓𝑛󵄩Ḃ𝑠,𝑘 R𝑑 󵄩𝑓𝑛󵄩Ḃ−𝑡,𝑘 R𝑑 𝑘 𝑞1,𝑞1 ( ) 𝑞2,𝑞2 ( ) 󵄩 󵄩 1−𝜃 󵄩 󵄩 𝜃 =(2𝑗𝑠󵄩Δ 𝑓󵄩 ) (2−𝑗𝑡󵄩Δ 𝑓󵄩 ) ≤𝐶𝑁1−𝜃𝑀𝜃2𝑛𝑎(1−𝜃), 󵄩 𝑗 󵄩𝐿𝑞1 (R𝑑) 󵄩 𝑗 󵄩𝐿𝑞2 (R𝑑) 𝑛 𝑛 𝑘 𝑘 (155) (148) 󵄩 󵄩 󵄩 󵄩1−𝜃 󵄩 󵄩𝜃 󵄩𝑓𝑛󵄩𝐿𝑝,𝑞2 (R𝑑) ≤𝐶󵄩𝑓𝑛󵄩Ḃ𝑠,𝑘 R𝑑 󵄩𝑓𝑛󵄩Ḃ−𝑡,𝑘 R𝑑 𝑘 𝑞1,𝑞2 ( ) 𝑞2,𝑞2 ( ) and get by Holder’s¨ inequality: 1−𝜃 𝜃 −𝑛(1−𝑎)(1−𝜃) ≤𝐶𝐿𝑛 𝑀𝑛2 . 󵄩 󵄩 󵄩 󵄩1−𝜃 󵄩 󵄩𝜃 𝑓=∑ 𝑓 󵄩𝑓󵄩 ̇0,𝑘 𝑑 ≤𝐶󵄩𝑓󵄩 ̇𝑠,𝑘 𝑑 󵄩𝑓󵄩 ̇−𝑡,𝑘 𝑑 . Sincewehave 𝑛∈Z 𝑛, with these two inequalities at 󵄩 󵄩B𝑝,𝑝(R ) 󵄩 󵄩B𝑞 ,𝑟 (R )󵄩 󵄩B𝑞 ,𝑟 (R ) (149) 1 1 2 2 𝑝 𝑑 𝑝,𝑞2 𝑑 hand, and using (57), we find that 𝑓∈[𝐿𝑘(R ), 𝐿𝑘 (R )]𝑎,𝑟, 1/𝑟 = (1−𝑎)/𝑝+𝑎/𝑞 1/𝑟 = (1−𝑎)/𝑞 +𝑎/𝑞 Ḃ0,𝑘 (R𝑑)⊂𝐿𝑝(R𝑑)=𝐿𝑝,𝑝(R𝑑) with 2,but,since 1 1 2 We then use the embedding 𝑝,𝑝 𝑘 𝑘 𝑝 𝑑 𝑝,𝑞2 𝑑 and 1/𝑝 = (1−𝜃)/𝑞1+𝜃/𝑞2,weobtain[𝐿𝑘(R ), 𝐿𝑘 (R )]𝑎,𝑟 = which is valid for 𝑝≤2. 𝑟 𝑑 𝐿𝑘(R ) with 1/𝑟 = (1 − 𝜃)/𝑟1 +𝜃/𝑞2. Theorem 53. Let 𝑠, 𝑡,let >0 𝑞1,𝑞2 ∈[1,∞]with 𝑞1 <𝑞2.Let Theorem 54. Let 𝑠, 𝑡,andlet >0 𝑞1,𝑞2 ∈[1,∞]with 𝑞1 <𝑞2. 𝜃 = 𝑠/(𝑠 + 𝑡) ∈ (0,1),andlet1/𝑝 = (1 − 𝜃)/𝑞1 +𝜃/𝑞2. Let 𝜃 = 𝑠/(𝑠 + 𝑡) ∈ (0,1),andlet1/𝑝 = (1 − 𝜃)/𝑞1 +𝜃/𝑞2.Let 𝑞1 ≤𝑟1 ≤𝑟2 ≤𝑞2,andlet1/𝑟 = (1 − 𝜃)/𝑟1 +𝜃/𝑟2.Then,one (i) If 𝑞1 ≤𝑟1 ≤𝑞2 and let 1/𝑟 = (1 − 𝜃)/𝑟1 +𝜃/𝑞2.Then, has one has 󵄩 󵄩 󵄩 󵄩1−𝜃 󵄩 󵄩𝜃 󵄩𝑓󵄩 𝑝,𝑟 ≤𝐶󵄩𝑓󵄩 󵄩𝑓󵄩 . 󵄩 󵄩𝐿 (R𝑑) 󵄩 󵄩Ḃ𝑠,𝑘 (R𝑑)󵄩 󵄩Ḃ−𝑡,𝑘 (R𝑑) (156) 󵄩 󵄩 󵄩 󵄩1−𝜃 󵄩 󵄩𝜃 𝑘 𝑞1,𝑟1 𝑞2,𝑟2 󵄩𝑓󵄩𝐿𝑝,𝑟(R𝑑) ≤𝐶󵄩𝑓󵄩Ḃ𝑠,𝑘 R𝑑 󵄩𝑓󵄩Ḃ−𝑡,𝑘 R𝑑 . (150) 𝑘 𝑞1,𝑟1 ( ) 𝑞2,𝑞2 ( ) Proof. Once the previous theorem is proved, it is enough to reapply similar arguments to obtain Theorem 54.As 𝑞 <𝑟 <𝑟 <𝑞 (ii) If 𝑞1 ≤𝑟2 ≤𝑞2,andlet1/𝑟 = (1 − 𝜃)/𝑞1 +𝜃/𝑟2.Then, 1 1 2 2,westartusing one has 𝑟 𝑞 𝑟 𝑙 1 =[𝑙 1 ,𝑙 2 ] 𝑎,𝑟1 (157) 󵄩 󵄩 󵄩 󵄩1−𝜃 󵄩 󵄩𝜃 󵄩𝑓󵄩𝐿𝑝,𝑟(R𝑑) ≤𝐶󵄩𝑓󵄩Ḃ𝑠,𝑘 R𝑑 󵄩𝑓󵄩Ḃ−𝑡,𝑘 R𝑑 . (151) Z =∑ 𝑍 𝑘 𝑞1,𝑞1 ( ) 𝑞2,𝑟2 ( ) instead of (152), and we obtain a partition 𝑗∈Z 𝑗 such that

󵄩 1/𝑞1 󵄩 󵄩 1/𝑟2 󵄩 Proof. We only prove the first inequality, as the proof for the 󵄩 󵄩 󵄩 󵄩 󵄩 −𝑎𝑗 𝑞1 󵄩 󵄩 (1−𝑎)𝑗 𝑟2 󵄩 ̇𝑠,𝑘 𝑑 󵄩2 ( ∑ 𝜆𝑛 ) 󵄩 + 󵄩2 ( ∑ 𝜆𝑛 ) 󵄩 second one is similar. Since 𝑓∈B𝑞 ,𝑟 (R ),notingthat 󵄩 󵄩 󵄩 󵄩 1 1 󵄩 𝑛∈𝑍 󵄩 󵄩 𝑛∈𝑍 󵄩 𝑠𝑗 𝑟 󵄩 𝑗 󵄩 𝑟1 󵄩 𝑗 󵄩 𝑟1 (158) 𝜆 =2‖Δ 𝑓‖ (𝜆 ) ∈𝑙1 𝑙 𝑙 𝑗 𝑗 𝐿𝑞1 (R𝑑),wehave 𝑗 𝑗∈Z .Thus,using 𝑘 󵄩 󵄩 ≤𝐶󵄩𝜆 󵄩 , Proposition 26 (i) for the interpolation 󵄩 𝑗󵄩𝑙𝑟1

𝑠𝑗 𝑟1 𝑞1 𝑞2 1/𝑟 = (1 − 𝑎)/𝑞 + 𝑎/𝑟 𝜆 =2 ‖Δ 𝑓‖ 𝑙 =[𝑙 ,𝑙 ] , with 1 1 2 and where 𝑗 𝑗 𝐿𝑞1 (R𝑑) 𝑎,𝑟 (152) 𝑘 𝑟 𝑠,𝑘 𝑑 𝑙 1 𝑓∈Ḃ(R ) 𝑓∈ belongs to ,since 𝑞1,𝑟1 .Moreover,since with 1/𝑟1 = (1 − 𝑎)/𝑞1 + 𝑎/𝑞2, we see that we have a partition Ḃ−𝑡,𝑘 (R𝑑) 𝑞2,𝑟2 ,wehave Z = ∑𝑗∈Z 𝑍𝑗 such that

1/𝑞2 𝑞 󵄩 󵄩 󵄩 󵄩 −𝑗𝑞 𝑡󵄩 󵄩 2 𝑞 󵄩 1/𝑞1 󵄩 󵄩 1/𝑞2 󵄩 (( ∑ 2 2 󵄩Δ 𝑓󵄩 ) ) ∈𝑙 2 . 󵄩 󵄩 󵄩 󵄩 󵄩 𝑗 󵄩𝐿𝑞2 (R𝑑) (159) 󵄩 −𝑎𝑗 𝑞1 󵄩 󵄩 (1−𝑎)𝑗 𝑞2 󵄩 𝑗∈𝑍 𝑘 󵄩2 ( ∑ 𝜆𝑛 ) 󵄩 + 󵄩2 ( ∑ 𝜆𝑛 ) 󵄩 𝑛 𝑛∈Z 󵄩 󵄩 󵄩 󵄩 󵄩 𝑛∈𝑍𝑗 󵄩 󵄩 𝑛∈𝑍𝑗 󵄩 󵄩 󵄩𝑙𝑟1 󵄩 󵄩𝑙𝑟1 (153) 1/𝑞 −𝑗𝑞2𝑡 𝑞2 2 𝑀 =(∑ 2 ‖Δ 𝑓‖ 𝑞 ) 𝑁 = 󵄩 󵄩 Let us note that 𝑛 𝑗∈𝑍𝑛 𝑗 𝐿 2 (R𝑑) , 𝑛 ≤𝐶󵄩𝜆 󵄩 . 𝑘 󵄩 𝑗󵄩 𝑟1 𝑞 1/𝑞 𝑞 1/𝑞 𝑙 2−𝑎𝑛(∑ 𝜆 1 ) 1 ,𝐿 =2(1−𝑎)𝑛(∑ 𝜆 2 ) 2 𝑓 = 𝑗∈𝑍𝑛 𝑗 𝑛 𝑗∈𝑍𝑛 𝑗 ,and 𝑛 ∑ Δ 𝑓 ̇−𝑡,𝑘 𝑑 𝑗∈𝑍𝑛 𝑗 . We apply now (151)andTheorem 48 instead of Moreover, since 𝑓∈B𝑞 ,𝑞 (R ),wehave 2 2 (155)toobtain 󵄩 󵄩 󵄩 󵄩1−𝜃 󵄩 󵄩𝜃 1/𝑞2 󵄩𝑓 󵄩 ≤𝐶󵄩𝑓 󵄩 󵄩𝑓 󵄩 󵄩 𝑛󵄩𝐿𝑝,𝑏(R𝑑) 󵄩 𝑛󵄩Ḃ𝑠,𝑘 R𝑑 󵄩 𝑛󵄩Ḃ−𝑡,𝑘 R𝑑 󵄩 󵄩𝑞 𝑘 𝑞1,𝑞1 ( ) 𝑞2,𝑟2 ( ) −𝑗𝑞2𝑡󵄩 󵄩 2 𝑞2 (( ∑ 2 󵄩Δ 𝑓󵄩 𝑞 ) ) ∈𝑙 . (160) 󵄩 𝑗 󵄩𝐿 2 (R𝑑) (154) 𝑗∈𝑍 𝑘 1−𝜃 𝜃 𝑛𝑎(1−𝜃) 𝑛 𝑛∈Z ≤𝐶𝑁𝑛 𝑀𝑛 2 , 14 Journal of Function Spaces and Applications

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Research Article Density in Spaces of Interpolation by Hankel Translates of a Basis Function

Cristian Arteaga and Isabel Marrero

Departamento de Analisis´ Matematico,´ Universidad de La Laguna, 38271 La Laguna (Tenerife), Spain

Correspondence should be addressed to Isabel Marrero; [email protected]

Received 3 May 2013; Accepted 24 June 2013

Academic Editor: Jozef´ Bana´s

Copyright © 2013 C. Arteaga and I. Marrero. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The function spaces 𝑌𝑚 (𝑚 ∈ Z+) arising in the theory of interpolation by Hankel translates of a basis function, as developed by the authors elsewhere, are defined through a seminorm which is expressed in terms of the Hankel transform of each function and involves a weight 𝑤. At least two special classes of weights allow to write these indirect seminorms in direct form, that is, in terms of the function itself rather than its Hankel transform. In this paper, we give fairly general conditions on 𝑤 which ensure that the Zemanian spaces B𝜇 and H𝜇 (𝜇 > −1/2) are dense in 𝑌𝑚 (𝑚 ∈ Z+). These conditions are shown to be satisfied by the weights giving rise to direct seminorms of the so-called type II.

󸀠 ∗ 1. Introduction H𝜇 when this latter space is endowed with either its weak or its strong topology. The Hankel integral transformation is usually defined by Zemanian [2]alsoconstructedthespaceB𝜇 as follows. ∞ For every 𝑎>0, B𝜇,𝑎 consists of all those 𝜑∈H𝜇 such (ℎ𝜇𝜑) (𝑥) = ∫ 𝜑 (𝑡) J𝜇 (𝑥𝑡) 𝑑𝑡 (𝑥∈𝐼) , (1) 𝜑(𝑥) =0 𝑥>𝑎 0 that whenever .Thisspaceisendowedwith the topology generated by the family of seminorms {𝛾𝜇,𝑟}𝑟∈Z , 1/2 + where 𝐼=]0,∞[, J𝜇(𝑧) = 𝑧 𝐽𝜇(𝑧) (𝑧 ,and∈ 𝐼) 𝐽𝜇 denotes given by the Bessel function of the first kind and order 𝜇∈R. 󵄨 󵄨 󵄨 −1 𝑟 −𝜇−1/2 󵄨 𝛾𝜇,𝑟 (𝜑) = sup 󵄨(𝑥 𝐷) 𝑥 𝜑 (𝑥)󵄨 𝑥∈]0,𝑎[ 󵄨 󵄨 1.1. The Distributional Hankel Transformation. Aiming to (3) define the Hankel transformation in spaces of distributions, (𝜑 ∈ B𝜇,𝑎,𝑟∈Z+). Zemanian [1] introduced the space H𝜇 of all those smooth, complex-valued functions 𝜑 = 𝜑(𝑥) (𝑥 ∈𝐼) such that In this way B𝜇,𝑎 becomes a Frechet´ space. Moreover, if 0< 󵄨 𝑟 𝑘 󵄨 𝑎<𝑏 B ⊂ B B 󵄨 2 −1 −𝜇−1/2 󵄨 ,then 𝜇,𝑎 𝜇,𝑏 and the topology of 𝜇,𝑎 coincides ]𝜇,𝑟 (𝜑) = max sup 󵄨(1 + 𝑥 ) (𝑥 𝐷) 𝑥 𝜑 (𝑥)󵄨 <∞ 0≤𝑘≤𝑟 𝑥∈𝐼 󵄨 󵄨 with that inherited from B𝜇,𝑏. This allows to consider the inductive limit B𝜇 =⋃𝑎>0 B𝜇,𝑎. As usual, the dual spaces of (𝑟∈Z+) . 󸀠 󸀠 B𝜇 and B𝜇,𝑎 are respectively denoted by B𝜇 and B𝜇,𝑎 (𝑎 > (2) 󸀠 0).SinceB𝜇 is a dense subspace of H𝜇, H𝜇 can be regarded 𝐷=𝐷 =𝑑/𝑑𝑥 󸀠 Here, and in the sequel, 𝑥 .Whentopologized as a subspace of B𝜇. {] } H by the family of norms 𝜇,𝑟 𝑟∈Z+ , 𝜇 becomes a Frechet´ space where ℎ𝜇 is an automorphism provided that 𝜇≥−1/2. 1.2. The Delsarte Kernel and the Hankel Convolution. Hir- ℎ󸀠 Then the generalized Hankel transformation 𝜇, defined by schman [3], Cholewinski [4], and Haimo [5]developeda 󸀠 transposition on the dual H𝜇 of H𝜇, is an automorphism of convolution theory on Lebesgue spaces for a variant of the 2 Journal of Function Spaces and Applications

Hankel transformation closely connected to (1). For 𝜇> The study of the Hankel convolution on compactly supp- −1/2, straightforward manipulations of the results in [3–5] orted distributions was initiated by de Sousa Pinto [6], only allow us to define a convolution for ℎ𝜇 as follows. Whenever for 𝜇=0. In a series of papers, Betancor and the second the integrals involved exist, the Hankel convolution of the namedauthorinvestigatedsystematicallythegeneralized#- functions 𝜑=𝜑(𝑥)and 𝜙=𝜙(𝑥)(𝑥∈𝐼)is defined as the convolution in wider spaces of distributions, allowing 𝜇> function −1/2. In this context (cf. [7]), the Hankel translation was ∞ shown to be a continuous operator from H𝜇 into itself. Thus, 󸀠 󸀠 (𝜑#𝜙) (𝑥) = ∫ 𝜑(𝑦)(𝜏𝑥𝜙) (𝑦) 𝑑𝑦 (𝑥∈𝐼) , (4) 𝑓 𝜑∈H 𝑓∈H 𝜑∈H 0 the Hankel convolution # 𝜇 of 𝜇 and 𝜇 can be defined through where the Hankel translate 𝜏𝑥𝜙 of 𝜙 is given by ∞ (𝑓#𝜑) (𝑥) = ⟨𝑓, 𝜏𝑥𝜑⟩ (𝑥∈𝐼) (13) (𝜏𝑥𝜙) (𝑦) = ∫ 𝜙 (𝑧) 𝐷𝜇 (𝑥,𝑦,𝑧)𝑑𝑧 (𝑥,𝑦∈𝐼). (5) 0 [7, Definition 3.1]. The formulas

Here 󸀠 −𝜇−1/2 󸀠 ∞ ℎ𝜇 (𝜏𝑦𝑓) (𝑥) =𝑥 J𝜇 (𝑥𝑦)𝜇 (ℎ 𝑓) (𝑥) (𝑦 ∈ 𝐼) , (14) −𝜇−1/2 𝐷𝜇 (𝑥,𝑦,𝑧)=∫ 𝑡 J𝜇 (𝑥𝑡) J𝜇 (𝑦𝑡) J𝜇 (𝑧𝑡) 𝑑𝑡 0 󸀠 −𝜇−1/2 󸀠 (6) ℎ𝜇 (𝑓#𝜑) (𝑥) =𝑥 (ℎ𝜇𝜑) (𝑥) (ℎ𝜇𝑓) (𝑥) , (15) (𝑥, 𝑦,) 𝑧∈𝐼 respectively extending (11)and(12), hold in the sense of 󸀠 is the so-called Delsarte kernel. Recall that equality in H𝜇 (cf. [7, Proposition 3.5]). 2𝜇−1 𝐷 (𝑥, 𝑦, 𝑧) = (𝑥𝑦𝑧)1/2−𝜇Δ(𝑥, 𝑦, 𝑧)2𝜇−1 𝜇 Γ (𝜇 + 1/2) 𝜋1/2 1.3. Interpolation by Hankel Translates of a Basis Function. In approximation theory, radially symmetric, (conditionally) (𝑥,𝑦,𝑧∈𝐼), positive definite functions are used to solve scattered data (7) interpolation problems in Euclidean space. The setting for a variational approach to such interpolation problems, the so- Δ(𝑥, 𝑦, 𝑧) being the area of the triangle with sides of length called native spaces, was constructed by several authors upon 𝑥, 𝑦, 𝑧 if such a triangle exists, and zero otherwise [3,p.308]. seminalworkofMicchelli[8]andMadychandNelson[9–11]. We note that 𝐷𝜇 (𝑥,𝑦,𝑧)=0̸ only when |𝑥−𝑦|< 𝑧 <𝑥+ Later, Light and Wayne [12]ideatedanalternativeapproachin 𝑦 (𝑥, 𝑦, .Furthermore,𝑧 ∈𝐼) which the distributional theory of the Fourier transformation ∞ plays a prominent role. 𝜇+1/2 −1 𝜇+1/2 ∫ 𝐷𝜇 (𝑥, 𝑦, 𝑧) 𝑧 𝑑𝑧𝜇 =𝑐 (𝑥𝑦) (𝑥, 𝑦 ∈ 𝐼) , (8) When dealing with interpolation by radial basis func- 0 tions, one can either (i) keep treating the involved functions 𝜇 R𝑑 (𝑑 ∈ N) with 𝑐𝜇 =2 Γ(𝜇 + 1). as radially symmetric functions on , or (ii) 𝑝 identify them with functions on the positive real half-axis. For 1≤𝑝<∞, denote by 𝐿𝜇 the space of Lebesgue mea- For instance, Schaback and Wu [13]devisedageneraltheory surable functions whose 𝑝th power is absolutely integrable on 𝐼 𝑡𝜇+1/2 which allows to write multivariate Fourier transforms or with respect to the weight ,normedwith convolutions of radial functions as very simple univariate ∞ 1/𝑝 operations. Motivated by [12], in [14] we benefited from the 𝑝 𝜇+1/2 𝑝 ‖𝑢‖𝜇,𝑝 =(∫ |𝑢(𝑡)| 𝑡 𝑑𝑡) (𝑢 ∈ 𝐿𝜇). (9) Hankel transformation and the Hankel convolution in order 0 to provide (ii) with an adequate theoretical support. This new ∞ By 𝐿𝜇 we will represent the space of Lebesgue measurable approach generalizes and improves (i) in a sense that is made 𝑢=𝑢(𝑡)(𝑡∈𝐼) 𝑡−𝜇−1/2𝑢(𝑡)(𝑡∈𝐼) precise next. functions such that is 𝑑∈N 𝑓(𝑥) =𝑓 (|𝑥|) 𝑥∈R𝑑 essentially bounded, normed with Recall that if and 0 (a.e. )is an integrable radial function, then its 𝑑-dimensional Fourier 󵄨 󵄨 ‖𝑢‖ = 󵄨𝑡−𝜇−1/2𝑢 (𝑡)󵄨 (𝑢 ∈ 𝐿∞). transform 𝜇,∞ esssup𝑡∈𝐼 󵄨 󵄨 𝜇 (10) 1 ∞ ̂ −2𝜋𝑖𝑥⋅𝜉 𝑑 𝜑∈𝐿 𝜙∈𝐿 𝑓 (𝜉) = ∫ 𝑓 (𝑥) 𝑒 𝑑𝑥 (𝜉∈R ) (16) If 𝜇 and 𝜇 ,then(5)and(4) exist as continuous 𝑑 1 R functions on 𝐼.If𝜑, 𝜙 ∈𝐿𝜇,then(5)and(4) exist as functions 1 is also radial and reduces to a 1-dimensional Hankel trans- in 𝐿𝜇, and, moreover, the formula form of order 𝑑/2 − 1 [15, Theorem 3.3]: ℎ (𝜏 𝜙) (𝑥) =𝑥−𝜇−1/2J (𝑥𝑦)(ℎ 𝜙) (𝑥) (𝑥, 𝑦 ∈𝐼) 𝜇 𝑦 𝜇 𝜇 (11) ̂ 󵄨 󵄨 󵄨 󵄨−(𝑑/2−1) 𝑓 (𝜉) =𝐹0 (󵄨𝜉󵄨)=2𝜋󵄨𝜉󵄨 and the exchange formula ∞ 󵄨 󵄨 𝑑/2 −𝜇−1/2 × ∫ 𝑓0 (𝑠) 𝐽𝑑/2−1 (2𝜋 󵄨𝜉󵄨 𝑠) 𝑠 𝑑𝑠 (17) ℎ𝜇 (𝜑#𝜙) (𝑥) =𝑥 (ℎ𝜇𝜑) (𝑥) (ℎ𝜇𝜙) (𝑥)(𝑥∈𝐼) (12) 0 𝑑 hold. (𝜉 ∈ R ). Journal of Function Spaces and Applications 3

2𝑗+𝜇+1/2 Actually, since 𝑝𝜇,𝑗(𝑥) = 𝑥 (𝑗 ∈ Z+, 0≤𝑗≤𝑚−1)are Muntz¨ monomials; 𝜏𝑧 (𝑧 ∈ 𝐼) denotes the Hankel translation 1/2 2 operator of order 𝜇;and𝛼𝑖,𝛽𝑗 (𝑖, 𝑗 ∈ Z+,1≤𝑖≤𝑛,0≤𝑗≤ 𝐽−1/2 (𝑧) =( ) cos 𝑧 (𝑧∈𝐼) , (18) 𝜋𝑧 𝑚−1)are complex coefficients. When applied to scattered data interpolation the previous it turns out that on radial univariate -even- functions, the scheme leaves a greater variety of manageable kernels at our Fourier transformation, which agrees with the Fourier-cosine disposal,whichcouldbeusefulinhandlingmathematical transformation, coincides (up to a multiplicative constant) models built upon a class of radial basis functions depending with the Hankel transform as well. Similarly, the above- on the order 𝜇 and whose performance is expected to 𝜇= mentioned variant of the Hankel convolution of order improve by adjusting 𝜇, as it happens with the family of −1/2 can be seen to coincide with the usual convolution on Matern´ kernels in [17, Supplement, p. 6]; the examples and R 2𝜇 + 2 ∉ N (cf. [16, Example 3.2]). Thus, for the Hankel numerical experiments exhibited in [14]seemtosupportthis convolution structure provides a strict generalization of the view. Other potential applications of interpolation by Hankel Fourier one. translates of a basis function are in the field of radial basis 𝐿1 Denote by 𝜇,𝑙 the class of all those Lebesgue measurable function neural networks [18–20]. functions 𝑢=𝑢(𝑡)(𝑡∈𝐼)such that Itmaybeobservedthattheseminormin(23)iswrittenin 𝑎 termsoftheHankeltransformofthefunction𝑓 (an indirect 𝜇+1/2 ∫ |𝑢 (𝑡)| 𝑡 𝑑𝑡 <∞ (𝑎>0) . (19) seminorm) rather than 𝑓 itself (a direct seminorm). The 0 latter is more convenient for the purpose of obtaining error estimates, however. Motivated by [21], in [22] we expressed The following spaces were introduced in14 [ ]. the indirect seminorm (23)intwoequivalentdirectforms, Definition 1. Let 𝑤=𝑤(𝑡)>0(𝑡∈𝐼)be a continuous fun- which were referred to as seminorms of type I and type II. ction, let HerewewanttousetypeIIseminormstogainadeeper understanding of the spaces 𝑌𝑚 (𝑚 ∈ Z+).Weshowthat, −𝜇−1/2 2𝜇+1 −𝜇−1/2 𝑤 𝑆𝜇 =𝑆𝜇,𝑡 =𝑡 𝐷𝑡𝑡 𝐷𝑡𝑡 (20) under rather general conditions on the weight ,whichare satisfied by those weights giving rise to seminorms of type II, B H 𝑌 (𝑚 ∈ Z ) be the Bessel differential operator, and let theZemanianspaces 𝜇 and 𝜇 are dense in 𝑚 + . 󸀠 󸀠 𝑚 1 2 1.4. Structure and Notation. This paper is organized as foll- 𝑌𝑚 ={𝑓∈H𝜇 :ℎ𝜇𝑆𝜇 𝑓∈𝐿𝜇,𝑙 ∩𝐿𝜇,𝑤}(𝑚∈Z+), (21) ows. In Section 2 we recall the definition of a seminorm of 0 𝑚 type II and introduce the notion of strong type II seminorm. where 𝑆 is the identity operator, 𝑆 (𝑚 ∈ N) is the operator 𝜇 𝜇 Also, we prove that those weights giving rise to type II 𝑆 𝑚 𝐿2 𝜇 iterated times, and 𝜇,𝑤 stands for the class of all seminorms are integrable near zero and exhibit polynomial measurable functions 𝑢 = 𝑢(𝑡) (𝑡 ∈𝐼) satisfying growth at infinity. With the aid of some preliminary lemmas concerning Hankel approximate identities, the density of B𝜇 ∞ 1/2 2 𝜇+1/2 and H𝜇 in 𝑌𝑚 (𝑚 ∈ Z+) is finally proved in Section 3. ‖𝑢‖𝜇,𝑤 =(∫ |𝑢 (𝑡)| 𝑤 (𝑡) 𝑡 𝑑𝑡) <∞. (22) 0 Throughout the rest of this paper, the positive real axis will be always denoted by 𝐼,while𝜇 will stand for a real A seminorm (norm if 𝑚=0) is defined on 𝑌𝑚 by setting number strictly greater than −1/2,and𝐶 will represent a

1/2 suitable positive constant, depending only on the opportune 󵄨 󵄨 ∞ 󵄨 󵄨2 󵄨𝑓󵄨 =(∫ 󵄨(ℎ󸀠 𝑆𝑚𝑓) (𝑡)󵄨 𝑤 (𝑡) 𝑡𝜇+1/2𝑑𝑡) (𝑓 ∈ 𝑌 ). subscripts (if any), whose value may vary from line to line. 󵄨 󵄨𝑚 󵄨 𝜇 𝜇 󵄨 𝑚 0 Moreover, we shall adhere to the notations Z+ = N ∪{0}for 1/2 (23) the set of nonnegative integers and J𝜇(𝑧) = 𝑧 𝐽𝜇(𝑧) (𝑧 ∈ 𝐼) for the function giving the kernel of the Hankel transforma- 𝑚∈N 𝑤 In [14], for and suitable conditions on the weight tion ℎ𝜇. The following classes of functions will be occasionally related to the values of 𝑚,thespaces𝑌𝑚 were shown to consist used: C, formed by the continuous functions on 𝐼,andE, of continuous functions on 𝐼. Also, interpolants to 𝑓∈𝑌𝑚 of consisting of all those infinitely differentiable functions on the form 𝐼. For the operational rules of the Hankel transformation 𝑛 𝑚−1 and further properties of the Hankel translation and Hankel (𝑈𝑓) (𝑥) = ∑𝛼 (𝜏 Φ) (𝑥) + ∑ 𝛽 𝑝 (𝑥)(𝑥∈𝐼) convolution that eventually might be required, both in the 𝑖 𝑎𝑖 𝑗 𝜇,𝑗 (24) 𝑖=1 𝑗=0 classical and the generalized senses, the reader is mainly referred to [3–5, 7, 23, 24]. were obtained, where {𝑎1,...,𝑎𝑛}⊂𝐼is the set of interpo- 󸀠 lation points; Φ∈H𝜇 is a complex function defined on 𝐼 2. Seminorms of Type II (the so-called basis function), connected with 𝑤 through the 1 Denote by 𝐿 the class of all those measurable functions 𝑢= distributional identity 𝜇,𝑐 𝑢(𝑡) (𝑡 ∈𝐼) such that ∞ 4𝑚 󸀠 1 𝜇+1/2 𝑡 (ℎ𝜇Φ) (𝑡) = ; (25) ∫ |𝑢 (𝑡)| 𝑡 𝑑𝑡 <∞ (𝑎>0) . (26) 𝑤 (𝑡) 𝑎 4 Journal of Function Spaces and Applications

Definition 2. Aseminormoftheformgivenin(23)iscalled Proof. It suffices to observe that atypeIIseminormprovidedthat ∞ 𝑥−𝜇−1/2 (ℎ 𝑓) (𝑥) = ∫ 𝑓 (𝑡)(𝑥𝑡)−𝜇𝐽 (𝑥𝑡) 𝑡𝜇+1/2𝑑𝑡 𝑤 (𝑡) =2𝑐−1𝑡−2𝜇−1𝑊 (𝑡) −𝑡−3𝜇−3/2 (𝜏 𝑊) (𝑡) 𝜇 𝜇 𝜇 𝑡 0 (30) (27) (𝑥∈𝐼) , −𝑐−1𝑡−𝜇−1/2 𝑧−𝜇−1/2𝑊 (𝑧) >0 (𝑡∈𝐼) , 𝜇 𝑧→0+lim 1 −𝜇 where 𝑓∈𝐿𝜇 and the function 𝑧 𝐽𝜇(𝑧) (𝑧 ∈ 𝐼) is con- where |𝑧−𝜇𝐽 (𝑧)| ≤−1 𝑐 󸀠 tinuous, with sup𝑧∈𝐼 𝜇 𝜇 [26, Equation 9.1.62], (i) the distribution 𝑊∈H𝜇 is regular, generated by a −𝜇 −1 lim𝑧→0+𝑧 𝐽𝜇(𝑧) = 𝑐 [26, Equation 9.1.7], and lim𝑧→∞ 𝐼 𝑊∈𝐿∞ 𝜇 continuous function on ,suchthat 𝜇 and the 𝑧−𝜇𝐽 (𝑧) = 0 −𝜇−1/2 𝜇 [26,Equation9.2.1]. limit lim𝑡→0+𝑡 𝑊(𝑡) exists, 󸀠 1 The space E𝜇 consists of all those 𝑢∈E such that (ii) ℎ𝜇𝑊∈𝐿𝜇,𝑐, 󵄨 𝑟 󵄨 󸀠 𝛾 𝜅 (𝑢) = 󵄨(𝑧−1𝐷) 𝑧−𝜇−1/2𝑢 (𝑧)󵄨 <∞ (iii) (ℎ𝜇𝑊)(𝑥) = O(𝑥 ) as 𝑥→0+,where𝛾+𝜇+7/2>, 0 𝜇,𝑙,𝑟 sup 󵄨 󵄨 𝑧∈]0,𝑙[ 󵄨 󵄨 and (31) 󸀠 (iv) (ℎ𝜇𝑊)(𝑥) ≤0 a.e. 𝑥∈𝐼. (𝑙∈N,𝑟∈Z+) . If condition 𝛾+𝜇+7/2>0in part (iii) is replaced with Endowed with the topology generated by the family of 𝛾+𝜇+3/2>0 {𝜅 } E the stronger one ,thenwecall(23)astrong seminorms 𝜇,𝑙,𝑟 (𝑙,𝑟)∈N×Z+ , 𝜇 becomes a Frechet´ space [23, type II seminorm. Proposition 4.3]. As usual, its dual space will be denoted by 󸀠 E𝜇.TheinclusionsB𝜇 ⊂ H𝜇 ⊂ E𝜇 being dense, we have 𝜇+1/2 Example 3 (strongtypeIIseminorm).Set𝑊(𝑡) = −𝑡 󸀠 󸀠 󸀠 E𝜇 ⊂ H𝜇 ⊂ B𝜇 [23, Proposition 4.4]. −𝑡2/2 2 𝑒 (𝑡 ∈ 𝐼).Thechangeofvariables𝑡 /2 = 𝑥 leads to 1 𝛾 ∞ ∞ Theorem 5. 𝑓∈𝐿 𝑓(𝑡) = O(𝑡 ) 𝑡→ 𝜇+1/2 −𝑡2/2 2𝜇+1 Let 𝜇,𝑐 and assume as ∫ |𝑊 (𝑡)| 𝑡 𝑑𝑡 = ∫ 𝑒 𝑡 𝑑𝑡 0+ 𝛾+𝜇+3/2>0 0 0 ,forsome .Then 󸀠 ∞ (i) 𝑓∈H𝜇, 𝜇 −𝑥 𝜇 𝜇 =2 ∫ 𝑒 𝑥 𝑑𝑥=2 Γ(𝜇+1)=𝑐𝜇, 󸀠 0 (ii) ℎ𝜇𝑓∈C, (28) −𝜇−1/2 󸀠 (iii) the limit lim𝑥→0+𝑥 (ℎ𝜇𝑓)(𝑥) exists, and 𝑊∈𝐿1 ⊂ H󸀠 𝑡−𝜇−1/2𝑊(𝑡) = −1 so that 𝜇 𝜇.Moreover,lim𝑡→0+ , (iv) some 𝑟∈N is such that −𝜇−1/2 󸀠 lim𝑡→∞𝑡 𝑊(𝑡),and =0 (ℎ 𝑊)(𝑥) = 𝑊(𝑥) (𝑥∈𝐼) 𝜇 (ℎ󸀠 𝑓) (𝑥) = O (𝑥2𝑟+𝜇+1/2) 𝑥󳨀→∞. [25,Equation8.6(10)].Thus,𝑊 satisfies the strong condition 𝜇 as (32) 𝛾=𝜇+1/2 (iii), hence the weak one (with ), as well as the 𝑓(𝑡) = O(𝑡𝛾) 𝑡→0+ remaining conditions (i), (ii), and (iv) in Definition.For 2 Proof. From the condition as ,itfollows 𝜇=1/2,withtheaidofMaple14,theweightdefinedby(27) that 󵄨 󵄨 𝛾 is found to be 󵄨𝑓 (𝑡)󵄨 ≤𝐶𝑡 (0<𝑡<𝑎) (33) 2 2 𝑤 (𝑡) = (2𝜋)−1/2𝑡−3 (1 − 𝑒−2𝑡 +2𝑡2 −4𝑡2𝑒−𝑡 /2) (𝑡∈𝐼) , for some 𝑎, 𝐶.Define >0 𝑔(𝑡)=0(0<𝑡<𝑎), 𝑔(𝑡) = (29) 𝑓(𝑡) (𝑡 ≥𝑎). Clearly, and the expression in parentheses can be seen to be positive 𝑓 (𝑡) =𝑔(𝑡) + (𝑓−𝑔) (𝑡)(𝑡∈𝐼) . (34) 𝑡 =0̸ 𝑤(𝑡) > 0 (𝑡 ∈𝐼) for .Consequently, . 1 1 As 𝑓∈𝐿𝜇,𝑐,wehave𝑔∈𝐿𝜇, and hence 𝑔 defines a 󸀠 Theorem 5 later will show that condition (i) in distribution in H𝜇. Definition 2 above is somewhat redundant and, at the 󸀠 We claim that 𝑓−𝑔 ∈ E𝜇.Firstweobservethat(𝑓−𝑔)(𝑡) = same time, will shed some light on how to construct 𝑓(𝑡) (0 < 𝑡 <𝑎) (𝑓 − 𝑔)(𝑡) = 0 (𝑡 ≥𝑎) weights giving rise to seminorms of type II. The following while ,sothat,from preliminary result is well known; we include it for the sake of (33), ∞ 𝑎 completeness. 󵄨 󵄨 𝜇+1/2 𝛾+𝜇+1/2 𝐵=∫ 󵄨(𝑓 − 𝑔) (𝑡)󵄨 𝑡 𝑑𝑡 ≤𝐶 ∫ 𝑡 𝑑𝑡 < ∞, (35) 1 0 0 Lemma 4. Let 𝑓∈𝐿𝜇.Then because 𝛾+𝜇+3/2.Denotingby >0 𝑙 the least positive integer ℎ 𝑓∈C (i) 𝜇 , greater than 𝑎,wemaythenwrite 𝑥−𝜇−1/2(ℎ 𝑓)(𝑥) 󵄨 ∞ 󵄨 (ii) the limit lim𝑥→0+ 𝜇 exists, 󵄨 󵄨 󵄨 󵄨 󵄨⟨𝑓 − 𝑔, 𝜙⟩󵄨 = 󵄨∫ (𝑓 − 𝑔) (𝑡) 𝜙 (𝑡) 𝑑𝑡󵄨 ≤𝐵𝜅𝜇,𝑙,0 (𝜙) 𝑥−𝜇−1/2(ℎ 𝑓)(𝑥) =0 󵄨 0 󵄨 (iii) lim𝑥→∞ 𝜇 ,and (36) −1 𝜇 (iv) ‖ℎ𝜇𝑓‖𝜇,∞ ≤𝑐𝜇 ‖𝑓‖𝜇,1,where𝑐𝜇 =2 Γ(𝜇 + 1). (𝜙 ∈ E𝜇), Journal of Function Spaces and Applications 5

󸀠 which proves our claim. In particular, 𝑓−𝑔∈H𝜇,andfrom Consequently, there exists 𝐶>0for which 󸀠 (34)weconcludethat𝑓∈H𝜇. 󵄨 −1 −𝜇−1/2 󵄨 −𝜇−1/2 −2𝜇−1 󸀠 𝑤 (𝑡) =𝑐𝜇 𝑡 󵄨2𝑡 𝑊 (𝑡) −𝑐𝜇𝑡 (𝜏𝑡𝑊) (𝑡) Since 𝑓−𝑔 ∈ E𝜇,[23, Propositions 4.5 and 4.6] yield 󵄨 𝑏, 𝐶 >0 𝑟∈N , for which 󵄨 −𝜇−1/2 󵄨 − lim 𝑧 𝑊 (𝑧)󵄨 󵄨 󵄨 𝑧→0+ 󵄨 (41) 󵄨𝑥−𝜇−1/2ℎ󸀠 (𝑓−𝑔) (𝑥)󵄨 ≤𝐶𝑥2𝑟 (𝑥>𝑏) . 󵄨 𝜇 󵄨 (37) −1 −𝜇−1/2 ≤𝑐𝜇 𝑡 (2‖𝑊‖𝜇,∞ + ‖𝑊‖𝜇,∞ + ‖𝑊‖𝜇,∞)

1 −𝜇−1/2 =𝐶𝑡−𝜇−1/2 (𝑡∈𝐼) . And since 𝑔∈𝐿𝜇, Lemma 4 entails that lim𝑥→∞𝑥 (ℎ󸀠 𝑔)(𝑥) =0 𝑐>0 𝜇 . Therefore, for some , This establishes the proposition.

󵄨 󵄨 󵄨𝑥−𝜇−1/2 (ℎ󸀠 𝑔) (𝑥)󵄨 ≤𝐶𝑥2𝑟 (𝑥>𝑐) . 3. Density Results 󵄨 𝜇 󵄨 (38) In this section we prove two density results for weights 𝑤= 𝑤(𝑥) > 0 (𝑥 ∈𝐼) A combination of (37)and(38)establishes(32). satisfying the conditions in the thesis of 󸀠 Proposition 7; namely, the Zemanian spaces B𝜇 and H𝜇 are Applying [7, Lemma 3.2] to 𝑓−𝑔 ∈ E𝜇 ⊂ E,wefind 󸀠 −𝜇−1/2 󸀠 dense in 𝑌𝑚 (𝑚 ∈ Z+). Both of these results will therefore that ℎ𝜇(𝑓 − 𝑔) ∈ C and the limit lim𝑥→0+𝑥 ℎ𝜇(𝑓 − 1 hold true for spaces endowed with type II seminorms. 𝑔)(𝑥) exists. Moreover, as 𝑔∈𝐿𝜇,fromLemma 4 we obtain In this way, direct seminorms allow us to establish direct 󸀠 −𝜇−1/2 󸀠 {𝑢 ∈ 𝑌 : that ℎ 𝑔∈C and the limit lim𝑥→0+𝑥 (ℎ 𝑔)(𝑥) exists counterparts of [14, Theorem 2.23], where the space 𝑚 𝜇 𝜇 ℎ󸀠 𝑢∈B } {𝑢 ∈ 𝑌 :ℎ󸀠 𝑢∈H } ℎ󸀠 𝑓∈C 𝜇 𝜇 and hence 𝑚 𝜇 𝜇 were shown to as well. This shows that 𝜇 and the limit lim𝑥→0+ 𝑌 (𝑚 ∈ Z ) −𝜇−1/2 󸀠 be dense in 𝑚 + . 𝑥 (ℎ𝜇𝑓)(𝑥) exists. The proof is thus complete. 1 Proposition 8. If 𝑤∈𝐿𝜇,𝑙 and there exists 𝛾∈R such that Corollary 6. 𝑓∈𝐿1 𝑓(𝑡) = O(𝑡𝛾) 𝑡→0+ 𝛾 Assume 𝜇,𝑐 and as 𝑤(𝑥) = O(𝑥 ) as 𝑥→∞,thenH𝜇 ⊂𝑌𝑚 (𝑚 ∈ Z+). 󸀠 for some 𝛾+𝜇+3/2>,sothat,by 0 Theorem 5, 𝑓∈H𝜇. 󸀠 ∞ Proof. For 𝑚=0,thisis[14,Theorem2.12];theproofbelow If 𝑓(𝑡) ≤0 a.e. 𝑡∈𝐼and 𝑊=ℎ𝜇𝑓∈𝐿𝜇 ,then𝑊 satisfies runs along similar lines, and we include it for completeness. conditions (i) to (iv) of Definition. 2 The following operational rule of the Hankel transformation 󸀠 will be used [24, Equation 5.4(5)]: Proof. From Theorem 5 we find that 𝑊∈H𝜇 ∩ C and −𝜇−1/2 2 the limit lim𝑡→0+𝑡 𝑊(𝑡) exists. This yields condition (i) ℎ (𝑆 𝜑) (𝑡) =−𝑡 (ℎ 𝜑) (𝑡) (𝑡 ∈ 𝐼, 𝜑∈ H ). 󸀠 𝜇 𝜇 𝜇 𝜇 (42) in Definition.Since 2 𝑓=ℎ𝜇𝑊, conditions (ii) to (iv) are trivially satisfied. From the hypothesis, there exist 𝑎, 𝐶 >0 such that 𝑤(𝑥) ≤ 𝛾 𝐶𝑥 (𝑥>𝑎).Fix𝜑∈H𝜇.Then, Proposition 7. If the weight 𝑤 gives rise to a type II seminorm 𝑤∈𝐿1 𝑤(𝑡) = O(𝑡𝛾) 𝑡→∞ ∞ 󵄨 󵄨2 as in Definition,then 2 𝜇,𝑙 and as ∫ 󵄨(ℎ 𝑆𝑚𝜑) (𝑡)󵄨 𝑤 (𝑡) 𝑡𝜇+1/2𝑑𝑡 󵄨 𝜇 𝜇 󵄨 for some 𝛾∈R. 0 𝑎 ∞ (43) 󸀠 󵄨 󵄨2 𝑊 𝑤 𝑓=ℎ 𝑊 ={∫ + ∫ } 󵄨(ℎ 𝑆𝑚𝜑) (𝑡)󵄨 𝑤 (𝑡) 𝑡𝜇+1/2𝑑𝑡. Proof. Associate to as in Definition,andlet 2 𝜇 . 󵄨 𝜇 𝜇 󵄨 From (5)and(8)wehave 0 𝑎 The first integral on the right-hand side of this identity is finite 󵄨 󵄨 1 󵄨 󵄨 󵄨 ∞ 󵄨 because 𝑤∈𝐿 : 󵄨(𝜏 𝑊) (𝑡)󵄨 = 󵄨∫ 𝑊 (𝑧) 𝐷 (𝑡, 𝑡, 𝑧) 𝑑𝑧󵄨 𝜇,𝑙 󵄨 𝑡 󵄨 󵄨 𝜇 󵄨 󵄨 0 󵄨 𝑎 󵄨 𝑚 󵄨2 𝜇+1/2 ∞ ∫ 󵄨(ℎ 𝑆 𝜑) (𝑡)󵄨 𝑤 (𝑡) 𝑡 𝑑𝑡 󵄨 󵄨 󵄨 𝜇 𝜇 󵄨 󵄨 −𝜇−1/2 󵄨 𝜇+1/2 0 ≤ ∫ 󵄨𝑧 𝑊 (𝑧)󵄨 𝐷𝜇 (𝑡, 𝑡, 𝑧) 𝑧 𝑑𝑧 0 (39) 𝑎 󵄨 󵄨2 4𝑚󵄨 󵄨 𝜇+1/2 ∞ = ∫ 𝑡 󵄨(ℎ𝜇𝜑) (𝑡)󵄨 𝑤 (𝑡) 𝑡 𝑑𝑡 𝜇+1/2 0 ≤ ‖𝑊‖𝜇,∞ ∫ 𝐷𝜇 (𝑡, 𝑡, 𝑧) 𝑧 𝑑𝑧 0 𝑎 󵄨 󵄨2 4𝑚+2𝜇+1󵄨 −𝜇−1/2 󵄨 𝜇+1/2 −1 2𝜇+1 = ∫ 𝑡 󵄨𝑡 (ℎ𝜇𝜑) (𝑡)󵄨 𝑤 (𝑡) 𝑡 𝑑𝑡 (44) =𝑐𝜇 ‖𝑊‖𝜇,∞𝑡 (𝑡∈𝐼) , 0 󵄨 󵄨2 4𝑚+2𝜇+1 󵄨 −𝜇−1/2 󵄨 ≤𝑎 sup󵄨𝑧 (ℎ𝜇𝜑) (𝑧)󵄨 or 𝑧∈𝐼 𝑎 󵄨 󵄨 𝜇+1/2 󵄨𝑡−2𝜇−1 (𝜏 𝑊) (𝑡)󵄨 ≤𝑐−1‖𝑊‖ (𝑡∈𝐼) . × ∫ 𝑤 (𝑡) 𝑡 𝑑𝑡 < ∞. 󵄨 𝑡 󵄨 𝜇 𝜇,∞ (40) 0 6 Journal of Function Spaces and Applications

On the other hand, From (7)wefindthat𝐷𝜇(𝑥0,𝑥,𝑦)≠ 0 (0 < 𝑦 < 𝛿) forces |𝑥−𝑥0|<𝛿;inthiscase,wehave|𝜑(𝑥) −0 𝜑(𝑥 )| < 𝜀.Along ∞ 󵄨 󵄨2 with (48)and(8), this leads us to ∫ 󵄨(ℎ 𝑆𝑚𝜑) (𝑡)󵄨 𝑤 (𝑡) 𝑡𝜇+1/2𝑑𝑡 󵄨 𝜇 𝜇 󵄨 𝑎 󵄨 𝛿 ∞ 󵄨 󵄨 𝜇+1/2 󵄨 󵄨∫ ∫ [𝜑 (𝑥) −𝜑(𝑥0)] 𝜓𝑘 (𝑦)𝜇 𝐷 (𝑥0,𝑥,𝑦)𝑥 𝑑𝑥 𝑑𝑦󵄨 ∞ 󵄨 󵄨2 󵄨 󵄨 4𝑚󵄨 󵄨 𝜇+1/2 󵄨 0 0 󵄨 = ∫ 𝑡 󵄨(ℎ𝜇𝜑) (𝑡)󵄨 𝑤 (𝑡) 𝑡 𝑑𝑡 𝑎 𝛿 ∞ 𝜇+1/2 ∞ 󵄨 󵄨2 ≤𝜀∫ ∫ 𝜓𝑘 (𝑦)𝜇 𝐷 (𝑥0,𝑥,𝑦)𝑥 𝑑𝑥 𝑑𝑦 4𝑚−2𝑛+2𝜇+1󵄨 𝑛−𝜇−1/2 󵄨 0 0 = ∫ 𝑡 󵄨𝑡 (ℎ𝜇𝜑) (𝑡)󵄨 𝑎 (45) 𝛿 −1 𝜇+1/2 𝜇+1/2 ≤𝜀𝑐𝜇 ∫ 𝜓𝑘 (𝑦) (𝑥0𝑦) 𝑑𝑦 ×𝑤(𝑡) 𝑡 𝑑𝑡 0 󵄨 󵄨2 𝜇+1/2 󵄨 𝑛−𝜇−1/2 󵄨 ≤𝜀𝑥 . ≤𝐶sup󵄨𝑧 (ℎ𝜇𝜑) (𝑧)󵄨 0 𝑧∈𝐼 (51) ∞ 4𝑚−2𝑛+𝛾+3𝜇+3/2 × ∫ 𝑡 𝑑𝑡 < ∞, On the other hand, in view of (49)and(8), there exists 𝑘0 ∈ N 𝑎 for which 󵄨 ∞ ∞ 󵄨 󵄨 𝜇+1/2 󵄨 provided that 2𝑛 > 4𝑚 + 𝛾 + 3𝜇. +5/2 󵄨∫ ∫ [𝜑 (𝑥) −𝜑(𝑥0)] 𝜓𝑘 (𝑦)𝜇 𝐷 (𝑥0,𝑥,𝑦)𝑥 𝑑𝑥 𝑑𝑦󵄨 󵄨 𝛿 0 󵄨 Lemma 9. 𝜓∈𝐿1 𝜓(𝑥) ≥ 0 (𝑥 ∈𝐼) ∞ ∞ Let 𝜇 satisfy and 󵄩 󵄩 𝜇+1/2 ∞ 𝜇+1/2 𝜇 ≤2󵄩𝜙󵄩𝜇,∞ ∫ ∫ 𝜓𝑘 (𝑦)𝜇 𝐷 (𝑥0,𝑥,𝑦)𝑥 𝑑𝑥 𝑑𝑦 ∫ 𝜓(𝑥)𝑥 𝑑𝑥=𝑐 𝑐 =2 Γ(𝜇 + 1) 𝜓 (𝑥) = 𝛿 0 0 𝜇,where 𝜇 .Define 𝑘 𝑘𝜇+3/2𝜓(𝑘𝑥) (𝑥 ∈ 𝐼,𝑘∈ N) 𝜙∈𝐿∞ 𝑥 ∈𝐼 ∞ .If 𝜇 is continuous at 0 , −1󵄩 󵄩 𝜇+1/2 =2𝑐𝜇 󵄩𝜙󵄩𝜇,∞ ∫ 𝜓𝑘 (𝑦) (𝑥0𝑦) 𝑑𝑦 then 𝛿 <𝜀𝑥𝜇+1/2 (𝑘 ∈ N, 𝑘≥𝑘). lim (𝜙#𝜓𝑘)(𝑥0)=𝜙(𝑥0). 0 0 𝑘→∞ (46) (52)

−𝜇−1/2 If, moreover, 𝑥 𝜙(𝑥) (𝑥 ∈𝐼) is uniformly continuous, then At this point, (46)followsfrom(50), (51), and (52). Moreover, if 𝜑 is uniformly continuous on 𝐼,then𝛿 and hence 𝑘0 in the 𝑥 󵄩 󵄩 previous argument do not depend on 0,sothat(47)holds. lim 󵄩𝜙#𝜓𝑘 −𝜙󵄩 =0. 𝑘→∞󵄩 󵄩𝜇,∞ (47)

𝛾 Lemma 10. Assume 𝑤(𝑥) = O(𝑥 ) as 𝑥→∞for some 𝛾∈ ∞ Proof. Note that R 𝜓∈H 𝜓(𝑡) ≥ 0 (𝑡 ∈𝐼) ∫ 𝜓(𝑡)𝑡𝜇+1/2𝑑𝑡 = .Let 𝜇 satisfy and 0 󵄩 󵄩 𝑐 𝜓 (𝑡) = 𝑘𝜇+3/2𝜓(𝑘𝑡)(𝑡∈𝐼,𝑘∈N) 𝑓∈H 󵄩𝜓 󵄩 =𝑐 𝑘∈N , 𝜇.Define 𝑘 .Given 𝜇 󵄩 𝑘󵄩𝜇,1 𝜇 ( ) (48) and 𝜀>0, there exists 𝑎>0such that ∞ ∞ ∫ 𝜓 (𝑥) 𝑥𝜇+1/2𝑑𝑥=0 (𝛿>0) . 󵄨 󵄨2 𝜇+1/2 lim 𝑘 (49) ∫ 󵄨(𝑓#𝜓𝑘) (𝑡)󵄨 𝑤 (𝑡) 𝑡 𝑑𝑡 <𝜀 (𝑘∈N) . (53) 𝑘→∞ 𝛿 𝑎 1 −𝜇−1/2 Proof. Fix 𝑟, 𝑘 ∈ N and 𝑥∈𝐼.Since𝜓∈𝐿𝜇,some𝑏>0is Put 𝜑(𝑥) =𝑥 𝜙(𝑥) (𝑥.Since ∈𝐼) 𝜑 is continuous such that at 𝑥0,given𝜀>0, there exists 𝛿=𝛿(𝑥0,𝜀) > 0 such that ∞ 1 |𝑥−𝑥0|<𝛿implies |𝜑(𝑥)−𝜑(𝑥0)| < 𝜀.Inviewof(4), (5), and 𝜇+1/2 ∫ 𝜓 (𝑡) 𝑡 𝑑𝑡 ≤ 𝑟 . (54) (8), we are allowed to write 𝑏 (1 + 𝑥2)

Clearly, (𝜙#𝜓𝑘)(𝑥0)−𝜙(𝑥0) 󵄨 󵄨 󵄨 󵄨 󵄨 ∞ 󵄨 ∞ ∞ 󵄨(𝑓 𝜓 ) (𝑥)󵄨 = 󵄨∫ (𝜏 𝑓) (𝑡) 𝜓 (𝑡) 𝑑𝑡󵄨 󵄨 # 𝑘 󵄨 󵄨 𝑥 𝑘 󵄨 = ∫ ∫ [𝜑 (𝑥) −𝜑(𝑥0)] 󵄨 0 󵄨 0 0 󵄨 󵄨 󵄨 𝑏 ∞ 󵄨 𝜇+1/2 = 󵄨{∫ + ∫ }(𝜏 𝑓) (𝑡) 𝜓 (𝑡) 𝑑𝑡󵄨 ×𝜓𝑘 (𝑦)𝜇 𝐷 (𝑥0,𝑥,𝑦)𝑥 𝑑𝑥 𝑑𝑦 󵄨 𝑥 𝑘 󵄨 󵄨 0 𝑏 󵄨 𝛿 ∞ ∞ 󵄨 󵄨 (55) ={∫ + ∫ } ∫ [𝜑 𝑥 −𝜑(𝑥 )] 󵄨 𝑏 󵄨 ( ) 0 ≤ 󵄨∫ (𝜏 𝑓) (𝑡) 𝜓 (𝑡) 𝑑𝑡󵄨 0 𝛿 0 󵄨 𝑥 𝑘 󵄨 󵄨 0 󵄨 ×𝜓 (𝑦) 𝐷 (𝑥 ,𝑥,𝑦)𝑥𝜇+1/2𝑑𝑥 𝑑𝑦. 󵄨 ∞ 󵄨 𝑘 𝜇 0 󵄨 󵄨 + 󵄨∫ (𝜏𝑥𝑓) (𝑡) 𝜓𝑘 (𝑡) 𝑑𝑡󵄨 . (50) 󵄨 𝑏 󵄨 Journal of Function Spaces and Applications 7

2𝑖+𝜇+1/2 −1 𝑛+𝑖−𝑗 −𝜇−1/2 As in the proof of Proposition 7, for the second integral × [𝑧 (𝑧 𝐷𝑧) 𝑧 (ℎ𝜇𝑓) (𝑧)] on the right-hand side of (55) we arrive at 𝑛+𝑖 𝑛+𝑖 𝑗 󵄨 ∞ 󵄨 = ∑ ( )[(−𝑡2) (𝑧𝑡)−𝜇−𝑗𝐽 (𝑧𝑡)] 󵄨 󵄨 𝑗 𝜇+𝑗 󵄨∫ (𝜏𝑥𝑓) (𝑡) 𝜓𝑘 (𝑡) 𝑑𝑡󵄨 𝑗=0 󵄨 𝑏 󵄨 ∞ 2𝑖+𝜇+1/2 −1 𝑛+𝑖−𝑗 −𝜇−1/2 −1󵄩 󵄩 𝜇+1/2 𝜇+1/2 ×[𝑧 (𝑧 𝐷𝑧) 𝑧 (ℎ𝜇𝑓) (𝑧)]. ≤𝑐𝜇 󵄩𝑓󵄩𝜇,∞𝑥 ∫ 𝜓𝑘 (𝑡) 𝑡 𝑑𝑡 𝑏 (59) ∞ (56) 𝜇+1/2 𝜇+1/2 =𝐶𝑥 ∫ 𝜓 (𝑡) 𝑡 𝑑𝑡 |𝑦−]𝐽 (𝑦)| ≤−1 𝑐 (] ≥ −1/2) ℎ 𝑓∈H 𝑘𝑏 Since sup𝑦∈𝐼 ] ] and 𝜇 𝜇,we (1 + 𝑡2)2𝑟 ∞ 𝜇+1/2 find that the right-hand side of (59)isboundedby 𝜇+1/2 𝜇+1/2 𝑥 𝐿1 𝑧∈𝐼 ≤𝐶𝑥 ∫ 𝜓 (𝑡) 𝑡 𝑑𝑡 ≤𝐶 𝑟 . times an 𝜇 function of .Itthenfollowsfrom(57)that 𝑏 (1 + 𝑥2) 𝜇+1/2 󵄨 −𝜇−1/2 󵄨 𝑥 2 2𝑟 󵄨𝑡 (𝜏 𝑓) (𝑡)󵄨 ≤𝐶 (1+𝑡 ) (𝑡∈] 0, 𝑏 [) . In order to estimate the first integral on the right-hand 󵄨 𝑥 󵄨 (1 + 𝑥2)𝑟 𝑡∈]0,𝑏[ ℎ side of (55), fix and apply the self-reciprocity of 𝜇 (60) on H𝜇,alongwith(42)and(14), to get Consequently, 󵄨 󵄨 󵄨𝑡−𝜇−1/2 (𝜏 𝑓) (𝑡)󵄨 󵄨 󵄨 󵄨 𝑥 󵄨 󵄨 𝑏 󵄨 󵄨 󵄨 󵄨∫ (𝜏𝑥𝑓) (𝑡) 𝜓𝑘 (𝑡) 𝑑𝑡󵄨 𝜇+1/2 󵄨 󵄨 0 󵄨 𝑥 󵄨 −𝜇−1/2 𝑟 = 𝑟 󵄨𝑥 ℎ𝜇 [(1 − 𝑆𝜇,𝑧) (1 + 𝑥2) 󵄨 𝑏 󵄨 󵄨 ≤ ∫ 󵄨𝑡−𝜇−1/2 (𝜏 𝑓) (𝑡)󵄨 𝜓 (𝑡) 𝑡𝜇+1/2𝑑𝑡 󵄨 󵄨 𝑥 󵄨 𝑘 (61) −𝜇 󵄨 0 × (𝑧𝑡) 𝐽𝜇 (𝑧𝑡) (ℎ𝜇𝑓) (𝑧) ] (𝑥) 󵄨 󵄨 𝜇+1/2 𝑏 𝑥 2 2𝑟 𝜇+1/2 𝜇+1/2 ≤𝐶 ∫ (1 + 𝑡 ) 𝜓 (𝑡) 𝑡 𝑑𝑡. 𝑥 ∞ 󵄨 𝑟 2 𝑟 𝑘 −1 󵄨 −𝜇 (1 + 𝑥 ) 0 ≤𝑐𝜇 𝑟 ∫ 󵄨(1 − 𝑆𝜇,𝑧) (𝑧𝑡) (1 + 𝑥2) 0 󵄨 A change of variables leads to 󵄨 󵄨 𝜇+1/2 ×𝐽𝜇 (𝑧𝑡) (ℎ𝜇𝑓) (𝑧) 󵄨 𝑧 𝑑𝑧. 𝑏 󵄨 2 2𝑟 𝜇+1/2 ∫ (1 + 𝑡 ) 𝜓𝑘 (𝑡) 𝑡 𝑑𝑡 (57) 0 𝑧∈𝐼 𝑎 (𝑛, 𝑖 ∈ N 0≤ 𝑘𝑏 𝑡2 2𝑟 For any fixed and suitable constants 𝑛,𝑖 , = ∫ (1 + ) 𝜓 (𝑡) 𝑡𝜇+1/2𝑑𝑡 𝑖≤𝑛≤𝑟) 2 (62) ,wemaywrite 0 𝑘

𝑟 2𝑟 (1 − 𝑆 ) (𝑧𝑡)−𝜇𝐽 (𝑧𝑡) (ℎ 𝑓) (𝑧) ∞ 𝑡2 𝜇,𝑧 𝜇 𝜇 ≤ ∫ (1 + ) 𝜓 (𝑡) 𝑡𝜇+1/2𝑑𝑡. 2 0 𝑘 𝑟 𝑟 = ∑(−1)𝑛 ( )𝑆𝑛 (𝑧𝑡)−𝜇𝐽 (𝑧𝑡) (ℎ 𝑓) (𝑧) 𝑛 𝜇,𝑧 𝜇 𝜇 The sequence of integrals on the right-hand side of(62) 𝑛=0 converges to 𝑐𝜇 as 𝑘→∞and is therefore bounded. 𝑟 𝑟 This follows from Lebesgue’s monotone convergence theorem =𝑧𝜇+1/2 ∑(−1)𝑛 ( )𝑧−𝜇−1/2𝑆𝑛 (𝑧𝑡)−𝜇𝐽 (𝑧𝑡) (ℎ 𝑓) (𝑧) 𝑛 𝜇,𝑧 𝜇 𝜇 applied to 𝑛=0 𝑡2 2𝑟 𝑟 𝑟 𝜑 (𝑡) =(1+ ) 𝜓 (𝑡)(𝑡∈𝐼,𝑛∈N) . =𝑧𝜇+1/2 ∑(−1)𝑛 ( ) 𝑛 2 (63) 𝑛 𝑛 𝑛=0 {𝜑 } 𝜑 (𝑡) = 𝑛 Indeed, the sequence 𝑛 𝑛∈N is nonincreasing and 1 2𝑖 −1 𝑛+𝑖 −𝜇 −𝜇−1/2 (1 + 𝑡2)2𝑟𝜓(𝑡) ∈ H ⊂𝐿1 𝜓∈H × ∑𝑎𝑛,𝑖𝑧 (𝑧 𝐷𝑧) (𝑧𝑡) 𝐽𝜇 (𝑧𝑡) 𝑧 (ℎ𝜇𝑓) (𝑧) , 𝜇 𝜇,because 𝜇 and even 𝑖=0 polynomials are multipliers of H𝜇 [24, Lemma 5.3.1]. Thus, (58) from (61), 󵄨 󵄨 󵄨 𝑏 󵄨 𝑥𝜇+1/2 with 󵄨 󵄨 󵄨∫ (𝜏𝑥𝑓) (𝑡) 𝜓𝑘 (𝑡) 𝑑𝑡󵄨 ≤𝐶 𝑟 . (64) 󵄨 0 󵄨 (1 + 𝑥2) 𝑛+𝑖 𝑧2𝑖+𝜇+1/2(𝑧−1𝐷 ) (𝑧𝑡)−𝜇𝐽 (𝑧𝑡) 𝑧−𝜇−1/2 (ℎ 𝑓) (𝑧) 𝑧 𝜇 𝜇 Plugging (56)and(64)into(55)finallyyields 𝑛+𝑖 2𝜇+1 𝑛+𝑖 −1 𝑗 −𝜇 󵄨 󵄨2 𝑥 = ∑ ( )[(𝑧 𝐷 ) (𝑧𝑡) 𝐽 (𝑧𝑡)] 󵄨(𝑓 𝜓 ) (𝑥)󵄨 ≤𝐶 (𝑥∈𝐼) . 𝑗 𝑧 𝜇 󵄨 # 𝑘 󵄨 2𝑟 (65) 𝑗=0 (1 + 𝑥2) 8 Journal of Function Spaces and Applications

𝛾 1/2 𝑤(𝑥) = O(𝑥 ) 𝑥→∞ ∞ 󵄨 󵄨2 The hypothesis that as furnishes +[{∫ 󵄨(ℎ󸀠 𝑓) (𝑡)󵄨 𝑤 (𝑡) 𝑡4𝑚+𝜇+1/2𝑑𝑡} 𝛾 󵄨 𝜇 󵄨 𝑠, 𝐶 >0 for which 𝑤(𝑥) ≤ 𝐶𝑥 (𝑥>𝑠).Combiningthis 𝑎 estimate with (65), we obtain ∞ 1/2 2 󵄨 󵄨2 4𝑚+𝜇+1/2 +{∫ 󵄨𝑔𝑘 (𝑡)󵄨 𝑤 (𝑡) 𝑡 𝑑𝑡} ] ∞ ∞ 𝛾+3𝜇+3/2 𝑎 󵄨 󵄨2 𝜇+1/2 𝑥 ∫ 󵄨(𝑓 𝜓𝑘) (𝑥)󵄨 𝑤 (𝑥) 𝑥 𝑑𝑥≤𝐶∫ 𝑑𝑥 <∞ 󵄨 # 󵄨 2 2𝑟 𝑎 𝑎 𝑎 (1 + 𝑥 ) 󵄩 󸀠 󵄩2 4𝑚+3𝜇+3/2 𝜀 < 󵄩ℎ 𝑓−𝑔 󵄩 ∫ 𝑤 (𝑡) 𝑡 𝑑𝑡 + . 󵄩 𝜇 𝑘󵄩𝜇,∞ 2 (𝑎>𝑠) , 0 (69) (66) Note that 𝑎 provided that 𝑟∈N is chosen so that 4𝑟 > 𝛾 + 3𝜇 + 5/2.This 4𝑚+3𝜇+3/2 ∫ 𝑤 (𝑡) 𝑡 𝑑𝑡 < ∞, (70) entails 0 𝑤∈𝐿1 ℎ󸀠 𝑓∈B ∞ because 𝜇,𝑙.Moreover,since 𝜇 𝜇,wehavethat 󵄨 󵄨2 ∫ 󵄨(𝑓 𝜓 ) (𝑥)󵄨 𝑤 (𝑥) 𝑥𝜇+1/2𝑑𝑥=0 󸀠 ∞ −𝜇−1/2 󸀠 lim 󵄨 # 𝑘 󵄨 (67) ℎ𝜇𝑓∈𝐿𝜇 and 𝑡 (ℎ𝜇𝑓)(𝑡) (𝑡 ∈𝐼) is uniformly conti- 𝑎→∞ 𝑎 nuous. Apply then Lemma 9 to obtain 𝑘∈N satisfying

󵄩 󸀠 󵄩2 𝜀 uniformly in 𝑘∈N and completes the proof. 󵄩ℎ 𝑓−𝑔𝑘󵄩 < 𝑎 . 󵄩 𝜇 󵄩𝜇,∞ 2∫ 𝑤 (𝑡) 𝑡4𝑚+3𝜇+3/2𝑑𝑡 (71) 0 1 Theorem 11. Let 𝑚∈Z+,andlet𝑤∈𝐿𝜇,𝑙 satisfy 𝑤(𝑥) = 𝛾 Hence, for that 𝑘, O(𝑥 ) as 𝑥→∞for some 𝛾∈R.ThespaceB𝜇 is a 𝑌 B 𝑌 ∞ 󵄨 󵄨2 subspace of 𝑚.Furthermore, 𝜇 is a dense subspace of 𝑚, ∫ 󵄨(ℎ󸀠 𝑓−𝑔 ) (𝑡)󵄨 𝑤 (𝑡) 𝑡4𝑚+𝜇+1/2𝑑𝑡 < 𝜀. 𝑓∈𝑌 𝜀>0 󵄨 𝜇 𝑘 󵄨 (72) in the following sense: given 𝑚 and , there exists 0 𝑢∈B𝜇 such that |𝑓 − 𝑢|𝑚 <𝜀. 󸀠 Finally, define 𝑢∈H𝜇 through Proof. From Proposition 8, H𝜇 ⊂𝑌𝑚.AsB𝜇 ⊂ H𝜇, B𝜇 is a 𝑌 󸀠 󸀠 󸀠 subspace of 𝑚. 𝑢 (𝑡) =(ℎ𝜇𝑔𝑘) (𝑡) =ℎ𝜇 (ℎ𝜇𝑓#𝜓𝑘) (𝑡) ∞ 𝜇+1/2 Take 𝜓∈H𝜇 such that 𝜓(𝑡)≥0(𝑡∈𝐼), ∫ 𝜓(𝑡)𝑡 (73) 0 −𝜇−1/2 𝑑𝑡𝜇 =𝑐 ,andℎ𝜇𝜓∈B𝜇.Foreach𝑘∈N, define 𝜓𝑘(𝑡) = =𝑡 𝑓 (𝑡) (ℎ𝜇𝜓𝑘) (𝑡)(𝑡∈𝐼) , 𝜇+3/2 𝑘 𝜓(𝑘𝑡)(𝑡∈𝐼).Let𝑓∈𝑌𝑚 and 𝜀>0.From[14, 󸀠 where the exchange formula (15)hasbeenused.Sinceℎ𝜇𝜓∈ Theorem 2.23], the set {𝑢 ∈𝑚 𝑌 :ℎ𝑢∈B𝜇} is dense in 𝜇 B (ℎ 𝜓 )(𝑡) = 𝑘𝜇+1/2(ℎ 𝜓)(𝑡/𝑘) (𝑡 ∈𝐼) 𝑌 ℎ󸀠 𝑓∈B 𝑔 (𝑡) = 𝜇,sodoes 𝜇 𝑘 𝜇 [25, 𝑚; therefore, we may assume that 𝜇 𝜇.Set 𝑘 󸀠 󸀠 Equation 8.1(2)]. And since ℎ𝜇𝑓∈B𝜇 ⊂ H𝜇,wehave (ℎ𝜇𝑓#𝜓𝑘)(𝑡) (𝑡 ∈ 𝐼, 𝑘∈ N).Then𝑔𝑘 ∈ H𝜇 (𝑘 ∈ N),and 4𝑚 𝑓∈H𝜇 ⊂ E𝜇.Thus𝑢∈B𝜇,with Lemma 10,appliedtotheweightV(𝑡) = 𝑡 𝑤(𝑡) (𝑡 ∈𝐼),gives ∞ 𝑎>0such that 󵄨 󵄨2 󵄨 󵄨2 󵄨𝑓−𝑢󵄨 = ∫ 󵄨ℎ󸀠 𝑆𝑚 (𝑓 − 𝑢) (𝑡)󵄨 𝑤 (𝑡) 𝑡𝜇+1/2𝑑𝑡 󵄨 󵄨𝑚 󵄨 𝜇 𝜇 󵄨 0 ∞ (74) 󵄨 󸀠 󵄨2 4𝑚+𝜇+1/2 ∞ ∫ 󵄨(ℎ 𝑓) (𝑡)󵄨 𝑤 (𝑡) 𝑡 𝑑𝑡 = 0, 󵄨 󸀠 󵄨2 4𝑚+𝜇+1/2 󵄨 𝜇 󵄨 = ∫ 󵄨(ℎ 𝑓−𝑔 ) (𝑡)󵄨 𝑤 (𝑡) 𝑡 𝑑𝑡 < 𝜀, 𝑎 󵄨 𝜇 𝑘 󵄨 0 ∞ (68) 󵄨 󵄨2 4𝑚+𝜇+1/2 𝜀 ∫ 󵄨𝑔𝑘 (𝑡)󵄨 𝑤 (𝑡) 𝑡 𝑑𝑡 < (𝑘∈N) . as required. 𝑎 2 1 Corollary 12. Let 𝑚∈Z+,andlet𝑤∈𝐿𝜇,𝑙 satisfy 𝑤(𝑥) = O(𝑥𝛾) 𝑥→∞ 𝛾∈R H From (68) and Minkowski’s inequality, as for some .Thespace 𝜇 is a dense subspace of 𝑌𝑚, in the following sense: given 𝑓∈𝑌𝑚 and 𝜀>0, there exists 𝑢∈H𝜇 such that |𝑓 − 𝑢|𝑚 <𝜀. ∞ 󵄨 󵄨2 ∫ 󵄨(ℎ󸀠 𝑓−𝑔 ) (𝑡)󵄨 𝑤 (𝑡) 𝑡4𝑚+𝜇+1/2𝑑𝑡 󵄨 𝜇 𝑘 󵄨 B ⊂ H 0 Proof. It suffices to recall that 𝜇 𝜇 and apply Theorem 11. 𝑎 󵄨 󵄨2 = ∫ 󵄨𝑡−𝜇−1/2 (ℎ󸀠 𝑓−𝑔 ) (𝑡)󵄨 𝑤 (𝑡) 𝑡4𝑚+3𝜇+3/2𝑑𝑡 󵄨 𝜇 𝑘 󵄨 0 Acknowledgments ∞ 󵄨 󵄨2 + ∫ 󵄨(ℎ󸀠 𝑓−𝑔 ) (𝑡)󵄨 𝑤 (𝑡) 𝑡4𝑚+𝜇+1/2𝑑𝑡 󵄨 𝜇 𝑘 󵄨 Cristian Arteaga is supported by a 2011 CajaCanarias 𝑎 Research Grant for Postgraduates. Both authors are par- 󵄩 󵄩2 𝑎 ≤ 󵄩ℎ󸀠 𝑓−𝑔 󵄩 ∫ 𝑤 (𝑡) 𝑡4𝑚+3𝜇+3/2𝑑𝑡 tially supported by MICINN-FEDER Grant MTM2011-28781 󵄩 𝜇 𝑘󵄩 𝜇,∞ 0 (Spain). Journal of Function Spaces and Applications 9

References [21] J. Levesley and W. Light, “Direct form seminorms arising in the theory of interpolation by translates of a basis function,” [1] A. H. Zemanian, “A distributional Hankel transformation,” Advances in Computational Mathematics,vol.11,no.2-3,pp.161– SIAM Journal on Applied Mathematics,vol.14,pp.561–576,1966. 182, 1999. [2] A. H. Zemanian, “The Hankel transformation of certain distri- [22] C. Arteaga and I. Marrero, “Direct form seminorms arising butions of rapid growth,” SIAM Journal on Applied Mathematics, in the theory of interpolation by Hankel translates of a basis vol. 14, pp. 678–690, 1966. function,” Advances in Computational Mathematics,2013. [3] I. I. Hirschman, Jr., “Variation diminishing Hankel transforms,” [23] J. J. Betancor and I. Marrero, “The Hankel convolution and the 󸀠 Journal d’Analyse Mathematique´ ,vol.8,pp.307–336,1961. Zemanian spaces 𝛽𝜇 and 𝛽𝜇,” Mathematische Nachrichten,vol. [4] F. M. Cholewinski, “A Hankel convolution complex inversion 160, pp. 277–298, 1993. theory,” Memoirs of the American Mathematical Society,vol.58, [24] A. H. Zemanian, Generalized Integral Transformations,Inter- p. 67, 1965. science Publishers, 1968. [5]D.T.Haimo,“IntegralequationsassociatedwithHankelcon- [25] A. Erdelyi,´ W. Magnus, F. Oberhettinger, and F. Tricomi, Tables volutions,” Transactions of the American Mathematical Society, of Integral Transforms, McGraw-Hill, 1954. vol. 116, pp. 330–375, 1965. [26] M. Abramowitz and I. A. Stegun, Handbook of Mathematical [6] J. de Sousa Pinto, “A generalised Hankel convolution,” SIAM Functions with Formulas, Graphs, and Mathematical Tables,vol. JournalonMathematicalAnalysis,vol.16,no.6,pp.1335–1346, 55 of Applied Mathematics Series, 9th printing, National Bureau 1985. of Standards, 1964. [7] I. Marrero and J. J. Betancor, “Hankel convolution of generalized functions,” Rendiconti di Matematica e delle sue Applicazioni, vol. 15, no. 3, pp. 351–380, 1995. [8] C.A.Micchelli,“Interpolationofscattereddata:distancematri- ces and conditionally positive definite functions,” Constructive Approximation,vol.2,no.1,pp.11–22,1986. [9] W. Madych and S. Nelson, “Multivariate interpolation: a variational theory,” Unpublished Manuscript,1983. [10]W.R.MadychandS.A.Nelson,“Multivariateinterpolation and conditionally positive definite functions,” Approximation Theory and Its Applications, vol. 4, no. 4, pp. 77–89, 1988. [11] W.R. Madych and S. A. Nelson, “Multivariate interpolation and conditionally positive definite functions—II,” Mathematics of Computation,vol.54,no.189,pp.211–230,1990. [12] W. Light and H. Wayne, “Spaces of distributions, interpolation by translates of a basis function and error estimates,” Numerische Mathematik,vol.81,no.3,pp.415–450,1999. [13] R. Schaback and Z. Wu, “Operators on radial functions,” Journal of Computational and Applied Mathematics,vol.73,no.1-2,pp. 257–270, 1996. [14] C. Arteaga and I. Marrero, “A scheme for interpolation by Hankel translates of a basis function,” Journal of Approximation Theory,vol.164,no.12,pp.1540–1576,2012. [15]E.M.SteinandG.Weiss,Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton, NJ, USA, 1971. [16] G. Gigante, “Transference for hypergroups,” Collectanea Math- ematica,vol.52,no.2,pp.127–155,2001. [17] H. Corrada, K. Leeb, B. Klein, R. Klein, S. Iyengarc, and G. Wah- bad, “Examining the relative influence of familial, genetic, and environmental covariate information in flexible risk models,” Proceedings of the National Academy of Sciences of the United States of America,vol.106,no.20,pp.8128–8133,2009. [18] C. Arteaga and I. Marrero, “Universal approximation by radial basis function networks of Delsarte translates,” Neural Net- works,vol.46,pp.299–305,2013. [19] C. Arteaga and I. Marrero, “Approximation in weighted p-mean by RBF networks of Delsarte translates,” Submitted Preprint, 2013. [20] C. Arteaga and I. Marrero, “Wiener’s tauberian theorems for the Fourier-Bessel transformation and uniform approximation by RBF networks of Delsarte translates,” Submitted Preprint,2013. Hindawi Publishing Corporation Journal of Function Spaces and Applications Volume 2013, Article ID 509613, 8 pages http://dx.doi.org/10.1155/2013/509613

Research Article 2 On a New Space 𝑚 (𝑀,𝐴,𝜙,𝑝)of Double Sequences

Cenap Duyar and OLuz OLur

Department of Mathematics, Art and Science Faculty, Ondokuz Mayıs University, Kurupelit Campus, Samsun, Turkey

Correspondence should be addressed to Cenap Duyar; [email protected]

Received 3 May 2013; Accepted 17 June 2013

Academic Editor: Jozef´ Bana´s

Copyright © 2013 C. Duyar and O. Ogur.˘ This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

2 We introduce a new space 𝑚 (𝑀,𝐴,𝜙,𝑝)of double sequences related to 𝑝-absolute convergent double sequence space, combining 2 an Orlicz function and an infinity double matrix. We study some properties of 𝑚 (𝑀,𝐴,𝜙,𝑝)and obtain some inclusion relations 2 involving 𝑚 (𝑀,𝐴,𝜙,𝑝).

1. Introduction An Orlicz function 𝑀 can always be represented in the 𝑥 following integral form: 𝑀(𝑥) =∫ 𝜂(𝑡)𝑑𝑡 ,where𝜂 is Throughout this work, 𝑁 and 𝐶 denote the set of positive inte- 0 known as the kernel of 𝑀,isrightdifferentiablefor𝑡≥0, gers and complex numbers, respectively. A complex double 𝜂(0) = 0 , 𝜂(𝑡) >0 for 𝑡>0, 𝜂 is nondecreasing, and sequence is a function 𝑥 from 𝑁×𝑁into 𝐶 and briefly 2 𝜂(𝑡) →∞ as 𝑡→∞. denoted by {𝑥𝑘,𝑙}.Throughoutthiswork,𝑤 and 𝑤 denote An Orlicz function is said to be satisfied Δ 2-condition the spaces of single complex sequences and double complex 𝑢 𝑇>0 𝜀>0 𝑛 ∈𝑁 for all values of , if there exists a constant such that sequences, respectively. If, for all ,thereis 𝜀 such 𝑀 (2𝑢) ≤ 𝑇𝑀 (𝑢) 𝑢≥0 ‖𝑥 −𝑎‖ <𝜀 𝑘>𝑛 𝑙>𝑛 for all . that 𝑘,𝑙 𝑋 where 𝜀 and 𝜀,thenadouble {𝑥 } Lindenstrauss and Tzafriri [5]usedtheideaofOrlicz sequence 𝑘,𝑙 is said to be converging (in terms of Pring- functions to construct Orlicz sequence space sheim) to 𝑎∈𝐶. A real double sequence {𝑥𝑘,𝑙} is nondecreas- 󵄨 󵄨 ing, if 𝑥𝑘,𝑙 ≤𝑥𝑝,𝑞 for (𝑘, 𝑙) < (𝑝, 𝑞). A double series is infinity 󵄨𝑥 󵄨 ∞ 󵄨 𝑘󵄨 ∑ 𝑥 ℓ𝑀 ={𝑥={𝑥𝑘}∈𝑤: ∑𝑀( )<∞ sum 𝑘,𝑙=1 𝑘,𝑙, and its convergence implies the convergence 𝜌 𝑘≥1 by |⋅|of partial sums sequence {𝑆𝑛,𝑚},where𝑆𝑛,𝑚 = 𝑛 𝑚 (2) ∑𝑘=1 ∑𝑙=1 𝑥𝑘,𝑙 (see [1–4]). Adoublesequencespace𝐸 is said to be solid if {𝑥𝑘,𝑙𝑦𝑘,𝑙}∈ for some 𝜌>0}. 𝐸 for all double sequences {𝑦𝑘,𝑙} of scalars such that |𝑦𝑘,𝑙|≤1 for all 𝑘, 𝑙 ∈𝑁 whenever {𝑥𝑘,𝑙}∈𝐸. ℓ ℘ 𝜎 The sequence space 𝑀 is a Banach space according to the Now let 𝑠 be a family of subsets having most elements norm defined by 𝑠 in 𝑁.Also℘𝑠,𝑡 denotes the class of subsets 𝜎=𝜎1 ×𝜎2 in 𝑁×𝑁 𝜎 𝜎 𝑠 󵄨 󵄨 such that the element numbers of 1 and 2 are most 󵄨𝑥𝑘󵄨 𝑡 {𝜙 } ‖𝑥‖ = {𝜌 > 0 : ∑𝑀( )≤1}. and , respectively. Besides 𝑘,𝑙 is taken as a nondecreasing inf 𝜌 (3) double sequence of the positive real numbers such that 𝑘≥1

𝑘𝜙𝑘+1,𝑙 ≤ (𝑘+1) 𝜙𝑘,𝑙, This space (ℓ𝑀,‖⋅‖)is called an Orlicz sequence space. The (1) space ℓ𝑀 is closely related to the space ℓ𝑝,whichisanOrlicz 𝑙𝜙 ≤ (𝑙+1) 𝜙 . 𝑝 𝑘,𝑙+1 𝑘,𝑙 sequence space with 𝑀(𝑥)=𝑥 for 1≤𝑝<∞. An Orlicz function is a function 𝑀 : [0, ∞) → [0, ∞) which The double sequence spaces in the various forms defined is continuous, nondecreasing, and convex with 𝑀(0) =, 0 by Orlicz functions were introduced and studied by Khan and 𝑀(𝑥)>0for 𝑥>0,and𝑀(𝑥) →∞as 𝑥→∞. Tabassum in [6–12]andbyKhanetal.in[13]. 2 Journal of Function Spaces and Applications

The space 𝑚(𝜙), introduced by Sargent in [14], is in the Also, we introduce and investigate the following space: form 𝑚2 (𝑀, 𝜙, 𝑝)

𝑚(𝜙)={𝑥={𝑥𝑘}∈𝑤:‖𝑥‖𝑚(𝜙) { 2 (4) = {𝑥=(𝑥𝑘𝑙)∈𝑤 : 1 󵄨 󵄨 { = sup ∑ 󵄨𝑥𝑘󵄨 <∞}. 𝑠≥1,𝜎∈℘ 𝜙 𝑠 𝑠 𝑘∈𝜎 󵄨 󵄨 𝑝 { 1 󵄨𝑥 󵄨 𝑘𝑙 ∑ ∑𝑀(󵄨 𝑘,𝑙󵄨) : (𝑠,) 𝑡 Sargent studied some properties of this space and exam- sup {𝜙 𝜌 𝑠𝑡 𝑘∈𝜎 𝑙∈𝜎 ined relationship between this space and 𝑙𝑝-space. Similar { 1 2 (8) sequence classes were studied by many mathematicians using Orlicz functions (see [15–17]). } ≥ (1, 1) ,𝜎 ×𝜎 ∈℘ Later on, this space was investigated from sequence space 1 2 𝑠𝑡} point of view by Rath [18], Rath and Tripathy [19], Tripathy } and Sen [20], Tripathy and Mahanta [17], and others. Recently } Altun and Bilgin [15] introduced and studied the following <∞ 𝜌>0 . sequence space 𝑚(𝑀, 𝐴, 𝜙, 𝑝). for some } } Let 𝐴=(𝑎𝑖𝑘) beaninfinitematrixofcomplexnumbers, 𝑀 an Orlicz space, and 𝑝=(𝑝𝑖) aboundedsequenceof In this work, we also use the following sequence spaces: positive real numbers such that 0<ℎ=inf 𝑝𝑖 ≤𝑝𝑖 ≤𝐻= 𝑙(2)(𝑀,𝐴,𝜙,𝑝) sup 𝑝𝑖 <∞.Thenthespace𝑚(𝑀, 𝐴, 𝜙, 𝑝) is defined by 0 2 󵄨 󵄨 𝑝 𝑚 (𝑀,𝐴,𝜙,𝑝) { ∞ 󵄨𝐴 (𝑥)󵄨 𝑖𝑗 2 󵄨 𝑖𝑗 󵄨 󵄨 󵄨 𝑝 = 𝑥={𝑥 }∈𝑤 : ∑ 𝑀( ) 1 󵄨𝐴 (𝑥)󵄨 𝑖 { 𝑘,𝑙 𝜌 ={{𝑥}∈𝑤: ∑𝑀(󵄨 𝑖 󵄨) <∞ { 𝑖,𝑗=1 𝑘 sup 𝜙 𝜌 𝑠≥1,𝜎∈℘𝑠 𝑠 𝑘∈𝜎 (5) } <∞ 𝜌>0 , for some 𝜌>0}, for some } } (9) 𝐴𝑥=(𝐴(𝑥)) 𝐴 (𝑥) =∞ ∑ 𝑎 𝑥 (2) where 𝑖 if 𝑖 𝑘=1 𝑖𝑘 𝑘 converges for each 𝑙 (𝑀,𝐴,𝜙,𝑝)∞ 𝑖. 󵄨 󵄨 𝑝 Let 𝑥={𝑥𝑘,𝑙} be a double sequence. A set 𝑆(𝑥) is defined 󵄨 󵄨 𝑖𝑗 { 󵄨𝐴𝑖𝑗 (𝑥)󵄨 by = 𝑥={𝑥 }∈𝑤2 : 𝑀(󵄨 󵄨) { 𝑘,𝑙 sup 𝜌 (𝑖,𝑗)∈𝑁×𝑁 𝑆 (𝑥) = {{𝑥 }:𝜋 𝜋 𝑁} . { 𝜋1(𝑘),𝜋2(𝑘) 1 and 2 are permutations of (6) } If 𝑆(𝑥) ⊆𝐸 for all 𝑥∈𝐸,then𝐸 is said to be symmetric. <∞for some 𝜌>0} . In this work, we introduce the following sequence space. } Let 𝐴=(𝑎𝑖𝑗𝑘𝑙 ) be an infinite double matrix of complex 𝑀 𝑝=(𝑝) The following inequality will be used throughout this numbers, an Orlicz function and 𝑖𝑗 bounded paper: double sequence of positive real numbers such that 0< 𝑝𝑞𝑟 𝐻−1 𝑝𝑞𝑟 𝑝𝑞𝑟 ℎ=inf 𝑝𝑖𝑗 ≤𝑝𝑖 ≤𝐻=sup 𝑝𝑖𝑗 <∞.Thenthespace |𝑎+𝑏| ≤ max (1, 2 )(|𝑎| + |𝑏| ), (10) 2 𝑚 (𝑀,𝐴,𝜙,𝑝)is defined by where 𝑎, 𝑏 ∈𝐶 and 𝐻=sup{𝑝𝑞𝑟 :(𝑞,𝑟)∈𝑁×𝑁}. 𝑚2 (𝑀,𝐴,𝜙,𝑝)

󵄨 󵄨 𝑝𝑖𝑗 2. Main Results { { 1 󵄨𝐴 (𝑥)󵄨 = 𝑥=(𝑥 )∈𝑤2 : ∑ ∑ 𝑀(󵄨 𝑖𝑗 󵄨) : { 𝑘𝑙 sup { 𝐾(2) 𝜙𝑠𝑡 𝑖∈𝜎 𝑗∈𝜎 𝜌 Definition 1. Let be a set of increasing positive integer { { 1 2 binaries, namely, (𝑘1,𝑘2)<(𝑙1,𝑙2) if and only if 𝑘1 <𝑙1 and (2) } 𝑘2 <𝑙2,and𝐸 beadoublesequencespace.A𝐾 -set space is a (𝑠,) 𝑡 ≥(1, 1) ,𝜎1 ×𝜎2 ∈℘𝑠𝑡} double sequence space, defined by } 𝐸 2 (2) 𝜆𝐾(2) = {{𝑥𝑘 ,𝑙 } ∈𝑤 : {𝑥𝑘,𝑙} ∈𝐸,(𝑘𝑚,𝑙𝑚) ∈𝐾 } . (11) } 𝑚 𝑚 𝐸 <∞,for some 𝜌>0 , {𝑥 }∈𝜆 (2) } The canonical preimage of a double sequence 𝑘𝑚,𝑙𝑚 𝐾 2 } is a double sequence {𝑦𝑘,𝑙}∈𝑤 with (7) 𝑥 , (𝑘, 𝑙) ∈𝐾(2) 𝐴𝑥=(𝐴 (𝑥)) 𝐴 (𝑥) = ∑∞ 𝑎 𝑥 𝑦 ={ 𝑘,𝑙 where 𝑖𝑗 if 𝑖𝑗 𝑘,𝑙=1 𝑖𝑗𝑘𝑙 𝑘𝑙 converges for 𝑘,𝑙 0, (𝑘, 𝑙) ∉𝐾(2). (12) each (𝑖, 𝑗) ∈ 𝑁 ×𝑁. Journal of Function Spaces and Applications 3

𝜆𝐸 The canonical preimage of a set space 𝐾(2) is a set of canonical Thus we can write 𝐸 𝜆 󵄨 󵄨 𝑝𝑖𝑗 preimages of all elements in 𝐾(2) . 󵄨 󵄨 { 1 󵄨𝐴𝑖𝑗 (𝛼𝑥+𝛽𝑦)󵄨 sup { ∑ ∑ 𝑀( ) : (𝑠,) 𝑡 Definition 2. If a double sequence space 𝐸 contains the 𝜙𝑠,𝑡 𝑖∈𝜎 𝑗∈𝜎 𝜌3 { 1 2 canonical preimages of all set spaces, then 𝐸 is said to be monotone. } ≥ (1, 1) ,𝜎1 ×𝜎2 ∈℘𝑠,𝑡} The following lemma is an easy result of the definitions. } 𝐻−1 ≤ max (1, 2 ) Lemma 3. If a double sequence space 𝐸 is solid, then 𝐸 is mon- 󵄨 󵄨 𝑝𝑖𝑗 otone. { { 1 󵄨𝐴 (𝑥)󵄨 × ∑ ∑ 𝑀(󵄨 𝑖𝑗 󵄨) : (𝑠,) 𝑡 2 {sup { Proposition 4. 𝑚 (𝑀,𝐴,𝜙,𝑝) 𝐶 𝜙𝑠,𝑡 𝑖∈𝜎 𝑗∈𝜎 𝜌1 The space is a -linear space. { { 1 2 (15) 2 Proof. Let 𝑥={𝑥𝑘,𝑙}, 𝑦={𝑦𝑘,𝑙} be in 𝑚 (𝑀,𝐴,𝜙,𝑝)and 𝛼, 𝛽 } ≥ (1, 1) ,𝜎 ×𝜎 ∈℘ in 𝐶\{0}. Then there exist positive numbers 𝜌1 and 𝜌2 such 1 2 𝑠,𝑡} that } 󵄨 󵄨 𝑝 { 󵄨𝐴 (𝑦)󵄨 𝑖𝑗 󵄨 󵄨 𝑝 1 󵄨 𝑖𝑗 󵄨 󵄨𝐴 (𝑥)󵄨 𝑖𝑗 + ∑ ∑𝑀( ) : (𝑠,) 𝑡 { 1 󵄨 𝑖𝑗 󵄨 sup {𝜙 𝜌 ∑ ∑ 𝑀( ) : (𝑠,) 𝑡 𝑠,𝑡 𝑘∈𝜎 𝑙∈𝜎 2 sup { { 1 2 𝜙𝑠,𝑡 𝑖∈𝜎 𝑗∈𝜎 𝜌1 { 1 2 }} ≥ (1, 1) ,𝜎 ×𝜎 ∈℘ <∞. } 1 2 𝑠,𝑡}} ≥ (1, 1) ,𝜎1 ×𝜎2 ∈℘𝑠,𝑡} <∞, }} } 2 This shows that {𝛼𝑥𝑘,𝑙 +𝛽𝑦𝑘,𝑙}∈𝑚(𝑀,𝐴,𝜙,𝑝).Hence (13) 2 󵄨 󵄨 𝑝𝑖𝑗 𝑚 (𝑀,𝐴,𝜙,𝑝) { 1 󵄨𝐴 (𝑦)󵄨 is a linear space. ∑ ∑ 𝑀(󵄨 𝑖𝑗 󵄨) : (𝑠,) 𝑡 sup { 2 𝜙𝑠,𝑡 𝑖∈𝜎 𝑗∈𝜎 𝜌2 Proposition 5. 𝑚 (𝑀,𝐴,𝜙,𝑝) { 1 2 The space is a paranormed space with the paranorm } 𝑔 (𝑥) ≥ (1, 1) ,𝜎1 ×𝜎2 ∈℘𝑠,𝑡} <∞. 󵄨 󵄨 𝑝𝑖𝑗 } { { 1 󵄨𝐴𝑖𝑗 (𝑥)󵄨 𝑝𝑞𝑟/𝐻 [ 󵄨 󵄨 = inf {𝜌 : sup { ∑ ∑ 𝑀( ) : (𝑠,) 𝑡 𝜙𝑠,𝑡 𝜌 𝑖∈𝜎1 𝑗∈𝜎2 Let 𝜌3 = max{2|𝛼|𝜌1, 2|𝛽|𝜌2}.Usingthat𝑀 is nondecreasing { [ { convex function, we have 1/𝐻 } ≥ (1, 1) ,𝜎 ×𝜎 ∈℘ ≤1] , 󵄨 󵄨 𝑝 1 2 𝑠,𝑡} 󵄨 󵄨 𝑖𝑗 󵄨𝐴𝑖𝑗 (𝛼𝑥+𝛽𝑦)󵄨 } ] ∑ ∑ 𝑀(󵄨 󵄨) 𝜌3 𝑖∈𝜎1 𝑗∈𝜎2 } 𝑞∈𝑁,𝑟∈𝑁} . 󵄨 󵄨 𝑝 󵄨∑∞ (𝛼𝑎 𝑥 +𝛽𝑎 𝑦 )󵄨 𝑖𝑗 󵄨 𝑘,𝑙=1 𝑖𝑗𝑘𝑙 𝑘,𝑙 𝑖𝑗𝑘𝑙 𝑘,𝑙 󵄨 } ≤ ∑ ∑ 𝑀( ) (16) 𝜌3 𝑖∈𝜎1 𝑗∈𝜎2 Proof. It is clear that 𝑔(𝑥) = 𝑔(−𝑥) and 𝑔(𝑥) =0 if 𝑥=0.If 󵄨 󵄨 󵄨 󵄨 𝑝 󵄨 󵄨 󵄨 󵄨 𝑖𝑗 𝜌 >0 𝜌 >0 󵄨𝛼𝐴 𝑖𝑗 (𝑥)󵄨 󵄨𝛽𝐴 𝑖𝑗 (𝑦)󵄨 there are 1 and 2 such that ≤ ∑ ∑ 𝑀( + 󵄨 󵄨 ) 󵄨 󵄨 𝑝 2 |𝛼| 𝜌 2 󵄨𝛽󵄨 𝜌 󵄨 󵄨 𝑖𝑗 𝑖∈𝜎 𝑗∈𝜎 1 󵄨 󵄨 2 { 1 󵄨𝐴 (𝑥)󵄨 1 2 ∑ ∑ 𝑀(󵄨 𝑖𝑗 󵄨) : (𝑠,) 𝑡 sup {𝜙 𝜌 󵄨 󵄨 󵄨 󵄨 𝑝𝑖𝑗 𝑠,𝑡 𝑖∈𝜎 𝑗∈𝜎 1 󵄨𝐴 (𝑥)󵄨 󵄨𝐴 (𝑦)󵄨 { 1 2 = ∑ ∑𝑀(󵄨 𝑖𝑗 󵄨 + 󵄨 𝑖𝑗 󵄨) 2𝜌1 2𝜌2 } 𝑘∈𝜎1 𝑙∈𝜎2 ≥ (1, 1) ,𝜎1 ×𝜎2 ∈℘𝑠,𝑡} ≤1, 󵄨 󵄨 𝑝 󵄨𝐴 (𝑥)󵄨 𝑖𝑗 } 𝐻−1 󵄨 𝑖𝑗 󵄨 (17) ≤ (1, 2 )(∑ ∑ 𝑀( ) 󵄨 󵄨 𝑝𝑖𝑗 max { 󵄨𝐴 (𝑦)󵄨 𝑖∈𝜎 𝑖∈𝜎 𝜌1 1 󵄨 𝑖𝑗 󵄨 1 2 sup { ∑ ∑ 𝑀( ) : (𝑠,) 𝑡 𝜙𝑠,𝑡 𝑖∈𝜎 𝑗∈𝜎 𝜌2 󵄨 󵄨 𝑝𝑖𝑗 { 1 2 󵄨𝐴 (𝑦)󵄨 + ∑ ∑𝑀(󵄨 𝑖𝑗 󵄨) ). 𝜌 } 𝑘∈𝜎 𝑙∈𝜎 2 1 2 ≥ (1, 1) ,𝜎1 ×𝜎2 ∈℘𝑠,𝑡} ≤1, (14) } 4 Journal of Function Spaces and Applications

󵄨 𝑛 󵄨 𝑝𝑖𝑗 then { 𝑝 /𝐻 { 1 󵄨𝐴𝑖𝑗 (𝑥 −𝑥)󵄨 = 𝜌 𝑞𝑟 : [ ∑ ∑ 𝑀(󵄨 󵄨) : inf { 𝑛 sup { 𝑛 𝜙𝑠,𝑡 𝑖∈𝜎 𝑗∈𝜎 (𝜌𝑛/ |𝜆 |) 󵄨 󵄨 𝑝𝑖𝑗 { [ { 1 2 { 1 󵄨𝐴 (𝑥 + 𝑦)󵄨 ∑ ∑ 𝑀(󵄨 𝑖𝑗 󵄨) : 𝑠, 𝑡 sup { ( ) } 𝜙𝑠,𝑡 𝑖∈𝜎 𝑗∈𝜎 𝜌1 +𝜌2 { 1 2 (𝑠,) 𝑡 ≥ (1, 1) ,𝜎1 ×𝜎2 ∈℘𝑠,𝑡} } } ≥ (1, 1) ,𝜎 ×𝜎 ∈℘ 1/𝐻 1 2 𝑠,𝑡} } ≤1] ≤1,𝑞∈𝑁,𝑟∈𝑁 } } 󵄨 󵄨 󵄨 󵄨 𝑝𝑖𝑗 ] } { 1 󵄨𝐴 (𝑥)󵄨 󵄨𝐴 (𝑦)󵄨 ≤ ∑ ∑ 𝑀(󵄨 𝑖𝑗 󵄨 + 󵄨 𝑖𝑗 󵄨) : (𝑠,) 𝑡 sup {𝜙 𝜌 +𝜌 𝜌 +𝜌 { 𝑠,𝑡 𝑖∈𝜎 𝑗∈𝜎 1 2 1 2 𝑛 𝑝𝑞𝑟/𝐻 { 1 2 = inf {(𝜆 𝜌𝑛) : { } 󵄨 󵄨 𝑝 ≥ (1, 1) ,𝜎 ×𝜎 ∈℘ { 󵄨𝐴 (𝑥𝑛 −𝑥)󵄨 𝑖𝑗 1 2 𝑠,𝑡} [ 1 󵄨 𝑖𝑗 󵄨 } sup { ∑ ∑ 𝑀( ) : (𝑠,) 𝑡 𝜙𝑠,𝑡 𝜌𝑛 𝑖∈𝜎1 𝑗∈𝜎2 󵄨 󵄨 𝑝 [ { ℎ { 󵄨𝐴 (𝑥)󵄨 𝑖𝑗 𝜌1 1 󵄨 𝑖𝑗 󵄨 1/𝐻 ≤( ) sup { ∑ ∑ 𝑀( ) : (𝑠,) 𝑡 } 𝜌1 +𝜌2 𝜙𝑠,𝑡 𝑖∈𝜎 𝑗∈𝜎 𝜌1 ≥ (1, 1) ,𝜎 ×𝜎 ∈℘ ] { 1 2 1 2 𝑠,𝑡} }] } ≥ (1, 1) ,𝜎 ×𝜎 ∈℘ } 1 2 𝑠,𝑡} ≤1,𝑞∈𝑁,𝑟∈𝑁 } } } 󵄨 󵄨 𝑝𝑖𝑗 𝜌 ℎ { 󵄨𝐴 (𝑥)󵄨 󵄨 𝑛󵄨ℎ/𝐻 󵄨 𝑛󵄨 2 1 󵄨 𝑖𝑗 󵄨 ≤ max {󵄨𝜆 󵄨 , 󵄨𝜆 󵄨}. +( ) sup { ∑ ∑ 𝑀( ) : (𝑠,) 𝑡 𝜌1 +𝜌2 𝜙𝑠,𝑡 𝑖∈𝜎 𝑗∈𝜎 𝜌1 { 1 2 { 󵄨 𝑛󵄨 𝑝𝑞𝑟/𝐻 inf {(󵄨𝜆 󵄨 𝜌𝑛) : } { ≥ (1, 1) ,𝜎1 ×𝜎2 ∈℘𝑠,𝑡} , 󵄨 󵄨 𝑝 } { 󵄨𝐴 (𝑥𝑛 −𝑥)󵄨 𝑖𝑗 [ 1 󵄨 𝑖𝑗 󵄨 (18) sup { ∑ ∑ 𝑀( ) : (𝑠,) 𝑡 𝜙𝑠,𝑡 𝑖∈𝜎 𝑗∈𝜎 𝜌𝑛 [ { 1 2 ℎ= 𝑝 𝑔(𝑥 + 𝑦) ≤ 𝑔(𝑥) +𝑔(𝑦) 1/𝐻 where inf 𝑖𝑗 .Thisshowsthat . } Using this triangle inequality we can write ] ≥ (1, 1) ,𝜎1 ×𝜎2 ∈℘𝑠,𝑡} }] 𝑛 𝑛 𝑛 𝑛 𝑛 𝑛 𝑔(𝜆 𝑥 −𝜆𝑥)≤𝑔(𝜆 𝑥 −𝜆 𝑥) + 𝑔 (𝜆 𝑥−𝜆𝑥). (19) } ≤1,𝑞∈𝑁,𝑟∈𝑁} Separately, we obtain } 󵄨 󵄨ℎ/𝐻 󵄨 󵄨 = {󵄨𝜆𝑛󵄨 , 󵄨𝜆𝑛󵄨}⋅𝑔(𝑥𝑛 −𝑥), 𝑔(𝜆𝑛𝑥𝑛 −𝜆𝑛𝑥) max 󵄨 󵄨 󵄨 󵄨 𝑔(𝜆𝑛𝑥−𝜆𝑥) { 𝑝𝑞𝑟/𝐻 = inf 𝜌𝑛 : { { 𝑝𝑞𝑟/𝐻 { = inf {𝜌𝑛 : { 󵄨 𝑛 𝑛 𝑛 󵄨 𝑝𝑖𝑗 { 1 󵄨𝐴 (𝜆 𝑥 −𝜆 𝑥)󵄨 󵄨 󵄨 𝑝 [ ∑ ∑ 𝑀(󵄨 𝑖𝑗 󵄨) : { 󵄨𝐴 ((𝜆𝑛 −𝜆)𝑥)󵄨 𝑖𝑗 sup { [ 1 󵄨 𝑖𝑗 󵄨 𝜙𝑠,𝑡 𝑖∈𝜎 𝑗∈𝜎 𝜌𝑛 sup ∑ ∑ 𝑀( ) : [ { 1 2 { 𝜙𝑠,𝑡 𝑖∈𝜎 𝑗∈𝜎 𝜌𝑛 [ { 1 2 } 1/𝐻 (𝑠,) 𝑡 ≥ (1, 1) ,𝜎 ×𝜎 ∈℘ } 1 2 𝑠,𝑡} (𝑠, 𝑡) ≥ (1, 1) ,𝜎 ×𝜎 ∈℘ ] } 1 2 𝑠,𝑡} }] 1/𝐻 } } ≤1] ≤1,𝑞∈𝑁,𝑟∈𝑁 ≤1,𝑞∈𝑁,𝑟∈𝑁 } } ] } } Journal of Function Spaces and Applications 5

󵄨 󵄨 𝑝𝑘𝑙 { 𝑝 /𝐻 { 1 󵄨𝑦 󵄨 = 𝜌 𝑞𝑟 : ≤ ∑ ∑𝑀(󵄨 𝑘,𝑙󵄨) : (𝑠,) 𝑡 inf { 𝑛 sup {𝜙 𝜌 𝑠,𝑡 𝑘∈𝜎 𝑙∈𝜎 { { 1 2

󵄨 󵄨 𝑝𝑖𝑗 { 1 󵄨𝐴 (𝑥)󵄨 } [ ∑ ∑ 𝑀( 󵄨 𝑖𝑗 󵄨 ) : (𝑠,) 𝑡 ≥ (1, 1) ,𝜎 ×𝜎 ∈℘ . sup { 𝑛 1 2 𝑠,𝑡} 𝜙𝑠,𝑡 𝑖∈𝜎 𝑗∈𝜎 (𝜌𝑛/ |𝜆 −𝜆|) [ { 1 2 } 1/𝐻 (21) } ≥ (1, 1) ,𝜎 ×𝜎 ∈℘ ] 2 1 2 𝑠,𝑡} This implies that {𝛼𝑘,𝑙𝑦𝑘,𝑙}∈𝑚(𝑀,𝜙,𝑝), and hence the 2 }] class 𝑚 (𝑀,𝜙,𝑝)is solid. } 2 Corollary 7. The space 𝑚 (𝑀, 𝜙, 𝑝) is monotone. ≤1,𝑞∈𝑁,𝑟∈𝑁} } Theorem 8. Let 𝜓 be another double sequence like 𝜙. 2 2 Then 𝑚 (𝑀,𝐴,𝜙,𝑝) ⊆𝑚 (𝑀,𝐴,𝜓,𝑝) if and only if { 󵄨 𝑛 󵄨 𝑝 /𝐻 󵄨 󵄨 𝑞𝑟 sup (𝜙𝑠,𝑡/𝜓𝑠,𝑡)<∞. = inf {(󵄨𝜆 −𝜆󵄨 𝜌𝑛) : (𝑠,𝑡)≥(1,1) { Proof. Let 𝐾=sup(𝑠,𝑡)≥(1,1)(𝜙𝑠,𝑡/𝜓𝑠,𝑡)<∞.Then𝜙𝑠,𝑡 ≤𝐾⋅𝜓𝑠,𝑡 2 󵄨 󵄨 𝑝𝑖𝑗 { 󵄨𝐴 (𝑥)󵄨 for all (𝑠, 𝑡) ≥ (1, 1).If𝑥={𝑥𝑘,𝑙}∈𝑚(𝑀,𝐴,𝜙,𝑝),then [ 1 󵄨 𝑖𝑗 󵄨 sup { ∑ ∑ 𝑀( ) : (𝑠,) 𝑡 𝜙 𝜌 󵄨 󵄨 𝑝𝑖𝑗 𝑠,𝑡 𝑖∈𝜎 𝑗∈𝜎 𝑛 󵄨 󵄨 [ { 1 2 { 1 󵄨𝐴𝑖𝑗 (𝑥)󵄨 sup { ∑ ∑ 𝑀( ) : (𝑠,) 𝑡 1/𝐻 𝜙𝑠,𝑡 𝑖∈𝜎 𝑗∈𝜎 𝜌 } { 1 2 ≥ (1, 1) ,𝜎 ×𝜎 ∈℘ ] (22) 1 2 𝑠,𝑡} } }] ≥ (1, 1) ,𝜎1 ×𝜎2 ∈℘𝑠,𝑡} <∞, } } ≤1,𝑞∈𝑁,𝑟∈𝑁 } for some 𝜌>0.Thus } 󵄨 󵄨 𝑝 󵄨 󵄨 𝑖𝑗 󵄨 󵄨ℎ/𝐻 󵄨 󵄨 { 1 󵄨𝐴𝑖𝑗 (𝑥)󵄨 ≤ {󵄨𝜆𝑛 −𝜆󵄨 , 󵄨𝜆𝑛 −𝜆󵄨}𝑔(𝑥) . ∑ ∑𝑀(󵄨 󵄨) : (𝑠,) 𝑡 max 󵄨 󵄨 󵄨 󵄨 sup {𝐾𝜓 𝜌 𝑠,𝑡 𝑘∈𝜎 𝑙∈𝜎 { 1 2 (20) (23) } 𝑛 𝑛 𝑛 𝑛 Hence 𝑔(𝜆 𝑥 −𝜆𝑥)→0where 𝜆 →𝜆and 𝑥 →𝑥as ≥ (1, 1) ,𝜎1 ×𝜎2 ∈℘𝑠,𝑡} <∞, 2 𝑛→∞.Consequently𝑔 is a paranom on 𝑚 (𝑀,𝐴,𝜙,𝑝). } 2 for some 𝜌>0, and hence 𝑥={𝑥𝑘,𝑙}∈𝑚(𝑀,𝐴,𝜓,𝑝).This 2 2 2 Proposition 6. The class 𝑚 (𝑀, 𝜙, 𝑝) of double sequences is shows that 𝑚 (𝑀,𝐴,𝜙,𝑝)⊆𝑚 (𝑀,𝐴,𝜓,𝑝). 2 2 solid. Conversely, let 𝑚 (𝑀,𝐴,𝜙,𝑝) ⊆𝑚 (𝑀,𝐴,𝜓,𝑝).We say 𝛼𝑠,𝑡 =𝜙𝑠,𝑡/𝜓𝑠,𝑡 for all (𝑠, 𝑡) ≥ (1, 1) and suppose 𝛼={𝛼 } Proof. Let 𝑘,𝑙 beadoublesequenceofscalarssuchthat sup(𝑠,𝑡)≥(1,1)𝛼𝑠,𝑡 =∞. Then there exists a subsequence {𝛼𝑠 ,𝑡 } |𝛼 |≤1 𝑦={𝑦 }∈ 𝑚2(𝑀,𝜙,𝑝) {𝛼 } 𝛼 =∞ 𝑥={𝑥}∈𝑖 𝑖 𝑘,𝑙 and 𝑘,𝑙 .Thenwecanwrite of 𝑠,𝑡 such that lim𝑖→∞ 𝑠𝑖,𝑡𝑖 .If 𝑘,𝑙 2 𝑚 (𝑀,𝐴,𝜙,𝑝),thenwehave 󵄨 󵄨 𝑝 { 󵄨𝛼 𝑥 󵄨 𝑘𝑙 1 󵄨 𝑘,𝑙 𝑘,𝑙󵄨 󵄨 󵄨 𝑝𝑖𝑗 ∑ ∑𝑀( ) : (𝑠,) 𝑡 { 󵄨𝐴 (𝑥)󵄨 sup {𝜙 𝜌 1 󵄨 𝑖𝑗 󵄨 𝑠,𝑡 𝑘∈𝜎 𝑙∈𝜎 ∑ ∑ 𝑀( ) : (𝑠,) 𝑡 { 1 2 sup { 𝜓𝑠,𝑡 𝑖∈𝜎 𝑗∈𝜎 𝜌 { 1 2 } } ≥ (1, 1) ,𝜎1 ×𝜎2 ∈℘𝑠,𝑡} ≥ (1, 1) ,𝜎1 ×𝜎2 ∈℘𝑠,𝑡} } } 󵄨 󵄨 󵄨 󵄨 𝑝 󵄨 󵄨 󵄨 󵄨 𝑘𝑙 󵄨 󵄨 𝑝𝑖𝑗 { 1 󵄨𝛼𝑘,𝑙󵄨 󵄨𝑦𝑘,𝑙󵄨 { 󵄨𝐴 (𝑥)󵄨 ≤ ∑ ∑𝑀(󵄨 󵄨 󵄨 󵄨) : (𝑠,) 𝑡 1 󵄨 𝑖𝑗 󵄨 sup { = sup 𝛼𝑠,𝑡 ∑ ∑ 𝑀( ) : (𝑠,) 𝑡 𝜙𝑠,𝑡 𝜌 { 𝜙 𝜌 𝑘∈𝜎1 𝑙∈𝜎2 𝑠,𝑡 𝑖∈𝜎 𝑗∈𝜎 { { 1 2 } } ≥ (1, 1) ,𝜎1 ×𝜎2 ∈℘𝑠,𝑡} ≥ (1, 1) ,𝜎1 ×𝜎2 ∈℘𝑠,𝑡} } } 6 Journal of Function Spaces and Applications

≥{ 𝛼 } Henceweget sup 𝑠𝑚,𝑡𝑚 𝑚≥1 󵄨 󵄨 𝑝𝑖𝑗 { 1 󵄨𝐴𝑖𝑗 (𝑥)󵄨 󵄨 󵄨 𝑝 󵄨 󵄨 󵄨 󵄨 𝑖𝑗 sup ∑ ∑ 𝑀(𝑀1 ( )) : (𝑠,) 𝑡 { 1 󵄨𝐴𝑖𝑗 (𝑥)󵄨 { 󵄨 󵄨 𝜙𝑠,𝑡 𝑖∈𝜎 𝑗∈𝜎 𝜌 ⋅ sup { ∑ ∑ 𝑀( ) : { 1 2 𝜙𝑠 ,𝑡 𝑖∈𝜎 𝑗∈𝜎 𝜌 { 𝑚 𝑚 1 2 } ≥ (1, 1) ,𝜎 ×𝜎 ∈℘ } 1 2 𝑠,𝑡} 𝜎 ×𝜎 ∈℘ =∞. } 1 2 𝑠𝑚,𝑡𝑚 } 𝐻 } ≤ max {1, 𝑀(1) } (24) { 1 𝑝 2 × ∑ ∑ (𝑦 ) 𝑖𝑗 : (𝑠,) 𝑡 Thisisacontradictionas𝑥={𝑥𝑘,𝑙}∉𝑚(𝑀,𝐴,𝜙,𝑝).This sup { 𝑖,𝑗 𝜙𝑠,𝑡 𝑖∈𝜎 𝑗∈𝜎 completes the proof. { 1 2

2 2 } (31) Corollary 9. 𝑚 (𝑀,𝐴,𝜙,𝑝)= 𝑚 (𝑀,𝐴,𝜓,𝑝)if and only if ≥ (1, 1) ,𝜎 ×𝜎 ∈℘ 𝛼 <∞ 𝛼−1 <∞ 1 2 𝑠,𝑡} sup(𝑠,𝑡)≥(1,1) 𝑠,𝑡 and sup(𝑠,𝑡)≥(1,1) 𝑠,𝑡 . } 𝑇 𝐻 Theorem 10. Let 𝑀, 𝑀1,and𝑀2 be Orlicz functions satisfying + max (1, ( 𝑀 (2)) ) Δ 2-condition. Then 𝛿 2 2 (a) 𝑚 (𝑀1,𝐴,𝜙,𝑝)⊆𝑚 (𝑀 ∘1 𝑀 ,𝐴,𝜙,𝑝), { 1 𝑝𝑖𝑗 2 2 2 × sup ∑ ∑ (𝑦𝑖,𝑗) : (𝑠,) 𝑡 (b) 𝑚 (𝑀1,𝐴,𝜙,𝑝)∩ 𝑚 (𝑀2,𝐴,𝜙,𝑝) ⊆𝑚 (𝑀1 +𝑀2, { 𝜙𝑠,𝑡 𝑖∈𝜎 𝑗∈𝜎 𝐴, 𝜙, 𝑝). { 1 2 } 𝑥={𝑥 }∈𝑚2(𝑀 ,𝐴,𝜙,𝑝) Proof. (a) Let 𝑘,𝑙 1 . Then there exists ≥ (1, 1) ,𝜎1 ×𝜎2 ∈℘𝑠,𝑡} . 𝜌>0such that } 󵄨 󵄨 𝑝 󵄨 󵄨 𝑖𝑗 {𝑥 }∈𝑚2(𝑀 ∘ 𝑀 ,𝜙) { 1 󵄨𝐴𝑖𝑗 (𝑥)󵄨 Finally, we have 𝑘,𝑙 1 , and hence ∑ ∑ 𝑀 (󵄨 󵄨) : (𝑠,) 𝑡 2 2 sup { 1 𝑚 (𝑀1,𝜙)⊆𝑚 (𝑀 ∘1 𝑀 ,𝜙). 𝜙𝑠,𝑡 𝑖∈𝜎 𝑗∈𝜎 𝜌 { 1 2 2 2 (25) (b) Let {𝑥𝑘,𝑙}∈𝑚(𝑀1,𝐴,𝜙,𝑝)∩𝑚 (𝑀2,𝐴,𝜙,𝑝).Then } there exists 𝜌>0such that ≥ (1, 1) ,𝜎 ×𝜎 ∈℘ <∞. 1 2 𝑠,𝑡} 󵄨 󵄨 𝑝 󵄨 󵄨 𝑖𝑗 } { 1 󵄨𝐴𝑖𝑗 (𝑥)󵄨 sup { ∑ ∑ 𝑀1( ) : (𝑠,) 𝑡 𝑀 𝛿 0<𝛿<1 𝜙𝑠,𝑡 𝑖∈𝜎 𝑗∈𝜎 𝜌 By the continuity of ,weselectanumber with { 1 2 such that 𝑀(𝑡) <𝜀,whenever0≤𝑡<𝛿, for arbitrary 0<𝜀< } 1.Nowlet𝑦𝑖,𝑗 =𝑀1(|𝐴𝑖𝑗 (𝑥)|/𝜌).Wecanwrite ≥ (1, 1) ,𝜎1 ×𝜎2 ∈℘𝑠,𝑡} <∞, 𝑝 𝑝 𝑝 ∑ ∑ 𝑀(𝑦 ) 𝑖𝑗 = ∑ 𝑀(𝑦 ) 𝑖𝑗 + ∑ 𝑀(𝑦 ) 𝑖𝑗 . } 𝑖,𝑗 & 𝑖,𝑗 𝑖,𝑗 (26) (32) 𝑖∈𝜎 𝑗∈𝜎 𝑦 ≤𝛿 𝑦 >𝛿 󵄨 󵄨 𝑝𝑖𝑗 1 2 𝑖,𝑗 𝑖,𝑗 { 1 󵄨𝐴 (𝑥)󵄨 ∑ ∑ 𝑀 (󵄨 𝑖𝑗 󵄨) : (𝑠,) 𝑡 𝑀 sup { 2 By the properties of ,wehave 𝜙𝑠,𝑡 𝑖∈𝜎 𝑗∈𝜎 𝜌 { 1 2 𝑝 𝑝 ∑ 𝑀(𝑦 ) 𝑖𝑗 ≤ {1, 𝑀(1)𝐻} ∑ (𝑦 ) 𝑖𝑗 . 𝑖,𝑗 max 𝑖,𝑗 (27) } 𝑦 ≤𝛿 𝑦 ≤𝛿 𝑖,𝑗 𝑖,𝑗 ≥ (1, 1) ,𝜎1 ×𝜎2 ∈℘𝑠,𝑡} <∞. } Again we can write 𝑦 1 1 2𝑦 By the inequality 𝑀(𝑦 )<𝑀(1+ 𝑖,𝑗 )< 𝑀 2 + 𝑀( 𝑖,𝑗 ), 𝑖,𝑗 ( ) (28) 󵄨 󵄨 𝑝𝑖𝑗 𝛿 2 2 𝛿 󵄨𝐴 (𝑥)󵄨 ∑ ∑ (𝑀 +𝑀)(󵄨 𝑖𝑗 󵄨) 𝑦 >𝛿 𝑀 Δ 1 2 𝜌 for 𝑖,𝑗 .Ifweusethat satisfies 2-condition, then we 𝑖∈𝜎1 𝑗∈𝜎2 find 󵄨 󵄨 𝑝𝑖𝑗 󵄨𝐴𝑖𝑗 (𝑥)󵄨 1 𝑦𝑖,𝑗 1 𝑦𝑖,𝑗 𝑦𝑖,𝑗 𝐻−1 󵄨 󵄨 ≤ max (1, 2 ) ∑ ∑ 𝑀1( ) (33) 𝑀(𝑦𝑖,𝑗)< 𝑇 𝑀 (2) + 𝑇 𝑀 (2) =𝑇 𝑀 (2) , 𝜌 2 𝛿 2 𝛿 𝛿 𝑖∈𝜎1 𝑗∈𝜎2 (29) 󵄨 󵄨 𝑝 󵄨𝐴 (𝑥)󵄨 𝑖𝑗 𝐻−1 󵄨 𝑖𝑗 󵄨 and so + max (1, 2 ) ∑ ∑ 𝑀2( ) , 𝑖∈𝜎 𝑗∈𝜎 𝜌 𝐻 1 2 𝑝𝑖𝑗 𝑇 ∑ 𝑀(𝑦𝑖,𝑗) ≤ max (1, ( 𝑀 (2)) ) ∑ 𝑦𝑖,𝑗. 2 2 2 𝛿 (30) we have 𝑚 (𝑀1,𝐴,𝜙,𝑝) ∩𝑚 (𝑀2,𝐴,𝜙,𝑝) ⊆𝑚 (𝑀1 + 𝑦𝑖,𝑗>𝛿 𝑦𝑖,𝑗>𝛿 𝑀2,𝐴,𝜙,𝑝). Journal of Function Spaces and Applications 7

(2) 2 2 (2) Theorem 11. (a) 𝑙 (𝑀,𝐴,𝜙,𝑝) ⊆𝑚 (𝑀,𝐴,𝜙,𝑝) ⊆ (c) Firstly we show that 𝑚 (𝑀,𝐴,𝜓,𝑝) =𝑙 (𝑀, 𝐴, 𝜙, (2) 𝑙 (𝑀,𝐴,𝜙,∞). ∞) if 𝜓𝑠,𝑡 =𝑠.𝑡for all (𝑠,𝑡)∈𝑁×𝑁.Let𝑥={𝑥𝑘,𝑙}∈ (2) 𝑙 (𝑀,𝐴,𝜙,∞) (𝑠, 𝑡) ≥ (1, 1) 𝜎1 ×𝜎2 ∈ 2 (2) .Thenfor and (b) 𝑚 (𝑀,𝐴,𝜙,𝑝) =𝑙 (𝑀,𝐴,𝜙,𝑝) if and only if ℘𝑠,𝑡,wecanfindsome𝜌>0such that sup(𝑠,𝑡)≥(1,1)𝜙𝑠,𝑡 <∞. 󵄨 󵄨 𝑝𝑖𝑗 2 (2) 1 󵄨𝐴 (𝑥)󵄨 (c) 𝑚 (𝑀,𝐴,𝜙,𝑝) =𝑙 (𝑀,𝐴,𝜙,∞) if and only if ∑ ∑ 𝑀(󵄨 𝑖𝑗 󵄨) (𝜙 /𝑠.𝑡) >0 𝑠.𝑡 𝜌 sup(𝑠,𝑡)≥(1,1) 𝑠,𝑡 . 𝑖∈𝜎1 𝑗∈𝜎2 (40) (2) 󵄨 󵄨 𝑝𝑖𝑗 Proof. (a) Let 𝑥={𝑥𝑘,𝑙}∈𝑙 (𝑀,𝐴,𝜙,𝑝),andletaset𝐵 be 1 󵄨𝐴 (𝑥)󵄨 ≤ 𝑠.𝑡. 𝑀(󵄨 𝑖𝑗 󵄨) <∞. defined as follows: 𝑠.𝑡 sup 𝜌 (𝑖,𝑗)∈𝑁×𝑁 󵄨 󵄨 𝑝 󵄨 󵄨 𝑖𝑗 { 1 󵄨𝐴𝑖𝑗 (𝑥)󵄨 𝐵={ ∑ ∑ 𝑀( ) : (𝑠,) 𝑡 𝑙(2)(𝑀,𝐴,𝜙,∞)2 ⊆𝑚 (𝑀,𝐴,𝜓,𝑝) 𝜙𝑠,𝑡 𝑖∈𝜎 𝑗∈𝜎 𝜌 This gives the inclusion . { 1 2 2 Conversely let 𝑥={𝑥𝑘,𝑙}∈𝑚(𝑀,𝐴,𝜓,𝑝).Thenfor(𝑖, 𝑗) ∈ (34) 𝑁×𝑁 𝜌>0 } ,wecanfindsome such that ≥ (1, 1) ,𝜎 ×𝜎 ∈℘ . 1 2 𝑠,𝑡} 󵄨 󵄨 𝑝𝑖𝑗 󵄨𝐴 (𝑥)󵄨 } 𝑀(󵄨 𝑖𝑗 󵄨) 𝜌 Since {𝜙𝑘,𝑙} is a nondecreasing double sequence, {1/𝜙𝑘,𝑙} is a 󵄨 󵄨 𝑝𝑖𝑗 nonincreasing double sequence. So we obtain 1 󵄨𝐴 (𝑥)󵄨 = 𝑠.𝑡.𝑀(󵄨 𝑖𝑗 󵄨) 󵄨 󵄨 𝑝 𝑠.𝑡 𝜌 ∞ 󵄨 󵄨 𝑖𝑗 1 󵄨𝐴𝑖𝑗 (𝑥)󵄨 ∑ 𝑀( ) ≥𝑏, (35) 󵄨 󵄨 𝑝 (41) 𝜙 𝜌 󵄨 󵄨 𝑖𝑗 1,1 𝑖,𝑗=1 { 1 󵄨𝐴𝑖𝑗 (𝑥)󵄨 ≤ sup { ∑ ∑ 𝑀( ) : (𝑠,) 𝑡 𝜙𝑠,𝑡 𝑖∈𝜎 𝑗∈𝜎 𝜌 for all 𝑏∈𝐵and hence { 1 2

󵄨 󵄨 𝑝𝑖𝑗 } 󵄨𝐴 (𝑥)󵄨 󵄨 𝑖𝑗 󵄨 ≥ (1, 1) ,𝜎1 ×𝜎2 ∈℘𝑠,𝑡} <∞. ∞>𝑀( ) ≥𝜙1,1 ⋅ sup 𝐵. (36) 𝜌 }

2 (2) (2) 2 𝑚 (𝑀,𝐴,𝜓,𝑝) ⊆𝑙 (𝑀,𝐴,𝜙,∞) Thus we have 𝑙 (𝑀,𝐴,𝜙,𝑝)⊆𝑚 (𝑀,𝐴,𝜙,𝑝). This shows that .By 𝑚2(𝑀,𝐴,𝜙,𝑝)= 2 Theorem 8 and alternative (a), we can write Conversely if 𝑥={𝑥𝑘,𝑙}∈𝑚(𝑀,𝐴,𝜙,𝑝),thenitisclear (2) 𝑙 (𝑀,𝐴,𝜙,∞)if and only if sup(𝑠,𝑡)≥(1,1)(𝑠.𝑡/𝜙𝑠,𝑡)<∞.This that completes the proof. 󵄨 󵄨 𝑝 󵄨 󵄨 𝑖𝑗 1 󵄨𝐴𝑖𝑗 (𝑥)󵄨 sup 𝐵≥ 𝑀( ) , (37) References 𝜙1,1 𝜌 𝑚 [1] V. A. Khan, “On Δ V -Cesaro´ summable double sequences,” Thai for all (𝑖, 𝑗) ∈ 𝑁 ×𝑁, and hence Journal of Mathematics,vol.10,no.3,pp.535–539,2012. [2] V. A. Khan and S. Tabassum, “Statistically pre-Cauchy double 󵄨 󵄨 𝑝 󵄨 󵄨 𝑖𝑗 sequences,” Southeast Asian Bulletin of Mathematics,vol.36,no. 1 󵄨𝐴𝑖𝑗 (𝑥)󵄨 𝐵≥ 𝑀(󵄨 󵄨) . 2,pp.249–254,2012. sup 𝜙 sup 𝜌 (38) 1,1 (𝑖,𝑗)∈𝑁×𝑁 [3]V.A.Khan,S.Tabassum,andA.Esi,“Statisticallyconvergent double sequence spaces in 𝑛-normed spaces,” ARPN Journal of 2 Science and Technology,vol.2,no.10,pp.991–995,2012. This shows that if 𝑥={𝑥𝑘,𝑙}∈𝑚(𝑀,𝐴,𝜙,𝑝),then (2) 2 [4] B. V.Limaye and M. Zeltser, “On the Pringsheim convergence of {𝑥𝑘,𝑙}∈𝑙 (𝑀,𝐴,𝜙,∞).Thuswehave𝑚 (𝑀,𝐴,𝜙,𝑝) ⊆ (2) double series,” Proceedings of the Estonian Academy of Sciences, 𝑙 (𝑀,𝐴,𝜙,∞). vol.58,no.2,pp.108–121,2009.

2 (2) [5] J. Lindenstrauss and L. Tzafriri, “On Orlicz sequence spaces,” (b) It is clear that 𝑚 (𝑀,𝐴,𝜓,𝑝) =𝑙 (𝑀,𝐴,𝜙,∞) Israel Journal of Mathematics,vol.10,pp.379–390,1971. where 𝜓𝑠,𝑡 =1for all (𝑠, 𝑡) ∈ 𝑁×𝑁.Thenwecanwrite [6] V. A. Khan and S. Tabassum, “Statistically convergent double sup(𝑠,𝑡)≤(1,1)𝜙𝑠,𝑡 = sup(𝑠,𝑡)≤(1,1)(𝜙𝑠,𝑡/𝜓𝑠,𝑡)<∞.ByThe- sequence spaces in 2-normed spaces defined by Orlicz func- 2 2 orem 8,wehave𝑚 (𝑀,𝐴,𝜙,𝑝) ⊆𝑚 (𝑀,𝐴,𝜓,𝑝), tion,” Applied Mathematics, vol. 2, no. 4, pp. 398–402, 2011. Δ𝑚 and according to alternative (a) [7]V.A.KhanandS.Tabassum,“Onsomenewquasialmost - lacunary strongly P-convergent double sequences defined by 𝑚2 (𝑀,𝐴,𝜙,𝑝)=𝑙(2) (𝑀,𝐴,𝜙,𝑝). Orlicz functions,” Journal of Mathematics and Applications,vol. (39) 34, pp. 45–52, 2011. 8 Journal of Function Spaces and Applications

[8] V. A. Khan and S. Tabassum, “On ideal convergent difference double sequence spaces in 2-normed spaces defined by Orlicz functions,” JMI International Journal of Mathematics and Appli- cations,vol.1,no.2,pp.26–34,2010. [9] V.A. Khan and S. Tabassum, “Some vector valued multiplier difference double sequence spaces in 2-normed spaces defined by Orlicz functions,” Journal of Mathematical and Computational Science,vol.1,no.1,pp.126–139,2011. [10] V. A. Khan and S. Tabassum, “On some new double sequence spaces of invariant means defined by Orlicz functions,” Com- munications de la Faculte´ des Sciences de l’Universited’Ankara´ A1, vol. 60, no. 2, pp. 11–21, 2011. [11] V. A. Khan and S. Tabassum, “The strongly summable generalized difference double sequence spaces in 2-normed spaces defined by Orlicz functions,” Journal of Mathematical Notes,vol. 7, no. 2, pp. 45–58, 2011. [12] V. A. Khan and S. Tabassum, “On some new almost double 𝑚 Lacunary Δ -squence spaces dened by Orlicz functions,” Jour- nal of Mathematical Notes,vol.6,no.2,pp.80–94,2011.

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[14] W.L. C. Sargent, “Some sequence spaces related to the 𝑙𝑝 spaces,” Journal of the London Mathematical Society,vol.35,pp.161–171, 1960. [15] Y. Altun and T. Bilgin, “On a new class of sequences related to the 𝑙𝑝 space defined by Orlicz function,” Taiwanese Journal of Mathematics,vol.13,no.4,pp.1189–1196,2009. [16] A. Esi, “On a class of new type difference sequence spaces related to the space 𝑙𝑝,” Far East Journal of Mathematical Sciences,vol. 13,no.2,pp.167–172,2004. [17] B. C. Tripathy and S. Mahanta, “On a class of sequences related to the 𝑙𝑝 space defined by Orlicz functions,” Soochow Journal of Mathematics,vol.29,no.4,pp.379–391,2003. [18] D. Rath, “Spaces of 𝑟-convex sequences and matrix transformations,” Indian Journal of Mathematics,vol.41,no.2,pp.265–280, 1999. [19] D. Rath and B. C. Tripathy, “Characterization of certain matrix operators,” Orissa Mathematical Society,vol.8,pp.121–134,1989. [20] B. C. Tripathy and M. Sen, “On a new class of sequences related to the space 𝑙𝑝,” Tamkang Journal of Mathematics,vol.33,no.2, pp. 167–171, 2002. Hindawi Publishing Corporation Journal of Function Spaces and Applications Volume 2013, Article ID 692879, 12 pages http://dx.doi.org/10.1155/2013/692879

Research Article Schauder-Tychonoff Fixed-Point Theorem in Theory of Superconductivity

Mariusz Gil and StanisBaw Wwdrychowicz

Department of Mathematics, Rzeszow´ University of Technology, al.Powstanc´ ow´ Warszawy 6, 35-959 Rzeszow,´ Poland

Correspondence should be addressed to Stanisław Wędrychowicz; [email protected]

Received 25 April 2013; Accepted 5 June 2013

Academic Editor: Jozef´ Bana´s

Copyright © 2013 M. Gil and S. Wędrychowicz. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We study the existence of mild solutions to the time-dependent Ginzburg-Landau ((TDGL), for short) equations on an unbounded interval. The rapidity of the growth of those solutions is characterized. We investigate the local and global attractivity of solutions of TDGL equations and we describe their asymptotic behaviour. The TDGL equations model the state of a superconducting sample in a magnetic field near critical temperature. This paper is based on the theory of Banach space,echet Fr´ space, and Sobolew space.

1. Introduction The system of (1)–(3)mustbesatisfiedeverywhereinΩ,the region occupied by the superconducting material, and at all The objective of the paper is to investigate the existence times 𝑡>0. The boundary conditions associated with the andasymptoticbehaviourofmildsolutionsonanunbound- differential equations have the form ed interval of time-dependent Ginzburg-Landau equations 𝑖 𝑖 (TDGL, for short) in superconductivity. n ⋅( ∇+A)𝜓+ 𝛾𝜓 = 0, (4) In the Ginzburg-Landau theory of phase transitions [1], 𝜅 𝜅 the state of a superconducting material near the critical tem- n × (∇×A − H) = 0 (5) perature is described by a complex-valued order parameter 𝜓, a real vector-valued vector potential A, and, when the on 𝜕Ω,where𝜕Ω is the boundary of Ω and n is the local outer system changes with time, a real-valued scalar potential 𝜙. unit normal to 𝜕Ω. They must be satisfied at all times 𝑡>0. The latter is a diagnostic variable; 𝜓 and A are prognostic We prove that the systems of (1)–(5)canbereduced variables, whose evolution is governed by a system of coupled to a semilinear equation; to use the appropriate theorem, differential equations: we investigate the local and global attractivity of solutions of equations in question and describe their asymptotic 2 𝜕 𝑖 󵄨 󵄨2 behaviour. 𝜂( + 𝑖𝜅𝜙) 𝜓 =−( ∇+A) 𝜓+(1−󵄨𝜓󵄨 )𝜓, (1) 𝜕𝑡 𝜅 󵄨 󵄨 In this paper, we consider the existence and asymptotic behaviour of mild solutions on an unbounded interval of the 𝜕A +∇𝜙=−∇×∇×A + J +∇×H. (2) semilinear evolution equation of the following form: 𝜕𝑡 𝑠 𝑑𝑢 + A𝑢=F (𝑡, 𝑢 (𝑡)) ,𝑡∈R+ = [0, +∞) , (6) The supercurrent density J𝑠 is a nonlinear function of 𝜓 and 𝑑𝑡 A, 𝑢 (0) =𝑢0 ∈ X, (7) 1 ∗ ∗ 󵄨 󵄨2 A :𝐷(A)⊂X → X 𝐶 J𝑠 ≡ J𝑠 (𝜓, A)= (𝜓 ∇𝜓 − 𝜓∇𝜓 )−󵄨𝜓󵄨 A where the operator generates a 0- 2𝑖𝜅 {𝑒−A(𝑡−𝑠)} X (3) semigroup 𝑡≥0 and is a real Banach space. 𝑖 Recently,alotofpapershaveappearedthatdealwiththe =− [𝜓∗ ( ∇+A)𝜓]. Re 𝜅 same or similar equations on a bounded interval (see [2–21]). 2 Journal of Function Spaces and Applications

However, only in a few papers, problem (6)-(7)wascon- aBanachspaceX.Thespace𝐶(R+, X) is furnished with the sidered on an unbounded interval [10, 22]. Additionally, family of seminorms; −A(𝑡−𝑠) in assumptions concerning the semigroup {𝑒 }𝑡≥0 or ‖𝑢‖ := {‖𝑢 (𝑡)‖ :𝑡∈[0, 𝑛]} the function F(𝑡, 𝑢), rather restrictive conditions have been 𝑛 sup imposed which frequently require the compactness of F(𝑡, 𝑢) (9) −A(𝑡−𝑠) −A(𝑡−𝑠) for 𝑛=1,2,..., 𝑢∈𝐶(R+, X). or {𝑒 }𝑡≥0 or equicontinuity of semigroup {𝑒 }𝑡≥0 [2–11, 13–22]. It is worthwhile mentioning that only a few Let us recall two facts: papers have discussed asymptotic behaviour of solutions, ∗ ∞ mostly without the formulation of existence theorems [10, 23, ( )asequence{𝑢𝑛}𝑛=1 is convergent to 𝑢 in 𝐶(R+, X) if ∞ 24]. and only if {𝑢𝑛}𝑛=1 is uniformly convergent to 𝑥 on In this paper, we present conditions guaranteeing the compact subsets of R+; existence of mild solutions on an unbounded interval of ∗∗ ( )asubset𝑈⊂𝐶(R+, X) is relatively compact if and problem (6)-(7). We dispense with assumptions on the [0, 𝑇] 𝑈 F(𝑡, 𝑢) {𝑒−A(𝑡−𝑠)} only if the restrictions to of all functions from compactness of or 𝑡≥0. form an equicontinuous set for each 𝑇>0and 𝑈(𝑡) is Moreover, we formulate theorems about asymptotic prop- relatively compact in X for each 𝑡∈R+,where𝑈(𝑡) = erties and both local and global attractivity of solutions of {𝑢(𝑡) : 𝑢. ∈𝑈} problem (6)-(7). The existence theorems concerning that problemwillbeprovedwiththehelpofthetechniqueofa Moreover, we recall that a nonempty subset 𝑈⊂ family of measures of noncompactness in the Frechet´ space 𝐶(R+, X) is said to be bounded if 𝐶(R+, X) and Schauder-Tychonoff fixed-point principle. The approach applied here was introduced and developed sup {‖𝑢‖𝑛 :𝑢∈𝑈} <∞ for 𝑛=1,2,.... (10) in [20, 25–35], for instance. The paper is organized as follows. In Section 2,thereare Further, the family of all nonempty and bounded subsets given notation and auxiliary facts that are needed further on. of 𝐶(R+, X) will be denoted by M𝐶, while the family of In Section 3, we formulate and prove a theorem on the exis- all nonempty and relatively compact subsets of 𝐶(R+, X) is tence of mild solution of (6) with condition (7). Moreover, denoted by N𝐶.ObviouslyN𝐶 ⊂ M𝐶. the rate of the growth of those solutions is characterized. We will use a family of measures of noncompactness Section 4 contains a theorem on local and global attrac- {𝜇𝑇}𝑇≥0 in the Frechet´ space 𝐶(R+, X) which was introduced tivity of solutions of problem (6)-(7). In Section 5,wegivea in [20, 35]. In order to define these measures, recall some theorem describing the asymptotic behaviour of solutions of quantities [25–27]. Let us fix a nonempty bounded subset 𝑈 (6)-(7). of the space 𝐶(R+, X).For𝑢∈𝑈, 𝜀≥0,and𝑇≥0denote 𝑇 Finally, in Section 6,weformulatethegaugedTDGL by 𝜔 (𝑢, 𝜀) the modulus of continuity of the function 𝑢 on the equation as an abstract evolution equation in a Hilbert interval [0, 𝑇],thatis, space. Moreover, this section is devoted to present examples 𝑇 󵄩 󵄩 of application of previously obtained theorems for TDGL 𝜔 (𝑢,) 𝜀 := sup {󵄩𝑢(𝑡2)−𝑢(𝑡1)󵄩 : (11) equations. 󵄨 󵄨 In order to convert the systems equations (1)–(5)tothe 𝑡1,𝑡2 ∈ [0, 𝑇] , 󵄨𝑡2 −𝑡1󵄨 ≤𝜀}. Cauchy problem (6)-(7)wearebasedonthepapersand monographs (see [1, 30, 36–69]). Further, let us put 𝑇 𝑇 𝜔 (𝑈, 𝜀) := sup {𝜔 (𝑢,) 𝜀 :𝑢∈𝑈}, 2. Preliminaries 𝑇 𝑇 (12) 𝜔0 (𝑈) := lim 𝜔 (𝑈, 𝜀) . Let (X,‖ ⋅ ‖)be a real Banach space with the zero element 𝜃. 𝜀→0+ Denote by 𝐵(𝑥, 𝑟) the closed ball in X centered at 𝑥 and with Obviously, the set 𝑈 is equicontinuous on [0, 𝑇] if and only if radius 𝑟.If𝑋 is a subset of a linear topological space, then the 𝑇 𝜔 (𝑈) =. 0 symbols 𝑋 and Conv𝑋 stand for the closure and the convex 0 Next, let us define the functions {𝜇𝑇}𝑇≥0 on the family of closure of 𝑋,respectively. bounded subsets of 𝐶(R+, X) by the following formula: Let 𝜒 denote the Hausdorff measure of noncompactness in X, defined on bounded subsets 𝑋 of X in the following way 𝑇 𝜇 (𝑈) := 𝜔 (𝑈) + {𝜒 (𝑈 (𝑡)) :𝑡∈[0, 𝑇]}. (13) (see [27]): 𝑇 0 sup

Itcanbeshownthatthefamily{𝜇𝑇}𝑇≥0 has the following 𝜒 (𝑋) := inf {𝑟>0:𝑋can be covered with finite properties: (8) 𝑜 numbers of balls of radius 𝑟} . 1 the family ker{𝜇𝑇}:={𝑈∈M𝐶 :𝜇𝑇(𝑈) = 0 for 𝑇≥ 0} is nonempty and ker{𝜇𝑇}=N𝐶; 𝑜 2 𝑈⊂𝑉⇒𝜇𝑇(𝑈) ≤𝑇 𝜇 (𝑉) for 𝑇≥0; Further, denote by 𝐶(R+, X) the Frechet´ space consisting 𝑜 of all functions defined and continuous on R+ with values in 3 𝜇𝑇(Conv𝑈) =𝑇 𝜇 (𝑈) for 𝑇≥0; Journal of Function Spaces and Applications 3

𝑜 ∞ 𝑇 4 if {𝑈𝑛}𝑛=1 isasequenceofclosedsetsfromM𝐶 such (i) sup𝑡∈[0,𝑇]𝜒(𝑈(𝑡)) ≤ 𝜒(𝑈([0,0 𝑇]))≤𝜔 (𝑈) + sup𝑡∈[0,𝑇] that 𝑈𝑛+1 ⊂𝑈𝑛 (𝑛=1,2,...)and if lim𝑛→∞𝜇𝑇(𝑈𝑛)= 𝜒(𝑈(𝑡)), 0 𝑇≥0 𝑈 := for each , then the intersection set ∞ 𝜒(𝑈(𝑡)) ≤𝜒 (𝑈) ≤𝑇 𝜔 (𝑈) + ⋂∞ 𝑈 (ii) sup𝑡∈[0,𝑇] 𝐶 0 sup𝑡∈[0,𝑇] 𝑛=1 𝑛 is nonempty. 𝜒(𝑈(𝑡)), Remark 1. Observe that in contrast to the definition of the for 𝑇≥0,where𝑈([0, 𝑇]) := {𝑢(𝑡) : 𝑡 ∈ [0,. 𝑇],𝑢∈𝑈} conceptofameasureofnoncompactnessgivenin[27], our mapping 𝜇𝑇 may take the value ∞.Moreover,asinglemap- Lemma 8 (Gronwall’s inequality [71]). Assume that the func- ping 𝜇𝑇 is not the measure of noncompactness in 𝐶(R+, X) tion 𝜙1 : R+ → R+ is measurable and the function 𝜙2 : but the whole family {𝜇𝑇}𝑇≥0 canbecalledfamilyofmeasures R+ → R+ is locally integrable. Moreover, we assume that 𝑡 of noncompactness. ∫ 𝜙 (𝑠)𝜙 (𝑠)𝑑𝑠 <∞ 𝑡≥0 0 1 2 for and a measurable function 𝜓:R+ → R+ satisfies the following inequality: Remark 2. Letusnoticethattheintersectionset𝑈∞ de- 𝑜 scribed in axiom 4 isamemberofthekernelofthefamily 𝑡 {𝜇 } 𝑈 𝜓 (𝑡) ≤𝜙1 (𝑡) + ∫ 𝜙2 (𝑠) 𝜓 (𝑠) 𝑑𝑠 𝑓𝑜𝑟 𝑡≥0. (18) of measure of noncompactness 𝑇 𝑇≥0, and therefore, ∞ 0 is compact in 𝐶(R+, X). In fact, the inequality 𝜇𝑇(𝑈∞)≤ 𝜇𝑇(𝑈𝑛) for 𝑛 = 1, 2, . . and 𝑇≥0implies that 𝜇𝑇(𝑈∞)=0. Then, Hence, 𝑈∞ ∈ ker{𝜇𝑇}. This property of the set 𝑈∞ will be 𝑡 𝑡 very important in our further investigations. 𝜓 (𝑡) ≤𝜙1 (𝑡) + ∫ 𝜙1 (𝑠) 𝜙2 (𝑠) exp (∫ 𝜙2 (𝜏) 𝑑𝜏) 𝑑𝑠 0 𝑠 (19) −A𝑡 Definition 3. Aset{𝑒 }𝑡≥0 of bounded linear operators on 𝑓𝑜𝑟 𝑡 ≥0. X is called 𝐶0-semigroup if Next, we consider the operator 𝐹:𝐶(R+, X)→ −A0 −A(𝑡+𝑠) −A𝑡 −A𝑠 (i) 𝑒 =𝐼, 𝑒 =𝑒 𝑒 for 𝑡, 𝑠 ≥0; 𝐶(R+, X) defined by the following formula: −A𝑡 𝑥∈X R ∋𝑡󳨃→𝑒 𝑥 𝑡 (ii) for all ,thefunction + is −A(𝑡−𝑠) continuous in R+. (𝐹𝑢)(𝑡) := ∫ 𝑒 F (𝑠, 𝑢 (𝑠)) 𝑑𝑠, 𝑡∈ R+, 0 (20)

Further, denote 𝑢∈𝐶(R+, X). 󵄩 󵄩 𝑁 (𝑇) := {󵄩𝑒−A𝑡󵄩 :𝑡≤𝑇} 𝑇≥0. sup 󵄩 󵄩 for (14) Lemma 9 (see [20]). Assume that conditions (𝐻1), (𝐻2),and (𝐻4) are satisfied (see Section 3), a set 𝑈⊂𝐶(R+, X) is Definition 4. Afunction𝑥∈𝐶(R+, X) is said to be a mild bounded, 𝑇>0,afunctionℎ:[0,𝑇]→ R+ is measurable, solution of the nonlocal initial-value problem (6)-(7)iffor and every 𝑡∈R+, 2𝑁 (𝑇) 𝜒 (F (𝑠, 𝑈 (𝑠))) ≤ℎ(𝑠) (21) 𝑡 −A𝑡 −A(𝑡−𝑠) 𝑠∈[0,𝑇] 𝑢 (𝑡) =𝑒 𝑢0 + ∫ 𝑒 F (𝑠, 𝑢 (𝑠)) 𝑑𝑠. (15) for a.e. .Then, 0 𝑇 𝜔𝑇 𝐹𝑈 ≤ ∫ ℎ 𝑠 𝑑𝑠. To prove the existence results in this paper, we need the 0 ( ) ( ) (22) 0 following lemmas.

Lemma 5 (see [21]). If 𝑌 is a bounded subset of Banach space 3. Main Result ∞ X,thenforeach𝜀>0there is a sequence {𝑦𝑘} ⊂𝑌such that 𝑘=1 In this section, we give an existence result for the semilinear 𝜒 (𝑌) ≤2𝜒({𝑦}∞ )+𝜀. equation of evolution (6)-(7), and we describe the asymptotic 𝑘 𝑘=1 (16) behaviour of those solutions. 1 First, we will assume that the functions involved in (6) We call a set 𝑈⊂𝐿([0, 𝑇], X) uniformly integrable if there 1 satisfy the following conditions: exists 𝜂∈𝐿([0, 𝑇], R+) such that ‖𝑢(𝑠)‖ ≤ 𝜂(𝑠) for 𝑢∈𝑈and 𝑠∈[0,𝑇] (𝐻1) A is a linear operator acting from 𝐷(A)⊂X and A a.e. . −A(𝑡−𝑠) generates 𝐶0-semigroup {𝑒 }𝑡≥0; Lemma 6 {𝑢 }∞ ⊂𝐿1([0, 𝑇], X) (see [70]). If 𝑘 𝑘=1 is uniformly (𝐻2) the mapping F : R+ × X → X satisfies the 𝜒({𝑢 (𝑡)}∞ ) integrable, then 𝑘 𝑘=1 is measurable and Caratheodory´ condition; that is, F(⋅, 𝑢) is measurable 𝑢∈X F(𝑡, ⋅) 𝑡∈R 𝑡 ∞ 𝑡 for and is continuous for a.e. +; ∞ (𝐻 )𝑎 : R → R 𝑏:R → R 𝜒({∫ 𝑢𝑘 (𝑠) 𝑑𝑠} )≤2∫ 𝜒({𝑢𝑘 (𝑠)}𝑘=1)𝑑𝑠 𝑓𝑜𝑟𝑡∈[0, 𝑇] . 3 + + and + + are locally 0 𝑘=1 0 integrable functions such that (17) ‖F (𝑡, 𝑢)‖ ≤𝑎(𝑡) +𝑏(𝑡) ‖𝑢‖ (23) Lemma 7 (see [34]). Assume that a set 𝑈⊂𝐶(R+, X) is bounded. Then, for any 𝑢∈X and a.e. 𝑡∈R+. 4 Journal of Function Spaces and Applications

(𝐻4) There exists a locally integrable function 𝐿:R+ → In the space 𝐶(R+, X), let us consider the following set: R+ such that for any bounded 𝐷⊂X, Γ:={𝑢∈𝐶(R , X) : ‖𝑢 (𝑡)‖ ≤ P (𝑡) , 𝑡≥0} . 𝜒 (F (𝑡, 𝐷)) ≤𝐿(𝑡) 𝜒 (𝐷) (24) + for (34)

𝑡∈R for a.e. +. Obviously, the set Γ is convex and closed. Moreover, in view of (32), we have that 𝐺 is a self-mapping of Γ. Theorem 10. 𝐻 𝐻 ∗ Under assumptions ( 1)–( 4), (6) with initial Using the criterion of convergence ( )in𝐶(R+, X) and 𝑢 ∈ X condition (7) has, for every 0 , at least one mild solution standard techniques (see [31–33, 35]), we can show that the 𝑢=𝑢(𝑡) which satisfied the following estimate: operator 𝐺 is continuous on 𝐶(R+, X). {Γ } ‖𝑢 (𝑡)‖ ≤𝑀(𝑡) Now, we consider the sequence of sets 𝑛 defined by induction as follows: 𝑡 𝑡 +𝑁(𝑡) ∫ 𝑏 (𝑠) 𝑀 (𝑠) exp (∫ 𝑏 (𝜏) 𝑁 (𝜏) 𝑑𝜏) 𝑑𝑠, 0 𝑠 Γ0 := Γ, Γ𝑛+1 := Conv (𝐺Γ𝑛) for 𝑛 = 0, 1, . . . (35) (25) where Thissequenceisdecreasing,thatis,Γ𝑛 ⊃Γ𝑛+1 for 𝑛= 0, 1, 2 . . . 𝑡 . 󵄩 󵄩 𝑇>0 𝑡∈[0,𝑇] 𝑀 (𝑡) =𝑁(𝑡) (󵄩𝑢0󵄩 + ∫ 𝑎 (𝑠) 𝑑𝑠) . (26) Further, let us fix and for ,letusput 0

Proof. Consider the operator 𝐺:𝐶(R+, X)→𝐶(R+, X) 𝑔𝑛 (𝑡) := sup {𝜒 (Γ𝑛 (𝑠)) :𝑠∈[0, 𝑡]} , (36) defined by the following formula: 𝑤 := 𝜔𝑇 (Γ ). 𝑡 𝑛 0 𝑛 (37) −A𝑡 −A(𝑡−𝑠) 𝐺𝑢 (𝑡) := 𝑒 𝑢0 + ∫ 𝑒 F (𝑠, 𝑢 (𝑠)) 𝑑𝑠. (27) 0 The sequences {𝑤𝑛} and {𝑔𝑛(𝑡)} are nonincreasing for all 𝑡∈ Now, let us observe that for any continuous function 𝑢: [0, 𝑇],sotheyhavelimits R+ → X, in view of condition (𝐻3),wegetthefollowing estimate: 𝑔 (𝑡) := 𝑔 (𝑡) ,𝑡∈[0, 𝑇] , ∞ 𝑛→∞lim 𝑛 𝑡 󵄩 󵄩 (38) ‖𝐺𝑢 (𝑡)‖ ≤𝑁(𝑡) 󵄩𝑢0󵄩 +𝑁(𝑡) ∫ (𝑎 (𝑠) +𝑏(𝑠) ‖𝑢 (𝑠)‖) 𝑑𝑠, 𝑤 := 𝑤 . 0 ∞ 𝑛→∞lim 𝑛 (28) 𝑔 𝑔 which yields Moreover, each function 𝑛 is nondecreasing; therefore 𝑛, and 𝑢∞ are measurable on [0, 𝑇] for 𝑛 = 1, 2, .. 𝑡 Now, we apply the family of measures of noncompactness ‖𝐺𝑢 (𝑡)‖ ≤𝑀(𝑡) +𝑁(𝑡) ∫ 𝑏 (𝑠) ‖𝑢 (𝑠)‖ 𝑑𝑠. (29) 0 {𝜇𝑇}𝑇≥0 defined in 𝐶(R+, X) by formula (13). In view of the above notation, we have Next, consider the following integral equation: 𝑡 𝜇𝑇 (Γ𝑛) =𝑤𝑛 +𝑔𝑛 (𝑇) . (39) P (𝑡) =𝑀(𝑡) +𝑁(𝑡) ∫ 𝑏 (𝑠) P (𝑠) 𝑑𝑠. (30) 0 Solving this equation by standard methods, we get We show that 𝑡 𝑡 P (𝑡)=𝑀(𝑡)+𝑁(𝑡)∫ 𝑏 (𝑠) 𝑀 (𝑠) exp(∫ 𝑏 (𝜏) 𝑁 (𝜏) 𝑑𝜏) 𝑑𝑠. 𝑤∞ =𝑔∞ (𝑇) =0. (40) 0 𝑠 (31) To fix a number 𝑡∈[0,𝑇], 𝑛∈N andtakeanarbitrary The function P(𝑡) is continuous, nonnegative, and nonde- number 𝑠∈[0,𝑡]. creasing. Observe that the following implication is true: We know from Lemma 5 that for any 𝜀>0,thereisa ∞ sequence {𝑦𝑘}𝑘=1 ⊂𝐺Γ𝑛(𝑠),suchthat ‖𝑢 (𝑡)‖ ≤ P (𝑡) 󳨐⇒ ‖𝐺𝑢 (𝑡)‖ ≤ P (𝑡) for 𝑡≥0. (32) Indeed, linking (29)and(32), we have ∞ 𝜒(𝐺Γ𝑛 (𝑠))≤2𝜒({𝑦𝑘}𝑘=1)+𝜀. (41) 𝑡 ‖𝐺𝑢 (𝑡)‖ ≤𝑀(𝑡) +𝑁(𝑡) ∫ 𝑏 (𝑠) ‖𝑢 (𝑠)‖ 𝑑𝑠 ∞ 0 This implies that there is a sequence {𝑢𝑘}𝑘=1 ⊂Γ𝑛,suchthat 𝑡 (33) ≤𝑀(𝑡) +𝑁(𝑡) ∫ 𝑏 (𝑠) P (𝑠) 𝑑𝑠 = P (𝑡) . 𝑦 =(𝐺𝑢) (𝑠) 𝑘=1,2,.... 0 𝑘 𝑘 for (42) Journal of Function Spaces and Applications 5

Hence, in view of Lemma 6, (𝐻4),and(36), we obtain Letting 𝑛→∞,weget

∞ 𝑇 𝜒(𝐺Γ𝑛 (𝑠)) ≤ 2𝜒 ({𝑦𝑘}𝑘=1)+𝜀 𝑤∞ ≤2𝑁(𝑇) ∫ 𝐿 (𝑠) 𝑔∞ (𝑠) 𝑑𝑠. (52) 0 ∞ −A𝑠 =2𝜒({(𝐺𝑢𝑘) (𝑠)}𝑘=1)+𝜀≤2𝜒(𝑒 𝑢0) Keeping in mind (47), we deduce that 𝑠 ∞ −A(𝑠−𝜏) +2𝜒({∫ 𝑒 F (𝜏,𝑘 𝑢 (𝜏))𝑑𝜏} ) 𝑤 =0. 0 𝑘=1 ∞ (53) 𝑠 −A(𝑠−𝜏) ∞ (43) This together with39 ( )and(47) yields +𝜀≤4∫ 𝜒(𝑒 F (𝜏, {𝑢𝑘 (𝜏)}𝑘=1)) 𝑑𝜏 +𝜀 0 𝑠 lim 𝜇𝑇 (Γ𝑛)= lim (𝑤𝑛 +𝑔𝑛 (𝑇))=𝑤∞ +𝑔∞ =0. ∞ 𝑛→∞ 𝑛→∞ (54) ≤4𝑁(𝑇) ∫ 𝐿 (𝜏) 𝜒({𝑢𝑘 (𝜏)}𝑘=1)𝑑𝜏+𝜀 0 Finally, using Remark 2 for the measure 𝜇𝑇,wededuce 𝑡 ∞ that the set Γ∞ := ⋂𝑛=0 Γ𝑛 is nonempty, convex, and compact. ≤4𝑁(𝑇) ∫ 𝐿 (𝜏) 𝑔𝑛 (𝜏) 𝑑𝜏 + 𝜀. 0 Then, by the Schauder-Tychonoff theorem, we conclude that operator 𝐹:Γ∞ →Γ∞ has at least one fixed-point 𝑢=𝑢(𝑡). Since 𝜀>0is arbitrary, it follows from the above inequalities Obviously, the function 𝑢=𝑢(𝑡)is a solution of problem (6)- that (7), and, in view of the definition of the set Γ∞,theestimate ‖𝑢(𝑡)‖ ≤ P(𝑡) 𝑡 holds to be true. This completes the proof. 𝜒(𝐺Γ𝑛 (𝑠))≤4𝑁(𝑇) ∫ 𝐿 (𝜏) 𝑔𝑛 (𝜏) 𝑑𝜏, 𝑠∈ [0, 𝑡] . (44) 0 4. Local and Global Attractivity Using (36), we have Following the concepts introduced in [36], we introduce first afewdefinitionsofvariouskindsoftheconceptofattractivity 𝑔𝑛+1 (𝑡) = sup {𝜒 (𝐺Γ𝑛 (𝑠)):𝑠∈[0, 𝑡]} of mild solution of (6). 𝑡 (45) ≤4𝑁(𝑇) ∫ 𝐿 (𝜏) 𝑔𝑛 (𝜏) 𝑑𝜏, 𝑡∈ [0, 𝑇] . Definition 11. The mild solution 𝑢=𝑢(𝑡)of (6)withinitial 0 condition (7)issaidtobegloballyattractiveifforeachmild V = V(𝑡) V(0) = V Letting 𝑛→∞, we derive the following inequality: solution of (6) with initial condition 0 we have that 𝑡 𝑔 (𝑡) ≤4𝑁(𝑇) ∫ 𝐿 (𝜏) 𝑔 (𝜏) 𝑑𝜏, 𝑡∈ [0, 𝑇] . (𝑢 (𝑡) − V (𝑡)) =𝜃. ∞ ∞ (46) 𝑡→∞lim (55) 0

This inequality, together with Gronwall’s Lemma 8,implies In other words, we may say that solutions of (6)areglobally that attractive if for arbitrary solutions 𝑢(𝑡) and V(𝑡) of this equation condition (55) are satisfied. 𝑔∞ (𝑡) =0, 𝑡∈[0, 𝑇] . (47) Definition 12. We say that mild solution 𝑢=𝑢(𝑡)of (6)with Next let us notice that in view of (𝐻4) and (36), we have initial condition (7) is locally attractive if there exists a ball 𝐵(𝑢(0), 𝑟) in the space X such that for arbitrary solution V(𝑡) V(0) ∈ 𝐵(𝑢(0), 𝑟) 𝜒 (F (𝑠,𝑛 Γ (𝑠))) ≤𝐿(𝑠) 𝜒 (Γ𝑛 (𝑠)) ≤𝐿(𝑠) 𝑔𝑛 (𝑠) (48) of (1) with initial-value , condition (55)does hold. for a.e. 𝑠∈[0,𝑇],andthefunction Inthecasewhenthelimit(55)isuniformwithrespectto V(𝑡) 𝜀>0 𝑇>0 [0, 𝑇] ∋𝑠󳨃󳨀→ 2 𝑁 (𝑇) 𝐿 (𝑠) 𝑔𝑛 (𝑠) (49) all solutions ,thatis,whenforeach there exist such that 𝜔𝑇(𝐺Γ )=𝜔𝑇(𝐹Γ ) is measurable. Then, in virtue of equality 0 𝑛 0 𝑛 ‖𝑢 (𝑡) − V (𝑡)‖ ≤𝜀 and Lemma 9,weget (56)

𝑇 V(𝑡) V(0) ∈ 𝑇 for all being solutions of (6) with initial-value 𝜔0 (𝐺Γ𝑛) ≤2𝑁(𝑇) ∫ 𝐿 (𝑠) 𝑔𝑛 (𝑠) 𝑑𝑠. (50) 𝐵(𝑢(0),,andfor 𝑟) 𝑡≥𝑇,wewillsaythatsolution𝑢=𝑢(𝑡) 0 is uniformly locally attractive on R+. Hence, we derive Now, we formulate the main result of this section. We will consider (6) under the following conditions: 𝑇 𝑇 𝑤𝑛+1 =𝜔0 (Γ𝑛+1)=𝜔0 (𝐺Γ𝑛) 󸀠 (𝐻1) A is the infinitesimal generator of an exponentially 𝑇 −A𝑡 (51) stable 𝐶0-semigroup {𝑒 }𝑡≥0; that is, there exist 𝑀> ≤2𝑁(𝑇) ∫ 𝐿 (𝑠) 𝑔𝑛 (𝑠) 𝑑𝑠. 0 𝜌>0 ‖𝑒−A𝑡‖≤𝑀𝑒−𝜌𝑡 𝑡≥0 0 , such that for all ; 6 Journal of Function Spaces and Applications

(𝐻5) there exist locally integrable functions 𝑚:R+ → R+, or, equivalently,

such that 𝑡 𝜌𝑡 𝜌𝑠 ‖F (𝑡, 𝑢) − F (𝑡, V)‖ ≤𝑚(𝑡) ‖𝑢−V‖ 𝑒 𝑔 (𝑡) ≤𝑀𝑟+∫ 𝑀𝑚 (𝑠) 𝑒 𝑔 (𝑠) 𝑑𝑠. (64) (57) 0

for 𝑡≥0and 𝑢, V ∈ X. Moreover, we assume that Using again Lemma 5 for the above estimate (where ℎ(𝑡) = 𝜌𝑡 𝑡 𝜌 𝑒 𝑔(𝑡)), we obtain lim (∫ 𝑚 (𝑠) 𝑑𝑠 − 𝑡) = −∞. (58) 𝑡→∞ 0 𝑀 𝑒𝜌𝑡𝑔 (𝑡) ≤𝑀𝑟

𝐻󸀠 𝑡 𝑡 (65) Remark 13. The property ( 1) is generally satisfied in diffu- 2 𝐻󸀠 + ∫ 𝑀 𝑟𝑚 (𝑠) exp (∫ 𝑀𝑚 (𝜏) 𝑑𝜏) 𝑑𝑠. sion problem. A necessary and sufficient condition for ( 1) 0 𝑠 is presented in [72]. Elementary calculations lead to the following equality: The main result of this section is shown in the given 𝑡 𝑡 theorem below. 2 ∫ 𝑀 𝑟𝑚 (𝑠) exp (∫ 𝑀𝑚 (𝜏) 𝑑𝜏) 𝑑𝑠 󸀠 0 𝑠 Theorem 14. Under assumptions (𝐻1)and(𝐻2)–(𝐻5), prob- 𝑡 (66) lem (6)-(7) has a mild solution 𝑢=𝑢(𝑡)for each 𝑢0 ∈ X,which =𝑀𝑟(exp (∫ 𝑀𝑚 (𝑠) 𝑑𝑠) − 1). is globally attractive and locally uniformly attractive. 0

Proof. Existence of a solution 𝑢=𝑢(𝑡)is a consequence of Hence, 𝑟>0 V ∈ 𝐵(𝑢(0), 𝑟) Theorem 10.Letusfix and 0 . 𝑡 Let V = V(𝑡) denote a mild solution of (6) with the initial 𝑔 (𝑡) ≤𝑀𝑟 (−𝜌𝑡) + 𝑀𝑟 (∫ 𝑀𝑚 (𝑠) 𝑑𝑠 − 𝜌𝑡) 󸀠 exp exp 0 condition V(0) = V0.Using(𝐻1)and(𝐻3), we get (67) 󵄩 󵄩 𝑡 󵄩 󵄩 −𝑀𝑟exp (−𝜌𝑡) . 󵄩 −A𝑡 󵄩 󵄩 −A(𝑡−𝑠)󵄩 ‖V (𝑡)‖ ≤ 󵄩𝑒 V0󵄩 + ∫ 󵄩𝑒 󵄩 ⋅ ‖F (𝑠, V (𝑠))‖ 𝑑𝑠 0 Applying assumption (𝐻5), we derive 󵄩 󵄩 𝑡 ≤𝑀(󵄩𝑢 󵄩 +𝑟)+𝑀 𝑒𝜌𝑠𝑎 𝑠 𝑑𝑠 󵄩 0󵄩 ∫ ( ) (59) lim 𝑔 (𝑡) =0, 0 𝑡→∞ (68) 𝑡 𝑢(𝑡) + ∫ 𝑀𝑒𝜌𝑠𝑏 (𝑠) ‖V (𝑠)‖ 𝑑𝑠. and this proves that is locally attractive. Finally, this 0 equality together with definition of the function 𝑔(𝑡) implies that 𝑢(𝑡) is globally attractive. The proof is complete. Now, let us put 𝑡 𝑚(𝑡) ≡𝑚 󵄩 󵄩 𝜌𝑠 Remark 15. In the case when is constant, the fol- P1 (𝑡) =𝑀(󵄩V0󵄩 +𝑟)+𝑀∫ 𝑒 𝑎 (𝑠) 𝑑𝑠, lowing condition 0 (60) P (𝑡) =𝑀𝑒𝜌𝑠𝑏 (𝑡) . 𝑡 𝜌 2 lim (∫ 𝑚 (𝑠) 𝑑𝑠 − 𝑡) = −∞ (69) 𝑡→∞ 0 𝑀 Taking into account Lemma 5,weobtain 𝑡 𝑠 means that 𝑚<𝜌𝑀. Observe that this condition cannot ‖V (𝑡)‖ ≤ P1 (𝑡) + ∫ P2 (𝑠) P1 (𝑠) exp (∫ P2 (𝜏) 𝑑𝜏) 𝑑𝑠. be weakened. This observation is illustrated by the following 0 0 exampled. (61) Example 16. Let X = R, 𝑚(𝑡) ≡𝑚, F(𝑡, 𝑢) = sin 𝑡+𝑚𝑢, Further, let 𝑆 be the set of all mild solutions V(𝑡) of (6)with −A𝑡 −𝜌𝑡 𝑀=1,and𝑒 =𝑒 .Thentheequation the initial-value V0 = V(0) ∈ 𝐵(𝑢(0), 𝑟). Put 𝑡 −A𝑡 −A(𝑡−𝑠) 𝑢 (𝑡) =𝑒 𝑢0 + ∫ 𝑒 F (𝑠, 𝑢 (𝑠)) 𝑑𝑠 (70) 𝑔 (𝑡) = sup {‖𝑢 (𝑡) − V (𝑡)‖ : V ∈𝑆} . (62) 0

The estimate (61) implies that the function 𝑔(𝑡) is well defined. (for any fixed 𝑢0 ∈ R)hasthesolution𝑢(𝑡) expressed by the 󸀠 Applying (𝐻5)and(𝐻1), we get following formula: 󵄩 󵄩 󵄩 −A𝑡󵄩 󵄩 󵄩 ‖𝑢 (𝑡) − V (𝑡)‖ ≤ 󵄩𝑒 󵄩 ⋅ 󵄩𝑢0 − V0󵄩 (𝜌 − 𝑚) 𝑡− 𝑡 1 𝑢 (𝑡) = sin cos + (𝑢 + ) 𝑒(𝑚−𝜌)𝑡. 2 0 2 𝑡 󵄩 󵄩 (𝜌 − 𝑚) +1 (𝜌 − 𝑚) +1 󵄩 −A(𝑡−𝑠)󵄩 + ∫ 󵄩𝑒 󵄩 𝑚 (𝑠) ‖𝑢 (𝑠) − V (𝑠)‖ 𝑑𝑠 0 (63) (71) 𝑡 𝑚≥𝜌 𝑢(𝑡) ≤𝑀𝑒−𝜌𝑡𝑟+∫ 𝑀𝑒−𝑔(𝑡−𝑠)𝑚 (𝑠) 𝑔 (𝑠) 𝑑𝑠, Notice that for ,thesolution is neither globally 0 attractive nor locally uniformly attractive, because for each Journal of Function Spaces and Applications 7

󸀠 other solution V(𝑡) with initial condition V(0) = V0,obviously Linking the above equality with (𝐻1), we obtain described by similar formula as 𝑢(𝑡),wewouldhavea 𝑔 (𝑡) contradiction: 󵄩 󵄩 󵄨 󵄨 (𝑚−𝜌)𝑡 󵄩 𝑡 ∞ 󵄩 {󵄨𝑢0 − V0󵄨 𝑒 󳨀→ ∞ = 󵄩𝑒−A𝑡𝑢 + ∫ 𝑒−A(𝑡−𝑠)F (𝑠, 𝑢 (𝑠)) 𝑑𝑠 − ∫ 𝑒−A𝑠F 𝑑𝑠󵄩 { 󵄩 0 ∞ 󵄩 |𝑢 (𝑡) − V (𝑡)| = {or (72) 󵄩 0 0 󵄩 {󵄨 󵄨 󵄨𝑢 − V 󵄨 . 󵄩 ∞ 󵄩 {󵄨 0 0󵄨 −𝜌𝑡 󵄩 󵄩 󵄩 −A𝑠 󵄩 ≤𝑀𝑒 󵄩𝑢0󵄩 + 󵄩∫ 𝑒 F∞𝑑𝑠󵄩 󵄩 𝑡 󵄩 5. Asymptotic Behaviour 󵄩 𝑡 𝑡 󵄩 󵄩 −A(𝑡−𝑠) −A𝑠 󵄩 + 󵄩∫ 𝑒 F (𝑠, 𝑢 (𝑠)) 𝑑𝑠 − ∫ 𝑒 F∞𝑑𝑠󵄩 In this section, we will give a theorem describing asymptotic 󵄩 0 0 󵄩 behaviourofmildsolutionsof(6) with condition (7). This −𝜌𝑡 󵄩 󵄩 𝑀 theorem generalizes the result included in [72,Theorem4.4]. ≤𝑀𝑒 󵄩𝑢0󵄩 + 󵄩 󵄩 󵄩 󵄩 𝑒−𝜌𝑡𝜌 󵄩F 󵄩 First, we formulate the assumptions. 󵄩 ∞󵄩 󵄩 󵄩 󸀠 󵄩 𝑡 𝑡 󵄩 (𝐻 ) This condition is almost identical with𝐻 ( 3)andthe 󵄩 −A(𝑡−𝑠) −A(𝑡−𝑠) 󵄩 3 + 󵄩∫ 𝑒 F (𝑠, 𝑢 (𝑠)) 𝑑𝑠 − ∫ 𝑒 F∞𝑑𝑠󵄩 onlydifferenceisthatweassumethefunctions𝑎 and 󵄩 0 0 󵄩 𝑏 are locally essentially bounded on R+. 󵄩 󵄩 −𝜌𝑡 󵄩 󵄩 󵄩F∞󵄩 (𝐻󸀠) 𝑢 ∈ X ≤𝑀𝑒 (󵄩𝑢 󵄩 + ) 5 There exists ∞ such that there exists the limit 󵄩 0󵄩 𝜌 lim𝑡→∞F(𝑡,∞ 𝑢 ) and 𝑡 󵄩 󵄩 −1 + ∫ 𝑀𝑒−𝜌(𝑡−𝑠) 󵄩F (𝑠, 𝑢 (𝑠)) − F 󵄩 𝑑𝑠 lim F (𝑡,∞ 𝑢 )=A 𝑢∞. (73) 󵄩 ∞󵄩 𝑡→∞ 0 󵄩 󵄩 󵄩 󵄩 󵄩F 󵄩 Moreover, there exists a number 𝑚<𝜌/𝑀such that ≤𝑀𝑒−𝜌𝑡 (󵄩𝑢 󵄩 + 󵄩 ∞󵄩) 󵄩 0󵄩 𝜌 󵄩 󵄩 󵄩 󵄩 󵄩F (𝑡, 𝑢) − F (𝑡,∞ 𝑢 )󵄩 ≤𝑚󵄩𝑢−𝑢∞󵄩 (74) 𝑡 −𝜌𝑡 𝜌𝑠 󵄩 󵄩 +𝑀𝑒 ∫ 𝑒 󵄩F (𝑠,∞ 𝑢 )−F∞󵄩 𝑑𝑠 for 𝑡≥0and 𝑢∈X. 0

󸀠 󸀠 𝑡 𝐻 𝐻 −𝜌𝑡 𝜌𝑠 󵄩 󵄩 Remark 17. The condition ( 5) in conjunction with ( 1) +𝑀𝑒 ∫ 𝑒 󵄩F (𝑠, 𝑢 (𝑠)) − F (𝑠, 𝑢 )󵄩 𝑑𝑠. −1 󸀠 󵄩 ∞ 󵄩 ensures the existence of A (see [72]).Clearly,(𝐻3)implies 0 (𝐻3). (79) 󸀠 󸀠 Next, putting Theorem 18. Under assumptions (𝐻1), (𝐻2), (𝐻3), (𝐻4), and 󸀠 󵄩 󵄩 (𝐻5), (6) with condition (7) has a mild solution 𝑢=𝑢(𝑡)for 󵄩 󵄩 󵄩F 󵄩 𝑐 (𝑡) =𝑀𝑒−𝜌(𝑡) (󵄩𝑢 󵄩 + 󵄩 ∞󵄩) each 𝑢0 ∈ X such that lim𝑡→∞𝑢(𝑡)∞ =𝑢 . 󵄩 0󵄩 𝜌 (80) 𝑢(𝑡) 𝑡 Proof. The existence of a mild solution is guaranteed by −𝜌𝑡 𝜌𝑠 󵄩 󵄩 +𝑀𝑒 ∫ 𝑒 󵄩F (𝑠,∞ 𝑢 )−F∞󵄩 𝑑𝑠 Theorem 10.Letusput 0 󵄩 󵄩 𝑔 (𝑡) = 󵄩𝑢 (𝑡) −𝑢 󵄩 , 󸀠 󵄩 ∞󵄩 and applying (𝐻5), we derive the following inequality: (75) F = F (𝑡, 𝑢 ). 𝑡 ∞ lim ∞ −𝜌𝑡 𝜌𝑠 𝑡→∞ 𝑔 (𝑡) ≤𝑐(𝑡) +𝑚𝑀𝑒 ∫ 𝑒 𝑔 (𝑠) 𝑑𝑠. (81) 0 We show that Hence, 𝑔 (𝑡) =0. lim (76) 𝑡 𝑡→∞ 𝜌𝑡 𝜌(𝑡) 𝜌𝑠 𝑒 𝑔 (𝑡) ≤𝑒 𝑐 (𝑡) +𝑚𝑀∫ 𝑒 𝑔 (𝑠) 𝑑𝑠. (82) 󸀠 0 Recall that if assumption (𝐻1) is fulfilled, then for each 𝑧∈X we have (see [72]) The above inequality in conjunction with Lemma 5 gives ∞ 𝜌𝑡 𝜌𝑡 −A𝑠 −1 𝑒 𝑔 (𝑡) ≤𝑒 𝑐 (𝑡) ∫ 𝑒 𝑧𝑑𝑠=−A 𝑧. (77) 0 𝑡 𝜌𝑠 𝐻󸀠 + ∫ 𝑒 𝑐 (𝑠) 𝑚𝑀 ⋅ exp (𝑚𝑀 (𝑡−𝑠)) 𝑑𝑠 Using the above fact and ( 5), we get 0 −1 −1 −1 𝜌𝑡 (83) 𝑢∞ =(−A (−A 𝑢∞)) = −A F∞ =𝑒 𝑐 (𝑡) ∞ (78) 𝑡 −A𝑠 𝑚𝑀𝑡 = ∫ 𝑒 F∞𝑑𝑠. +𝑚𝑀𝑒 ∫ 𝑐 (𝑠) exp ((𝜌 − 𝑚𝑀) 𝑠) 𝑑𝑠. 0 0 8 Journal of Function Spaces and Applications

𝑛 Hence, values in R .ThevectorH will represent the (externally) 𝑡 applied magnetic field, which is a function of space and 𝑚𝑀 ∫ 𝑐 (𝑠) 𝑒(𝑔−𝑚𝑀)𝑠𝑑𝑠 A R𝑛 0 time; similarly to , it takes its values in .Thefunction 𝑔 (𝑡) ≤𝑐(𝑡) + . (84) 𝛾 𝜕Ω exp ((𝜌 − 𝑚𝑀) 𝑡) is defined and satisfies Lipschitz condition on ,and 𝛾(𝑥) ≥0 for 𝑥∈𝜕Ω. The parameters in the TDGL Before proving (76), we first show that equations are 𝜂, a (dimensionless) friction coefficient, and 𝜅, the (dimensionless) Ginzburg-Landau parameter. 𝑐 (𝑡) =0. 𝑡→∞lim (85) The order parameter should be thought of as the wave function of the center-of-mass motion of the “superelectrons” 2 To prove this equality, it is sufficient to show that the last (Cooper pairs), whose density is 𝑛𝑠 =|𝜓| and whose flux −𝜌𝑡 component in the formula expressing 𝑐(𝑡),thatis,𝑀𝑒 is J𝑠.ThevectorpotentialA determines the electromagnetic 𝑡 ∫ 𝑒𝜌𝑠‖F(𝑠, 𝑢 )−F ‖𝑑𝑠 𝑡→∞ field; E =−𝜕𝑡A −∇𝜙is the electric field and B =∇×A 0 ∞ ∞ ,tendstozeroas . 󸀠 is the magnetic induction, where J, the total current, is the To this end, let us fix 𝜀>0. Assumption (𝐻5)impliesthat sum of a “normal” current J𝑛 = E, the supercurrent J𝑠,and there exists 𝑡0 such that the transport current J𝑡 =∇×H.Thenormalcurrentobeys 󵄩 󵄩 𝜀𝜌 Ohm’s law J𝑛 =𝜎𝑛E; the “normal conductivity” coefficient 𝜎𝑛 {󵄩F (𝑠, 𝑢 )−F 󵄩 :𝑠≥𝑡}≤ . sup 󵄩 ∞ ∞󵄩 0 2𝑀 (86) is equal to one in the adopted system of units. The difference M = B−H is known as the magnetization. The trivial solution 󸀠 This inequality together with𝐻 ( 3)implies (𝜓=0, B = H, E = 0) represents the normal state, where all 󵄩 󵄩 superconducting properties have been lost. {󵄩F (𝑠, 𝑢 )−F 󵄩 :𝑠∈R}<∞. ess sup 󵄩 ∞ ∞󵄩 (87) Nowweacceptthefollowingnotion:allBanachspacesare 󸀠 real; the (real) dual of a Banach space 𝑋 is denoted by 𝑋 .The 𝑡 ∈ R 𝑡 ≥𝑡 𝑝 Now, fix 1 such that 1 and symbol 𝐿 (Ω),for1≤𝑝≤∞, denotes the usual Lebesgue 󵄩 󵄩 2 {󵄩F (𝑠, 𝑢 )−F 󵄩 :𝑠∈R} space, with norm ‖⋅‖𝐿𝑝 ; (⋅, ⋅) is the inner product in 𝐿 (Ω). ess sup 󵄩 ∞ ∞󵄩 𝑚,2 𝑊 (Ω), for nonnegative integer 𝑚,istheusualSobolev 𝑚,2 𝜀𝜌 exp (𝜌𝑡1) (88) space, with norm ‖⋅‖𝑊𝑚,2 ; 𝑊 (Ω) is a Hilbert space for the ≤ . (⋅, ⋅) (𝑢, V) = ∑ (𝜕𝛼𝑢,𝛼 𝜕 V) 2𝑀 ( (𝜌𝑡0)−1) inner product 𝑚,2 given by 𝑚,2 |𝛼|≤𝑚 exp 𝑚,2 𝑠,2 for 𝑢, V ∈𝑊 (Ω). Fractional Sobolev space 𝑊 (Ω),witha Then using (86)and(88), we conclude that for 𝑡≥𝑡1 we have fractional 𝑠, is defined by interpolation ([40, Chap. VII], and 𝐶](Ω) ] ≥0 ] =𝑚+𝜆 0≤𝜆<1 𝑡 [41, 49, 50]). ,for , with ,is −𝜌𝑡 𝜌𝑠 󵄩 󵄩 𝑚 𝑀𝑒 ∫ 𝑒 󵄩F (𝑠,∞ 𝑢 )−F∞󵄩 𝑑𝑠 the space of times continuous differentiable functions on 0 Ω;those𝑚th-order derivatives satisfy the Holder¨ condition 𝑡 with exponent 𝜆 if ] is a proper fraction; the norm ‖⋅‖𝐶] is −𝜌𝑡 0 𝜌𝑠 󵄩 󵄩 ≤𝑀𝑒 ∫ 𝑒 󵄩F (𝑠,∞ 𝑢 )−F∞󵄩 𝑑𝑠 definedintheusualway. 0 The definitions extend to the space of vector-valued 𝑡 𝜀 𝜀 functions in the standard way, with the caveat that the inner −𝜌𝑡 𝜌𝑠 󵄩 󵄩 𝑛 +𝑀𝑒 ∫ 𝑒 󵄩F (𝑠,∞ 𝑢 )−F∞󵄩 𝑑𝑠 ≤ + =𝜀. [𝐿2(Ω)] (𝑢, V)=∫ 𝑢⋅V 𝑡 2 2 product in is defined by Ω ,wherethe 0 𝑛 (89) symbol (𝑢, V) indicates the scalar product in R . Complex- valued functions are interpreted as vector-valued functions This fact proves (85). with two real components. 󸀠 Further, using (𝐻5)and(85)andemployingde Functions that vary in space and time, like the order L’Hospital’s rule for the fraction on the right-hand side parameter and the vector potential, are considered as map- of inequality (84), we obtain that condition (76) is satisfied. pings from the time domain, which is a subinterval of [0, ∞), This fact completes the proof. into spaces of complex- or vector-valued functions defined in Ω.Let𝑋=(𝑋,‖⋅‖𝑋) be a Banach space of functions Ω Ω 6. An Application to the Ginzburg-Landau defined in . Then, the functions are defined in .Then,the functions of space and time defined on Ω × (0, 𝑇),for𝑇>0, 𝑝 Equations of Superconductivity may be considered as elements of 𝐿 (0, 𝑇; 𝑋),for1≤𝑝≤∞, 𝑊2,𝑚(0, 𝑇; 𝑋) 𝑚 𝐶](0, 𝑇; 𝑋) ] ≥0 In this section, we formulate the gauged time-dependent or ,fornonnegative ,or ,for , ] =𝑚+𝜆 0≤𝜆<1 Ginzburg-Landau (TDGL) equations as an abstract evolution with . Detailed definitions can be found, equation in a Hilbert space. Moreover, we show applications for example, in [43]. (𝜓, A) of the above theorems to TDGL equations. Obviously, function spaces of ordered pairs ,where 𝑛 2 𝑛 We assume that Ω is a bounded domain in R with 𝜓:Ω → R and A :Ω → R (𝑛 = 2, 3),playanimportant 𝜕Ω 𝐶1,1 Ω role in the study of the gauged TDGL equations. We therefore boundary of class .Thatis, is an open and connected 2 𝑛 set whose boundary 𝜕Ω is a compact (𝑛−1)-manifold adapt the following special notation: H = [𝑋(Ω)] ×[𝑋(Ω)] , described by Lipschitz continuous differentiable charts. We for any Banach space for the order parameter 𝜓 and the consider two- and three-dimensional problems (𝑛=2and vector potential A,respectively.Asuitableframeworkfor 𝑛=3, resp.). Assume that the vector potential A takes its the functional analysis of the gauged TDGL equations is the Journal of Function Spaces and Applications 9

1+𝛼,2 1+𝛼,2 2 1+𝛼,2 𝑛 Cartesian product W =[𝑊 (Ω)] ×[𝑊 (Ω)] , Thefollowinganalysisisrestrictedtothecase𝜔>0and where 𝛼 ∈ (1/2, 1).Thisspaceiscontinuouslyimbeddedin the case 𝜔=0(see [73]). 1,2 ∞ 2 󸀠 W ∩ L . Let vector 𝑢:[0,∞)→ L represent the pair (𝜓, A ), 2 𝑛 Assume H ≠0 and H ∈[𝐿(Ω)] .LetAH be a minimizer 𝑢=(𝜓,A󸀠)≡(𝜓,A − A ), of the convex quadratic form 𝐽𝜔 ≡ J𝜔[𝐴], H (99) 2 2 2 and let A be the linear self-adjoint operator in L associated 𝐽𝜔 [A] = ∫ [𝜔(∇⋅A) + |∇×A − H| ] 𝑑𝑥, (90) Ω with the quadratic form 𝑄𝜔 ≡𝑄𝜔[𝑢] given by the following formula: on the domain 1 󵄨 󵄨2 󸀠 2 󵄨 󸀠󵄨2 𝑛 𝑄 [𝑢] = ∫ [ 󵄨∇𝜓󵄨 +𝜔(∇⋅A ) + 󵄨∇×A 󵄨 ]𝑑𝑥 1,2 𝜔 2 󵄨 󵄨 󵄨 󵄨 D (𝐽𝜔)={A ∈[𝑊 (Ω)] : n ⋅ A =0 on 𝜕Ω} . (91) Ω 𝜂𝜅

󸀠 𝛾 󵄨 󵄨2 A + ∫ 󵄨𝜓󵄨 𝑑𝜎 (𝑥) , We now introduce the reduced vector potential : 2 󵄨 󵄨 𝜕Ω 𝜂𝜅 󸀠 A = A − AH. (92) (100)

󸀠 In terms of 𝜓 and A the gauged TDGL equations have the on the domain following form: 1/2 D (𝑄𝜔)=D (A ) 𝜕𝜓 1 − Δ𝜓 = 𝜑 Ω×(0, ∞) , ={𝑢=(𝜓,A󸀠)∈W1,2 : n ⋅ A󸀠 =0 𝜕Ω} . 𝜕𝑡 𝜂𝜅2 in on (101) 󸀠 𝜕A 󸀠 󸀠 +∇×∇×A −𝜔∇(∇⋅A )=F in Ω×(0, ∞) The quadratic form 𝑄𝜔 is nonnegative. Furthermore, since 𝜕𝑡 (93) 󸀠 1,2 𝜔>0, 𝑄𝜔[𝜓, A ]+𝑐 ‖𝜓‖𝐿2 is coercive on W for any constant 2 n ⋅∇+𝛾𝜓=0, n ⋅ A󸀠 =0, 𝑐>0.Hence,A is positively definite in L ([44,Chap.1], equation (5,45)). If it does not lead to confusion, we use the 󸀠 n ×(∇⋅A )=0 on 𝜕Ω × (0, ∞) . same symbol A for the restriction A𝜓 and AA of A to the 2 2 2 2 respective linear subspace [𝐿 (Ω)] =[𝐿(Ω)] ×{0}(for 𝜓) 𝜑 F 𝜓 A󸀠 2 𝑛 2 𝑛 2 Here, and are nonlinear functions of and : and [𝐿 (Ω)] ≡{0}×[𝐿(Ω)] (for A)ofL . 1 2𝑖 Now, consider the initial-value problems (6)and(7)in 𝜑≡𝜑(𝑡,𝜓,A󸀠)= [− (∇𝜓) ⋅A ( 󸀠 + A ) L2 F(𝑡, 𝑢) = (𝜑, F) 𝜑 F 𝜂 𝜅 H ,where , and is given by (94)and(95) and 𝑢0 =(𝜓0, A0 − AH(0)). 1+𝜆,2 𝑖 With 𝜆 ∈ (1/2, 1) and 𝑢0 ∈ W ,wesaythat𝑢 are a − (1 − 𝜂𝜅2𝜔) 𝜓 (∇ ⋅ A󸀠) 𝜅 solution of (6)and(7)ontheinterval[0, 𝑇],forsome𝑇>0, 1+𝜆,2 if 𝑢:[0,𝑇]→ W is continuous and 󵄨 󸀠 󵄨2 󵄨 󵄨2 −𝜓󵄨A + AH󵄨 +(1−󵄨𝜓󵄨 )𝜓], 𝑡 󵄨 󵄨 󵄨 󵄨 −A(𝑡) −A(𝑡−𝑠) 𝑢 (𝑡) =𝑒 + ∫ 𝑒 F (𝑠, 𝑢 (𝑠)) 𝑑𝑠 for 𝑡∈[0, 𝑇] , 0 (94) (102)

1 2 F ≡ F (𝑡, 𝜓, A󸀠)= (𝜓∗∇𝜓 − 𝜓∇𝜓∗) in L . A mild solution of the initial-value problems (6)and 2𝑖𝑘 (𝜓, A󸀠) (95) (7) defines a weak solution of the boundary value 󵄨 󵄨2 𝜕A problem (93), which in turn defines a weak solution (𝜓, A) − 󵄨𝜓󵄨 (A󸀠 + A )− H . 󵄨 󵄨 H 𝜕𝑡 of the gauged TDGL equations, provided AH is sufficiently regular. 2 2 The equations are supplemented by initial data, which isin Namely, let us assume that X = L and 𝑢0 ∈ L ,and 𝑛 𝑛 the followimg form: 𝑢:R+ × R → R for 𝑛∈N is an unknown function, 𝑢= 𝑢(𝑡, 𝑥) 𝜓=𝜓, A = A Ω×{0} , . In order to apply Theorems 10 and 14,wearenotgoing 0 0 on (96) to consider 𝑢 as a function of 𝑡 and 𝑥 together, but rather as 2 𝜓 A amapping𝑢 of variable 𝑡 into the space X = L of functions where 0 and 0 are given, and by (92), we have 2 𝑛 𝑥,thatis,𝑢:R+ → L , 𝑢(𝑡)(𝑥) = 𝑢(𝑡, 𝑥), 𝑥∈R , 𝑡∈R+. 𝜓=𝜓, A󸀠 = A − A (0) Ω×{0} . 0 0 H on (97) 2 Remark 19. Since A is the linear self-adjoint operator in L In the next part we connect the evolution of the solution associated with the quadratic form (100), then A generates a (𝜓, A󸀠) (𝜓 , A − −A𝑡 of the system of (93) with the initial data 0 0 semigroup {𝑒 }𝑡≥0 (see [74]). AH(0)) with the dynamics of a vector 𝑢 in a Hilbert, space Below, we formulate the principal theorem of this paper. 2 𝑝 2 𝑝 𝑛 L =[𝐿 (Ω)] ×[𝐿 (Ω)] . (98) This theorem is a simple consequence of Theorem 10. 10 Journal of Function Spaces and Applications

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Research Article 𝑠 Frequency-Uniform Decomposition, Function Spaces 𝑋𝑝,𝑞,and Applications to Nonlinear Evolution Equations

Shaolei Ru1 and Jiecheng Chen2

1 Department of Mathematics, Zhejiang University, Hangzhou 310027, China 2 Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China

Correspondence should be addressed to Jiecheng Chen; [email protected]

Received 3 February 2013; Revised 30 May 2013; Accepted 30 May 2013

Academic Editor: Janusz Matkowski

Copyright © 2013 S. Ru and J. Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

𝑝 𝑞 𝑠 By combining frequency-uniform decomposition with 𝐿 (ℓ ), we introduce a new class of function spaces (denoted by 𝑋𝑝,𝑞). 𝑟 𝑠 Moreover, we study the Cauchy problem for the generalized NLS equations and Ginzburg-Landau equations in 𝐿 (0, 𝑇;𝑝,1 𝑋 ).

∞ 1. Introduction and Notation andtodefinethesequence{𝜑𝑘}0 by 𝑋𝑠 −𝑘 −𝑘+1 In this paper, we introduce a new space (denoted by 𝑝,𝑞)by 𝜑𝑘 (𝜉) =𝜑(2 𝜉) − 𝜑 (2 𝜉) , 𝑘 ∈ N; 𝑝 𝑞 combining frequency-uniform decomposition with 𝐿 (ℓ ). ∞ (2) In this new space, we discuss the semilinear estimates, 𝜑 (𝜉) =1−∑𝜑 (𝜉) =𝜓(𝜉) . 𝑋𝑠 0 𝑘 Schwartz estimates, embedding properties between 𝑝,𝑞 and 𝑘=1 𝑠 Triebel-Lizorkin space 𝐹 ,andsoforth.Moreover,westudy 𝑝,𝑞 −1 the well-posedness of NLS equations and Ginzburg-Landau Since supp 𝜑⊂{𝜉:2 ≤|𝜉|≤2},wecaneasilyseethat 𝑟 𝑠 𝑘−1 𝑘+1 equations in 𝐿 (0,𝑇;𝑋𝑝,1). supp 𝜑𝑘 ⊂{𝜉:2 ≤|𝜉|≤2 }, 𝑘∈N,andsupp𝜑0 ⊂{𝜉: In recent decades, the well-posedness of Schrodinger¨ |𝜉| ≤ 2}.Define equations and Ginzburg-Landau equations is developed −1 Δ =𝐹 𝜑 𝐹, 𝑘 ∈ N ∪ {0} , (3) quickly. In this paper, we mainly study the local Cauchy prob- 𝑘 𝑘 lem of NLS equations and Ginzburg-Landau equations in this −1 𝑠 where 𝐹 is the Fourier transform and 𝐹 is the inverse Fou- new space 𝑋𝑝,𝑞. Many people have studied these problems ∞ rier transform. {Δ 𝑘}0 is said to be Littlewood-Paley decom- in Bessel potential and Besov spaces, for example, see [1–9], position operator. Formally, we see that andsoforth.WeknowthatBesovspacesandTriebelspaces ∞ are generated by combining Littlewood-Paley decomposition 󸀠 𝑞 𝑝 𝑝 𝑞 ∑Δ =𝐼, S . with ℓ (𝐿 ) and 𝐿 (ℓ ) (one can refer to [10]fortheproperties 𝑘 in (4) 𝑘=0 of Besov and Triebel spaces.). Next, we give a brief introduc- 𝑠 𝑛 tion of these spaces (see [10, 11]). We now define the Besov space 𝐵𝑝,𝑞(R ) for −∞<𝑠<∞and To define the Littlewood-Paley decomposition, it is con- 0<𝑝,𝑞≤∞ 𝑛 by venient to consider a radial function 𝜓:R → [0, 1] such 𝑠 𝑛 󸀠 𝑛 󵄩 󵄩 that 𝐵𝑝,𝑞 (R )={𝑓∈𝑆 (R ):󵄩𝑓󵄩𝐵𝑠 <∞}, 󵄨 󵄨 𝑝,𝑞 1, 󵄨𝜉󵄨 ≤1; { 󵄨 󵄨 1/𝑞 󵄨 󵄨 ∞ (5) 𝜓 (𝜉) = {smooth,1<󵄨𝜉󵄨 <2; (1) 󵄩 󵄩 𝑘𝑠𝑞󵄩 󵄩𝑞 { 󵄨 󵄨 󵄩𝑓󵄩 𝑠 := (∑2 󵄩Δ 𝑘𝑓󵄩 ) , 󵄨 󵄨 󵄩 󵄩𝐵𝑝,𝑞 󵄩 󵄩𝑝 {0, 󵄨𝜉󵄨 ≥2, 𝑘=0 2 Journal of Function Spaces and Applications

𝑠 𝑛 and Triebel space 𝐹𝑝,𝑞(R ) for −∞ < 𝑠 < ∞ and 0<𝑝<∞, Hence, the set 0<𝑞≤∞by Υ𝑛 ={{𝜎𝑘}𝑘∈Z𝑛 :{𝜎𝑘}𝑘∈Z𝑛 satisfy (∗∗∗)} (10) 𝑠 𝑛 󸀠 𝑛 󵄩 󵄩 𝐹 (R )={𝑓∈𝑆 (R ):󵄩𝑓󵄩 𝑠 <∞}, 𝑝,𝑞 󵄩 󵄩𝐹𝑝,𝑞 is nonvoid. Let {𝜎𝑘}𝑘∈Z𝑛 ∈Υ𝑛 be a function sequence. Denote 󵄩 1/𝑞󵄩 󵄩 ∞ 󵄩 (6) 󵄩 󵄩 󵄩 𝑘𝑠𝑞󵄨 󵄨𝑞 󵄩 ◻ := 𝐹−1𝜎 𝐹, 𝑘 ∈ Z𝑛, 󵄩𝑓󵄩𝐹𝑠 := 󵄩(∑2 󵄨Δ 𝑘𝑓󵄨 ) 󵄩 . 𝑘 𝑘 (11) 𝑝,𝑞 󵄩 󵄩 󵄩 𝑘=0 󵄩𝑝 which is said to be the frequency-uniform decomposition 𝑛 Besov and Triebel spaces had been formulated during operator. For any 𝑘∈Z ,weset|𝑘| = 1|𝑘 | + ⋅⋅⋅ + |𝑘𝑛|, 1960s–1980s, which have been widely applied in recent years. ⟨𝑘⟩ = 1 + |𝑘| 𝑠∈R 0<𝑝,𝑞≤∞ 𝑠 𝑠 .Forany , , we denote Besov spaces 𝐵𝑝,𝑞 and Triebel spaces 𝐹𝑝,𝑞 are introduced by combining Littlewood-Paly decomposition operator with the { 𝑞 𝑝 𝑝 𝑞 𝑠 𝑛 󸀠 𝑛 function spaces ℓ (𝐿 ) and 𝐿 (ℓ ).SimilartoBesovspaces 𝑀𝑝,𝑞 (R )={𝑓∈𝑆 (R ): and Triebel spaces, one can use frequency-uniform decom- { ℓ𝑞(𝐿𝑝) 𝐿𝑝(ℓ𝑞) position and , to generate a new class of func- 1/𝑞 tion spaces. Actually, the spaces introduced by frequency- 𝑞 } 𝑞 𝑝 󵄩 󵄩 𝑠𝑞󵄩 󵄩 ℓ (𝐿 ) 󵄩𝑓󵄩 𝑠 =(∑ ⟨𝑘⟩ 󵄩◻𝑘𝑓󵄩 𝑝 ) <∞ . uniform decomposition and are modulation spaces. 󵄩 󵄩𝑀𝑝,𝑞 󵄩 󵄩𝐿 } 𝑛 Many people have studied the well-posedness of evolution 𝑘∈Z } equations in these spaces (e.g., see [12–16], etc.). In this (12) paper, we will consider the spaces generated by the frequency- 𝑝 𝑞 𝑠 𝑠 𝑛 uniform decomposition and 𝐿 (ℓ ).Firstly,wewillrecallthe 𝑀𝑝,𝑞 := 𝑀𝑝,𝑞(R ) is said to be a modulation space, which definition of the frequency-uniform decomposition and was first introduced by Feichtinger19 [ ]inthecase1⩽𝑝,𝑞⩽ 𝑠 𝑛 modulation spaces. In the 1930s, Wiener [17]firstintroduced ∞ (Appendix Theorem E shows the properties on 𝑀𝑝,𝑞(R )). the frequency-uniform decomposition. So, sometimes we call 0⩽𝑠<∞ 0<𝑝,𝑞⩽∞ 𝑛 For any , , we denote it Wiener decomposition of R that is roughly denoted by

−1 𝑛 𝑠 𝑛 { 󸀠 𝑛 ◻𝑘 ∼𝐹 𝜒𝑄 𝐹, 𝑘 ∈ Z , 𝑘 (7) 𝐸𝑝,𝑞 (R )={𝑓∈𝑆 (R ): { where 𝜒𝐸 denotes the characteristic function on the set 𝐸. 𝑄 𝑄 ◻ 1/𝑞 Since 𝑘 is a translation of 0, 𝑘 havethesamelocalized } 󵄩 󵄩 |𝑘|𝑠𝑞󵄩 󵄩𝑞 structures in frequency space, which are said to be the 󵄩𝑓󵄩 𝑠 =(∑ 2 󵄩◻𝑘𝑓󵄩 𝑝 ) <∞ . 󵄩 󵄩𝐸𝑝,𝑞 󵄩 󵄩𝐿 } 𝜒𝑄 𝑘∈Z𝑛 frequency-uniform decomposition operators. Since 𝑘 can } not make differential operations, one needs to redefine it by (13) smooth truncation function (see [12, 14, 18, 19]). Now we give an exact definition on modulation spaces. 𝑠 𝑠 𝑛 𝑛 𝑛 𝐸𝑝,𝑞 := 𝐸𝑝,𝑞(R ) was first introduced by Baoxiang et al. [12]. We first denote |𝜉|∞ := max𝑖=1,...,𝑛|𝜉𝑖|.Let𝜌∈𝑆(R ):R → Wenow introduce a new class of function spaces (denoted [0, 1] be a smooth radial bump function adapted to {𝜉 : 𝑠 by 𝑋𝑝,𝑞)bycombiningthefrequency-uniformdecomposi- |𝜉𝑖|∞ ⩽1}with 𝜌(𝜉) =1 if |𝜉|∞ ⩽1/2and 𝜌(𝜉) =0 if |𝜉|∞ ⩾1. 𝑝 𝑞 tion with 𝐿 (ℓ ).Forany0<𝑝<∞, 0<𝑞⩽∞, 0⩽𝑠<∞, Let 𝜌𝑘 be a translation of 𝜌: we denote 𝑛 𝜌𝑘 (𝜉) =𝜌(𝜉−𝑘) ,𝑘∈Z . (8) { 𝑛 𝑋𝑠 (R𝑛):= 𝑓∈𝑆󸀠 (R𝑛): Since 𝜌𝑘(𝜉) = 1 in the unit closed cube 𝑄𝑘 :{𝜉∈R :|𝜉−𝑘|∞ 𝑝,𝑞 { 𝑛 ⩽1/2}and {𝑄𝑘}𝑘∈Z𝑛 is a covering of R , one has that for all { 𝜉∈R𝑛 ∑ 𝜌 (𝜉) ⩾ 1 , 𝑘∈Z𝑛 𝑘 .Then,wewrite 󵄩 󵄩 󵄩 1/𝑞󵄩 󵄩 󵄩 󵄩 󵄨 󵄨𝑞 󵄩 } −1 󵄩 󵄩 󵄩 |𝑘|𝑠𝑞󵄨 󵄨 󵄩 󵄩𝑓󵄩𝑋𝑠 = 󵄩( ∑ 2 󵄨◻𝑘𝑓󵄨 ) 󵄩 <∞} . 𝑛 𝑝,𝑞 󵄩 󵄩 󵄩 𝑘∈Z𝑛 󵄩 𝑝 𝜎𝑘 (𝜉) =𝜌𝑘 (𝜉) ( ∑ 𝜌𝑘 (𝜉)) ,𝑘∈Z . (9) 󵄩 󵄩𝐿 } 𝑘∈Z𝑛 (14)

It is easy to see that Moreover, we denote 󵄨 󵄨 󵄨𝜎𝑘 (𝜉)󵄨 ≥𝑐, 𝜉∈𝑄𝑘; 1/𝑟 𝑟 𝑠 󸀠 󵄩 󵄩𝑟 󵄨 󵄨 𝐿 (R,𝑋 ):={𝑓∈S :(∫ 󵄩𝑓󵄩 𝑠 𝑑𝑡) <∞}. (15) 󵄨 󵄨 𝑝,𝑞 󵄩 󵄩𝑋𝑝,𝑞 supp 𝜎𝑘 ⊂{𝜉:󵄨𝜉−𝑘󵄨∞ ≤1}; R 𝑛 (∗∗∗) ∑ 𝜎𝑘 (𝜉) ≡1. ∀𝜉∈R ; Main Results 𝑘∈Z𝑛 Theorem 1. 𝑓(𝑢) = 𝑢|𝑢|𝑘 𝑘=2𝑚𝑚∈Z+ 󵄨 𝛼 󵄨 𝑛 𝑛 Assume , , ,and 󵄨𝐷 𝜎𝑘 (𝜉)󵄨 ≤𝐶|𝛼|,∀𝜉∈R ,𝛼∈(𝑁∪{0}) . 𝑠 󵄨 󵄨 𝑟⩾𝑘+2;thenforanyinitialdata𝑢0 ∈𝑋1,1, 0⩽𝑠<∞,there Journal of Function Spaces and Applications 3

∗ ∗ 𝑇 := 𝑇 (‖𝑢0‖ 𝑠 )>0 𝑝<∞ exists 𝑋1,1 such that the initial value prob- (3) if ,then lem 𝑠 𝑠 𝑠 𝐸𝑝,𝑝∧𝑞 ⊂𝑋𝑝,𝑞 ⊂𝐸𝑝,𝑝∨𝑞,(𝑝∧𝑞=min (𝑝, 𝑞) , 𝑡 (24) 𝑢 (𝑡) =𝑆(𝑡) 𝑢0 −𝑖∫ 𝑆 (𝑡−𝜏) 𝑓 (𝑢 (𝜏)) 𝑑𝜏, (𝑁𝐿𝑆 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛) 0 𝑝∨𝑞=max (𝑝, 𝑞)) . (16) 𝑝 𝑝+𝑎 Proof. Because ℓ ⊂ℓ , 𝑎≥0,wecangettheresultsof(1) has a unique solution directly. For (2), we know that 𝑢∈𝐿𝑟 (0, 𝑇∗;𝑋𝑠 ). loc 1,1 (17) 1/𝑞2 |𝑘|𝑠𝑞 󵄨 󵄨𝑞 ( ∑ 2 2 󵄨𝑎 󵄨 2 ) 𝑇∗ <∞ 󵄨 𝑘󵄨 Moreover, if ,then 𝑘∈Z𝑛

1/𝑞 ‖𝑢‖𝐿𝑟(0,𝑇∗;𝑋𝑠 ) =∞, (18) 2 1,1 𝑞 (25) −|𝑘|𝜀𝑞2 |𝑘|(𝑠+𝜀)𝑞2 󵄨 󵄨 2 =(∑ 2 2 󵄨𝑎𝑘󵄨 ) −1 −𝑖𝑡|𝜉|2 where 𝑆(𝑡) =𝐹 𝑒 𝐹. 𝑘∈Z𝑛 |𝑘|(𝑠+𝜀) 󵄨 󵄨 𝑘 ≲ 𝑛 2 󵄨𝑎 󵄨 , (𝜀>0) . Theorem 2. Assume 𝑓(𝑢) = (𝑏 + 𝑖𝛽)𝑢|𝑢| +𝜇𝑢, 𝑘=2𝑚, 𝑚∈ sup𝑘∈Z 󵄨 𝑘󵄨 + Z ,and𝑟⩾𝑘+2. Assume also 𝑎, 𝑏, >0 𝛼, 𝛽, 𝜇∈ R.Then,for 𝑢 ∈𝑋𝑠 1⩽𝑝<∞ 0⩽𝑠<∞ Then, letting 𝑎𝑘 =◻𝑘𝑓, we get (2). any initial data 0 𝑝,1, , , there exists |𝑘|𝑠 ∗ ∗ Finally, we prove (3). Let 𝑏𝑘 =2 ◻𝑘𝑓.Theproofcanbe 𝑇 := 𝑇 (‖𝑢0‖𝑋𝑠 )>0such that the initial value problem 𝑝,1 classified into the following two kinds of cases. (complex Ginzburg-Landau equation) 𝑞 𝑝 𝑡 Case 1 (𝑞 ≤ 𝑝). In this case, ℓ ⊂ℓ .Wehave 𝑢 (𝑡) =𝑈(𝑡) 𝑢0 −𝑖∫ 𝑈 (𝑡−𝜏) 𝑓 (𝑢 (𝜏)) 𝑑𝜏 (19) 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 0 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩𝑏𝑘󵄩ℓ𝑝(𝐿𝑝) ≤ 󵄩𝑏𝑘󵄩𝐿𝑝(ℓ𝑞) ≤ 󵄩𝑏𝑘󵄩ℓ𝑞(𝐿𝑝). (26) has a unique solution 𝑞 𝑝 Actually, noticing that ℓ ⊂ℓ,wehave‖𝑏𝑘‖ℓ𝑝(𝐿𝑝) = 𝑟 ∗ 𝑠 ‖𝑏𝑘‖𝐿𝑝(ℓ𝑝) ≤‖𝑏𝑘‖𝐿𝑝(ℓ𝑞).So,thefirstpartoftheaboveinequality 𝑢∈𝐿loc (0, 𝑇 ;𝑋𝑝,1). (20) holds. On the other hand, by Minkowski’s inequality, we have 𝑇∗ <∞ Moreover, if ,then 󵄩 󵄩1/𝑞 󵄩 ∞ 󵄩 󵄩 󵄩 󵄩 󵄨 󵄨𝑞󵄩 󵄩𝑏 󵄩 𝑝 𝑞 = 󵄩∑󵄨𝑏 󵄨 󵄩 ‖𝑢‖𝐿𝑟(0,𝑇∗;𝑋𝑠 ) =∞, 󵄩 𝑘󵄩𝐿 (ℓ ) 󵄩 󵄨 𝑘󵄨 󵄩 𝑝,1 (21) 󵄩 󵄩 󵄩𝑘=0 󵄩𝐿𝑝/𝑞 −1 −𝑡𝑃(𝜉) 2 (27) where 𝑈(𝑡) =𝐹 𝑒 𝐹, 𝑃(𝜉) = 1 + (𝑎 + 𝑖𝛼)|𝜉| . ∞ 1/𝑞 󵄩󵄨 󵄨𝑞󵄩 󵄩 󵄩 ⩽(∑󵄩󵄨𝑏 󵄨 󵄩 ) = 󵄩𝑏 󵄩 . 󵄩󵄨 𝑘󵄨 󵄩𝐿𝑝/𝑞 󵄩 𝑘󵄩ℓ𝑞(𝐿𝑝) Thispaperisorganizedasfollows.InSection 2,wewill 𝑘=0 𝑠 statesomepropertieson𝑋𝑝,𝑞, which is useful to establish the following inclusions. Section 3 is devoted to considering This proves the second part. 𝑠 the multiplication algebra of 𝑋 .Somedispersivesmooth 𝑝 𝑞 𝑝,𝑞 Case 2 (𝑝 ≤ 𝑞). By Minkowski’s inequality and ℓ ⊂ℓ,we ¨ effects for the Schrodinger and Ginzburg-Landau semigroups have will be given in Section 4.Theorems1 and 2 will be proved in 󵄩 󵄩 Section 5. 󵄩𝑏𝑘󵄩ℓ𝑞(𝐿𝑝)

𝑠 󵄩 󵄩 1/𝑝 1/𝑝 󵄩 󵄨 󵄨𝑝 󵄩 󵄩󵄨 󵄨𝑝󵄩 2. The Properties on 𝑋𝑝,𝑞 =(󵄩∫ 󵄨𝑏 󵄨 𝑑𝑥󵄩 ) ⩽(∫ 󵄩󵄨𝑏 󵄨 󵄩 𝑑𝑥) 󵄩 󵄨 𝑘󵄨 󵄩 󵄩󵄨 𝑘󵄨 󵄩ℓ𝑞/𝑝 (28) 󵄩 R𝑛 󵄩ℓ𝑞/𝑝 R𝑛 𝑋𝑠 In order to study the Cauchy problem in 𝑝,𝑞,wefirstgive 󵄩 󵄩 󵄩 󵄩 𝑠 = 󵄩𝑏𝑘󵄩𝐿𝑝(ℓ𝑞) ⩽ 󵄩𝑏𝑘󵄩ℓ𝑝(𝐿𝑝). some properties on 𝑋𝑝,𝑞 as follows. This finishes the proof of this proposition. Proposition 3. Letting 0⩽𝑠<∞, 0<𝑝<∞, 0<𝑞⩽∞, one has Proposition 4 (completeness). For any 0⩽𝑠<∞, 0<𝑝< ∞ 0<𝑞⩽∞ (1) if 𝑞1 ≤𝑞2,then , , one has the following: 𝑠 𝑠 𝑠 𝑋 ⊂𝑋 ; (1) 𝑋𝑝,𝑞 is a quasi-Banach space. Moreover, if 1⩽𝑝<∞, 𝑝,𝑞1 𝑝,𝑞2 (22) 𝑠 1⩽𝑞⩽∞,then𝑋𝑝,𝑞 is a Banach space; 𝜀>0 𝑛 0 󸀠 𝑛 (2) if ,then (2) 𝑆(R )⊂𝑋𝑝,𝑞 ⊂𝑆(R ); 𝑋𝑠+𝜀 ⊂𝑋𝑠 ; 0<𝑝,𝑞<∞ S(R𝑛) 𝑋0 𝑝,𝑞1 𝑝,𝑞2 (23) (3) if ,then is dense in 𝑝,𝑞. 4 Journal of Function Spaces and Applications

Proof For the sake of convenience, we denote

𝑛 0 󸀠 𝑛 Step 1. In this step, we prove that 𝑆(R )⊂𝑋 ⊂𝑆(R ). 𝜎 −1 𝜙 −1 𝑝,𝑞 ◻𝑘 =𝐹 𝜎𝑘𝐹, ◻𝑘 =𝐹 𝜙𝑘𝐹. (32) 𝑛 0 󸀠 𝑛 𝑠 𝑠 Actually, by 𝑆(R )⊂𝐸𝑝,𝑞 ⊂𝑆(R ) [12]and𝐸𝑝,𝑝∧𝑞 ⊂𝑋𝑝,𝑞 ⊂ 𝑠 𝐸𝑝,𝑝∨𝑞 (Proposition 3), we immediately obtain the result. Then, we have

𝑋𝑠 𝜎 𝜎 𝜙 Step 2. 𝑝,𝑞 is a quasinormed space. We prove the complete- ◻𝑘 = ∑ ◻𝑘 ◻𝑘+ℓ, ∞ 𝑠 |ℓ| ≤1 ness. Let {𝑓ℓ}ℓ=1 be a Cauchy sequence in 𝑋𝑝,𝑞 (with respect to ∞ 𝑋𝑠 {𝑓 }∞ 󵄩 󵄩 afixedquasinormin 𝑝,𝑞). Proposition 3 shows that ℓ ℓ=1 is 󵄩 󵄩 (33) 𝑠 𝑠 󵄩 󵄩 󵄩 𝜙 󵄩 𝐸 𝐸 󵄩◻𝜎𝑓󵄩 ≤ 󵄩 ∑ ◻𝜎 (◻ 𝑓)󵄩 . also a Cauchy sequence in 𝑝,𝑝∨𝑞.Because 𝑝,𝑝∨𝑞 is a complete 󵄩 𝑘 󵄩𝐿𝑝(ℓ𝑞) 󵄩 𝑘 𝑘+ℓ 󵄩 󵄩 󵄩 topological linear space, we can find a limit element 𝑓∈ 󵄩|ℓ|∞≤1 󵄩𝐿𝑝(ℓ𝑞) 𝑠 −1 −1 𝑠 𝐸𝑝,𝑝∨𝑞.Then,𝐹 𝜎𝑘𝐹𝑓ℓ converges to 𝐹 𝜎𝑘𝐹𝑓 in 𝐸𝑝,𝑝∨𝑞 if ℓ→ −1 ∞ ∞. On the other hand, {𝐹 𝜎𝑘𝐹𝑓ℓ}ℓ=1 is a Cauchy sequence By Appendix Theorem F, we have 𝑝 𝑛 𝑠 𝑠 in 𝐿 (R ) (𝑋𝑝,𝑞 ⊂𝐸𝑝,𝑝∨𝑞, 𝑠⩾0.). By Appendix Theorem B, 󵄩 𝜙 󵄩 󵄩 󵄩 󵄩 𝜙 󵄩 𝐿∞(R𝑛) 󵄩◻𝜎 (◻ 𝑓)󵄩 ≲ 󵄩𝜎 󵄩 󵄩◻ 𝑓󵄩 , it is also a Cauchy sequence in .Thisshowsthatthe 󵄩 𝑘 𝑘+ℓ 󵄩 𝑝 𝑞 sup𝑘󵄩 𝑘󵄩𝐻𝑠 󵄩 𝑘+ℓ 󵄩 𝑝 𝑞 −1 ∞ 𝑝 𝑛 󵄩 󵄩𝐿 (ℓ ) 󵄩 󵄩𝐿 (ℓ ) limiting element of {𝐹 𝜎𝑘𝐹𝑓ℓ}ℓ=1 in 𝐿 (R ) (which is the ∞ 𝑛 −1 1 1 (34) same as in 𝐿 (R )) coincides with {𝐹 𝜎𝑘𝐹𝑓 } . Now it follows 𝑠 𝑠>𝑛( − ). by standard arguments that 𝑓 belongs to 𝑋𝑝,𝑞 and that 𝑓ℓ 1∧𝑝 2 𝑠 𝑠 converges in 𝑋𝑝,𝑞 to 𝑓.Hence,𝑋𝑝,𝑞 is complete. Then, by ‖𝜎𝑘‖𝐻𝑠 <𝐶,wehave S(R𝑛) 𝑋0 (R𝑛) 0<𝑝<∞ Step 3. We prove that is dense in 𝑝,𝑞 if 󵄩 󵄩 0 󵄩 𝜎 󵄩 󵄩 𝜙 󵄩 0<𝑞<∞ 𝑓∈𝑋 󵄩◻ 𝑓󵄩 𝑝 𝑞 ≲ ∑ 󵄩◻ 𝑓󵄩 . and .Let 𝑝,𝑞;thenweput 󵄩 𝑘 󵄩𝐿 (ℓ ) 󵄩 𝑘+ℓ 󵄩𝐿𝑝(ℓ𝑞) (35) |ℓ|∞≤1 𝑁 𝑓 (𝑥) = ∑𝐹−1𝜎 𝐹𝑓. 𝑁 𝑘 (29) {𝜎𝑘} {𝜙𝑘} 𝑘=0 By the above discussion, we get ‖𝑓‖ 0 ≲‖𝑓‖ 0 . 𝑋𝑝,𝑞 𝑋𝑝,𝑞 0 𝑛 Of course, 𝑓𝑁 ∈𝑋𝑝,𝑞(R ). Consequently (by Appendix Theo- rem F), The inverse inequality can be proved similarly. Based on 󵄩 󵄩 {𝜎𝑘} 󵄩𝑓−𝑓𝑁󵄩 0 𝑛 the above observations, one can also obtain that ‖𝑓‖ 𝑠 ∼ 𝑋𝑝,𝑞(R ) 𝑋𝑝,𝑞 {𝜙𝑘} 󵄩 1/𝑞󵄩 ‖𝑓‖𝑋𝑠 . 󵄩 ∞ 1 󵄩 𝑝,𝑞 󵄩 󵄨 −1 󵄨𝑞 󵄩 ⩽𝑐󵄩( ∑ ∑ 󵄨𝐹 𝜎 𝜎 𝐹𝑓 󵄨 ) 󵄩 󵄩 󵄨 𝑘 𝑘+𝑟 󵄨 󵄩 Theorem 6. 1⩽𝑝 ⩽𝑝 <∞ 1⩽𝑞⩽∞ 0⩽𝑠< 󵄩 𝑘=𝑁 𝑟=−1 󵄩𝐿𝑝(R𝑛) (30) Assume 2 1 , , ∞; then one has 󵄩 󵄩 󵄩 ∞ 1/𝑞󵄩 󵄩 󵄨 −1 󵄨𝑞 󵄩 ⩽𝑐󵄩( ∑ 󵄨𝐹 𝜎𝑘𝐹𝑓 󵄨 ) 󵄩 . ‖𝑢‖𝑋𝑠 ≲ ‖𝑢‖𝑋𝑠 . 󵄩 󵄨 󵄨 󵄩 𝑝1,𝑞 𝑝2,𝑞 (36) 󵄩 𝑘=𝑁 󵄩𝐿𝑝(R𝑛) 𝑠 Lebesgue’s bounded convergence theorem proves that the Proof. By the definition of 𝑋𝑝,𝑞,wehave right-hand side of the above inequality tends to zero if 𝑁→ ∞ 𝑓 𝑓 𝑋0 (R𝑛) 𝜑∈S .Hence, 𝑁 approximates in 𝑝,𝑞 . Next, we let ‖𝑢‖𝑋𝑠 𝑝1,𝑞 with 𝜑(0) = 1 and supp 𝐹𝜑 ⊂ {𝑦 : |𝑦| ⩽1}.Let𝑓𝛿(𝑥) 𝑛 󵄩 1/𝑞󵄩 = 𝜑(𝛿𝑥)𝑓(𝑥) with 0<𝛿<1.Then,(𝑓𝑁)𝛿 ∈ S(R ) approx- 󵄩 󵄩 Ω 𝑝 󵄩 |𝑖|𝑞𝑠󵄨 󵄨𝑞 󵄩 𝑓 𝐿 := {𝑓 ∈ 𝐿 : 𝑓⊂Ω}̂ Ω={𝑦: = 󵄩( ∑ 2 󵄨◻ 𝑢󵄨 ) 󵄩 imates 𝑁 in 𝑝 supp with 󵄩 󵄨 𝑖 󵄨 󵄩 𝑁+2 󵄩 𝑖∈Z𝑛 󵄩 𝑝 |𝑦| ⩽ 2 } if 𝛿→0. However this is also an approximation 󵄩 󵄩𝐿 1 0 𝑛 𝑛 of 𝑓𝑁 in 𝑋 (R ).ThisprovesthatS(R ) is dense in 󵄩 󵄨 󵄨𝑞 1/𝑞󵄩 𝑝,𝑞 󵄩 󵄨 󵄨 󵄩 0 𝑛 󵄩 |𝑖|𝑞𝑠󵄨 󵄨 󵄩 𝑋 (R ). = 󵄩( ∑ 2 󵄨◻ ∑ ◻ 𝑢󵄨 ) 󵄩 𝑝,𝑞 󵄩 󵄨 𝑖 𝑖+ℓ 󵄨 󵄩 󵄩 𝑖∈Z𝑛 󵄨 |ℓ| ⩽1 󵄨 󵄩 󵄩 󵄨 ∞ 󵄨 󵄩𝐿𝑝1 Proposition 5 (equivalent norm). Assume {𝜎𝑘}𝑘∈Z𝑛 , {𝜙𝑘}𝑘∈Z𝑛 󵄩 󵄨 󵄨𝑞 1/𝑞󵄩 ∈Υ𝑛, 0<𝑝<∞, 0<𝑞⩽∞.Then,{𝜎𝑘}𝑘∈Z𝑛 and {𝜙𝑘}𝑘∈Z𝑛 󵄩 󵄨 󵄨 󵄩 𝑠 󵄩 |𝑖|𝑞𝑠󵄨 ∨ 󵄨 󵄩 𝑋 = 󵄩( ∑ 2 󵄨𝜎 ∗( ∑ ◻ 𝑢)󵄨 ) 󵄩 generate equivalent norms on 𝑝,𝑞. 󵄩 󵄨 𝑖 𝑖+ℓ 󵄨 󵄩 󵄩 𝑖∈Z𝑛 󵄨 |ℓ| ⩽1 󵄨 󵄩 󵄩 󵄨 ∞ 󵄨 󵄩𝐿𝑝1 Proof. Firstly, we have the following translation equality: 󵄩 󵄩 󵄩𝑞 1/𝑞󵄩 −1 𝑖𝑥𝑘 −1 −𝑖𝑘𝑦 󵄩 󵄩 󵄩 󵄩 (𝐹 𝑚𝐹𝑓) (𝑥) =𝑒 [𝐹 𝑚 (⋅+𝑘) 𝐹(𝑒 𝑓(𝑦))](𝑥) . 󵄩 |𝑖|𝑞𝑠󵄩 ∨ 󵄩 󵄩 ⩽ 󵄩( ∑ 2 󵄩𝜎 |∗| ( ∑ ◻ 𝑢)󵄩 ) 󵄩 󵄩 󵄩 𝑖 𝑖+ℓ 󵄩 󵄩 (31) 󵄩 𝑖∈Z𝑛 󵄩 |ℓ| ⩽1 󵄩 󵄩 󵄩 󵄩 ∞ 󵄩 󵄩𝐿𝑝1 Journal of Function Spaces and Applications 5

󵄩 󵄩 󵄩𝑞 1/𝑞󵄩 𝑘−1 𝑘+1 󵄩 󵄩 󵄩 󵄩 Proof. Let 𝑎𝑘 = max(0, 2 − √𝑛), 𝑏𝑘 =2 + √𝑛.Wecan 󵄩 |𝑖|𝑞𝑠󵄩 ∨ 󵄩 󵄩 = 󵄩( ∑ 2 󵄩𝜎 |∗| ( ∑ ◻ 𝑢)󵄩 ) 󵄩 |𝑖|∈[𝑎 ,𝑏 ] Δ ◻ 𝑓=0 𝑝>0 󵄩 󵄩 0 𝑖+ℓ 󵄩 󵄩 easily find that if 𝑘 𝑘 ,then 𝑘 𝑖 .For , 󵄩 𝑖∈Z𝑛 󵄩 |ℓ| ⩽1 󵄩 󵄩 𝑞⩽1 󵄩 󵄩 ∞ 󵄩 󵄩𝐿𝑝1 , by Appendix Theorem F, we have 󵄩 󵄩 󵄩 1/𝑞󵄩 󵄩Δ 𝑓󵄩 󵄩 󵄨 󵄨𝑞 󵄩 󵄩 𝑘 󵄩𝐿𝑝(ℓ𝑞) 󵄩 󵄨 󵄨 󵄩 󵄩 |𝑖|𝑞𝑠󵄨 󵄨 󵄩 ≲ 󵄩( ∑ 2 󵄨( ∑ ◻𝑖+ℓ𝑢)󵄨 ) 󵄩 󵄩 1/𝑞󵄩 󵄩 𝑛 󵄨 󵄨 󵄩 󵄩 󵄩 󵄩 𝑖∈Z 󵄨 |ℓ|∞⩽1 󵄨 󵄩 󵄩 󵄨 󵄨𝑞 󵄩 󵄩 󵄩𝐿𝑝2 = 󵄩(∑󵄨Δ 𝑓󵄨 ) 󵄩 󵄩 󵄨 𝑘 󵄨 󵄩 󵄩 󵄩 󵄩 𝑘 󵄩 𝑝 󵄩 1/𝑞󵄩 𝐿 󵄩 |𝑖|𝑞𝑠󵄨 󵄨𝑞 󵄩 ≲ 󵄩( ∑ 2 󵄨◻ 𝑢󵄨 ) 󵄩 . 󵄩 󵄨 󵄨𝑞 1/𝑞󵄩 󵄩 󵄨 𝑖 󵄨 󵄩 󵄩 󵄨 󵄨 󵄩 󵄩 𝑖∈Z𝑛 󵄩 𝑝 󵄩 󵄨 󵄨 󵄩 󵄩 󵄩𝐿 2 ⩽ 󵄩(∑󵄨 ∑ Δ ◻ ◻ 𝑓󵄨 ) 󵄩 󵄩 󵄨 𝑘 𝑖 𝑖 󵄨 󵄩 󵄩 𝑘 󵄨𝑖∈Z𝑛,|𝑖|∈[𝑎 ,𝑏 ] 󵄨 󵄩 (37) 󵄩 󵄨 𝑘 𝑘 󵄨 󵄩𝐿𝑝 󵄩 󵄩 ‖|𝜎∨|∗𝑓‖ ⩽‖𝑓‖ 𝑝 ⩾𝑝 󵄩 1/𝑞󵄩 In this proof, we used 0 𝐿𝑝1 𝐿𝑝2 ( 1 2, 󵄩 󵄨 󵄨𝑞 󵄩 (42) ≲ 󵄩(∑ ∑ 󵄨𝐹−1 (𝜑 𝜎 )𝐹◻𝑓󵄨 ) 󵄩 Young’s inequality) and Appendix Theorem G. 󵄩 󵄨 𝑘 𝑖 𝑖 󵄨 󵄩 󵄩 𝑘 𝑖∈Z𝑛,|𝑖|∈[𝑎 ,𝑏 ] 󵄩 󵄩 𝑘 𝑘 󵄩𝐿𝑝 2 Theorem 7. Assume 𝑓∈𝐿; then one has 󵄩 1/𝑞󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄨 −1 󵄨𝑞 󵄩 󵄩𝑓󵄩 ∼ 󵄩𝑓󵄩 . ≲ 󵄩( ∑ ∑ 󵄨𝐹 (𝜑 𝜎 )𝐹◻𝑓󵄨 ) 󵄩 󵄩 󵄩𝐿2 󵄩 󵄩𝑋0 (38) 󵄩 󵄨 𝑘 𝑖 𝑖 󵄨 󵄩 2,2 󵄩 𝑖∈Z𝑛𝑘,|𝑖|∈[𝑎 ,𝑏 ] 󵄩 󵄩 𝑘 𝑘 󵄩𝐿𝑝

Proof. We first prove ‖𝑓‖𝐿2 ≲‖𝑓‖𝑋0 . Consider the following: 󵄩 1/𝑞󵄩 2,2 󵄩 󵄩 󵄩 󵄨 󵄨𝑞 󵄩 ≲ 󵄩( ∑ 󵄨◻𝑖𝑓󵄨 ) 󵄩 . 󵄨 󵄨2 1/2 󵄩 𝑛 󵄩 󵄨 󵄨 󵄩 𝑖∈Z 󵄩𝐿𝑝 󵄩 󵄩 󵄨 󵄨 󵄩𝑓󵄩 2 =(∫ 󵄨 ∑ ◻𝑖𝑓󵄨 𝑑𝑥) 𝐿 𝑛 󵄨 󵄨 R 󵄨𝑖∈Z𝑛 󵄨 This proves the theorem. 𝑠 𝑛 󵄨 󵄨2 1/2 Theorem 10. 𝐹 (R ) 󵄨 󵄨 Let 𝑝,𝑞 be Triebel space; then one has 󵄨 󵄨 =(∫ 󵄨 ∑ 𝜎𝑖𝐹𝑓 󵄨 𝑑𝑥) 𝑛 󵄨 󵄨 𝑠 𝑛 0 𝑛 1 1 R 󵄨𝑖∈Z𝑛 󵄨 𝐹 (R )⊂𝑋 (R ), 𝑠>𝑛( − ), 2,𝑞 2,𝑞 𝑞 2 1/2 󵄨 󵄨2 0<𝑞<2, ≲(∫ ∑ 󵄨𝜎𝑖𝐹𝑓 󵄨 𝑑𝑥) 𝑛 R 𝑛 (43) 𝑖∈Z 1 1 𝑋0 (R𝑛)⊂𝐹𝑠 (R𝑛), 𝑠<𝑛( − ), 1/2 2,𝑞 2,𝑞 𝑞 2 󵄨 −1 󵄨2 =(∫ ∑ 󵄨𝐹 𝜎𝑖𝐹𝑓 󵄨 𝑑𝑥) 𝑛 󵄨 󵄨 R 𝑖∈Z𝑛 2<𝑞⩽∞.

1/2 1/2 2 Proof. By Appendix Theorem A, we know that 󵄨 󵄨2 󵄩 󵄩 =(∫ (( ∑ 󵄨𝐹−1𝜎 𝐹𝑓 󵄨 ) ) 𝑑𝑥) = 󵄩𝑓󵄩 . 󵄨 𝑖 󵄨 󵄩 󵄩𝑋0 𝑠 0 1 1 R𝑛 2,2 𝐻 ⊂𝐸 ;𝑠>𝑛(− ), 0<𝑞<2. 𝑖∈Z𝑛 2,𝑞 𝑞 2 (44) (39) 𝑠 𝑠 Then, by the embedding between 𝐸𝑝,𝑞 and 𝑋𝑝,𝑞 The inverse inequality can be proved similarly (Proposition 3), we have (for any given 𝜀>0) 2 2 (∑ 𝑛 |𝜎𝑖𝐹𝑓 | ≲|∑ 𝑛 𝜎𝑖𝐹𝑓 | ). Moreover, by the similar 𝑖∈Z 𝑖∈Z 𝐹𝑛(1/𝑞−1/2)+𝜀 ⊂𝐹𝑛(1/𝑞−1/2)+𝜀 discussion as mentioned previously, we can also obtain that 2,𝑞 2,2 −1 2 𝑠/2 2 1/2 (45) ‖(∑ 𝑛 |𝐹 (1 + |𝜉| ) 𝜎𝑘𝐹𝑓 | ) ‖ 2 ∼‖𝑓‖ 𝑠 . 𝑛(1/𝑞−1/2)+𝜀 0 0 𝑘∈Z 𝐿 𝐻2 =𝐻 ⊂𝐸2,𝑞 ⊂𝑋2,𝑞. Theorem 8. Let 0<𝑠<∞, 0<𝑝<∞, 0<𝑞⩽∞.Then, 𝑠 𝑠 This proves the first result (𝐹2,2 ∼𝐻,[11]). one has On the other hand, by Appendix Theorem A, we have 𝑋𝑠 (R𝑛)⊂𝐶∞ (R𝑛). 1 1 𝑝,𝑞 (40) 𝐸0 ⊂𝐻𝑠;𝑠<𝑛(− ), 2<𝑞⩽∞. 2,𝑞 𝑞 2 (46) Proof. One can obtain the result directly by Appendix Theo- 𝐸𝑠 𝑋𝑠 remDandProposition 3. Then, by the embedding between 𝑝,𝑞 and 𝑝,𝑞 (Proposition 3), we have Theorem 9. 0<𝑝<∞ 0<𝑞⩽1 Let , .Onehas 0 0 𝑛(1/𝑞−1/2)−𝜀 𝑛(1/𝑞−1/2)−𝜀 𝑋2,𝑞 ⊂𝐸2,𝑞 ⊂𝐻 ⊂𝐹2,𝑞 . (47) 𝑋0 (R𝑛)⊂𝐹0 (R𝑛). 𝑝,𝑞 𝑝,𝑞 (41) This implies the theorem. 6 Journal of Function Spaces and Applications

Theorem 11. 𝑠 > 𝑛(1/𝑞 − 1/2) 0<𝑞<2 (1/𝑞−1/2) Let , .Then,onehas −2𝑠𝑞/(2−𝑞) 󵄩 󵄩 ≲(∑ ∑ (1+𝑖) ) 󵄩𝑓󵄩𝐻𝑠 󵄩 󵄩 󵄩 󵄩 (1/𝑞−1/2)󵄩 󵄩 𝑖>𝑇|𝑘|=𝑖 󵄩𝑓󵄩 0 ⩽𝐶𝑠 (1+[ln (1+󵄩𝑓󵄩 𝑠 )] 󵄩𝑓󵄩 𝑛(1/𝑞−1/2) ) . 󵄩 󵄩𝑋2,𝑞 󵄩 󵄩𝐻 󵄩 󵄩𝐻 (1/𝑞−1/2) (48) −2𝑠𝑞/(2−𝑞)+𝑛−1 󵄩 󵄩 ≲(∑(1+𝑖) ) 󵄩𝑓󵄩𝐻𝑠 𝑠 𝑖>𝑇 Proof. By the definition of 𝑋𝑝,𝑞,wehave 󵄩 󵄩 ⩽𝐶𝑇−𝑠+𝑛(1/𝑞−1/2)󵄩𝑓󵄩 . 󵄩 1/𝑞󵄩 𝑠 󵄩 󵄩𝐻𝑠 󵄩 󵄩 󵄩 󵄩 󵄩 󵄨 󵄨𝑞 󵄩 (51) 󵄩𝑓󵄩𝑋0 = 󵄩( ∑ 󵄨◻𝑘𝑓󵄨 ) 󵄩 2,𝑞 󵄩 󵄩 󵄩 𝑘∈Z𝑛 󵄩 2 𝐿 Let 󵄩 1/𝑞󵄩 󵄩 󵄩 󵄩 󵄩1/(𝑠−𝑛(1/𝑞−1/2)) 󵄩 󵄨 󵄨𝑞 󵄩 𝑇= (1, 󵄩𝑓󵄩 ). ≲ 󵄩( ∑ 󵄨◻ 𝑓󵄨 ) 󵄩 max 󵄩 󵄩𝐻𝑠 (52) 󵄩 󵄨 𝑘 󵄨 󵄩 󵄩 |𝑘|⩽𝑇 󵄩𝐿2 (49) 󵄩 󵄩 From the above discussion, we get the result. 󵄩 1 󵄩 󵄩 󵄩 󵄩 𝑞 󵄩 󵄩 󵄨 󵄨𝑞 󵄩 Remark 12. By Theorem 9,wehave + 󵄩( ∑ 󵄨◻𝑘𝑓󵄨 ) 󵄩 =𝐴+𝐵. 󵄩 󵄩 󵄩 |𝑘|>𝑇 󵄩 𝑋0 (R𝑛)⊂𝐹0 (R𝑛). 󵄩 󵄩𝐿2 𝑝,1 𝑝,1 (53)

Then, by Appendix Theorem F, Theorem 7 and Holder’s¨ Then, by Theorem 11,wehave 𝐴 inequality, the term canbedominatedby 𝑛 𝐻𝑠 (R𝑛)⊂𝑋0 (R𝑛)⊂𝐹0 (R𝑛), 𝑠> . 󵄩 2,1 2,1 (54) 󵄩 󵄨 𝑛(1/𝑞−1/2)/2 2 󵄩 󵄨 2 󵄩 󵄨 −1 (1+|𝑘|) 𝐴=󵄩( ∑ 󵄨𝐹 ( ) 0 𝑠 󵄩 󵄨 󵄨 󵄨2 𝑋 𝐻 (𝑠 > 𝑛/2) 󵄩 󵄨 1+󵄨𝜉󵄨 So, 2,1 is an intermediate space between and 󵄩 |𝑘|⩽𝑇 󵄨 󵄨 󵄨 0 𝐹2,1. 𝑞 1/𝑞󵄩 2 𝑛(1/𝑞−1/2)/2 󵄨 󵄩 1+|𝜉| 󵄨 󵄩 𝑠 ×( ) 𝜎 𝐹𝑓 󵄨 ) 󵄩 𝑋 2 𝑘 󵄨 󵄩 3. Multilinear Estimate in 𝑝,𝑞 (1 + |𝑘|) 󵄨 󵄩 󵄩𝐿2 𝑠 It is well known that 𝐵𝑝,𝑞 is a multiplication algebra if 𝑠>𝑛/𝑝, 󵄩 󵄨 󵄨𝑞 1/𝑞󵄩 𝑠 󵄩 󵄨 󵄨 󵄨2 𝑛(1/𝑞−1/2)/2 󵄨 󵄩 𝑋 󵄩 󵄨 1+󵄨𝜉󵄨 󵄨 󵄩 see [2]. The regularity indices, for which 𝑝,𝑞 constitutes a ≲ 󵄩( ∑ 󵄨𝐹−1( 󵄨 󵄨 ) 𝜎 𝐹𝑓 󵄨 ) 󵄩 󵄩 󵄨 2 𝑘 󵄨 󵄩 multiplication algebra, are quite different from those of Besov 󵄩 |𝑘|⩽𝑇󵄨 (1+|𝑘|) 󵄨 󵄩 󵄩 󵄨 󵄨 󵄩𝐿2 space. (1/𝑞−1/2) Theorem 13. Let 0⩽𝑠<∞, 0<𝑝⩽𝑝1, 𝑝2 <∞, 0<𝑞⩽ −𝑛 ≲(∑ (1 + |𝑘|) ) ∞.If1/𝑝 = 1/𝑝1 +1/𝑝2, then one has |𝑘|⩽𝑇 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩𝑓𝑔 󵄩 ≲𝐶2𝐶𝑠󵄩𝑓󵄩 󵄩𝑔󵄩 , 󵄩 1/2󵄩 󵄩 󵄩𝑋𝑠 󵄩 󵄩𝑋𝑠 󵄩 󵄩𝑋𝑠 (55) 󵄩 󵄨 󵄨2 󵄩 𝑝,𝑞 𝑝1,𝑞∧1 𝑝2,𝑞∧1 󵄩 󵄨 −1 2 𝑛(1/𝑞−1/2)/2 󵄨 󵄩 × 󵄩( ∑ 󵄨𝐹 (1 + |𝜉| ) 𝜎𝑘𝐹𝑓 󵄨 ) 󵄩 󵄩 󵄨 󵄨 󵄩 where 𝐶 is independent of 𝑠, 𝑞. 󵄩 |𝑘|⩽𝑇 󵄩𝐿2 𝑛 (1/𝑞−1/2) Proof. Recall that we denote 𝑘∈Z and |𝑘| = 1|𝑘 |+⋅⋅⋅+|𝑘𝑛|. −1 󵄩 󵄩 ≲( ∑ (1+𝑖) ) 󵄩𝑓󵄩𝐻𝑛(1/𝑞−1/2) We have 0⩽𝑖⩽𝑇 󵄩 󵄩 󵄩 1/𝑞󵄩 󵄩 󵄩 󵄩 󵄨 󵄨𝑞 󵄩 (1/𝑞−1/2)󵄩 󵄩 󵄩 󵄩 󵄩 |𝑘|𝑠𝑞󵄨 󵄨 󵄩 ≲ [ (1+𝑇)] 󵄩𝑓󵄩 . 󵄩𝑓𝑔 󵄩𝑋𝑠 = 󵄩( ∑ 2 󵄨◻𝑘 (𝑓𝑔 ) 󵄨 ) 󵄩 . (56) ln 󵄩 󵄩𝐻(𝑛(1/𝑞−1/2)) 𝑝,𝑞 󵄩 󵄩 󵄩 𝑘∈Z𝑛 󵄩 𝑝 (50) 󵄩 󵄩𝐿

By ∑ 𝑛 𝜎𝑘 =1, we can obtain the decomposition of 𝑓𝑔 as On the other hand, by the similar discussion as the above, we 𝑘∈Z follows: have

(1/𝑞−1/2) 𝑓𝑔 = ∑ (◻𝑖𝑓) (◻𝑗𝑔) . 𝑛 (57) 𝐵≲ ( ∑ (1+|𝑘|)−2𝑠𝑞/(2−𝑞)) 𝑖,𝑗∈Z |𝑘|>𝑇 Then, we have 󵄩 1/2󵄩 󵄩 󵄨 󵄨2 󵄩 󵄩 󵄨 󵄨 󵄨2 𝑠/2 󵄨 󵄩 󵄩 󵄨 −1 󵄨 󵄨 󵄨 󵄩 ◻𝑘 (𝑓𝑔 )= ∑ ◻𝑘 (◻𝑖𝑓) (◻𝑗𝑔) . × 󵄩( ∑ 󵄨𝐹 (1 + 󵄨𝜉󵄨 ) 𝜎𝑘𝐹𝑓 󵄨 ) 󵄩 (58) 󵄩 󵄨 󵄨 󵄩 𝑖,𝑗∈Z𝑛 󵄩 |𝑘|>𝑇 󵄩𝐿2 Journal of Function Spaces and Applications 7

̂ 𝑠 It is easy to see that 𝐹(◻𝑖𝑓◻𝑗𝑔) = (𝜎𝑖𝑓) ∗ (𝜎𝑗𝑔)̂ and Case 2 (𝑞 > 1). By the definition of 𝑋𝑝,𝑞 and Appendix Theorem F, we have ̂ 󵄨 󵄨 √ 󵄩 󵄩 (𝜎𝑖𝑓) ∗ (𝜎𝑗𝑔)̂ ⊂ {𝜉 : 󵄨𝜉−𝑖−𝑗󵄨 ⩽2 2𝑛} . 󵄩𝑓𝑔 󵄩 𝑠 supp 󵄨 󵄨 (59) 󵄩 󵄩𝑋𝑝,𝑞 󵄩 1/𝑞󵄩 󵄩 󵄩 Then, we have 󵄩 |𝑘|𝑠𝑞󵄨 󵄨𝑞 󵄩 = 󵄩( ∑ 2 󵄨◻𝑘 (𝑓𝑔 ) 󵄨 ) 󵄩 󵄩 𝑛 󵄩 󵄩 𝑘∈Z 󵄩𝐿𝑝 󵄩 󵄩 ◻ (◻ 𝑓◻ 𝑔) = ◻ ( ∑ ◻ 𝑓◻ 𝑔) , 󵄩 󵄩 𝑘 𝑖 𝑗 𝑘 𝑖 𝑗(𝑖,𝑘) 󵄩 |𝑘|𝑠 󵄨 󵄨󵄩 1 𝑞 𝑛 ⩽ 󵄩 ∑ 2 󵄨◻ (𝑓𝑔 ) 󵄨󵄩 ,(ℓ⊂ℓ, 𝑞⩾1) 𝑖∈Z 󵄩 󵄨 𝑘 󵄨󵄩 if (60) 󵄩 𝑛 󵄩 󵄩𝑘∈Z 󵄩𝐿𝑝 󵄨 󵄨 󵄨𝑘−𝑖−𝑗(𝑖, 𝑘)󵄨 ⩽𝐶(𝑛) . 󵄩 󵄨 󵄨󵄩 󵄩 󵄨 󵄨󵄩 ⩽ 󵄩 ∑ 2|𝑘|𝑠 󵄨◻ ( ∑ ◻ 𝑓◻ 𝑔)󵄨󵄩 󵄩 󵄨 𝑘 𝑖 𝑗(𝑖,𝑘) 󵄨󵄩 󵄩 𝑛 󵄨 𝑛 󵄨󵄩 󵄩𝑘∈Z 󵄨 𝑖∈Z 󵄨󵄩𝐿𝑝 󵄩 󵄨 󵄨󵄩 Case 1 (𝑞 ⩽ 1). Based on the previous observations and 󵄩 󵄨 󵄨󵄩 ⩽ 󵄩 ∑ 2|𝑘|𝑠 󵄨 ∑ ◻ 𝑓◻ 𝑔󵄨󵄩 Appendix Theorem F, we have 󵄩 󵄨 𝑖 𝑗(𝑖,𝑘) 󵄨󵄩 󵄩 𝑛 󵄨 𝑛 󵄨󵄩 󵄩𝑘∈Z 󵄨𝑖∈Z 󵄨󵄩𝐿𝑝

󵄩 󵄩 𝐶(𝑛)󵄩 󵄨 󵄨󵄩 󵄩𝑓𝑔 󵄩 𝑠 󵄩 󵄨 󵄨󵄩 󵄩 󵄩𝑋𝑝,𝑞 󵄩 |𝑘|𝑠 󵄨 󵄨󵄩 ≲ ∑ 󵄩 ∑ 2 󵄨 ∑ (◻ 𝑓◻ 𝑔)󵄨󵄩 󵄩 󵄨 𝑖 𝑗 󵄨󵄩 𝑡=0 󵄩𝑘∈Z𝑛 󵄨𝑖∈Z𝑛,𝑗=𝑘−𝑖±𝑡 󵄨󵄩 󵄩 1/𝑞󵄩 󵄩 󵄨 󵄨󵄩𝐿𝑝 󵄩 󵄩 󵄩 |𝑘|𝑠𝑞󵄨 󵄨𝑞 󵄩 󵄩 󵄨 󵄨󵄩 = 󵄩( ∑ 2 󵄨◻𝑘 (𝑓𝑔 ) 󵄨 ) 󵄩 𝐶(𝑛)󵄩 󵄨 󵄨󵄩 󵄩 󵄩 󵄩 |𝑘−𝑖−𝑗|𝑠 󵄨 |𝑖|𝑠 |𝑗|𝑠 󵄨󵄩 󵄩 𝑛 󵄩 󵄩 󵄨 󵄨󵄩 󵄩 𝑘∈Z 󵄩𝐿𝑝 ⩽ ∑ 󵄩 ∑ 2 󵄨 ∑ (2 ◻𝑖𝑓) (2 ◻𝑗𝑔)󵄨󵄩 󵄩 𝑛 󵄨 𝑛 󵄨󵄩 𝑡=0 󵄩𝑘∈Z 󵄨𝑖∈Z ,𝑗=𝑘−𝑖±𝑡 󵄨󵄩 𝑝 󵄩 󵄩 𝐿 󵄩 󵄨 󵄨𝑞 1/𝑞󵄩 󵄩 󵄨 󵄨 󵄩 󵄩 󵄩 ⩽ 󵄩( ∑ 2|𝑘|𝑠𝑞󵄨◻ ( ∑ ◻ 𝑓◻ 𝑔)󵄨 ) 󵄩 󵄩 󵄨 󵄨 󵄨 󵄨󵄩 󵄩 󵄨 𝑘 𝑖 𝑗(𝑖,𝑘) 󵄨 󵄩 𝐶𝑠󵄩 󵄨 𝑖𝑠 󵄨 󵄨 𝑗𝑠 󵄨󵄩 󵄩 𝑛 󵄨 𝑛 󵄨 󵄩 ⩽𝐶2 󵄩 ∑ 󵄨(2 ◻𝑖𝑓)󵄨 ∑ 󵄨(2 ◻𝑗𝑔)󵄨󵄩 󵄩 𝑘∈Z 󵄨 𝑖∈Z 󵄨 󵄩 𝑝 󵄩 󵄨 󵄨 󵄨 󵄨󵄩 󵄩 󵄩𝐿 󵄩 𝑛 𝑛 󵄩 󵄩𝑖∈Z 𝑗∈Z 󵄩𝐿𝑝 󵄩 󵄩 󵄩 1/𝑞󵄩 𝐶𝑠󵄩 󵄩 󵄩 󵄩 󵄩 |𝑘|𝑠𝑞󵄨 󵄨𝑞 󵄩 ⩽𝐶2 󵄩𝑓󵄩𝑋𝑠 󵄩𝑔󵄩𝑋𝑠 . 󵄩 󵄨 󵄨 󵄩 𝑝 ,1 𝑝 ,1 ⩽ 󵄩( ∑ ∑ 2 󵄨◻𝑖𝑓◻𝑗(𝑖,𝑘)𝑔󵄨 ) 󵄩 1 2 󵄩 𝑛 𝑛 󵄩 󵄩 𝑘∈Z 𝑖∈Z 󵄩𝐿𝑝 (63)

󵄩 1/𝑞󵄩 By the above discussion, we can obtain the claimed results. 󵄩 󵄨 󵄨𝑞 󵄩 󵄩 |𝑘−𝑖−𝑗(𝑖,𝑘)|𝑠𝑞󵄨 |𝑖|𝑠 |𝑗(𝑖,𝑘)|𝑠 󵄨 󵄩 ⩽ 󵄩( ∑ ∑ 2 󵄨(2 ◻𝑖𝑓) (2 ◻𝑗(𝑖,𝑘)𝑔)󵄨 ) 󵄩 󵄩 𝑛 𝑛 󵄩 󵄩 𝑘∈Z 𝑖∈Z 󵄩𝐿𝑝 Corollary 14. Let 0⩽𝑠<∞, 0<𝑝⩽𝑝𝑖 <∞, 𝑖 = 1,...,𝑁, 󵄩 0<𝑞⩽∞.If1/𝑝 = 1/𝑝1 +⋅⋅⋅+1/𝑝𝑁, then one has 𝐶(𝑛) 󵄩 󵄩 |𝑘−𝑖−𝑗|𝑠𝑞 󵄩 󵄩 ≲ ∑ 󵄩( ∑ ∑ 2 󵄩 𝑁 󵄩 𝑁 󵄩 󵄩 󵄩 𝑁 𝐶𝑁𝑠 󵄩 󵄩 𝑡=0 󵄩 𝑘∈Z𝑛 𝑖∈Z𝑛,𝑗=𝑘−𝑖±𝑡 󵄩∏𝑢 󵄩 ≲𝐶 2 ∏󵄩𝑢 󵄩 , 󵄩 󵄩 𝑖󵄩 󵄩 𝑖󵄩𝑋𝑠 (64) 󵄩 󵄩 𝑝𝑖,𝑞∧1 󵄩 𝑖=1 󵄩𝑋𝑠 𝑖=1 󵄩 𝑝,𝑞 1/𝑞󵄩 󵄨 󵄨𝑞 󵄩 󵄨 |𝑖|𝑠 |𝑗|𝑠 󵄨 󵄩 where 𝐶 is independent of 𝑠, 𝑞. × 󵄨(2 ◻𝑖𝑓) (2 ◻𝑗𝑔)󵄨 ) 󵄩 󵄨 󵄨 󵄩 󵄩𝐿𝑝 ∗ ∗ Proof. Let 1/𝑝 = 1/𝑝1 +1/𝑝 ,where1/𝑝 =1/𝑝2 +⋅⋅⋅+1/𝑝𝑁. Then, we have 󵄩 1/𝑞 1/𝑞󵄩 󵄩 󵄩 𝐶𝑠󵄩 |𝑖|𝑠𝑞󵄨 󵄨𝑞 |𝑗|𝑠𝑞󵄨 󵄨𝑞 󵄩 󵄩 𝑁 󵄩 󵄩 𝑁 󵄩 ⩽𝐶2 󵄩( ∑ 2 󵄨◻ 𝑓󵄨 ) ( ∑ 2 󵄨◻ 𝑔󵄨 ) 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄨 𝑖 󵄨 󵄨 𝑗 󵄨 󵄩 󵄩∏𝑢 󵄩 ⩽𝐶2𝐶𝑠󵄩𝑢 󵄩 󵄩∏𝑢 󵄩 . 󵄩 𝑖∈Z𝑛 𝑗∈Z𝑛 󵄩 󵄩 𝑖󵄩 󵄩 1󵄩𝑋𝑠 󵄩 𝑖󵄩 󵄩 󵄩𝐿𝑝 󵄩 󵄩 𝑝1,𝑞∧1 󵄩 󵄩 (65) 󵄩 𝑖=1 󵄩𝑋𝑠 󵄩 𝑖=2 󵄩𝑋𝑠 𝑝,𝑞 𝑝∗,𝑞∧1 𝐶𝑠󵄩 󵄩 󵄩 󵄩 ⩽𝐶2 󵄩𝑓󵄩𝑋𝑠 󵄩𝑔󵄩𝑋𝑠 , 𝑝1,𝑞 𝑝2,𝑞 By induction, we can easily see that the claimed result holds. (61) where we used |𝑘−𝑖−𝑗(𝑖, 𝑘)| <𝐶(𝑛) and the following Young’s 4. Smooth Effects of the Schrödinger and inequality: Ginzburg-Landau Semigroups In this section, we will discuss a kind of Strichartz’s estimates. 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 1 1 1 󵄩𝑎 ∗𝑏󵄩 ⩽ 󵄩𝑎 󵄩 𝑝 󵄩𝑏 󵄩 ,1+ = + . This kind of estimates is first introduced by Strichartz20 in[ ], 󵄩 𝑖 𝑗󵄩ℓ𝑝 󵄩 𝑖󵄩ℓ 1 󵄩 𝑗󵄩ℓ𝑝2 𝑝 𝑝 𝑝 (62) 1 2 then developed by [18, 21, 22], and so forth. 8 Journal of Function Spaces and Applications

Put Theorem 17. Letting 0⩽𝑠<∞,onehas −1 −𝑖𝑡|𝜉|2 𝑆 (𝑡) =𝐹 𝑒 𝐹, 󵄩 󵄩 𝑛/2 󵄩 󵄩 󵄩𝑆 (𝑡) 𝑓󵄩 𝑠 ⩽ (1+|𝑡|) 󵄩𝑓󵄩 𝑠 , 󵄩 󵄩𝑋1,1 󵄩 󵄩𝑋1,1 (73) −1 −𝑡𝑃(𝜉) 󵄨 󵄨2 (66) 𝑈 (𝑡) =𝐹 𝑒 𝐹, 𝑃 (𝜉) =1+(𝑎+𝑖𝛼) 󵄨𝜉󵄨 , 𝑠 which holds for all 𝑓∈𝑋1,1. 𝑎>0,𝛼∈R. Proof. By [18] (Appendix Theorem H), we have Our aim is to derive the estimates of 𝑆(𝑡), 𝑈(𝑡) in the spaces 𝑠 󵄩 󵄩 𝑛|1/2−1/𝑝|󵄩 󵄩 𝑋𝑝,𝑞. 󵄩 󵄩 󵄩 󵄩 󵄩◻𝑘𝑆 (𝑡) 𝑓󵄩𝐿𝑝 ≲ (1+|𝑡|) 󵄩◻𝑘𝑓󵄩𝐿𝑝 , 1⩽𝑝⩽∞. 󸀠 󸀠 (74) Theorem 15. Assume 2⩽𝑝<∞, 𝑝 ⩽𝑞⩽𝑝, 1/𝑝+1/𝑝 =1, 0⩽𝑠<∞ ; then one has Then, we have 󵄩 󵄩 −𝑛(1/2−1/𝑝)󵄩 󵄩 󵄩𝑆 (𝑡) 𝑓󵄩 𝑠 ≲ (1+|𝑡|) 󵄩𝑓󵄩 𝑠 . 󵄩 󵄩 󵄩 󵄩𝑋𝑝,𝑞 󵄩 󵄩𝑋 󸀠 (67) 󵄩𝑆 (𝑡) 𝑓󵄩 𝑠 𝑝 ,𝑞 󵄩 󵄩𝑋1,1 󵄩 󵄩 Proof 󵄩 󵄨 󵄨󵄩 = 󵄩 ∑ 2|𝑘|𝑠 󵄨◻ 𝑆 (𝑡) 𝑓󵄨󵄩 2⩽𝑝<∞ 󵄩 󵄨 𝑘 󵄨󵄩 Step 1. By Appendix Theorem C, for ,wehave 󵄩 𝑛 󵄩 󵄩𝑘∈Z 󵄩𝐿1 󵄩 󵄩 −𝑛(1/2−1/𝑝)󵄩 󵄩 󵄩◻ 𝑆 (𝑡) 𝑓󵄩 ≲ (1+|𝑡|) 󵄩◻ 𝑓󵄩 󸀠 . |𝑘|𝑠󵄩 󵄩 󵄩 𝑘 󵄩𝐿𝑝 󵄩 𝑘 󵄩𝐿𝑝 (68) 󵄩 󵄩 = ∑ 2 󵄩◻𝑘𝑆 (𝑡) 𝑓󵄩𝐿1 𝑛 Thus, for 0⩽𝑠<∞,wehave 𝑘∈Z 󵄩 󵄩 −𝑛(1/2−1/𝑝)󵄩 󵄩 𝑛/2 |𝑘|𝑠󵄩 󵄩 𝑛/2󵄩 󵄩 󵄩𝑆 (𝑡) 𝑓󵄩 ≲ (1+|𝑡|) 󵄩𝑓󵄩 . ≲ (1+|𝑡|) ∑ 2 󵄩◻𝑘𝑓󵄩𝐿1 ≲ (1+|𝑡|) 󵄩𝑓󵄩𝑋𝑠 . 󵄩 󵄩𝐸𝑠 󵄩 󵄩𝐸𝑠 (69) 1,1 𝑝,𝑞 𝑝󸀠,𝑞 𝑘∈Z𝑛 󸀠 (75) Step 2.ByProposition 3,for𝑝 ⩽𝑞⩽𝑝,wehave 󵄩 󵄩 󵄩 󵄩 This proves the theorem. 󵄩𝑆 (𝑡) 𝑓󵄩 𝑠 ⩽ 󵄩𝑆 (𝑡) 𝑓󵄩 𝑠 󵄩 󵄩𝑋𝑝,𝑞 󵄩 󵄩𝐸𝑝,𝑞 𝑡 Theorem 18. 𝑟⩾1,0⩽𝑠<∞𝐴𝑓 =∫ 𝑆(𝑡 − −𝑛(1/2−1/𝑝)󵄩 󵄩 Assume , 0 ⩽ (1+𝑡) 󵄩𝑓󵄩𝐸𝑠 𝑝󸀠,𝑞 (70) 𝜏)𝑓(𝜏)𝑑𝜏; then one has −𝑛(1/2−1/𝑝)󵄩 󵄩 ⩽ (1+𝑡) 󵄩𝑓󵄩 . 󵄩 󵄩 𝑛/2 2/𝑟󵄩 󵄩 󵄩 󵄩𝑋𝑠 󵄩𝐴𝑓󵄩 ≲ (1+𝑇) 𝑇 󵄩𝑓󵄩 󸀠 . 󸀠 󵄩 󵄩𝐿𝑟(−𝑇,𝑇;𝑋𝑠 ) 󵄩 󵄩 𝑟 𝑠 𝑝 ,𝑞 1,1 𝐿 (−𝑇,𝑇;𝑋1,1) (76)

This proves the theorem. Proof. By Theorem 17,wehave 󸀠 Theorem 16. Assume 𝑟⩾1, 0⩽𝑠<∞, 𝑝 ⩽𝑞⩽𝑝, 2⩽𝑝< 󵄩 󵄩 𝑡 󵄩𝐴𝑓󵄩𝐿𝑟(−𝑇,𝑇;𝑋𝑠 ) ∞ 𝐴𝑓 =∫ 𝑆(𝑡 − 𝜏)𝑓(𝜏)𝑑𝜏 1,1 , 0 ; then one has 󵄩 󵄩 󵄩 𝑡 󵄩 󵄩 󵄩 󵄩 󵄩 2/𝑟󵄩 󵄩 ⩽ 󵄩∫ 󵄩𝑆 (𝑡−𝜏) 𝑓 (𝜏)󵄩 𝑑𝜏󵄩 󵄩𝐴𝑓󵄩 𝑟 𝑠 ≲𝑇 󵄩𝑓󵄩 𝑟󸀠 𝑠 . 󵄩 󵄩 󵄩𝑋𝑠 󵄩 󵄩 󵄩𝐿 (−𝑇,𝑇;𝑋𝑝,𝑞) 󵄩 󵄩𝐿 (−𝑇,𝑇;𝑋 ) (71) 󵄩 1,1 󵄩 𝑟 𝑝󸀠,𝑞 󵄩 0 󵄩𝐿 (−𝑇,𝑇) 󵄩 𝑡 󵄩 Proof. By Theorem 15,wehave 󵄩 𝑛/2󵄩 󵄩 󵄩 ⩽ 󵄩∫ (1+|𝑡−𝜏|) 󵄩𝑓 (𝜏)󵄩 𝑠 𝑑𝜏󵄩 󵄩 𝑋1,1 󵄩 󵄩 󵄩 󵄩 0 󵄩𝐿𝑟(−𝑇,𝑇) 󵄩𝐴𝑓󵄩𝐿𝑟(−𝑇,𝑇;𝑋𝑠 ) 𝑝,𝑞 󵄩 ∞ 󵄩 (77) 𝑛/2󵄩 󵄩 󵄩 󵄩 󵄩 𝑡 󵄩 ≲ (1+𝑇) 󵄩∫ 𝜒 󵄩𝑓 (𝜏)󵄩 𝑑𝜏󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 𝜏∈[0,𝑡]󵄩 󵄩𝑋𝑠 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 0 1,1 󵄩𝐿𝑟(−𝑇,𝑇) ⩽ 󵄩∫ 󵄩𝑆 (𝑡−𝜏) 𝑓 (𝜏)󵄩𝑋𝑠 𝑑𝜏󵄩 󵄩 0 𝑝,𝑞 󵄩𝐿𝑟(−𝑇,𝑇) 𝑇 1/𝑟 󵄩 𝑡 󵄩 𝑛/2 1/𝑟 󵄩 󵄩𝑟 󵄩 󵄩 ≲ (1+𝑇) 𝑇 (∫ 󵄩𝑓 (𝜏)󵄩 󸀠 𝑑𝑡) 󵄩 −𝑛(1/2−1/𝑝)󵄩 󵄩 󵄩 󵄩 󵄩𝐿𝑟 (−𝑇,𝑇;𝑋𝑠 ) ⩽ 󵄩∫ (1 + |𝑡−𝜏| ) 󵄩𝑓 (𝜏)󵄩𝑋𝑠 𝑑𝜏󵄩 −𝑇 1,1 󵄩 0 𝑝󸀠,𝑞 󵄩𝐿𝑟(−𝑇,𝑇) 𝑛/2 2/𝑟󵄩 󵄩 ≲ (1+𝑇) 𝑇 󵄩𝑓󵄩 󸀠 . 󵄩 ∞ 󵄩 (72) 󵄩 󵄩𝐿𝑟 (−𝑇,𝑇;𝑋𝑠 ) 󵄩 󵄩 󵄩 󵄩 1,1 ≲ 󵄩∫ 𝜒𝜏∈[0,𝑡]󵄩𝑓 (𝜏)󵄩𝑋𝑠 𝑑𝜏󵄩 󵄩 0 𝑝󸀠,𝑞 󵄩𝐿𝑟(−𝑇,𝑇) This proves the theorem. 𝑇 1/𝑟 1/𝑟 󵄩 󵄩𝑟 ≲𝑇 (∫ 󵄩𝑓 (𝜏)󵄩 󸀠 𝑑𝑡) Theorem 19. Assume 𝑟⩾𝑘+2, 0⩽𝑠<∞. Assume also 󵄩 󵄩𝐿𝑟 (−𝑇,𝑇;𝑋𝑠 ) 𝑡 −𝑇 𝑝󸀠,𝑞 𝑓(𝑢) = 𝑢|𝑢|𝑘 𝑘=2𝑚 𝑚∈Z+ 𝐴(𝑓) =∫ 𝑆(𝑡 − 𝜏)𝑓(𝑢(𝜏))𝑑𝜏 , , , 0 . 2/𝑟󵄩 󵄩 ≲𝑇 󵄩𝑓󵄩 󸀠 . Then, one has 󵄩 󵄩𝐿𝑟 (−𝑇,𝑇;𝑋𝑠 ) 𝑝󸀠,𝑞 󵄩 󵄩 𝑛/2 1−𝑘/𝑟 𝑘+1 󵄩𝐴(𝑓)󵄩𝐿𝑟(−𝑇,𝑇;𝑋𝑠 ) ≲ (1+𝑇) 𝑇 ‖𝑢‖𝐿𝑟(−𝑇,𝑇;𝑋𝑠 ). (78) This proves the theorem. 1,1 1,1 Journal of Function Spaces and Applications 9

Proof. By the definition, Theorems 18 and 13,wehave Combining the previous discussion, we have 󵄩 󵄩 󵄩𝐴(𝑓)󵄩 𝑟 𝑠 󵄩 󵄩𝐿 (−𝑇,𝑇;𝑋1,1) 󵄩 󵄩 2 󵄩𝜎 𝑒−𝑡𝑃(𝜉)󵄩 ≲𝐶𝑒−𝑐𝑡(1+|𝑘| ). 󵄩 𝑘 󵄩 𝐿 (85) 󵄩 𝑡 󵄩 𝐻 󵄩 󵄩 = 󵄩∫ 𝑆 (𝑡−𝜏) 𝑓 (𝑢 (𝜏)) 𝑑𝜏󵄩 󵄩 0 󵄩 𝑟 𝑠 󵄩 󵄩𝐿 (−𝑇,𝑇;𝑋1,1) −𝑡𝑃(𝜉) Step 2.ByStep1,wehavesup𝑘‖𝜎𝑘𝑒 ‖𝐻𝐿 ≲1.Thenby 𝑛/2 2/𝑟󵄩 󵄩 Appendix Theorem F, we can obtain that ≲ (1+𝑇) 𝑇 󵄩𝑓󵄩 𝑟󸀠 𝑠 𝐿 (−𝑇,𝑇;𝑋1,1)

1/𝑟󸀠 󵄩 󵄩 󵄩 −𝑡𝑃(𝜉)󵄩 󵄩 󵄩 󵄩 󵄩 𝑇 󸀠 󵄩𝑈 𝑡 𝑓󵄩 ⩽𝐶 󵄩𝜎 𝑒 󵄩 󵄩𝑓󵄩 ≲ 󵄩𝑓󵄩 . 𝑛/2 2/𝑟 󵄩 󵄩𝑟 󵄩 ( ) 󵄩 𝑠 sup𝑘󵄩 𝑘 󵄩 𝑠 󵄩 󵄩 𝑠 󵄩 󵄩 𝑠 󵄩 󵄩 󵄩 󵄩𝑋𝑝,𝑞 󵄩 󵄩𝐻 󵄩 󵄩𝑋𝑝,𝑞 󵄩 󵄩𝑋𝑝,𝑞 (86) ≲ (1+𝑇) 𝑇 (∫ 󵄩𝑓 (𝑢 (𝜏))󵄩𝑋𝑠 𝑑𝜏) −𝑇 1,1 (79)

󸀠 𝑇 1/𝑟 This proves the theorem. 𝑛/2 2/𝑟 (𝑘+1)𝑟󸀠 ≲ (1+𝑇) 𝑇 (∫ ‖𝑢 (𝜏)‖𝑋𝑠 𝑑𝜏) −𝑇 𝑘+1,1 Theorem 21. Assume 1⩽𝑝<∞, 1⩽𝑞⩽∞, 𝑟⩾1,0⩽𝑠< 𝑡 𝑇 (𝑘+1)/𝑟 ∞, 𝐴𝑓 =∫ 𝑈(𝑡 − 𝜏)𝑓(𝜏)𝑑𝜏; then one has 𝑛/2 2/𝑟 𝑟 (𝑟−𝑘−2)/𝑟 0 ≲ (1+𝑇) 𝑇 (∫ ‖𝑢 (𝜏)‖𝑋𝑠 𝑑𝜏) 𝑇 −𝑇 1,1 󵄩 󵄩 2/𝑟󵄩 󵄩 󵄩𝐴𝑓󵄩 ≲𝑇 󵄩𝑓󵄩 󸀠 . 𝑛/2 1−𝑘/𝑟 𝑘+1 󵄩 󵄩𝐿𝑟(−𝑇,𝑇;𝑋𝑠 ) 󵄩 󵄩 𝑟 𝑠 (87) ≲ (1+𝑇) 𝑇 ‖𝑢‖ 𝑟 𝑠 . 𝑝,𝑞 𝐿 (−𝑇,𝑇;𝑋𝑝,𝑞) 𝐿 (−𝑇,𝑇;𝑋1,1)

This proves the theorem. Proof. By Theorem 20,wehave Theorem 20. Let 0<𝑝<∞, 0<𝑞⩽∞, 0⩽𝑠<∞;then 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩𝐴𝑓󵄩𝐿𝑟(−𝑇,𝑇;𝑋𝑠 ) 󵄩𝑈 (𝑡) 𝑓󵄩 ⩽𝐶󵄩𝑓󵄩 , 𝑝,𝑞 󵄩 󵄩𝑋𝑠 󵄩 󵄩𝑋𝑠 (80) 𝑝,𝑞 𝑝,𝑞 󵄩 󵄩 󵄩 𝑡 󵄩 󵄩 󵄩 𝑠 ⩽ 󵄩∫ 󵄩𝑈 (𝑡−𝜏) 𝑓 (𝜏)󵄩 𝑑𝜏󵄩 which holds for all 𝑓∈𝑋𝑝,𝑞. 󵄩 󵄩 󵄩𝑋𝑠 󵄩 󵄩 0 𝑝,𝑞 󵄩𝐿𝑟(−𝑇,𝑇) 󵄩 𝑡 󵄩 Proof 󵄩 󵄩 󵄩 󵄩 ⩽ 󵄩∫ 󵄩𝑓 (𝜏)󵄩𝑋𝑠 𝑑𝜏󵄩 (88) 󵄩 0 𝑝,𝑞 󵄩 𝑟 Step 1 (see [12]). Let 𝐿 > 𝑛(𝑛/ min(1,𝑝,𝑞)− 1/2) and 𝐿∈ 𝐿 (−𝑇,𝑇) Z+ ‖𝜎 𝑒−𝑡𝑃(𝜉)‖ ≲1 󵄩 ∞ 󵄩 ;thenwehave 𝑘 𝐻𝐿 .Actually,inviewofthe 󵄩 󵄩 󵄩 󵄩 ≲ 󵄩∫ 𝜒𝜏∈[0,𝑡]󵄩𝑓 (𝜏)󵄩𝑋𝑠 󵄩 Leibnitz rule, we have 󵄩 0 𝑝,𝑞 󵄩𝐿𝑟(−𝑇,𝑇) 󵄨 𝐿 −𝑡𝑃(𝜉) 󵄨 󵄨 󵄨 2/𝑟󵄩 󵄩 󵄨𝐷 (𝜎𝑘𝑒 )󵄨 󵄩 󵄩 󵄨 󵄨 ≲𝑇 󵄩𝑓󵄩 𝑟󸀠 𝑠 . 𝐿 (−𝑇,𝑇;𝑋𝑝,𝑞) 󵄨 𝐿 𝐿 −𝑡𝑃(𝜉)󵄨 󵄨 1 2 󵄨 ⩽𝐶 ∑ 󵄨𝐷 𝜎𝑘𝐷 𝑒 󵄨 1 2 (81) 𝐿 +𝐿 =𝐿 This proves the theorem. 󵄨 𝑠 −𝑡𝑃(𝜉+𝑘)󵄨 ⩽𝐶 ∑ 󵄨𝐷 𝑒 󵄨 𝜒|𝜉| ⩽1. 𝑘 󵄨 󵄨 ∞ Theorem 22. Assume 𝑏>0, 𝛽, 𝜇 ∈ R, 𝑓(𝑢) = (𝑏 + 𝑖𝛽)𝑢|𝑢| + 0⩽𝑠⩽𝐿 𝑡 𝜇𝑢, 𝐴(𝑓) =∫ 𝑈(𝑡 − 𝜏)𝑓(𝑢(𝜏))𝑑𝜏 𝑘=2𝑚𝑟⩾ 0 . Assume also , 𝛽=(𝛽, ..., 𝛽 ) Λ𝑠 ={𝛽:𝛽+⋅⋅⋅+𝛽 = + Let 1 𝑞 ,andlet 𝑞 1 𝑞 𝑘+2, 𝑚∈Z , 1⩽𝑝<∞, 1⩽𝑞⩽∞.Then,onehas 𝑠,1 𝛽 , ..., 𝛽𝑞 ⩾1}.Itiseasytoseethat 󵄩 󵄩 1−𝑘/𝑟 𝑘+1 𝑠 𝑞 󵄩𝐴(𝑓)󵄩 𝑟 𝑠 ≲𝑇 ‖𝑢‖ 𝑟 𝑠 +𝑇‖𝑢‖ 𝑟 𝑠 . { } 󵄩 󵄩𝐿 (0,𝑇;𝑋 ) 𝐿 (0,𝑇;𝑋 ) 𝐿 (0,𝑇;𝑋𝑝,𝑞) 𝑠 𝑓(𝜂) 𝑓(𝜂) 𝛽 𝑝,𝑞 𝑝,1 𝐷 𝑒 =𝑒 ∑ ∑𝐶 ∏𝐷 𝑖 𝑓 (𝜉) . { 𝛽 } (82) (89) 𝑞=1 Λ𝑠 𝑖=1 { 𝑞 }

Let 𝑓(𝜉) = −𝑡𝑃(𝜉);oneeasilyseesthat Proof. By Theorems 20 and 21 and the multilinear estimate, 󵄨 𝛽 󵄨 󵄨 󵄨2 we have 󵄨 𝑖 󵄨 󵄨 󵄨 󵄨𝐷 𝑓 (𝜉)󵄨 ⩽𝐶𝑡(1+󵄨𝜉+𝑘󵄨 ), 𝑘∈Z. (83) 󵄩 󵄩 󵄩𝐴𝑓󵄩 𝑟 𝑠 Hence, we have 󵄩 󵄩𝐿 (−𝑇,𝑇;𝑋𝑝,𝑞) 𝑠 󵄨 󵄨 2 𝑞 2/𝑟󵄩 󵄩 󵄨 𝑠 𝑓(𝜂)󵄨 −𝑐𝑡(1+|𝜉+𝑘| ) 󵄨 󵄨2 ≲𝑇 󵄩𝑓󵄩 󸀠 󵄨𝐷 𝑒 󵄨 ⩽𝑒 ∑(𝑡 (1 + 󵄨𝜉+𝑘󵄨 )) 󵄩 󵄩𝐿𝑟 (−𝑇,𝑇;𝑋𝑠 ) 󵄨 󵄨 󵄨 󵄨 𝑝,𝑞 𝑞=1 (84) 1/𝑟󸀠 𝑇 󸀠 −𝑐𝑡(1+|𝜉+𝑘|2) 2/𝑟 󵄩 󵄩𝑟 ⩽𝑒 . =𝑇 (∫ 󵄩𝑓 (𝑢 (𝜏))󵄩𝑋𝑠 𝑑𝜏) −𝑇 𝑝,𝑞 10 Journal of Function Spaces and Applications

1/𝑟󸀠 𝑇 󵄩 󵄩𝑟󸀠 be equipped with the distance ≲𝑇2/𝑟(∫ 󵄩𝑢|𝑢|𝑘󵄩 𝑑𝜏) 󵄩 󵄩𝑋𝑠 −𝑇 𝑝,𝑞 𝑑 (𝑢,) 𝜐 = ‖𝑢−𝜐‖ 𝑟 𝑠 . 𝐿 (0,𝑇;𝑋1,1) (96) 󸀠 𝑇 1/𝑟 2/𝑟 𝑟󸀠 (D,𝑑) +𝑇 (∫ ‖𝑢 (𝜏)‖𝑋𝑠 𝑑𝜏) It is easy to know that is a complete metric space. Now −𝑇 𝑝,𝑞 we consider the following map:

󸀠 𝑇 1/𝑟 𝑡 2/𝑟 (𝑘+1)𝑟󸀠 I :𝑢(𝑡) 󳨀→ 𝑆 (𝑡) 𝑢0 −𝑖∫ 𝑆 (𝑡−𝜏) 𝑓 (𝑢 (𝜏)) 𝑑𝜏. (97) ≲𝑇 (∫ ‖𝑢 (𝜏)‖𝑋𝑠 𝑑𝜏) −𝑇 (𝑘+1)𝑝,𝑞∧1 0

󸀠 𝑇, 𝛿 >0 I : 𝑇 1/𝑟 We will prove that there exists ,suchthat 2/𝑟 𝑟󸀠 (D,𝑑) →D ( ,𝑑)is a strict contraction map. +𝑇 (∫ ‖𝑢 (𝜏)‖𝑋𝑠 𝑑𝜏) −𝑇 𝑝,𝑞 By the nonlinear term estimate and the semigroup estimate, 𝑇 (𝑘+1)/𝑟 2/𝑟 𝑟 (𝑟−𝑘−2)/𝑟 󵄩 𝑘 󵄩 𝑘+1 ≲𝑇 (∫ ‖𝑢 (𝜏)‖ 𝑠 𝑑𝜏) 𝑇 󵄩|𝑢| 𝑢󵄩 ≲ ‖𝑢‖ 𝑠 , 𝑋𝑝,1 󵄩 󵄩 𝑠 𝑋 −𝑇 󵄩 󵄩𝑋1,1 (𝑘+1),1 󵄩 󵄩 𝑛/2󵄩 󵄩 +𝑇‖𝑢‖ 𝑟 𝑠 󵄩𝑆 (𝑡) 𝑢0󵄩𝑋𝑠 ≲ (1+|𝑡|) 󵄩𝑢0󵄩𝑋𝑠 , (98) 𝐿 (0,𝑇;𝑋𝑝,𝑞) 1,1 1,1 󵄩 󵄩 1−𝑘/𝑟 𝑘+1 󵄩𝐴(𝑓)󵄩 ≲ (1+𝑇)𝑛/2𝑇1−𝑘/𝑟‖𝑢‖𝑘+1 , ≲𝑇 ‖𝑢‖ 𝑟 𝑠 +𝑇‖𝑢‖𝐿𝑟(0,𝑇;𝑋𝑠 ). 󵄩 󵄩𝐿𝑟(0,𝑇;𝑋𝑠 ) 𝐿𝑟(0,𝑇;𝑋𝑠 ) 𝐿 (0,𝑇;𝑋𝑝,1) 𝑝,𝑞 1,1 1,1 (90) we have This proves the theorem. 𝑛/2 1/𝑟󵄩 󵄩 ‖I𝑢‖𝐿𝑟(0,𝑇;𝑋𝑠 ) ≲ (1+𝑇) 𝑇 󵄩𝑢0󵄩 𝑠 1,1 󵄩 󵄩𝑋1,1 (99) 5. Well-Posedness of Nonlinear Schrödinger + (1+𝑇)𝑛/2𝑇1−𝑘/𝑟‖𝑢‖𝑘+1 . 𝐿𝑟(0,𝑇;𝑋𝑠 ) Equation and Ginzburg-Landau Equation 1,1 In this section, we first study the well-posedness of the Then, we let following Schrodinger¨ equation: 󵄩 󵄩 𝛿=2𝐶󵄩𝑢0󵄩 𝑠 ,𝑇<1. 󵄩 󵄩𝑋1,1 (100) 𝑖𝑢𝑡 +Δ𝑢=𝑓(𝑢) ;𝑢(0, 𝑥) =𝑢0 (𝑥) . (91) Then, we choose 𝑇 such that The solution, 𝑢(𝑥, 𝑡),oftheaboveCauchyproblemisgiven by: (1−𝑘/𝑟) 𝑘 1 2𝐶𝑇 𝛿 ⩽ . (101) 𝑡 2 𝑢 (𝑡) =𝑆(𝑡) 𝑢0 −𝑖∫ 𝑆 (𝑡−𝜏) 𝑓 (𝑢 (𝜏)) 𝑑𝜏, (92) 0 Then, we have

−1 𝑖𝑡|𝜉|2 𝑆(𝑡) =𝐹 𝑒 𝐹 ‖I𝑢‖ 𝑟 𝑠 <𝛿. where .Then,westudythewell-posedness 𝐿 (0,𝑇;𝑋1,1) (102) of the following complex Ginzburg-Landau equation: Moreover, we have that 𝑘 𝑢𝑡 − (𝑎+𝑖𝛼) Δ𝑢 + (𝑏 + 𝑖𝛽) |𝑢| 𝑢 (93) 1 ‖I𝑢−I𝜐‖ 𝑟 𝑠 ⩽ . 𝐿 (0,𝑇;𝑋1,1) (103) + (𝜇 + 1) 𝑢 = 0,𝑢 (0, 𝑥) =𝑢0 (𝑥) . 2‖𝑢−𝜐‖ 𝑟 𝑠 𝐿 (0,𝑇;𝑋1,1) The solution, 𝑢(𝑥, 𝑡),oftheaboveCauchyproblemisgivenby In this way, we prove that I :(D,𝑑) →D ( ,𝑑) is a strict 𝑡 contraction map. By Banach’s fixed-point theorem, there 𝑢 (𝑡) =𝑈(𝑡) 𝑢0 −𝑖∫ 𝑈 (𝑡−𝜏) 𝑓 (𝑢 (𝜏)) 𝑑𝜏, (94) exists a unique solution 𝑢∈D that satisfies the conditions 0 in the theorem. Using standard argument, we can extend the −1 −𝑡𝑃(𝜉) where 𝑎, 𝑏, >0 𝛼, 𝛽, 𝜇∈ R, 𝑈(𝑡) =𝐹 𝑒 𝐹,and𝑃(𝜉) = solution, considering the map 1 + (𝑎 + 𝑖𝛼)|𝜉|2 . 𝑡 I :𝑢(𝑡) 󳨀→ 𝑆 (𝑡−𝑇) 𝑢𝑇 −𝑖∫ 𝑆 (𝑡−𝜏) 𝑓 (𝑢 (𝜏)) 𝑑𝜏, (104) 6. Proof of Theorem 1 𝑇 where 𝑢(𝑇) is the solution that we have chosen. We claim Proof. Fix 𝑇>0, 𝛿>0to be chosen later. Let 𝑠 𝑢(𝑇)1,1 ∈𝑋 ; one can refer to [23] for the details. In this way, ∗ 𝑟 𝑠 we can find a maximum 𝑇 >0which satisfies the conditions D ={𝑢∈𝐿 (0, 𝑇;1,1 𝑋 ):‖𝑢‖𝐿𝑟(0,𝑇;𝑋𝑠 ) <𝛿} (95) 1,1 in the theorem. Journal of Function Spaces and Applications 11

7. Proof of Theorem 2 in the theorem. Using standard argument, we can extend the solution considering the map Proof. Fix 𝑇>0, 𝛿>0to be chosen later. Let 𝑡 𝑟 𝑠 I :𝑢(𝑡) 󳨀→ 𝑈 (𝑡−𝑇) 𝑢𝑇 −𝑖∫ 𝑈 (𝑡−𝜏) 𝑓 (𝑢 (𝜏)) 𝑑𝜏, (114) D ={𝑢∈𝐿 (0, 𝑇; 𝑋 ):‖𝑢‖ 𝑟 𝑠 <𝛿} 𝑇 𝑝,1 𝐿 (0,𝑇;𝑋𝑝,1) (105) where 𝑢(𝑇) is the solution that we have chosen. We claim be equipped with the distance 𝑠 𝑢(𝑇)𝑝,1 ∈𝑋 ; one can refer to [23] for the details. In this way, ∗ we can find a maximum 𝑇 >0which satisfies the conditions 𝑑 (𝑢,) 𝜐 = ‖𝑢−𝜐‖𝐿𝑟(0,𝑇;𝑋𝑠 ). (106) 𝑝,1 in the theorem. It is easy to know that (D,𝑑)is a complete metric space. Now, we consider the following map: Appendix 𝑡 Theorem A. I :𝑢(𝑡) 󳨀→ 𝑈 (𝑡) 𝑢0 −𝑖∫ 𝑈 (𝑡−𝜏) 𝑓 (𝑢 (𝜏)) 𝑑𝜏. (107) We have 0 1 1 𝐻𝑠 (R𝑛)⊂𝐸0 (R𝑛), 𝑠>𝑛( − ), 0<𝑞<2, We shall prove that there exists 𝑇, 𝛿 >0,suchthatI : 2,𝑞 𝑞 2 (D,𝑑) →D ( ,𝑑)is a strict contraction map. 2 𝑛 0 𝑛 By the nonlinear term estimate and the semigroup esti- 𝐿 (R )=𝐸2,2 (R ), (A.1) mate, 1 1 󵄩 󵄩 𝐸0 (R𝑛)⊂𝐻𝑠 (R𝑛), 𝑠<𝑛( − ), 2<𝑞⩽∞. 󵄩 𝑢 𝑘𝑢󵄩 ≲ 𝑢 𝑘+1 , 2,𝑞 󵄩| | 󵄩 𝑠 ‖ ‖𝑋𝑠 𝑞 2 󵄩 󵄩𝑋𝑝,1 𝑝(𝑘+1),1 󵄩 󵄩 󵄩 󵄩 Proof. One can refer to [12]. 󵄩𝑈 (𝑡) 𝑢0󵄩 𝑠 ≲ 󵄩𝑢0󵄩 𝑠 , 󵄩 󵄩𝑋𝑝,1 󵄩 󵄩𝑋𝑝,1 (108) Theorem B. 0<𝑝⩽𝑞⩽∞ Ω⊂R𝑛 󵄩 󵄩 1−𝑘/𝑟 𝑘+1 Assume Let be compact 󵄩𝐴(𝑓)󵄩 𝑟 𝑠 ≲𝑇 ‖𝑢‖𝐿𝑟(0,𝑇;𝑋𝑠 ) Ω<2𝑅 𝐶(𝑝, 𝑞, 𝑅) >0 󵄩 󵄩𝐿 (0,𝑇;𝑋𝑝,1) 𝑝,1 set, diam . Then, there exists ,suchthat 󵄩 󵄩 󵄩 󵄩 𝑝 +𝑇‖𝑢‖ 𝑟 𝑠 , 󵄩𝑓󵄩 𝑞 ⩽𝐶󵄩𝑓󵄩 𝑝 ,∀𝑓∈𝐿, 𝐿 (0,𝑇;𝑋𝑝,1) 󵄩 󵄩𝐿 󵄩 󵄩𝐿 Ω (A.2) 𝐿𝑝 ={𝑓∈𝐿𝑝 : 𝑓⊂Ω}̂ we have where Ω supp .

‖I𝑢‖ 𝑟 𝑠 𝐿 (0,𝑇;𝑋𝑝,1) Proof. One can refer to [15, 18]. 󵄩 󵄩 󸀠 ≲𝑇1/𝑟󵄩𝑢 󵄩 +𝑇1−𝑘/𝑟 𝑢 𝑘+1 Theorem C. Assume 𝑠∈R, 2⩽𝑝<∞, 1/𝑝 + 1/𝑝 =1, 󵄩 0󵄩𝑋𝑠 ‖ ‖𝐿𝑟(0,𝑇;𝑋𝑠 ) (109) 𝑝,1 𝑝,1 Then we have

+𝑇‖𝑢‖ 𝑟 𝑠 . 󵄩 󵄩 −𝑛(1/2−1/𝑝)󵄩 󵄩 𝐿 (0,𝑇;𝑋𝑝,1) 󵄩◻ 𝑆 (𝑡) 𝑓󵄩 ≲ (1+|𝑡|) 󵄩◻ 𝑓󵄩 󸀠 󵄩 𝑘 󵄩𝐿𝑝 󵄩 𝑘 󵄩𝐿𝑝 −𝑛(1/2−1/𝑝)󵄩 󵄩 (A.3) Then, we let ‖𝑆(𝑡) 𝑓‖ 𝑠 ≲ (1+|𝑡|) 󵄩𝑓󵄩 𝑠 . 𝑀𝑝,𝑞 󵄩 󵄩𝑀 𝑝󸀠,𝑞 󵄩 󵄩 1 𝛿=3𝐶󵄩𝑢0󵄩𝑋𝑠 ,𝑇<. (110) 𝑝,1 3 Proof. One can refer to [18].

Then, we choose 𝑇 such that Theorem D (infinite smoothness). Let 0<𝜆<∞, 0<𝑝,𝑞⩽ ∞.Then,onehas (1−𝑘/𝑟) 𝑘 1 𝐶𝑇 𝛿 ⩽ . (111) 𝜆 𝑛 {𝑠𝑘} 𝑛 ∞ 𝑛 3 𝐸𝑝,𝑞 (R )⊂𝐵𝑝,𝑞 (R )⊂𝐶 (R ), (A.4)

𝑘 Then, we have if 𝑠𝑘 ⩽𝜆2/3𝑘, 𝑘∈𝑁.

‖I𝑢‖𝐿𝑟(0,𝑇;𝑋𝑠 ) <𝛿. 𝑝,1 (112) Proof. One can refer to [12].

Moreover, we have that Theorem E (embedding). Assume 𝑠1,𝑠2 ∈ R, 0<𝑝1,𝑝2, 𝑞 ,𝑞 ⩽∞ 2 1 2 . ‖I𝑢−I𝜐‖ 𝑟 𝑠 ⩽ . 𝑠1 𝑠2 𝐿 (0,𝑇;𝑋𝑝,1) 𝑠 ⩽𝑠 𝑝 ⩽𝑝,𝑞 ⩽𝑞 𝑀 ⊂𝑀 3‖𝑢−𝜐‖ 𝑟 𝑠 (113) (a) If 2 1, 1 2 1 2,then 𝑝 ,𝑞 𝑝 ,𝑞 . 𝐿 (0,𝑇;𝑋𝑝,1) 1 1 2 2 𝑠 𝑠 𝑞 <𝑞 𝑠 −𝑠 >𝑛/𝑞 −𝑛/𝑞 𝑀 1 ⊂𝑀2 (b) If 2 1, 1 2 2 1,then 𝑝,𝑞1 𝑝,𝑞2 . In this way, we prove that I :(D,𝑑)→(D,𝑑) is a strict (c) If 0⩽𝑠<∞, 0<𝑝1 ⩽𝑝2 ⩽∞, 0<𝑞1 ⩽𝑞2 ⩽∞, contraction map. By Banach’s fixed-point theorem, there 𝑠 𝑠 then 𝐸𝑝 ,𝑞 ⊂𝐸𝑝 ,𝑞 . exists a unique solution 𝑢∈D that satisfies the conditions 1 1 2 2 12 Journal of Function Spaces and Applications

Proof. One can refer to [12, 18, 24]. [8] M. Nakamura and T. Ozawa, “Nonlinear Schrodinger¨ equations in the Sobolev space of critical order,” Journal of Functional ∞ Theorem F. Assume 0<𝑝<∞, 0<𝑞⩽∞; Ω={Ω𝑘}𝑘=0 Analysis,vol.155,no.2,pp.364–380,1998. 𝑛 is a sequence with compact support in R ;let𝑑𝑘 >0be the [9] C. E. Kenig, G. Ponce, and L. Vega, “On the IVP for the nonlin- diameter of Ω𝑘.If𝑥>𝑛(𝑛/min(1, 𝑝, 𝑞)−1/2), then there exists ear Schrodinger¨ equations,” in Harmonic Analysis and Operator aconstant𝐶 such that Theory (Caracas, 1994),vol.189ofContemporary Math,pp. 󵄩 −1 󵄩 󵄩 󵄩 󵄩 󵄩 353–367, AMS, Providence, RI, USA, 1995. 󵄩𝐹 𝑀 𝐹𝑓 󵄩 ⩽𝐶 󵄩𝑀 (𝑑 ⋅)󵄩 𝑥 󵄩𝑓 󵄩 𝑝 𝑞 , 󵄩 𝑘 𝑘󵄩𝐿𝑝(ℓ𝑞) sup𝑖󵄩 𝑖 𝑖 󵄩𝐻 󵄩 𝑘󵄩𝐿 (ℓ ) (A.5) [10] H. Triebel, TheoryofFunctionSpace,Birkhauser,¨ Basel, Switzer- ∞ 𝑝 𝑞 ∞ 𝑥 land, 1992. where {𝑓𝑘} ∈𝐿 (ℓ ), {𝑀𝑘(𝑥)} ⊂𝐻 . 𝑘=0 Ω 𝑘=0 [11] H. Triebel, Function Space Theory,Birkhauser,¨ Basel, Switzer- 𝑝 𝑞 land, 1983. Proof. One can refer to [11]. Note that 𝐿Ω(ℓ )={𝑓|𝑓= ∞ 󸀠 [12] W. Baoxiang, Z. Lifeng, and G. Boling, “Isometric decomposi- {𝑓𝑘}𝑘=0 ⊂𝑆,supp𝐹𝑓𝑘 ⊂Ω𝑘 if 𝑘=0,1,2...and‖𝑓𝑘‖𝐿𝑝(ℓ𝑞) < 𝑋𝜆 ∞} tion operators, function spaces 𝑝,𝑞 and applications to nonlin- . ear evolution equations,” Journal of Functional Analysis,vol.233, no.1,pp.1–39,2006. Theorem G. Let 0<𝑝,𝑞⩽∞and (𝑋, 𝜇), (𝑌,𝜐) be two [13] B. Wang and H. Hudzik, “The global Cauchy problem for the measure space. Let 𝑇 be a positive linear operator mapping 𝑝 𝑞 𝑞,∞ NLS and NLKG with small rough data,” Journal of Differential 𝐿 (𝑋) into 𝐿 (𝑌) (resp., into 𝐿 (𝑌)) with norm 𝐴.Let𝐵 be ⃗ Equations,vol.231,no.1,pp.36–73,2007. aBanachspace.Then,𝑇 has a 𝐵-valued extension 𝑇 that maps 𝑝 𝑞 𝑞,∞ [14] B. Wang and C. Huang, “Frequency-uniform decomposition 𝐿 (𝑋, 𝐵) into 𝐿 (𝑌,𝐵) (resp., into 𝐿 (𝑌,𝐵))withthesame method for the generalized BO, KdV and NLS equations,” norm. Journal of Differential Equations,vol.239,no.1,pp.213–250, 2007. Proof. One can refer to [25]. [15] B. Wang, L. Han, and C. Huang, “Global well-posedness and scattering for the derivative nonlinear Schrodinger¨ equation Theorem H. 𝑠∈R 1⩽𝑝⩽∞ 0<𝑞<∞ Let , , ; then one has with small rough data,” Annales de l’Institut Henri Poincare´,vol. 󵄩 󵄩 𝑛|1/2−1/𝑝|󵄩 󵄩 26,no.6,pp.2253–2281,2009. 󵄩◻𝑘𝑆 (𝑡) 𝑓󵄩 𝑝 ≲ (1+|𝑡|) 󵄩◻𝑘𝑓󵄩 𝑝 𝐿 𝐿 [16] B. Wang, The Cauchy problem for the nonlinear Schrodinger (A.6) 󵄩 󵄩 𝑛|1/2−1/𝑝|󵄩 󵄩 equation, nonlinear Klein-Gordon equation and their coupled 󵄩𝑆 (𝑡) 𝑓󵄩𝑀𝑠 ≲ (1+|𝑡|) 󵄩𝑓󵄩𝑀𝑠 . 𝑝,𝑞 𝑝,𝑞 equations [doctoral thesis], Institute of Applied Physics and Computational Mathematics, Beijing, China, 1993. Proof. One can refer to [18]. [17] N. Wiener, “Tauberian theorems,” Annals of Mathematics. Sec- ond Series,vol.33,no.1,pp.1–100,1932. Acknowledgment [18] B. Wang, C. Hao, and Z. Huo, Harmonic Analysis Techniques for Nonlinear Evolution Equations, World Scientific, River Edge, NJ, This paper is supported by NSF of China (Grant no. USA, 2011. 11271330). [19] H. G. Feichtinger, “Modulation spaces on locally compact abelian groups,” Tech. Rep., University of Vienna, Vienna, Aus- References tria, 1983, published in: Proceedings of the International Con- ference on Wave let and Applications, New Delhi Allied Pub- [1] T. Kato, “On nonlinear Schrodinger¨ equations,” Annales de lishers, India, 2003. l’Institut Henri Poincare´,vol.46,no.1,pp.113–129,1987. [20] R. S. Strichartz, “Restrictions of Fourier transforms to quadratic [2] T. Cazenave and F. B. Weissler, “The Cauchy problem for the surfaces and decay of solutions of wave equations,” Duke 𝑠 critical nonlinear Schrodinger¨ equation in 𝐻 ,” Nonlinear Anal- Mathematical Journal,vol.44,no.3,pp.705–714,1977. ysis:Theory,MethodsandApplicationsA, vol. 14, no. 10, pp. 807– [21] H. Pecher, “Nonlinear small data scattering for the wave and 836, 1990. Klein-Gordon equation,” Mathematische Zeitschrift,vol.185,no. [3] T. Cazenave, Semilinear Schrodinger¨ Equations,American 2,pp.261–270,1984. Mathematical Society, New York, NY, USA, 2004. [22] J. Ginibre and G. Velo, “Generalized Strichartz inequalities for [4] T. Cazenave and F. B. Weissler, “Some remarks on the nonlinear the wave equation,” Journal of Functional Analysis,vol.133,no. Schrodinger¨ equation in the critical case,” in Nonlinear Semi- 1,pp.50–68,1995. groups, Partial Differential Equations and Attractors (Washing- [23] D. Li and Y. Chen, Nonlinear Evolution Equation, Science press, ton, DC, 1987),vol.1394ofLecture Notes in Mathematics,pp. Beijing, China, 1989, (Chinese). 18–29, Springer, Berlin, Germany, 1989. 𝑠 [24] J. Han and B. Wang, “𝛼 modulation space (I) ,” http://arxiv.org/ [5] T. Kato, “On nonlinear Schrodinger¨ equations. II. 𝐻 -solutions abs/1108.0460. and unconditional well-posedness,” Journal d’Analyse Mathe-´ matique,vol.67,pp.281–306,1995. [25] L. Grafakos, Classical and Modern Fourier Analysis,Pearson/ Prentice Hall, Upper Saddle River, NJ, 2004. [6] T. 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Research Article The Use of an Isometric Isomorphism on the Completion of the Space of Henstock-Kurzweil Integrable Functions

Luis Ángel Gutiérrez Méndez, Juan Alberto Escamilla Reyna, Francisco Javier Mendoza Torres, and María Guadalupe Morales Macías Facultad de Ciencias F´ısico Matematicas,´ Benemerita´ Universidad, Autonoma´ de Puebla, Avenida San Claudio y 18 Sur, Colonia San Manuel, 72570, Puebla, Mexico

Correspondence should be addressed to Luis Angel´ Gutierrez´ Mendez;´ [email protected]

Received 19 April 2013; Accepted 5 June 2013

Academic Editor: Nelson Merentes

Copyright © 2013 Luis Angel´ Gutierrez´ Mendez´ et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Employing an isometrically isomorphic space, we determine new properties for the completion of the space of the Henstock- Kurzweil integrable functions with the Alexiewicz norm.

1. Introduction space of the distributions that are derivatives of the continuous functions on [𝑎, 𝑏], which is an isometrically isomorphic Let [𝑎, 𝑏] be a compact interval in R.Inthevectorspaceof (HK̂[𝑎, 𝑏], ‖ ⋅‖ ) Henstock-Kurzweil integrable functions on [𝑎, 𝑏] with values space to 𝐴 . Making use of this same isometri- in R, the Alexiewicz seminorm is defined as cally isomorphic space, Bongiorno and Panchapagesan in [4] 󵄨 󵄨 establish characterizations for the relatively weakly compact 󵄩 󵄩 󵄨 𝑟 󵄨 󵄩𝑓󵄩 = 󵄨∫ 𝑓󵄨 . (HK[𝑎, 𝑏], ‖ ⋅‖ ) (HK̂[𝑎, 𝑏], ‖ ⋅‖ ) 󵄩 󵄩𝐴 sup 󵄨 󵄨 (1) subsets of 𝐴 and 𝐴 . 𝑎≤𝑟≤𝑏 󵄨 𝑎 󵄨 ̂ In this paper, we make an analysis on (HK[𝑎, 𝑏], ‖𝐴 ⋅‖ ) The corresponding normed space is built using the by means of another isometrically isomorphic space to prove 𝑓∼𝑔 ̂ quotient space determined by the relation if and only if that (HK[𝑎, 𝑏], ‖𝐴 ⋅‖ )has the Dunford-Pettis property, it has 𝑓=𝑔 , except in a set of Lebesgue measure zero or, equivalently, a complemented subspace isomorphic to 𝑐0, it does not have the if they have the same indefinite integral. This normed space Radon-Riesz property, it is not weakly sequentially complete, (HK[𝑎, 𝑏], ‖ ⋅‖ ) will be denoted by 𝐴 . and it is not isometrically isomorphic to the dual of any normed It is known that (HK[𝑎, 𝑏], ‖𝐴 ⋅‖ ) is neither complete nor ̂ space; hence, we will also prove that (HK[𝑎, 𝑏], ‖𝐴 ⋅‖ )is of the second category [1]. However, (HK[𝑎, 𝑏], ‖𝐴 ⋅‖ ) is a neither reflexive nor has the Schur property.Then,asanappli- separable space [1] and, consequently, its completion also has ̂ cation of the above results, we prove that (HK[𝑎, 𝑏], ‖𝐴 ⋅‖ ) the same property. In addition, (HK[𝑎, 𝑏], ‖𝐴 ⋅‖ ) has “nice” properties, from the point of view of functional analysis, since is not isomorphic to the dual of any normed space and that (HK[𝑎, 𝑏], ‖ ⋅‖ ) the space of all bounded, linear, weakly compact operators it is an ultrabornological space [2]. As 𝐴 is ̂ not complete, it is natural to study its completion, which will from (HK[𝑎, 𝑏], ‖𝐴 ⋅‖ ) into itself is not a complemented (HK̂[𝑎, 𝑏], ‖ ⋅‖ ) subspace in the space of all bounded, linear operators from be denoted by 𝐴 . ̂ Talvila in [3] makes an analysis to determine some pro- (HK[𝑎, 𝑏], ‖𝐴 ⋅‖ ) into itself. ̂ perties of the Henstock-Kurzweil integral on (HK[𝑎, 𝑏], ‖⋅‖ ) 𝐴 ,suchasintegrationbyparts,Holder¨ inequality, change 2. Preliminaries of variables, convergence theorems, the Banach lattice structure, the Hake theorem, the Taylor theorem, and second mean In this section, we restate the conventions, notations, and value theorem. Talvila makes this analysis by means of the concepts that will be used throughout this paper. 2 Journal of Function Spaces and Applications

All the vector spaces are considered over the field of the Definition 4. Let 𝑌 be a subspace of a normed space 𝑋.Itis real numbers or complex numbers. said that 𝑌 is complemented in 𝑋 if it is closed in 𝑋 and there ∗ Let 𝑋 be a normed space. By 𝑋 , we denote the dual exists a closed subspace 𝑊 in 𝑋 such that 𝑋=𝑌⊕𝑊. space of 𝑋. A topological property that holds with respect totheweaktopologyof𝑋 is said to be a weak property or Theorem 5 (see [5]). Let 𝑋 be a Banach space with the to hold weakly. On the other hand, if a topological property Dunford-Pettis property. If 𝑌 is a complemented subspace in holds without specifying the topology, the norm topology is 𝑋,then𝑌 has the Dunford-Pettis property. implied. ̂ Let 𝑋, 𝑌 be two Banach spaces. We denote by 𝐿(𝑋, 𝑌) To prove that (HK[𝑎, 𝑏], ‖𝐴 ⋅‖ ) has a complemented sub- (𝑊(𝑋,, 𝑌) resp.) we denote the Banach space of all bounded, space isomorphic to 𝑐0 the following result is essential. linear (bounded, linear, weakly compact resp.), operators from 𝑋 into 𝑌.If𝑋=𝑌,thenwewrite𝐿(𝑋) (resp. 𝑊(𝑋)) Theorem 6 (see [6]). Let 𝐾 be a compact metric space. If 𝑋 is instead of 𝐿(𝑋, 𝑋) (resp. 𝑊(𝑋,). 𝑋) an infinite-dimensional complemented subspace of C(𝐾),then 𝑋 𝑐 The symbols 𝑐0, 𝑙1,and𝑙∞ represent, as usual, the contains a complemented subspace isomorphic to 0. vector spaces of all sequences of scalars convergent to 0, all On the other hand, making use again of Theorem 2 we sequences of scalars absolutely convergent, and all bounded ̂ sequences of scalars, respectively, neither one with nor usual will prove that (HK[𝑎, 𝑏], ‖𝐴 ⋅‖ ) is neither weakly sequen- norm. tially complete nor has the Radon-Riesz property. Let 𝐾 be a compact metric space. We denote by C(𝐾) the 𝑋 {𝑥 } vector space of all continuous functions of scalar-values on 𝐾 Definition 7. Let be a normed space, and let 𝑛 be a 𝑋 𝑥∈𝑋 together with the norm defined by ‖𝐹‖∞ = sup{|𝐹(𝑥)| : 𝑥∈ sequence in and . 𝐾} . (i) If {𝑥𝑛} weakly converges whenever {𝑥𝑛} is weakly B [𝑎, 𝑏] By 𝑐 we denote the following collection: Cauchy, then it is said that 𝑋 is weakly sequentially complete. {𝐹:[𝑎,] 𝑏 󳨀→ R |𝐹is continuous on [𝑎,] 𝑏 and 𝐹 (𝑎) =0} {𝑥 } 𝑥 {𝑥 } (2) (ii) If 𝑛 converges to whenever 𝑛 weakly converges to 𝑥 and ‖𝑥𝑛‖ → ‖𝑥‖,thenitissaidthat𝑋 has the which is a closed subspace of C[𝑎, 𝑏] and (B𝑐[𝑎, 𝑏], ‖∞ ⋅‖ ) is Radon-Riesz property or the Kadets-Klee property. therefore a Banach space. (iii) If {𝑥𝑛} converges to 𝑥 whenever {𝑥𝑛} weakly converges to 𝑥,thenitissaidthat𝑋 has the Schur property. Definition 1. Let 𝑋𝑎𝑛𝑑𝑌be normed spaces and let 𝑇:𝑋 → 𝑌 be a lineal operator. We have the following. The following result establishes a characterization of (i) 𝑇 is an isomorphism if it is one-to-one and continuous weakly Cauchy sequences and weakly convergent sequences −1 and its inverse mapping 𝑇 is continuous on the of the space C[𝑎, 𝑏]. range of 𝑇.Moreover,if‖𝑇(𝑥)‖ = ,forall‖𝑥‖ 𝑥∈𝑋, Theorem 8 {𝐹 } 𝐹 it is said that 𝑇 is an isometric isomorphism. (see [7]). Let 𝑛 and be a sequence and an C[𝑎, 𝑏] 𝑋 𝑌 𝑋≅𝑌 element, respectively, in the space .Thenwehavethe (ii) and are isomorphic, which is denoted by , following. if there exists an isomorphism from 𝑋 onto 𝑌. (1) The sequence {𝐹𝑛} is weakly convergent to 𝐹 if and only (iii) 𝑋 and 𝑌 are isometrically isomorphic if there exists an if isometric isomorphism from 𝑋 onto 𝑌. (i) lim𝑛→∞𝐹𝑛(𝑥) = 𝐹(𝑥),forall𝑥∈[𝑎,𝑏], The following result is key to our principal results. (ii) there exists 𝑀>0such that ‖𝐹𝑛‖∞ ≤𝑀,forall 𝑛∈N. Theorem 2 (see [4]). The space (B𝑐[𝑎, 𝑏], ‖∞ ⋅‖ ) is isometri- ̂ cally isomorphic to (HK[𝑎, 𝑏], ‖𝐴 ⋅‖ ). (2) The sequence {𝐹𝑛} isweaklyCauchyifandonlyif ̂ 𝐹 (𝑥) 𝑥∈[𝑎,𝑏] According to Theorem 2,wewillprovethat(HK[𝑎, 𝑏], (i) lim𝑛→∞ 𝑛 exists, for all , 𝑀>0 ‖𝐹 ‖ ≤𝑀 ‖⋅‖𝐴) has the Dunford-Pettis property. (ii) there exists such that 𝑛 ∞ ,forall 𝑛∈N. Definition 3. Let 𝑋 be a Banach space. It is said that 𝑋 has ̂ We will also prove that (HK[𝑎, 𝑏], ‖𝐴 ⋅‖ ) is not isomet- the Dunford-Pettis property if for every sequence {𝑥𝑛} in ∗ ∗ ricallyisomorphictothedualofanynormedspaceand,as 𝑋 converging weakly to 0 and every sequence {𝑥𝑛 } in 𝑋 ∗ (HK̂[𝑎, 𝑏], ‖ ⋅‖ ) converging weakly to 0, the sequence {𝑥𝑛 (𝑥𝑛)} converges to aconsequence,wewillprovethat 𝐴 is not 0. reflexive, for which we will use again Theorem 2 and the concept of extremal point. If a Banach space 𝑋 has the Dunford-Pettis property, then not necessarily every closed subspace of 𝑋 inherits such Definition 9. Let 𝑋 be a vector space, 𝐾⊆𝑋,and𝑧∈𝐾.It property, except when the subspace is complemented in 𝑋 is said that 𝑧 is an extremal point of 𝐾 if for all 𝑥, 𝑦 ∈𝐾 such [5]. that 𝑧 = (1/2)(𝑥 +𝑦) it holds that 𝑧=𝑥=𝑦. Journal of Function Spaces and Applications 3

If 𝑋 is a normed space, then its closed unit ball will be However, since 𝐺 is not continuous, it follows that the denoted by 𝐵𝑋 and the collection of all extremal points of 𝐵𝑋 sequence {𝐹𝑛} does not converge weakly in (B𝑐[𝑎, 𝑏], ‖∞ ⋅‖ ). as ext(𝐵𝑋). The following theorem establishes that the extremal points are preserved under isometric isomorphisms. Lemma 15. The space (B𝑐[𝑎, 𝑏], ‖∞ ⋅‖ ) does not have the Radon-Riesz property. Theorem 10 (see [8]). Let 𝑋𝑎𝑛𝑑𝑌be Banach spaces, 𝐾⊆𝑋 and let 𝑇:𝑋be →𝑌 an isometric isomorphism. Then 𝑥 is an Proof. Without loss of generality, suppose that |𝑏−𝑎|.Let ≥1 extremal point of 𝐾 if and only if 𝑇(𝑥) is an extremal point of 𝐹𝑛 be the function defined by 𝑇(𝐾). 𝐹𝑛 (𝑥) Corollary 11 (see [9]). An infinite-dimensional normed space 𝑛 1 whoseclosedunitballhasonlyfinitelymanyextremepointsis {( ) (𝑥−𝑎) , 𝑥∈[𝑎,𝑏− ], { 𝑛 (𝑏−𝑎) −1 if 𝑛 not isometrically isomorphic to the dual of any normed space. { { 1 1 = {2𝑛 (𝑏−𝑥) −1, if 𝑥∈(𝑏− ,𝑏− ], { 𝑛 2𝑛 3. Principal Results { { 1 2𝑛 (𝑥−𝑏) +1, if 𝑥∈(𝑏− ,𝑏], Lemma 12. The space (B𝑐[𝑎, 𝑏], ‖∞ ⋅‖ ) has the Dunford- { 2𝑛 Pettis property. (6)

Proof. Let 𝐻:C[𝑎, 𝑏] → R be the functional defined by for all 𝑛≥2.Thus, 𝐻(𝐹) = 𝐹(𝑎).Itisclearthat𝐻 is bounded and therefore {𝐹 } 𝐹 ker(𝐻) is a hyperplane of the space C[𝑎, 𝑏],thatis, (i) the sequence 𝑛 converges pointwise to ,where 𝐹(𝑥) = (𝑥 − 𝑎)/(𝑏 −𝑎) for all 𝑥∈[𝑎,𝑏], C [𝑎,] 𝑏 = ker (𝐻) ⊕𝑊, (3) (ii) ‖𝐹𝑛‖∞ =1,forall𝑛≥2. where ker(𝐻) = {𝐹 ∈ C[𝑎, 𝑏] : 𝐹(𝑎) =0} B𝑐[𝑎, 𝑏] and 𝑊 is a one-dimension, subspace in C[𝑎, 𝑏].Then,as Therefore, according to Theorem 8 item (1),itfollowsthat C[𝑎, 𝑏] has the Dunford-Pettis property [10] and according the sequence {𝐹𝑛} converges weakly to 𝐹 in (B𝑐[𝑎, 𝑏], ‖∞ ⋅‖ ); to Theorem 5, we obtain the desired conclusion. in addition, as ‖𝐹‖∞ =1,itholdsthat‖𝐹𝑛‖∞ →‖𝐹‖∞. However, the sequence {𝐹𝑛} does not converge to 𝐹 in Lemma 13. The space (B𝑐[𝑎, 𝑏], ‖∞ ⋅‖ ) has a complemented (B𝑐[𝑎, 𝑏], ‖∞ ⋅‖ ). subspace isomorphic to 𝑐0. It is not difficult to prove that if a Banach space has Proof. Using the proof of Lemma 12,wecanseethatB𝑐[𝑎, 𝑏] the Dunford-Pettis property, or if it has a complemented is a complemented subspace in C[𝑎, 𝑏].Then,accordingto subspace isomorphic to 𝑐0,orifitisnotweaklysequentially Theorem 6, we obtain the desired conclusion. complete,orifithastheRadon-Rieszproperty,thenthese properties are preserved under isometric isomorphisms. Lemma 14. (B [𝑎, 𝑏], ‖ ⋅‖ ) The space 𝑐 ∞ is not weakly sequen- Therefore, according to Theorem 2 and Lemmas 12, 13, 14 and tially complete. 15, we obtain the following result. Proof. Without loss of generality, suppose that |𝑏−𝑎|.Let ≥1 ̂ Proposition 16. The space (HK[𝑎, 𝑏], ‖𝐴 ⋅‖ ) 𝐹𝑛 be the function defined by 1 (1) has the Dunford-Pettis property, {0, if 𝑥∈[𝑎,𝑏− ], 𝐹 (𝑥) = 𝑛 (2) has a complemented subspace isomorphic to 𝑐0, 𝑛 { 1 (4) 𝑛 (𝑥−𝑏) +1, 𝑥∈(𝑏− ,𝑏], { if 𝑛 (3) is not weakly sequentially complete, (4) does not have the Radon-Riesz property. for all 𝑛∈N.Thus,

(i) lim𝑛→∞𝐹𝑛(𝑥) exists, for all 𝑥∈[𝑎,𝑏], Remark 17. According to Definition 7 and Proposition 16 ̂ (ii) ‖𝐹𝑛‖∞ =1,forall𝑛∈N. item (4),itfollowsthat(HK[𝑎, 𝑏], ‖𝐴 ⋅‖ ) does not have the Schur property. Therefore, according to Theorem 8 item (2),itfollowsthat {𝐹 } (B [𝑎, 𝑏], ‖ ⋅‖ ) the sequence 𝑛 is weakly Cauchy in 𝑐 ∞ . Lemma 18. The collection of all extremal points of the closed Now, suppose that there exists a function 𝐺∈ unit ball of the space (B𝑐[𝑎, 𝑏], ‖∞ ⋅‖ ) is empty. (B𝑐[𝑎, 𝑏], ‖∞ ⋅‖ ) such that the sequence {𝐹𝑛} converges 𝐺 (1) weakly to . Then according to Theorem 8 item ,itholds Proof. Let 𝐹∈𝐵B [𝑎,𝑏].Since𝐹 is continuous on [𝑎, 𝑏],it 𝐺(𝑥) = 𝐹 (𝑥) 𝑥∈[𝑎,𝑏] 𝑐 that lim𝑛→∞ 𝑛 ,forall ,thatis, holds that for 𝜀=1/2there exits 𝛿>0such that 0, if 𝑥∈[𝑎,) 𝑏 , 1 𝐺 (𝑥) ={ (5) |𝐹 (𝑥)| < ,∀𝑥∈[𝑎, 𝑎) +𝛿 . 1, if 𝑥=𝑏. 2 (7) 4 Journal of Function Spaces and Applications

Now, define the following functions: space. However, we can ask ourselves the following. Is there a ̂ normed space X such that (HK[𝑎, 𝑏], ‖𝐴 ⋅‖ ) is isomorphic 𝐹 (𝑥) +𝑟(𝑥) , 𝑥∈[𝑎, 𝑎] +𝛿 , 𝐺 (𝑥) ={ if to the dual of 𝑋? To answer this question, we need of the 𝐹 (𝑥) , if 𝑥∈[𝑎+𝛿,𝑏] , following result. (8) 𝐹 (𝑥) −𝑟(𝑥) , 𝑥∈[𝑎, 𝑎] +𝛿 , Lemma 23 𝑋 𝑌 𝐻 (𝑥) ={ if (see [7]). Let be a normed space and let be a 𝐹 (𝑥) , 𝑥∈[𝑎+𝛿,𝑏] , Banach space with the Dunford-Pettis property that does not if ∗ have the Schur property. If 𝑋 contains a copy of 𝑌,then𝑋 where the function 𝑟 can be any continuous function defined contains a copy of 𝑙1. over the interval [𝑎, 𝑎 + 𝛿] such that 𝑟(𝑎) = 0 = 𝑟(𝑎 +𝛿) and ‖𝑟‖ <1/2 ̂ ∞ . Proposition 24. The space (HK[𝑎, 𝑏], ‖𝐴 ⋅‖ ) is not isomor- Since the functions 𝐺 and 𝐻 are continuous, ‖𝐺‖∞ =1= phic to the dual of any normed space. ‖𝐻‖∞,and𝐹 = (1/2)(𝐺,itholdsthat +𝐻) 𝐹 cannot be an (𝐵 )=0 𝑋 extremal point; therefore, ext B𝑐[𝑎,𝑏] . Proof. Suppose that there exists a normed space such that (𝐵 ) ̂ ∗ It is a known fact that the collection ext C[𝑎,𝑏] is formed (HK [𝑎,] 𝑏 , ‖⋅‖𝐴)≅𝑋 . (9) only by the constant functions ±1. However, since in general there is not a relationship between the extremal points of the Since (HK[𝑎, 𝑏], ‖𝐴 ⋅‖ ) is separable [1]andconsequently ̂ closed unit ball of a subspace with the extremal points of (HK[𝑎, 𝑏], ‖𝐴 ⋅‖ ) also is separable, it follows from (9)that ∗ theclosedunitballofallspace,weneedLemma 18 for the 𝑋 is separable. following result. On the other hand, by Proposition 16 item (2) and ∗ according to the isomorphism from (9), it holds that 𝑋 has Proposition 19. (HK̂[𝑎, 𝑏], ‖ ⋅‖ ) The space 𝐴 is not isometri- a complemented subspace isomorphic to 𝑐0.Since𝑐0 has the callyisomorphictothedualofanynormedspace. Dunford-Pettis property [9] and does not have the Schur property [9], it holds that 𝑋 has a copy of 𝑙1,byLemma 23. ∗ Proof. By Lemma 18 and Theorems 2 and 10,itholdsthat As 𝑋 has a copy of 𝑙1,itholdsthat𝑙1 is isometrically ̂ ∗ ⊥ ⊥ ext(𝐵HK̂[𝑎,𝑏])=0. Then, as the space (HK[𝑎, 𝑏], ‖𝐴 ⋅‖ ) is isomorphic to 𝑋 /𝑙1 ,where𝑙1 denotes the annihilator of 𝑙1. ∗ dimensionality infinite and according to Corollary 11,we Since 𝑙1 is isometrically isomorphic to 𝑙∞ it holds that, in obtain the desired conclusion. particular, ̂ 𝑋∗ Corollary 20. The space (HK[𝑎, 𝑏], ‖𝐴 ⋅‖ ) is not reflexive. 𝑙 ≅ . ∞ ⊥ (10) 𝑙1 ̂ Proof. Suppose that the space (HK[𝑎, 𝑏], ‖𝐴 ⋅‖ ) is reflex- ∗ ̂ Therefore, since 𝑋 is separable and according to the ive. Then (HK[𝑎, 𝑏], ‖𝐴 ⋅‖ ) coincides under the canoni- isomorphism from (10), we obtain that 𝑙∞ is separable, which cal imbedding with its second dual. Therefore, the space ̂ is a contradiction. (HK[𝑎, 𝑏], ‖𝐴 ⋅‖ ) is isometrically isomorphic to the dual of ̂ ∗ On this way, we can see that Proposition 19 and the space (HK[𝑎, 𝑏], ‖𝐴 ⋅‖ ) , which is a contradiction by Proposition 19. Corollary 20 are consequences of the above result. We did not do it this way because one of the principal objectives of In general, it is important to know when a Banach space this paper is to show the importance of knowing explicitly a enjoys certain functional analysis properties. However, in closed subspace of C[𝑎, 𝑏] which is isometrically isomorphic ̂ certain contexts, also it is useful to know when a Banach space to (HK[𝑎, 𝑏], ‖𝐴 ⋅‖ ) anditisaknownfactthatevery does not have certain properties; Propositions 22 and 24 are separable Banach space of infinite dimension is isometrically examples of both facts. isomorphic to a closed subspace of C[𝑎, 𝑏];however,this information is not sufficient to prove, in particular, the results Lemma 21 𝑋, 𝑌 (see [11]). Let be two Banach spaces. Assume that we have shown in this paper. that 𝑋 and 𝑌 contain a complemented copy of 𝑐0.Then𝑊(𝑋, 𝑌) is uncomplemented in 𝐿(𝑋,. 𝑌) References Proposition 22. 𝑊(HK̂[𝑎, 𝑏]) The space is uncomplemented [1] C. Swartz, Introduction to Gauge Integrals, World Scientific, ̂ in the space 𝐿(HK[𝑎, 𝑏]). Singapore, 2001. [2] J. L. Gamez,´ Integracionesdedenjoydefuncionesconvaloresen Proof. According to Proposition 16 item (2),itholdsthat espacios de banach [Ph.D. thesis], Universidad Complutense de ̂ (HK[𝑎, 𝑏], ‖𝐴 ⋅‖ ) contains a complemented copy of 𝑐0. Madrid, 1997. Therefore, on the basis of Lemma 21, we obtain the desired [3] E. Talvila, “The distributional Denjoy integral,” Real Analysis conclusion. Exchange,vol.33,no.1,pp.51–82,2008. ̂ [4] B. Bongiorno and T. V. Panchapagesan, “On the Alexiewicz By Proposition 19,wecanseethat(HK[𝑎, 𝑏], ‖𝐴 ⋅‖ ) is topology of the Denjoy space,” Real Analysis Exchange,vol.21, not isometrically isomorphic to the dual of any normed no. 2, pp. 604–614, 1995-1996. Journal of Function Spaces and Applications 5

[5] J. Diestel, “A survey of results related to the Dunford-Pettis property,” AMS Contemporary Mathematics,vol.2,pp.15–60, 1980. [6] F. Albiac and N. J. Kalton, Topics in Banach Space Theory, Springer,NewYork,NY,USA,2006. [7] J. Diestel, Sequences and Series in Banach Spaces, Springer, New York, NY, USA, 1984. [8] A. Curnock, “An introduction to extreme points and applications in isometric Banach space theory,” in Proceedings of the the Analysis Group, Goldsmiths College, University of London, May 1998, part of early doctoral work. [9]R.E.Megginson,An Introduction to Banach Space Theory, Springer,NewYork,NY,USA,1998. [10] R. G. Bartle, N. Dunford, and J. Schwartz, “Weak compactness and vector measures,” Canadian Journal of Mathematics,vol.7, pp.289–305,1955. [11] G. Emmanuele, “Remarks on the uncomplemented subspace 𝑊(𝐸,,” 𝐹) Journal of Functional Analysis,vol.99,no.1,pp.125– 130, 1991. Hindawi Publishing Corporation Journal of Function Spaces and Applications Volume 2013, Article ID 237858, 8 pages http://dx.doi.org/10.1155/2013/237858

Research Article Generalized Virtually Stable Maps and Their Associated Sequences

P. Chaoha,1,2 S. Iampiboonvatana,1 and J. Intrakul1

1 Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand 2 Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand

Correspondence should be addressed to P. Chaoha; [email protected]

Received 18 April 2013; Accepted 8 June 2013

Academic Editor: Janusz Matkowski

Copyright © 2013 P. Chaoha et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We extend the concept of virtual stability of continuous self-maps to arbitrary selfmaps and investigate the structure of sequences associated with uniformly virtually stable selfmaps. We also obtain a necessary and sufficient condition for a uniformly virtually stable selfmap to have the largest possible associated sequence. Examples of a uniformly virtually stable selfmap having the prescribed largest sequence and a uniformly virtually stable selfmap having no largest sequence are given.

1. Introduction stabilityaswellastotakecareofthepreviosquestion.We will first redefine the notion of virtual stability to include Virtual stability of a self-map was first introduced in1 [ ], discontinuous self-maps and investigate some properties of where it was proved to unify various (continuous) types of these newly defined virtually stable self-maps and prove nonexpansiveness in fixed-point theory. The main feature of the expected connection between the convergence set and a virtually stable self-map is that its fixed-point set is always a the fixed-point set. Then we will show that the sequence retract of its convergence set, and this fact immediately allows associated with a uniformly virtually stable self-map can be us to connect topological structures (e.g., connectedness made, in some sense, largest possible provided that the set of and contractibility) of the fixed-point set to those of the all associated sequences has a certain structure. We will also convergence set. Although the original definition of virtual see that when the largest sequence exists, it may not be (𝑛) stability requires the continuity of a self-map, it seems that the unless the map is continuous. For the sake of completeness, connection between the fixed-point set and the convergence since the existence of the largest possible sequence associated setalsoremainsvalidtosomedegreeindiscontinuous with a given self-map cannot be guaranteed in general, we settings as hinted in [2, Corollary 2.6] and the work on will provide two constructive examples of uniformly virtually mean-type mappings in [3–5](seeExample4 below). A stable self-maps: one having the prescribed largest sequence careful investigation of this fact will definitely result in a and the other having no largest sequence. more powerful notion of virtual stability that enables us to study fixed-point sets of interesting discontinuous self-maps (e.g., Suzuki generalized nonexpansive maps [6]). Moreover, 2. Virtual Stability without Continuity since all well-known self-maps in metric fixed-point theory Let 𝑋 be a (nonempty) Hausdorff space and 𝑓:𝑋a →𝑋 are always uniformly virtually stable with respect to the self-map (that may not be continuous). We will also let N(𝑥) same sequence (𝑛) of all natural numbers, it is quite natural and N𝑌(𝑥) denote the sets of (open) neighborhoods of 𝑥 in to ask whether this also holds in general for a uniformly 𝑋 and in 𝑌⊆𝑋, respectively. Recall that the fixed-point set virtually stable self-map. It turns out that the answer to and the convergence set of 𝑓 are defined, respectively, to be this simple question is highly nontrivial and interestingly 𝐹(𝑓)={𝑥:𝑓(𝑥)=𝑥}and involves algebraic properties of a certain object. Therefore, 𝑛 the purpose of this work is to revise the concept of virtual 𝐶 (𝑓) = {𝑥:the sequence (𝑓 (𝑥)) converges} , (1) 2 Journal of Function Spaces and Applications

𝑛 where 𝑓 denotes the 𝑛-th iterate of 𝑓.Wealsodefinethemap (2) If 𝑓 is uniformly quasi-lipschitzian, then it is uniformly ∞ 𝑓 : 𝐶(𝑓) →𝑋 by virtually stable with respect to (𝑛). 𝑓∞ (𝑥) = 𝑓𝑛 (𝑥) , 𝑓 (𝑋, 𝑑) 𝑛→∞lim (2) Proof. Suppose is a self-map on a metric space .

∞ (1) Let 𝑝∈𝐹(𝑓)and 𝜖>0. Then there is 𝐻∈N(𝑝) 𝑥∈𝐶(𝑓) 𝑓 𝑛 for all .Noticethat may not be continuous, and such that 𝑓 (𝐻) ⊆ 𝐵(𝑝; 𝜖) for all 𝑛∈N.Thus,for since we do not assume the continuity of 𝑓,weonlyhave 𝑛 ∞ 𝛿>0whose 𝐵(𝑝; 𝛿) ⊆𝐻,wehave𝑓 (𝐵(𝑝; 𝛿)) ⊆ 𝐹(𝑓) ⊆𝑓 (𝐶(𝑓)) 𝑛 𝑛 . However, in this work, we will always 𝑓 (𝐻) ⊆ 𝐵(𝑝; 𝜖) for all 𝑛∈N. This implies that {𝑓 } assume that is equicontinuous on 𝐹(𝑓) and hence 𝑓 is virtually 0 =𝐹(𝑓)=𝑓̸ ∞ (𝐶 (𝑓)) . nonexpansive. (3) 𝑛 (2) Suppose there is 𝐿>0such that 𝑑(𝑓 (𝑥), 𝑝) ≤ This condition is quite natural in the sense that it is automati- 𝐿𝑑(𝑥, 𝑝) for all (𝑥, 𝑝, 𝑛) ∈ 𝑋 × 𝐹(𝑓)× N.Itfollows 𝑛 cally satisfied whenever 𝑓 is continuous with 𝐹(𝑓) =0̸ ,andit that 𝑓 (𝐵(𝑝; 𝑟/𝐿)) ⊆ 𝐵(𝑝; 𝑟) for all 𝑝∈𝐹(𝑓), 𝑛∈N 𝑛 is also required for the iterative sequence (𝑓 ) to be a scheme and 𝑟>0.Thus,𝑓 is uniformly virtually stable with according to [2,Definition2.1].Withtheaboveassumption respect to (𝑛). in mind, we are able to previous define the concept of virtual stability just as in [1,Definition2.1]. It is not difficult to see that the class of uniformly quasi- Definition 1. Afixedpoint𝑥 of 𝑓 is said to be virtually 𝑓- lipschitzian self-maps includes various important (possibly stable if for each 𝑈∈N(𝑥), there exist 𝑉∈N(𝑥) together discontinuous) self-maps in metric fixed-point theory such with an increasing sequence (𝑎𝑛) of positive integers such that 𝑎 as nonexpansive maps, Kannan maps [8], Suzuki gener- 𝑓 𝑛 (𝑉) ⊆ 𝑈 for all 𝑛∈N.Wesimplycall𝑓virtually stable if alized nonexpansive maps [6], quasi-nonexpansive maps, every fixed point of 𝑓 isvirtuallystable.Moreover,wewillcall and even asymptotically quasi-nonexpansive maps. Hence, afixedpoint𝑥 of 𝑓uniformly virtually 𝑓-stable with respect the previous proposition immediately implies that those to an increasing sequence (𝑎𝑛) of positive integers if for each 𝑎 maps are all uniformly virtually stable with respect to (𝑛). 𝑈∈N(𝑥), there exists 𝑉∈N(𝑥) such that 𝑓 𝑛 (𝑉) ⊆ 𝑈 for all The following theorem directly extends [1,Theorem2.6]to 𝑛∈N.Wheneveryfixedpointof𝑓 is uniformly virtually 𝑓- include discontinuous maps. stable with respect to the same sequence, we will simply call 𝑓 uniformly virtually stable. Theorem 3. Suppose that 𝑋 is a regular space. If 𝑓 is virtually 𝑘 ∞ stable and 𝑓 is continuous for some 𝑘∈N,then𝑓 is Clearly, a uniformly virtually stable self-map is always 𝐹(𝑓) 𝐶(𝑓) virtually stable, and a continuously (uniformly) virtually continuous and hence is a retract of . stable self-map as defined in [1]isalso(uniformly)virtually ∞ Proof. Let 𝑥∈𝐶(𝑓)and 𝑈∈N𝐹(𝑓)(𝑓 (𝑥)).Since𝑋 is stable according to our new definition. Moreover, if 𝑓 is uni- ∞ regular, so is 𝐹(𝑓) andthenthereis𝑊∈N(𝑓 (𝑥)) such formly virtually stable with respect to (𝑎𝑛),itisimmediately that 𝑊∩𝐹(𝑓)⊆ 𝑊∩𝐹(𝑓)⊆.Byvirtualstabilityof 𝑈 uniformly virtually stable with respect to any subsequence of 𝑎 𝑓 𝑓 𝑛 (𝑉) ⊆ 𝑊 𝑛∈N (𝑎 ) (𝑎𝑛). , for all , for some sequence 𝑛 and 𝑉∈N(𝑓∞(𝑥)) 𝑉∈N(𝑓∞(𝑥)) Let us recall that, when (𝑋, 𝑑) is a metric space, a self-map .Alsothereis,bythefactthat , 𝑛 ∈ N 𝑓𝑛(𝑥) ∈ 𝑉 𝑛≥𝑛 𝐴= 𝑓:𝑋 →𝑋is called 0 such that for all 0.Then −𝑘𝑛0 𝑓 (𝑉) ∩𝐶(𝑓) ∈ N𝐶(𝑓)(𝑥) and for each 𝑎∈𝐴, (1) virtually nonexpansive [7]if𝑓 is continuous and the {𝑓𝑛} 𝑓 𝑓∞ (𝑎) = 𝑓𝑛 (𝑎) family of all iterates of is equicontinuous on 𝑛→∞lim 𝐹(𝑓) (equivalently, on 𝐶(𝑓)), 𝑎 𝑘𝑛 (5) 𝐿>0 = 𝑓 𝑛 (𝑓 0 (𝑎))∈𝑊∩𝐹(𝑓)⊆𝑈. (2) uniformly quasi-Lipschitzian if there is such that 𝑛→∞lim

𝑛 ∞ 𝑑(𝑓 (𝑥) ,𝑝)≤𝐿𝑑(𝑥,𝑝), (4) Thus, 𝑓 is continuous and hence 𝐹(𝑓) is a retract of 𝐶(𝑓).

for all (𝑥, 𝑝, 𝑛) ∈ 𝑋 × 𝐹(𝑓)× N. Example 4. Let 𝑝∈N and 𝑝≥2, and let 𝐼⊆R be an 𝑝 𝑝 Notice that a virtually nonexpansive self-map is always interval. Consider the subspace 𝐼 of R equipped with the continuousanduniformlyvirtuallystablewithrespecttothe maximum norm ‖(𝑥1,...,𝑥𝑝)‖ = max{|𝑥1|,...,|𝑥𝑝|}.Since (𝑛) 𝑝 𝑝 sequence of all natural numbers, while a uniformly quasi- the maximum norm on R also induces the usual topology, 𝐼 𝑝 𝑝 lipschitzian self-map may not be continuous in general. is a regular space. Recall that a self-map 𝑓:𝐼 →𝐼 is called 𝑝 mean-type if 𝑓=(𝑀1,...,𝑀𝑝),whereeach𝑀𝑖 :𝐼 → R is Proposition 2. Suppose 𝑓 is a self-map on a metric space 𝑝 a mean on 𝐼; that is, for each (𝑥1,...,𝑥𝑝)∈𝐼 , (𝑋, 𝑑).

(1) If 𝑓 is continuous and uniformly virtually stable with min {𝑥1,...,𝑥𝑝}≤𝑀𝑖 (𝑥1,...,𝑥𝑝)≤max {𝑥1,...,𝑥𝑝}. respect to (𝑛), then it is virtually nonexpansive. (6) Journal of Function Spaces and Applications 3

Notice that 𝑓 may not be continuous in general. However, if It is straightforward to verify that 𝑓 is quasi-nonexpansive 𝑛 𝑓 satisfies (hence, uniformly virtually stable) and 𝑓 is not continuous 𝑛∈N 𝑓∞ 𝑓∞(𝑥, 𝑦) = 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 for all .However, is continuous because 󵄨𝑀 (𝑥 ,...,𝑥 )−𝑧󵄨 ≤ {󵄨𝑥 −𝑧 󵄨 ,...,󵄨𝑥 −𝑧 󵄨} 2 󵄨 𝑖 1 𝑝 𝑖󵄨 max 󵄨 1 1󵄨 󵄨 𝑝 𝑝󵄨 (𝑥, |𝑥|) for all (𝑥, 𝑦) ∈ R . (∗) The next theorem gives a condition allowing us to enlarge (𝑥 ,...,𝑥 )∈𝐼𝑝 (𝑧 ,...,𝑧 )∈𝐹(𝑓) 𝑖= for all 1 𝑝 , 1 𝑝 and thesequenceassociatedwithagivenuniformlyvirtuallysta- 1,...,𝑝 , it is always quasi-nonexpansive (with respect to the ble self-map. As a result, we immediately obtain a refinement maximum norm), and hence uniformly virtually stable (with 𝑝 and a generalization of [1,Theorem2.17]. respect to the sequence (𝑛)). For if (𝑥1,...,𝑥𝑝)∈𝐼 and (𝑧 ,...,𝑧 )∈𝐹(𝑓) 1 𝑝 ,onehas Definition 8. An increasing sequence (𝑎𝑛) of natural numbers (𝑎 −𝑎):= 󵄩 󵄩 is said to satisfy the sup-finite condition if sup𝑛 𝑛+1 𝑛 󵄩𝑓(𝑥 ,...,𝑥 )−𝑓(𝑧 ,...,𝑧 )󵄩 󵄩 1 𝑝 1 𝑝 󵄩 sup{𝑎𝑛+1 −𝑎𝑛 :𝑛∈N}<∞. 󵄨 󵄨 = {󵄨𝑀 (𝑥 ,...,𝑥 )−𝑧󵄨 :𝑖=1,...,𝑝} 𝑘 max 󵄨 𝑖 1 𝑝 𝑖󵄨 Theorem 9. If, for some 𝑘∈N, 𝑓 is continuous on 𝐹(𝑓) and 󵄨 󵄨 󵄨 󵄨 (7) 𝑓 is uniformly virtually stable with respect to (𝑎𝑛𝑘) where (𝑎𝑛) ≤ {󵄨𝑥 −𝑧 󵄨 ,...,󵄨𝑥 −𝑧 󵄨}[ (∗)] max 󵄨 1 1󵄨 󵄨 𝑝 𝑝󵄨 by satisfies the sup-finite condition, then 𝑓 is uniformly virtually 󵄩 󵄩 (𝑛𝑘) 󵄩 󵄩 stable with respect to . = 󵄩(𝑥1,...,𝑥𝑝)−(𝑧1,...,𝑧𝑝)󵄩 . Proof. Let 𝑠=sup (𝑎𝑛+1 −𝑎𝑛), 𝑝∈𝐹(𝑓),and𝑈∈N(𝑝). 𝑓 : (0,2 ∞) →(0 𝑛 In particular, the mean-type mapping , Because 𝑓 is uniformly virtually stable with respect to (𝑎𝑛𝑘), ∞)2 𝑓(𝑥, 𝑦) = ((𝑥 + 𝑦)/2, 2𝑥𝑦/(𝑥 +𝑦)) 𝑎 𝑘 defined by satisfies there is 𝑊∈N(𝑝) such that 𝑓 𝑛 (𝑊) ⊆ 𝑈 for all 𝑛∈N.By (∗) 𝑓 𝑖𝑘 , and hence it is uniformly virtually stable. Since is also continuity at 𝑝 of 𝑓 , 𝑖∈N ∪{0}, there exists 𝑉∈N(𝑝) for continuous, 𝐹(𝑓) is a retract of 𝐶(𝑓) by the previous theorem. 𝑖𝑘 which 𝑓 (𝑉) ⊆ 𝑊 ∩𝑈 for all 𝑖 = 0,...,max{𝑠,1 𝑎 }.Nowlet Infact,itfollowsfrom[4]that𝐹(𝑓) = {(𝑥, 𝑥) :𝑥>0}, 𝐶(𝑓) = 𝑛∈N 𝑛≤𝑎 𝑓𝑛𝑘(𝑉) ⊆ 𝑊 ∩𝑈 ⊆𝑈 𝑛>𝑎 (0, ∞)2 𝑓∞(𝑥, 𝑦)√ =( 𝑥𝑦, √𝑥𝑦) (𝑥, 𝑦) ∈ (0,2 ∞) .If 1, ,whereasif 1, ,and for . 𝑗∈N 𝑛=𝑎 +𝑖 𝑖=0,...,𝑠 Therefore, 𝐹(𝑓) is clearly a retract of 𝐶(𝑓). there is such that 𝑗 for some and so 𝑛𝑘 𝑎 𝑘 𝑖𝑘 𝑎 𝑘 2 𝑓 (𝑉) =𝑓𝑗 (𝑓 (𝑉))⊆𝑓𝑗 (𝑊∩𝑈) Example 5. Consider that R equipped with the maximum 𝑓:R2 → R2 (11) norm, and a discontinuous self-map given by 𝑎 𝑘 ⊆𝑓𝑗 (𝑊) ⊆𝑈. (0, 1) if (𝑥, 𝑦) = (0, 3) ; 𝑓(𝑥,𝑦)={ (8) (𝑥, |𝑥|) otherwise. Corollary 10. Suppose that 𝑋 is a metric space. If 𝑓 is It is straightforward to verify that 𝑓 is quasi-nonexpansive continuous and uniformly virtually stable with respect to (𝑎𝑛), 𝑓2 (hence, uniformly virtually stable) and is continuous. where (𝑎𝑛) satisfies the sup-finite condition, then 𝑓 is virtually 𝑓∞ Therefore, by the previous theorem, is continuous and nonexpansive. In particular, 𝐶(𝑓) is a 𝐺𝛿-set when 𝑋 is 𝐹(𝑓) 𝐶(𝑓) 𝐹(𝑓) = hence is a retract of . Notice also that complete. {(𝑥, |𝑥|) :𝑥∈ R} is not convex. Proof. By setting 𝑘=1in the previous theorem, 𝑓 is The following examples show that if we drop the continu- (𝑛) 𝑘 ∞ uniformly virtually stable with respect to and hence ity assumption of some iterate 𝑓 in Theorem 3,themap𝑓 virtually nonexpansive by Proposition 2 (1). For, if 𝑋 is may or may not be continuous. complete, by [9,Theorem1.2],𝐶(𝑓) is then a 𝐺𝛿-set.

Example 6. Let 𝑓 : [0, 1] → [0, 1] be defined by Corollary 11. Suppose that 𝑋 is a complete metric space. If, for 1 some 𝑘∈N, 𝑓 is uniformly virtually stable with respect to (𝑛𝑘) {𝑥 𝑥< ; 𝑘 𝑘 if and 𝑓 is continuous, then 𝐶(𝑓 ) is a 𝐺𝛿-set. 𝑓 (𝑥) = { 2 (9) 1 . { otherwise 𝑘 Proof. Let 𝑔=𝑓.Then𝑔 is uniformly virtually stable with 𝑓 𝑓𝑛 =𝑓 𝑘 Observe that is uniformly virtually stable and is not respect to (𝑛).Thus,𝐶(𝑓 )=𝐶(𝑔)is a 𝐺𝛿-set by the previous continuous for all 𝑛∈N.Moreover,𝐹(𝑓) = [0, 1/2) ∪{1} and ∞ corollary. 𝐶(𝑓) = [0,.Hence 1] 𝑓 is not continuous. 2 3. Sequences Associated with Uniformly Example 7. Consider that R equipped with the supremum 2 2 norm. Define 𝑓:R → R by Virtually Stable Self-Maps 𝑛−1 𝑛 (𝑎 ) (0, 3 ) if (𝑥, 𝑦) = (0, 3 ) for some 𝑛∈N; In view of Proposition 2 (2) and Theorem 9,thesequence 𝑛 𝑓(𝑥,𝑦)={ associated with a given uniformly virtually stable self-map 𝑓 (𝑥, |𝑥|) otherwise. seems to have an implicit structure. In fact, some questions (10) may naturally arise. 4 Journal of Function Spaces and Applications

(i) Can we replace (𝑎𝑛) with the larger (in some sense) Proposition 12. Suppose 𝑓 is uniformly virtually stable with 󸀠 sequence (𝑎𝑛) so that 𝑓 is still uniformly virtually respect to (𝑎𝑛). one has the following: (𝑎󸀠 ) stable with respect to 𝑛 ? (𝑏 )⊆(𝑎) (𝑏 )∈S 󸀠 󸀠 (1) If 𝑛 𝑛 ,then 𝑛 𝑓. (ii) If (𝑎𝑛) canbeenlargedto(𝑎 ),canwemake(𝑎 ) largest 𝑛 𝑛 (𝑏 )∈S (𝑎 )∪(𝑏)∈S possible? and how does it look like? (2) If 𝑛 𝑓,then 𝑛 𝑛 𝑓. (𝑎󸀠 ) 𝑘 (iii) Does the largest possible sequence 𝑛 satisfy the sup- (3) ⊕𝑖=1(𝑎𝑛)∈S𝑓 for any 𝑘∈N. finite condition? (𝑎 + 𝑐), (𝑎 )∪{𝑐}∈S 𝑐∈⟨𝑎⟩ 󸀠 (4) 𝑛 𝑛 𝑓 for any 𝑛 . (iv) Is the largest possible sequence (𝑎𝑛) always (𝑛)? 𝑥∈𝐹(𝑓) 𝑈∈N(𝑥) As we will see later on in this section, it turns out that Proof. Let and . theanswerstothesequestionsareinterestinglynontrivialand (1) It is straightforward from the definition of virtual require some careful considerations in both continuous and stability. discontinuous settings. 𝑎𝑛 Throughout this section, 𝑋 isaHausdorffspaceand𝑓 is a (2) There exist 𝑉, 𝑊 ∈ N(𝑥) such that 𝑓 (𝑉) ⊆ 𝑈 and 𝑏 𝑐 self-map of 𝑋.Everysequenceisassumedtobeanincreasing 𝑓 𝑛 (𝑊) ⊆ 𝑈 for all 𝑛∈N.Hence𝑓 𝑛 (𝑉 ∩𝑊) ⊆𝑈 for sequence of natural numbers, so it is uniquely determined by all 𝑛∈N where (𝑐𝑛)=(𝑎𝑛)∪(𝑏𝑛). its image. We then always identify a sequence (𝑎𝑛) with the (𝑎 )⊕(𝑎 )∈S {𝑎 :𝑛∈N} (3) By induction, it suffices to show that 𝑛 𝑛 𝑓. infinite set 𝑛 , and with this identification in mind, There are, by virtual stability of 𝑓, 𝑉, 𝑊 ∈ N(𝑥) such 𝑎 𝑎 the following notations become natural: that 𝑓 𝑛 (𝑉) ⊆ 𝑈 and 𝑓 𝑛 (𝑊) ⊆ 𝑉 for all 𝑛.This 𝑎 +𝑎 𝑎 𝑓 𝑛 𝑚 (𝑊) ⊆ 𝑓 𝑛 (𝑉) ⊆ 𝑈 𝑛, 𝑚 ∈ N (i) (𝑎𝑛)⊆(𝑏𝑛) if (𝑎𝑛) is a subsequence of (𝑏𝑛); implies that for all . 𝑘 (ii) (𝑎𝑛)∪{𝑐}represents the sequence whose image is {𝑎𝑛 : 𝑐∈⟨𝑎⟩ 𝑐=𝑎 +⋅⋅⋅+𝑎 ∈⊕ (𝑎 ) (4) By letting 𝑛 ,wehave 𝑖1 𝑖𝑘 𝑖=1 𝑛 𝑛∈N}∪{𝑐}; 𝑘+1 for some 𝑘∈N, and hence, (𝑎𝑛 +𝑐)⊆⊕𝑖=1 (𝑎𝑛)∈S𝑓 (iii) (𝑎𝑛+𝑐)representsthesequencewhoseimageis{𝑎𝑛+𝑐 : 𝑘 𝑗 and (𝑎𝑛)∪{𝑐}⊆⋃ ⊕ (𝑎𝑛)∈S𝑓 by (1) and (2). 𝑛∈N}; 𝑗=1 𝑖=1

(iv) (𝑎𝑛)∪(𝑏𝑛) representsthesequencewhoseimageis {𝑎𝑛 :𝑛∈N}∪{𝑏𝑛 :𝑛∈N},andso(𝑎𝑛)∪⋅⋅⋅∪(𝑏𝑛) (𝑎 ) represents the sequence whose image is {𝑎𝑛 :𝑛∈N}∪ Remark 13. From (2) in the previous proposition, if 𝑛 and (𝑏 ) (S ,⊆) (𝑎 )=(𝑎)∪(𝑏 )= ⋅⋅⋅∪{𝑏𝑛 :𝑛∈N}; 𝑛 are maximal elements in 𝑓 ,then 𝑛 𝑛 𝑛 (𝑏𝑛). This implies that a maximal sequence associated with 𝑓, (v) (𝑎𝑛)⊕(𝑏𝑛) representsthesequencewhoseimageis{𝑎𝑖+ 𝑘 if exists, is always unique. 𝑏𝑗 :𝑖,𝑗∈N},and⊕𝑖=1(𝑎𝑛)=(𝑎⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟𝑛)⊕⋅⋅⋅⊕(𝑎𝑛). 𝑘 (𝑎 ) copies of 𝑛 Before considering the existence, we will first investigate For a nonempty subset 𝐴 of N,thenaturalnumber𝑑 the structure of the maximal sequence associated with a given satisfying self-map. 𝑑|𝑎 𝑎∈𝐴 (1) for all and Lemma 14. Suppose 𝑓 is uniformly virtually stable. For a (2) if 𝑐|𝑎for all 𝑎∈𝐴,then𝑑|𝑐 sequence (𝑎𝑛),onehas 𝐴 will be called the greatest common divisor of and denoted (1) gcd (𝑎𝑛)= gcd ⟨𝑎𝑛⟩= gcd {𝑎𝑖 :𝑖≤𝑚}for some 𝐴 ⟨𝐴⟩ by gcd .Also,welet denotethesetofallfinitesumsof 𝑚∈N, elements in 𝐴;thatis, (2) there are 𝑏∈⟨𝑎𝑛⟩ and 𝑚∈N such that 𝑘 gcd (𝑎𝑛)+𝑏 ∈ ⟨𝐴⟩ ={𝑥:𝑥=𝑥 +⋅⋅⋅+𝑥 𝑥 ,...,𝑥 ∈𝐴}. 1 𝑘 for some 1 𝑘 ⟨𝑎1,...,𝑎𝑚⟩ for all 𝑘≥0, (12) (3) if (𝑎𝑛)∈S𝑓 and (𝑎𝑛) satisfies the sup-finite condition, Notice that 𝐴 ⊆ ⟨𝐴⟩ and if 1∈𝐴,wehave⟨𝐴⟩ = N.As then ⟨𝑎𝑛⟩∈S𝑓. usual, gcd(𝑎𝑛) will represent gcd{𝑎𝑛 :𝑛∈N},and⟨𝑎𝑛⟩ will (𝑎 )= ⟨𝑎 ⟩ represent at the same time both the set ⟨{𝑎𝑛 :𝑛∈N}⟩ and the Proof. (1) The fact that gcd 𝑛 gcd 𝑛 follows directly 𝑑= { {𝑎 :𝑖≤𝑘}:𝑘∈N} sequence whose image is ⟨{𝑎𝑛 :𝑛∈N}⟩. We may simply write from the definition. Now, let min gcd 𝑖 . 𝑑= {𝑎 :𝑖≤𝑚} 𝑚∈N 𝑑∤𝑎 ⟨𝑎1,...,𝑎𝑛⟩ for ⟨{𝑎1,...,𝑎𝑛}⟩. Then, gcd 𝑖 for some .If 𝑛 Moreover, for a uniformly virtually stable self-map 𝑓,we for some 𝑛>𝑚,thengcd{𝑎1,...,𝑎𝑛}<𝑑, a contradiction. let Also, for any 𝑐>0satisfying 𝑐|𝑎𝑛 for all 𝑛∈N,wehave 𝑐|𝑎𝑖 for all 1≤𝑖≤𝑚, and hence, 𝑐≤𝑑. This implies S𝑓 ={(𝑎𝑛):𝑓is uniformly virtually stable gcd(𝑎𝑛)=gcd{𝑎𝑖 :𝑖≤𝑚}. (13) (2) If gcd(𝑎𝑛)=𝑎1, the proof is clear. So, we assume with respect to (𝑎𝑛)} . that gcd(𝑎𝑛) =𝑎̸ 1.By(1),gcd(𝑎𝑛)=gcd{𝑎𝑖 :𝑖≤𝑚}for Clearly, S𝑓 is partially ordered by ⊆, and a maximal element some 𝑚∈N.Then,forsomepartition{𝑃, 𝑁} of {1,...,𝑚}, in (S𝑓,⊆) will be called a maximal sequence associated with 𝑓. gcd(𝑎𝑛)=∑𝑖∈𝑃 𝑛𝑖𝑎𝑖 −∑𝑖∈𝑁 𝑛𝑖𝑎𝑖 where 𝑛𝑖 ≥0.Set𝐴=∑𝑖∈𝑃 𝑛𝑖𝑎𝑖 The following are some basic properties of S𝑓. and 𝐵=∑𝑖∈𝑁 𝑛𝑖𝑎𝑖.Noticethat𝐵 =0̸ because gcd(𝑎𝑛) =𝑎̸ 1, Journal of Function Spaces and Applications 5

𝑙𝐵 ∈ ⟨𝑎 ,...,𝑎 ⟩ 𝑙∈N 𝑙∈N and 1 𝑚 for all . Then for each and 𝜋 0 0≤𝑟≤𝐵, 𝛿0 ··· 𝛿4 𝑟 (𝑎 )+𝐵2 +𝑙𝐵=𝑟𝐴+(𝐵−𝑟+𝑙) 𝐵∈⟨𝑎,...,𝑎 ⟩. 𝛿3 gcd 𝑛 1 𝑚 ··· (14) 𝛿2 Since for 𝑘≥0,wehave𝑘=𝑞𝐵+𝑟for some 𝑞≥0and 0≤𝑟<𝐵, 𝛿1 2 𝑘 gcd (𝑎𝑛)+(𝐵 +𝐵) Figure 1: The map 𝑓 in Example 17. 2 =𝑟gcd (𝑎𝑛)+𝐵 +(1+𝑞gcd (𝑎𝑛)) 𝐵 ∈ 1⟨𝑎 ,...,𝑎𝑚⟩. (15) Theorem 16. Suppose 𝑓 is uniformly virtually stable. Then the 2 Therefore, the result follows by letting 𝑏=𝐵 +𝐵. maximal sequence associated with 𝑓 exists if and only if there (3) Assume that (𝑎𝑛)∈S𝑓 and Δ:=sup𝑛(𝑎𝑛+1 −𝑎𝑛)<∞. is (𝑎𝑛)∈S𝑓 satisfying the sup-finite condition. By (2), there is 𝑏∈⟨𝑎𝑛⟩ for which 𝑘 gcd(𝑎𝑛)+𝑏∈⟨𝑎𝑛⟩ for all 𝑘≥0.ByProposition12,wehave Proof. (⇒) If (𝑎𝑛) is maximal, then (𝑛𝑎1)⊆(𝑎𝑛) and hence

(𝑏𝑛):=(𝑎𝑛)∪(𝑎𝑛 + gcd (𝑎𝑛)+𝑏) sup (𝑎𝑛+1 −𝑎𝑛)≤sup ((𝑛+1) 𝑎1 −𝑛𝑎1)=𝑎1 <∞. 𝑛 𝑛 (17)

∪(𝑎𝑛 +2gcd (𝑎𝑛)+𝑏) (16) (⇐) Assume that Δ:=sup𝑛(𝑎𝑛+1 −𝑎𝑛)<∞for some (𝑎𝑛)∈ S ⟨𝑎 ⟩∈S 𝑎 =1 ∪⋅⋅⋅∪(𝑎𝑛 +Δgcd (𝑎𝑛)+𝑏)∈S𝑓, 𝑓.ByLemma14 (3), 𝑛 𝑓.If 1 ,wearedone. Suppose that 𝑎1 >1.For0≤𝑖<𝑎1 = min⟨𝑎𝑛⟩,let and hence (𝑐𝑛):=(𝑏𝑛)∪{𝑐∈⟨𝑎𝑛⟩:𝑐≤𝑎1 +𝑏}∈ S𝑓.The proof will immediately follow by Proposition 12 (1) once we 𝐴𝑖 ={𝑚:𝑚≡𝑖mod 𝑎1 and ⟨𝑎𝑛⟩∪{𝑚} ∈ S𝑓}, (18) can show that ⟨𝑎𝑛⟩⊆(𝑐𝑛).Toseethis,let𝑐∈⟨𝑎𝑛⟩.Thecase 𝑐≤𝑎 +𝑏 𝑎 +𝑏 ≤ 𝑐 <𝑎 +𝑏 of 1 is clear. Otherwise, 𝑚 𝑚+1 and 𝑚𝑖 = min 𝐴𝑖 if 𝐴𝑖 =0̸ ,otherwise𝑚𝑖 =𝑎1.Then{𝑐 : ⟨𝑎𝑛⟩∪ 𝑚≥1 (𝑎 )|𝑐−𝑎 −𝑏 𝑐= for some .Sincegcd 𝑛 𝑚 ,wehave {𝑐} ∈ S𝑓}=𝐴0 ∪⋅⋅⋅∪𝐴𝑎 −1.ByProposition12 (2), ⟨𝑎𝑛⟩∪{𝑚𝑖 : 𝑎 +𝑏+𝑟 (𝑎 ) 0≤𝑟<𝑎 −𝑎 ≤Δ 1 𝑚 gcd 𝑛 for some 𝑚+1 𝑚 .This 0≤𝑖<𝑎1}∈S𝑓 and again by Lemma 14 (3), since sup𝑛(𝑏𝑛+1− 𝑐∈(𝑎 +𝑟 (𝑎 )+𝑏)⊆(𝑏)⊆(𝑐) implies that 𝑛 gcd 𝑛 𝑛 𝑛 . 𝑏𝑛)≤sup𝑛(𝑎𝑛+1 −𝑎𝑛)<∞where (𝑏𝑛):=⟨⟨𝑎𝑛⟩∪{𝑚𝑖 :0≤𝑖< 𝑎1}⟩, (𝑏𝑛)∈S𝑓.Noticethat(𝑏𝑛) is maximal because whenever Theorem 15. 𝑓 (𝑎 ) Suppose is uniformly virtually stable and 𝑛 (𝑏𝑛)∪{𝑐}∈ S𝑓 for some 𝑐,thereis0≤𝑖<𝑎1 such that is the maximal sequence associated with 𝑓.Onehas 𝑐∈𝐴 ⊆𝐴 ∪⋅⋅⋅∪𝐴 ⊆(𝑏) 𝑖 0 𝑎1−1 𝑛 .

(1) (𝑎𝑛)=⟨𝑎𝑛⟩, Towardstheendofthissection,wewillconstructsome (2) (𝑎𝑛) = ⟨𝐴⟩ for some finite subset 𝐴 of (𝑎𝑛), interesting examples including a uniformly virtually stable (3) if 𝑓 is continuous, then (𝑎𝑛) = (𝑛). self-map whose maximal sequence is prescribed as well as a uniformly virtually stable self-map having no maximal ⟨𝑎 ⟩⊆(𝑎) 𝑐∈⟨𝑎⟩ Proof. (1)Itsufficestoshowthat 𝑛 𝑛 .Let 𝑛 . sequence. By Proposition 12, (𝑎𝑛)∪{𝑐}∈ S𝑓, and hence, 𝑐∈(𝑎𝑛) by (𝑎 ) 𝑘 𝑖 maximality of 𝑛 . Example 17. For 𝑘∈N ∪{0},let𝜃𝑘 =𝜋∑ (1/2 ) .Then (𝑎 )=⟨𝑎⟩ 𝑖=0 (2) From (1), we have 𝑛 𝑛 .ThenbyLemma14 (2), 𝜋≤𝜃𝑘 <2𝜋for all 𝑘. Consider the sequence (𝑎𝑛)=⟨𝑎𝑛⟩.Set 𝑙, 𝑚 ∈ N 𝑘 (𝑎 )+𝑎 ∈⟨𝑎,...,𝑎 ⟩ there exist such that gcd 𝑛 𝑙 1 𝑚 for 𝛿0 =𝑎1 and 𝛿𝑛 =𝑎𝑛+1 −𝑎𝑛 for any 𝑛∈N.Define𝑓:C → C 𝑘≥0 𝐴={𝑎 :𝑖≤ {𝑙, 𝑚}} all .Bysetting 𝑖 max ,weclearlyhave by ⟨𝐴⟩ ⊆ ⟨𝑎𝑛⟩=(𝑎𝑛). On the other hand, for each 𝑖∈N,notice that 𝑓 (𝑧) =𝑓(𝑟𝑒𝑖𝜙)

(i) if 𝑖≤max{𝑙, 𝑚},then𝑎𝑖 ∈𝐴,and { 𝑖𝜙 if 𝜙=𝜃𝑘,0<𝑟<𝛿𝑘 −1 (ii) if 𝑖>max{𝑙, 𝑚},then𝑎𝑖 =𝑎𝑙 +𝑘gcd(𝑎𝑛)∈ {(𝑟+1) 𝑒 { for some 𝑘; ⟨𝑎1,...,𝑎𝑚⟩ ⊆ ⟨𝐴⟩ for some 𝑘∈N. { = 𝜙=𝜃,𝑟≥𝛿 −1 {(𝑟−⌊𝑟⌋) 𝑒𝑖(𝜋+𝜙/2) if 𝑘 𝑘 (𝑎 ) ⊆ ⟨𝐴⟩ { Therefore, 𝑛 . { for some 𝑘; (3) Suppose that 𝑓 is continuous. It follows that (𝑎𝑛)∪{1} ∈ { {0 otherwise. S𝑓, and by maximality of (𝑎𝑛), 1∈(𝑎𝑛). Therefore, by (1), we (19) have (𝑎𝑛)=⟨𝑎𝑛⟩ = (𝑛). The next theorem gives a necessary and sufficient condi- Then, 𝑓 can be illustrated in Figure 1 and it is uniformly tion for the existence of the maximal sequence. virtually stable with respect to the maximal sequence (𝑎𝑛). 6 Journal of Function Spaces and Applications

To see this, note that 𝐹(𝑓) = ,{0} 𝜋+𝜃𝑘/2 = 𝜋(1 + Example 19. For each 𝑚∈N,let𝐴𝑚 be defined as in the 𝑘+1 𝑖 𝑘+1 𝑖 𝑝 =3 𝑛>1 𝑝 ∑ (1/2 )) = 𝜋 ∑ (1/2 )=𝜃𝑘+1 for all 𝑘≥0, and for each previous lemma, 1 and for each ,let 𝑛 be the 𝑖=1 𝑖=0 2 𝑛∈N and 0<𝑟<1, smallest prime number such that 𝑝𝑛 > max{𝑛 +𝑛,𝑝𝑛−1}.For examples, 𝑝2 =7and 𝑝3 =13. Clearly, 𝑝𝑛 =𝑝̸ 𝑚 whenever 𝑎 𝑖𝜃 𝛿 +⋅⋅⋅+𝛿 𝑖𝜃 2 𝑓 𝑛 (𝑟𝑒 0 )=𝑓 𝑛−1 0 (𝑟𝑒 0 ) 𝑛 =𝑚̸ .Now,let𝑋 = [0, 2]×[0, 2] be a subspace of R equipped (20) 0 := (0, 0) ∈ 𝑋 𝐵(𝑎; 𝑟) 𝛿 +⋅⋅⋅+𝛿 𝑖𝜃 𝑖𝜃 with the maximum norm, and let =𝑓 𝑛−1 1 (𝑟𝑒 1 )=⋅⋅⋅=𝑟𝑒 𝑛 . denote the open ball {𝑥 ∈ 𝑋 : ‖𝑥 − 𝑎‖ <𝑟}. The goal of this example is to find a uniformly virtually To show the virtual stability of 𝑓 with respect to (𝑎𝑛),it 𝑎 stable self-map having no maximal sequence. This can be suffices to prove that 𝑓 𝑛 (𝐵(0; 𝑟)) ⊆ 𝐵(0; 𝑟) for all 0<𝑟<1. done by constructing a uniformly virtually stable self-map 0<𝑟<1 𝑧=𝑠𝑒𝑖𝜙 ∈ 𝐵(0; 𝑟) 𝑛 Let .Foreach , 𝑓:𝑋with →𝑋 respect to (2 ) but not (2𝑛).Forif(𝑎𝑛) is the maximal sequence associated with such an 𝑓,wemust 󵄨 𝑎 󵄨 󵄨 𝑎 𝑖𝜃 󵄨 󵄨 𝑎 𝑎 𝑖𝜃 󵄨 󵄨𝑓 𝑛 (𝑧)󵄨 ≤ 󵄨𝑓 𝑛 (𝑠𝑒 𝑘 )󵄨 = 󵄨𝑓 𝑛 𝑓 𝑘 (𝑠𝑒 0 )󵄨 𝑛 𝑛 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 have (2 )⊆(𝑎𝑛) and hence (2𝑛) = ⟨2 ⟩⊆⟨𝑎𝑛⟩=(𝑎𝑛),but (21) 𝑓 󵄨 𝑎 𝑖𝜃 󵄨 󵄨 𝑖𝜃 󵄨 this contradicts the property that is not uniformly virtually 󵄨 𝑚 0 󵄨 󵄨 𝑚 󵄨 = 󵄨𝑓 (𝑠𝑒 )󵄨 = 󵄨𝑠𝑒 󵄨 =𝑠<𝑟 stable with respect to (2𝑛). To obtain the desired self-map, it suffices to require the for some 𝑘,and𝑎𝑚 =𝑎𝑛 +𝑎𝑘.Thus,wegettheclaim.Moreover map 𝑓:𝑋 →𝑋to satisfy the following three conditions: (𝑎𝑛) is maximal because whenever 𝑏∉(𝑎𝑛),either𝑏<𝑎1 or 𝑎𝑚 <𝑏<𝑎𝑚+1 for some 𝑚∈N (and so 𝑏=𝑎𝑚 +𝑘for 𝐹(𝑓)0 ={ } some 1≤𝑘<𝛿𝑚)holds,whichthefirstcaseimpliesthat (C1) , 𝑏 𝑖𝜃 𝑖𝜃 𝑓 (𝑟𝑒 0 ) = (𝑟 + 𝑏)𝑒 0 ∉ 𝐵(0; 1) for all 0<𝑟<1, while the 2𝑛 latter implies that (C2) 𝑓 (𝐵(0; 1/(𝑖 + 1))) ⊆ 𝐵(0;1/𝑖)for each 𝑖, 𝑛 ∈ N,

󵄨 𝑏 𝑖𝜃 󵄨 󵄨 𝑘 𝑎 𝑖𝜃 󵄨 𝑎 󵄨 0 󵄨 󵄨 𝑚 0 󵄨 𝑘∈N 𝑎∈2N 𝑓 (1/(𝑘+ 󵄨𝑓 (𝑟𝑒 )󵄨 = 󵄨𝑓 𝑓 (𝑟𝑒 )󵄨 (C3) For each , there exists such that 2), 0) ∉ 𝐵(0;1). 󵄨 𝑘 𝑖𝜃 󵄨 󵄨 𝑖𝜃 󵄨 󵄨 𝑚 󵄨 󵄨 𝑚 󵄨 = 󵄨𝑓 (𝑟𝑒 )󵄨 = 󵄨(𝑟+𝑘) 𝑒 󵄨 (22) Notice that (C1) and (C2) imply the virtual stability of 𝑓 =𝑟+𝑘>1 𝑛 with respect to (2 ), while (C1) and (C3) imply that, for each 𝑘∈N 𝑎∈2N 𝑓𝑎(𝐵(0; 1/(𝑘 + 1))) ⊈ 𝑏 𝑖𝜃0 , there exists such that for all 0<𝑟<1;thatis,𝑓 (𝑟𝑒 ) ∉ 𝐵(0; 1).Thus(𝑎𝑛) ∪ {𝑏} ∉ 𝐵(0;1) 𝑓 S 𝑏∉(𝑎) (𝑎 ) , and hence is not uniformly virtually stable with 𝑓 for all 𝑛 , which induces the maximality of 𝑛 as respect to (2𝑛). desired. We are now ready to give the explicit description of such 𝑛 𝑓 𝑋 Lemma 18. Let 𝐴1 ={2 :𝑛∈N} and for 𝑚>1, an . Consider the following subsets of :

𝑛 𝑛 𝐴 ={2 1 +⋅⋅⋅+2 𝑚 :𝑛 ∈ N}−⋃𝐴 . 𝑇 ={(1/(𝑘+2),0):𝑘∈N} 𝑚 𝑖 𝑖 (23) 1 , 𝑖<𝑚 𝑙 𝑇2 ={(1,1/𝑝):𝑘,𝑙∈N and 𝑙+1 ∈𝑚 𝐴 for some Then {𝐴𝑛 :𝑛∈N} forms a partition of 2N :={2𝑛:𝑛∈N}, 𝑘 𝑚≤𝑘}, and 𝐴𝑛 is infinite for each 𝑛∈N. 𝑛 𝑛 𝑛+1 𝑇 = {(1/(𝑘 + 2 − 𝑚), 𝑙1/𝑝 ):𝑘,𝑙∈N 𝑙∈𝐴 Proof. Following from the fact that 2 +2 =2 for all 𝑛∈ 3 𝑘 and 𝑚 for 𝑚 𝑛𝑖 𝑚≤𝑘} N,if∑𝑖=1 2 ∈𝐴𝑚,then𝑛𝑖 =𝑛̸ 𝑗 for all 𝑖 =𝑗̸ .Alsoforeach some , 𝑛 𝑛 𝑛∈N,since2𝑛 = ∑𝑖=1 2∈⋃𝑖=1 𝐴𝑖,wehave2𝑛 ∈ 𝐴𝑖 for 𝑇 ={(1,1/𝑝𝑙 ):𝑘,𝑙∈N 𝑙∈𝐴 𝑙+1∈ some 𝑖≤𝑛.Hence,{𝐴𝑛 :𝑛∈N} forms a partition of 2N.For 4 𝑘 and either 𝑚 or 𝐴 𝑚>𝑘} the next implication, notice that 𝐴1 is clearly infinite. Now, 𝑚 for some . 𝑛 assume that 𝑛>1.Wefirstclaimthat2 −2 ∈ 𝐴𝑛−1.Since 2𝑛 −2 ∈ 2N 2𝑛 −2 ∈ 𝐴 𝑘∈N ,wehave 𝑘 for some .Itfollowsthat Observe that 𝑛 𝑘 𝑛𝑖 𝑘 𝑛𝑖 𝑘<𝑛and for if 2 −2=∑𝑖=1 2 ,then𝑛𝑖 <𝑛and ∑𝑖=1 2 = 𝑛 𝑛−1 𝑖 2 −2 = ∑𝑖=1 2 , and hence 𝑘=𝑛−1.Next,weclaimthat 𝑛+𝑚 𝑛 (O1) 𝑇𝑖 ∩𝑇𝑗 =0whenever 𝑖 =𝑗̸ .Allcasesaretrivialexcept 2 +2 −2∈𝐴𝑛 for all 𝑚∈N.Let𝑚∈N.Bytheprevious 𝑛+𝑚 𝑛 𝑛 𝑛+𝑚 𝑛 for the case of 𝑇2 ∩𝑇4, where it follows from the fact claim, we have 2 +(2 −2) ∈ ⋃𝑖=1 𝐴𝑖.Hence,2 +2 −2 ∈ that {𝐴𝑚 :𝑚∈N} forms a partition of 2N; 𝑛+𝑚 𝑛 𝑘 𝑛𝑖 𝐴𝑘 for some 1≤𝑘≤𝑛;thatis,2 +2 −2=∑𝑖=1 2 .Since 𝑛+𝑚−1 𝑖 𝑛+𝑚 𝑛+𝑚 𝑛 𝑛+𝑚+1 ∑ 2 =2 −2<2 +2 −2<2 4 𝑖=1 ,wemusthave ‖(𝑥, 𝑦)‖ =𝑥 (𝑥, 𝑦) ∈⋃ 𝑇 𝑛𝑖 𝑛+𝑚 𝑛+𝑚 𝑛 (O2) for all 𝑖=1 𝑖, because for each 2 =2 for some 1≤𝑖≤𝑘.Thus,2 +2 −2∈𝐴𝑛 as 𝑙 2 𝑛+𝑚 𝑛 𝑘, 𝑚, 𝑙∈ N with 𝑚≤𝑘,wehave𝑝𝑘 ≥𝑝𝑘 >𝑘 +𝑘≥ claimed. Finally, since {2 +2 −2:𝑚∈N}⊆𝐴𝑛,then𝐴𝑛 𝑘+1 1/𝑝𝑙 <1/(𝑘+1)≤1/(𝑘+2−𝑚) is infinite as desired. and hence 𝑘 . Journal of Function Spaces and Applications 7

Define 𝑓:𝑋 →𝑋by (1) If (𝑥, 𝑦) = (1/(𝑘 + 2),1 0)∈𝑇 ∩𝐵( 0;1/(𝑖+1)),then 1/(𝑘 + 2) < 1/(𝑖 +1), and hence 󵄩 󵄩 󵄩 𝑛 󵄩 󵄩 𝑛 1 󵄩 (𝑥, 𝑦) 󵄩𝑓2 (𝑥, 𝑦)󵄩 = 󵄩𝑓2 ( ,0)󵄩 { 1 if 󵄩 󵄩 󵄩 󵄩 { (1, ) 1 󵄩 𝑘+2 󵄩 { 𝑝 =( ,0)∈𝑇; { 𝑘 𝑘+2 1 󵄩 󵄩 (27) { ( 2) 󵄩 1 1 󵄩 ( 2) 1 1 { P 󵄩 󵄩 O { 1 = 󵄩( , 𝑛 )󵄩 = < . { 󵄩 𝑘+1 𝑝2 󵄩 𝑘+1 𝑖 { 1 1 if (𝑥, 𝑦)=(1, )∈𝑇2 󵄩 𝑘 󵄩 { ( , ) 𝑝𝑙 { 𝑘+2−𝑚 𝑝𝑙+1 𝑘 { 𝑘 { where 𝑙+1∈𝐴𝑚; 𝑓(𝑥,𝑦)= (𝑥, 𝑦) = (1/(𝑘 + 2𝑙 −𝑚),1/𝑝 )∈𝑇 ∩𝐵( 0;1/(𝑖+1)) { if (𝑥, 𝑦) (2) If 𝑘 3 , { ( 2) { 1 1 𝑙∈𝐴 𝑚≤𝑘 1/(𝑘+2−𝑚) O= { =( , )∈𝑇3 then 𝑚 for some ,and { 𝑘+2−𝑚 𝑝𝑙 𝑙 { 1 𝑘 ‖(1/(𝑘 + 2 − 𝑚), 1/𝑝 )‖ < 1/(𝑖 + 1) 𝑚=𝑘 { (1, ) 𝑘 .If ,wehave { 𝑝𝑙+1 or (𝑥, 𝑦) { 𝑘 1/2 = 1/(𝑘 + 2 − 𝑚) < 1/(𝑖, +1) a contradiction. Thus { 1 { =(1, )∈𝑇; 𝑚<𝑘, and by (P3), there is some 𝑗≤𝑚+1such that { 𝑙 4 { 𝑝𝑘 { 0 otherwise. 󵄩 󵄩 󵄩 󵄩 󵄩 𝑛 1 1 󵄩 ( 3) 󵄩 1 1 󵄩 (24) 󵄩𝑓2 ( , )󵄩 P= 󵄩( , )󵄩 󵄩 𝑘+2−𝑚 𝑝𝑙 󵄩 󵄩 𝑘+2−𝑗 𝑝2𝑛+𝑙 󵄩 󵄩 𝑘 󵄩 󵄩 𝑘 󵄩 Clearly, 𝑓 is well-defined by (O1). The following properties ( 2) 1 arealsosatisfied. O= (28) 𝑘+2−𝑗 (P1) 𝑓(𝑇1)⊆𝑇2, 𝑓(𝑇2)⊆𝑇3 and 𝑓(𝑇3 ∪𝑇4)⊆𝑇2 ∪𝑇4. 4 1 1 So, ⋃𝑖=1 𝑇𝑖 is 𝑓-invariant. Moreover, since 𝑝𝑖’s are all ≤ < . 𝑘 𝑙 𝑘+2−(𝑚+1) 𝑖 distinct primes, we have 𝑝𝑖 =𝑝𝑗 if and only if (𝑖, 𝑘) = (𝑗, 𝑙) (𝜋 ∘𝑓)| 4 𝜋 , and hence 2 ⋃ 𝑇 is injective, where 2 𝑛 𝑖=1 𝑖 𝑓2 (𝐵(0; 1/(𝑖 + 1))) ⊆ denotes the second-coordinate projection 𝜋2(𝑥, 𝑦) = Fromabovecases,itfollowsthat 𝑦. 𝐵(0;1/𝑖). Finally, for (C3), let 𝑘∈N and 𝑐=min 𝐴𝑘+1 ∈2N.Then (P2) For each 𝑘, 𝑙 ∈ N,onecanverifythat 𝑐 𝑘+1 by (P2), we have 𝑓 (1/(𝑘 + 2), 0) = (1,𝑘 1/𝑝 )∉𝐵(0;1).

{ 1 1 if 𝑙∈𝐴𝑚 { ( , ) Remark 20. From the previous example, we can easily check 1 { 𝑘+2−𝑚 𝑝𝑙 for some 𝑚≤𝑘; 𝑙 𝑘 that 𝑓 is uniformly virtually stable with respect to (𝑎𝑛)=𝐴1 ∪ 𝑓 ( ,0)={ 𝑎 𝑘+2 { 1 ⋅⋅⋅∪𝐴 𝑚∈N 𝑓 𝑛 (𝐵(0;1/(𝑘+ { (1, ) . 𝑚 for any ,justbyshowingthat 𝑙 otherwise 𝑚))) ⊆ 𝐵(0;1/𝑘) 𝑘, 𝑛 ∈ N { 𝑝𝑘 for any .Hence,Proposition12 (2) (25) does not hold in general for an arbitrary union of sequences in S𝑓. In particular, we have the following. Acknowledgments 2𝑙 2𝑙 (i) 𝑓 (1/(𝑘+2),0) = (1/(𝑘+2−1),1/𝑝𝑘 )=(1/(𝑘+ 𝑙 1), 1/𝑝2 ) 2𝑙 ∈(2𝑛)=𝐴 1≤𝑘 The first author is (partially) supported by the Centre of 𝑘 since 1 and . Excellence in Mathematics, the Commission on Higher 𝑓𝑐(1/(𝑘+2), 0) = (1, 𝑐1/𝑝 ) 𝑐= 𝐴 (ii) 𝑘 ,where min 𝑘+1, Education, Thailand. The third author wishes to thank the H. since 𝑐∈𝐴𝑘+1 and 𝑘+1>𝑘. M. King Bhumibol Adulyadej’s 72nd Birthday Anniversary Scholarship. The authors are grateful to Professor Pimpen (P3) For each 𝑘, 𝑙, 𝑚, 𝑛∈ N with 𝑚<𝑘and 𝑙∈𝐴𝑚,we Vejjajiva and the anonymous referee(s) for their valuable 𝑙 𝑙 have 𝑚+1 ≤, 𝑘 𝑓 (1/(𝑘+2), 0) = (1/(𝑘+2−𝑚), 𝑘1/𝑝 ) comments and suggestions for improving this paper. 𝑛 𝑚+1 by (P2), and 2 +𝑙 ∈ 𝐴𝑗 ⊆⋃𝑖=1 𝐴𝑖 for some 𝑗≤𝑚+1. Again by (P2), it follows that References

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[4] J. Matkowski, “Iterations of mean-type mappings and invariant means,” Annales Mathematicae Silesianae, no. 13, pp. 211–226, 1999. [5] J. Matkowski, “Fixed points and iterations of mean-type mappings,” Central European Journal of Mathematics,vol.10,no.6, pp. 2215–2228, 2012. [6] T. Suzuki, “Fixed point theorems and convergence theorems for some generalized nonexpansive mappings,” JournalofMathe- matical Analysis and Applications,vol.340,no.2,pp.1088–1095, 2008. [7] P.Chaoha, “Virtually nonexpansive maps and their convergence sets,” Journal of Mathematical Analysis and Applications,vol. 326, no. 1, pp. 390–397, 2007. [8] R. Kannan, “Some results on fixed points—II,” The American Mathematical Monthly,vol.76,pp.405–408,1969. [9] P. Chaoha and P. Chanthorn, “Convergence and fixed point sets of generalized homogeneous maps,” Thai Journal of Mathemat- ics,vol.5,no.2,pp.281–289,2007. Hindawi Publishing Corporation Journal of Function Spaces and Applications Volume 2013, Article ID 280970, 8 pages http://dx.doi.org/10.1155/2013/280970

Research Article Asymptotics of the Eigenvalues of a Self-Adjoint Fourth Order Boundary Value Problem with Four Eigenvalue Parameter Dependent Boundary Conditions

Manfred Möller and Bertin Zinsou

The John Knopfmacher Centre for Applicable Analysis and Number Theory, School of Mathematics, University of the Witwatersrand (Wits), Private Bag 3, Johannesburg 2050, South Africa

Correspondence should be addressed to Manfred Moller;¨ [email protected]

Received 9 April 2013; Revised 24 May 2013; Accepted 27 May 2013

Academic Editor: Janusz Matkowski

Copyright © 2013 M. Moller¨ and B. Zinsou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Considered is a regular fourth order ordinary differential equation which depends quadratically on the eigenvalue parameter 𝜆 and which has separable boundary conditions depending linearly on 𝜆. It is shown that the eigenvalues lie in the closed upper half plane or on the imaginary axis and are symmetric with respect to the imaginary axis. The first four terms in the asymptotic expansion of the eigenvalues are provided.

1. Introduction operator pencil consists of self-adjoint operators have been obtained. In [4, 5], we have derived eigenvalue asymptotics Sturm-Liouville problems have attracted extensive attention associated with particular boundary conditions. In this paper, due to their intrinsic mathematical challenges and their appli- we are considering the case of separable boundary conditions cations in physics and engineering. Classical Sturm-Liouville where all four of these boundary conditions depend on the problems have been extended to higher-order differential eigenvalue parameter. equations and to differential equations with eigenvalue para- Other recent results on fourth order differential opera- meter dependent boundary conditions. For example, the gen- tors whose boundary conditions depend on the eigenvalue eralized Regge problem is realised by a second order differen- parameter but which are represented by linear operator pen- tial operator which depends quadratically on the eigenvalue cils, include spectral asymptotics and basis properties, see [6– parameter and which has eigenvalue parameter dependent 8]. boundary conditions, see [1]. The particular feature of this problem is that the coefficient operators of this pencil are self- In Section 2, we introduce the operator pencil associated adjoint, and it is shown in [1]thatthisgivessomeapriori with the eigenvalue problem (1), (2), and we derive the knowledgeaboutthelocationofthespectrum.In[2], this boundary conditions such that the operators in the pencil approach has been extended to a fourth order differential are self-adjoint. In Section 3, we obtain the location of the equation describing small transversal vibrations of a homo- spectrum and the asymptotic distribution of the eigenvalues geneous beam compressed or stretched by a force 𝑔. Again, for the case 𝑔=0.InSection 4,weprovethatthebound- this problem is represented by a quadratic operator pencil, in ary value problem under investigation is Birkhoff regular, a suitably chosen Hilbert space, whose coefficient operators which implies that the eigenvalues for general 𝑔 are small are self-adjoint. In [3], we have considered this fourth order perturbations of the eigenvalues for 𝑔=0.Hence,in differential equation with general two-point boundary con- Section 5, we derive the first four terms of the asymptotics of ditions which depend linearly on the eigenvalue parameter. the eigenvalues and compare them to those obtained for the Necessary and sufficient conditions such that the associated boundary conditions considered in [5]. 2 Journal of Function Spaces and Applications

2. The Quadratic Operator Pencil 𝐿 given by

On the interval [0, 𝑎], we consider the boundary value 𝑦[4] (𝐴 (𝑈)) 𝑦=(̃ ) for 𝑦∈̃ D (𝐴 (𝑈)) , problem 𝑈0𝑦 (7) (4) 󸀠 󸀠 2 00 𝐼0 𝑦 −(𝑔𝑦) =𝜆𝑦, (1) 𝐾=( ), 𝑀=( ). 0𝐼 00

𝐵𝑗 (𝜆) 𝑦 = 0, 𝑗 = 1, 2, 3,4, (2) It is easy to check that 𝐾≥0, 𝑀≥0, 𝑀+𝐾=𝐼,and 𝑀| >0 1 D(𝐴(𝑈)) . We associate a quadratic operator pencil where 𝑔∈𝐶[0, 𝑎] is a real valued function and (2)issepa- 𝐵 2 rated boundary conditions with the 𝑗 depending linearly on 𝐿 (𝜆, 𝛼) =𝜆𝑀−𝑖𝛼𝜆𝐾−𝐴(𝑈) ,𝜆∈C, (8) the eigenvalue parameter 𝜆. The boundary conditions (2)are taken at the endpoint 0 for 𝑗=1,2and at the endpoint 𝑎>0 𝐿 (0, 𝑎) ⊕ C4 𝑗=3,4 in the space 2 with the problem (1), (2). for . Further, we assume for simplicity that We observe that (8) is an operator representation of the eigenvalue problem (1), (2)inthesensethatafunction𝑦 [𝑝𝑗] [𝑞𝑗] 𝐵𝑗 (𝜆) 𝑦=𝑦 (𝑎𝑗) +𝑖𝛼𝜀𝑗𝜆𝑦 (𝑎𝑗) , (3) satisfies (1), (2)ifandonlyif𝐿(𝜆,𝑦=0 𝛼) ̃ . For the boundary conditions (2) with the assumptions where 𝑎𝑗 =0for 𝑗=1,2and 𝑎𝑗 =𝑎for 𝑗=3,4, 0≤𝑞𝑗 <𝑝𝑗 ≤ made so far, [3, Theorem 1.2] leads to the following. 3, 𝛼>0,and𝜀𝑗 ∈ C \{0}. We recall that the quasi-derivatives Proposition 1. 𝐴(𝑈) associated with (1)aregivenby The differential operator associated with (1), (2) is self-adjoint if and only if for 𝑗=1,...,4;thenumbers 𝑝 𝑞 𝜀 𝑝 +𝑞 =3 𝜀 = 𝑦[0] =𝑦,[1] 𝑦 =𝑦󸀠,𝑦[2] =𝑦󸀠󸀠, 𝑗, 𝑗,and 𝑗 satisfy the following conditions: 𝑗 𝑗 , 𝑗 1 if 𝑞𝑗 is even in case 𝑎𝑗 =0or odd in case 𝑎𝑗 =𝑎,and𝜀𝑗 =−1, 󸀠 (4) otherwise. 𝑦[3] =𝑦(3) −𝑔𝑦󸀠,𝑦[4] =𝑦(4) −(𝑔𝑦󸀠) , Sinceweaimtoconsidertheboundaryeigenvalueprob- see [9, page 26]. In order to have independent boundary lem (1), (2)incasethat𝐴(𝑈) is self-adjoint, the boundary conditions, we will also assume that the numbers 𝑝1, 𝑞1, 𝑝2, operators are, up to permutation, 𝑞2 as well as the numbers 𝑝3, 𝑞3, 𝑝4, 𝑞4 are mutually disjoint. 𝐵 (𝜆) 𝑦=𝑦󸀠󸀠 (0) −𝑖𝛼𝜆𝑦󸀠 (0) , Recall that in applications, using separation of variables, 1 (9) the parameter 𝜆 emanates from derivatives with respect to 𝐵 (𝜆) 𝑦=𝑦[3] (0) +𝑖𝛼𝜆𝑦(0) , the time variable in the original partial differential equation, 2 (10) and it is reasonable that the highest space derivative occurs 𝐵 (𝜆) 𝑦=𝑦󸀠󸀠 (𝑎) +𝑖𝛼𝜆𝑦󸀠 (𝑎) , in the term without time derivative. Thus, the most relevant 3 (11) boundary conditions would have 𝑞𝑗 <𝑝𝑗 for 𝑗=1,...,4. [3] 𝐵4 (𝜆) 𝑦=𝑦 (𝑎) −𝑖𝛼𝜆𝑦(𝑎) . (12) Further assumptions on the 𝑝𝑗, 𝑞𝑗,and𝜀𝑗 will be made later and will be justified by the requirements on the operator As in [4, Proposition 2.3], we obtain the following. pencil which we are going to define now. For rather generic boundary conditions, a quadratic Proposition 2. The operator pencil 𝐿(⋅, 𝛼) is a Fredholm operator pencil has been associated in [3], and we will now valued operator function with index 0.Thespectrumofthe recall notations and results from [3] which are relevant in our Fredholm operator 𝐿(⋅, 𝛼) consists of discrete eigenvalues of case. finite multiplicities, and all eigenvalues of 𝐿(⋅, 𝛼), 𝛼≥0,lie 𝑈 We denote by the collection of the boundary conditions in the closed upper half-plane and on the imaginary axis and 𝑈 (2) and define the following operators related to : are symmetric with respect to the imaginary axis.

[𝑝 ] 4 [𝑞 ] 4 𝑈 𝑦=(𝑦 𝑗 (𝑎 )) ,𝑈𝑦=(𝜀𝑦 𝑗 (𝑎 )) , Proof. As in [2,Section3],wecanarguethatforall𝜆∈ 0 𝑗 𝑗=1 1 𝑗 𝑗 𝑗=1 C 𝐿(𝜆, 𝛼) 𝐿(0, 0) (5) , is a relatively compact perturbation of , 𝐿(0, 0) 𝑦∈𝑊2 (0, 𝑎) , where is well known to be a Fredholm operator. The 4 statement on the location of the spectrum follows as in [2, 2 Lemma 3.1]. where 𝑊4 (0, 𝑎) is the Sobolev space of order 4 on the interval (0, 𝑎).Weconsiderthelinearoperators𝐴(𝑈), 𝐾,and𝑀 in 4 3. Asymptotics of Eigenvalues for 𝑔=0 the space 𝐿2(0, 𝑎) ⊕ C with domains In this section, we investigate the boundary value problem (1), 𝑦 D (𝐴 (𝑈)) ={𝑦=(̃ ): 𝑦∈𝑊2 (0, 𝑎)}, (2)with𝑔=0. We count all eigenvalues with their proper 𝑈 𝑦 4 1 (6) multiplicities and develop a formula for the asymptotic distribution of the eigenvalues, which we will use to obtain D (𝐾) = D (𝑀) =𝐿 (0, 𝑎) ⊕ C4, 2 the corresponding formula for general 𝑔.Observethatfor Journal of Function Spaces and Applications 3

[𝑗] 𝑔=0, the quasi-derivatives 𝑦 coincide with the standard In view of (11), (12), we get that (𝑗) derivatives 𝑦 . We take the canonical fundamental system (𝑚) Φ𝑗 (𝜇) 𝑦𝑗(⋅, 𝜆), 𝑗 = 1,...,4,of(1)with𝑦𝑗 (0) = 𝛿𝑗,𝑚+1 for 𝑚= 0,...,3. It is well known that the functions 𝑦𝑗(⋅, 𝜆) are analytic 2 󸀠 (3) =[𝑖𝛼𝜇 {𝑦𝜎 (𝑎, 𝜇)𝜎 𝑦 (𝑎, 𝜇) on C with respect to 𝜆.Putting 𝑗,1 𝑗,2 −𝑦󸀠 (𝑎, 𝜇) 𝑦(3) (𝑎, 𝜇) 4 𝜎𝑗,2 𝜎𝑗,1 𝑀 (𝜆) =(𝐵(𝜆) 𝑦 (⋅,𝜆)) , 𝑖 𝑗 𝑖,𝑗=1 (13) +𝑦󸀠󸀠 (𝑎, 𝜇) 𝑦 (𝑎, 𝜇) 𝜎𝑗,2 𝜎𝑗,1 the eigenvalues of the boundary value problem (1), (2)are 󸀠󸀠 𝑀 −𝑦 (𝑎, 𝜇)𝜎 𝑦 (𝑎, 𝜇)} the eigenvalues of the analytic matrix function ,where 𝜎𝑗,1 𝑗,2 (20) the corresponding geometric and algebraic multiplicities 2 4 󸀠 coincide, see [10,Theorem6.3.2]. +𝛼 𝜇 {𝑦𝜎 (𝑎, 𝜇)𝜎 𝑦 (𝑎, 𝜇) 2 𝑗,1 𝑗,2 Setting 𝜆=𝜇 and 󸀠 −𝑦𝜎 (𝑎, 𝜇)𝜎 𝑦 (𝑎, 𝜇)} 1 1 𝑗,2 𝑗,1 𝑦(𝑥,𝜇)= sinh (𝜇𝑥) − sin (𝜇𝑥) , 2𝜇3 2𝜇3 (14) +𝑦󸀠󸀠 (𝑎, 𝜇) 𝑦(3) (𝑎, 𝜇) 𝜎𝑗,1 𝜎𝑗,2 it is easy to see that −𝑦󸀠󸀠 (𝑎, 𝜇) 𝑦(3) (𝑎, 𝜇)] . 𝜎𝑗,2 𝜎𝑗,1

(4−𝑗) Observing that the 𝑦𝑗 are given by (14)and(15) a straightfor- 𝑦𝑗 (𝑥,) 𝜆 =𝑦 (𝑥,𝜇), 𝑗=1,...,4. (15) ward calculation leads to 𝑀(𝜆) 1 The first and the second rows of have exactly Φ1 (𝜇) = 𝑖𝛼𝜇 (sin (𝜇𝑎) cosh (𝜇𝑎) + cos (𝜇𝑎) sinh (𝜇𝑎)) two nonzero entries (for 𝜆 =0̸), and these nonzero entries 2 2 are: 𝐵1(𝜆)𝑦2(⋅, 𝜆) = −𝑖𝛼𝜇 and 𝐵1(𝜆)𝑦3(⋅, 𝜆) = 1 while 1 2 + 𝑖𝛼sin ( (𝜇𝑎) cosh (𝜇𝑎) 𝐵2(𝜆)𝑦1(⋅, 𝜆) = 𝑖𝛼𝜇 and 𝐵2(𝜆)𝑦4(⋅, 𝜆) =. 1 Therefore, 2𝜇

𝑀 (𝜆) − cos (𝜇𝑎) sinh (𝜇𝑎)) 1 0−𝑖𝛼𝜇2 10 − 𝛼2 (1 − (𝜇𝑎) (𝜇𝑎)) 2 cos cosh 𝑖𝛼𝜇2 001 =( ). 1 𝐵 (𝜇2)𝑦 𝐵 (𝜇2)𝑦 𝐵 (𝜇2)𝑦 𝐵 (𝜇2)𝑦 + (1 + (𝜇𝑎) (𝜇𝑎)) , 3 1 3 2 3 3 3 4 2 cos cosh 2 2 2 2 𝐵4 (𝜇 )𝑦1 𝐵4 (𝜇 )𝑦2 𝐵4 (𝜇 )𝑦3 𝐵4 (𝜇 )𝑦4 4 Φ2 (𝜇) = −𝑖𝛼𝜇 sin (𝜇𝑎) sinh (𝜇𝑎) (16) 2 1 2 +𝑖𝛼𝜇 cos (𝜇𝑎) cosh (𝜇𝑎) − (1 + 𝛼 ) An expansion of det 𝑀(𝜆) = 𝜙(𝜇) gives 2 3 ×𝜇 (sin (𝜇𝑎) cosh (𝜇𝑎) + cos (𝜇𝑎) sinh (𝜇𝑎)) , 𝜙(𝜇)=−𝛼2𝜇4Φ (𝜇) + 𝑖𝛼𝜇2Φ (𝜇) + 𝑖𝛼𝜇2Φ (𝜇) + Φ (𝜇) , 1 2 3 4 Φ (𝜇) = 𝑖𝛼𝜇2 (𝜇𝑎) (𝜇𝑎) + 𝑖𝛼 (𝜇𝑎) (𝜇𝑎) (17) 3 cos cosh sin sinh 1 − (1 + 𝛼2) where 2

2 2 ×𝜇(sin (𝜇𝑎) cosh (𝜇𝑎) − cos (𝜇𝑎) sinh (𝜇𝑎)) , 𝐵3 (𝜇 )𝑦𝜎 (⋅, 𝜇)3 𝐵 (𝜇 )𝑦𝜎 (⋅, 𝜇) Φ (𝜇) = ( 𝑗,1 𝑗,2 ), 𝑗 det 2 2 1 5 𝐵4 (𝜇 )𝑦𝜎 (⋅, 𝜇)4 𝐵 (𝜇 )𝑦𝜎 (⋅, 𝜇) Φ (𝜇) = − 𝑖𝛼𝜇 ( (𝜇𝑎) (𝜇𝑎) + (𝜇𝑎) (𝜇𝑎)) 𝑗,1 𝑗,2 4 2 sin cosh cos sinh (18) 1 − 𝑖𝛼𝜇3 ( (𝜇𝑎) (𝜇𝑎) − (𝜇𝑎) (𝜇𝑎)) 2 sin cosh cos sinh with 1 − 𝛼2𝜇4 (1 + (𝜇𝑎) (𝜇𝑎)) (3, 4) 𝑗=1, 2 cos cosh { if {(1, 3) 𝑗=2, (𝜎 ,𝜎 )= if 1 4 𝑗,1 𝑗,2 { 2, 4 𝑗=3, (19) + 𝜇 (1 − cos (𝜇𝑎) cosh (𝜇𝑎)) . {( ) if 2 {(1, 2) if 𝑗=4. (21) 4 Journal of Function Spaces and Applications

It follows that the term with the highest 𝜇-power in 𝜙 comes for 𝜙0(𝜇) =0̸, that is, sin(𝜇𝑎) =0̸ and sinh(𝜇𝑎) =0̸.Itfollows from Φ2 andisanonzeromultipleof that

3 6 𝑖(𝛼 +𝛼) 𝜙0 (𝜇) := 𝜇 sin (𝜇𝑎) sinh (𝜇𝑎) . (22) 𝜙 (𝜇) = − ( (𝜇𝑎) + (𝜇𝑎)) 1 𝜇 cot coth

2 𝜙 𝛼 1 The following result on the zeros of 0,withpropercount- − [ +3cot (𝜇𝑎) coth (𝜇𝑎)] ing, is obvious. 𝜇2 sin (𝜇𝑎) sinh (𝜇𝑎) Lemma 3. 𝜙 8 0 𝛼2 0 has a zero of multiplicity at ,simplezerosat − 𝜇4

𝜋 4 𝜇̃ = (𝑘−2) , 𝑘=3,4,..., 𝛼 1 𝑘 𝑎 (23) + [ − cot (𝜇𝑎) coth (𝜇𝑎)] 2𝜇2 sin (𝜇𝑎) sinh (𝜇𝑎) 1 1 simple zeros at −𝜇̃𝑘, 𝜇̃−𝑘 =𝑖𝜇̃𝑘 and −𝑖𝜇̃𝑘 for 𝑘=3,4..., and no + [ − (𝜇𝑎) (𝜇𝑎)] 2𝜇2 (𝜇𝑎) (𝜇𝑎) cot coth other zeros. sin sinh 3 Proposition 4. 𝑔=0 𝑘 𝑖(𝛼 +𝛼) For , there exists a positive integer 0 such − ( (𝜇𝑎) − (𝜇𝑎)) . ̂ 3 cot coth that the eigenvalues 𝜆𝑘, 𝑘∈Z \{0},countedwithmultiplicity, 𝜇 of the problem (1), (9)–(12) can be enumerated in such a way (27) ̂ that the eigenvalues 𝜆𝑘 are pure imaginary for |𝑘| <0 𝑘 ,and ̂ ̂ ̂ 2 𝜆−𝑘 =−𝜆𝑘 for 𝑘≥𝑘0.For𝑘>0,wecanwritethat𝜆𝑘 = 𝜇̂𝑘, Fix 𝜀 ∈ (0, 𝜋/2𝑎),andfor𝑘 = 3, 4, .,let . 𝑅𝑘,𝜀 be the where the 𝜇̂𝑘 have the following asymptotic representation as squares determined by the vertices (𝑘 − 2)(𝜋/𝑎) ± 𝜀, ±𝑖𝜀 𝑘→∞: 𝑘 = 3,4,.... These squares do not intersect due to 𝜀<𝜋/2𝑎. There exists 𝐶1(𝜀) > 0 such that | coth(𝜇𝑎)|1 <𝐶 (𝜀) for 𝜇 𝑅 𝑘≥3 𝜋 all on the squares 𝑘,𝜀, . By periodicity, there are 𝜇̂ = (𝑘−2) +𝑜(1) . 𝐶 (𝜀) 𝐶 (𝜀) | (𝜇𝑎)| >𝐶 (𝜀) 𝑘 𝑎 (24) numbers 2 and 3 such that sin 2 and |cot(𝜇𝑎)|3 <𝐶 (𝜀) for all 𝜇 on the boundary of the squares 𝑅𝑘,𝜀 for all 𝑘≥3.Since| sinh(𝜇𝑎)| ≥| sinh((Re 𝜇)𝑎)|,itfollows 𝑘 (𝜀) ∈ N 𝜇 𝑅 In particular, the number of pure imaginary eigenvalues is even. that there is 1 such that for all on the squares 𝑘,𝜀, where 𝑘>𝑘1(𝜀),theestimate|𝜙1(𝜇)| < 1 holds. By Lemma 3, Proof. It follows from (17)and(21)that 𝜙0 has exactly one simple zero inside 𝑅𝑘,𝜀.Hence,itfollows from Rouche’s´ theorem that there is exactly one (simple) zero 𝜇̂𝑘 of 𝜙 in each 𝑅𝑘,𝜀 for 𝑘≥𝑘1(𝜀), which proves the existence 2 6 3 𝜙(𝜇)=𝛼 𝜇 sin (𝜇𝑎) sinh (𝜇𝑎) − 𝑖 (𝛼 +𝛼) of zeros 𝜇̂𝑘 of 𝜙 with 𝜇̂𝑘 = 𝜇̃𝑘 + 𝑜(1) as 𝑘→∞. Let 5 ×𝜇 (sin (𝜇𝑎) cosh (𝜇𝑎) + cos (𝜇𝑎) sinh (𝜇𝑎)) ̃ 2 6 2 4 𝜙0 (𝜇) = 𝛼 𝜇 sin (𝜇𝑎) sinh (𝜇𝑎) −𝛼 𝜇 (1 + 3 cos (𝜇𝑎) cosh (𝜇𝑎)) 2 4 2 2 −𝛼 𝜇 (1 + 3 cos (𝜇𝑎) cosh (𝜇𝑎)) −𝛼 𝜇 sin (𝜇𝑎) sinh (𝜇𝑎) (25) −𝛼2𝜇2 (𝜇𝑎) (𝜇𝑎) 1 4 4 sin sinh + 𝛼 𝜇 (1 − cos (𝜇𝑎) cosh (𝜇𝑎)) 2 1 4 4 + 𝛼 𝜇 (1 − cos (𝜇𝑎) cosh (𝜇𝑎)) 1 4 3 2 + 𝜇 (1 − cos (𝜇𝑎) cosh (𝜇𝑎)) − 𝑖 (𝛼 +𝛼) 2 1 4 + 𝜇 (1 − cos (𝜇𝑎) cosh (𝜇𝑎)) , 3 2 ×𝜇 (sin (𝜇𝑎) cosh (𝜇𝑎) − cos (𝜇𝑎) sinh (𝜇𝑎)) . ̃ 3 𝜙1 (𝜇) = −𝑖 (𝛼 +𝛼) 𝛼2 𝜙 (𝜇) 5 Up to the constant factor ,thefirsttermequals 0 . ×𝜇 (sin (𝜇𝑎) cosh (𝜇𝑎) + cos (𝜇𝑎) sinh (𝜇𝑎)) Considering that, therefore, −𝑖(𝛼3 +𝛼)

2 3 𝜙(𝜇)−𝛼 𝜙0 (𝜇) ×𝜇 (sin (𝜇𝑎) cosh (𝜇𝑎) − cos (𝜇𝑎) sinh (𝜇𝑎)) . 𝜙1 (𝜇) := (26) 𝜙0 (𝜇) (28) Journal of Function Spaces and Applications 5

̃ ̃ Then, 𝜙(𝜇) = 𝜙0(𝜇) + 𝜙1(𝜇),andforall𝜇∈C, 4. Birkhoff Regularity 7.3.1 𝜙 (𝑖𝜇) = 𝜙̃ (𝜇) − 𝜙̃ (𝜇) = 𝜙(𝜇). (29) We refer to [10,Definition ] for the definition of Birkhoff 0 1 regularity. 𝜙 Observing that is an even function, it follows altogether that Proposition 5. The boundary value problem (1), (9)–(12) is 𝜙 ±𝜇̂ ±𝜇̂ 𝑘>𝑘(𝜀) 𝜇̂ =𝑖𝜇̂ has zeros 𝑘, −𝑘 for 1 with −𝑘 𝑘. Birkhoff regular for 𝛼>0with respect to the eigenvalue 2 We still have to show that the stated counting and parameter 𝜇 given by 𝜆=𝜇. asymptotic behaviour describes all zeros of 𝜙.Tothisend, 𝜙 𝑆 𝑘∈N we are going to estimate 1 on the squares 𝑘, ,whose Proof. The characteristic function of1 ( ) as defined in [10, vertices are ±(𝑘 + (1/2))(𝜋/𝑎) ± 𝑖(𝑘 + (1/2))(𝜋/𝑎).For𝜇= (7.1.4) 𝜋(𝜌) =𝜌4 −1 𝑖𝑘−1 𝑘 = 1,...,4 𝑥+𝑖𝑦𝑥 𝑦∈R 𝑥 =0̸ ]is ,anditszerosare , . , , and ,wehave We can choose

𝑒(𝑎𝑥+𝑖𝑎𝑦) +𝑒−(𝑎𝑥+𝑖𝑎𝑦) 2 3 (𝑘−1)(𝑗−1) 4 (𝜇𝑎) = 󳨀→ ± 1 (30) 𝐶(𝑥,𝜇)=diag (1, 𝜇, 𝜇 ,𝜇 )(𝑖 ) (35) coth 𝑒(𝑎𝑥+𝑖𝑎𝑦) −𝑒−(𝑎𝑥+𝑖𝑎𝑦) 𝑘,𝑗=1 according to [10,Theorem7.2.4.A]. The boundary condition uniformly in 𝑦 as 𝑥→±∞.Hence,thereare𝜉≥1and ̃ ̃ (9)–(12)canbewrittenintheform 𝑘0 >0such that for all 𝑘∈N, 𝑘≥𝑘0 and 𝛾∈R, 𝐵 (𝜆) 𝑦=𝐵̂ (𝜇) (𝑦 (𝑎 ),𝑦󸀠 (𝑎 ),𝑦󸀠󸀠 (𝑎 ),𝑦(3) (𝑎 )) , 󵄨 󵄨 𝑗 𝑗 𝑗 𝑗 𝑗 𝑗 󵄨 (𝑘+(1/2)) 𝜋 󵄨 󵄨coth (( +𝑖𝛾)𝑎)󵄨 ≤𝜉. (31) 󵄨 𝑎 󵄨 𝑗 = 1, 2, 3, 4. (36) Note that for 𝑘∈N and 𝛾∈R, Thus, the boundary matrices defined in[10, (7.3.1)]aregiven (𝑘+(1/2)) 𝜋 (( +𝑖𝛾)𝑎)=− (𝑖𝛾𝑎) =−𝑖 (𝛾𝑎) . by cot 𝑎 tan tanh ̂ (32) 𝐵1 (𝜇) (0) 𝐵̂ (𝜇) Hence, we have shown that for 𝑘∈N and 𝛾∈R, 𝑊 (𝜇) = ( 2 )𝐶(0,𝜇), 0 󵄨 󵄨 󵄨 (𝑘+(1/2)) 𝜋 󵄨 0 󵄨 (( +𝑖𝛾)𝑎)󵄨 ≤1. 󵄨cot 𝑎 󵄨 (33) (37) 󵄨 󵄨 0 0 (1) Furthermore, we will make use of the estimates 𝑊 (𝜇) = ( ̂ )𝐶(𝑎,𝜇), 𝐵3 (𝜇) 󵄨 󵄨 󵄨 󵄨 󵄨 (𝑘+(1/2)) 𝜋 󵄨 󵄨 1 󵄨 𝐵̂ (𝜇) 󵄨sinh (( +𝑖𝛾)𝑎)󵄨 ≥ 󵄨sinh ((𝑘 + )𝜋)󵄨 , 4 󵄨 𝑎 󵄨 󵄨 2 󵄨 󵄨 󵄨 󵄨 (𝑘+(1/2)) 𝜋 󵄨 where 󵄨 (( +𝑖𝛾)𝑎)󵄨 = (𝛾𝑎) ≥ 1, 󵄨sin 𝑎 󵄨 cosh ̂ 2 󵄨 󵄨 𝐵1 (𝜇) = (0, −𝑖𝛼𝜇 ,1,0), (34) 𝐵̂ (𝜇) = (𝑖𝛼𝜇2,−𝑔(0) ,0,1), 𝑘∈Z 𝛾∈R 2 which hold for all and all . Therefore, it follows (38) from (31), (33)-(34), and the symmetry of 𝜙,see(29), that 𝐵̂ (𝜇) = (0, 𝑖𝛼𝜇2,1,0), ̃ ̃ 3 there is 𝑘1 ≥ 𝑘0 such that |𝜙1(𝜇)| < 1 for all 𝜇∈𝑆𝑘 with ̃ ̂ 2 𝑘>𝑘1. Again from the definition of 𝜙1 in (27)andRouche’s´ 𝐵4 (𝜇) = (−𝑖𝛼𝜇 ,−𝑔(𝑎) ,0,1). theorem, we conclude that the functions 𝜙0 and 𝜙 have the ̃ 3 3 3 3 same number of zeros in the square 𝑆𝑘,for𝑘∈N with 𝑘≥𝑘1. Choosing 𝐶2(𝜇) = diag(𝜇 ,𝜇 ,𝜇 ,𝜇 ),itfollowsthat 𝜙 4𝑘+8 𝑆 4𝑘+8+4 −1 (𝑗) (𝑗) −1 Since 0 has zeros inside 𝑘 and thus zeros 𝐶2(𝜇) 𝑊 (𝜇) =0 𝑊 +𝑂(𝜇 ),where inside 𝑆𝑘+1,itfollowsthatallsuchzerosof𝜙 are accounted for by ±𝜇̂𝑗 for 0<|𝑗|≤𝑘+2and sufficiently large 𝑘. −𝑖𝛼 𝛼 𝑖𝛼−𝛼 ̂ 2 1−𝑖−1𝑖 Finally, 𝜆𝑘 = 𝜇̂𝑘 account for all eigenvalues of the problem (0) 𝑊0 =( ), (1)-(2) since each of these eigenvalues gives rise to two zeros of 0000 𝜙,countedwithmultiplicity.ByProposition 2,alleigenvalues 0000 ̂ ̂ (39) with nonzero real part occur in pairs 𝜆𝑘, −𝜆𝑘,whichshows 00 00 𝜆̂ =−𝜆̂ 00 00 that we can index all such eigenvalues as −𝑘 𝑘.Since 𝑊(1) =( ). there is an even number of remaining indices, the number of 0 𝑖𝛼 −𝛼 −𝑖𝛼 𝛼 pure imaginary eigenvalues must be even. 1−𝑖−1𝑖 6 Journal of Function Spaces and Applications

4 The Birkhoff matrices are more convenient to replace (𝑦𝑗)𝑗=1 with the asymptotic fund- 4 amental system (𝜂]) obtained in [10,Theorem8.2.1], (1), 𝑊(0)Δ +𝑊(1) (𝐼−Δ ) , ]=1 0 𝑗 0 𝑗 (40) which can be written as Δ 𝑗 = 1, 2, 3, 4 4×4 (𝑗) 𝑖]−1𝜇𝑥 where 𝑗, are the diagonal matrices with 2 𝜂] (𝑥,) 𝜇 =𝛿],𝑗 (𝑥,) 𝜇 𝑒 ; ] =1,...,4; 𝑗=0,...,3, consecutive ones and 2 consecutive zeros in the diagonal in a (43) cyclic arrangement, see [10,Definition7.3.1 and Proposition 4.1.7]. It is easy to see that after a permutation of columns, the where matrices (40) are block diagonal matrices consisting of 2×2 𝑗 4 𝑑 −𝑟 ]−1 ]−1 blocks taken from two consecutive columns (in the sense of 𝛿 (𝑥, 𝜇) =[ ]{∑(𝜇𝑖]−1) 𝜑 (𝑥) 𝑒𝑖 𝜇𝑥}𝑒−𝑖 𝜇𝑥 𝑊(0) ],𝑗 𝑗 𝑟 cyclic arrangement) of the first two rows of 0 and the last 𝑑𝑥 𝑟=0 (1) two rows of 𝑊 , respectively. Hence, the determinants of the 0 −4+𝑗 Birkhoff matrices (40)are +𝑜(𝜇 ), 󵄨 󵄨 󵄨 󵄨 (44) 󵄨 −𝑖𝑗𝛼−𝑖𝑗+1𝛼󵄨 󵄨 𝑖𝑗+2𝛼𝑖𝑗+3𝛼 󵄨 −1 𝑗−1 󵄨 󵄨 󵄨 󵄨 ( ) 󵄨 𝑗−1 𝑗 󵄨 󵄨 𝑗+1 𝑗+2󵄨 𝑗 𝑗 󵄨(−𝑖) (−𝑖) 󵄨 󵄨(−𝑖) (−𝑖) 󵄨 and [𝑑 /𝑑𝑥 ] meansthatweomitthosetermsoftheLeibniz 𝜑(𝑘) 𝑘>4−𝑟 𝑗−1 (41) expansion which contain a function 𝑟 with .Since = (−1) (−2𝛼)(2𝛼) (3) the coefficient of 𝑦 in (1)iszero,wehave𝜑0(𝑥) =,see[ 1 10, (8.2.3) = (−1)𝑗4𝛼2 =0.̸ ]. We will now determine the functions 𝜑1 and 𝜑2.Inthis 𝑛 =0 𝑙=4 𝛼>0 regard, observe that 0 and in the notation of Thus, the problem (1), (9)–(12) is Birkhoff regular for . [10, (8.1.2) and (8.1.3)], see [10,Theorem8.1.2]. From [10, (8.2.45)], we know that 𝜑 =𝜑 =𝜀⊤𝑉𝑄[𝑟]𝜀 , 5. Asymptotic Expansions of Eigenvalues 𝑟 1,𝑟 1 1 (45) 2 2 4 4 (𝑗−1)(𝑘−1) 4 With 𝜆=𝜇, 𝐷(𝜇) = det(𝐵𝑖(𝜇 )𝑦𝑗(⋅, 𝜇))𝑖,𝑗=1 defines a char- where 𝜀] is the ]th unit vector in C , 𝑉=(𝑖 )𝑗,𝑘=1, [𝑟] acteristic determinant of the problem (1), (9)–(12)with and 𝑄 are 4×4matrices given by [10, (8.2.28), (8.2.33),and 𝑦𝑗 𝑗 = 1, 2, 3, 4 [0] respect to the fundamental system , con- (8.2.34)], that is, 𝑄 =𝐼4, sidered in Section 3.Observethat𝜙 is the corresponding 𝑔=0 󸀠 characteristic determinant for .DuetotheBirkhoff Ω 𝑄[1] −𝑄[1]Ω =𝑄[0] =0, regularity, 𝑔 only influences lower order terms in 𝐷.Together 4 4 with the estimates in Section 3,itcanbeinferredthat [2] [2] [1]󸀠 1 ⊤ −2 [0] Ω4𝑄 −𝑄 Ω4 =𝑄 − 𝑔Ω4𝜀𝜀 Ω 𝑄 , outside the interior of the small squares 𝑅𝑘,𝜀, −𝑅𝑘,𝜀, 𝑖𝑅𝑘,𝜀, 4 4 −𝑖𝑅𝑘,𝜀 around the zeros of 𝜙0, |𝐷(𝜇)0 −𝜙 (𝜇)| < |𝜙0(𝜇)| 2 (46) if |𝜇| is sufficiently large. Since the fundamental system 𝑦𝑗, 󸀠 1 −1−𝑗 0=𝜀⊤ (𝑄[2] + ∑𝑘 Ω 𝜀𝜀⊤Ω 𝑄[2−𝑗])𝜀 𝑗 = 1, 2, 3, 4, depends analytically on 𝜇,also𝐷 depends ] 3−𝑗 4 4 ] 4 𝑗=1 analytically on 𝜇. Hence, applying Rouche’s´ theorem both to the large squares 𝑆𝑘 and to the small squares which are (] =1,2,3,4) , sufficiently far away from the origin, it follows that the eigenvalues of the boundary value problem for general 𝑔 󸀠 where 𝑘2 =−𝑔, 𝑘1 =−𝑔, Ω4 = diag(1, 𝑖, −1, −𝑖) and havethesameasymptoticdistributionasfor𝑔=0.Hence ⊤ 𝑥 𝜀 = (1, 1, 1, 1).Letting𝐺(𝑥) =∫ 𝑔(𝑡)𝑑𝑡,alengthybut Proposition 4 leads to the following. 0 straightforward calculation gives 1 Proposition 6. For 𝑔∈𝐶[0, 𝑎], there exists a positive integer 1 1 1 𝑘 𝜆 𝜑 = 𝐺, 𝜑 = 𝐺2 − 𝑔, 0 such that the eigenvalues 𝑘,countedwithmultiplicity,ofthe 1 4 2 32 8 (47) problem (1), (9)–(12),where𝑘∈Z \{0}can be enumerated in such a way that the eigenvalues 𝜆𝑘 are pure imaginary for |𝑘| < and thus 𝑘 𝜆 =−𝜆 𝑘≥𝑘 𝑘>0 𝜆 =𝜇2 0,and −𝑘 𝑘 for 0.For ,wecanwrite 𝑘 𝑘, 1 𝜇 𝜂 (𝑥, 𝜇) = [(1 + 𝑖−]+1𝐺 (𝑥) 𝜇−1 where the 𝑘 have the following asymptotic representation as ] 4 𝑘→∞: 1 1 𝜋 +(−1)]−1 ( 𝐺(𝑥)2 − 𝑔 (𝑥))𝜇−2) 𝜇 = (𝑘−2) +𝑜(1) . 32 8 (48) 𝑘 𝑎 (42) ]−1 +{𝑜 (𝜇−2)} ]𝑒𝑖 𝜇𝑥 In particular, the number of pure imaginary eigenvalues is even. ∞

In the remainder of the section, we establish more pre- for ] = 1,...,4,where{𝑜(⋅)}∞ means that the estimate is cise asymptotic expansions of the eigenvalues. For this, it is uniform in 𝑥. Journal of Function Spaces and Applications 7

In view of (44), the characteristic determinant of (1), (9)– A straightforward calculation gives that 4 (12) with respect to the fundamental system (𝜂𝑘)𝑘=1 is

𝛾13𝛾24 −𝛾14𝛾23 4 4 𝐷(̃ 𝜇) = (𝐵 (𝜇2) 𝜂 (⋅,𝜇)) = (𝛾 (𝜀 )) , 6 2 5 4 4 det 𝑗 𝑘 𝑗,𝑘=1 det 𝑗𝑘 exp 𝑗𝑘 𝑗,𝑘=1 =−2𝛼𝜇 −(1+𝛼 ) (1−𝑖) 𝜇 +2𝑖𝛼𝜇 +𝑜(𝜇 ), (49) 𝛾31𝛾42 −𝛾32𝛾41 =2𝛼𝜇6 + 1−𝑖 𝜇5 (1 + 𝛼2 +2𝛼𝜑 𝑎 ) where ( ) 1 ( ) 4 2 2 4 −2𝑖𝜇 (𝛼 (11 +𝜑 (𝑎) )+(1+𝛼 )𝜑1 (𝑎))+𝑜(𝜇 ), 𝑘−1 𝜀1𝑘 =𝜀2𝑘 =0, 𝜀3𝑘 =𝜀4𝑘 =𝑖 𝜇𝑎, 𝛾12𝛾23 −𝛾13𝛾22 2 𝛾1𝑘 =𝛿𝑘,2 (0, 𝜇) − 𝑖𝛼𝜇 𝛿𝑘,1 (0, 𝜇) , =−2𝛼𝜇6 +(1+𝛼2) (1+𝑖) 𝜇5 −2𝑖𝛼𝜇4 +𝑜(𝜇4), 2 𝛾2𝑘 =𝛿𝑘,3 (0, 𝜇) − 𝑔 (0) 𝛿𝑘,1 (0, 𝜇) + 𝑖𝛼𝜇 𝛿𝑘,0 (0, 𝜇) , 𝛾31𝛾44 −𝛾34𝛾41 𝛾 =𝛿 (𝑎, 𝜇) +2 𝑖𝛼𝜇 𝛿 (𝑎, 𝜇) , 3𝑘 𝑘,2 𝑘,1 6 5 2 =−2𝛼𝜇 + (1+𝑖) 𝜇 (1 + 𝛼 −2𝛼𝜑1 (𝑎)) 2 𝛾4𝑘 =𝛿𝑘,3 (𝑎, 𝜇) −𝑔 (𝑎) 𝛿𝑘,1 (𝑎, 𝜇) − 𝑖𝛼𝜇 𝛿𝑘,0 (𝑎, 𝜇) . 4 2 2 4 (50) −2𝑖𝜇 (𝛼 (11 +𝜑 (𝑎) )−(1+𝛼 )𝜑1 (𝑎))+𝑜(𝜇 ). (55)

An expansion of 𝐷̃ leads to Hence,

5 −12 2 2 −1 ̃ 𝜔𝑚𝜇𝑎 𝜇 𝜓1 (𝜇) = −4𝛼 − (1−𝑖) 𝛼[4(1+𝛼 )+𝛼𝐺(𝑎)]𝜇 𝐷(𝜇)= ∑ 𝜓𝑚 (𝜇) 𝑒 , (51) 𝑚=1 1 +𝑖[ 𝛼2𝐺2 (𝑎) +2𝛼(1+𝛼2)𝐺(𝑎) 4 where 𝜔1 =1+𝑖, 𝜔2 =−1+𝑖, 𝜔3 =−1−𝑖, 𝜔4 =1−𝑖, 2 +2(1 + 𝛼2) +8𝛼2]𝜇−2 +𝑜(𝜇−2), 𝜔5 =0,andeachofthefunctions𝜓1,...,𝜓5 has asymptotic 𝑘 𝑘−1 𝑘 𝑘 𝑐 𝜇 +𝑐 𝜇 +⋅⋅⋅+𝑐 𝜇 0 +𝑜(𝜇 0 ) representations of the form 𝑘 𝑘−1 𝑘0 . (56) It follows from (51)that −12 2 2 −1 𝜇 𝜓4 (𝜇) = 4𝛼 − (1+𝑖) 𝛼[4(1+𝛼 )−𝛼𝐺(𝑎)]𝜇

5 1 2 2 2 ̃ −𝜔1𝜇𝑎 (𝜔𝑚−𝜔1)𝜇𝑎 +𝑖[ 𝛼 𝐺 (𝑎) −2𝛼(1+𝛼 )𝐺(𝑎) 𝐷1 (𝜇) := 𝐷(𝜇) 𝑒 =𝜓1 (𝜇) + ∑ 𝜓𝑚 (𝜇) 𝑒 , 4 (57) 𝑚=2 2 (52) +2(1 + 𝛼2) +8𝛼2]𝜇−2 +𝑜(𝜇−2).

where 𝜔2 −𝜔1 =−2, 𝜔3 −𝜔1 =−2−2𝑖, 𝜔4 −𝜔1 =−2𝑖, 𝜔5 − (𝜔 −𝜔 )𝜇𝑎 𝜇 𝐷̃ 𝜔 =−1−𝑖 𝜇 ∈ [−(3𝜋/8), 𝜋/8] |𝑒 𝑚 1 |≤ We know by Proposition 6 that the zeros 𝑘 of satisfy 1 .Ifarg ,wehave 𝜇 = 𝑘(𝜋/𝑎)+𝜏 +𝑜(1) 𝑘→ − sin(𝜋/8)|𝜇|𝑎 (𝜔𝑚−𝜔1)𝜇𝑎 the asymptotic representations 𝑘 0 as 𝑒 for 𝑚 = 2, 3, 5 and the terms 𝜓𝑚(𝜇)𝑒 for ∞. In order to improve on these asymptotic representations, 𝑚 = 2, 3,,canbeabsorbedby 5 𝜓1(𝜇) as they are of the form −𝑠 write 𝑜(𝜇 ) for any integer 𝑠.Hence,forarg𝜇 ∈ [−(3𝜋/8), 𝜋/8],

2 𝜋 −𝑚 −2 (𝜔 −𝜔 )𝜇𝑎 −2𝑖𝜇𝑎 𝜇 =𝑘 +𝜏(𝑘) ,𝜏(𝑘) = ∑ 𝜏 𝑘 +𝑜(𝑘 ), 𝐷 (𝜇) = 𝜓 (𝜇) + 𝜓 (𝜇) 𝑒 4 1 =𝜓 (𝜇) + 𝜓 (𝜇) 𝑒 , 𝑘 𝑎 𝑚 1 1 4 1 4 𝑚=0 (58) (53) 𝑘=1,2,.... where Because of the symmetry of the eigenvalues, we will only need to find the asymptotic expansions as 𝑘→∞.Weknowfrom 𝜓1 (𝜇) = 13[𝛾 𝛾24 −𝛾14𝛾23][𝛾31𝛾42 −𝛾32𝛾41], Proposition 6 that 𝜏0 =−(2𝜋/𝑘),anditisouraimtofind𝜏1 (54) and 𝜏2.Tothisend,wewillsubstitute(58)into𝐷1(𝜇𝑘)=0, 𝜓 (𝜇) = [𝛾 𝛾 −𝛾 𝛾 ][𝛾 𝛾 −𝛾 𝛾 ]. 0 −1 −2 4 12 23 13 22 31 44 34 41 andwewillthencomparethecoefficientsof𝑘 , 𝑘 ,and𝑘 . 8 Journal of Function Spaces and Applications

Observe that References

−2𝑖𝜇 𝑎 −2𝑖𝜏(𝑘)𝑎 −2𝑖𝜏 𝑎 𝜏 𝜏 −2 𝑒 𝑘 =𝑒 =𝑒 0 (−2𝑖𝑎 ( 1 + 2 +𝑜(𝑘 ))) [1] V. Pivovarchik and C. van der Mee, “The inverse generalized exp 2 𝑘 𝑘 Regge problem,” Inverse Problems,vol.17,no.6,pp.1831–1845, −2𝑖𝜏 𝑎 2001. =𝑒 0 [2] M. Moller¨ and V. Pivovarchik, “Spectral properties of a fourth 1 1 order differential equation,” Journal of Analysis and its Applica- ×(1−2𝑖𝑎𝜏 − (2𝑎2𝜏2 +2𝑖𝑎𝜏) +𝑜(𝑘−2)) , 1 𝑘 1 2 𝑘2 tions,vol.25,no.3,pp.341–366,2006. (59) [3] M. Moller¨ and B. Zinsou, “Self-adjoint fourth order differential operators with eigenvalue parameter dependent boundary con- while ditions,” Quaestiones Mathematicae,vol.34,no.3,pp.393–406, 2011. 1 𝑎 𝑎𝜏 (𝑘) −1 𝑎 𝑎2𝜏 = (1 + ) = − 0 +𝑜(𝑘−2). [4] M. Moller¨ and B. Zinsou, “Spectral asymptotics of self-adjoint 2 2 (60) 𝜇𝑘 𝜋𝑘 𝑘𝜋 𝑘𝜋 𝑘 𝜋 fourth order differential operators with eigenvalue parameter dependent boundary conditions,” Complex Analysis and Oper- Using (53), 𝐷1(𝜇𝑘)=0canbewrittenas ator Theory,vol.6,no.3,pp.799–818,2012. −12 −12 −2𝑖𝜇 𝑎 [5] M. Moller¨ and B. Zinsou, “Spectral asymptotics of self-adjoint 𝜇 𝜓 (𝜇 )+𝜇 𝜓 (𝜇 )𝑒 𝑘 =0. 𝑘 1 𝑘 𝑘 4 𝑘 (61) fourth order boundary value problem with eigenvalue parameter dependent boundary conditions,” Boundary Value Problems, Substituting (56), (57), (59), and (60)into(61)andcomparing vol. 2012, article 106, 2012. 𝑘0 𝑘−1 𝑘−2 the coefficients of , ,and ,weget [6] Z. S. Aliyev and N. B. Kerimov, “Spectral properties of the differ- Theorem 7. 𝑔∈𝐶1[0, 𝑎] 𝑘 ential operators of the fourth-order with eigenvalue parameter For , there exists a positive integer 0 dependent boundary condition,” International Journal of Math- such that the eigenvalues 𝜆𝑘, 𝑘∈Z,countedwithmultiplicity, ematics and Mathematical Sciences,vol.2012,ArticleID456517, of the problem (1), (9)–(12),where𝑘∈Z \{0}can be enu- 28 pages, 2012. 𝜆 merated in such a way that the eigenvalues 𝑘 are pure [7]N.B.KerimovandZ.S.Aliev,“Basispropertiesofaspectral imaginary for |𝑘| <0 𝑘 ,and𝜆−𝑘 =−𝜆𝑘 for 𝑘≥𝑘0,where problem with a spectral parameter in the boundary condition,” 2 𝜆𝑘 =𝜇𝑘 and the 𝜇𝑘 have the asymptotic representations Matematicheski˘ıSbornik,vol.197,no.10,pp.65–86,2006 (Russian), Translation in Sbornik: Mathematics,vol.197,pp. 𝜋 𝜏 𝜏 𝜇 =𝑘 +𝜏 + 1 + 2 +𝑜(𝑘−2), 1467–1487, 2006. 𝑘 0 2 (62) 𝑎 𝑘 𝑘 [8]N.B.KerimovandZ.S.Aliev,“Onthebasispropertyofthe system of eigenfunctions of a spectral problem with a spectral and the numbers 𝜏0, 𝜏1, 𝜏2 are as follows: parameter in the boundary condition,” Differentsial’nye Urav- 2 neniya,vol.43,no.7,pp.886–895,2007(Russian),Translation 2𝜋 (1 + 𝛼 )𝑖 1 𝐺 𝑎 ( ) in Differential Equations,vol.43,pp.905–915,2007. 𝜏0 =− ,𝜏1 = + , 𝑎 𝜋𝛼 4 𝜋 [9] J. Weidmann, Spectral Theory of Ordinary Differential Operators, (63) 2 2 2 vol. 1258 of Lecture Notes in Mathematics, Springer, Berlin, 2(1+𝛼 )𝑖 1 𝑎(1−𝛼 ) 1 𝐺 (𝑎) 𝜏 = − + . Germany, 1987. 2 𝜋𝛼 2 𝜋2𝛼2 2 𝜋 [10] R. Mennicken and M. Moller,¨ Non-Self-Adjoint Boundary Eigen- value Problems,vol.192ofNorth-Holland Mathematics Studies, In particular, the number of pure imaginary eigenvalues is Elsevier,Amsterdam,TheNetherlands,2003. even.

Remark 8. In [5], we have considered the differential equation (1) with the same boundary conditions 𝐵3, 𝐵4 at 𝑎 as in this paper but only one 𝜆-dependent boundary conditions at 0. We observe that the first two terms in the eigenvalue expansion coincide with those in Case 1 of [5], which differs from the present case that the 𝜆-term is absent in the boundary condition (10). However, the third and fourth terms are similar but different.

Acknowledgments Variousoftheabovecalculationshavebeenverifiedwith Sage. This work is based upon research supported by the National Research Foundation of South Africa under Grant number 80956. Any opinion, findings and conclusions or recommendations expressed in this material are those of the authors and therefore the NRF do not accept any liability in regard thereto. Hindawi Publishing Corporation Journal of Function Spaces and Applications Volume 2013, Article ID 409760, 7 pages http://dx.doi.org/10.1155/2013/409760

Research Article The Uniqueness of Strong Solutions for the Camassa-Holm Equation

Meng Wu1 and Chong Lai2

1 The School of Mathematics, Southwestern University of Finance and Economics, Chengdu 610074, China 2 The School of Finance, Southwestern University of Finance and Economics, Chengdu 610074, China

Correspondence should be addressed to Meng Wu; [email protected]

Received 28 February 2013; Accepted 10 May 2013

Academic Editor: Janusz Matkowski

Copyright © 2013 M. Wu and C. Lai. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Assume that there exists a strong solution of the Camassa-Holm equation and the initial value of the solution belongs to the Sobolev 1 space 𝐻 (𝑅). We provide a new proof of the uniqueness of the strong solution for the equation.

1. Introduction We consider the equivalent form of the Cauchy problem for (1) The integrable Camassa-Holm model [1] 1 𝑢 + (𝑢2) +𝜕 𝑃 (𝑡, 𝑥) =0, 𝑡 2 𝑥 𝑥 𝑢 (2)

𝑢 (0, 𝑥) =𝑢0, 𝑢𝑡 −𝑢𝑡𝑥𝑥 +3𝑢𝑢𝑥 =2𝑢𝑥𝑢𝑥𝑥 +𝑢𝑢𝑥𝑥𝑥 (1) ∞ 𝑃 (𝑡, 𝑥)=Λ−2(𝑢2 + (1/2)𝑢2 )=(1/2)∫ 𝑒|𝑥−𝑦|(𝑢2(𝑡, 𝑦)+ where 𝑢 𝑥 −∞ 2 (1/2)𝑢𝑦(𝑡, 𝑦))𝑑𝑦.If(1)hasasuitablesmoothstrongsolution, has been investigated by many scholars. Equation (1)has we have the conservation law −|𝑥−𝑐𝑡| peaked solitary wave solutions, which takes the form 𝑐𝑒 , ∞ 2 2 𝑐∈𝑅. The existence and uniqueness of the global weak ∫ (𝑢 (𝑡, 𝑥) +𝑢𝑥 (𝑡, 𝑥))𝑑𝑥 solutions for (1)havebeengivenbyConstantinandEscher[2] −∞ ∞ (3) and Constantin and Molinet [3]inwhichthe𝑚=𝑢0 −𝑢0𝑥𝑥 2 2 = ∫ (𝑢 (0, 𝑥) +𝑢𝑥 (0, 𝑥))𝑑𝑥, is a positive (or negative) Radon measure. The local well- −∞ posedness of strong solutions for the Camassa-Holm model and its various generalized forms are provided in [4–8]. For which derives the initial value 𝑢0 satisfying 𝑢0 −𝑢0𝑥𝑥 ≥0or 𝑢0 −𝑢0𝑥𝑥 ≤ 󵄩 󵄩 ‖𝑢‖𝐿∞(𝑅) ≤ ‖𝑢‖𝐻1(𝑅) = 󵄩𝑢0󵄩𝐻1(𝑅). (4) 0,itisshownin[9] that the Camassa-Holm equation has 𝑠 unique global strong solutions in the Sobolev space 𝐻 (𝑅) The objective of this work is to give a new proof of the with 𝑠>3/2. If the initial data satisfy certain conditions, uniqueness for the solutions of the Camassa-Holm equation we know that the local strong solutions blow up in finite (1). Firstly, we establish the following inequality: time [10, 11]. It means that the slope of the solution becomes unbounded while the solution itself remains bounded. For ‖𝑢 (𝑡,) ⋅ − V (𝑡,) ⋅ ‖𝐿1(𝑅) other techniques to obtain the dynamic properties for the ∞ 󵄨 󵄨 ∞ 󵄨 󵄨 ≤𝑐𝑒𝑐𝑡 (∫ 󵄨𝑢 (𝑥) − V (𝑥)󵄨 𝑑𝑥 + ∫ 󵄨𝑢2 − V2 󵄨 𝑑𝑥) , Camassa-Holm equation and other related shallow water 󵄨 0 0 󵄨 󵄨 0𝑥 0𝑥󵄨 equations, the reader is referred to [12–16] and the references −∞ −∞ therein. (5) 2 Journal of Function Spaces and Applications

where 𝑡∈[0,𝑇0),functions𝑢 and V are two local or global Lemma 2 (see [17]). If the function |𝜕𝐹(𝑢)/𝜕𝑢| is bounded, strong solutions of problem (2)withinitialdata𝑢(0, ⋅)0 =𝑢 ∈ then the function 𝐻(𝑢, V)=sign(𝑢 − V)(𝐹(𝑢) V− 𝐹( )) satisfies 1 1 𝑢 V 𝐻 (𝑅) and V(0, ⋅) = V0 ∈𝐻(𝑅),respectively.Constant𝑐 the Lipschitz condition in and ,respectively. ‖𝑢 ‖ 1 ‖V ‖ 1 depends on 0 𝐻 (𝑅), 0 𝐻 (𝑅), and the maximum existence 1 𝑇 Lemma 3. Let 𝑢0 ∈𝐻(𝑅). It holds that the function 𝑄𝑢(𝑡, time 0. Secondly, from (5), we immediately arrive at the goal −2 2 2 of the uniqueness. Here we state that the approach to establish 𝑥) =𝑥 𝜕 𝑃𝑢(𝑡, 𝑥)𝑥 =𝜕 Λ (𝑢 + (1/2)𝑢𝑥) satisfies (5) is the device of doubling variables which was presented in Kruzkov’s paper [17]. 󵄩 󵄩 󵄩𝑃𝑢 (𝑡, 𝑥)󵄩𝐿∞(𝑅) <∞, Thispaperisorganizedasfollows.Severallemmasare 󵄩 󵄩 given in Section 2, while the proofs of the main results are 󵄩𝑃𝑢 (𝑡, 𝑥)󵄩𝐿1(𝑅) <∞, established in Section 3. 󵄩 󵄩 󵄩𝑃𝑢 (𝑡, 𝑥)󵄩𝐿2(𝑅) <∞, (10) 2. Notations and Several Lemmas 󵄩 󵄩 󵄩𝑄𝑢 (𝑡, 𝑥)󵄩𝐿∞(𝑅) <∞, Set 𝜉𝑇 =[0,𝑇]×𝑅for an arbitrary 𝑇>0.Thespaceofall 󵄩 󵄩 󵄩𝑄 (𝑡, 𝑥)󵄩 <∞, infinitely differentiable functions 𝑓(𝑡, 𝑥) with compact sup- 󵄩 𝑢 󵄩𝐿1(𝑅) ∞ port in [0, 𝑇] × 𝑅 is denoted by 𝐶 (𝜉𝑇). We define 𝜌(𝜎) as a 󵄩 󵄩 0 󵄩𝑄 (𝑡, 𝑥)󵄩 <∞. function which is infinitely differentiable on (−∞, +∞) such 󵄩 𝑢 󵄩𝐿2(𝑅) ∞ 𝜌(𝜎) ≥0 𝜌(𝜎) =0 |𝜎| ≥ 1 ∫ 𝜌(𝜎)𝑑𝜎 =1 that , for and −∞ .For −1 any number ℎ>0,welet𝜌ℎ(𝜎) = 𝜌(ℎ 𝜎)/ℎ.Thenwehave The proof of Lemma 3 can be found in [13, 15](see[13,Lemma ∞ that 𝜌ℎ(𝜎) is a function in 𝐶 (−∞, ∞) and 5.1]).

𝜌ℎ (𝜎) ≥0, 𝜌ℎ (𝜎) =0 |𝜎| ≥ℎ, Lemma 4. Let 𝑢 be the strong solution of problem (2), 𝑓(𝑡, 𝑥) ∈ if ∞ 𝐶 (𝜉𝑇),and𝑓(0, 𝑥).Then =0 ∞ (6) 0 󵄨 󵄨 𝑐 󵄨𝜌ℎ (𝜎)󵄨 ≤ , ∫ 𝜌ℎ (𝜎) =1. ℎ −∞ 1 ∬ {|𝑢−𝑘| 𝑓 + (𝑢−𝑘) [𝑢2 −𝑘2]𝑓 V(𝑥) (−∞, 𝑡 sign 2 𝑥 Assume that the function is locally integrable in 𝜉𝑇 ∞). We define the approximation of function V(𝑥) as (11) ∞ − sign (𝑢−𝑘) 𝑄𝑢 (𝑡, 𝑥) 𝑓} 𝑑𝑥 𝑑𝑡 =0, ℎ 1 𝑥−𝑦 V (𝑥) = ∫ 𝜌( ) V (𝑦) 𝑑𝑦, ℎ >0. (7) ℎ −∞ ℎ where 𝑘 is an arbitrary constant. We call 𝑥0 a Lebesgue point of function V(𝑥) if 1 󵄨 󵄨 Proof. Let Φ(𝑢) be an arbitrary twice smooth function on lim ∫ 󵄨V (𝑥) − V (𝑥0)󵄨 𝑑𝑥=0. (8) ℎ→0ℎ |𝑥−𝑥 |≤ℎ the line −∞<𝑢<∞. We multiply (2)bythefunction 0 󸀠 ∞ Φ (𝑢)𝑓(𝑡,,where 𝑥) 𝑓(𝑡, 𝑥) ∈𝐶 (𝜉𝑇). Integrating over 𝜉𝑇 and 𝑥 V(𝑥) 0 At any Lebesgue point 0 of the function ,wehave transferring the derivatives with respect to 𝑡 and 𝑥 to the test ℎ limℎ→0V (𝑥0)=V(𝑥0). Since the set of points which are not function 𝑓, for any constant 𝑘,weobtain ℎ Lebesgue points of V(𝑥) has measured zero, we get V (𝑥) → V(𝑥) ℎ→0 𝑢 as almost everywhere. 󸀠 We introduce notation connected with the concept of ∬ {Φ (𝑢) 𝑓𝑡 +[∫ Φ (𝑧) 𝑧𝑑𝑧]𝑓𝑥 𝜉 𝑘 𝑀 >0 𝑁> 𝑇 acharacteristiccone.Forany 0 , we define (12) ‖𝑢‖ ∞ <∞ ℧ {(𝑡, 𝑥) : sup𝑡∈[0,∞) 𝐿 (𝑅) .Let designate the cone 󸀠 −1 −Φ (𝑢) 𝑄𝑢 (𝑡, 𝑥) 𝑓} 𝑑𝑥 𝑑𝑡 =0, |𝑥| ≤0 𝑀 −𝑁𝑡,0 < 𝑡0 <𝑇 = min(T,𝑀0𝑁 )}.Welet𝑆𝜏 designate the cross section of the cone ℧ by the plane 𝑡= 𝜏, 𝜏 ∈ [0,0 𝑇 ]. ∞ 𝑢 󸀠 in which we have used ∫ [∫ Φ (𝑧)𝑧 𝑑𝑧]𝑓𝑥𝑑𝑥 = Let 𝐻𝑟 = {𝑥 : |𝑥|,where ≤𝑟} 𝑟>0. −∞ 𝑘 ∞ 󸀠 −∫ [𝑓Φ (𝑢)𝑢𝑢𝑥]𝑑𝑥. Lemma 1 V(𝑡, 𝑥) −∞ (see [17]). Let the function be bounded and We have measurable in cylinder Ω=[0,𝑇]×𝐻𝑟.If𝛿 ∈ (0, min[𝑟, 𝑇]) and any number ℎ ∈ (0, 𝛿), then the function ∞ 𝑢 ∫ [∫ Φ󸀠 (𝑧) 𝑧𝑑𝑧]𝑓 𝑑𝑥 1 󵄨 󵄨 𝑥 𝑉 = ∭∫ 󵄨V (𝑡, 𝑥) − V (𝜏, 𝑦)󵄨 𝑑𝑥 𝑑𝑡 𝑑𝑦𝑑𝜏 −∞ 𝑘 ℎ 2 |(𝑡−𝜏)/2|≤ℎ, 󵄨 󵄨 ℎ 𝛿≤(𝑡+𝜏)/2≤𝑇−𝛿, ∞ 󸀠 1 2 1 2 |(𝑥−𝑦)/2|≤ℎ, = ∫ [Φ (𝑢) ( 𝑢 − 𝑘 ) (13) |(𝑥+𝑦)/2|≤𝑟−𝛿 −∞ 2 2 (9) 1 𝑢 − ∫ (𝑧2 −𝑘2)Φ󸀠󸀠 (𝑧) 𝑑𝑧] 𝑓 𝑑𝑥. 2 𝑥 satisfies limℎ→0𝑉ℎ =0. 𝑘 Journal of Function Spaces and Applications 3

ℎ ∞ 󵄨 󵄨 ∞ 󵄨 󵄨 Let Φ (𝑢) be an approximation of the function |𝑢 − 𝑘| and set ≤𝑐(∫ 󵄨𝑢2 − V2󵄨 𝑑𝑥 + ∫ 󵄨𝑢2 − V2 󵄨 𝑑𝑥 ℎ 󵄨 0 0󵄨 󵄨 0𝑥 0𝑥󵄨 Φ(𝑢) = Φ (𝑢). Using the properties of the sign(𝑢 − 𝑘),(12), −∞ −∞ and (13) and sending ℎ→0,wehave ∞ + ∫ |𝑢−V| 𝑑𝑥) , −∞ 1 ∬ {|𝑢−𝑘| 𝑓 + (𝑢−𝑘) [𝑢2 −𝑘2]𝑓 (16) 𝑡 sign 2 𝑥 𝜉𝑇 (14) in which we have used the Fubini theorem, ‖𝑢‖𝐿∞ ≤‖𝑢0‖ 1 − sign (𝑢−𝑘) 𝑄𝑢 (𝑡, 𝑥) 𝑓} 𝑑𝑥 𝑑𝑡 =0, 𝐻 (𝑅) and ‖V‖𝐿∞ ≤‖V0‖𝐻1(𝑅).Theproofiscompleted. which completes the proof. 3. Main Results In fact, the proof of (11)canalsobefoundin[17]. Theorem 6. Let 𝑢 and V be two local or global strong solutions 1 of problem (2) with initial data 𝑢(0, ⋅)0 =𝑢 ∈𝐻(𝑅) and Lemma 5. Assume 𝑢 and V are two strong solutions of problem 1 V(0, ⋅) = V0 ∈𝐻(𝑅),respectively.Let𝑇0 be the maximum (2).Ithas existence time of solutions 𝑢 and V.Forany𝑡∈[0,𝑇0),itholds that 󵄨 ∞ 󵄨 󵄨 󵄨 󵄨∫ sign (𝑢−V) [𝑄𝑢 (𝑡, 𝑥) −𝑄V (𝑡, 𝑥)]𝑓𝑑𝑥󵄨 󵄨 −∞ 󵄨 ‖𝑢 (𝑡,) ⋅ − V (𝑡,) ⋅ ‖𝐿1(𝑅) (15) ∞ 𝑐𝑡 󵄨 2 2󵄨 󵄨 2 2 󵄨 ≤𝑒 𝑐(1+𝑇 ) ≤𝑐∫ (󵄨𝑢 − V 󵄨 + 󵄨𝑢 − V 󵄨 + |𝑢−V|)𝑑𝑥. 0 󵄨 0 0󵄨 󵄨 0𝑥 0𝑥󵄨 −∞ ∞ 󵄨 󵄨 ∞ 󵄨 󵄨 ×(∫ 󵄨𝑢 (𝑥) − V (𝑥)󵄨 𝑑𝑥 + ∫ 󵄨𝑢2 − V2 󵄨 𝑑𝑥) , 󵄨 0 0 󵄨 󵄨 0𝑥 0𝑥󵄨 Proof. We have −∞ −∞ (17) ∞ ∞ 2 2 2 2 2 2 ∫ (𝑢𝑥 − V𝑥)𝑑𝑥=∫ [(𝑢0 +𝑢0𝑥)−(V0 + V0𝑥)] 𝑑𝑥 −∞ −∞ where 𝑐 depends on ‖𝑢0‖𝐻1(𝑅) and ‖V0‖𝐻1(𝑅).

∞ From Theorem 6, we immediately obtain the uniqueness − ∫ (𝑢2 − V2)𝑑𝑥, −∞ result.

󵄨 ∞ Theorem 7. Let 𝑢(𝑡, 𝑥) be a strong solution of (1) with 𝑢0 ∈ 󵄨 −2 2 1 2 2 1 2 1 󵄨∫ Λ (𝑢 + 𝑢𝑥 − V − V𝑥) 𝐻 (𝑅),andlet𝑇0 be the maximum existence time of solution 󵄨 −∞ 2 2 𝑢. Then any strong solution of (1) is unique. 󵄨 󵄨 ×𝜕 [ [𝑢 (𝑡, 𝑥) − V (𝑡, 𝑥)] 𝑓 (𝑡, 𝑥)]𝑑𝑥󵄨 𝑥 sign 󵄨 Proof of Theorem 6. For an arbitrary 𝑇>0,set𝜉𝑇 =[0,𝑇]×𝑅. 󵄨 ∞ Let 𝑓(𝑡, 𝑥)0 ∈𝐶 (𝜉𝑇). We assume that 𝑓(𝑡, 𝑥) =0 outside 󵄨 ∞ some cylinder 󵄨1 −|𝑥−𝑦| 2 1 2 2 1 2 = 󵄨 ∬ 𝑒 (𝑢 + 𝑢𝑦 − V − V𝑦)𝑑𝑦 󵄨2 −∞ 2 2 z = {(𝑡, 𝑥)} = [𝛿, 𝑇 − 2𝛿] ×𝐻𝑟−2𝛿,0<2𝛿≤min (𝑇, 𝑟) . 󵄨 󵄨 (18) ×𝜕𝑥 [sign [𝑢 (𝑡, 𝑥) − V (𝑡, 𝑥)] 𝑓 (𝑡, 𝑥)]𝑑𝑥󵄨 󵄨

∞ 󵄨 ∞ 󵄨 We define 1 󵄨 2 1 2 2 1 2 󵄨 ≤ ∫ 󵄨∫ (𝑢 + 𝑢𝑦 − V − V𝑦)𝑑𝑦󵄨 2 −∞ 󵄨 −∞ 2 2 󵄨 𝑡+𝜏 𝑥+𝑦 𝑡−𝜏 𝑥−𝑦 𝑔=𝑓( , )𝜌ℎ ( )𝜌ℎ ( ) −|𝑥−𝑦| 󵄨 󵄨 2 2 2 2 ×𝑒 󵄨𝜕𝑥 [sign [𝑢 (𝑡, 𝑥) − V (𝑡, 𝑥)] 𝑓 (𝑡, 𝑥)]󵄨 𝑑𝑥 (19) =𝑓(⋅⋅⋅) 𝜆ℎ (∗) , 󵄨 ∞ 󵄨 1 󵄨 2 1 2 2 1 2 󵄨 ≤ 󵄨∫ (𝑢 + 𝑢𝑦 − V − V𝑦)𝑑𝑦󵄨 2 󵄨 −∞ 2 2 󵄨 where (⋅ ⋅ ⋅) = ((𝑡 + 𝜏)/2, (𝑥 + 𝑦)/2) and (∗) = ((𝑡 − 𝜏)/2, (𝑥 − ∞ 𝑦)/2).Thefunction𝜌(𝜎) isdefinedin(6). Note that 󵄨 󵄨 × ∫ 󵄨𝜕𝑥 [sign [𝑢 (𝑡, 𝑥) − V (𝑡, 𝑥)] 𝑓 (𝑡, 𝑥)]󵄨 𝑑𝑥 −∞ 𝑔𝑡 +𝑔𝜏 =𝑓𝑡 (⋅⋅⋅) 𝜆ℎ (∗) , 󵄨 ∞ 󵄨 󵄨 2 1 2 2 1 2 󵄨 (20) ≤𝑐󵄨∫ (𝑢 + 𝑢𝑦 − V − V𝑦)𝑑𝑦󵄨 𝑔𝑥 +𝑔𝑦 =𝑓𝑥 (⋅⋅⋅) 𝜆ℎ (∗) . 󵄨 −∞ 2 2 󵄨 4 Journal of Function Spaces and Applications

󵄨 𝑘=V(𝜏, 𝑦) 𝑓(𝑡, 𝑥) =0 󵄨 Taking and assuming outside the + 󵄨∬ (𝑢 (𝑡, 𝑥) − V (𝑡, 𝑥)) z 󵄨 sign cylinder ,fromLemma 4,wehave 󵄨 𝜉𝑇 󵄨 󵄨 󵄨 󵄨 ×[𝑄 𝑡, 𝑥 −𝑄 𝑡, 𝑥 ] 𝑓 𝑑𝑥󵄨 𝑑𝑡 . ∭∫ { 󵄨𝑢 (𝑡, 𝑥) − V (𝜏, 𝑦)󵄨 𝑔 + (𝑢 (𝑡, 𝑥) − V (𝜏, 𝑦)) 𝑢 ( ) V ( ) 󵄨 󵄨 󵄨 𝑡 sign 󵄨 𝜉𝑇×𝜉𝑇 (24) 𝑢2 (𝑡, 𝑥) V2 (𝜏, 𝑦) ×( − )𝑔 2 2 𝑥 We note that the first two terms in the integrand of(23)can be represented in the form + (𝑢 (𝑡, 𝑥) − V (𝜏, 𝑦)) sign 𝑌ℎ =𝑌(𝑡, 𝑥, 𝜏, 𝑦,𝑢 (𝑡, 𝑥) , V (𝜏,)) 𝑦 𝜆ℎ (∗) . (25)

Since ‖𝑢‖𝐿∞ ≤‖𝑢0‖ 1 and ‖V‖𝐿∞ ≤‖V0‖ 1 ,from ×𝑄 (𝑡, 𝑥) 𝑔} 𝑑𝑥 𝑑𝑡 𝑑𝑦 𝑑𝜏=0. 𝐻 (𝑅) 𝐻 (𝑅) 𝑢 Lemma 2,weknow𝑌ℎ satisfies the Lipschitz condition in 𝑢 and V,respectively.Bythechoiceof𝑔,wehave𝑌ℎ =0outside (21) the region Similarly, it has 𝑡+𝜏 |𝑡−𝜏| {(𝑡, 𝑥; 𝜏, 𝑦)} ={𝛿≤ ≤ 𝑇 − 2𝛿, ≤ℎ, 2 2 󵄨 󵄨 ∭∫ { 󵄨V (𝜏, 𝑦) −𝑢 (𝑡, 𝑥) 𝑔 󵄨 + (V (𝜏, 𝑦) −𝑢 (𝑡, 𝑥)) 󵄨 󵄨 󵄨 󵄨 󵄨 𝜏󵄨 sign 󵄨𝑥+𝑦󵄨 󵄨𝑥−𝑦󵄨 𝜉𝑇×𝜉𝑇 ≤ 𝑟 − 2𝛿, ≤ℎ}, 2 2 V2 (𝜏, 𝑦) 𝑢2 (𝑡, 𝑥) ×( − )𝑔𝑦 2 2 ∭∫ 𝑌ℎ𝑑𝑥 𝑑𝑡 𝑑𝑦𝑑𝜏 𝜉𝑇×𝜉𝑇 + sign (V (𝜏, 𝑦) −𝑢 (𝑡, 𝑥)) = ∭∫ [𝑌(𝑡,𝑥,𝜏,𝑦,𝑢(𝑡, 𝑥) , V (𝜏, 𝑦)) 𝜉𝑇×𝜉𝑇 (26) ×𝑄V (𝜏, 𝑦) 𝑔} 𝑑𝑥 𝑑𝑡 𝑑𝑦𝑑𝜏=0, −𝑌 (𝑡, 𝑥, 𝑡, 𝑥,𝑢 (𝑡, 𝑥) , V (𝑡, 𝑥)) ] (22) ×𝜆ℎ (∗) 𝑑𝑥 𝑑𝑡 𝑑𝑦𝑑𝜏 from which we obtain + ∭∫ 𝑌 (𝑡, 𝑥, 𝑡, 𝑥,𝑢 (𝑡, 𝑥) , V (𝑡, 𝑥)) 𝜉 ×𝜉 󵄨 󵄨 𝑇 𝑇 0≤∭∫ { 󵄨𝑢 (𝑡, 𝑥) − V (𝜏, 𝑦)󵄨 (𝑔𝑡 +𝑔𝜏) 𝜉 ×𝜉 𝑇 𝑇 ×𝜆ℎ (∗) 𝑑𝑥 𝑑𝑡 𝑑𝑦𝑑𝜏 + (𝑢 (𝑡, 𝑥) − V (𝜏, 𝑦)) sign =𝐽1 (ℎ) +𝐽2. 𝑢2 (𝑡, 𝑥) V2 (𝜏, 𝑦) |𝜆(∗)| ≤ 𝑐/ℎ2 ×( − ) Considering the estimate and the expression of 2 2 function 𝑌ℎ,wehave 󵄨 󵄨 󵄨𝐽1 (ℎ)󵄨 ×(𝑔 +𝑔 )}𝑑𝑥𝑑𝑡𝑑𝑦𝑑𝜏 𝑥 𝑦 (23) [ 󵄨 [ 󵄨 [ 1 + 󵄨∭∫ (𝑢 (𝑡, 𝑥) − V (𝑡, 𝑥)) ≤𝑐[ℎ+ 󵄨 sign 2 󵄨 𝜉 ×𝜉 [ ℎ 󵄨 𝑇 𝑇 [ 󵄨 󵄨 󵄨 [ ×(𝑄𝑢 (𝑡, 𝑥) −𝑄V (𝜏, 𝑦)) 𝑔 𝑑𝑥 𝑑𝑡 𝑑𝑦𝑑𝜏󵄨 󵄨 󵄨 × ∭∫ |(𝑡−𝜏)/2|≤ℎ, 󵄨V (𝑡, 𝑥) = ∭∫ (𝐼 +𝐼 +𝐼)𝑑𝑥𝑑𝑡𝑑𝑦𝑑𝜏. 𝛿≤(𝑡+𝜏)/2≤𝑇−𝛿, 1 2 3 |(𝑥−𝑦)/2|≤ℎ, 𝜉𝑇×𝜉𝑇 |(𝑥+𝑦)/2|≤𝑟−𝛿 We will show that ] ] 󵄨 ] −V (𝜏, 𝑦)󵄨 𝑑𝑥 𝑑𝑡 𝑑𝑦 𝑑𝜏, ] , 0≤∬ { |𝑢 (𝑡, 𝑥) − V (𝑡, 𝑥)| 𝑓𝑡 + sign (𝑢 (𝑡, 𝑥) − V (𝑡, 𝑥)) 󵄨 ] 𝜉𝑇 ] 𝑢2 (𝑡, 𝑥) V2 (𝑡, 𝑥) ] ×( − )𝑓 }𝑑𝑥𝑑𝑡 2 2 𝑥 (27) Journal of Function Spaces and Applications 5

where the constant 𝑐 does not depend on ℎ.UsingLemma 1, By Lemmas 1 and 3,wehave𝐾1(ℎ) → 0 as ℎ→0.Using we obtain 𝐽1(ℎ)→0as ℎ→0.Theintegral𝐽2 does not (28), we have depend on ℎ. In fact, substituting 𝑡=𝛼, (𝑡 − 𝜏)/2, =𝛽 𝑥=𝜂, (𝑥 − 𝑦)/2 =𝜉 2 and noting that 𝐾2 =2∬ 𝐼3 (𝛼,𝜂,𝛼,𝜂,𝑢(𝛼,𝜂),V (𝛼, 𝜂)) ℎ ∞ 𝜉𝑇 ∫ ∫ 𝜆ℎ (𝛽, 𝜉) 𝑑𝜉 𝑑𝛽 =1, (28) ℎ ∞ −ℎ −∞ ×{∫ ∫ 𝜆ℎ (𝛽, 𝜉) 𝑑𝜉 𝑑𝛽} 𝑑𝜂𝑑𝛼 we have −ℎ −∞ 𝐽 =22∬ 𝑌 (𝛼,𝜂,𝛼,𝜂,𝑢(𝛼,𝜂),V (𝛼, 𝜂)) (33) 2 ℎ =4∬ 𝐼3 (𝑡, 𝑥, 𝑡, 𝑥,𝑢 (𝑡, 𝑥) , V (𝑡, 𝑥)) 𝑑𝑥 𝑑𝑡 𝜉𝑇 𝜉𝑇 ℎ ∞ ×{∫ ∫ 𝜆ℎ (𝛽, 𝜉) 𝑑𝜉 𝑑𝛽} 𝑑𝜂𝑑𝛼 (29) =4∬ sign (𝑢 (𝑡, 𝑥) − V (𝑡, 𝑥)) −ℎ −∞ 𝜉𝑇

×(𝑄𝑢 (𝑡, 𝑥) −𝑄V (𝑡, 𝑥))𝑓(𝑡, 𝑥) 𝑑𝑥 𝑑𝑡. =4∬ 𝑌ℎ (𝑡, 𝑥, 𝑡, 𝑥,𝑢 (𝑡, 𝑥) , V (𝑡, 𝑥)) 𝑑𝑥 𝑑𝑡. 𝜉𝑇 From (30)and(33), we prove that inequality (24)holds. Hence Let lim ∭∫ 𝑌ℎ𝑑𝑥 𝑑𝑡 𝑑𝑦𝑑𝜏 ∞ ℎ→0 𝜉 ×𝜉 𝑇 𝑇 𝜔 (𝑡) = ∫ |𝑢 (𝑡, 𝑥) − V (𝑡, 𝑥)| 𝑑𝑥. (34) (30) −∞

=4∬ 𝑌ℎ (𝑡, 𝑥, 𝑡, 𝑥,𝑢 (𝑡, 𝑥) , V (𝑡, 𝑥)) 𝑑𝑥 𝑑𝑡. 𝜉𝑇 We define 𝜎 Since 󸀠 𝜃ℎ = ∫ 𝜌ℎ (𝜎) 𝑑𝜎 (𝜃ℎ (𝜎) =𝜌ℎ (𝜎) ≥0) (35) 𝐼3 = sign (𝑢 (𝑡, 𝑥) − V (𝜏, 𝑦))𝑢 (𝑄 (𝑡, 𝑥) −𝑄V (𝜏, 𝑦)) 𝑓𝜆ℎ (∗) , −∞

and choose two numbers 𝜏1 and 𝜏2 ∈(0,𝑇0), 𝜏1 <𝜏2.In(24), ∭∫ 𝐼3𝑑𝑥 𝑑𝑡 𝑑𝑦𝑑𝜏 𝜉𝑇×𝜉𝑇 we choose

𝑓= [𝜃ℎ (𝑡 −1 𝜏 )−𝜃ℎ (𝑡 −2 𝜏 )] 𝜒 (𝑡, 𝑥) , = ∭∫ [𝐼3 (𝑡,𝑥,𝜏,𝑦)−𝐼3 (𝑡, 𝑥, 𝑡,𝑥)] 𝜉𝑇×𝜉𝑇 (36) ℎ0. =𝐾1 (ℎ) +𝐾2, 𝜀 0 (31) We note that function 𝜒(𝑡, 𝑥) =0 outside the cone ℧ and we obtain 𝑓(𝑡, 𝑥) =0 outside the set z.For(𝑡, 𝑥) ∈℧,wehavethe 󵄨 󵄨 󵄨𝐾1 (ℎ)󵄨 relations 󵄨 󵄨 0=𝜒𝑡 +𝑁󵄨𝜒𝑥󵄨 ≥𝜒𝑡 +𝑁𝜒𝑥. (38) 1 ≤𝑐(ℎ+ Applying (24)and(35)–(38), we have the inequality ℎ2

0≤∬ {[𝜌ℎ (𝑡 −1 𝜏 )−𝜌ℎ (𝑡 −2 𝜏 )] 𝜉𝑇 󵄨 0 × ∭∫ 󵄨𝑄 (𝑡, 𝑥) |(𝑡−𝜏)/2|≤ℎ, 󵄨 V ×𝜒 𝑢 𝑡, 𝑥 − V 𝑡, 𝑥 }𝑑𝑥𝑑𝑡 𝛿≤(𝑡+𝜏)/2≤𝑇−𝛿, 𝜀 | ( ) ( )| |(𝑥−𝑦)/2|≤ℎ, 󵄨 󵄨 |(𝑥+𝑦)/2|≤𝑟−𝛿 󵄨 + 󵄨∬ [𝜃ℎ (𝑡 −1 𝜏 )−𝜃ℎ (𝑡 −2 𝜏 )] 󵄨 𝜉 󵄨 𝑇0 󵄨 󵄨 󵄨 󵄨 −𝑄 (𝜏, 𝑦)󵄨 𝑑𝑥 𝑑𝑡 𝑑𝑦 𝑑𝜏). ×[𝑄𝑢 (𝑡, 𝑥) −𝑄V (𝑡, 𝑥)]𝐵(𝑡, 𝑥) 𝜒 (𝑡, 𝑥) 𝑑𝑥󵄨 𝑑𝑡 , V 󵄨 󵄨 (39)

(32) where 𝐵(𝑡, 𝑥) = sign[𝑢(𝑡, 𝑥) − V(𝑡, 𝑥)]. 6 Journal of Function Spaces and Applications

From (39), we obtain Using the similar proof of (43), we get

𝑇 󸀠 0 0≤∬ {[𝜌ℎ (𝑡 −1 𝜏 )−𝜌ℎ (𝑡 −2 𝜏 )] 𝐿 (𝜏 )=−∫ 𝜌 (𝑡 − 𝜏 )𝜔(𝑡) 𝑑𝑡 󳨀→ −𝜔 (𝜏 ) ℎ󳨀→0, 𝜉 1 ℎ 1 1 as 𝑇0 0 (45) ×𝜒𝜀 |𝑢 (𝑡, 𝑥) − V (𝑡, 𝑥)|}𝑑𝑥𝑑𝑡 from which we obtain 𝑇0 𝜏 + ∫ (𝜃ℎ (𝑡 −1 𝜏 )−𝜃ℎ (𝑡 −2 𝜏 )) 1 0 𝐿(𝜏1)󳨀→𝐿(0) − ∫ 𝜔 (𝜎) 𝑑𝜎 as ℎ󳨀→0. (46) 0 󵄨 ∞ 󵄨 󵄨 󵄨 × 󵄨∫ [𝑄𝑢 (𝑡, 𝑥) −𝑄V (𝑡, 𝑥)]𝐵(𝑡, 𝑥) 𝜒 (𝑡, 𝑥) 𝑑𝑥󵄨 𝑑𝑡. 󵄨 −∞ 󵄨 Similarly, we have

(40) 𝜏2 𝐿(𝜏2)󳨀→𝐿(0) − ∫ 𝜔 (𝜎) 𝑑𝜎 as ℎ󳨀→0. (47) Using Lemma 5,wehave 0 Then, we get 0≤∬ {[𝜌ℎ (𝑡 −1 𝜏 )−𝜌ℎ (𝑡 −2 𝜏 )] 𝜉 𝜏2 𝑇0 𝐿 (𝜏1) −𝐿(𝜏2) 󳨀→ ∫ 𝜔 (𝜎) 𝑑𝜎 as ℎ󳨀→0. (48) 𝜏1 ×𝜒𝜀 |𝑢 (𝑡, 𝑥) − V (𝑡, 𝑥)|}𝑑𝑥𝑑𝑡 ℎ→0 𝑇0 (41) Furthermore, if ,wehave +𝑐∫ (𝜃ℎ (𝑡 −1 𝜏 )−𝜃ℎ (𝑡 −2 𝜏 )) 𝑇 𝜏 0 0 2 ∫ (𝜃ℎ (𝑡 −1 𝜏 )−𝜃ℎ (𝑡 −2 𝜏 ))𝐺𝑑𝑡󳨀→𝐺∫ 𝑑𝑡 ∞ 0 𝜏1 ×(𝐺+∫ |𝑢−V| 𝑑𝑥) 𝑑𝑡, ∞ ∞ −∞ 󵄨 󵄨 󵄨 󵄨 =(∫ 󵄨𝑢2 − V2󵄨 𝑑𝑥 + ∫ 󵄨𝑢2 − V2 󵄨 𝑑𝑥) (𝜏 −𝜏). 󵄨 0 0󵄨 󵄨 0𝑥 0𝑥󵄨 2 1 ∞ ∞ −∞ −∞ 𝐺=∫ |𝑢2 −V2|𝑑𝑥+∫ |𝑢2 −V2 |𝑑𝑥 𝑐 where −∞ 0 0 −∞ 0𝑥 0𝑥 and is defined (49) in Lemma 5. Letting 𝜀→0in (41) and sending 𝑀0 →∞,wehave Let 𝜏1 →0and 𝜏2 →𝑡, and note that 𝑇 󵄨 󵄨 󵄨 󵄨 0 󵄨𝑢(𝜏1,𝑥)−V (𝜏1,𝑥)󵄨 ≤ 󵄨𝑢(𝜏1,𝑥)−𝑢0 (𝑥)󵄨 0≤ ∫ {[𝜌ℎ (𝑡 −1 𝜏 )−𝜌ℎ (𝑡 −2 𝜏 )] 0 󵄨 󵄨 + 󵄨V (𝜏1,𝑥)−V0 (𝑥)󵄨 (50) ∞ 󵄨 󵄨 × ∫ |𝑢 (𝑡, 𝑥) − V (𝑡, 𝑥)| 𝑑𝑥} 𝑑𝑡 + 󵄨𝑢 (𝑥) − V (𝑥)󵄨 . −∞ 󵄨 0 0 󵄨 (42) 𝑇0 Thus, from (42), (43), (48), (49), and (50), for any 𝑡∈ +𝑐∫ (𝜃 (𝑡 − 𝜏 )−𝜃 (𝑡 − 𝜏 )) ℎ 1 ℎ 2 [0, 𝑇0],wehave 0 ∞ ∞ ×(𝐺+∫ |𝑢−V| 𝑑𝑥) 𝑑𝑡. ∫ |𝑢 (𝑡, 𝑥) − V (𝑡, 𝑥)| 𝑑𝑥 −∞ −∞ ∞ By the properties of the function 𝜌ℎ(𝜎) for ℎ≤min(𝜏1, ≤ ∫ |𝑢 (0, 𝑥) − V (0, 𝑥)| 𝑑𝑥 −∞ 𝑇0 −𝜏1),wehave (51) ∞ 󵄨 󵄨 ∞ 󵄨 󵄨 󵄨 𝑇 󵄨 󵄨 2 2󵄨 󵄨 2 2 󵄨 󵄨 0 󵄨 +𝑐𝑇 (∫ 󵄨𝑢 − V 󵄨 𝑑𝑥 + ∫ 󵄨𝑢 − V 󵄨 𝑑𝑥) 󵄨 󵄨 0 󵄨 0 0󵄨 󵄨 0𝑥 0𝑥󵄨 󵄨∫ 𝜌ℎ (𝑡 −1 𝜏 )𝜔(𝑡) 𝑑𝑡 − 1𝜔 (𝜏 )󵄨 −∞ −∞ 󵄨 0 󵄨 𝑡 ∞ 󵄨 𝑇 󵄨 󵄨 0 󵄨 + ∫ ∫ |𝑢−V| 𝑑𝑥, 󵄨 󵄨 0 −∞ = 󵄨∫ 𝜌ℎ (𝑡 −1 𝜏 )[𝜔(𝑡) −𝜔(𝜏1)] 𝑑𝑡󵄨 (43) 󵄨 0 󵄨 from which we complete the proof of Theorem 6 by using the 𝜏 +ℎ 1 1 󵄨 󵄨 ≤𝑐 ∫ 󵄨𝜔 (𝑡) −𝜔(𝜏)󵄨 𝑑𝑡 󳨀→ 0 ℎ󳨀→0, Gronwall inequality. ℎ 󵄨 1 󵄨 as 𝜏1−ℎ Acknowledgments where 𝑐 is independent of ℎ. Set Thanks are given to referees whose suggestions are very

𝑇0 𝑇0 𝑡−𝜏1 helpful to the paper. This work is supported by both the 𝐿(𝜏1)=∫ 𝜃ℎ (𝑡 −1 𝜏 )𝜇(𝑡) 𝑑𝑡 = ∫ ∫ 𝜌ℎ (𝜎) 𝑑𝜎𝜔 (𝑡) 𝑑𝑡. Fundamental Research Funds for the Central Universities 0 0 −∞ (JBK120504) and the Applied and Basic Project of Sichuan (44) Province (2012JY0020). Journal of Function Spaces and Applications 7

References

[1] R. Camassa and D. D. Holm, “Anintegrable shallow water equation with peaked solitons,” Physical Review Letters, vol. 71, no. 11, pp. 1661–1664, 1993. [2] A. Constantin and J. Escher, “Global weak solutions for a shallow water equation,” Indiana University Mathematics Journal, vol.47,no.4,pp.1527–1545,1998. [3] A. Constantin and L. Molinet, “Global weak solutions for a shallow water equation,” Communications in Mathematical Physics, vol. 211, no. 1, pp. 45–61, 2000. [4] A. Constantin and J. Escher, “Global existence and blow-up for a shallow water equation,” Annali della Scuola Normale Superiore di Pisa. Classe di Scienze, vol. 26, no. 2, pp. 303–328, 1998. [5] A. A. Himonas and G. Misiołek, “The Cauchy problem for an integrable shallow-water equation,” Differential and Integral Equations,vol.14,no.7,pp.821–831,2001. [6] S. Lai and Y. Wu, “The local well-posedness and existence of weak solutions for a generalized Camassa-Holm equation,” Journal of Differential Equations,vol.248,no.8,pp.2038–2063, 2010. [7]Y.A.LiandP.J.Olver,“Well-posednessandblow-upsolutions for an integrable nonlinearly dispersive model wave equation,” Journal of Differential Equations,vol.162,no.1,pp.27–63,2000. [8] G. Rodr´ıguez-Blanco, “On the Cauchy problem for the Ca- massa-Holm equation,” Nonlinear Analysis: Theory, Methods & Applications,vol.46,no.3,pp.309–327,2001. [9] A. Constantin and J. Escher, “Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation,” Communications on Pure and Applied Mathematics, vol. 51, no. 5, pp. 475–504, 1998. [10] A. Constantin and J. Escher, “Wave breaking for nonlinear nonlocal shallow water equations,” Acta Mathematica,vol.181,no. 2, pp. 229–243, 1998. [11] A. Constantin and J. Escher, “On the Cauchy problem for a family of quasilinear hyperbolic equations,” Communications in Partial Differential Equations,vol.23,no.7-8,pp.1449–1458, 1998. [12] A. Bressan and A. Constantin, “Global conservative solutions of the Camassa-Holm equation,” Archive for Rational Mechanics and Analysis,vol.183,no.2,pp.215–239,2007. [13]G.M.Coclite,H.Holden,andK.H.Karlsen,“Globalweak solutions to a generalized hyperelastic-rod wave equation,” SIAM Journal on Mathematical Analysis,vol.37,no.4,pp.1044– 1069, 2005. [14] H. Holden and X. Raynaud, “Global conservative solutions of the Camassa-Holm equation—a Lagrangian point of view,” Communications in Partial Differential Equations,vol.32,no. 10–12,pp.1511–1549,2007. [15] Z. Xin and P. Zhang, “On the weak solutions to a shallow water equation,” Communications on Pure and Applied Mathematics, vol.53,no.11,pp.1411–1433,2000. [16] Z. Yin, “On the Cauchy problem for an integrable equation with peakon solutions,” Illinois Journal of Mathematics,vol.47,no.3, pp.649–666,2003. [17] S. Kruzkov, “First order quasi-linear equations in several independent variables,” Mathematics of the USSR-Sbornik,vol.10, no. 2, pp. 217–243, 1970. Hindawi Publishing Corporation Journal of Function Spaces and Applications Volume 2013, Article ID 921828, 6 pages http://dx.doi.org/10.1155/2013/921828

Research Article On the Hermite-Hadamard Inequality and Other Integral Inequalities Involving Several Functions

Banyat Sroysang

Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University, Pathumthani 12121, Thailand

Correspondence should be addressed to Banyat Sroysang; [email protected]

Received 18 March 2013; Accepted 14 May 2013

Academic Editor: Nelson Merentes

Copyright © 2013 Banyat Sroysang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We present some new Hermite-Hadamard-type inequalities and other integral inequalities involving several functions.

1. Introduction and Preliminaries Hermite-Hadamard’s Inequality (see [6–8]). If 𝑓 is a convex function on [𝑎, 𝑏],then In 2003, Pachpatte [1] gave some Hermite-Hadamard-type inequalities involving two convex functions, and then Pach- 𝑎+𝑏 1 𝑏 𝑓 (𝑎) +𝑓(𝑏) patte [2] also gave, in 2004, some Hermite-Hadamard-type 𝑓( )≤ ∫ 𝑓 (𝑥) 𝑑𝑥 ≤ . (2) inequalities involving two log-convex functions. In 2007, 2 𝑏−𝑎 𝑎 2 Kirmaci et al. [3]gavesomeHadamard-typeinequalities involving 𝑠-convex functions. In 2008, Bakula et al. [4] If 𝑓 is a concave function on [𝑎, 𝑏],then presented some Hadamard-type inequalities involving 𝑚- convex functions and (𝛼, 𝑚)-convex functions. In 2010, Set et al. [5] gave some new Hermite-Hadamard-type inequalities 𝑓 (𝑎) +𝑓(𝑏) 1 𝑏 𝑎+𝑏 ≤ ∫ 𝑓 (𝑥) 𝑑𝑥≤𝑓( ). (3) and other integral inequalities involving two functions. More 2 𝑏−𝑎 𝑎 2 details about results proved in [5]willbegiveninSection 2.In this paper, we present more general Hermite-Hadamard-type inequalities and some integral inequalities involving several Barnes-Gugunova-Levin Inequality (see [9–11]). If 𝑓 and 𝑔 are functions. nonnegative concave functions on [𝑎, 𝑏],andif𝑝, 𝑞 >1,then Let 𝑓:[𝑎,𝑏] → R and 𝑝≥1.The𝑝-normofthefunction 𝑓 on [𝑎, 𝑏] is defined by 𝑏 1/𝑝 𝑏 1/𝑞 (∫ 𝑓𝑝 (𝑥) 𝑑𝑥) (∫ 𝑔𝑞 (𝑥) 𝑑𝑥) 𝑎 𝑎 (4) 𝑏 1/𝑝 𝑏 { 󵄨 󵄨𝑝 {(∫ 󵄨𝑓 (𝑥)󵄨 𝑑𝑥) ;1≤𝑝<∞, ≤𝐵(𝑝,𝑞)∫ 𝑓 (𝑥) 𝑔 (𝑥) 𝑑𝑥, 󵄩 󵄩 󵄨 󵄨 󵄩𝑓󵄩 = 𝑎 (1) 𝑎 𝑝 { 󵄨 󵄨 { sup 󵄨𝑓 (𝑥)󵄨 ;𝑝=+∞. {𝑥∈[𝑎,𝑏] where

6(𝑏−𝑎)1/𝑝+1/𝑞−1 Below we recall few well-known inequalities that will be 𝐵(𝑝,𝑞)= . (5) (𝑝 + 1)1/𝑝(𝑞 + 1)1/𝑞 useful in the proofs of our results. 2 Journal of Function Spaces and Applications

The Power-Mean Inequality (see [12]). Let 𝑥1,𝑥2,...,𝑥𝑛,𝑝1, Then 𝑝2,...,𝑝𝑛 >0and let 𝑟∈R ∪{−∞,+∞}.Then (𝑀+1)𝑝𝑓𝑝 ≤𝑀𝑝(𝑓 + 𝑔)𝑝, 𝑛 𝑟 1/𝑟 { ∑𝑖=1 𝑝𝑖𝑥𝑖 𝑝 𝑝 {( ) ;𝑟=0,±∞,̸ 1 1 𝑝 (13) { ∑𝑛 𝑝 ( +1) 𝑔𝑝 ≤( ) (𝑓 + 𝑔) . { 𝑖=1 𝑖 { 1/ ∑𝑛 𝑝 𝑚 𝑚 { 𝑛 𝑖=1 𝑖 [𝑟] 𝑝 𝑀 = (∏𝑥 𝑖 ) ;𝑟=0, 𝑛 { 𝑖 (6) Thus, { { 𝑖=1 { 1/𝑝 1/𝑝 {max{𝑥1,𝑥2,...,𝑥𝑛}; 𝑟=∞, 𝑏 𝑏 { 𝑀 𝑝 (∫ 𝑓𝑝 (𝑥) 𝑑𝑥) ≤ (∫ (𝑓 (𝑥) +𝑔(𝑥)) 𝑑𝑥) , min{𝑥1,𝑥2,...,𝑥𝑛}; 𝑟=−∞. { 𝑎 𝑀+1 𝑎 Notice that if −∞≤𝑟<𝑠≤+∞then 𝑏 1/𝑝 1 𝑏 1/𝑝 (∫ 𝑔𝑝 (𝑥) 𝑑𝑥) ≤ (∫ (𝑓 (𝑥) +𝑔(𝑥))𝑝𝑑𝑥) . [𝑟] [𝑠] 𝑎 𝑚+1 𝑎 𝑀𝑛 ≤𝑀𝑛 . (7) (14)

A Generalization of Holder¨ Integral Inequality. For any 𝑝1, 𝑛 𝑝2,...,𝑝𝑛 >1, ∑𝑖=1(1/𝑝𝑖)=1,if𝑓1,𝑓2,...,𝑓𝑛 are nonnega- 𝑝1 𝑝2 𝑝𝑛 tive functions on [𝑎, 𝑏] and if 𝑓1 ,𝑓2 ,...,𝑓𝑛 are integrable 2. Main Results functions on [𝑎, 𝑏],then We start this section with the following. 𝑏 𝑛 𝑛 𝑏 1/𝑝𝑖 𝑝𝑖 Theorem 2. 𝑛 𝑝 ,𝑝 ,..., ∫ (∏𝑓𝑖 (𝑥))𝑑𝑥≤∏(∫ 𝑓𝑖 (𝑥) 𝑑𝑥) . (8) Let be a positive even integer and 1 2 𝑎 𝑎 𝑖=1 𝑖=1 𝑝𝑛 >1and let 𝑓1,𝑓2,...,𝑓𝑛 be non-negative functions on [𝑎, 𝑏] 𝑝1 𝑝2 𝑝𝑛 such that 𝑓1 ,𝑓2 ,...,𝑓𝑛 are concave on [𝑎, 𝑏].Then A Generalization of Minkowski Integral Inequality. If 𝑝≥1 𝑛 and if 𝑓1,𝑓2,...,𝑓𝑛 are non-negative functions on [𝑎, 𝑏] such 𝑓𝑖 (𝑎) +𝑓𝑖 (𝑏) 𝑏 ∏ ( ) 0<∫ 𝑓𝑝(𝑥)𝑑𝑥 <∞ 𝑖=1,2,...,𝑛 2 that 𝑎 𝑖 for all ,then 𝑖=1

𝑝 1/𝑝 𝑛/2 𝑏 𝑏 𝑛 𝑛 𝑏 1/𝑝 1 𝑝 ≤ (∏𝐵(𝑝 ,𝑝 ) ∫ 𝑓 (𝑥) 𝑓 (𝑥) 𝑑𝑥) , (∫ (∑𝑓 (𝑥)) 𝑑𝑥) ≤ ∑(∫ 𝑓 (𝑥) 𝑑𝑥) . (9) ∑𝑛 (1/𝑝 ) 2𝑖−1 2𝑖 2𝑖−1 2𝑖 𝑖 𝑖 (𝑏−𝑎) 𝑖=1 𝑖 𝑖=1 𝑎 𝑎 𝑖=1 𝑖=1 𝑎 (15)

A Generalization of Young Inequality. If 𝑥1,𝑥2,...,𝑥𝑛 ≥0and 𝑛 where 𝑝1,𝑝2,...,𝑝𝑛 >1, ∑𝑖=1(1/𝑝𝑖)=1,then 1/𝑝 +1/𝑝 −1 6 𝑏−𝑎 2𝑖−1 2𝑖 𝑛 𝑛 ( ) 𝑝𝑖 𝐵(𝑝 ,𝑝 )= 𝑥 2𝑖−1 2𝑖 1/𝑝 1/𝑝 (16) ∏𝑥 ≤ ∑ . 2𝑖−1 2𝑖 𝑖 (10) (𝑝2𝑖−1 +1) (𝑝2𝑖 +1) 𝑖=1 𝑖=1 𝑝𝑖 𝑖=1,...,𝑛/2 ∑𝑛 (1/𝑝 )=1 To prove results, we refer to the following lemma. for all .Moreover,if 𝑖=1 𝑖 ,then Lemma 1 𝑝≥1 𝑓 𝑔 1 𝑏 𝑛 𝑛 𝑎+𝑏 (see [13]). If and if and are positive ∫ (∏𝑓 (𝑥))𝑑𝑥≤∏𝑓 ( ). [𝑎, 𝑏] 0 < 𝑚 ≤ 𝑓(𝑥)/𝑔(𝑥) ≤𝑀 𝑖 𝑖 (17) functions on such that for 𝑏−𝑎 𝑎 𝑖=1 𝑖=1 2 all 𝑥∈[𝑎,𝑏],then 𝑝 𝑓 𝑖 𝑖= 𝑏 1/𝑝 𝑀 𝑏 1/𝑝 Proof. Applying the inequality (3)with 𝑖 ,forany (∫ 𝑓𝑝 (𝑥) 𝑑𝑥) ≤ (∫ (𝑓 (𝑥) +𝑔(𝑥))𝑝𝑑𝑥) , 1,...,𝑛,weget 𝑎 𝑀+1 𝑎 𝑝 𝑝 𝑓 𝑖 (𝑎) +𝑓 𝑖 (𝑏) 1 𝑏 𝑎+𝑏 𝑏 1/𝑝 𝑏 1/𝑝 𝑖 𝑖 𝑝𝑖 𝑝𝑖 𝑝 1 𝑝 ≤ ∫ 𝑓𝑖 (𝑥) 𝑑𝑥𝑖 ≤𝑓 ( ) , (∫ 𝑔 (𝑥) 𝑑𝑥) ≤ (∫ (𝑓 (𝑥) +𝑔(𝑥)) 𝑑𝑥) . 2 𝑏−𝑎 𝑎 2 𝑚+1 𝑎 𝑎 (18) (11) and, consequently, Proof. Let 𝑝≥1. Assume that 𝑓 and 𝑔 are positive functions [𝑎, 𝑏] 0 < 𝑚 ≤ 𝑓(𝑥)/𝑔(𝑥) ≤𝑀 𝑥∈[𝑎,𝑏] 𝑝 𝑝 1/𝑝 on such that for all . 𝑓 𝑖 (𝑎) +𝑓 𝑖 (𝑏) 𝑖 Then ( 𝑖 𝑖 ) 2 𝑓≤𝑀𝑔=𝑀(𝑓+𝑔)−𝑀𝑓, (19) 𝑏 1/𝑝𝑖 (12) 1 𝑝 𝑎+𝑏 1 1 1 ≤ (∫ 𝑓 𝑖 (𝑥) 𝑑𝑥) ≤𝑓( ). 𝑔≤ 𝑓= (𝑓 + 𝑔) − 𝑔. 1/𝑝 𝑖 𝑖 𝑚 𝑚 𝑚 (𝑏−𝑎) 𝑖 𝑎 2 Journal of Function Spaces and Applications 3

By the Barnes-Gudunova-Levin inequality (4), it follows where that 6(𝑏−𝑎)1/𝑝+1/𝑞−1 𝑝 𝑝 1/𝑝 𝑛 𝑓 𝑖 (𝑎) +𝑓 𝑖 (𝑏) 𝑖 𝐵 (𝑝, 𝑞) = . (25) ∏( 𝑖 𝑖 ) (𝑝 + 1)1/𝑝(𝑞 + 1)1/𝑞 𝑖=1 2 Moreover, if 1/𝑝 + 1/𝑞,then =1 𝑛 𝑏 1/𝑝𝑖 1 𝑝 ≤ ∏(∫ 𝑓 𝑖 (𝑥) 𝑑𝑥) ∑𝑛 (1/𝑝 ) 𝑖 𝑏 (𝑏−𝑎) 𝑖=1 𝑖 𝑎 1 𝑎+𝑏 𝑎+𝑏 𝑖=1 ∫ 𝑓 (𝑥) 𝑔 (𝑥) 𝑑𝑥 ≤ 𝑓( )𝑔( ). (26) 𝑏−𝑎 𝑎 2 2 𝑛/2 𝑏 1/𝑝2𝑖−1 1 𝑝 = ∏(∫ 𝑓 2𝑖−1 (𝑥) 𝑑𝑥) ∑𝑛 (1/𝑝 ) 2𝑖−1 Theorem 4. Let 𝑝≥1and 𝑛 be a positive integer such that (𝑏−𝑎) 𝑖=1 𝑖 𝑖=1 𝑎 𝑛≥2and let 𝑓1,𝑓2,...,𝑓𝑛 be positive functions on [𝑎, 𝑏] such 𝑝 𝑝 1/𝑝 𝑝 𝑏 2𝑖 that the functions 𝑓 ,𝑓 ,...,𝑓 are integrable functions on 𝑝 1 2 𝑛 ×(∫ 𝑓 2𝑖 𝑥 𝑑𝑥) 𝑏 𝑝 2𝑖 ( ) [𝑎, 𝑏] 0<∫ 𝑓 (𝑥)𝑑𝑥 <∞ 𝑖=1,2,...,𝑛 𝑎 , 𝑎 𝑖 for all ,and

𝑛/2 𝑏 1 𝑓𝑖 (𝑥) ≤ (∏𝐵(𝑝 ,𝑝 ) ∫ 𝑓 (𝑥) 𝑓 (𝑥) 𝑑𝑥), 0<𝑚𝑖 ≤ ≤𝑀𝑖 (27) ∑𝑛 (1/𝑝 ) 2𝑖−1 2𝑖 2𝑖−1 2𝑖 𝑓 (𝑥) (𝑏−𝑎) 𝑖=1 𝑖 𝑖=1 𝑎 𝑖+1 (20) for all 𝑥∈[𝑎,𝑏]and for all 𝑖=1,...,𝑛−1.Then where 𝑛 󵄩 󵄩 𝑛 1/𝑝 +1/𝑝 −1 (∑ 󵄩𝑓󵄩 ) 6(𝑏−𝑎) 2𝑖−1 2𝑖 𝑖=1 󵄩 𝑖󵄩𝑝 1 󵄩 󵄩 ≥ , (28) 𝐵(𝑝2𝑖−1,𝑝2𝑖)= (21) ∏𝑛 󵄩𝑓󵄩 ∏𝑛 𝑠 1/𝑝2𝑖−1 1/𝑝2𝑖 𝑖=1󵄩 𝑖󵄩𝑝 𝑖=1 𝑖 (𝑝2𝑖−1 +1) (𝑝2𝑖 +1) for all 𝑖=1,...,𝑛/2. where By the power-mean inequality (7), we have 𝑀1 1 𝑝 𝑝 1/𝑝 𝑠1 = ,𝑠𝑛 = , 𝑖 𝑖 𝑖 𝑀 +1 𝑚 +1 𝑓𝑖 (𝑎) +𝑓𝑖 (𝑏) 𝑓𝑖 (𝑎) +𝑓𝑖 (𝑏) 1 𝑛−1 ( ) ≥ (22) (29) 2 2 1 𝑀 𝑠 = { , 𝑖 } 𝑖 min 𝑚 +1 𝑀 +1 for all 𝑖=1,2,...,𝑛. 𝑖−1 𝑖 This implies the inequality (15). 𝑛 1<𝑖<𝑛 Next, we assume that ∑𝑖=1(1/𝑝𝑖)=1.Bytheinequality for all . (19) and the generalized Holder¨ inequality, we obtain that Proof. By Lemma 1,wehave 𝑛 𝑛 𝑏 1/𝑝𝑖 𝑎+𝑏 1 𝑝 ∏𝑓 ( )≥ ∏(∫ 𝑓 𝑖 (𝑥) 𝑑𝑥) 𝑏 1/𝑝 𝑏 1/𝑝 𝑖 ∑𝑛 (1/𝑝 ) 𝑖 𝑀 𝑝 2 (𝑏−𝑎) 𝑖=1 𝑖 𝑎 𝑝 𝑖 𝑖=1 𝑖=1 (∫ 𝑓𝑖 (𝑥) 𝑑𝑥) ≤ (∫ (𝑓𝑖 (𝑥) +𝑓𝑖+1 (𝑥)) 𝑑𝑥) , 𝑎 𝑀𝑖 +1 𝑎 𝑛 𝑏 1/𝑝𝑖 1 𝑝 = ∏(∫ 𝑓 𝑖 (𝑥) 𝑑𝑥) 𝑏 1/𝑝 𝑏 1/𝑝 𝑖 𝑝 1 𝑝 𝑏−𝑎 𝑎 𝑖=1 (∫ 𝑓𝑖+1 (𝑥) 𝑑𝑥) ≤ (∫ (𝑓𝑖 (𝑥) +𝑓𝑖+1 (𝑥)) 𝑑𝑥) 𝑎 𝑚𝑖 +1 𝑎 1 𝑏 𝑛 (30) ≥ ∫ (∏𝑓𝑖 (𝑥))𝑑𝑥. 𝑏−𝑎 𝑎 𝑖=1 for all 𝑖=1,...,𝑛−1. (23) Then

This proof is completed. 𝑝 1/𝑝 𝑏 1/𝑝 𝑏 𝑛 𝑝 𝑀 It is easy to notice that if we put 𝑛=2in Theorem 2 then (∫ 𝑓 (𝑥) 𝑑𝑥) ≤ 𝑖 (∫ (∑𝑓 (𝑥)) 𝑑𝑥) , 𝑖 𝑀 +1 𝑗 we get the following. 𝑎 𝑖 𝑎 𝑗=1

1/𝑝 Corollary 3 𝑝, 𝑞 >1 𝑓, 𝑔 1/𝑝 𝑝 (see [5]). Let and let be non-negative 𝑏 1 𝑏 𝑛 [𝑎, 𝑏] 𝑓𝑝,𝑔𝑞 [𝑎, 𝑏] 𝑝 functions on such that are concave on .Then (∫ 𝑓𝑖+1 (𝑥) 𝑑𝑥) ≤ (∫ (∑𝑓𝑗 (𝑥)) 𝑑𝑥) 𝑎 𝑚𝑖 +1 𝑎 𝑗=1 𝑓 (𝑎) +𝑓(𝑏) 𝑔 (𝑎) +𝑔(𝑏) ( )( ) (31) 2 2 (24) 𝑖=1,...,𝑛−1 1 𝑏 for all . ≤ (𝐵 (𝑝, 𝑞) ∫ 𝑓 (𝑥) 𝑔 (𝑥) 𝑑𝑥) , 𝑠 =𝑀/(𝑀 +1) 𝑠 =1/(𝑚 +1) 𝑠 = 1/𝑝+1/𝑞 Let 1 1 1 , 𝑛 𝑛−1 ,and 𝑖 (𝑏−𝑎) 𝑎 min{1/(𝑚𝑖−1 + 1), 𝑀𝑖/(𝑀𝑖 +1)}for all 1<𝑖<𝑛. 4 Journal of Function Spaces and Applications

It follows that where

𝑝 1/𝑝 𝑀 1 𝑏 1/𝑝 𝑏 𝑛 𝑠 = ,𝑠= . 𝑝 1 𝑀+1 2 𝑚+1 (38) (∫ 𝑓1 (𝑥) 𝑑𝑥) ≤𝑠1(∫ (∑𝑓𝑗 (𝑥)) 𝑑𝑥) , 𝑎 𝑎 𝑗=1 Let 𝑠=𝑠1𝑠2.Then 𝑝 1/𝑝 𝑏 1/𝑝 𝑏 𝑛 󵄩 󵄩 󵄩 󵄩 2 𝑝 (󵄩𝑓󵄩 + 󵄩𝑔󵄩 ) (∫ 𝑓 (𝑥) 𝑑𝑥) ≤𝑠(∫ (∑𝑓 (𝑥)) 𝑑𝑥) , (32) 1 󵄩 󵄩𝑝 󵄩 󵄩𝑝 𝑛 𝑛 𝑗 ≤ 󵄩 󵄩 󵄩 󵄩 𝑎 𝑎 𝑗=1 𝑠 󵄩 󵄩 󵄩 󵄩 󵄩𝑓󵄩𝑝󵄩𝑔󵄩𝑝

𝑝 1/𝑝 2 2 𝑏 1/𝑝 𝑏 𝑛 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 𝑝 󵄩𝑓󵄩𝑝 +2󵄩𝑓󵄩𝑝󵄩𝑔󵄩𝑝 + 󵄩𝑔󵄩𝑝 (∫ 𝑓𝑖 (𝑥) 𝑑𝑥) ≤𝑠𝑖(∫ (∑𝑓𝑗 (𝑥)) 𝑑𝑥) = 󵄩 󵄩 󵄩 󵄩 (39) 𝑎 𝑎 󵄩 󵄩 󵄩 󵄩 𝑗=1 󵄩𝑓󵄩𝑝󵄩𝑔󵄩𝑝 1<𝑖<𝑛 󵄩 󵄩2 󵄩 󵄩2 for all . 󵄩𝑓󵄩𝑝 + 󵄩𝑔󵄩𝑝 = 󵄩 󵄩 󵄩 󵄩 +2. By multiplying the above inequalities and the generalized 󵄩𝑓󵄩 󵄩𝑔󵄩 Minkowski inequality, we obtain that 󵄩 󵄩𝑝󵄩 󵄩𝑝

𝑛 𝑏 1/𝑝 This implies the inequality (36). 𝑝 ∏(∫ 𝑓𝑖 (𝑥) 𝑑𝑥) 𝑖=1 𝑎 Theorem 6. Let 𝑛 be a positive integer such that 𝑛≥2 𝑝 ,𝑝 ,...,𝑝 >1 𝑓 ,𝑓,...,𝑓 𝑛 and 1 2 𝑛 and let 1 2 𝑛 be non-negative 𝑝 1/𝑝 𝑝1 𝑝2 𝑝𝑛 𝑛 𝑏 𝑛 functions on [𝑎, 𝑏] such that 𝑓1 ,𝑓2 ,...,𝑓𝑛 are concave on [𝑎, 𝑏] ≤(∏𝑠𝑖)((∫ (∑𝑓𝑗 (𝑥)) 𝑑𝑥) ) (33) .Then 𝑖=1 𝑎 𝑗=1 𝑛 ∑𝑛 𝑝 𝑛 2 𝑖=1 𝑖 𝑝𝑖 󵄩 󵄩𝑝𝑖 𝑛 ∏(𝑓𝑖 (𝑎) +𝑓𝑖 (𝑏)) ≤ 𝑛 ∏󵄩𝑓𝑖󵄩𝑝 . (40) 𝑛 𝑛 𝑏 1/𝑝 (𝑏−𝑎) 𝑖 𝑝 𝑖=1 𝑖=1 ≤(∏𝑠𝑖)(∑(∫ 𝑓𝑗 (𝑥) 𝑑𝑥) ) . 𝑖=1 𝑗=1 𝑎 𝑝𝑖 Proof. Using the inequality (3)with𝑓𝑖 ,forany𝑖 = 1,...,𝑛, we obtain Then 𝑝𝑖 𝑝𝑖 𝑏 𝑛 𝑓𝑖 (𝑎) +𝑓𝑖 (𝑏) 1 𝑝 𝑛 𝑛 𝑛 𝑖 ≤ ∫ 𝑓𝑖 (𝑥) 𝑑𝑥. (41) 󵄩 󵄩 󵄩 󵄩 2 𝑏−𝑎 𝑎 ∏󵄩𝑓𝑖󵄩𝑝 ≤(∏𝑠𝑖)(∑󵄩𝑓𝑖󵄩𝑝) . (34) 𝑖=1 𝑖=1 𝑗=1 Then

𝑛 𝑝𝑖 𝑝𝑖 𝑛 𝑏 This implies the inequality (28). 𝑓 (𝑎) +𝑓 (𝑏) 1 𝑝 ∏ ( 𝑖 𝑖 )≤ ∏ (∫ 𝑓 𝑖 (𝑥) 𝑑𝑥) . 2 (𝑏−𝑎)𝑛 𝑖 Notice that from above theorem one can easily get the 𝑖=1 𝑖=1 𝑎 following. (42)

Corollary 5 (see [5]). Let 𝑝≥1and let 𝑓, 𝑔 be positive By the power-mean inequality (7), we have 𝑏 𝑝 [𝑎, 𝑏] 0<∫𝑓 (𝑥)𝑑𝑥 <∞ 0< 1/𝑝 functions on such that 𝑎 , 𝑝𝑖 𝑝𝑖 𝑖 𝑏 𝑓 (𝑎) +𝑓 (𝑏) 𝑓 (𝑎) +𝑓(𝑏) ∫ 𝑔𝑝(𝑥)𝑑𝑥 <∞ ( 𝑖 𝑖 ) ≥ 𝑖 𝑖 , (43) 𝑎 ,and 2 2

𝑓 (𝑥) so 0<𝑚≤ ≤𝑀 (35) 𝑔 (𝑥) 𝑝 𝑝 𝑝 𝑓 𝑖 (𝑎) +𝑓 𝑖 (𝑏) (𝑓 (𝑎) +𝑓(𝑏)) 𝑖 𝑖 𝑖 ≥ 𝑖 𝑖 , 𝑝 (44) for all 𝑥∈[𝑎,𝑏].Then 2 2 𝑖 󵄩 󵄩2 󵄩 󵄩2 for all 𝑖=1,2,...,𝑛. 󵄩𝑓󵄩𝑝 + 󵄩𝑔󵄩𝑝 1 ≥ −2, (36) Then 󵄩 󵄩 󵄩 󵄩 𝑠 󵄩𝑓󵄩𝑝󵄩𝑔󵄩𝑝 𝑝 1 𝑛 𝑏 𝑛 (𝑓 (𝑎) +𝑓(𝑏)) 𝑖 𝑝𝑖 𝑖 𝑖 𝑛 ∏ (∫ 𝑓𝑖 (𝑥) 𝑑𝑥) ≥ ∏ ( ) 𝑠 = 𝑀/(𝑀 + 1)(𝑚 +1) 𝑝𝑖 where . (𝑏−𝑎) 𝑖=1 𝑎 𝑖=1 2 𝑝 Proof. By Theorem 4 where 𝑛=2,wehave ∏𝑛 (𝑓 (𝑎) +𝑓(𝑏)) 𝑖 = 𝑖=1 𝑖 𝑖 . ∑𝑛 𝑝 2 𝑖=1 𝑖 󵄩 󵄩 󵄩 󵄩 2 (󵄩𝑓󵄩𝑝 + 󵄩𝑔󵄩𝑝) 1 (45) 󵄩 󵄩 󵄩 󵄩 ≥ , (37) 󵄩𝑓󵄩 󵄩𝑔󵄩 𝑠 𝑠 󵄩 󵄩𝑝󵄩 󵄩𝑝 1 2 This implies the inequality (40). Journal of Function Spaces and Applications 5

𝑝 𝑝−1 𝑝 𝑝 One can easily check that if we put 𝑛=2in Theorem 6 Using the elementary inequality (𝛼 + 𝛽) ≤2 (𝛼 +𝛽 ) then we get the following. where 𝑝>1and 𝛼, 𝛽,weget >0

Corollary 7 𝑝, 𝑞 >1 𝑓, 𝑔 𝑝 (see [5]). Let and let be non-negative 𝑏 𝑀 𝑖 𝑏 𝑝 𝑞 𝑝𝑖 𝑖 𝑝𝑖 functions on [𝑎, 𝑏] such that 𝑓 ,𝑔 are concave on [𝑎, 𝑏].Then ∫ 𝑓𝑖 (𝑥) 𝑑𝑥≤( ) ∫ (𝑓𝑖 (𝑥) +𝑓𝑖+1 (𝑥)) 𝑑𝑥 𝑎 𝑀𝑖 +1 𝑎

𝑝𝑖 𝑏 𝑝 𝑞 𝑀 𝑝 −1 𝑝 𝑝 (𝑓 (𝑎) +𝑓(𝑏)) (𝑔 (𝑎) +𝑔(𝑏)) ≤( 𝑖 ) ∫ 2 𝑖 (𝑓 𝑖 (𝑥) +𝑓 𝑖 (𝑥))𝑑𝑥, 1 󵄩 󵄩𝑝󵄩 󵄩𝑞 𝑖 𝑖+1 ≤ 󵄩𝑓󵄩 󵄩𝑔󵄩 . (46) 𝑀𝑖 +1 𝑎 2𝑝+𝑞 (𝑏−𝑎)2 𝑝 𝑞 𝑝 𝑏 1 𝑖+1 𝑏 𝑝𝑖+1 𝑝𝑖+1 ∫ 𝑓𝑖+1 (𝑥) 𝑑𝑥≤( ) ∫ (𝑓𝑖 (𝑥) +𝑓𝑖+1 (𝑥)) 𝑑𝑥 𝑎 𝑚 +1 𝑎 Theorem 8. Let 𝑛 be a positive integer such that 𝑛≥2and 𝑖 𝑛 𝑝 ,𝑝 ,...,𝑝 >1∏ (1/𝑝 )=1 𝑓 ,𝑓,...,𝑓 𝑝𝑖+1 𝑏 1 2 𝑛 , 𝑖=1 𝑖 ,andlet 1 2 𝑛 1 𝑝 −1 𝑝𝑖 ≤( ) ∫ 2 𝑖+1 be positive functions on [𝑎, 𝑏] such that the function 𝑓𝑖 is 𝑚𝑖 +1 𝑎 𝑏 𝑝 [𝑎, 𝑏] 0<∫ 𝑓 𝑖 (𝑥)𝑑𝑥 <∞ 𝑖 = 1,2,..., integrable on , 𝑎 𝑖 for all 𝑝 𝑝 ×(𝑓 𝑖+1 (𝑥) +𝑓 𝑖+1 (𝑥))𝑑𝑥 𝑛,and 𝑖 𝑖+1 (51) 𝑓 (𝑥) 0<𝑚 ≤ 𝑖 ≤𝑀 for all 𝑖=1,...,𝑛−1. 𝑖 𝑓 (𝑥) 𝑖 (47) 𝑖+1 Then

𝑝 𝑛 𝑏 𝑀 𝑖 𝑏 for all 𝑥∈[𝑎,𝑏]and 𝑖=1,...,𝑛−1.Then 𝑝𝑖 𝑝𝑖−1 𝑖 𝑝𝑖 ∫ 𝑓𝑖 (𝑥) 𝑑𝑥≤2 ( ) ∫ (∑𝑓𝑗 (𝑥))𝑑𝑥 𝑎 𝑀𝑖 +1 𝑎 𝑗=1

𝑏 𝑛 𝑛 𝑛 𝑝𝑖 𝑛 𝑏 𝑠𝑖 󵄩 󵄩𝑝𝑖 𝑝 −1 𝑀 𝑝 ∫ (∏𝑓 (𝑥))𝑑𝑥≤∑ ( (∑󵄩𝑓 󵄩 )) , 𝑖 𝑖 𝑖 𝑖 󵄩 𝑗󵄩 (48) =2 ( ) ∑ (∫ 𝑓𝑗 (𝑥) 𝑑𝑥) , 𝑝 󵄩 󵄩𝑝𝑖 𝑎 𝑖=1 𝑖=1 𝑖 𝑗=1 𝑀𝑖 +1 𝑗=1 𝑎

𝑝 𝑛 𝑏 1 𝑖+1 𝑏 𝑝𝑖+1 𝑝𝑖+1−1 𝑝𝑖+1 where ∫ 𝑓𝑖+1 (𝑥) 𝑑𝑥≤2 ( ) ∫ (∑𝑓𝑗 (𝑥))𝑑𝑥 𝑎 𝑚𝑖 +1 𝑎 𝑗=1

𝑝 𝑝 𝑝 𝑛 𝑀 1 1 𝑛 1 𝑖+1 𝑏 𝑝1−1 1 𝑝𝑛−1 𝑝𝑖+1−1 𝑝𝑖+1 𝑠1 =2 ( ) ,𝑠𝑛 =2 ( ) , =2 ( ) ∑ (∫ 𝑓𝑗 (𝑥) 𝑑𝑥) 𝑀1 +1 𝑚𝑛−1 +1 𝑚𝑖 +1 𝑗=1 𝑎 𝑝 𝑝 (49) 1 𝑖 𝑀 𝑖 (52) 𝑝𝑖−1 𝑝𝑖−1 𝑖 𝑠𝑖 = min{2 ( ) ,2 ( ) } 𝑚𝑖−1 +1 𝑀𝑖 +1 for all 𝑖=1,...,𝑛−1. 𝑝1−1 𝑝1 𝑝𝑛−1 𝑝𝑛 Let 𝑠1 =2 (𝑀1/(𝑀1 +1)) , 𝑠𝑛 =2 (1/(𝑚𝑛−1 +1)) , for all 1<𝑖<𝑛. and

𝑝 𝑝 1 𝑖 𝑀 𝑖 Proof. By Lemma 1,wehave 𝑝𝑖−1 𝑝𝑖−1 𝑖 𝑠𝑖 = min{2 ( ) ,2 ( ) } (53) 𝑚𝑖−1 +1 𝑀𝑖 +1

𝑏 1/𝑝𝑖 𝑝 𝑖 for all 1<𝑖<𝑛. (∫ 𝑓𝑖 (𝑥) 𝑑𝑥) 𝑎 It follows that 1/𝑝 𝑀 𝑏 𝑖 𝑏 𝑛 𝑏 𝑖 𝑝𝑖 𝑝 𝑝 ≤ (∫ (𝑓 (𝑥) +𝑓 (𝑥)) 𝑑𝑥) , 1 1 𝑖 𝑖+1 ∫ 𝑓 (𝑥) 𝑑𝑥≤𝑠1∑ (∫ 𝑓 (𝑥) 𝑑𝑥) , 𝑀 +1 𝑎 1 𝑗 𝑖 𝑎 𝑗=1 𝑎 (50) 1/𝑝 𝑏 𝑖+1 𝑏 𝑛 𝑏 𝑝𝑖+1 𝑝 𝑝 (∫ 𝑓𝑖+1 (𝑥) 𝑑𝑥) ∫ 𝑓 𝑛 (𝑥) 𝑑𝑥≤𝑠 ∑ (∫ 𝑓 𝑛 (𝑥) 𝑑𝑥) , 𝑎 𝑛 𝑛 𝑗 (54) 𝑎 𝑗=1 𝑎

𝑏 1/𝑝𝑖+1 1 𝑝 𝑏 𝑛 𝑏 𝑖+1 𝑝 𝑝 ≤ (∫ (𝑓𝑖 (𝑥) +𝑓𝑖+1 (𝑥)) 𝑑𝑥) ∫ 𝑓 𝑖 (𝑥) 𝑑𝑥≤𝑠∑ (∫ 𝑓 𝑖 (𝑥) 𝑑𝑥) 𝑚𝑖 +1 𝑎 𝑖 𝑖 𝑗 𝑎 𝑗=1 𝑎 for all 𝑖=1,...,𝑛−1. for all 1<𝑖<𝑛. 6 Journal of Function Spaces and Applications

By the generalized Young inequality, we obtain that [5] E. Set, M. E. Ozdemir,¨ and S. S. Dragomir, “On the Hermite- Hadamard inequality and other integral inequalities involving 𝑏 𝑛 𝑏 𝑛 1 two functions,” Journal of Inequalities and Applications,vol. 𝑝𝑖 ∫ (∏𝑓𝑖 (𝑥))𝑑𝑥≤∫ (∑ 𝑓𝑖 (𝑥))𝑑𝑥 2010,ArticleID148102,9pages,2010. 𝑎 𝑎 𝑝𝑖 𝑖=1 𝑖=1 [6]D.S.Mitrinovic,´ J. E. Pecariˇ c,´ and A. M. Fink, Classical 𝑛 1 𝑏 and New Inequalities in Analysis,vol.61ofMathematics and 𝑝𝑖 = ∑ (∫ 𝑓𝑖 (𝑥) 𝑑𝑥) Its Applications, Kluwer Academic Publishers, Dodrecht, The 𝑖=1 𝑝𝑖 𝑎 Netherlands, 1993. [7] M. Alomari and M. Darus, “On the Hadamard’s inequality for 𝑛 𝑛 𝑏 (55) 1 𝑝 log-convex functions on the coordinates,” Journal of Inequalities ≤ ∑ (𝑠 ∑ (∫ 𝑓 𝑖 (𝑥) 𝑑𝑥)) 𝑖 𝑗 and Applications,vol.2009,ArticleID283147,13pages,2009. 𝑖=1 𝑝𝑖 𝑗=1 𝑎 [8] C. Dinu, “Hermite-Hadamard inequality on time scales,” Jour- 𝑛 𝑛 nal of Inequalities and Applications,vol.2008,ArticleID287947, 𝑠 󵄩 󵄩𝑝 ≤ ∑ ( 𝑖 (∑󵄩𝑓 󵄩 𝑖 )) . 24 pages, 2008. 󵄩 𝑗󵄩𝑝 𝑖=1 𝑝𝑖 𝑗=1 𝑖 [9] J. Pecariˇ c´ and T. Pejkovic,´ “On an integral inequality,” Journal of Inequalities in Pure and Applied Mathematics,vol.5,no.2,article This proof is completed. 47, 6 pages, 2004. [10] J. E. Pecariˇ c,´ F. Proschan, and Y. L. Tong, Convex Functions, Applying Theorem 8 with 𝑛=2and putting there 𝑝1 = PartialOrderings,andStatisticalApplications,vol.187ofMath- 𝑝,2 𝑝 =𝑞, 2𝑠1/𝑝1 =𝑐1,and2𝑠2/𝑝2 =𝑐2,wegetthefollowing. ematics in Science and Engineering,1992. [11] T. K. Pogany,“OnanopenproblemofF.Qi,”´ Journal of Corollary 9 (see [5]). Let 𝑝, 𝑞>1, 1/𝑝 + 1/𝑞,and =1 Inequalities in Pure and Applied Mathematics,vol.3,no.4,article 𝑏 𝑓 𝑔 [𝑎, 𝑏] 0<∫ 54, 5 pages, 2002. let and be positive functions on such that 𝑎 𝑏 [12]P.S.Bullen,D.S.Mitrinovic,andP.M.Vasi´ c,´ Means and 𝑓𝑝(𝑥)𝑑𝑥 <∞ 0<∫ 𝑔𝑞(𝑥)𝑑𝑥 <∞ , 𝑎 ,and Their Inequalities,vol.31ofMathematicsandItsApplications, D. Reidel Publishing, Dordrecht, The Netherlands, 1988. 𝑓 (𝑥) 0<𝑚≤ ≤𝑀 [13]L.Bougoffa,“OnMinkowskiandHardyintegralinequalities,” 𝑔 (𝑥) (56) Journal of Inequalities in Pure and Applied Mathematics,vol.7, no.2,article60,3pages,2006. for all 𝑥∈[𝑎,𝑏].Then

󵄩 󵄩𝑝 󵄩 󵄩𝑝 󵄩 󵄩𝑞 󵄩 󵄩𝑞 𝑏 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩𝑓󵄩𝑝 + 󵄩𝑔󵄩𝑝 󵄩𝑓󵄩𝑞 + 󵄩𝑔󵄩𝑞 ∫ 𝑓 (𝑥) 𝑔 (𝑥) 𝑑𝑥1 ≤𝑐 ( )+𝑐2( ), 𝑎 2 2 (57) where 2𝑝 𝑀 𝑝 2𝑞 1 𝑞 𝑐 = ( ) ,𝑐= ( ) . 1 𝑝 𝑀+1 2 𝑞 𝑚+1 (58)

Acknowledgment The author would like to thank the referees for their useful comments and suggestions.

References

[1] B. G. Pachpatte, “On some inequalities for convex functions,” RGMIA Research Report Collection,vol.6,9pages,2003. [2] B. G. Pachpatte, “A note on integral inequalities involving two log-convex functions,” Mathematical Inequalities & Applica- tions,vol.7,no.4,pp.511–515,2004. [3] U.S.Kirmaci,M.Klariciˇ cBakula,M.E.´ Ozdemir,¨ and J. Pecariˇ c,´ “Hadamard-type inequalities for 𝑠-convex functions,” Applied Mathematics and Computation,vol.193,no.1,pp.26–35,2007. [4]M.K.Bakula,M.E.Ozdemir,¨ and J. Pecariˇ c,´ “Hadamard type inequalities for 𝑚-convex and (𝛼, 𝑚)-convex functions,” Journal of Inequalities in Pure and Applied Mathematics,vol.9,no.4, article 96, 12 pages, 2008. Hindawi Publishing Corporation Journal of Function Spaces and Applications Volume 2013, Article ID 140130, 6 pages http://dx.doi.org/10.1155/2013/140130

Research Article Positive Solutions for Some Competitive Fractional Systems in Bounded Domains

Imed Bachar,1 Habib Mâagli,2 and Noureddine Zeddini2

1 Mathematics Department, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia 2 Department of Mathematics, College of Sciences and Arts, King Abdulaziz University, Rabigh Campus, P.O. Box 344, Rabigh 21911, Saudi Arabia

Correspondence should be addressed to Noureddine Zeddini; [email protected]

Received 18 January 2013; Accepted 19 March 2013

Academic Editor: Nelson Merentes

Copyright © 2013 Imed Bachar et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Using some potential theory tools and the Schauder fixed point theorem, we prove the existence and precise global behavior of 𝛼/2 𝜎 𝑟 𝛼/2 𝑠 𝛽 positive continuous solutions for the competitive fractional system (−Δ |𝐷) 𝑢+𝑝(𝑥)𝑢 V =0, (−Δ |𝐷) V + 𝑞(𝑥)𝑢 V =0in a 1,1 𝑛 bounded 𝐶 -domain 𝐷 in R (𝑛 ≥ 3), subject to some Dirichlet conditions, where 0<𝛼<2, 𝜎,𝛽≥1,𝑠,𝑟≥0.The potential functions 𝑝, 𝑞 are nonnegative and required to satisfy some adequate hypotheses related to the Kato class of functions 𝐾𝛼(𝐷).

𝑢 (𝑥) 1. Introduction and Statement of Main Results lim =𝜑(𝑧) , 𝑥→𝑧∈𝜕𝐷𝑀𝐷1 (𝑥) 1,1 𝑛 𝛼 Let 𝐷 be a bounded 𝐶 -domain in R (𝑛 ≥ 3) and Δ |𝐷 be the 𝛼/2 V (𝑥) Dirichlet Laplacian in 𝐷. The fractional power −(−Δ |𝐷) , lim =𝜓(𝑧) , 𝑥→𝑧∈𝜕𝐷𝑀𝐷1 (𝑥) 0<𝛼<2, of the negative Dirichlet Laplacian is a very useful 𝛼 object in analysis and partial differential equations; see, for (1) 𝐷 instance, [1, 2]. There is a Markov process 𝑍 corresponding 𝛼 where 𝜎, 𝛽 ≥1, 𝑠, 𝑟, ≥0 0<𝛼<2and the nonnegative −(−Δ )𝛼/2 to |𝐷 which can be obtained as follows: we first kill potential functions 𝑝, 𝑞 are required to satisfy some adequate 𝑋 𝜏 𝑋 theBrownianmotion at 𝐷, the first exit time of from hypotheses related to the Kato class of functions 𝐾𝛼(𝐷) (see the domain 𝐷, and then we subordinate this killed Brownian 𝑀𝐷1(𝑥) 𝛼/2 𝑇 Definition). 1 The function 𝛼 is defined by motion using the -stable subordinator 𝑡 starting at zero. ∞ 𝑍𝐷 𝐷 1−𝛼/2 −2+(𝛼/2) 𝐷 For more description of the process 𝛼 and the development 𝑀𝛼 1 (𝑥) = ∫ 𝑡 (1 − 𝑃𝑡 1 (𝑥))𝑑𝑡, (2) of its potential theory, we refer to [3–6]. Γ (𝛼/2) 0 In this paper, we will exploit these potential theory tools 𝐷 where (𝑃𝑡 )𝑡>0 is the semigroup corresponding to the killed to study the existence of positive solutions for some nonlinear Brownian motion upon exiting 𝐷. systems of fractional differential equations. More precisely, 3.3 𝜑 𝜓 𝜕𝐷 We recall that in [6,Remark ], the authors have proved we fix two positive continuous functions and on , the existence of a constant 𝐶>0such that for each 𝑥∈𝐷, and we will deal with the existence of positive continuous 1 solutions (in the sense of distributions) for the following (𝛿 (𝑥))𝛼−2 ≤𝑀𝐷1 (𝑥) ≤𝐶(𝛿 (𝑥))𝛼−2, 𝐶 𝛼 (3) competitive fractional system: where 𝛿(𝑥) denotes the Euclidian distance from 𝑥 to the boundary of 𝐷. 𝛼/2 𝜎 𝑟 (−Δ |𝐷) 𝑢+𝑝(𝑥) 𝑢 V =0 in 𝐷, In the classical case (i.e., 𝛼=2), there is a large amount of literature dealing with the existence, nonexistence, and qual- 𝛼/2 𝑠 𝛽 (−Δ |𝐷) V +𝑞(𝑥) 𝑢 V =0 in 𝐷, itative analysis of positive solutions for the problems related 2 Journal of Function Spaces and Applications

to (1); see for example, the papers of Cˆırstea and Radulescu˘ Theorem 2. Under hypothesis (H1),theproblem(5) has a [7], Ghanmi et al. [8], Ghergu and Radulescu˘ [9], Lair and unique positive continuous solution satisfying for each 𝑥∈𝐷, Wood [10, 11], Mu et al. [12], and references therein. In these 𝑐 𝑀𝐷𝜑 (𝑥) ≤𝑢(𝑥) ≤𝑀𝐷𝜑 (𝑥) , works, various existence results of positive bounded solutions 0 𝛼 𝛼 (9) orpositiveblowingupones(calledalsolargesolutions)have where the constant 𝑐0 ∈ (0, 1]. been established, and a precise global behavior is given. We note also that several methods have been used to treat these Using (7), hypothesis (H1) is satisfied if 𝑝0 verifies the systems such as sub- and supersolutions method, variational following condition: there exists a constant 𝐶>0,suchthat method, and topological methods. In [11], the authors studied for each 𝑥∈𝐷, the system (1)with𝛼=2in the case 𝜎=𝛽=0, 𝑠>0, 𝑟> 0,and𝑝, 𝑞 are nonnegative continuous and not necessarily 𝐶 𝑝0 (𝑥) ≤ 𝜏 , with 𝜏+(2−𝛼) (𝛾−1) <𝛼. (10) radial. They showed that entire positive bounded solutions (𝛿 (𝑥)) exist if 𝑝 and 𝑞 satisfy the following condition: Next, we exploit the result of Theorem 2, to prove the exis- −(2+𝛿) 𝑝 (𝑥) +𝑞(𝑥) ≤𝐶|𝑥| , tence of a positive continuous solution (𝑢, V) to the system (4) (1). To this end, we assume the following hypothesis. for some positive constant 𝛿 and |𝑥| large. (H2) The functions 𝑝, 𝑞 are two nonnegative Borel measur- These results have been extended recently by Alsaedi et al. able functions such that in [13], in the case 𝛼=2, 𝜎, 𝛽 ≥1, 𝑠>0, 𝑟>0,where 𝑥󳨀→(𝛿 (𝑥))(𝛼−2)(𝜎+𝑟−1)𝑝 (𝑥) ∈𝐾 (𝐷) , the authors established the existence of a positive continuous 𝛼 bounded solution for (1). (11) 𝑥󳨀→(𝛿 (𝑥))(𝛼−2)(𝛽+𝑠−1)𝑞 (𝑥) ∈𝐾 (𝐷) . In this paper, first, we aim at proving the existence and 𝛼 uniqueness of a positive continuous solution (in the sense of Then, by using Schauder’s fixed point theorem, we prove the distributions) for the following scalar equation: following. 𝛼/2 𝛾 (−Δ |𝐷) 𝑢+𝑝0 (𝑥) 𝑢 =0 in 𝐷, Theorem 3. Under assumption (H2),theproblem(1) has a 𝑢>0 𝐷, positive continuous solution (𝑢, V) satisfying for each 𝑥∈𝐷, in (5) 𝑢 (𝑥) 𝑐 𝑀𝐷𝜑 (𝑥) ≤𝑢(𝑥) ≤𝑀𝐷𝜑 (𝑥) , =𝜑(𝑧) , 1 𝛼 𝛼 lim 𝐷 (12) 𝑥→𝑧∈𝜕𝐷𝑀𝛼 1 (𝑥) 𝐷 𝐷 𝑐2𝑀𝛼 𝜓 (𝑥) ≤ V (𝑥) ≤𝑀𝛼 𝜓 (𝑥) , where 𝛾≥1and 𝑝0 is a nonnegative Borel measurable function in 𝐷 satisfying the following. where 𝑐1 ∈ (0, 1] and 𝑐2 ∈ (0, 1]. (𝛼−2)(𝛾−1) (H1) The function 𝑥 → (𝛿(𝑥)) 𝑝0(𝑥) ∈𝛼 𝐾 (𝐷). We note that contrary to the classical case 𝛼=2,inour The class of functions 𝐾𝛼(𝐷),isdefinedbymeansofthe situation,thesolutionblowsupontheboundaryof𝐷. 𝐷 𝐷 Green function 𝐺𝛼 of 𝑍𝛼 as follows. The content of this paper is organized as follows. In Section 2, we collect some properties of functions belonging 𝑞 𝐷 Definition 1 (see [14]). A Borel measurable function in to the Kato class of functions 𝐾𝛼(𝐷),whichareusefulto belongs to the Kato class of functions 𝐾𝛼(𝐷) if establish our results. Our main results are proved in Section 3. + 𝛿(𝑦) As usual, let 𝐵 (𝐷) be the set of nonnegative Borel 𝐷 󵄨 󵄨 lim (sup ∫ 𝐺𝛼 (𝑥,) 𝑦 󵄨𝑞 (𝑦)󵄨 𝑑𝑦) =0. (6) measurable functions in 𝐷. We denote by 𝐶0(𝐷) the set of 𝑟→0 𝑥∈𝐷 (|𝑥−𝑦|≤𝑟)∩𝐷 𝛿 (𝑥) continuous functions in 𝐷 vanishing continuously on 𝜕𝐷. It has been shown in [14], that Note that 𝐶0(𝐷) is a Banach space with respect to the uniform −𝜆 norm ‖𝑢‖∞ = sup |𝑢(𝑥)|. When two positive functions 𝑥󳨀→(𝛿 (𝑥)) ∈𝐾 (𝐷) , 𝜆<𝛼. (7) 𝑥∈𝐷 𝛼 for 𝑓 and 𝑔 are defined on a set 𝑆,wewrite𝑓≈𝑔if the two- For more examples of functions belonging to 𝐾𝛼(𝐷), we refer sided inequality (1/𝐶)𝑔 ≤ 𝑓 ≤𝐶𝑔 holds on 𝑆. We define the 𝐷 𝐷 to [14]. Note that for the classical case (i.e., 𝛼=2), the class potential kernel 𝐺𝛼 of 𝑍𝛼 by of functions 𝐾2(𝐷) was introduced and studied in [15]. 𝑀𝐷𝜑 𝐷 𝐷 + In order to state our existence result, we denote by 𝛼 𝐺𝛼 𝑓 (𝑥) :=∫ 𝐺𝛼 (𝑥,𝑦)𝑓(𝑦)𝑑𝑦, for 𝑓∈𝐵 (𝐷) ,𝑥∈𝐷. (see [3])theuniquepositivecontinuoussolutionof 𝐷 (13) 𝛼/2 (−Δ |𝐷) 𝑢=0 in 𝐷 (in the sense of distributions) . Finally, let us recall some potential theory tools that are 𝑢 (𝑥) needed, and we refer to [14, 16, 17] for more details. For 𝑞∈ =𝜑(𝑧) . + + lim 𝐷 𝐵 (𝐷) 𝑉 𝐵 (𝐷) 𝑥→𝑧∈𝜕𝐷𝑀𝛼 1 (𝑥) , we define the kernel 𝑞 on by

(8) ∞ 𝑡 𝑥 −∫ 𝑞(𝑍𝐷(𝑠))𝑑𝑠 𝐷 ̃ 0 𝛼 𝑉𝑞𝑓 (𝑥) := ∫ 𝐸 (𝑒 𝑓(𝑍𝛼 (𝑡))) 𝑑𝑡, 𝑥 ∈𝐷, Using some potential theory tools and an approximating 0 sequence, we establish the following. (14) Journal of Function Spaces and Applications 3

𝐷 ̃𝑥 with 𝑉0 :=𝑉=𝐺𝛼 ,where𝐸 stands for the expectation with That is 𝐷 respect to 𝑍𝛼 starting from 𝑥.If𝑞 satisfies 𝑉𝑞 <∞,wehave the following resolvent equation: 𝑉(𝑞𝑉𝑓𝑘) (𝑥) 𝑉𝑓 𝑘 (𝑥) ≤𝑉𝑞𝑓𝑘 (𝑥) exp ( ). (21) 𝑉𝑓 𝑘 (𝑥) 𝑉=𝑉𝑞 +𝑉𝑞 (𝑞𝑉)𝑞 =𝑉 +𝑉(𝑞𝑉𝑞). (15)

+ Hence, it follows from (17)that In particular, if 𝑢∈𝐵(𝐷) is such that 𝑉(𝑞𝑢),thenwe <∞ have 𝑉𝑓 𝑘 (𝑥) ≤𝑉𝑞𝑓𝑘 (𝑥) exp (𝑎𝛼 (𝑞)) . (22)

(𝐼−𝑉𝑞 (𝑞.)) (𝐼 + 𝑉 (𝑞.)) 𝑢 = (𝐼 + 𝑉𝑞 (𝑞.))(𝐼−𝑉 (𝑞.)) 𝑢 = 𝑢. (16) Consequently, from (15)weobtain

(−𝑎 (𝑞)) 𝑉𝑓 (𝑥) ≤𝑉𝑓 (𝑥) −𝑉 (𝑞𝑉𝑓 ) (𝑥) ≤𝑉𝑓 (𝑥) . 2. The Kato Class of Functions 𝐾𝛼(𝐷) exp 𝛼 𝑘 𝑘 𝑞 𝑘 𝑘 (23) Proposition 4 (see [14]). Let 𝑞 be a function in 𝐾𝛼(𝐷),then we have the following. The result holds by letting 𝑘→∞. 𝑎 (𝑞) := ∫ (𝐺𝐷(𝑥, 𝑧)𝐺𝐷(𝑧, 𝑦)/𝐺𝐷(𝑥, 𝑦)) (i) 𝛼 sup𝑥,𝑦∈𝐷 𝐷 𝛼 𝛼 𝛼 Lemma 6. 𝑞 𝐾 (𝐷) |𝑞(𝑧)|𝑑𝑧 <∞ Let be a nonnegative function in 𝛼 , then the . family of functions (ii) Let ℎ be a positive excessive function on 𝐷 with respect 𝑍𝐷 to 𝛼 .Then,wehave 1 󵄨 󵄨 Λ ={ ∫ 𝐺𝐷 (𝑥, 𝑦)𝐷 𝑀 𝜑 (𝑦) 𝑓 (𝑦) 𝑑𝑦, 󵄨𝑓󵄨 ≤𝑞} 𝑞 𝐷 𝛼 𝛼 󵄨 󵄨 𝑀𝛼 𝜑 (𝑥) 𝐷 𝐷 󵄨 󵄨 ∫ 𝐺𝛼 (𝑥,) 𝑦 ℎ (𝑦) 󵄨𝑞 (𝑦)󵄨 𝑑𝑦 ≤𝑎𝛼 (𝑞) ℎ (𝑥) . (17) (24) 𝐷

is uniformly bounded and equicontinuous in 𝐷.Consequently, Λ 𝐶 (𝐷) Furthermore, for each 𝑥0 ∈ 𝐷,wehave 𝑞 is relatively compact in 0 .

ℎ≡𝑀𝐷𝜑 |𝑓| ≤ 𝑞 1 𝐷 󵄨 󵄨 Proof. Taking 𝛼 in (17), we deduce that for lim (sup ∫ 𝐺 (𝑥,𝑦)ℎ(𝑦)󵄨𝑞(𝑦)󵄨 𝑑𝑦) = 0. 𝑥∈𝐷 𝑟→0 ℎ (𝑥) 𝛼 󵄨 󵄨 and ,wehave 𝑥∈𝐷 𝐵(𝑥0,𝑟)∩𝐷 (18) 󵄨 󵄨 󵄨 𝐺𝐷 (𝑥, 𝑦) 󵄨 󵄨∫ 𝛼 𝑀𝐷𝜑 (𝑦) 𝑓 (𝑦)󵄨 𝑑𝑦 󵄨 𝐷 𝛼 󵄨 󵄨 𝐷 𝑀𝛼 𝜑 (𝑥) 󵄨 𝛼−1 1 󵄨 󵄨 (iii) The function 𝑥 → (𝛿(𝑥)) 𝑞(𝑥) is in 𝐿 (𝐷). (25) 𝐺𝐷 (𝑥, 𝑦) ≤ ∫ 𝛼 𝑀𝐷𝜑 (𝑦) 𝑞 (𝑦) 𝑑𝑦≤𝑎 (𝑞) < ∞. 𝐷 𝛼 𝛼 The next two lemmas will play a special role. 𝐷 𝑀𝛼 𝜑 (𝑥)

Lemma 5. Let 𝑞 be a nonnegative function in 𝐾𝛼(𝐷) and ℎ 𝐷 Λ be a positive finite excessive function on 𝐷 with respect to 𝑍𝛼 . So, the family 𝑞 is uniformly bounded. Then, for all 𝑥∈𝐷,wehave Next, we aim at proving that the family Λ 𝑞 is equicontinuous in 𝐷. (−𝑎 (𝑞)) ℎ (𝑥) ≤ℎ(𝑥) −𝑉 (𝑞ℎ) (𝑥) ≤ℎ(𝑥) . exp 𝛼 𝑞 (19) First, we recall the following interesting sharp estimates 𝐷 on 𝐺𝛼 ,whichisprovedin[5]: Proof. Let ℎ be a positive finite excessive function on 𝐷 𝐷 with respect to 𝑍 .Then,by[18, Chapter II, proposition 𝛼 𝛿 (𝑥) 𝛿(𝑦) 3.11 (𝑓 ) 𝐷 󵄨 󵄨𝛼−𝑛 ], there exists a sequence 𝑘 𝑘 of nonnegative measurable 𝐺 (𝑥, 𝑦) ≈ 󵄨𝑥−𝑦󵄨 min (1, ). (26) 𝐷 ℎ= 𝑉𝑓 𝑥∈𝐷 𝑘∈N 𝛼 󵄨 󵄨 󵄨 󵄨2 functions in such that sup𝑘 𝑘.Let and 󵄨𝑥−𝑦󵄨 such that 0<𝑉𝑓𝑘 <∞. Consider 𝜃(𝑡)𝑡𝑞 =𝑉 𝑓𝑘(𝑥),for 𝑡≥0.Then,by(14), the function 𝜃 is completely monotone on [0, ∞),andsofromtheHolder¨ inequality and [19,Theorem Let 𝑥0 ∈ 𝐷 and 𝜀>0.By(18), there exists 𝑟>0such that 12a], the function log 𝜃 is convex on [0, ∞). This implies that 1 𝜀 ∫ 𝐺𝐷 (𝑧, 𝑦)𝐷 𝑀 𝜑 (𝑦) 𝑞 (𝑦) 𝑑𝑦≤ . 󸀠 sup 𝐷 𝛼 𝛼 𝜃 (0) 𝑀 𝜑 (𝑧) 𝐵(𝑥 ,2𝑟)∩𝐷 2 𝜃 (0) ≤𝜃(1) exp (− ). (20) 𝑧∈𝐷 𝛼 0 𝜃 (0) (27) 4 Journal of Function Spaces and Applications

󸀠 If 𝑥0 ∈𝐷and 𝑥, 𝑥 ∈𝐵(𝑥0,𝑟)∩𝐷,thenfor|𝑓| ≤ 𝑞,wehave 3. Proofs of Theorems 2 and 3 󵄨 𝐷 󵄨 𝐺𝛼 (𝑥, 𝑦) 𝐷 The next Lemma will be used for uniqueness. 󵄨∫ 𝑀 𝜑 (𝑦) 𝑓 (𝑦) 𝑑𝑦 󵄨 𝑀𝐷𝜑 (𝑥) 𝛼 󵄨 𝐷 𝛼 + Lemma 7 (see [14, Proposition 6]). Let ℎ∈𝐵(𝐷) and 𝜐 be 𝐷 󸀠 󵄨 𝐷 𝑍𝐷 𝐺𝛼 (𝑥 ,𝑦) 󵄨 a nonnegative excessive function on with respect to 𝛼 .Let − ∫ 𝑀𝐷𝜑 (𝑦) 𝑓 (𝑦)󵄨 𝑑𝑦 𝑧 𝐷 𝑉(ℎ|𝑧|) <∞ 𝐷 󸀠 𝛼 󵄨 be a Borel measurable function in such that 𝐷 𝑀 𝜑(𝑥 ) 󵄨 𝛼 󵄨 and 𝜐 = 𝑧 + 𝑉(ℎ𝑧).Then,𝑧 satisfies 󵄨 󵄨 󵄨 𝐷 𝐺𝐷 (𝑥󸀠,𝑦)󵄨 0≤𝑧≤𝜐. 󵄨𝐺𝛼 (𝑥, 𝑦) 𝛼 󵄨 𝐷 (33) ≤ ∫ 󵄨 − 󵄨 𝑀 𝜑 (𝑦) 𝑞 (𝑦) 𝑑𝑦 󵄨 𝑀𝐷𝜑 (𝑥) 𝑀𝐷𝜑(𝑥󸀠) 󵄨 𝛼 𝐷 󵄨 𝛼 𝛼 󵄨 Proof of Theorem 2. Let 𝜑 be a positive continuous function on 𝜕𝐷.Werecallthaton𝐷,wehave 1 𝐷 𝐷 ≤2 ∫ 𝐺 (𝑧, 𝑦) 𝑀 𝜑 (𝑦) 𝑞 (𝑦) 𝑑𝑦 𝐷 𝐷 𝛼−2 sup 𝑀𝐷𝜑 (𝑧) 𝛼 𝛼 𝑀 𝜑 (𝑥) ≈𝑀 1 (𝑥) ≈ (𝛿 (𝑥)) . (34) 𝑧∈𝐷 𝐵(𝑥0,2𝑟)∩𝐷 𝛼 𝛼 𝛼 󵄨 󵄨 𝐷 𝛾−1 −𝑎 (𝑝̃) 󵄨 𝐷 𝐺𝐷 (𝑥󸀠,𝑦)󵄨 𝑝̃ =𝛾(𝑀𝜑) 𝑝 𝑐 =𝑒 𝛼 0 𝑎 (𝑝̃) 󵄨𝐺𝛼 (𝑥, 𝑦) 𝛼 󵄨 𝐷 Let 0 𝛼 0 and put 0 ,where 𝛼 0 + ∫ 󵄨 − 󵄨 𝑀 𝜑 (𝑦) 𝑞 (𝑦) 𝑑𝑦 (H ) 𝑝̃ ∈𝐾(𝐷) 󵄨 𝑀𝐷𝜑 (𝑥) 𝑀𝐷𝜑(𝑥󸀠) 󵄨 𝛼 is given by Proposition 4(i). Since by 1 , 0 𝛼 ,it (|𝑥0−𝑦|≥2𝑟)∩𝐷 󵄨 𝛼 𝛼 󵄨 󵄨 󵄨 follows from Proposition 4 that 𝑉(𝑝̃0)≤𝑎𝛼(𝑝̃0)<∞.Define Λ 󵄨 𝐷 𝐷 󸀠 󵄨 the nonempty closed bounded convex by 󵄨𝐺 (𝑥 , 𝑦) 𝐺𝛼 (𝑥 ,𝑦)󵄨 ≤𝜀+∫ 󵄨 𝛼 − 󵄨 𝑀𝐷𝜑(𝑦) 𝑞 (𝑦) 𝑑𝑦. + 󵄨 𝐷 𝐷 󸀠 󵄨 𝛼 Λ={𝜔∈𝐵 (𝐷) :𝑐 ≤𝜔≤1} . (|𝑥 −𝑦|≥2𝑟)∩𝐷 󵄨 𝑀 𝜑 (𝑥) 𝑀 𝜑(𝑥) 󵄨 0 (35) 0 󵄨 𝛼 𝛼 󵄨 (28) Let 𝑇 be the operator defined on Λ by 𝑐 󸀠 1 On the other hand, for every 𝑦∈𝐵(𝑥0,2𝑟)∩𝐷and 𝑥, 𝑥 ∈ 𝐷 𝑇𝜔 := 1− 𝑉𝑝̃ (𝑝̃0𝑀 𝜑) 𝐷 𝑀𝐷𝜑 0 𝛼 𝐵(𝑥0,𝑟) ∩,byusing( 𝐷 26) and the fact that 𝑀𝛼 𝜑(𝑧) ≈ 𝛼 𝛼−2 (36) (𝛿(𝑧)) ,wehave 1 𝛾 + 𝑉 (𝑝̃𝜔𝑀𝐷𝜑−𝑝 (𝜔𝑀𝐷𝜑) ). 󵄨 󵄨 𝐷 𝑝̃0 0 𝛼 0 𝛼 󵄨 1 𝐷 1 𝐷 󸀠 󵄨 𝐷 𝑀𝛼 𝜑 󵄨 𝐺 (𝑥, 𝑦) − 𝐺 (𝑥 ,𝑦)󵄨 𝑀 𝜑(𝑦) 󵄨𝑀𝐷𝜑 (𝑥) 𝛼 𝑀𝐷𝜑(𝑥󸀠) 𝛼 󵄨 𝛼 󵄨 𝛼 𝛼 󵄨 We claim that 𝑇 maps Λ to itself. Indeed, for each 𝜔∈Λ,we ≤ 𝐶(𝛿 (𝑦))𝛼−1. have 1 𝐷 𝛾 (29) 𝑇𝜔 ≤ 1− 𝑉𝑝̃ (𝑝0(𝜔𝑀 𝜑) )≤1. 𝑀𝐷𝜑 0 𝛼 (37) 𝐷 𝐷 𝛼 Now since 𝑥→(1/𝑀𝛼 𝜑(𝑥))𝐺𝛼 (𝑥, 𝑦) is continuous outside 𝑝̃𝜔𝑀𝐷𝜑−𝑝 the diagonal and 𝑞∈𝐾𝛼(𝐷),wededucebythedominated On the other hand, since the function 0 𝛼 0 𝐷 𝛾 𝐷 convergence theorem and Proposition 4 (iii) that (𝜔𝑀𝛼 𝜑) ≥0,wededucebyLemma 5 with ℎ=𝑀𝛼 𝜑,that 𝐷 ̃ 𝐷 󵄨 𝐷 󸀠 󵄨 𝑇𝜔 ≥ 1 − (1/𝑀𝛼 𝜑)𝑉𝑝̃(𝑝0𝑀𝛼 𝜑) ≥0 𝑐 .Hence,𝑇Λ ⊂. Λ Next, 󵄨𝐺𝐷 (𝑥, 𝑦) 𝐺 (𝑥 ,𝑦)󵄨 0 ∫ 󵄨 𝛼 − 𝛼 󵄨 𝑀𝐷𝜑 (𝑦) 𝑞 (𝑦) 𝑑𝑦 weaimatprovingthat𝑇 is nondecreasing on Λ.Tothisend, 󵄨 𝐷 𝐷 󸀠 󵄨 𝛼 (|𝑥 −𝑦|≥2𝑟)∩𝐷 󵄨 𝑀 𝜑 (𝑥) 𝑀 𝜑(𝑥 ) 󵄨 𝜔 𝜔 ∈Λ 𝜔 ≤𝜔 0 󵄨 𝛼 𝛼 󵄨 we let 1, 2 such that 1 2.Usingthefactthatthe 𝑡→𝛾𝑡−𝑡𝛾 [0, 1] 󵄨 󵄨 function is nondecreasing on ,wededuce 󵄨 󸀠󵄨 󳨀→ 0 as 󵄨𝑥−𝑥󵄨 󳨀→ 0. that (30) 𝑇𝜔2 −𝑇𝜔1 𝑥 ∈𝜕𝐷 𝑥∈𝐵(𝑥,𝑟)∩𝐷 If 0 and 0 ,thenwehave 1 𝐷 𝐷 𝛾 = 𝑉̃ (𝑝̃𝜔 𝑀 𝜑−𝑝 (𝜔 𝑀 𝜑) ) 󵄨 𝐷 󵄨 𝐷 𝑝0 0 2 𝛼 0 2 𝛼 󵄨 𝐺 (𝑥, 𝑦) 󵄨 𝑀𝛼 𝜑 󵄨∫ 𝛼 𝑀𝐷𝜑 (𝑦) 𝑓 (𝑦)󵄨 𝑑𝑦 󵄨 𝐷 𝛼 󵄨 󵄨 𝐷 𝑀𝛼 𝜑 (𝑥) 󵄨 1 𝐷 𝐷 𝛾 − 𝑉𝑝̃ (𝑝̃0𝜔1𝑀 𝜑−𝑝0(𝜔1𝑀 𝜑) ) (31) 𝑀𝐷𝜑 0 𝛼 𝛼 𝜀 𝐺𝐷 (𝑥, 𝑦) 𝛼 ≤ + ∫ 𝛼 𝑀𝐷𝜑 (𝑦) 𝑞 (𝑦) 𝑑𝑦. 𝐷 𝛼 2 (|𝑥 −𝑦|≥2𝑟)∩𝐷 𝑀 𝜑 (𝑥) 1 𝐷 𝛾 𝛾 𝛾 0 𝛼 = 𝑉 (𝑝 (𝑀 𝜑) [(𝛾𝜔 −𝜔 )−(𝛾𝜔 −𝜔 )]) 𝐷 𝑝̃0 0 𝛼 2 2 1 1 𝐷 𝐷 𝑀𝛼 𝜑 Now, since 𝐺𝛼 (𝑥, 𝑦)/𝑀𝛼 𝜑(𝑥)→0as |𝑥−𝑥0|→0,for |𝑥0 −𝑦|≥2𝑟, then by the same argument as above, we get ≥0. 𝐺𝐷 (𝑥, 𝑦) (38) ∫ 𝛼 𝑀𝐷𝜑 (𝑦) 𝑞 (𝑦) 𝑑𝑦 󳨀→0 𝐷 𝛼 (|𝑥 −𝑦|≥2𝑟)∩𝐷 𝑀 𝜑 (𝑥) (𝜔 ) 0 𝛼 (32) Next, we define the sequence 𝑘 𝑘≥0 by 󵄨 󵄨 󵄨𝑥−𝑥 󵄨 󳨀→ 0. 1 𝐷 as 󵄨 0󵄨 𝜔0 =1− 𝑉𝑝̃ (𝑝̃0𝑀 𝜑) , 𝑀𝐷𝜑 0 𝛼 𝛼 (39) Consequently,byAscoli’stheorem,wededucethatΛ 𝑞 is relatively compact in 𝐶0(𝐷). 𝜔𝑘+1 =𝑇𝜔𝑘. Journal of Function Spaces and Applications 5

𝜔 ∈Λ 𝜔 =𝑇𝜔 ≥𝜔 ̃ 𝐷 𝜎−1 𝐷 𝑟 Clearly 0 and 1 0 0. Thus, from the monoton- Proof of Theorem 3. Let 𝑝=𝜎(𝑀𝛼 𝜑) (𝑀𝛼 𝜓) 𝑝 and 𝑞=̃ icity of 𝑇,wededucethat 𝐷 𝛽−1 𝐷 𝑠 𝛽(𝑀𝛼 𝜓) (𝑀𝛼 𝜑) 𝑞. −𝑎𝛼(𝑝)̃ −𝑎𝛼(𝑞)̃ 𝑐0 ≤𝜔0 ≤𝜔1 ≤⋅⋅⋅≤𝜔𝑘 ≤1. (40) Put 𝑐1 =𝑒 , 𝑐2 =𝑒 .Notethatfrom(H2) and Proposition 4,wehave𝑎𝛼(𝑝)̃ < ∞ and 𝑎𝛼(𝑞)̃ <. ∞ Consider (𝜔 ) So, the sequence 𝑘 𝑘≥0 converges to a measurable function the nonempty closed convex set Γ defined by 𝜔∈Λ. Therefore, by applying the monotone convergence theorem, we obtain Γ = {(𝑦, 𝑧) ∈𝐶(𝐷) × 𝐶 (𝐷) :1 𝑐 ≤𝑦≤1,𝑐2 ≤𝑧≤1}. 1 𝜔=1− 𝑉 (𝑝̃𝑀𝐷𝜑) (49) 𝐷 𝑝̃0 0 𝛼 𝑀𝛼 𝜑 (41) Let 𝑇 be the operator defined on Γ by 𝑇(𝑦, 𝑧) := (𝜔,, 𝜃) 1 𝐷 𝐷 𝛾 𝐷 𝐷 + 𝑉𝑝̃ (𝑝̃0𝜔𝑀 𝜑−𝑝0(𝜔𝑀 𝜑) ). such that (𝑢=𝜔𝑀̃ 𝜑, ̃V =𝜃𝑀𝜓) istheuniquepositive 𝑀𝐷𝜑 0 𝛼 𝛼 𝛼 𝛼 𝛼 continuous solution of the following problem: 𝑢=𝜔𝑀𝐷𝜑 Put 𝛼 .Then,wehave 𝛼/2 𝐷 𝑟 𝑟 𝜎 (−Δ |𝐷) 𝑢+((𝑀̃ 𝛼 𝜓) 𝑧 𝑝) (𝑥) 𝑢̃ =0 in 𝐷, 𝐷 ̃ 𝐷 ̃ 𝛾 𝑢=𝑀𝛼 𝜑−𝑉𝑝̃ (𝑝0𝑀𝛼 𝜑) +𝑝̃ 𝑉 (𝑝0𝑢−𝑝0𝑢 ) (42) 0 0 𝛼/2 𝐷 𝑠 𝑠 𝛽 (−Δ |𝐷) ̃V +((𝑀𝛼 𝜑) 𝑦 𝑞) (𝑥) ̃V =0 in 𝐷, or equivalently 𝑢̃ (𝑥) (50) ̃ 𝐷 ̃ 𝐷 𝛾 =𝜑(𝑧) , 𝑢−𝑉𝑝̃ (𝑝0𝑢) =𝛼 𝑀 𝜑−𝑉𝑝̃ (𝑝0𝑀𝛼 𝜑) −𝑝̃ 𝑉 (𝑝0𝑢 ). (43) lim 𝐷 0 0 0 𝑥→𝑧∈𝜕𝐷𝑀𝛼 1 (𝑥)

Observe that by Proposition 4(ii), we have 𝑉(𝑝̃0𝑢) <.So, ∞ ̃V (𝑥) (𝐼+𝑉(𝑝̃.)) lim =𝜓(𝑧) . applying the operator 0 on both sides of (43), we 𝑥→𝑧∈𝜕𝐷𝑀𝐷1 (𝑥) deduce by using (15)and(16)that 𝛼 𝐷 𝛾 According to Theorem 2,wehave 𝑢=𝑀𝛼 𝜑−𝑉(𝑝0𝑢 ) . (44) (H ) 1 𝑟 𝜎 Now, using 1 and similar argument as in the proof of 𝜔=1− 𝑉(𝑧𝑟𝜔𝜎(𝑀𝐷𝜓) (𝑀𝐷𝜑) 𝑝) 𝐷 𝛾 𝐷 𝛼 𝛼 Lemma 6,weprovethat𝑥→(1/𝑀𝛼 𝜑)𝑉(𝑝0𝑢 )∈𝐶0(𝐷). 𝑀𝛼 𝜑 So, 𝑢 is a continuous function in 𝐷,and𝑢 is a solution of (51) 1 𝑠 𝛽 (5). It remains to prove the uniqueness of such a solution. 𝜃=1− 𝑉(𝑦𝑠𝜔𝛽(𝑀𝐷𝜑) (𝑀𝐷𝜓) 𝑞) . 𝑀𝐷𝜓 𝛼 𝛼 Let 𝑢 be a continuous solution of (5). Since the function 𝛼 𝐷 𝑥→𝑢(𝑥)/𝑀𝛼 1(𝑥) is continuous and positive in 𝐷 such 𝐷 Moreover, we have 𝑐1 ≤𝜔≤1and 𝑐2 ≤𝜃≤1and by that lim𝑥→𝑧∈𝜕𝐷(𝑢(𝑥)/𝑀𝛼 1(𝑥)) = 𝜑(𝑧),itfollowsthat𝑢(𝑥) ≈ 𝐷 𝐷 𝑇(Γ) 𝐷 𝑇(Γ) 𝑀 1(𝑥) ≈ 𝑀 𝜑(𝑥).Then,byusingthisfactandLemma 6, Lemma 6, is equicontinuous on .Since is also 𝛼 𝛼 𝑇(Γ) we have bounded, then we deduce that is relatively compact in 𝐶(𝐷) × 𝐶(𝐷) 𝑇(Γ) ⊂ Γ 𝛼/2 𝛾 . This implies in particular that . (−Δ |𝐷) (𝑢 + 𝑉0 (𝑝 𝑢 )) = 0 in 𝐷, Next,weshallprovethecontinuityoftheoperator𝑇 in Γ in the supremum norm. Let (𝑦𝑘,𝑧𝑘)𝑘 be a sequence in Γ which (𝑢 + 𝑉 (𝑝 𝑢𝛾)) (𝑥) (45) 0 =𝜑(𝑧) 𝐷. converges uniformly to a function (𝑦, 𝑧) in Γ.Put(𝜔𝑘,𝜃𝑘)= lim 𝐷 in 𝑥→𝑧∈𝜕𝐷 𝑀𝛼 1 (𝑥) 𝑇(𝑦𝑘,𝑧𝑘) and (𝜔, 𝜃) = 𝑇(𝑦, 𝑧).Then,wehave ( 󵄨 So, from the uniqueness of the problem (8) see [3]), we 󵄨 󵄨 󵄨 1 𝑟 𝜎 󵄨𝜔 −𝜔󵄨 = 󵄨 𝑉(𝑧𝑟𝜔𝜎(𝑀𝐷𝜓) (𝑀𝐷𝜑) 𝑝) deduce that 󵄨 𝑘 󵄨 󵄨 𝐷 𝛼 𝛼 󵄨𝑀𝛼 𝜑 𝛾 𝐷 𝑢+𝑉(𝑝0𝑢 )=𝑀𝛼 𝜑 in 𝐷. (46) 󵄨 1 𝑟 𝜎 󵄨 − 𝑉(𝑧𝑟𝜔𝜎(𝑀𝐷𝜓) (𝑀𝐷𝜑) 𝑝)󵄨 𝑢 V 𝐷 𝑘 𝑘 𝛼 𝛼 󵄨 (52) It follows that if and are two continuous solution of (5), 𝑀𝛼 𝜑 󵄨 then 𝑧=V −𝑢satisfies 1 󵄨 󵄨 ≤ 𝑉(󵄨𝑧𝑟𝜔𝜎 −𝑧𝑟𝜔𝜎󵄨 (𝑀𝐷𝜑) 𝑝)̃ . 𝑧+𝑉(𝑝0ℎ𝑧) =0 in 𝐷, (47) 𝐷 󵄨 𝑘 𝑘 󵄨 𝛼 𝜎𝑀𝛼 𝜑 where ℎ is the nonnegative measurable function defined in 𝐷 |𝑧𝑟𝜔𝜎 −𝑧𝑟𝜔𝜎|≤2 𝑝∈𝐾̃ (𝐷) by Using the fact that 𝑘 𝑘 and that 𝛼 , we deduce by Proposition 4 and the dominated convergence 𝛾 𝛾 {V −𝑢 theorem, that 𝜔𝑘 →𝜔as 𝑘→∞.Similarly,weprovethat { , if 𝑢 (𝑥) ≠V (𝑥) , ℎ (𝑥) = V −𝑢 𝜃𝑘 →𝜃as 𝑘→∞.So,𝑇(𝑦𝑘,𝑧𝑘)→𝑇(𝑦,𝑧)as 𝑘→∞. { (48) 0, 𝑢 𝑥 = V 𝑥 . 𝑇(Γ) 𝐶(𝐷) × 𝐶(𝐷) { if ( ) ( ) Since is relatively compact in ,wededuce that Since 𝑉(𝑝0ℎ|𝑧|) < ∞,wededucebyLemma 7 that 𝑧=0,and 󵄩 󵄩 so 𝑢=V. 󵄩𝑇(𝑦𝑘,𝑧𝑘)−𝑇(𝑦,𝑧)󵄩∞ 󳨀→ 0 as 𝑘󳨀→∞. (53) 6 Journal of Function Spaces and Applications

From the Schauder fixed point theorem there exists (𝑦, 𝑧) ∈Γ [13]R.S.Alsaedi,H.M.Maagli,ˆ and N. Zeddini:, “Positive solutions such that 𝑇(𝑦, 𝑧) = (𝑦, 𝑧) or equivalently for some competetive elliptic systems,” to appear in Mathemat- ica Slovaca. 𝐷 𝜎 𝑟 𝑢=𝑀𝛼 𝜑−𝑉(𝑝𝑢 V ), [14] A. Dhifli, H.aagli, Mˆ and M. Zribi, “On the subordinate killed (54) 𝐷 𝑠 𝛽 B.M in bounded domains and existence results for nonlinear V =𝑀𝛼 𝜓−𝑉(𝑞𝑢V ), fractional Dirichlet problems,” Mathematische Annalen,vol. 352, no. 2, pp. 259–291, 2012. 𝐷 𝐷 where (𝑢, V) = (𝑦𝑀𝛼 𝜑, 𝑧𝑀𝛼 𝜓).Thepair(𝑢, V) is a required [15] H. Maagliˆ and M. Zribi, “On a new Kato class and singular solution of (1) in the sense of distributions. This completes solutions of a nonlinear elliptic equation in bounded domains 𝑛 the proof. of R ,” Positivity,vol.9,no.4,pp.667–686,2005. [16] K. L. Chung and Z. X. Zhao, From Brownian Motion to Sch- Acknowledgment rodinger’s¨ Equation,vol.312ofFundamental Principles of Math- ematical Sciences, Springer, Berlin, Germany, 1995. The research of Imed Bachar is supported by NPST Program [17] H. M. Maagli,ˆ “Perturbation Semi-Lineaire´ des Resolvantes´ et of King Saud University; Project no. 11-MAT1716-02. des Semi-Groupes,” Potential Analysis,vol.3,pp.61–87,1994. [18] J. Bliedtner and W. Hansen, PotentialTheory.AnAnalytic and Probabilistic Approach to Balayage, Universitext, Springer, References Berlin, Germany, 1986. [1] A. Pazy, Semigroups of Linear Operators and Applications to [19] D. V. Widder, The Laplace Transform,vol.6ofPrinceton Partial Differential Equations,vol.44ofApplied Mathematical Mathematical Series, Princeton University Press, Princeton, NJ, Sciences, Springer, New York, NY, USA, 1983. USA, 1941. [2] K. Yosida, Functional Analysis,vol.123ofFundamental Prin- ciples of Mathematical Sciences, Springer, Berlin, Germany, 6th edition, 1980. [3] J. Glover, Z. Pop-Stojanovic, M. Rao, H. Sikiˇ c,´ R. Song, and Z. Vondracek,ˇ “Harmonic functions of subordinate killed Brown- ian motion,” Journal of Functional Analysis,vol.215,no.2,pp. 399–426, 2004. [4] J. Glover, M. Rao, H. Sikiˇ c,´ and R. M. Song, “Γ-potentials,” in Classical and Modern Potential Theory and Applications (Chateau de Bonas, 1993),vol.430ofNATO Adv. Sci. Inst. Ser. CMath.Phys.Sci., pp. 217–232, Kluwer Acad. Publ., Dordrecht, The Netherlands, 1994. [5] R. Song, “Sharp bounds on the density, Green function and jumping function of subordinate killed BM,” Probability Theory and Related Fields,vol.128,no.4,pp.606–628,2004. [6] R. Song and Z. Vondracek,ˇ “Potential theory of subordinate killed Brownian motion in a domain,” Probability Theory and Related Fields,vol.125,no.4,pp.578–592,2003. [7] F.-C. S¸t. Cˆırstea and V. D. Radulescu,˘ “Entire solutions blow- ing up at infinity for semilinear elliptic systems,” Journal de Mathematiques´ Pures et Appliquees´ ,vol.81,no.9,pp.827–846, 2002. [8]A.Ghanmi,H.Maagli,ˆ S. Turki, and N. Zeddini, “Existence of positive bounded solutions for some nonlinear elliptic systems,” Journal of Mathematical Analysis and Applications,vol.352,no. 1,pp.440–448,2009. [9] M. Ghergu and V. Radulescu,˘ “On a class of singular Gierer- Meinhardt systems arising in morphogenesis,” Comptes Rendus Mathematique´ ,vol.344,no.3,pp.163–168,2007. [10] A. V. Lair and A. W. Wood, “Large solutions of semilinear elliptic equations with nonlinear gradient terms,” International Journal of Mathematics and Mathematical Sciences,vol.22,no. 4, pp. 869–883, 1999. [11] A. V. Lair and A. W. Wood, “Existence of entire large positive solutions of semilinear elliptic systems,” Journal of Differential Equations,vol.164,no.2,pp.380–394,2000. [12]C.Mu,S.Huang,Q.Tian,andL.Liu,“Largesolutionsforan elliptic system of competitive type: existence, uniqueness and asymptotic behavior,” Nonlinear Analysis: Theory, Methods & Applications,vol.71,no.10,pp.4544–4552,2009. Hindawi Publishing Corporation Journal of Function Spaces and Applications Volume 2013, Article ID 842375, 15 pages http://dx.doi.org/10.1155/2013/842375

Research Article Commutators of Higher Order Riesz Transform Associated with Schrödinger Operators

Yu Liu,1 Lijuan Wang,1 and Jianfeng Dong2

1 School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China 2 Department of Mathematics, Shanghai University, Shanghai 200444, China

Correspondence should be addressed to Yu Liu; [email protected]

Received 25 February 2013; Accepted 3 April 2013

Academic Editor: Jozef´ Bana´s

Copyright © 2013 Yu Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

𝑛 Let 𝐿=−Δ+𝑉beaSchrodingeroperatoron¨ R (𝑛≥3), where 𝑉 ≡0̸ is a nonnegative potential belonging to certain reverse 𝐻 𝐻 𝐻 Holder¨ class 𝐵𝑠 for 𝑠≥𝑛/2. In this paper, we prove the boundedness of commutators R𝑏 𝑓=𝑏R 𝑓−R (𝑏𝑓) generated by the 𝐻 2 −1 𝑛 higher order Riesz transform R =∇(−Δ + 𝑉) ,where𝑏∈BMO𝜃(𝜌), which is larger than the space BMO(R ).Moreover,we R𝐻 𝐻1(R𝑛) 𝐿1 (R𝑛) prove that 𝑏 is bounded from the Hardy space 𝐿 into weak weak .

𝑛 1. Introduction Euclidean spaces R and [4–6]onspacesofhomogeneous type. 𝐿=−Δ+𝑉 R𝑛 𝑛≥3 Let be a Schrodinger¨ operator on , ,where In recent years, singular integral operators related to 𝑉 ≡0̸ is a nonnegative potential belonging to the reverse Schrodinger¨ operators and their commutators have been 𝐵 𝑠≥𝑛/2 Holder¨ class 𝑠 for some .Inthispaper,wewill brought to many scholars attention. See, for example, [7–19] consider the higher order Riesz transforms associated with and their references. Especially, Guo et al. [12]investigatedthe 𝐿 R𝐻 ≐∇2𝐿−1 𝐻 𝑛 the Schrodinger¨ operator defined by and the boundedness of the commutators R𝑏 when 𝑏∈BMO(R ). commutator But their method is not valid to prove the boundedness of the 𝐻 𝐻 𝐻 𝐻 𝑛 commutators R𝑏 when 𝑏∈BMO∞(𝜌). In fact, since R𝑏 (𝑓) (𝑥) = R (𝑏𝑓) (𝑥) −𝑏(𝑥) R 𝑓 (𝑥) ,𝑥∈R . (1) R𝐻 =∇2(−Δ)−1 (−Δ)(−Δ + 𝑉)−1 (3) We also consider its dual higher order transforms associated 2 −1 −1 ̃𝐻 −1 2 =∇(−Δ) (𝐼−𝑉(−Δ + 𝑉) ), with the Schrodinger¨ operator 𝐿 defined by R ≐𝐿 ∇ and the commutator 𝐻 then R𝑏 may be written as follows: R̃𝐻 (𝑓) (𝑥) = R̃𝐻 (𝑏𝑓) (𝑥) −𝑏(𝑥) R̃𝐻𝑓 (𝑥) ,𝑥∈R𝑛, 𝑏 R𝐻 (𝑓) = [𝑏, 𝑇 𝑇 ]𝑓=[𝑏,𝑇 ](𝑇 𝑓) + 𝑇 [𝑏, 𝑇 ]𝑓, (2) 𝑏 1 2 1 2 1 2 (4) 𝑛 2 −1 −1 𝑏∈ (𝜌) (R ) where 𝑇1 =∇(−Δ) and 𝑇2 = 𝐼−𝑉(−Δ+𝑉) .If𝑏∈ where BMO∞ , which is larger than the space BMO . 𝑛 𝑝 BMO(R ),byusingCorollary1in[12], we obtain the 𝐿 Because the investigation of commutators of singular 𝐻 𝑛 boundedness of R𝑏 .Butif𝑏∈BMO𝜃(𝜌) and 𝑏∉BMO(R ), integral operators plays an important role in Harmonic 𝑝 it follows from [1]that[𝑏,1 𝑇 ] is not bounded on 𝐿 ,andthen analysis and PDE, many authors concentrate on this topic. It 𝑝 𝐻 is well known that Coifman et al. [1]provedthat[𝑏, 𝑇] is a we cannot obtain the 𝐿 boundedness of R𝑏 . 𝑝 bounded operator on 𝐿 for 1<𝑝<∞if and only if 𝑏∈ Motivated by [12, 15, 17], our aim in this paper is to inves- 𝑛 𝑝 𝐻 BMO(R ) when 𝑇 is a Calderon-Zygmund´ operator. See tigate the 𝐿 estimates and endpoint estimates for R𝑏 when [2, 3] for the research development of the commutator 𝑇𝑏 on 𝑏∈BMO∞(𝜌). Different from the classical higher order Riesz 2 Journal of Function Spaces and Applications

1/𝑞−1 transform, there exist some new problems for the higher (ii) ‖𝑎‖𝐿𝑞(R𝑛) ≤|𝐵(𝑥,𝑟)| , 𝐻 order Riesz transform R . We need to obtain some new 𝐻 (iii) when 𝑟<𝜌(𝑥)/4, ∫ 𝑛 𝑎(𝑥)𝑑𝑥=0. estimates for R when the potential 𝑉 satisfies more stronger R conditions. 1 𝑛 𝑠 The space 𝐻 (R ) admits the following atomic decompo- A nonnegative locally 𝐿 -integrable function 𝑉 (1<𝑠< 𝐿 sitions (cf. [21]). ∞)iscalledtobelongto𝐵𝑠 if there exists a constant 𝐶>0 such that the reverse Holder¨ inequality 1 𝑛 1 𝑛 Proposition 3. Let 𝑓∈𝐿(R ).Then,𝑓∈𝐻𝐿(R ) if and only 1/𝑠 if 𝑓 can be written as 𝑓=∑𝑗 𝜆𝑗𝑎𝑗,where𝑎𝑗 are (1, 𝑞)𝜌-atoms 1 𝑠 1 ( ∫ 𝑉 𝑑𝑦) ≤𝐶( ∫ 𝑉𝑑𝑦) and ∑𝑗 |𝜆𝑗|<∞.Moreover, |𝐵 (𝑥,) 𝑟 | 𝐵(𝑥,𝑟) |𝐵 (𝑥,) 𝑟 | 𝐵(𝑥,𝑟) (5) 󵄩 󵄩 { 󵄨 󵄨} 󵄩𝑓󵄩 ∼ ∑ 󵄨𝜆 󵄨 , 󵄩 󵄩𝐻1 inf { 󵄨 𝑗󵄨} (10) 𝐵 R𝑛 𝐿 holds for every ball in . { 𝑗 } Moreover, a locally bounded nonnegative function 𝑉∈ 𝐵∞, if there exists a positive constant 𝐶 such that where the infimum is taken over all atomic decompositions of 1 𝑓 into 𝐻𝐿-atoms. 𝐶 ‖𝑉‖𝐿∞(𝐵(𝑥,𝑟)) ≤ ∫ 𝑉(𝑦)𝑑𝑦 (6) |𝐵 (𝑥,) 𝑟 | 𝐵(𝑥,𝑟) Following [17], the class BMO𝜃(𝜌) of locally integrable function 𝑏 is defined as follows: 𝑛 holds for every 𝐵(𝑥, 𝑟) in R and 0<𝑟<∞. 𝜃 1 󵄨 󵄨 𝑟 𝐵 ⊂𝐵 𝑠 >𝑠 ∫ 󵄨𝑏(𝑦)−𝑏 󵄨 𝑑𝑦 ≤ 𝐶(1 + ) , Obviously, 𝑠2 𝑠1 ,if 2 1.Butitisimportantthat 󵄨 𝐵󵄨 (11) |𝐵 (𝑥,) 𝑟 | 𝐵(𝑥,𝑟) 𝜌 (𝑥) the 𝐵𝑠 class has a property of “self-improvement”; that is, if 𝑉∈𝐵 𝑉∈𝐵 𝜀>0 𝑠,then 𝑠+𝜀 for some .Furthermore,itiseasy 𝑥∈R𝑛 𝑟>0 𝜃>0 𝑏 = 𝐵 ⊆𝐵 1<𝑠<∞ for all and ,where and 𝐵 to see that ∞ 𝑠 for any . (1/|𝐵|) ∫ 𝑏(𝑦)𝑑𝑦 𝑏∈ (𝜌) [𝑏] 𝑛/2 𝑛 𝐵 .Anormfor BMO𝜃 , denoted by 𝜃, Assume that 𝑉≥0and 𝑉∈𝐿loc (R ).TheSchrodinger¨ 𝐿 is given by the infimum of the constants satisfying (11), after operator 𝐿=−Δ+𝑉generates a (𝐶0)semigroup{𝑇𝑡 }𝑡>0 = −𝑡𝐿 identifying functions that differ upon a constant. If we let {𝑒 }𝑡>0. The maximal function with respect to the semi- 𝜃=0in (11), then BMO𝜃(𝜌) is exactly the John-Nirenberg 𝐿 𝑛 group {𝑇𝑡 }𝑡>0 is given by space BMO(R ).DenotethatBMO∞(𝜌) =𝜃>0 ⋃ BMO𝜃(𝜌). 𝑛 (R )⊂ (𝜌) ⊂ 󸀠 (𝜌) 󵄨 󵄨 It is easy to see that BMO BMO𝜃 BMO𝜃 for 𝑀𝐿𝑓 (𝑥) = 󵄨𝑇𝐿𝑓 (𝑥)󵄨 . 0<𝜃≤𝜃󸀠 sup 󵄨 𝑡 󵄨 .Bongioannietal.[17] gave some examples to 𝑡>0 (7) 𝑛 clarify that the space BMO(R ) is a subspace of BMO∞(𝜌). Let 𝜌1(𝑥) be the auxiliary function of |∇𝑉(𝑥)|.Ourmain 𝐻1(R𝑛) The Hardy space 𝐿 associated with the Schrodinger¨ results are given as follows. operator 𝐿 is defined as follows in terms of the maximal function mentioned earlier (cf. [20]). Theorem 4. Suppose that 𝑉∈𝐵𝑠 for some 𝑠≥𝑛, |∇𝑉| ∈𝑠 𝐵 𝑠 ≥𝑛/2 𝜌(𝑥) ≲𝜌 (𝑥) 𝜌(𝑥) ≲1 𝑏∈𝐵𝑀𝑂 (𝜌)1 1 𝑛 1 𝑛 ( 1 ), 1 ,and .Let ∞ . Definition 1. Afunction𝑓∈𝐿(R ) is said to be in 𝐻𝐿(R ) R̃𝐻(𝑓) 𝐿𝑝(R𝑛) 𝑠󸀠 <𝑝< 𝐿 1 𝑛 The commutator 𝑏 is bounded on for 1 if the semigroup maximal function 𝑀 𝑓 belongs to 𝐿 (R ). 󸀠 ∞,where(1/𝑠1)+(1/𝑠1)=1. The norm of such a function is defined by 󵄩 󵄩 󵄩 󵄩 By duality, we immediately have the following. 󵄩𝑓󵄩 = 󵄩𝑀𝐿𝑓󵄩 . 󵄩 󵄩𝐻1 󵄩 󵄩𝐿1 (8) 𝐿 Corollary 5. 𝑉∈𝐵 𝑠≥𝑛 |∇𝑉| ∈ 𝐵 Suppose that 𝑠 for some , 𝑠1 We introduce the auxiliary function 𝜌(𝑥, 𝑉) = 𝜌(𝑥) (𝑠1 ≥𝑛/2), 𝜌(𝑥)1 ≲𝜌 (𝑥),and𝜌(𝑥).Let ≲1 𝑏∈𝐵𝑀𝑂∞(𝜌). 𝐻 𝑝 𝑛 defined by The commutator R𝑏 is bounded on 𝐿 (R ) for 1<𝑝<𝑠1.

1 Furthermore, we get the endpoint estimate for the com- 𝜌 (𝑥) = 𝐻 𝑚 (𝑥,) 𝑉 mutator R𝑏 . (9) 1 𝑛 ≐ {𝑟 : ∫ 𝑉(𝑦)𝑑𝑦≤1}, 𝑥∈R . Theorem 6. Suppose that 𝑉∈𝐵𝑠 for some 𝑠≥𝑛, |∇𝑉| ∈𝑠 𝐵 sup 𝑟𝑛−2 1 𝑟>0 𝐵(𝑥,𝑟) (𝑠1 ≥𝑛/2), 𝜌(𝑥)1 ≲𝜌 (𝑥),and𝜌(𝑥).Let ≲1 𝑏∈𝐵𝑀𝑂∞(𝜌). 𝜆>0 𝑛 Then, for any , It is known that 0<𝜌(𝑥)<∞for any 𝑥∈R (from Lemma 8 󵄨 󵄨 󵄨 󵄨 [𝑏] in Section 2). 󵄨 𝑛 󵄨 𝐻 󵄨 󵄨 𝜃 󵄩 󵄩 󵄨{𝑥 ∈ R : 󵄨R𝑏 (𝑓) (𝑥)󵄨 >𝜆}󵄨 ≲ 󵄩𝑓󵄩𝐻1(R𝑛), 󵄨 󵄨 󵄨 󵄨 𝜆 𝐿 (12) Definition 2. Let 1<𝑞≤∞. A measurable function 𝑎 is ∀𝑓 ∈ 𝐻1 (R𝑛). called a (1, 𝑞)𝜌-atom associated to the ball 𝐵(𝑥, 𝑟) if 𝑟<𝜌(𝑥) 𝐿 and the following conditions hold: 𝐻 1 𝑛 Namely, the commutator R𝑏 is bounded from 𝐻𝐿(R ) into 𝑛 1 𝑛 (i) supp 𝑎⊂𝐵(𝑥,𝑟)for some 𝑥∈R and 𝑟>0, 𝐿𝑤𝑒𝑎𝑘(R ). Journal of Function Spaces and Applications 3

󸀠 This paper is organized as follows. In Section 2,wecollect (2) There exist 𝐶>0and 𝑘0 >0such that some known facts about the auxiliary function 𝜌(𝑥) and some 𝑘󸀠 necessary estimates for the kernel of the higher order Riesz 1 𝑅 0 𝐻 ∫ 𝑉(𝑦)𝑑𝑦≤𝐶(1+ ) . transform R .InSection 3,wegivetheproofofTheorems 𝑛−2 (19) 𝑅 𝐵(𝑥,𝑅) 𝜌 (𝑥) 4 and 6. Section 4 gives the corresponding results when the potential 𝑉 satisfies stronger conditions. In Section 5,wegive Let Γ(𝑥, 𝑦) be the fundamental solution of 𝐿.Then,there some examples for the potentials 𝑉 in Theorems 4 and 6. exists 𝐶𝑙 >0such that for each 𝑙>0, Throughout this paper, unless otherwise indicated, we 0 ≡𝑉∈𝐵̸ 𝑠>𝑛 always assume that 𝑠 for some .Wewilluse 󵄨 󵄨 𝐶 1 𝐶 󵄨Γ(𝑥,𝑦)󵄨 ≤ 𝑙 . to denote the positive constants, which are not necessarily 󵄨 󵄨 󵄨 󵄨 𝑙 󵄨 󵄨𝑛−2 (20) same at each occurrence even be different in the same line, (1 + 󵄨𝑥−𝑦󵄨 /𝜌 (𝑥)) 󵄨𝑥−𝑦󵄨 and may depend on the dimension 𝑛 and the constant in (5) or (6). By 𝐴∼𝐵and 𝐴≲𝐵,wemeanthatthereexistsome In particular, Γ(𝑥,𝑦) = Γ(𝑥,𝑦,0) = Γ(𝑦,𝑥,0) is the funda- 󸀠 󸀠 constants 𝐶, 𝐶 such that 1/𝐶 ≤ 𝐴/𝐵 ≤𝐶 and 𝐴≤𝐶𝐵, mental solution of the Schrodinger¨ operator 𝐿.If𝑉∈𝐵𝑛, respectively. then there exists 𝐶𝑙 >0such that for each 𝑙>0,

󵄨 󵄨 𝐶 1 2. Some Lemmas 󵄨∇Γ (𝑥, 𝑦)󵄨 ≤ 𝑙 . 󵄨 󵄨 󵄨 󵄨 𝑙 󵄨 󵄨𝑛−1 (21) (1 + 󵄨𝑥−𝑦󵄨 /𝜌 (𝑥)) 󵄨𝑥−𝑦󵄨 In this section, we collect some known results about auxiliary 𝜌(𝑥) function and some necessary estimates for the kernel of The previous facts had been obtained by Shen8 in[ ]. the higher order Riesz transform in the paper. We denote the fundamental solution of −Δ by Γ0(𝑥, 𝑦), which satisfies the following. Lemma 7. 𝑉∈𝐵𝑠 (𝑠 ≥ 𝑛/2) is a doubling measure; that is, 𝐶>0 there exists a constant such that (i) There exists 𝐶>0such that ∫ 𝑉(𝑦)𝑑𝑦≤𝐶∫ 𝑉(𝑦)𝑑𝑦. (13) 󵄨 󵄨 𝐶 𝐵(𝑥,2𝑟) 𝐵(𝑥,𝑟) 󵄨Γ (𝑥, 𝑦)󵄨 ≤ . 󵄨 0 󵄨 󵄨 󵄨𝑛−2 󵄨𝑥−𝑦󵄨 (22) Especially, there exist constants 𝜇≥1and 𝐶 such that 󵄨 󵄨

𝑛𝜇 ∫ 𝑉(𝑦)𝑑𝑦≤𝐶𝑡 ∫ 𝑉(𝑦)𝑑𝑦 (14) 𝐶>0 𝐵(𝑥,𝑡𝑟) 𝐵(𝑥,𝑟) (ii) There exists such that holds for every ball 𝐵(𝑥, 𝑟) and 𝑡>1. 󵄨 󵄨 𝐶 󵄨∇Γ (𝑥, 𝑦)󵄨 ≤ . 󵄨 0 󵄨 󵄨 󵄨𝑛−1 󵄨𝑥−𝑦󵄨 (23) Lemma 8. There exist constants 𝐶,0 𝑘 >0such that 󵄨 󵄨

󵄨 󵄨 −𝑘0 󵄨 󵄨 𝑘0/(𝑘0+1) 1 󵄨𝑥−𝑦󵄨 𝜌(𝑦) 󵄨𝑥−𝑦󵄨 (1 + ) ≤ ≤𝐶(1+ ) . Lemma 11. Suppose that 𝑉∈𝐵𝑠 for some 𝑠>𝑛and |∇𝑉| ∈𝑠 𝐵 𝐶 𝜌 (𝑥) 𝜌 (𝑥) 𝜌 (𝑥) 1 for some 𝑠1 >𝑛/2. Assume that (−Δ + 𝑉)𝑢 =0 in 𝐵(𝑥0,2𝑅). (15) Then,

In particular, 𝜌(𝑦) ∼ 𝜌(𝑥) if |𝑥−𝑦|<𝐶𝜌(𝑥). 󸀠 𝑘0 󵄨 2 󵄨 𝑅 󵄨∇ 𝑢 (𝑥)󵄨 ≲(1+ ) Using the Holder¨ inequality and 𝐵𝑠 condition, we have the 󵄨 󵄨 𝜌(𝑥0) following. 󵄨 󵄨 󵄨 󵄨 󵄨∇𝑉 (𝑦)󵄨 1 Lemma 9. × 󵄨𝑢 (𝑦)󵄨 (∫ 󵄨 󵄨 𝑑𝑦 + ) Let sup 󵄨 󵄨 󵄨 󵄨𝑛−1 𝑅2 𝐵(𝑥 ,2𝑅) 𝐵(𝑥0,2𝑅) 󵄨𝑥−𝑦󵄨 𝑉(𝑦)𝑑𝑦 𝐶 0 󵄨 󵄨 ∫ ≤ ∫ 𝑉 (𝑦) 𝑑𝑦. 󵄨 󵄨𝑛−2 𝑛−2 (16) 󸀠 𝐵(𝑥,𝑅) 󵄨𝑥−𝑦󵄨 𝑅 𝐵(𝑥,𝑅) 𝑘0 󵄨 󵄨 1 𝑅 󵄨 󵄨 + (1 + ) sup 󵄨∇𝑢 (𝑦)󵄨 . Moreover, if 𝑉∈𝐵𝑛, then there exists 𝐶>0such that 𝑅 𝜌(𝑥 ) 0 𝐵(𝑥0,2𝑅) 𝑉(𝑦)𝑑𝑦 𝐶 (24) ∫ ≤ ∫ 𝑉(𝑦)𝑑𝑦. 󵄨 󵄨𝑛−1 𝑛−1 (17) 𝐵(𝑥,𝑅) 󵄨 󵄨 𝑅 𝐵(𝑥,𝑅) 󵄨𝑥−𝑦󵄨 ∞ Proof. Let 𝜙∈𝐶𝑐 (𝐵(𝑥0, 2𝑅)) such that 𝜙≡1on 𝐵(𝑥0, Lemma 10. (1) 0<𝑟<𝑅<∞ −1 2 −2 For , (3/2)𝑅), 0<𝜙≤1, |∇𝜙| ≤ 𝐶𝑅 , |∇ 𝜙| ≤ 𝐶𝑅 ,and 3 −3 1 𝑟 2−𝑛/𝑠 1 |∇ 𝜙| ≤ 𝐶𝑅 .Since ∫ 𝑉(𝑦)𝑑𝑦≤𝐶( ) ∫ 𝑉(𝑦)𝑑𝑦, 𝑛−2 𝑛−2 𝑟 𝐵(𝑥,𝑟) 𝑅 𝑅 𝐵(𝑥,𝑅) 1 𝑢 (𝑥) 𝜙 (𝑥) = ∫ Γ0 (𝑥, 𝑦) (−Δ) (𝑢𝜙) (𝑦) 𝑑𝑦, (25) ∫ 𝑉(𝑦)𝑑𝑦∼ 1 𝑖𝑓𝑓𝑟 ∼𝜌(𝑥) . R𝑛 𝑛−2 𝑟 𝐵(𝑥,𝑟) (18) then, for 𝑥∈𝐵(𝑥0,𝑅), 4 Journal of Function Spaces and Applications

2 ∇ 𝑢 (𝑥) 󵄨 2 󵄨 𝐶𝑙 1 󵄨∇ Γ(𝑥,𝑦)󵄨 ≤ 󵄨 󵄨 󵄨 󵄨 𝑙 󵄨 󵄨𝑛−2 (1 + 󵄨𝑥−𝑦󵄨 /𝜌 (𝑥)) 󵄨𝑥−𝑦󵄨 = ∫ ∇Γ0 (𝑥, 𝑦) (−Δ) (∇ (𝑢𝜙)) (𝑦) 𝑑𝑦 R𝑛 |∇𝑉 (𝑧)| 1 ×(∫ 𝑑𝑧 + ). 󵄨 󵄨𝑛−1 󵄨 󵄨2 𝐵(𝑥,2|𝑥−𝑦|) 󵄨𝑥−𝑦󵄨 󵄨𝑥−𝑦󵄨 = ∫ ∇Γ0 (𝑥, 𝑦) (−Δ) (∇𝑢𝜙 + 𝑢∇𝜙) (𝑦)𝑑𝑦 󵄨 󵄨 󵄨 󵄨 R𝑛 (28) = ∫ ∇Γ (𝑥,𝑦)(−𝑉(𝑦)𝑢(𝑦)∇𝜙(𝑦)−2∇𝑢(𝑦)⋅∇2𝜙(𝑦) 0 Lemma 13. Suppose that 𝑉∈𝐵𝑠 for some 𝑠>𝑛and |∇𝑉| ∈ R𝑛 𝐵 𝑠 >𝑛/2 𝐶 >0 𝑠1 for some 1 .Thereexistsaconstant 𝑙 such that −𝑢 (𝑦) Δ∇𝜙 (𝑦) )𝑑𝑦 for each 𝑙>0, 󵄨 󵄨 󵄨 2 2 󵄨 󵄨∇𝑦Γ(𝑥+ℎ,𝑦)−∇𝑦Γ(𝑥,𝑦)󵄨 + ∫ ∇Γ0 (𝑥, 𝑦) (−∇𝑉𝑢𝜙 − 𝑉∇𝑢𝜙) 𝑑𝑦. 󵄨 󵄨 R𝑛 (26) 𝐶 |ℎ|𝛿 ≤ 𝑙 󵄨 󵄨 𝑙 󵄨 󵄨𝑛−2+𝛿 (1 + 󵄨𝑥−𝑦󵄨 /𝜌 (𝑥)) 󵄨𝑥−𝑦󵄨 (29) Therefore, we have, for 𝑥∈𝐵(𝑥0,𝑅), 󵄨 󵄨 |∇𝑉 (𝑧)| 1 󵄨∇2𝑢 (𝑥)󵄨 ×(∫ 𝑑𝑧 + ), 󵄨 󵄨 𝑛−1 󵄨 󵄨2 𝐵(𝑥,2|𝑥−𝑦|) |𝑥−𝑧| 󵄨𝑥−𝑦󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨∇𝑉 (𝑦)󵄨 󵄨𝑢(𝑦)󵄨 𝑉(𝑦)󵄨∇𝑢 (𝑦)󵄨 ≲∫ 󵄨 󵄨 󵄨 󵄨𝑑𝑦 + ∫ 󵄨 󵄨𝑑𝑦 󵄨 󵄨𝑛−1 󵄨 󵄨𝑛−1 𝛿=1−𝑛/𝑡>0 𝑡>𝑛 𝐵(𝑥 ,2𝑅) 󵄨 󵄨 𝐵(𝑥 ,2𝑅) 󵄨 󵄨 where for . 0 󵄨𝑥−𝑦󵄨 0 󵄨𝑥−𝑦󵄨 1 󵄨 󵄨 1 󵄨 󵄨 Proof. Let 𝑅 = |𝑥 −. 𝑦|/4 Assume that |ℎ| < 𝑅/2.Itfollows + ∫ 󵄨𝑉 (𝑦) 𝑢󵄨 (𝑦) 𝑑𝑦+ ∫ 󵄨∇𝑢 (𝑦)󵄨 𝑑𝑦 𝑛 󵄨 󵄨 𝑛+1 󵄨 󵄨 𝑅 𝐵(𝑥 ,2𝑅) 𝑅 𝐵(𝑥 ,2𝑅) from the embedding theorem of Morrey, Corollary 1, and 0 0 Remark 4.9 in [8]that 1 󵄨 󵄨 + ∫ 󵄨𝑢(𝑦)󵄨 𝑑𝑦 󵄨 2 2 󵄨 𝑅𝑛+2 󵄨 󵄨 󵄨∇ Γ(𝑥+ℎ,𝑦)−∇ Γ(𝑥,𝑦)󵄨 𝐵(𝑥0,2𝑅) 󵄨 𝑦 𝑦 󵄨 󵄨 󵄨 1/𝑡 󵄨 󵄨 󵄨∇𝑉 (𝑦)󵄨 󵄨 󵄨𝑡 ≲ 󵄨𝑢(𝑦)󵄨 (∫ 󵄨 󵄨 𝑑𝑦 ≤𝐶|ℎ|1−𝑛/𝑡(∫ 󵄨∇ ∇2Γ (𝑧, 󵄨𝑦) 𝑑𝑧) sup 󵄨 󵄨 󵄨 󵄨𝑛−1 󵄨 𝑥 𝑦 󵄨 𝐵(𝑥 ,2𝑅) 󵄨𝑥−𝑦󵄨 𝐵(𝑥,𝑅) 𝐵(𝑥0,2𝑅) 0 󵄨 󵄨 1−𝑛/𝑡 |ℎ| 𝑘 󵄨 2 󵄨 1 󵄨 󵄨 1 ≤𝐶( ) (1+𝑅𝑚(𝑥,)) 𝑉 0 󵄨∇ Γ(𝑧,𝑦)󵄨 + ∫ 󵄨𝑉(𝑦)󵄨 𝑑𝑦 + ) sup 󵄨 𝑦 󵄨 𝑅𝑛 󵄨 󵄨 𝑅2 𝑅 𝐵(𝑥,2𝑅) (30) 𝐵(𝑥0,2𝑅) 󵄨 󵄨 𝐶 |ℎ|𝛿 󵄨 󵄨 󵄨𝑉(𝑦)󵄨 1 ≤ 𝑙 + 󵄨∇𝑢 (𝑦)󵄨 (∫ 󵄨 󵄨 𝑑𝑦 + ) 󵄨 󵄨 𝑙 󵄨 󵄨𝑛−2+𝛿 sup 󵄨 󵄨 󵄨 󵄨𝑛−1 (1 + 󵄨𝑥−𝑦󵄨 /𝜌 (𝑥)) 󵄨𝑥−𝑦󵄨 𝐵(𝑥 ,2𝑅) 󵄨𝑥−𝑦󵄨 𝑅 󵄨 󵄨 󵄨 󵄨 𝐵(𝑥0,2𝑅) 0 󵄨 󵄨

󸀠 |∇𝑉 (𝑧)| 1 𝑘0 ×(∫ 𝑑𝑧 + ), 𝑅 󵄨 󵄨 𝑛−1 󵄨 󵄨2 ≲(1+ ) 󵄨𝑢(𝑦)󵄨 𝐵(𝑥,2|𝑥−𝑦|) |𝑥−𝑧| 󵄨𝑥−𝑦󵄨 𝜌(𝑥 ) sup 󵄨 󵄨 0 𝐵(𝑥0,2𝑅) 𝛿=1−𝑛/𝑡>0 󵄨 󵄨 where . 󵄨∇𝑉 (𝑦)󵄨 1 ×(∫ 󵄨 󵄨 𝑑𝑦 + ) 󵄨 󵄨𝑛−1 2 𝐵(𝑥 ,2𝑅) 󵄨 󵄨 𝑅 Similarly, we have the following. 0 󵄨𝑥−𝑦󵄨 󸀠 Lemma 14. 𝑉∈𝐵 𝑠>𝑛 |∇𝑉| ∈ 𝑘0 Suppose that 𝑠 for some and 1 𝑅 󵄨 󵄨 𝐵 𝑠 >𝑛/2 𝐶 >0 + (1 + ) sup 󵄨∇𝑢 (𝑦)󵄨 , 𝑠1 for some 1 .Thereexistsaconstant 𝑙 such that 𝑅 𝜌(𝑥 ) 𝑙>0 0 𝐵(𝑥0,2𝑅) for each , (27) 󵄨 󵄨 󵄨∇2Γ(𝑥,𝑦+ℎ)−∇2Γ(𝑥,𝑦)󵄨 󵄨 𝑥 𝑥 󵄨 where we use Lemma 9 and (2) in Lemma 10 in the last step. 𝐶 |ℎ|𝛿 Therefore, we complete the proof of the lemma. ≤ 𝑙 󵄨 󵄨 𝑙 󵄨 󵄨𝑛−2+𝛿 (1 + 󵄨𝑥−𝑦󵄨 /𝜌 (𝑥)) 󵄨𝑥−𝑦󵄨 (31) Furthermore, we get the following corollary via the proof 󵄨 󵄨 󵄨 󵄨 of Lemma 11. |∇𝑉 (𝑧)| 1 ×(∫ 𝑑𝑧 + ), 𝑥−𝑧𝑛−1 󵄨 󵄨2 Corollary 12. Suppose that 𝑉∈𝐵𝑠 for some 𝑠>𝑛and |∇𝑉| ∈ 𝐵(𝑥,2|𝑥−𝑦|) | | 󵄨𝑥−𝑦󵄨 𝐵 𝑠 >𝑛/2 𝐶 >0 𝑠1 for some 1 .Thereexistsaconstant 𝑙 such that for each 𝑙>0, where 𝛿=1−𝑛/𝑡>0for 𝑡>𝑛. Journal of Function Spaces and Applications 5

𝛿 Corollary 15. 𝑉∈𝐵𝑠 𝑠>𝑛 |∇𝑉| ∈𝑠 𝐵 𝐶𝑙 |ℎ| Suppose that for some , 1 ≤ , 𝑠 >𝑛/2 𝐶 󵄨 󵄨 𝑙 󵄨 󵄨𝑛+𝛿 for some 1 , and there exists a constant such that (1 + 󵄨𝑥−𝑦󵄨 /𝜌 (𝑥)) 󵄨𝑥−𝑦󵄨 3 |∇𝑉 (𝑥)| ≤𝐶𝑚(𝑥,) 𝑉 . (32) (36) where 𝛿=1−𝑛/𝑡>0for 𝑡>𝑛. There exists a constant 𝐶𝑙 >0such that for each 𝑙>0, 󵄨 󵄨 𝐶 1 Remark 17. Following Remark 5 in [22], we know that if 𝑉 is 󵄨∇2Γ(𝑥,𝑦)󵄨 ≤ 𝑙 . 󵄨 󵄨 󵄨 󵄨 𝑙 󵄨 󵄨𝑛 (33) a nonnegative polynomial, condition (32) holds true. There- (1 + 󵄨𝑥−𝑦󵄨 /𝜌 (𝑥)) 󵄨𝑥−𝑦󵄨 󵄨 󵄨 fore, Corollaries 15 and 16 also hold true. |∇𝑉(𝑧)| ≤ 𝐶𝑚(𝑧, 𝑉)3 𝑧∈𝐵(𝑥,2|𝑥−𝑦|) Proof. Since for ,then Lemma 18. Suppose that 𝑉∈𝐵𝑠 for some 𝑠>𝑛,and(1/𝑠) + 󸀠 by using Lemma 8, (1/𝑠 )=1. 𝑘 −1 −1 𝑝 𝑛 𝑚 (𝑧, 𝑉) ≲ (1+|𝑥−𝑧| 𝑚 (𝑥,)) 𝑉 0 𝑚 (𝑥,) 𝑉 (1) 𝐿 (−Δ) and 𝐿 𝑉 are bounded on the space 𝐿 (R ), (34) where 1≤𝑝≤𝑠. 󵄨 󵄨 𝑘 ≲(1+󵄨𝑥−𝑦󵄨 𝑚 (𝑥,) 𝑉 ) 0 𝑚 (𝑥,) 𝑉 . 𝐻 𝑝 𝑛 󵄨 󵄨 (2) R is bounded on the space 𝐿 (R ) for 1<𝑝≤𝑠. ̃𝐻 𝑝 𝑛 󸀠 Therefore, by Corollary 12, (3) R is bounded on the space 𝐿 (R ) for 𝑠 <𝑝<∞. 󵄨 󵄨 󵄨 2 󵄨 R𝐻 =∇2(−Δ)−1(−Δ)(−Δ + 𝑉)−1 (1) 󵄨∇ Γ(𝑥,𝑦)󵄨 Since ,byusing in Lemma 18,weobtainthefollowing. 𝐶 1 ≤ 𝑙 𝑛−2 󵄨 󵄨 𝑙 󵄨 󵄨 Lemma 19. Suppose that 𝑉∈𝐵𝑠 for some 𝑠>𝑛.Then,forany (1 + 󵄨𝑥−𝑦󵄨 /𝜌 (𝑥)) 󵄨𝑥−𝑦󵄨 𝜆>0, |∇𝑉 (𝑧)| 1 󵄨 󵄨 󵄨 󵄨 𝐶 ×(∫ 𝑑𝑧 + ) 󵄨 𝑛 󵄨 𝐻 󵄨 󵄨 󵄩 󵄩 𝑛−1 󵄨 󵄨2 󵄨{𝑥 ∈ R : 󵄨R (𝑓) (𝑥)󵄨 >𝜆}󵄨 ≤ 󵄩𝑓󵄩 1 𝑛 , 𝐵(𝑥,2|𝑥−𝑦|) |𝑥−𝑧| 󵄨𝑥−𝑦󵄨 󵄨 󵄨 󵄨 󵄨 𝜆 𝐿 (R ) 󵄨 󵄨 (37) 1 𝑛 󵄨 󵄨 3𝑘 ∀𝑓 ∈ 𝐿 (R ). 𝐶 (1 + 󵄨𝑥−𝑦󵄨 𝑚 (𝑥,) 𝑉 ) 0 𝑚(𝑥,) 𝑉 3 ≲ 𝑙 󵄨 󵄨 󵄨 󵄨 𝑙 󵄨 󵄨𝑛−2 (1 + 󵄨𝑥−𝑦󵄨 /𝜌 (𝑥)) 󵄨𝑥−𝑦󵄨 󵄨 󵄨 󵄨 󵄨 2.1.SomeLemmasRelatedtoBMOSpaces𝐵𝑀𝑂𝜃(𝜌). In this 1 1 section, we recall some propositions and lemmas for the ×(∫ 𝑑𝑧 + ) BMO spaces BMO𝜃(𝜌) in [17]. 𝑛−1 󵄨 󵄨2 𝐵(𝑥,2|𝑥−𝑦|) |𝑥−𝑧| 󵄨𝑥−𝑦󵄨 Aball𝐵(𝑥, 𝜌(𝑥)) is called critical. In [20], Dziubanski´ and 𝑛 Zienkiewicz gave the following covering lemma on R . 󵄨 󵄨 3𝑘0+3 𝐶𝑙 (1 + 󵄨𝑥−𝑦󵄨 𝑚 (𝑥,) 𝑉 ) ≲ Proposition 20. {𝑥 }∞ 󵄨 󵄨 𝑙 󵄨 󵄨𝑛+1 There exists a sequence of points 𝑘 𝑘=1 in (1 + 󵄨𝑥−𝑦󵄨 /𝜌 (𝑥)) 󵄨𝑥−𝑦󵄨 𝑛 R , such that the family of critical balls 𝑄𝑘 =𝐵(𝑥𝑘,𝜌(𝑥𝑘)), 𝑘≥1 1 1 ,satisfiesthefollowing: ×(∫ 𝑑𝑧 + ) 𝑛−1 󵄨 󵄨2 ⋃ 𝑄 = R𝑛 𝐵(𝑥,2|𝑥−𝑦|) |𝑥−𝑧| 󵄨𝑥−𝑦󵄨 (i) 𝑘 𝑘 ; (ii) there exists 𝑁 = 𝑁(𝜌) such that for every 𝑘∈N, 𝐶𝑙 1 {𝑗 : 4𝑄 ⋂ 4𝑄 =0}≤𝑁̸ ≲ 󵄨 󵄨𝑛 . card 𝑗 𝑘 . 󵄨 󵄨 𝑙−3𝑘0−3 󵄨 󵄨 (1 + 󵄨𝑥−𝑦󵄨 /𝜌 (𝑥)) 󵄨𝑥−𝑦󵄨 Lemma 21. 𝜃>0 1≤𝑝<∞ 𝑏∈𝐵𝑀𝑂(𝜌) (35) Let and .If 𝜃 ,then 1/𝑝 𝜃󸀠 1 𝑝 𝑟 ( ∫ |𝑏 (𝑦)𝐵 −𝑏 | 𝑑𝑦) ≤𝐶[𝑏]𝜃(1+ ) , Furthermore, we obtain the following corollary by using |𝐵 (𝑥,) 𝑟 | 𝐵(𝑥,𝑟) 𝜌 (𝑥) Corollary 12 and Lemma 14. (38) Corollary 16. 𝑉∈𝐵 𝑠>𝑛 |∇𝑉| ∈ 𝐵 𝑛 󸀠 Suppose that 𝑠 for some , 𝑠 for all 𝐵=𝐵(𝑥,𝑟),with𝑥∈R and 𝑟>0,where𝜃 =(1+𝑘0)𝜃 𝑠 >𝑛/2 1 for some 1 and satisfies (32).Thereexistsaconstant and 𝑘0 is the constant appearing in Lemma 8. 𝐶𝑙 >0such that for each 𝑙>0, Lemma 22. 𝑏∈𝐵𝑀𝑂(𝜌) 𝐵=𝐵(𝑥,𝑟) 𝑝≥1 󵄨 󵄨 Let 𝜃 , 0 ,and .Then, 󵄨∇2Γ(𝑥,𝑦+ℎ)−∇2Γ(𝑥,𝑦)󵄨 󵄨 𝑥 𝑥 󵄨 󸀠 1/𝑝 𝑘 𝜃 1 󵄨 󵄨𝑝 2 𝑟 𝛿 ( ∫ 󵄨𝑏(𝑦) −𝑏 󵄨 𝑑𝑦) ≤𝐶[𝑏] 𝑘(1 + ) , 𝐶𝑙 |ℎ| 𝑘 󵄨 𝐵󵄨 𝜃 ≤ , |2 𝐵| 2𝑘𝐵 𝜌(𝑥0) 󵄨 󵄨 𝑙 󵄨 󵄨𝑛+𝛿 (1 + 󵄨𝑥−𝑦󵄨 /𝜌 (𝑥)) 󵄨𝑥−𝑦󵄨 (39) 󵄨 󵄨 󵄨∇2Γ(𝑥+ℎ,𝑦)−∇2Γ(𝑥,𝑦)󵄨 󸀠 󵄨 𝑥 𝑥 󵄨 for all 𝑘∈N,with𝜃 as in (38). 6 Journal of Function Spaces and Applications

󸀠 Given that 𝛼>0, we define the following maximal func- Firstly, by the Holder¨ inequality with 𝑝>𝑠1 and Lemma 21, 1 𝑛 𝑛 tions for 𝑔∈𝐿loc(R ) and 𝑥∈R : 1 󵄨 𝐻 󵄨 ∫ 󵄨(𝑏 − 𝑏 ) R̃ 𝑓(𝑦)󵄨 𝑑𝑦 |𝑄| 󵄨 𝑄 󵄨 1 󵄨 󵄨 𝑄 𝑀𝜌,𝛼𝑔 (𝑥) = sup ∫ 󵄨𝑔󵄨 , |𝐵| 󸀠 𝑥∈𝐵∈B𝜌,𝛼 𝐵 1/𝑝 1/𝑝 1 󵄨 󵄨𝑝󸀠 1 󵄨 𝐻 󵄨𝑝 ≤( ∫ 󵄨𝑏−𝑏 󵄨 𝑑𝑦) ( ∫ 󵄨R̃ 𝑓(𝑦)󵄨 𝑑𝑦) (40) |𝑄| 󵄨 𝑄󵄨 |𝑄| 󵄨 󵄨 ♯ 1 󵄨 󵄨 𝑄 𝑄 𝑀𝜌,𝛼𝑔 (𝑥) = sup ∫ 󵄨𝑔−𝑔𝐵󵄨 , 𝑥∈𝐵∈B |𝐵| 𝐵 1/𝑝 𝜌,𝛼 1 󵄨 󵄨𝑝 󵄨̃𝐻 󵄨 ≤𝐶[𝑏]𝜃( ∫ 󵄨R 𝑓(𝑦)󵄨 𝑑𝑦) . |𝑄| 𝑄 𝑛 where B𝜌,𝛼 ={𝐵(𝑦,𝑟):𝑦∈R ,𝑟≤𝛼𝜌(𝑦)}. (46) 𝑛 1 Also, given a ball 𝑄⊂R ,for𝑔∈𝐿loc(𝑄) and 𝑥∈𝑄,we define If we write 𝑓=𝑓1 +𝑓2 with 𝑓1 =𝑓𝜒2𝑄,then

1 1/𝑝 1/𝑝 󵄨 󵄨 1 󵄨 𝐻 󵄨𝑝 1 󵄨 󵄨𝑝 𝑀𝑄𝑔 (𝑥) = sup 󵄨 󵄨 ∫ 󵄨𝑔󵄨 , 󵄨̃ 󵄨 󵄨 󵄨 󵄨𝐵 ⋂ 𝑄󵄨 󵄨 󵄨 ( ∫ 󵄨R 𝑓1 (𝑦)󵄨 𝑑𝑦) ≤𝐶( ∫ 󵄨𝑓(𝑦)󵄨 𝑑𝑦) 𝑥∈𝐵∈F(𝑄) 󵄨 󵄨 𝐵⋂𝑄 |𝑄| 𝑄 |𝑄| 2𝑄 (41) ♯ 1 󵄨 󵄨 𝑀 𝑔 (𝑥) = ∫ 󵄨𝑔−𝑔 󵄨 , ≤𝐶inf 𝑀𝑝𝑓(𝑦), 𝑄 sup 󵄨 󵄨 󵄨 𝐵󵄨 𝑦∈𝑄 𝑥∈𝐵∈F(𝑄) 󵄨𝐵⋂𝑄󵄨 𝐵⋂𝑄 (47) where F(𝑄) = {𝐵(𝑦, 𝑟) : 𝑦 ∈ 𝑄,𝑟>0}. ̃𝐻 𝑝 𝑛 where we use the fact that R is bounded on 𝐿 (R ) with 𝑠󸀠 <𝑠󸀠 <𝑝<∞ Lemma 23. For 1<𝑝<∞,thereexist𝛽 and 𝛾 such that if 1 . ∞ By Corollary 12 and the Holder¨ inequality, we have {𝑄𝑘}𝑘=1 is a sequence of balls as in Proposition 20,then 󵄨 󵄨 󵄨 𝐻 󵄨 󵄨 2 󵄨 󵄨R̃ 𝑓 (𝑥)󵄨 = 󵄨∫ ∇ Γ (𝑥,) 𝑧 𝑓 (𝑧) 𝑑𝑧󵄨 󵄨 󵄨𝑝 󵄨 ♯ 󵄨𝑝 󵄨 2 󵄨 󵄨 𝑧 󵄨 ∫ 󵄨𝑀 (𝑔)󵄨 ≤𝐶(∫ 󵄨𝑀 (𝑔)󵄨 󵄨 |𝑥0−𝑧|>2𝜌(𝑥0) 󵄨 󵄨 𝜌,𝛽 󵄨 󵄨 𝜌,𝛾 󵄨 (48) R𝑛 R𝑛 ≲𝐼1 (𝑥) +𝐼2 (𝑥) , 𝑝 (42) 󵄨 󵄨 1 󵄨 󵄨 +∑ 󵄨𝑄 󵄨 (󵄨 󵄨 ∫ 󵄨𝑔󵄨)) , 󵄨 𝑘󵄨 󵄨𝑄 󵄨 󵄨 󵄨 where 󵄨 𝑘󵄨 2𝑄𝑘 𝑘 󵄨 󵄨 󵄨𝑓 (𝑧)󵄨 𝐼 (𝑥) = ∫ 󵄨 󵄨 𝑑𝑧, 1 𝑛 1 𝑙 𝑛 for all 𝑔∈𝐿𝑙𝑜𝑐(R ). |𝑥0−𝑧|>2𝜌(𝑥0) (1 + |𝑥−𝑧| /𝜌 (𝑥)) |𝑥−𝑧| 󵄨 󵄨 󵄨𝑓 (𝑧)󵄨 𝐼 (𝑥) = ∫ (49) 3. Proofs of the Main Results 2 𝑙 𝑛−2 |𝑥0−𝑧|>2𝜌(𝑥0) (1 + |𝑥−𝑧| /𝜌 (𝑥)) |𝑥−𝑧| Firstly, in order to prove Theorem 4, we need the following |∇𝑉 (𝑢)| 𝑓∈𝐿1 (R𝑛) 𝑀 × ∫ 𝑑𝑢 𝑑𝑧. lemmas.Asusual,for loc , we denote by 𝑝 the 𝑛−1 𝑝-maximal function which is defined as 𝐵(𝑧,|𝑥−𝑧|/4) |𝑢−𝑧| 𝑥∈𝑄 𝜌(𝑥) ∼ 𝜌(𝑥 ) 1/𝑝 For ,notethat 0 by using Lemma 8.Wealso 1 󵄨 󵄨𝑝 󵄨 󵄨 note that |𝑥 − 𝑧| 0∼ |𝑥 −𝑧|.Then, 𝑀𝑝𝑓 (𝑥) = sup( ∫ 󵄨𝑓(𝑦)󵄨 𝑑𝑦) . (43) 𝑟>0 |𝐵 (𝑥,) 𝑟 | 𝐵(𝑥,𝑟) 󵄨 󵄨 󵄨𝐼1 (𝑥)󵄨 Lemma 24. Suppose that 𝑉∈𝐵𝑠 for some 𝑠≥𝑛, |∇𝑉| ∈𝑠 𝐵 1 2−𝑙𝑘 󵄨 󵄨 (𝑠1 ≥𝑛/2), 𝜌(𝑥)1 ≲𝜌 (𝑥),and𝜌(𝑥).Let ≲1 𝑏∈𝐵𝑀𝑂𝜃(𝜌). 󵄨 󵄨 ≤𝐶∑ 𝑛 ∫ 󵄨𝑓 (𝑧)󵄨 𝑑𝑧 𝑘 𝑘 𝑘+1 Then, there exists a constant 𝐶 such that 𝑘≥1 (2 𝜌(𝑥0)) 2 𝜌(𝑥0)≤|𝑥0−𝑧|<2 𝜌(𝑥0)

1 󵄨 󵄨 2−𝑙𝑘 󵄨 󵄨 󵄨̃𝐻 󵄨 ≤𝐶∑ ∫ 󵄨𝑓 (𝑧)󵄨 𝑑𝑧 ∫ 󵄨R𝑏 𝑓(𝑦)󵄨 𝑑𝑦 ≤𝐶[𝑏]𝜃 inf 𝑀𝑝𝑓(𝑦), (44) 𝑛 󵄨 󵄨 𝑄 󵄨 󵄨 𝑦∈𝑄 𝑘 𝑘+1 | | 𝑄 𝑘≥1 (2 𝜌(𝑥0)) |𝑥0−𝑧|<2 𝜌(𝑥0)

1/𝑝 𝑝 𝑛 󸀠 𝑓∈𝐿 (R ) 𝑝>𝑠 𝑄=𝐵(𝑥,𝜌(𝑥 )) −𝑙𝑘 1 󵄨 󵄨𝑝 for all 𝑙𝑜𝑐 for 1 and every ball 0 0 . ≤𝐶∑2 ( ∫ 󵄨𝑓 (𝑧)󵄨 𝑑𝑧) 󵄨 𝑘 󵄨 󵄨 󵄨 󵄨𝐵(𝑥 ,2 𝜌(𝑥 ))󵄨 𝑘+1 𝑘≥1 󵄨 0 0 󵄨 |𝑥0−𝑧|<2 𝜌(𝑥0) 𝑝 𝑛 ̃𝐻 Proof. Let 𝑓∈𝐿 (R ) and 𝑄=𝐵(𝑥0,𝜌(𝑥0)).WriteR𝑏 𝑓 as ≤𝐶inf 𝑀𝑝𝑓(𝑦). 𝑦∈𝑄 ̃𝐻 ̃𝐻 ̃𝐻 R𝑏 𝑓=(𝑏−𝑏𝑄) R 𝑓−R (𝑓 (𝑏𝑄 −𝑏 )) . (45) (50) Journal of Function Spaces and Applications 7

−𝑙𝑘+𝑘𝑘󸀠 +1 𝑥∈𝑄 0 Since ,then ≲ ∑2 𝜌(𝑥0) inf 𝑀𝑝𝑓(𝑦) 𝑦∈𝑄 𝑘≥1 𝐼2 (𝑥) 󵄨 󵄨 ≲ 𝑀 𝑓(𝑦), 󵄨𝑓 (𝑧)󵄨 inf 𝑝 ≲ ∫ 󵄨 󵄨 𝑦∈𝑄 𝑙 𝑛−2 |𝑥 −𝑧|>2𝜌(𝑥 ) 󵄨 󵄨 󵄨 󵄨 0 0 (1 + 󵄨𝑥0 −𝑧󵄨 /𝜌 0(𝑥 )) 󵄨𝑥0 −𝑧󵄨 (55) |∇𝑉 (𝑢)| wherewechoose𝑙 large enough such that the previous series × ∫ 𝑑𝑢 𝑑𝑧 𝑛−1 𝜌(𝑥 )≲1 𝐵(𝑥 ,4|𝑥 −𝑧|) |𝑢−𝑧| converges and we use the fact that 0 . 0 0 To deal with the second term of (45), we split again 𝑓= −𝑙𝑘 2 󵄨 󵄨 𝑓1 +𝑓2 with 𝑓1 =𝑓𝜒2𝑄. ≲ ∑ ∫ 󵄨𝑓 (𝑧)󵄨 𝑑𝑧 𝑛−2 󵄨 󵄨 𝑘 |𝑥 −𝑧|<2𝑘+1𝜌(𝑥 ) Firstly, using the Holder¨ inequality and the boundedness 𝑘≥1 (2 𝜌(𝑥0)) 0 0 ̃𝐻 𝑝 𝑛 of R on 𝐿 (R ), |∇𝑉 (𝑢)| × ∫ 1 󵄨 𝐻 󵄨 𝑛−1 ∫ 󵄨R̃ ((𝑏 − 𝑏 )𝑓)(𝑦)󵄨 𝑑𝑦 𝑘+3 󵄨 𝑄 1 󵄨 𝐵(𝑥0,2 𝜌(𝑥0)) |𝑢−𝑧| |𝑄| 𝑄

−𝑙𝑘 1/𝑝 2 1 󵄨 󵄨𝑝 1 ≲ ∑ ≤( ∫ 󵄨R̃𝐻 ((𝑏 − 𝑏 )𝑓)(𝑦)󵄨 1 𝑑𝑦) 𝑘 𝑛−2 󵄨 𝑄 1 󵄨 𝑘≥1 (2 𝜌(𝑥0)) |𝑄| 𝑄

1/𝑝 󵄨 󵄨 1 󵄨 󵄨𝑝 1 × ∫ 󵄨𝑓 (𝑧)󵄨 I1 (|∇𝑉| 𝜒𝐵(𝑥 ,2𝑘+3𝜌(𝑥 )))𝑑𝑧. ≤( ∫ 󵄨((𝑏 − 𝑏 )𝑓)(𝑦)󵄨 1 𝑑𝑦) (56) 𝑘 0 0 󵄨 𝑄 󵄨 |𝑥0−𝑧|<2 𝜌(𝑥0) |𝑄| 2𝑄 (51) 1/V 1/𝑝 1 󵄨 󵄨V 1 󵄨 󵄨𝑝 Using the Holder¨ inequality and the boundedness of the ≤( ∫ 󵄨𝑏−𝑏𝑄󵄨 𝑑𝑦) ( ∫ 󵄨𝑓(𝑦)󵄨 𝑑𝑦) 󸀠 |𝑄| 𝑄 |𝑄| 2𝑄 fractional integral I1 with 1/𝑠1 =1/𝑝 +1/𝑛,wehave

󵄨 󵄨 ≤𝐶[𝑏]𝜃 inf 𝑀𝑝𝑓(𝑦), ∫ 󵄨𝑓 (𝑧)󵄨 I1 (|∇𝑉| 𝜒𝐵(𝑥 ,2𝑘+3𝜌(𝑥 )))𝑑𝑧 𝑦∈𝑄 𝑘 0 0 |𝑥0−𝑧|<2 𝜌(𝑥0) 𝑝 /𝑝 + 𝑝 /V =1 𝑝>𝑝 󵄩 󵄩 󵄩 󵄩 where 1 1 , 1,andwehaveusedLemma 21 × 󵄩𝑓𝜒 𝑘+3 󵄩 󵄩I (|∇𝑉| 𝜒 𝑘+3 )󵄩 (52) 󵄩 𝐵(𝑥0,2 𝜌(𝑥0))󵄩𝑝󵄩 1 𝐵(𝑥0,2 𝜌(𝑥0)) 󵄩𝑝󸀠 in the last inequality. 󵄩 󵄩 󵄩 󵄩 For the remaining term, we firstly see the fact that 𝜌(𝑥) ∼ 󵄩 󵄩 󵄩 󵄩 ≤ 󵄩𝑓𝜒 𝐵(𝑥 ,2𝑘+3𝜌(𝑥 ))󵄩 󵄩|∇𝑉| 𝜒𝐵(𝑥 ,2𝑘+3𝜌(𝑥 ))󵄩 . 𝜌(𝑥0) and |𝑥 − 𝑧| 0∼ |𝑥 −𝑧|.Then,wedealwith 󵄩 0 0 󵄩𝑝󵄩 0 0 󵄩𝑠1 󵄨 󵄨 |∇𝑉| ∈ 𝐵 󵄨R̃𝐻 [𝑓 (𝑏 − 𝑏 )] (𝑥)󵄨 Since 𝑠1 ,weobtain 󵄨 2 𝑄 󵄨 󵄩 󵄩 󵄨 󵄨 󵄩|∇𝑉| 𝜒 𝑘+3 󵄩 󵄨 󵄨 󵄩 𝐵(𝑥0,2 𝜌(𝑥0))󵄩𝑠 󵄨 2 󵄨 1 = 󵄨∫ ∇ Γ (𝑥,) 𝑧 [𝑓 (𝑏 − 𝑏 )] (𝑧) 𝑑𝑧󵄨 (57) 󵄨 𝑧 2 𝑄 󵄨 󸀠 󵄨 |𝑥0−𝑧|>2𝜌(𝑥0) 󵄨 𝑘 −𝑛/𝑠1 ≲(2𝜌(𝑥0)) ∫ |∇𝑉 (𝑧)| 𝑑𝑧 𝑘 ̃ ̃ 𝐵(𝑥0,2 𝜌(𝑥0)) ≲ 𝐼1 (𝑥) + 𝐼2 (𝑥) , 𝑛−2−𝑛/𝑠󸀠 1 ≲(2𝑘𝜌(𝑥 )) 1 ∫ ∇𝑉 (𝑧) 𝑑𝑧 where 0 𝑛−2 | | 𝑘 𝐵(𝑥 ,2𝑘𝜌(𝑥 )) 󵄨 󵄨 (2 𝜌(𝑥0)) 0 0 󵄨𝑓 (𝑧) 𝑏−𝑏 󵄨 𝐼̃ (𝑥) = ∫ 󵄨 2 𝑄󵄨 𝑑𝑧, 1 󵄨 󵄨𝑛 󵄨 󵄨 𝑙 𝑛−2−𝑛/𝑠󸀠 𝑘󸀠 |𝑥 −𝑧|>2𝜌(𝑥 ) 󵄨 󵄨 󵄨 󵄨 𝑘 1 𝑘 0 0 0 󵄨𝑥0 −𝑧󵄨 (1 + 󵄨𝑥0 −𝑧󵄨 /𝜌 0(𝑥 )) ≲(2𝜌(𝑥0)) (2 𝜌(𝑥0)𝑚(𝑥0, |∇𝑉|)) 󵄨 󵄨 󵄨𝑓2 (𝑧) (𝑏 −𝑄 𝑏 )󵄨 𝑛−2−𝑛/𝑠󸀠 𝑘󸀠 𝐼̃ (𝑥) = ∫ 𝑘 1 𝑘 0 2 󵄨 󵄨 𝑙󵄨 󵄨𝑛−2 ≲(2𝜌(𝑥0)) (2 ) , |𝑥 −𝑧|>2𝜌(𝑥 ) 󵄨 󵄨 󵄨 󵄨 0 0 (1 + 󵄨𝑥0 −𝑧󵄨 /𝜌 0(𝑥 )) 󵄨𝑥0 −𝑧󵄨 (53) |∇𝑉 (𝑢)| × ∫ 𝑑𝑢 𝑑𝑧. whereweusetheassumptionthat𝑚(𝑥0,|∇𝑉|) ≲ 𝑚(𝑥0,𝑉) 𝑛−1 𝐵(𝑧,|𝑥 −𝑧|/4) |𝑢−𝑧| and (2) in Lemma 10. 0 We also have (58) 󵄩 󵄩 𝑛/𝑝 󵄩 󵄩 𝑘 By the Holder¨ inequality and Lemma 22,wehave 󵄩𝑓𝜒 𝐵(𝑥 ,2𝑘+3𝜌(𝑥 ))󵄩 ≲(2𝜌(𝑥0)) inf 𝑀𝑝𝑓(𝑦). 󵄩 0 0 󵄩𝑝 𝑦∈𝑄 (54) 𝐼̃ (𝑥) 󸀠 1 Therefore, using the fact that 𝑛/𝑝 −1 𝑛/𝑠 =1,weobtain −𝑙𝑘 −𝑙𝑘 2 󵄨 󵄨 2 𝑛−2−𝑛/𝑠󸀠 󵄨 󵄨 𝑘 1 ≲ ∑ 𝑛 ∫ 󵄨[𝑓2 (𝑏−𝑏𝑄)] (𝑧)󵄨 𝑑𝑧 𝐼2 (𝑥) ≲ ∑ (2 𝜌(𝑥0)) (2𝑘𝜌(𝑥 )) 2𝑘𝜌(𝑥 )≤|𝑥 −𝑧|<2𝑘+1𝜌(𝑥 ) 𝑘 𝑛−2 𝑘≥1 0 0 0 0 𝑘≥1 (2 𝜌(𝑥0)) −𝑙𝑘 󸀠 2 󵄨 󵄨 𝑘 𝑘0 𝑘 𝑛/𝑝 󵄨 󵄨 ×(2 ) (2 𝜌(𝑥 )) 𝑀 𝑓(𝑦) ≤𝐶∑ 𝑛 ∫ 󵄨[𝑓2 (𝑏 −𝑄 𝑏 )] (𝑧)󵄨 𝑑𝑧 0 inf 𝑝 𝑘 𝑘+1 𝑦∈𝑄 𝑘≥1 (2 𝜌(𝑥0)) |𝑥0−𝑧|<2 𝜌(𝑥0) 8 Journal of Function Spaces and Applications

1/𝑝 1 󵄨 󵄨𝑝 Also, ≤𝐶∑2−𝑙𝑘( ∫ 󵄨𝑓 (𝑧)󵄨 𝑑𝑧) 󵄨 𝑘 󵄨 󵄨 󵄨 󵄨𝐵(𝑥 ,2 𝜌(𝑥 ))󵄨 |𝑥 −𝑧|<2𝑘+1𝜌(𝑥 ) 𝑘≥1 󵄨 0 0 󵄨 0 0 󵄩 󵄩 󵄩𝑓 (𝑧) (𝑏 − 𝑏 )𝜒 𝑘+3 󵄩 󵄩 2 𝑄 𝐵(𝑥0,2 𝜌(𝑥0))󵄩𝑝̃ 1/𝑝󸀠 1 𝑝󸀠 󵄨 󵄨 𝑘 𝑛/𝑝̃ ×(󵄨 󵄨 ∫ 󵄨𝑏−𝑏𝑄󵄨 𝑑𝑧) ≲(2𝜌(𝑥 )) 󵄨𝐵(𝑥 ,2𝑘𝜌(𝑥 ))󵄨 𝑘+1 0 󵄨 0 0 󵄨 |𝑥0−𝑧|<2 𝜌(𝑥0)

󸀠 1/𝑝 −𝑙𝑘+𝜃 𝑘 1 󵄨 󵄨𝑝 ≤𝐶∑2 𝑘[𝑏]𝜃 inf 𝑀𝑝𝑓(𝑦) 󵄨 󵄨 𝑦∈𝑄 ×(󵄨 󵄨 ∫ 󵄨𝑓 (𝑧)󵄨 𝑑𝑧) 𝑘≥1 󵄨𝐵(𝑥 ,2𝑘𝜌(𝑥 ))󵄨 𝑘 󵄨 0 0 󵄨 |𝑥0−𝑧|<2 𝜌(𝑥0) ≤𝐶 𝑀 𝑓(𝑦), 1/] inf 𝑝 1 󵄨 󵄨] 𝑦∈𝑄 ×( ∫ 󵄨𝑏−𝑏 󵄨 𝑑𝑧) 󵄨 𝑘 󵄨 󵄨 𝑄󵄨 󵄨𝐵(𝑥 ,2 𝜌(𝑥 ))󵄨 |𝑥 −𝑧|<2𝑘𝜌(𝑥 ) (59) 󵄨 0 0 󵄨 0 0

󸀠 𝑘𝜃󸀠 𝑘 𝑛/𝑝̃ where 1/𝑝 + 1/𝑝 =1,andwechoose𝑙 large enough. The ≲2 𝑘(2 𝜌(𝑥0)) [𝑏]𝜃 inf 𝑀𝑝𝑓(𝑦), 𝑦∈𝑄 following estimate is similar to the estimate of 𝐼2(𝑥).We repeat the previous method. (63) Then, 𝑝/𝑝̃ + 𝑝/̃ ] =1 󵄨 󵄨 where . 󵄨𝑓2 (𝑧) (𝑏 −𝑄 𝑏 )󵄨 󸀠 𝐼̃ (𝑥) ≲ ∫ Therefore, using that 𝑛/𝑝 −1 𝑛/𝑠 =1,weobtain 2 𝑙 𝑛−2 |𝑥 −𝑧|>2𝜌(𝑥 ) 󵄨 󵄨 󵄨 󵄨 0 0 (1 + 󵄨𝑥0 −𝑧󵄨 /𝜌 0(𝑥 )) 󵄨𝑥0 −𝑧󵄨 −𝑙𝑘 󸀠 2 𝑘 𝑘 𝑛−2−𝑛/𝑠1 |∇𝑉 (𝑢)| 𝐼2 (𝑥) ≲ ∑ (2 𝜌(𝑥0)) × ∫ 𝑑𝑢 𝑑𝑧 𝑘 𝑛−2 𝑛−1 𝑘≥1 (2 𝜌(𝑥0)) 𝐵(𝑥0,4|𝑥0−𝑧|) |𝑢−𝑧| 𝑘󸀠 +𝜃󸀠 𝑛/𝑝̃ −𝑙𝑘 𝑘 0 𝑘 2 ×(2 ) (2 𝜌(𝑥0)) [𝑏]𝜃 inf 𝑀𝑝𝑓(𝑦) ≲ ∑ 𝑦∈𝑄 𝑘 𝑛−2 (64) 𝑘≥1 (2 𝜌(𝑥0)) 󸀠 󸀠 −𝑙𝑘+𝑘𝑘0+𝑘+𝑘𝜃 ≲ ∑𝑘2 [𝑏]𝜃𝜌(𝑥0) inf 𝑀𝑝𝑓(𝑦) 󵄨 󵄨 𝑦∈𝑄 󵄨 󵄨 𝑘≥1 × ∫ 󵄨𝑓2 (𝑧) (𝑏 −𝑄 𝑏 )󵄨 𝑑𝑧 𝑘+1 |𝑥0−𝑧|<2 𝜌(𝑥0) ≲ [𝑏] inf 𝑀𝑝𝑓(𝑦), |∇𝑉 (𝑢)| 𝑦∈𝑄 × ∫ 𝑛−1 𝐵(𝑥 ,2𝑘+3𝜌(𝑥 )) |𝑢−𝑧| 0 0 wherewechoose𝑙 large enough such that the previous series −𝑙𝑘 2 converges and we use the fact that 𝜌(𝑥0)≲1. ≲ ∑ 𝑘 𝑛−2 Therefore, this completes the proof. 𝑘≥1 (2 𝜌(𝑥0)) Remark 25. Similarly,wecanconcludethattheprevious 󵄨 󵄨 × ∫ 󵄨𝑓2 (𝑧) (𝑏 −𝑄 𝑏 )󵄨 𝑄 2𝑄 𝑘 lemmaalsoholdsifthecriticalball is replaced by . |𝑥0−𝑧|<2 𝜌(𝑥0) Lemma 26. 𝑉∈𝐵 𝑠≥𝑛 |∇𝑉| ∈ 𝐵 Suppose that 𝑠 for some , 𝑠1 × I1 (|∇𝑉| 𝜒𝐵(𝑥 ,2𝑘+3𝜌(𝑥 )))𝑑𝑧. 0 0 (𝑠1 ≥𝑛/2), 𝜌(𝑥)1 ≲𝜌 (𝑥),and𝜌(𝑥).Let ≲1 𝑏∈𝐵𝑀𝑂𝜃(𝜌). (60) Then, there exists a constant 𝐶 such that Using the Holder¨ inequality and the boundedness of the 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 ̃󸀠 ∫ 󵄨∇2Γ (𝑥,) 𝑧 −∇2Γ(𝑦,𝑧)󵄨 󵄨𝑏 (𝑧) −𝑏 󵄨 󵄨𝑓 (𝑧)󵄨 𝑑𝑧 fractional integral I1 with 1/𝑠1 =1/𝑝 +1/𝑛,wehave 󵄨 𝑧 𝑧 󵄨 󵄨 𝐵󵄨 󵄨 󵄨 (2𝐵)𝑐 (65) 󵄨 󵄨 ∫ 󵄨𝑓2 (𝑧) (𝑏 −𝑄 𝑏 )󵄨 I1 (|∇𝑉| 𝜒𝐵(𝑥 ,2𝑘+3𝜌(𝑥 )))𝑑𝑧 ≤𝐶[𝑏]𝜃 inf 𝑀𝑝𝑓(𝑦), 𝑘 0 0 𝑦∈𝑄 |𝑥0−𝑧|<2 𝜌(𝑥0) 󵄩 󵄩 󵄩 󵄩 𝑝 × 󵄩𝑓2 (𝑧) (𝑏 −𝑄 𝑏 )𝜒𝐵(𝑥 ,2𝑘+3𝜌(𝑥 ))󵄩 𝑛 󸀠 󵄩 0 0 󵄩𝑝̃ for all 𝑓∈𝐿𝑙𝑜𝑐(R ) for 𝑝>𝑠1 and 𝑥, 𝑦 ∈ 𝐵 0= 𝐵(𝑥 ,𝑟),with 𝑟<𝛾𝜌(𝑥) 𝛾≥1 󵄩 󵄩 0 ,where . × 󵄩I (|∇𝑉| 𝜒 𝑘+3 )󵄩 󵄩 1 𝐵(𝑥0,2 𝜌(𝑥0)) 󵄩𝑝̃󸀠 Proof. Denote that 𝑄=𝐵(𝑥0,𝛾𝜌(𝑥0)).Notethat𝜌(𝑥) ∼ 0𝜌(𝑥 ) 󵄩 󵄩 󵄩 󵄩 |𝑥 − 𝑧| ∼ |𝑥 −𝑧| ≤ 󵄩𝑓2 (𝑧) (𝑏 −𝑄 𝑏 )𝜒𝐵(𝑥 ,2𝑘+3𝜌(𝑥 ))󵄩 󵄩|∇𝑉| 𝜒𝐵(𝑥 ,2𝑘+3𝜌(𝑥 ))󵄩 . and 0 .Bytheestimate(29), we have 󵄩 0 0 󵄩𝑝̃󵄩 0 0 󵄩𝑠1 (61) 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 ∫ 󵄨∇2Γ (𝑥,) 𝑧 −∇2Γ(𝑦,𝑧)󵄨 󵄨𝑏 (𝑧) −𝑏 󵄨 󵄨𝑓 (𝑧)󵄨 𝑑𝑧 󵄨 𝑧 𝑧 󵄨 󵄨 𝐵󵄨 󵄨 󵄨 |∇𝑉| ∈ 𝐵 (2𝐵)𝑐 Since 𝑠1 ,wehavealreadyobtained 󵄨 󵄨 󵄨 󵄨 󵄩 󵄩 𝑛−2−𝑛/𝑠󸀠 𝑘󸀠 𝛿 󵄨𝑓 (𝑧)󵄨 󵄨𝑏 (𝑧) −𝑏𝐵󵄨 󵄩 󵄩 𝑘 1 𝑘 0 ≲𝑟 ∫ 𝑑𝑧 󵄩|∇𝑉| 𝜒 𝑘+3 󵄩 ≲(2𝜌(𝑥0)) (2 ) . (62) 󵄨 󵄨𝑛+𝛿 󵄩 𝐵(𝑥0,2 𝜌(𝑥0))󵄩𝑠 𝑄\2𝐵 󵄨 󵄨 1 󵄨𝑥0 −𝑧󵄨 Journal of Function Spaces and Applications 9 󵄨 󵄨 󵄨 󵄨 󵄨𝑓 (𝑧)󵄨 󵄨𝑏 (𝑧) −𝑏 󵄨 𝑗0 +𝑟𝛿𝜌(𝑥 )𝑙 ∫ 󵄨 󵄨 󵄨 𝐵󵄨𝑑𝑧 1 −𝑗(𝑛−2+𝛿) 󵄨 𝑗 󵄨1/𝑝̃ 0 𝑛+𝛿+𝑙 ≲ ∑2 𝑗󵄨2 𝐵󵄨 𝑄𝑐 󵄨 󵄨 𝑛−2 󵄨 󵄨 󵄨𝑥0 −𝑧󵄨 𝑟 𝑗=2 󵄨 󵄨 󵄨𝑓 (𝑧) (𝑏 − 𝑏 )󵄨 󵄩 󵄩 𝛿 󵄨 𝑄 󵄨 |∇𝑉 (𝑢)| × [𝑏] 𝑀 𝑓(𝑦)󵄩∇𝑉𝜒 𝑗+2 󵄩 , +𝑟 ∫ ∫ 𝑑𝑢 𝑑𝑧 𝜃inf 𝑝 󵄩 2 𝐵󵄩𝑠 󵄨 󵄨𝑛−2+𝛿 𝑛−1 𝑦∈𝐵 1 𝑄\2𝐵 󵄨 󵄨 𝐵(𝑥 ,|𝑥 −𝑧|/4) |𝑢−𝑧| 󵄨𝑥0 −𝑧󵄨 0 0 (69) 󵄨 󵄨 𝛿 𝑙 󵄨𝑓 (𝑧) (𝑏 −𝑄 𝑏 )󵄨 +𝑟 𝜌(𝑥 ) ∫ 󸀠 󸀠 0 𝑛−2+𝑙+𝛿 1/𝑠 =1/𝑝̃ +1/𝑛 1/𝑝 + 1/] +1/𝑝̃ =1 𝑄𝑐 󵄨 󵄨 where 1 and . 󵄨𝑥0 −𝑧󵄨 |∇𝑉| ∈ 𝐵 Since 𝑠1 ,then |∇𝑉 (𝑢)| × ∫ 𝑑𝑢 𝑑𝑧 󵄩 󵄩 󵄩 󵄩 󵄩∇𝑉𝜒 𝑗+2 󵄩 ≲ 󵄩∇𝑉𝜒 󵄩 𝑛−1 󵄩 2 𝐵󵄩𝑠 󵄩 𝑄󵄩𝑠 𝐵(𝑥0,|𝑥0−𝑧|/4) |𝑢−𝑧| 1 1

󸀠 =𝐼 +𝐼 +𝐼 +𝐼. −𝑛/𝑠1 1 2 3 4 ≲𝜌(𝑥0) ∫ |∇𝑉 (𝑧)| 𝑑𝑧 (70) (66) 𝑄

𝑛/𝑠1−2 For 𝐼1,byusingtheHolder¨ inequality and Lemma 22,wehave ≲𝜌(𝑥0) , 𝑗 0 2−𝑗𝛿 󵄨 󵄨 󵄨 󵄨 𝐼 ≲ ∑ ∫ 󵄨𝑓 (𝑧)󵄨 󵄨𝑏 (𝑧) −𝑏 󵄨 𝑑𝑧 𝑗≤𝑗 𝑚(𝑥 ,|∇𝑉|) ≲ 1 󵄨 𝑗 󵄨 󵄨 󵄨 󵄨 𝐵󵄨 for all 0, where we use the fact that 0 󵄨𝐵(𝑥0,2 𝑟)󵄨 2𝑗𝐵 𝑗=2 󵄨 󵄨 𝑚(𝑥0,𝑉).

󸀠 Therefore, 𝑗0 𝑗 𝜃 −𝑗𝛿 2 𝑟 ≲ ∑2 𝑗[𝑏]𝜃(1 + ) inf𝑀𝑝𝑓(𝑦) 𝑛/𝑝−𝑛+2̃ 𝑗0 𝑟 ̃ 𝜌(𝑥0) 𝑦∈𝐵 −𝑗(𝑛−2+𝛿−𝑛/𝑝) 𝑗=2 (67) 𝐼3 ≲ [𝑏]𝜃inf𝑀𝑝𝑓(𝑦) ∑𝑗2 𝑦∈𝐵 2−𝑛/𝑠1 𝜌(𝑥 ) 𝑗=2 ∞ 0 ≲ ∑2−𝑗𝛿𝑗 𝑏 𝑀 𝑓(𝑦) [ ]𝜃inf 𝑝 2−𝑛/𝑠1 𝑗 𝑦∈𝐵 0 𝑗=2 𝑟 −𝑗(𝑛/−3+𝛿) ≲ [𝑏]𝜃inf𝑀𝑝𝑓(𝑦)𝑟( ) ∑𝑗2 𝑦∈𝐵 𝜌(𝑥0) 𝑗=2 ≲ [𝑏]𝜃inf𝑀𝑝𝑓(𝑦), (71) 𝑦∈𝐵 2−𝑛/𝑠1 𝑗0 𝑟 𝑗 (3−𝑛/𝑠 ) −𝑗𝛿 𝑗0 0 1 𝑗 2 ≥(𝛾𝜌(𝑥))/𝑟 ≲ [𝑏]𝜃inf𝑀𝑝𝑓(𝑦)( ) 2 ∑𝑗2 where 0 is the least integer such that 0 . 𝑦∈𝐵 󸀠 𝜌(𝑥0) 𝑗=2 To deal with 𝐼2,usingLemma 22 and choosing 𝑙>𝜃,we have 𝑙 ∞ ≲ [𝑏]𝜃inf𝑀𝑝𝑓(𝑦), 𝜌(𝑥 ) −𝑗(𝛿+𝑙) 𝑦∈𝐵 0 2 󵄨 󵄨 󵄨 󵄨 𝐼2 ≲ ∑ 󵄨 󵄨 ∫ 󵄨𝑓 (𝑧)󵄨 󵄨𝑏 (𝑧) −𝑏𝐵󵄨 𝑑𝑧 𝑙 󵄨 𝑗 󵄨 𝑗 𝑟 𝑗=𝑗 −1 󵄨𝐵(𝑥0,2 𝑟)󵄨 2 𝐵 0 where we use that 𝑟≤𝛾𝜌(𝑥0)≲1. 󸀠 At last, for 𝐼4 we have, for 𝑗>𝑗0, 𝜌(𝑥 )𝑙 ∞ 2𝑗𝑟 𝜃 ≲ 0 ∑ 2−𝑗(𝛿+𝑙)𝑗[𝑏] (1 + ) 𝑀 𝑓(𝑦) 𝑙 𝜃 inf 𝑝 𝑟 𝜌(𝑥 ) 𝑦∈𝐵 𝐼4 𝑗=𝑗0 0

󸀠 𝑙 ∞ ∞ 𝑙−𝜃 𝜌(𝑥0) −𝑗(𝑛−2+𝛿+𝑙) −𝑗𝛿 𝜌(𝑥0) ≲ ∑ 2 ≲ ∑ 𝑗2 ( ) [𝑏] 𝑀 𝑓(𝑦) 𝑛−2+𝑙 𝑗 𝜃inf 𝑝 𝑟 2 𝑟 𝑦∈𝐵 𝑗=𝑗0−1 𝑗=𝑗0 󵄨 󵄨 󵄨 󵄨 ≲ [𝑏]𝜃inf𝑀𝑝𝑓(𝑦), × ∫ 󵄨𝑓 (𝑧)󵄨 󵄨𝑏 (𝑧) −𝑏𝐵󵄨 I1 (∇𝑉𝜒2𝑗+2𝐵) (𝑧) 𝑑𝑧 𝑦∈𝐵 2𝑗𝐵 (68) 𝜌(𝑥 )𝑙 ∞ 0 −𝑗(𝑛−2+𝛿+𝑙)󵄩 󵄩 𝑗 ≲ ∑ 2 󵄩𝑓𝜒 𝑗 󵄩 where we use the fact that 𝜌(𝑥0)/2 𝑟≤1/𝛾when 𝑗>𝑗0. 𝑛−2+𝑙 󵄩 2 𝐵󵄩𝑝 𝑟 𝑗=𝑗 −1 To deal with 𝐼3,byusingLemma 22 and 𝑗≤𝑗0, 0 𝑗0 󵄩 󵄩 󵄩 󵄩 1 × 󵄩(𝑏 − 𝑏 )𝜒 𝑗 󵄩 󵄩I (∇𝑉𝜒 𝑗+2 )󵄩 𝐼 ≲ ∑2−𝑗(𝑛−2+𝛿) 󵄩 𝐵 2 𝐵󵄩]󵄩 1 2 𝐵 󵄩𝑝̃󸀠 3 𝑟𝑛−2 𝑗=2 𝜃󸀠+𝑛/𝑝̃ 𝜌(𝑥 )𝑙 ∞ (2𝑗𝑟) ≲ 0 ∑ 2−𝑗(𝑛−2+𝛿+𝑙)𝑗 󵄨 󵄨 󵄨 󵄨 󸀠 × ∫ 󵄨𝑓 (𝑧)󵄨 󵄨𝑏 (𝑧) −𝑏 󵄨 I (∇𝑉𝜒 𝑗+2 ) (𝑧) 𝑑𝑧 𝑛−2+𝑙 𝜃 󵄨 󵄨 󵄨 𝐵󵄨 1 2 𝐵 𝑟 𝑗=𝑗 −1 𝜌(𝑥 ) 2𝑗𝐵 0 0

𝑗 󵄩 󵄩 0 × [𝑏]𝜃inf𝑀𝑝𝑓(𝑦)󵄩∇𝑉𝜒2𝑗+2𝐵󵄩 , 1 −𝑗(𝑛−2+𝛿)󵄩 󵄩 𝑦∈𝐵 󵄩 󵄩𝑠1 ≲ ∑2 󵄩𝑓𝜒 𝑗 󵄩 𝑛−2 󵄩 2 𝐵󵄩𝑝 𝑟 𝑗=2 (72)

󵄩 󵄩 󵄩 󵄩 󸀠 󸀠 × 󵄩(𝑏 − 𝑏 )𝜒 𝑗 󵄩 󵄩I (∇𝑉𝜒 𝑗+2 )󵄩 󵄩 𝐵 2 𝐵󵄩]󵄩 1 2 𝐵 󵄩𝑝̃󸀠 where 1/𝑠1 =1/𝑝̃ +1/𝑛and 1/𝑝 + 1/] +1/𝑝̃ =1. 10 Journal of Function Spaces and Applications

󵄩 󵄩𝑞 󵄨 󵄨𝑞 Furthermore, by using Lemma 7, 󵄩R̃𝐻𝑓󵄩 ≤ ∫ 󵄨𝑀 (R̃𝐻𝑓) (𝑥)󵄨 𝑑𝑥 󵄩 𝑏 󵄩 𝑞 󵄨 𝜌,𝛽 𝑏 󵄨 𝐿 R𝑛 󸀠 󵄩 󵄩 𝑗 −𝑛/𝑠1 󵄩∇𝑉𝜒 𝑗+2 󵄩 ≲(2𝑟) ∫ ∇𝑉 𝑧 𝑑𝑧 󵄩 2 𝐵󵄩𝑠 | ( )| 󵄨 ♯ 𝐻 󵄨𝑞 1 𝑗 ≲ ∫ 󵄨𝑀 (R̃ 𝑓) (𝑥)󵄨 𝑑𝑥 2 𝐵 󵄨 𝜌,𝛾 𝑏 󵄨 R𝑛 󸀠 𝑗 −𝑛/𝑠1 =(2𝑟) ∫ |∇𝑉 (𝑧)| 𝑑𝑧 1 󵄨 󵄨 𝑞 𝐵(𝑥 ,(2𝑗𝑟/𝜌(𝑥 ))𝜌(𝑥 )) 󵄨 󵄨 󵄨̃𝐻 󵄨 0 0 0 +𝐶∑ 󵄨𝑄 󵄨 (󵄨 󵄨 ∫ 󵄨R 𝑓 (𝑥)󵄨 𝑑𝑥) 󵄨 𝑘󵄨 󵄨𝑄 󵄨 󵄨 𝑏 󵄨 𝑘 󵄨 𝑘󵄨 2𝑄𝑘 𝑛𝜇−𝑛/𝑠󸀠 1 ≲(2𝑗𝑟) 1 ∫ |∇𝑉 (𝑧)| 𝑑𝑧 𝑛𝜇 󵄨 󵄨𝑞 𝜌(𝑥0) 𝑄 ≤𝐶∫ 󵄨𝑀♯ (R̃𝐻𝑓) (𝑥)󵄨 𝑑𝑥 󵄨 𝜌,𝛾 𝑏 󵄨 R𝑛 𝑛−2 󸀠 𝑗 𝑛𝜇−𝑛/𝑠1 1 𝜌(𝑥0) ≲(2𝑟) ∫ ∇𝑉 (𝑧) 𝑑𝑧 𝑝 󵄨 󵄨𝑞 𝑛𝜇 𝑛−2 | | 󵄨 󵄨 𝜌(𝑥 ) 𝑄 +𝐶[𝑏]𝜃 ∑ ∫ 󵄨𝑀𝑝𝑓 (𝑥)󵄨 𝑑𝑥 0 𝜌(𝑥0) 󵄨 󵄨 𝑘 2𝑄𝑘 𝑛−2 󸀠 𝑛𝜇−𝑛/𝑠 𝜌(𝑥 ) 𝑘󸀠 󵄨 󵄨𝑞 𝑞󵄩 󵄩𝑞 𝑗 1 0 0 󵄨 ♯ ̃𝐻 󵄨 󵄩 󵄩 ≲(2𝑟) 𝑛𝜇 (1+𝜌(𝑥0)𝑚(𝑥0, |∇𝑉|)) ≤𝐶∫ 󵄨𝑀𝜌,𝛾 (R𝑏 𝑓) (𝑥)󵄨 𝑑𝑥[ +𝐶 𝑏] 󵄩𝑓󵄩 𝑞 , 𝑛 󵄨 󵄨 𝜃 𝐿 𝜌(𝑥0) R (75) 𝑛−2 𝑛𝜇−𝑛/𝑠󸀠 𝜌(𝑥 ) ≲(2𝑗𝑟) 1 0 , where we use the finite overlapping property given by 𝜌(𝑥 )𝑛𝜇 𝑞 𝑛 0 Proposition 20 and the boundedness of 𝑀𝑝 in 𝐿 (R ) for (73) 𝑝<𝑞. ♯ ̃𝐻 𝑞 Next, we consider the term ∫ 𝑛 |𝑀𝜌,𝛾(R𝑏 𝑓)(𝑥)| 𝑑𝑥.Our where we use the fact that 𝑚(𝑥0,|∇𝑉|)≲𝑚(𝑥0,𝑉). Consider R ♯ ̃𝐻 goal is to find a pointwise estimate of 𝑀 (R 𝑓)(𝑥).Let𝑥∈ 󸀠 𝜌,𝛾 𝑏 𝑙 𝑗 𝜃 +𝑛/𝑝̃ 𝑛 𝜌(𝑥 ) ∞ (2 𝑟) R and 𝐵=𝐵(𝑥0,𝑟),with𝑟<𝛾𝜌(𝑥0) such that 𝑥∈𝐵.If 𝐼 ≲ 0 ∑ 2−𝑗(𝑛−2+𝛿+𝑙)𝑗 𝑓=𝑓 +𝑓 𝑓 =𝑓𝜒 4 𝑟𝑛−2+𝑙 𝜃󸀠 1 2,with 1 2𝐵,thenwewrite 𝑗=𝑗0−1 𝜌(𝑥0) R̃𝐻𝑓=(𝑏−𝑏) R̃𝐻𝑓−R̃𝐻 (𝑓 (𝑏 − 𝑏 )) 𝑛−2 𝑏 𝐵 1 𝐵 𝑛𝜇−𝑛/𝑠󸀠 𝜌(𝑥 ) (76) 𝑗 1 0 𝐻 × [𝑏]𝜃inf𝑀𝑝𝑓(𝑦)(2 𝑟) 𝑛𝜇 ̃ 𝑦∈𝐵 − R (𝑓2 (𝑏 −𝐵 𝑏 )) . 𝜌(𝑥0) 𝐵 𝜃󸀠+1 Therefore, we need to control the mean oscillation on of 𝑙 ∞ 𝑗 𝜌(𝑥 ) (2 𝑟) each term that we call 𝐽1, 𝐽2,and𝐽3.ByusingtheHolder¨ ≲ 0 ∑ 2−𝑗(𝑛−2+𝛿+𝑙)𝑗 𝑟𝑛−2+𝑙 𝜃󸀠 inequality and Lemma 21,weobtain 𝑗=𝑗0−1 𝜌(𝑥0) 2 󵄨 󵄨 󵄨 ̃𝐻 󵄨 𝑛−2 𝐽1 ≤ ∫ 󵄨(𝑏 −𝐵 𝑏 ) R 𝑓 (𝑥)󵄨 𝑑𝑥 𝑗 𝑛𝜇 𝜌(𝑥0) |𝐵| 𝐵 × [𝑏]𝜃inf𝑀𝑝𝑓(𝑦)(2 𝑟) 𝑛𝜇 𝑦∈𝐵 𝜌(𝑥0) 1/𝑝󸀠 1/𝑝 2 󸀠 1 󵄨 󵄨𝑝 󵄨 󵄨𝑝 󵄨̃𝐻 󵄨 󸀠 ≲( ∫ 󵄨𝑏−𝑏𝐵󵄨 𝑑𝑥) ( ∫ 󵄨R 𝑓 (𝑥)󵄨 𝑑𝑥) 𝑙 𝑗 𝜃 +1 |𝐵| |𝐵| 󵄨 󵄨 𝜌(𝑥 ) ∞ (2 𝑟) 𝐵 𝐵 ≲ 0 ∑ 2−𝑗(𝑛−2+𝛿+𝑙)𝑗 𝑟𝑛−2+𝑙 𝜃󸀠 ≤𝐶[𝑏] 𝑀 (R̃𝐻𝑓) (𝑥) , 𝑗=𝑗0−1 𝜌(𝑥0) 𝜃 𝑝

𝑛−2 (77) 𝑗 𝑛𝜇 𝜌(𝑥0) × [𝑏]𝜃inf𝑀𝑝𝑓(𝑦)(2 𝑟) 𝑛𝜇 since 𝑟/𝜌(𝑥0)<𝛾. 𝑦∈𝐵 𝜌(𝑥 ) 0 To estimate 𝐽2,let1<𝑝<𝑝̃ .Then, 𝑛−2+𝑙−𝜃󸀠−1−𝑛𝜇 𝜌(𝑥 ) 2 󵄨̃𝐻 󵄨 0 𝐽2 ≤ ∫ 󵄨R [(𝑏 −𝐵 𝑏 )𝑓1] (𝑥)󵄨 𝑑𝑥 ≲ [𝑏]𝜃inf𝑀𝑝𝑓(𝑦)( ) 𝐵 󵄨 󵄨 𝑦∈𝐵 𝑟 | | 𝐵 1/𝑝̃ ∞ 1 󵄨 󵄨𝑝̃ 󸀠 󵄨̃𝐻 󵄨 ×𝜌(𝑥 ) ∑ 2−𝑗(𝛿+𝑛−2+𝑙−𝑛𝜇−𝜃 −1)𝑗 ≲( ∫ 󵄨R [(𝑏 −𝐵 𝑏 )𝑓1] (𝑥)󵄨 𝑑𝑥) 0 |𝐵| 𝐵 𝑗=𝑗0−1 1/𝑠̃ 1 𝐻󵄨 󵄨𝑠̃ (78) ≲ [𝑏]𝜃inf𝑀𝑝𝑓(𝑦), ≲( ∈ R𝑏 󵄨(𝑏 −𝐵 𝑏 )𝑓1 (𝑥)󵄨 𝑑𝑥) 𝑦∈𝐵 |𝐵|

(74) 1/V 1/𝑝 1 󵄨 󵄨V 1 󵄨 󵄨𝑝 𝑙 ≲( ∫ 󵄨𝑏−𝑏𝐵󵄨 𝑑𝑥) ( ∫ 󵄨𝑓 (𝑥)󵄨 𝑑𝑥) wherewechoose large enough such that the previous series |𝐵| 2𝐵 |𝐵| 2𝐵 converges and we use the fact that 𝜌(𝑥0)≲1. ≲ [𝑏] 𝑀 (𝑓) (𝑥) , 𝑝 𝑛 𝜃 𝑝 Proof of Theorem 4. We start with a function 𝑓∈𝐿 (R ) for 󸀠 ̃ ̃ 𝑠1 <𝑝<∞.ByLemmas24 and 26 and Remark 25,wehave where V =𝑝𝑝/(𝑝 − 𝑝). Journal of Function Spaces and Applications 11

For 𝐽3,byLemma 26,weobtain When we consider the term 𝐴2(𝑥),wenotethat𝜌(𝑥𝑗)> 𝑟 ≥𝜌(𝑥)/4 1 󵄨 𝑗 𝑗 . Consider 𝐽 ≤𝐶 ∬ 󵄨R̃𝐻 (𝑓 (𝑏 − 𝑏 )) (𝑢) 󵄩 󵄩 3 2 󵄨 2 𝐵 󵄩 𝐻 󵄩 |𝐵| 𝐵 󵄩(𝑏 (𝑥) −𝑏𝐵 ) R 𝑎𝑗 (𝑥) 𝜒(8𝐵 )𝑐 (𝑥)󵄩 󵄩 𝑗 𝑗 󵄩𝐿1(R𝑛) 𝐻 󵄨 −(R̃ 𝑓 (𝑏 − 𝑏 )) (𝑦)󵄨 𝑑𝑢 𝑑𝑦 2 𝐵 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 ≲ ∫ 󵄨𝑎 (𝑦)󵄨 𝑑𝑦 {∫ 󵄨∇2Γ(𝑥,𝑦)󵄨 󵄨𝑏 (𝑥) −𝑏 󵄨 𝑑𝑥} 󵄨 𝑗 󵄨 󵄨 󵄨 󵄨 𝐵𝑗 󵄨 1 󵄨 󵄨 𝐵 |𝑥−𝑥 |≥8𝑟 󵄨 󵄨 ≤𝐶 ∬ ∫ 󵄨∇2Γ (𝑢,) 𝑧 −∇2Γ(𝑦,𝑧)󵄨 (79) 𝑗 𝑗 𝑗 2 󵄨 󵄨 |𝐵| 𝐵 (2𝐵)𝑐 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 ≲ ∫ 󵄨𝑎𝑗 (𝑦)󵄨 𝑑𝑦 󵄨 󵄨 󵄨 󵄨 𝐵 × 󵄨𝑏 (𝑧) −𝑏𝐵󵄨 󵄨𝑓 (𝑧)󵄨 𝑑𝑧 𝑑𝑢 𝑑𝑦 𝑗

≤𝐶[𝑏]𝜃𝑀𝑝𝑓 (𝑥) . 𝐶 1 ×{∫ 𝑙 󵄨 󵄨 𝑙 󵄨 󵄨𝑛−2 |𝑥−𝑥 |≥8𝑟 󵄨 󵄨 󵄨𝑥−𝑦󵄨 Therefore, we have proved that 𝑗 𝑗 (1 + 󵄨𝑥−𝑦󵄨 /𝜌 (𝑥)) 󵄨 󵄨 󵄨 󵄨 󵄨𝑀♯ (R̃𝐻𝑓) (𝑥)󵄨 ≤𝐶[𝑏] (𝑀 R̃𝐻𝑓 (𝑥) +𝑀 𝑓 (𝑥)). |∇𝑉 (𝑧)| 1 󵄨 𝜌,𝛾 𝑏 󵄨 𝜃 𝑝 𝑝 ×(∫ 𝑑𝑧 + ) 𝑛−1 󵄨 󵄨2 (80) 𝐵(𝑥,2|𝑥−𝑦|) |𝑥−𝑧| 󵄨𝑥−𝑦󵄨 Then, we have obtained the desired result. 󵄨 󵄨 󵄨 󵄨 × 󵄨𝑏 (𝑥) −𝑏𝐵 󵄨 𝑑𝑥} 1 𝑛 󵄨 𝑗 󵄨 Proof of Theorem 6. For 𝑓∈𝐻𝐿(R ),wecanwrite𝑓= ∞ ∞ ∑𝑗=−∞ 𝜆𝑗𝑎𝑗,whereeach𝑎𝑗 is a (1, 𝑞)𝜌 atom and ∑𝑗=−∞ |𝜆𝑗|≤ 󵄨 󵄨 2‖𝑓‖𝐻1 .Supposethatsupp𝑎𝑗 ⊆𝐵𝑗 =𝐵(𝑥𝑗,𝑟𝑗) with 𝑟𝑗 < ≲ ∫ 󵄨𝑎 (𝑦)󵄨 𝑑𝑦 {𝐼 +𝐼}. 𝐿 󵄨 𝑗 󵄨 1 2 𝐵 𝜌(𝑥𝑗).Write 𝑗 (83) ∞ 𝐻 𝐻 R 𝑓 (𝑥) = ∑ 𝜆 (𝑏 (𝑥) −𝑏 ) R 𝑎 (𝑥) 𝜒 (𝑥) Note that |𝑥−𝑥𝑗|∼|𝑥−𝑦|and 𝑏 𝑗 𝐵𝑗 𝑗 8𝐵𝑗 𝑗=−∞ 󵄨 󵄨 1/(𝑘 +1) 󵄨 󵄨 󵄨𝑥−𝑥󵄨 0 󵄨𝑥−𝑦󵄨 󵄨 𝑗󵄨 |𝑥−𝑥𝑗| 𝐻 (1+ )≥𝐶(1+ ) ≥ 𝐶(1+ ) . + ∑ 𝜆 (𝑏 (𝑥) −𝑏 ) R 𝑎 (𝑥) 𝜒 𝑐 (𝑥) 𝜌 (𝑥) 𝜌 (𝑥) 𝑗 𝐵𝑗 𝑗 (8𝐵𝑗) 𝜌(𝑥𝑗) 𝑗:𝑟 ≥𝜌(𝑥 )/4 𝑗 𝑗 (84) 𝐻 + ∑ 𝜆 (𝑏 (𝑥) −𝑏 ) R 𝑎 (𝑥) 𝜒 𝑐 (𝑥) Then, by Lemma 22, 𝑗 𝐵𝑗 𝑗 (8𝐵𝑗) ∞ 𝑗:𝑟𝑗<𝜌(𝑥𝑗)/4 1 𝐼 ≲ ∑ ∫ 2 𝑙/(𝑘 +1) 2𝑘+3𝑟 ≤|𝑥−𝑥 |<2𝑘+4𝑟 󵄨 󵄨 −1 0 ∞ 𝑘=1 𝑗 𝑗 𝑗 (1 + 󵄨𝑥−𝑥󵄨 𝜌(𝑥 ) ) − R𝐻 ( ∑ 𝜆 (𝑏 − 𝑏 )𝑎 ) (𝑥) 󵄨 𝑗󵄨 𝑗 𝑗 𝐵𝑗 𝑗 𝑗=−∞ 󵄨 󵄨 1 󵄨 󵄨 × 󵄨 󵄨𝑛 󵄨𝑏 (𝑥) −𝑏𝐵 󵄨 𝑑𝑥 󵄨𝑥−𝑥󵄨 󵄨 𝑗 󵄨 =𝐴1 (𝑥) +𝐴2 (𝑥) +𝐴3 (𝑥) +𝐴4 (𝑥) . 󵄨 𝑗󵄨 (81) ∞ −(𝑘+1)𝑙/(𝑘 +1) 1 𝑞 𝑞 ≲ ∑2 0 󵄨 󵄨 Using the Holder¨ inequality, the (𝐿 ,𝐿 ) boundedness of 󵄨𝐵(𝑥,2𝑘+3𝑟 )󵄨 𝐻 𝑘=1 󵄨 𝑗 󵄨 R with 1<𝑞<𝑠,andLemma 21, 󵄨 󵄨 󵄩 󵄩 󵄨 󵄨 󵄩 𝐻 󵄩 × ∫ 󵄨𝑏−𝑏𝐵 󵄨 𝑑𝑥 󵄩(𝑏 (𝑥) −𝑏𝐵) R 𝑎𝑗 (𝑥) 𝜒8𝐵 (𝑥)󵄩 |𝑥−𝑥 |<2𝑘+4𝑟 󵄨 𝑗 󵄨 󵄩 𝑗 󵄩𝐿1(R𝑛) 𝑗 𝑗

󸀠 ∞ 1/𝑞 −(𝑘+1)𝑙/(𝑘 +1) 1 󵄨 󵄨𝑞󸀠 󵄩 𝐻 󵄩 ≲ ∑2 0 󵄨 󵄨 ≤(∫ 󵄨𝑏 (𝑥) −𝑏 󵄨 𝑑𝑥) 󵄩R 𝑎 󵄩 󵄨 𝑘+4 󵄨 󵄨 𝐵󵄨 󵄩 𝑗󵄩𝐿𝑞 𝑘=1 󵄨𝐵(𝑥,2 𝑟𝑗)󵄨 8𝐵𝑗 󵄨 󵄨 󵄨 󵄨 1/𝑞󸀠 󵄨 󵄨 󸀠 󵄩 󵄩 × ∫ 󵄨𝑏−𝑏𝐵 󵄨 𝑑𝑥 󵄨 󵄨𝑞 󵄩 󵄩 𝑘+4 󵄨 𝑗 󵄨 ≤(∫ 󵄨𝑏 (𝑥) −𝑏 󵄨 𝑑𝑥) 󵄩𝑎 󵄩 (82) |𝑥−𝑥𝑗|<2 𝑟𝑗 󵄨 𝐵󵄨 󵄩 𝑗󵄩𝐿𝑞 8𝐵𝑗 (𝑘 +1)𝜃 ∞ 𝑘+4 0 󸀠 2 𝑟𝑗 1/𝑞 −(𝑘+1)𝑙/(𝑘0+1) 1 󸀠 ≲ ∑2 [𝑏]𝜃𝑘(1 + ) 󵄨 󵄨𝑞 𝜌(𝑥 ) ≲(󵄨 󵄨 ∫ 󵄨𝑏 (𝑥) −𝑏𝐵󵄨 𝑑𝑥) 𝑘=1 𝑗 󵄨𝐵 󵄨 8𝐵 󵄨 𝑗󵄨 𝑗 ≤ C[𝑏]𝜃, ≲𝐶[𝑏] , 𝜃 (85) since 𝑟𝑗 <𝜌(𝑥𝑗). where we choose 𝑙 large enough. 12 Journal of Function Spaces and Applications

Similarly, For 𝐴3, by using the vanishing condition of 𝑎𝑗 and Lemma 14,then 𝐼 1 󵄩 󵄩 󵄩 𝐻 󵄩 ∞ 󵄩(𝑏 (𝑥) −𝑏𝐵 ) R 𝑎𝑗 (𝑥) 𝜒(8𝐵 )𝑐 (𝑥)󵄩 1 󵄩 𝑗 𝑗 󵄩𝐿1(R𝑛) ≲ ∑ ∫ 𝑙/(𝑘 +1) 2𝑘+3𝑟 ≤|𝑥−𝑥 |<2𝑘+4𝑟 󵄨 󵄨 −1 0 𝑘=1 𝑗 𝑗 𝑗 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 (1 + 󵄨𝑥−𝑥𝑗󵄨 𝜌(𝑥𝑗) ) 󵄨 󵄨 󵄨 2 2 󵄨 󵄨 󵄨 ≲ ∫ 󵄨𝑎𝑗 (𝑦)󵄨 𝑑𝑦 {∫ 󵄨∇ Γ(𝑥,𝑦)−∇ Γ(𝑥,𝑥𝑗)󵄨 𝐵 |𝑥−𝑥 |≥8𝑟 1 𝑗 𝑗 𝑗 × I (∇𝑉𝜒 𝑘+4 ) 󵄨 󵄨𝑛−2 1 2 𝐵𝑗 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨𝑥−𝑥𝑗󵄨 × 󵄨𝑏 (𝑥) −𝑏𝐵 󵄨 𝑑𝑥} 󵄨 󵄨 󵄨 𝑗 󵄨 󵄨 󵄨 󵄨 󵄨 × (𝑥) 󵄨𝑏 (𝑥) −𝑏𝐵 󵄨 𝑑𝑥 󵄨 󵄨 󵄨 𝑗 󵄨 󵄨 󵄨 ≲ ∫ 󵄨𝑎𝑗 (𝑦)󵄨 𝑑𝑦 ∞ 𝐵𝑗 −(𝑘+1)𝑙/(𝑘 +1) 1 ≲ ∑2 0 𝑛−2 𝑘+3 󵄨 󵄨𝛿 𝑘=1 (2 𝑟𝑗) { 𝐶 󵄨𝑦−𝑥󵄨 × ∫ 𝑙 󵄨 𝑗󵄨 { 󵄨 󵄨 𝑙 󵄨 󵄨𝑛−2+𝛿 󵄨 󵄨 |𝑥−𝑥 |≥8𝑟 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 { 𝑗 𝑗 (1 + 󵄨𝑥−𝑦󵄨 /𝜌 (𝑥)) 󵄨𝑥−𝑦󵄨 × ∫ I1 (∇𝑉𝜒2𝑘+4𝐵 ) (𝑥) 󵄨𝑏−𝑏𝐵 󵄨 𝑑𝑥 |𝑥−𝑥 |<2𝑘+4𝑟 𝑗 󵄨 𝑗 󵄨 𝑗 𝑗 |∇𝑉 (𝑧)| 1 ∞ ×(∫ 𝑑𝑧 + ) 2 𝑛−1 󵄨 󵄨2 −(𝑘+1)𝑙/(𝑘0+1) 𝑘+3 𝐵(𝑥,2|𝑥−𝑦|) |𝑥−𝑧| 󵄨𝑥−𝑦󵄨 ≲ ∑2 (2 𝑟𝑗) 󵄨 󵄨 𝑘=1 󵄨 󵄨 } 1/𝑞 × 󵄨𝑏 (𝑥) −𝑏 󵄨 𝑑𝑥 󵄨 󵄨𝑞 󵄨 𝐵𝑗 󵄨 } 1 󵄨 󵄨 󵄨 󵄨 ×( 𝑛 ∫ 󵄨𝑏−𝑏𝐵 󵄨 𝑑𝑥) } 𝑘+3 |𝑥−𝑥 |<2𝑘+4𝑟 󵄨 𝑗 󵄨 (2 𝑟𝑗) 𝑗 𝑗 󵄨 󵄨 = ∫ 󵄨𝑎 (𝑦)󵄨 𝑑𝑦 {𝐼̃ + 𝐼̃ }. −𝑛/𝑞󸀠 󵄩 󵄩 󵄨 𝑗 󵄨 1 2 𝑘+3 󵄩 󵄩 𝐵𝑗 ×(2 𝑟𝑗) 󵄩I1 (∇𝑉𝜒2𝑘+4𝐵 )󵄩 󵄩 𝑗 󵄩𝑞󸀠 (88) ∞ 2 −(𝑘+1)𝑙/(𝑘0+1) 𝑘+3 First of all, we need to obtain the following new estimate: ≲ ∑2 (2 𝑟𝑗) 𝑘=1 󵄩 󵄩 󵄩 󵄩 󵄩|∇𝑉| 𝜒 𝑘+3 󵄩 󵄩 𝐵(𝑥𝑗,2 𝑟𝑗)󵄩 1/𝑞 󵄩 󵄩𝑠1 1 󵄨 󵄨𝑞 ×( ∫ 󵄨𝑏−𝑏 󵄨 𝑑𝑥) −𝑛/𝑠󸀠 𝑛 󵄨 𝐵 󵄨 𝑘 1 𝑘+3 𝑘+4 󵄨 𝑗 󵄨 (2 𝑟 ) |𝑥−𝑥𝑗|<2 𝑟𝑗 ≲(2𝑟𝑗) ∫ |∇𝑉 (𝑧)| 𝑑𝑧 𝑗 𝑘 𝐵(𝑥𝑗,2 𝑟𝑗) −𝑛/𝑞󸀠 󵄩 󵄩 𝑘+3 󵄩 󵄩 𝑛−2−𝑛/𝑠󸀠 ×(2 𝑟 ) 󵄩∇𝑉𝜒 𝑘+4 󵄩 𝑘 1 1 𝑗 󵄩 2 𝐵𝑗 󵄩 ≲(2𝑟 ) ∫ |∇𝑉 (𝑧)| 𝑑𝑧 󵄩 󵄩𝑠1 𝑗 𝑛−2 (89) 𝑘 𝐵(𝑥 ,2𝑘𝑟 ) (2 𝑟𝑗) 𝑗 𝑗 (𝑘 +1)𝜃 ∞ 𝑘+4 0 2 𝑟𝑗 −(𝑘+1)𝑙/(𝑘0+1) 𝑛−2−𝑛/𝑠󸀠 𝑘󸀠 ≲ ∑2 [𝑏]𝜃𝑘(1 + ) 𝑘 1 𝑘 0 ≲(2𝑟𝑗) (1 + 2 𝑟𝑗𝑚(𝑥𝑗, |∇𝑉|)) 𝑘=1 𝜌(𝑥𝑗)

󸀠 󸀠 󸀠 󸀠 󸀠 𝑘 𝑛−2−𝑛/𝑠1 𝑘 𝑘0 𝑘 𝑛−𝑛/𝑠1−𝑛/𝑞 𝑘 𝑘0 ≲(2𝑟𝑗) (1 + 2 𝑟𝑗𝑚(𝑥𝑗,𝑉)) , ×(2 𝜌(𝑥𝑗)) (2 )

(𝑘0+1)𝜃 whereweusetheassumptionthat𝑚(𝑥0,|∇𝑉|) ≲ 𝑚(𝑥0,𝑉) ∞ 2𝑘+4𝑟 −(𝑘+1)𝑙/(𝑘 +1) 𝑗 (2) ≲ ∑2 0 [𝑏] 𝑘(1 + ) and in Lemma 10. Consider 𝜃 𝜌(𝑥 ) 𝑘=1 𝑗 ̃ 𝐼1 𝑘󸀠 𝑘 𝑘 0 󵄨 󵄨 ×(2 𝜌(𝑥𝑗)) (2 ) 󵄨 󵄨 ≲ ∫ 󵄨𝑎𝑗 (𝑦)󵄨 𝑑𝑦 𝐵𝑗 ≤𝐶[𝑏]𝜃, ∞ (86) I1 (∇𝑉𝜒2𝑘+4𝐵 ) (𝑥) × ∑ ∫ 𝑗 𝑙 𝑘+3 𝑘+4 󵄨 󵄨 −1 𝑙/(𝑙0+1) where we choose large enough and we use the fact that 2 𝑟𝑗≤|𝑥−𝑥𝑗|<2 𝑟𝑗 󵄨 󵄨 𝑘=1 (1 + 󵄨𝑥−𝑥𝑗󵄨 𝜌(𝑥𝑗) ) 𝜌(𝑥𝑗)≲1. 𝜌(𝑥 )>𝑟 ≥𝜌(𝑥)/4 󵄨 󵄨𝛿 󵄨 󵄨 Therefore, if 𝑗 𝑗 𝑗 ,then 󵄨𝑥 −𝑦󵄨 󵄨𝑏 (𝑥) −𝑏 󵄨 󵄨 𝑗 󵄨 󵄨 𝐵𝑗 󵄨 󵄩 󵄩 × 󵄨 󵄨𝑑𝑥 󵄩 𝐻 󵄩 󵄨 󵄨𝑛−2󵄨 󵄨𝛿 󵄩 𝑐 󵄩 󵄨 󵄨 󵄨 󵄨 󵄩(𝑏 (𝑥) −𝑏𝐵 ) R 𝑎𝑗 (𝑥) 𝜒(8𝐵 ) (𝑥)󵄩 ≤𝐶[𝑏]𝜃. (87) 󵄨𝑥−𝑥󵄨 󵄨𝑥−𝑥󵄨 󵄩 𝑗 𝑗 󵄩𝐿1(R𝑛) 󵄨 𝑗󵄨 󵄨 𝑗󵄨 Journal of Function Spaces and Applications 13

2−𝑛/𝑞󸀠 ∞ 󵄨 󵄨 ∞ (2𝑘+3𝑟 ) −(𝑘+3)𝛿 1 󵄨 󵄨 −(𝑘+3)𝛿 𝑗 ≲ ∑2 󵄨 󵄨 ∫ 󵄨𝑏−𝑏𝐵 󵄨 𝑑𝑥 ≲ ∑2 󵄨𝐵(𝑥,2𝑘+3𝑟 )󵄨 |𝑥−𝑥 |<2𝑘+4𝑟 󵄨 𝑗 󵄨 −1 𝑙/(𝑙0+1) 𝑘=1 󵄨 𝑗 󵄨 𝑗 𝑗 𝑘=1 𝑘 (1 + 2 𝑟𝑗𝜌(𝑥𝑗) ) ∞ 󵄨 󵄨 −(𝑘+3)𝛿 1 󵄨 󵄨 1/𝑞 ≲ ∑2 󵄨 󵄨 ∫ 󵄨𝑏−𝑏𝐵 󵄨 𝑑𝑥 󵄨 󵄨𝑞 󵄨𝐵(𝑥,2𝑘+4𝑟 )󵄨 |𝑥−𝑥 |<2𝑘+4𝑟 󵄨 𝑗 󵄨 1 󵄨 󵄨 𝑘=1 󵄨 𝑗 󵄨 𝑗 𝑗 ×(󵄨 󵄨 ∫ 󵄨𝑏−𝑏𝐵 󵄨 𝑑𝑥) 󵄨 𝑘+3 󵄨 𝑘+4 󵄨 𝑗 󵄨 󵄨𝐵(𝑥,2 𝑟𝑗)󵄨 |𝑥−𝑥𝑗|<2 𝑟𝑗 (𝑘 +1)𝜃 󵄨 󵄨 ∞ 2𝑘+4𝑟 0 󵄩 󵄩 −(𝑘+3)𝛿 𝑗 󵄩 󵄩 ≲ ∑2 [𝑏]𝜃𝑘(1 + ) × 󵄩I1 (∇𝑉𝜒2𝑘+4𝐵 )󵄩 𝜌(𝑥 ) 󵄩 𝑗 󵄩𝑞󸀠 𝑘=1 𝑗

󸀠 ∞ 𝑘+3 2−𝑛/𝑞 −(𝑘+3)𝛿+𝑘(𝑘 +1)𝜃 ∞ (2 𝑟 ) 0 −(𝑘+3)𝛿 𝑗 ≲ ∑2 [𝑏]𝜃𝑘 ≲ ∑2 𝑘=1 −1 𝑙/(𝑙0+1) 𝑘=1 𝑘 (1 + 2 𝑟𝑗𝜌(𝑥𝑗) ) ≲ [𝑏]𝜃, 1/𝑞 󵄨 󵄨𝑞 (91) 1 󵄨 󵄨 ×(󵄨 󵄨 ∫ 󵄨𝑏−𝑏𝐵 󵄨 𝑑𝑥) 󵄨 𝑘+3 󵄨 𝑘+4 󵄨 𝑗 󵄨 󵄨𝐵(𝑥,2 𝑟𝑗)󵄨 |𝑥−𝑥𝑗|<2 𝑟𝑗 󵄨 󵄨 whereweusethefactthat𝛿>(𝑘0 +1)𝜃. 󵄩 󵄩 𝑟 ≤𝜌(𝑥)/4 󵄩 󵄩 Therefore, if 𝑗 𝑗 ,then × 󵄩∇𝑉𝜒 𝑘+4 󵄩 󵄩 2 𝐵𝑗 󵄩 󵄩 󵄩𝑠1 󵄩 𝐻 󵄩 󵄩(𝑏 (𝑥) −𝑏 ) R 𝑎 (𝑥) 𝜒 𝑐 (𝑥)󵄩 ≤𝐶[𝑏] . (𝑘0+1)𝜃 󵄩 𝐵 𝑗 (8𝐵 ) 󵄩 𝜃 (92) ∞ 2𝑘+4𝑟 󵄩 𝑗 𝑗 󵄩𝐿1(R𝑛) ≲ ∑2−(𝑘+3)𝛿[𝑏] 𝑘(1 + 𝑗 ) 𝜃 𝜌(𝑥 ) 𝑘=1 𝑗 Thus, we have 𝑛−2−𝑛/𝑠󸀠 +2−𝑛/𝑞󸀠 𝑘󸀠 −𝑙/(𝑙 +1) 𝑘 1 𝑘 0 0 󵄨 𝜆 󵄨 𝐶 ×(2 𝑟𝑗) (1 + 2 𝑟𝑗𝑚(𝑥𝑗,𝑉)) 󵄨 𝑛 󵄨 󵄨 󵄨 󵄩 󵄩 󵄨{𝑥 ∈ R : 󵄨𝐴𝑖 (𝑥)󵄨 > }󵄨 ≤ 󵄩𝐴𝑖 (𝑥)󵄩 1 󵄨 4 󵄨 𝜆 𝐿 ∞ −(𝑘+3)𝛿+𝑘(𝑘0+1)𝜃 𝑘 ∞ ≲ ∑2 [𝑏]𝜃𝑘(2 𝑟𝑗) 𝐶[𝑏] 󵄨 󵄨 𝜃 󵄨 󵄨 𝑘=1 ≤ ∑ 󵄨𝜆𝑗󵄨 ,𝑖=1,2,3. 𝜆 𝑗=−∞ 󸀠 𝑘 𝑘0−𝑙/(𝑙0+1)+1 𝑘 −1 ×(1+2 𝑟𝑗𝑚(𝑥𝑗,𝑉)) (1+2 𝑟𝑗𝑚(𝑥𝑗,𝑉)) (93)

∞ −1 −(𝑘+3)𝛿+𝑘(𝑘0+1)𝜃 𝑘 𝑘 Note that ≲ ∑2 [𝑏]𝜃𝑘(2 𝑟𝑗)(2 𝑟𝑗𝑚(𝑥𝑗,𝑉)) 𝑘=1 1/𝑞󸀠 󵄩 󵄩 󵄨 󵄨𝑞󸀠 󵄩 󵄩 ∞ 󵄩(𝑏 − 𝑏 )𝑎 󵄩 ≤(∫ 󵄨𝑏 (𝑥) −𝑏 󵄨 𝑑𝑥) 󵄩𝑎 󵄩 󵄩 𝐵𝑗 𝑗󵄩 1 󵄨 𝐵󵄨 󵄩 𝑗󵄩𝐿𝑞 −(𝑘+3)𝛿+𝑘(𝑘0+1)𝜃 󵄩 󵄩𝐿 𝐵 = ∑2 [𝑏]𝜃𝑘𝜌 (𝑥𝑗) 𝑗 𝑘=1 1/𝑞󸀠 1 󸀠 ≲ [𝑏] , 󵄨 󵄨𝑞 𝜃 ≤( ∫ 󵄨𝑏 (𝑥) −𝑏𝐵󵄨 𝑑𝑥) 𝜇(𝐵 ) 𝐵 (90) 𝑗 𝑗 (94) 󸀠 𝑟 𝜃 𝛿>(𝑘 +1)𝜃 𝜌(𝑥 )≲1 𝑗 whereweusethefactthat 0 , 𝑗 ,andwe ≤𝐶[𝑏]𝜃(1 + ) 󸀠 𝜌(𝑥𝑗) choose 𝑙>(𝑘0 + 1)/(𝑙0 +1). Secondly, ≤𝐶[𝑏]𝜃,

𝐼̃ 2 where 𝑟𝑗 <𝜌(𝑥𝑗). 𝐻 󵄨 󵄨 By the weak (1, 1) boundedness of R (cf. Lemma 19), we ≲ ∫ 󵄨𝑎 (𝑦)󵄨 𝑑𝑦 󵄨 𝑗 󵄨 get 𝐵𝑗 ∞ 󵄩 󵄩 𝐶 󵄨 𝜆 󵄨 𝐶󵄩 ∞ 󵄩 × ∑ ∫ 𝑙 󵄨 𝑛 󵄨 󵄨 󵄨 󵄩 󵄩 𝑙/(𝑙 +1) 󵄨{𝑥 ∈ R : 󵄨𝐴4 (𝑥)󵄨 > }󵄨 ≤ 󵄩 ∑ 𝜆𝑗 (𝑏 − 𝑏𝑗 (𝑥 )) 𝑎𝑗󵄩 2𝑘+3𝑟 ≤|𝑥−𝑥 |<2𝑘+4𝑟 󵄨 󵄨 −1 0 󵄨 4 󵄨 𝜆 󵄩 󵄩 𝑘=1 𝑗 𝑗 𝑗 󵄨 󵄨 󵄨 󵄨 󵄩𝑗=−∞ 󵄩 (1 + 󵄨𝑥−𝑥𝑗󵄨 𝜌(𝑥𝑗) ) 󵄩 󵄩𝐿1 ∞ 󵄨 󵄨𝛿 󵄨 󵄨 𝐶[𝑏] 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝜃 󵄨 󵄨 󵄨𝑥𝑗 −𝑦󵄨 󵄨𝑏 (𝑥) −𝑏𝐵 󵄨 ≤ ∑ 󵄨𝜆𝑗󵄨 . 󵄨 󵄨 󵄨 𝑗 󵄨 󵄨 󵄨 × 𝑑𝑥 𝜆 𝑗=−∞ 󵄨 󵄨𝑛󵄨 󵄨𝛿 󵄨 󵄨 󵄨 󵄨 󵄨𝑥−𝑥𝑗󵄨 󵄨𝑥−𝑥𝑗󵄨 (95) 14 Journal of Function Spaces and Applications

3 3 Therefore, For 1<𝑝<∞,itiseasytoseethat𝐵∞(R )⊆𝐵𝑝(R ). 󵄨 󵄨 Moreover, it follows from (0.14) in [26]that 󵄨 󵄨 󵄨 𝜆 󵄨 󵄨{𝑥 ∈ R𝑛 : 󵄨[𝑏,] 𝑇 𝑓 (𝑥)󵄨 > }󵄨 󵄨 󵄨𝛼/(𝛼|𝛽|+2) 󵄨 󵄨 󵄨 󵄨 𝑚 (𝑥,) 𝑉 ∼ ∑ 󵄨𝜕𝛽𝑃 (𝑥)󵄨 . 󵄨 4 󵄨 󵄨 𝑥 󵄨 (99) |𝛽|≤𝑘 4 󵄨 󵄨 󵄨 𝑛 󵄨 󵄨 𝜆 󵄨 ≤𝐶∑ 󵄨{𝑥 ∈ R : 󵄨𝐴 (𝑥)󵄨 > }󵄨 𝑉(𝑥) = 1+|𝑥| = 1+(𝑥2 +𝑥2 +𝑥2)1/2 ∈𝐵 (R3) 󵄨 󵄨 𝑖 󵄨 4 󵄨 Therefore, 1 2 3 ∞ . 𝑖=1 󵄨 󵄨 𝑉 (𝑥) ≲ 𝑉 (𝑥) (96) If 1 2 ,then ∞ 𝐶[𝑏]𝜃 󵄨 󵄨 1 1 1 ≤ ∑ 󵄨𝜆 󵄨 =̇ {𝑟 : ∫ 𝑉 (𝑦) 𝑑𝑦 ≤ 1}≲ 𝜆 󵄨 𝑗󵄨 sup 2 𝑗=−∞ 𝑚(𝑥,𝑉2) 𝑟>0 𝑟 𝐵(𝑥,𝑟) 𝑚(𝑥,𝑉1) 𝐶[𝑏] 1 𝜃 󵄩 󵄩 =̇ {𝑟 : ∫ 𝑉 (𝑦) 𝑑𝑦 ≤ 1}. ≤ 󵄩𝑓󵄩𝐻1 . sup 1 𝜆 𝐿 𝑟>0 𝑟 𝐵(𝑥,𝑟) (100) This completes the proof of Theorem 6. Thus, 𝑥 𝑥 4. Another Case ∇𝑉 (𝑥) =( 1 , 2 , (𝑥2 +𝑥2 +𝑥2)1/2 (𝑥2 +𝑥2 +𝑥2)1/2 In this section, we obtain same results for the commutator 1 2 3 1 2 3 R𝐻 𝑉 (101) if we impose another condition on .ViaCorollaries 𝑥 15 and 16 in Section 2 and Theorems 1.2 and 1.3 in[23], we 3 ). 2 2 2 1/2 obtain the following theorems. (𝑥1 +𝑥2 +𝑥3) |∇𝑉(𝑥)| =1 |∇𝑉(𝑥)| ∈𝐵 (R3) Theorem 27. Suppose that 𝑉∈𝐵𝑠 for some 𝑠>𝑛, |∇𝑉| ∈𝑠 𝐵 Therefore, .Clearly, ∞ .So, 1 𝑉(𝑥) ≥ |∇𝑉(𝑥)| 𝜌(𝑥) ≲𝜌 (𝑥) 𝑉(𝑥) ≥ for some 𝑠1 >𝑛/2,and|∇𝑉| satisfies (32).Let𝑏∈𝐵𝑀𝑂∞(𝜌). . Therefore, 1 .Also,since R𝐻 𝐿𝑝(R𝑛) 1<𝑝<𝑠 1,then𝜌(𝑥).Then,thepotential ≲1 𝑉(𝑥) = 1 + |𝑥| =1+ The commutator 𝑏 is bounded on for 1. 2 2 2 1/2 (𝑥1 +𝑥2 +𝑥3) satisfies the assumption of Theorems 4 and Theorem 28. Suppose that 𝑉∈𝐵𝑠 for some 𝑠>𝑛, |∇𝑉| ∈𝑠 𝐵 6. 𝑠 >𝑛/2 |∇𝑉| 𝑏∈𝐵𝑀𝑂 (𝜌)1 for some 1 ,and satisfies (32).Let ∞ . 3 2 2 2 3/2 𝜆>0 Example 2. Let 𝑉(𝑥) = 1 + |𝑥| =1+(𝑥1 +𝑥2 +𝑥3) .Bythe Then, for any , 𝑛 previous argument, we conclude that 𝑉∈𝐵∞(R ). 󵄨 󵄨 󵄨 󵄨 [𝑏] Then, 󵄨 𝑛 󵄨 𝐻 󵄨 󵄨 𝜃 󵄩 󵄩 󵄨{𝑥 ∈ R : 󵄨R𝑏 (𝑓) (𝑥)󵄨 >𝜆}󵄨 ≲ 󵄩𝑓󵄩𝐻1(R𝑛), 󵄨 󵄨 󵄨 󵄨 𝜆 𝐿 2 2 2 1/2 2 2 2 1/2 (97) ∇𝑉 (𝑥) = (3(𝑥1 +𝑥2 +𝑥3) 𝑥1, 3(𝑥1 +𝑥2 +𝑥3) 𝑥2, 1 𝑛 ∀𝑓 ∈𝐿 𝐻 (R ). 1/2 3(𝑥2 +𝑥2 +𝑥2) 𝑥 ). 𝐻 1 𝑛 1 2 3 3 Namely, the commutator R𝑏 is bounded from 𝐻𝐿(R ) into 1 𝑛 (102) 𝐿𝑤𝑒𝑎𝑘(R ). 2 2 2 2 Thus, |∇𝑉(𝑥)| = 3|𝑥| =3(𝑥1 +𝑥2 +𝑥3).Clearly,|∇𝑉(𝑥)| ∈ Remark 29. Following Remark 5 in [22], we know that if 𝐵 (R3) 𝑚(𝑥, 𝑉) ∼ 1 3+ |𝑥| 𝑉 ∞ .From(99), we know that and is a non-negative polynomial, condition (32)holdstrue. 𝑚(𝑥, |∇𝑉|) ∼ 1+|𝑥| 𝜌(𝑥) ≲𝜌 (𝑥) 𝑉(𝑥) = |𝑝(𝑥)|𝛼 𝑝(𝑥) . Therefore, 1 .Also,since Furthermore, we know that if ,where is a 𝑉(𝑥) ≥1 𝜌(𝑥) ≲1 𝑉(𝑥) = 1 + |𝑥| = 𝛼>0 ,then .Then,thepotential polynomial and , condition (32)holdstrue(seeRemark 1+(𝑥2 +𝑥2 +𝑥2)3/2 6in[24]). 1 2 3 satisfies the assumption of Theorems 4 and 6. 4 4 4 1/2 Example 3. Let 𝑉(𝑥) = 1 +1 (𝑥 +𝑥2 +𝑥3) .Bytheprevious 5. Examples 𝑛 argument, we conclude that 𝑉∈𝐵∞(R ). In this section, we give some examples for the potentials Then, which can satisfy the assumption in Theorems 4 and 6.We 4 4 4 −1/2 3 4 4 4 −1/2 3 always assume that 𝑛=3throughout this section. Denote the ∇𝑉 (𝑥) = (2(𝑥1 +𝑥2 +𝑥3) 𝑥1, 2(𝑥1 +𝑥2 +𝑥3) 𝑥2, 3 2 2 2 1/2 norm of R by |𝑥| = 1(𝑥 +𝑥2 +𝑥3) . −1/2 2(𝑥4 +𝑥4 +𝑥4) 𝑥3). Example 1. Let 1 2 3 3 (103) 1/2 𝑉 (𝑥) =1+|𝑥| =1+(𝑥2 +𝑥2 +𝑥2) . 1 2 3 (98) Thus, 1/2 𝑃(𝑥) 𝑥6 +𝑥6 +𝑥6 Following [25], we know that if is a polynomial of |∇𝑉 (𝑥)| =2( 1 2 3 ) . (104) 𝑘 𝛼>0 𝑉(𝑥) = |𝑃(𝑥)|𝛼 𝐵 (R3) 4 4 4 degree and ,then belongs to ∞ . 𝑥1 +𝑥2 +𝑥3 Journal of Function Spaces and Applications 15

From (99), we know that 𝑚(𝑥, 𝑉) ∼ 1 .Since+ |𝑥| [11] X. T. Duong and L. X. Yan, “Commutators of BMO functions and singular integral operators with non-smooth kernels,” 3 6 6 6 2 2 2 Bulletin of the Australian Mathematical Society,vol.67,no.2, (𝑥1 +𝑥2 +𝑥3)∼(𝑥1 +𝑥2 +𝑥3) , (105) pp.187–200,2003. 2 𝐿𝑝 (𝑥4 +𝑥4 +𝑥4)∼(𝑥2 +𝑥2 +𝑥2) , [12] Z. H. Guo, P. T. Li, and L. Z. Peng, “ boundedness of 1 2 3 1 2 3 commutators of Riesz transform associated to Schrodinger operator,” Journal of Mathematical Analysis and Applications, then vol.241,no.1,pp.421–432,2008. 1/2 [13]Y.LiuandJ.F.Dong,“SomeestimatesofhigherorderRiesz ∇𝑉 𝑥 ∼ (𝑥2 +𝑥2 +𝑥2) . | ( )| 1 2 3 (106) transform related to Schrodinger¨ type operators,” Potential Analysis,vol.32,no.1,pp.41–55,2010. 3 Thus, |∇𝑉(𝑥)|∞ ∈𝐵 (R ).From(99), we know that 𝑚(𝑥, [14] D. Yang, D. Yang, and Y. Zhou, “Endpoint properties of 1/2 |∇𝑉|) ∼ 1 + |𝑥| . Therefore, 𝜌(𝑥)1 ≲𝜌 (𝑥).Also,since localized Riesz transforms and fractional integrals associated 𝑉(𝑥),then ≥1 𝜌(𝑥).Then,thepotential ≲1 𝑉(𝑥) = 1+ to Schrodinger¨ operators,” Potential Analysis,vol.30,no.3,pp. 1/2 271–300, 2009. (𝑥4 +𝑥4 +𝑥4) 1 2 3 satisfies the assumption of Theorems 4 and [15] P. T. Li and L. Z. Peng, “Endpoint estimates for commutators 6. of Riesz transforms associated with Schrodinger¨ operators,” Bulletin of the Australian Mathematical Society,vol.82,no.3, Acknowledgments pp. 367–389, 2010. [16]L.TangandJ.F.Dong,“BoundednessforsomeSchrodinger¨ This paper is supported by Research Fund for the Doctoral type operators on Morrey spaces related to certain nonnegative Program of Higher Education of China under Grant (no. potentials,” Journal of Mathematical Analysis and Applications, 20113108120001), the Shanghai Leading Academic Discipline vol. 355, no. 1, pp. 101–109, 2009. Project (J50101), the National Natural Science Foundation [17] B. Bongioanni, E. 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Hindawi Publishing Corporation Journal of Function Spaces and Applications Volume 2013, Article ID 820437, 13 pages http://dx.doi.org/10.1155/2013/820437

Research Article On the Space of Functions with Growths Tempered by a Modulus of Continuity and Its Applications

Józef BanaV and RafaB Nalepa

Department of Mathematics, Rzeszow´ University of Technology, Powstanc´ ow´ Warszawy 8, 35-959 Rzeszow,´ Poland

Correspondence should be addressed to Jozef´ Bana´s; [email protected]

Received 7 February 2013; Accepted 20 March 2013

Academic Editor: Janusz Matkowski

Copyright © 2013 J. Bana´s and R. Nalepa. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We are going to study the space of real functions defined on a bounded metric space and having growths tempered by a modulus of continuity. We prove also a sufficient condition for the relative compactness in the mentioned function space. Using that condition and the classical Schauder fixed point theorem, we show the existence theorem for some quadratic integral equations of Fredholm type in the space of functions satisfying the Holder¨ condition. An example illustrating the mentioned existence result is also included.

1. Introduction It is also worthwhile mentioning that it is very difficult to encountersomeapplicationsofthespaceoffunctionswith The principal aim of the paper is to discuss the function space growths tempered by a modulus of continuity in the theory consisting of functions with growths tempered by a given of operator equations (functional, differential, integral, etc.), modulus of continuity. Functions belonging to such a space (cf. [5, 6]). have their moduli of continuity in some sense proportional to In this paper we are going to discuss in detail the above a given modulus of continuity. We assume here that the men- mentioned function space consisting of real functions having tioned function space consists of real functions defined on a growths tempered by a given modulus of continuity. We bounded metric space and, as we pointed out above, having describe a general form of such a space and we point out some growths tempered by a given modulus of continuity. Typical important particular cases of spaces of the described type. examplesoffunctionspacesofsuchatypearethespaceof Apart from this, we indicate some essential properties of functions satisfying the Lipschitz condition and the space of the considered function space. Namely, we show that such Holder¨ functions, that is, the space of functions satisfying a space is not a closed subspace of the space of all real the Holder¨ condition with a given exponent belonging to the continuous functions with the supremum norm, with respect interval (0, 1]. to this norm. The consequence of this property is the fact that It is a surprising fact that those spaces, although they thenormintroducedinthespaceoffunctionswithtempered arefrequentlyusedinmathematicalinvestigations,werenot growthsisnotequivalenttotheclassicalsupremumnorm. intensively studied and described in mathematical literature Thiscausesthatitisnotpossibletousethesupremumnorm (cf. [1–4]). It is probably caused by the fact that the norm in all considerations conducted in the mentioned space of in the mentioned function space is not convenient for use functions with tempered growths. in comparison with the norm in the classical space of real The main result proved in the paper is a sufficient functions being continuous on a compact set. On the other condition for relative compactness of bounded subset of the hand, there are no known convenient sufficient conditions discussed function space. We also point out several particular for relative compactness of bounded subsets of the function cases of that result, especially in the space of functions spaceinquestion. satisfying the Holder¨ condition. 2 Journal of Function Spaces and Applications

The last part of the paper contains an application of In the sequel, let us notice that if 𝑥∈𝐶𝜔(𝑋),theninview the obtained results in proving a theorem on the existence of (2) there exists a constant 𝐾𝑥 >0such that of a solution of a quadratic integral equation in the space ] (𝑥,) 𝜀 ⩽𝐾𝜔 (𝜀) of functions satisfying the Holder¨ condition. An example 𝑥 (4) showing the applicability of that theorem to a concrete for any 𝜀>0.Obviouslytheconverseimplicationisalsotrue. quadratic integral equation is also included. Thus, we can say equivalently that the set 𝐶𝜔(𝑋) consists of all real functions defined on X such that the moduli 2. The Space of Functions with Tempered ofcontinuityofthosefunctionsaretemperedbythegiven 𝜔=𝜔(𝜀) Moduli of Continuity modulus of continuity . Now, fix arbitrarily an element 𝑢0 ∈𝑋. For an arbitrary 𝑥∈𝐶 (𝑋) ‖𝑥‖ In this section we discuss the space of real functions defined function 𝜔 we define the quantity by the formula on a given bounded metric space and having the growths 󵄨 󵄨 |𝑥 (𝑢) −𝑥(V)| ‖𝑥‖ = 󵄨𝑥 (𝑢 )󵄨 + { :𝑢,V ∈𝑋,𝑢≠V} . tempered by a given modulus of continuity. Moreover, we will 󵄨 0 󵄨 sup 𝜔 (𝑑 (𝑢, V)) also study some important particular cases of that space. (5) To this end denote by R thesetofallrealnumbersand R =[0,∞) 𝜔:R → R put + .Afunction + + is said to be a Notice that ‖𝑥‖ < ∞ for any 𝑥∈𝐶𝜔(𝑋).Moreover,itcanbe 𝜔(0) = 0, 𝜔(𝜀) >0 𝜀>0 𝜔 modulus of continuity if for ,and shown that ‖⋅‖is a norm in the space 𝐶𝜔(𝑋);thatis,𝐶𝜔(𝑋) is nondecreasing on R+. is a normed space with the norm defined by5 ( ). We will also assume (but not always) that the modulus of We show that the norm defined by5 ( )iscomplete. 𝜔=𝜔(𝜀) 𝜀=0 𝜔(𝜀) →0 continuity is continuous at ,thatis, To this end, take a Cauchy sequence (𝑥𝑛) in the space 𝜀→0 as . 𝐶𝜔(𝑋). Fix arbitrarily a number 𝜀>0and denote 𝐷= Inordertoclarifytheconceptofthemodulusofcontinu- max{1, 𝜔(diam 𝑋)},wherediam𝑋 denotes the diameter of (𝑋, 𝑑) ity, let us assume that is a given bounded metric space. the metric space 𝑋.Then,wecanfindanaturalnumber𝑛0 𝐶(𝑋) Denote by thespaceofallrealfunctionsdefinedand such that for 𝑛, 𝑚0 ⩾𝑛 ,wehavethat‖𝑥𝑛 −𝑥𝑚‖⩽𝜀/2𝐷or, by continuous on the metric space 𝑋.For𝑥∈𝐶(𝑋)and for an (5), arbitrary fixed number 𝜀>0let us define the quantity ](𝑥, 𝜀) 󵄨 󵄨 󵄨𝑥 (𝑢 )−𝑥 (𝑢 )󵄨 by the formula 󵄨 𝑛 0 𝑚 0 󵄨 󵄨 󵄨 󵄨[𝑥𝑛 (𝑢) −𝑥𝑚 (𝑢)]−[𝑥𝑛 (V) −𝑥𝑚 (V)]󵄨 ] (𝑥,) 𝜀 = sup {|𝑥 (𝑢) −𝑥(V)| :𝑢,V ∈𝑋,𝑑(𝑢, V) ⩽𝜀} . (1) + sup { 𝜔 (𝑑 (𝑢, V)) (6) The function 𝜀→](𝑥, 𝜀) is called the modulus 𝜀 of continuity of the function 𝑥. Observe that in order to :𝑢,V ∈𝑋,𝑢≠V}⩽ . define this concept it is sufficient to consider the space 𝐵(𝑋) 2𝐷 consisting of real functions defined and bounded on 𝑋. Further, let us notice that from (6)weinferthat Observe that if 𝑥∈𝐵(𝑋), then the modulus of continuity 𝑥 𝜀→](𝑥, 𝜀) 󵄨 󵄨 𝜀 𝜀 of ,thatis,thefunction ,isthemodulus 󵄨𝑥𝑛 (𝑢0)−𝑥𝑚 (𝑢0)󵄨 ⩽ ⩽ . (7) of continuity in the above defined sense. Obviously, this 2𝐷 2 𝜀=0 𝑥 modulus is continuous at if and only if is uniformly This means that the real sequence {𝑥𝑛(𝑢0)} satisfies the continuous on 𝑋. Cauchy condition in R with natural metric. Hence we obtain 𝜔=𝜔(𝜀) In what follows let us fix a modulus of continuity that this sequence is convergent to a real number, say 𝑥(𝑢0), (𝑋, 𝑑) and assume, as before, that is a given bounded metric that is, lim𝑛→∞𝑥𝑛(𝑢0)=𝑥(𝑢0). 𝐶 (𝑋) space. Denote by 𝜔 the set of all real functions defined Next, keeping in mind (6)againwededucethat on 𝑋 such that their growths are tempered by the modulus 󵄨 󵄨 󵄨[𝑥 𝑢 −𝑥 𝑢 ]−[𝑥 V −𝑥 V ]󵄨 of continuity 𝜔. More precisely, a function 𝑥=𝑥(𝑢)belongs 󵄨 𝑛 ( ) 𝑚 ( ) 𝑛 ( ) 𝑚 ( ) 󵄨 sup { to the set 𝐶𝜔(𝑋) provided 𝑥:𝑋 → R and there exists a 𝜔 (𝑑 (𝑢, V)) constant 𝐾𝑥 >0such that (8) 𝜀 :𝑢,V ∈𝑋,𝑢≠V}⩽ . |𝑥 (𝑢) −𝑥(V)| ⩽𝐾𝑥𝜔 (𝑑 (𝑢, V)) (2) 2𝐷 V =𝑢 for all 𝑢, V ∈𝑋. In other words, we have that 𝑥∈𝐶𝜔(𝑋) if and Putting in the above inequality 0, we get that for each only if the quantity 𝑢∈𝑋,𝑢=𝑢̸ 0, the following estimate holds 󵄨 󵄨 󵄨[𝑥 (𝑢) −𝑥 (𝑢)]−[𝑥 (𝑢 )−𝑥 (𝑢 )]󵄨 𝜀 |𝑥 (𝑢) −𝑥(V)| 󵄨 𝑛 𝑚 𝑛 0 𝑚 0 󵄨 ⩽ . sup { :𝑢,V ∈𝑋,𝑢≠V} (3) 2𝐷 (9) 𝜔 (𝑑 (𝑢, V)) 𝜔(𝑑(𝑢,𝑢0)) This yields the inequality is finite. 󵄨 󵄨 𝜀 It is easy to check that the set 𝐶𝜔(𝑋) forms a linear space 󵄨[𝑥 (𝑢) −𝑥 (𝑢)]−[𝑥 (𝑢 )−𝑥 (𝑢 )]󵄨 ⩽ 𝜔(𝑑(𝑢,𝑢 )) 󵄨 𝑛 𝑚 𝑛 0 𝑚 0 󵄨 2𝐷 0 over the field of real numbers R.Obviously,𝐶𝜔(𝑋) is a linear subspace of the space 𝐶(𝑋). (10) Journal of Function Spaces and Applications 3

which is valid for 𝑢∈𝑋and 𝑚, 𝑛 ∈ N,𝑛,𝑚⩾𝑛0 (N denotes Now, fix arbitrarily 𝑢, V ∈𝑋.Then,passingintheabove the set of natural numbers). inequality with 𝑛→∞,weobtain From that last inequality we get |𝑥 (𝑢) −𝑥(V)| ⩽𝑀𝜔(𝑑 (𝑢, V)) (19) 󵄨 󵄨 󵄨 󵄨 𝜀 𝜀 󵄨𝑥 (𝑢)−𝑥 (𝑢)󵄨−󵄨𝑥 (𝑢 )−𝑥 (𝑢 )󵄨 ⩽ 𝜔(𝑑(𝑢,𝑢 )) ⩽ . for all 𝑢, V ∈𝑋.Butthismeansthat𝑥∈𝐶𝜔(𝑋). 󵄨 𝑛 𝑚 󵄨 󵄨 𝑛 0 𝑚 0 󵄨 2𝐷 0 2 In what follows, using the fact that the sequence (𝑥𝑛) (11) satisfies inequality (6), we derive that for arbitrary 𝑢, V ∈ 𝑋, 𝑢 ≠V and for arbitrary 𝑛, 𝑚 ∈ N,𝑛,𝑚⩾𝑛0,thefollowing Henceweobtain inequality holds 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝜀 󵄨[𝑥𝑛 (𝑢) −𝑥𝑚 (𝑢)]−[𝑥𝑛 (V) −𝑥𝑚 (V)]󵄨 𝜀 󵄨𝑥𝑛 (𝑢) −𝑥𝑚 (𝑢)󵄨 ⩽ 󵄨𝑥𝑛 (𝑢0)−𝑥𝑚 (𝑢0)󵄨 + ⩽ . (20) 󵄨 󵄨 󵄨 󵄨 2 (12) 𝜔 (𝑑 (𝑢, V)) 2𝐷

Fixing 𝑢, V and passing with 𝑚→∞,weget for an arbitrary 𝑢∈𝑋and 𝑛, 𝑚0 ⩾𝑛 . Linking the above inequality and (7), we deduce that 󵄨 󵄨 󵄨[𝑥𝑛 (𝑢) −𝑥(𝑢)]−[𝑥𝑛 (V) −𝑥(V)]󵄨 𝜀 ⩽ . (21) 𝜔 (𝑑 (𝑢, V)) 2𝐷 󵄨 󵄨 󵄨𝑥𝑛 (𝑢) −𝑥𝑚 (𝑢)󵄨 ⩽𝜀 (13) Hence, in virtue of the arbitrariness of the choice of points 𝑢, V ∈𝑋,𝑢≠V,wehavetheestimate for 𝑢∈𝑋and for 𝑚, 𝑛 ∈ N,𝑛,𝑚⩾𝑛0.Butthisfactmeans (𝑥 (𝑢)) 𝑢∈ 󵄨 󵄨 that 𝑛 is a real Cauchy sequence for arbitrarily fixed 󵄨[𝑥𝑛 (𝑢) −𝑥(𝑢)]−[𝑥𝑛 (V) −𝑥(V)]󵄨 𝑋 R { . Thus this sequence is convergent in .Denoteitslimitby sup 𝜔 (𝑑 (𝑢, V)) 𝑥(𝑢) ,thatis, (22) 𝜀 𝑥 (𝑢) = 𝑥 (𝑢) . :𝑢,V ∈𝑋,𝑢≠V}⩽ 𝑛→∞lim 𝑛 (14) 2𝐷 𝑛⩾𝑛 𝑚→∞ being valid for 0. Passing in (13)with ,weobtain In a similar way, keeping in mind inequality (6), we infer 󵄨 󵄨 that 󵄨𝑥 (𝑢) −𝑥(𝑢)󵄨 ⩽𝜀 󵄨 𝑛 󵄨 (15) 󵄨 󵄨 𝜀 󵄨𝑥 (𝑢 )−𝑥 (𝑢 )󵄨 ⩽ 󵄨 𝑛 0 𝑚 0 󵄨 2𝐷 (23) for any 𝑢∈𝑋and for 𝑛∈N,𝑛⩾𝑛0.Thismeansthatthe for 𝑛, 𝑚 ∈ N,𝑛,𝑚⩾𝑛0.Hence,for𝑚→∞,weobtain sequence (𝑥𝑛) converges uniformly to the function 𝑥 on the 𝑋 metric space . 󵄨 󵄨 𝜀 󵄨𝑥𝑛 (𝑢0)−𝑥(𝑢0)󵄨 ⩽ . Now, we show that the sequence (𝑥𝑛) converges to the 󵄨 󵄨 2𝐷 (24) function 𝑥 in the sense of norm (5). To this end we prove first that 𝑥∈𝐶𝜔(𝑋). Indeed, taking into account the fact that (𝑥𝑛) Now, combining (22), (24), and (5), we obtain that 𝐶 (𝑋) (𝑥 ) is a Cauchy sequence in the space 𝜔 ,weinferthat 𝑛 is 󵄩 󵄩 𝜀 𝐶 (𝑋) 𝑀>0 󵄩𝑥 −𝑥󵄩 ⩽ ⩽𝜀 bounded in 𝜔 ; that is, there exists a constant such 󵄩 𝑛 󵄩 𝐷 (25) that ‖𝑥𝑛‖⩽𝑀for 𝑛 = 1, 2, . . . Hence, in view of (5)wehave for all 𝑛∈N,𝑛⩾𝑛0.Thismeansthatthesequence(𝑥𝑛) is 󵄨 󵄨 𝐶 (𝑋) 𝑥 𝐶 (𝑋) 󵄨𝑥 (𝑢) −𝑥 (V)󵄨 convergent in the space 𝜔 to the function .Thus, 𝜔 {󵄨 𝑛 𝑛 󵄨 :𝑢,V ∈𝑋,𝑢≠V}⩽𝑀. sup 𝜔 (𝑑 (𝑢, V)) (16) is a Banach space. Now, we provide a few examples.

This implies that for arbitrarily fixed 𝑢, V ∈𝑋,𝑢≠V and for Example 1. Take 𝜔𝐿(𝜀) = 𝜀 for 𝜀∈R+.Obviously𝜔𝐿 is a 𝑥∈𝐶 (𝑋) 𝑛∈N,weget modulus of continuity. Moreover, notice that 𝜔𝐿 if and only if there exists a constant 𝐿𝑥 >0such that 󵄨 󵄨 󵄨𝑥𝑛 (𝑢) −𝑥𝑛 (V)󵄨 |𝑥 (𝑢) −𝑥(V)| ⩽𝐿 𝑑 (𝑢, V) ⩽𝑀, (17) 𝑥 (26) 𝜔 (𝑑 (𝑢, V)) 𝑢, V ∈𝑋 𝐶 (𝑋) for . This means that the space 𝜔𝐿 consists of functions 𝑥:𝑋 → R satisfying the Lipschitz condition. The and, consequently, 𝐶 (𝑋) norm in the space 𝜔𝐿 has the form 󵄨 󵄨 󵄨𝑥 (𝑢) −𝑥 (V)󵄨 ⩽𝑀𝜔(𝑑 (𝑢, V)) 󵄨 󵄨 |𝑥 (𝑢)−𝑥(V)| 󵄨 𝑛 𝑛 󵄨 (18) ‖𝑥‖ = 󵄨𝑥(𝑢 )󵄨+ { :𝑢,V ∈𝑋,𝑢≠V}. 𝐿 󵄨 0 󵄨 sup 𝑑 (𝑢, V) for all 𝑢, V ∈𝑋and for 𝑛∈N. (27) 4 Journal of Function Spaces and Applications

Example 2. Let 𝜔𝐻 denote the modulus of continuity which on [𝑎, 𝑏] andequippedwiththenorm||𝑥||∞ = sup{|𝑥(𝑡)| : 𝑡∈ corresponds to the Holder¨ condition. This means that [𝑎, 𝑏]}. 𝛼 𝜔𝐻(𝜀) = 𝜀 for 𝜀⩾0,where𝛼 is a fixed number from the Further, if 𝛼(0<𝛼⩽1)is a fixed number, then (0, 1] 𝐻 [𝑎, 𝑏] 𝐶 ([𝑎, 𝑏]) interval . the symbol 𝛼 will denote the space 𝜔𝐻 (see 𝑥∈𝐶 (𝑋) 𝜔 =𝜀𝛼 𝐻 [𝑎, 𝑏] Observe that 𝜔𝐻 if and only if there exists a Example 2)with 𝐻 ;thatis, 𝛼 is the collection constant 𝐻𝑥 >0such that of all real functions 𝑥 defined on [𝑎, 𝑏] and satisfying the Holder¨ condition. More precisely, 𝑥∈𝐻𝛼[𝑎, 𝑏] if there exists |𝑥 (𝑢) −𝑥(V)| ⩽𝐻(𝑑 (𝑢, V))𝛼 𝛼 𝑥 (28) aconstant𝐻𝑥 such that for all 𝑢, V ∈𝑋. In other words, the function 𝑥 satisfies the 𝐻 Holder¨ condition with a constant 𝑥 and with an exponent |𝑥 (𝑡) −𝑥(𝑠)| ⩽𝐻𝛼|𝑡−𝑠|𝛼 𝛼 𝐶 (𝑋) 𝑥 (35) . Notice that the norm in the space 𝜔𝐻 has the from 󵄨 󵄨 |𝑥 (𝑢) −𝑥(V)| ‖𝑥‖ = 󵄨𝑥 (𝑢 )󵄨 + { :𝑢,V ∈𝑋,𝑢≠V} . 𝐻 󵄨 0 󵄨 sup (𝑑 (𝑢, V))𝛼 for all 𝑡, 𝑠 ∈ [𝑎, 𝑏]. 𝐻 [𝑎, 𝑏] (29) Observe that 𝛼 forms a linear subspace of the linear space 𝐶[𝑎,. 𝑏] 𝛼 Example 3. Now, let us take into account the modulus of In what follows, for 𝑥∈𝐻𝛼[𝑎, 𝑏],thesymbol𝐻𝑥 will continuity 𝜔 having the form denote the least possible constant for which inequality (35) is satisfied. In other words, we can write 0 for 𝜀=0, 𝜔 (𝜀) ={ (30) 1 for 𝜀>0. 𝛼 |𝑥 (𝑡) −𝑥(𝑠)| 𝐻𝑥 = sup { 𝛼 :𝑡,𝑠∈[𝑎,] 𝑏 ,𝑡=𝑠}.̸ (36) Note that 𝑥∈𝐶𝜔(𝑋) if and only if there exists a constant 𝐾𝑥 > |𝑡−𝑠| 0 such that |𝑥(𝑢) V− 𝑥( )| ⩽ 𝐾𝑥 for 𝑢, V ∈𝑋.Obviously,this condition is equivalent to the boundedness of the function 𝑥. 𝑢 ∈𝑋 𝑢∈𝑋 Indeed, fix 0 .Then,forany we have Notice that in the case 𝛼=1, the linear space 𝐻1[𝑎, 𝑏] 󵄨 󵄨 󵄨 󵄨 coincides with the space of real functions defined on [𝑎, 𝑏] |𝑥 (𝑢)| − 󵄨𝑥 (𝑢 )󵄨 ⩽ 󵄨𝑥 (𝑢) −𝑥(𝑢 )󵄨 ⩽𝐾. 󵄨 0 󵄨 󵄨 0 󵄨 𝑥 (31) and satisfying the Lipschitz condition, that is, This yields 󵄨 󵄨 |𝑥 (𝑡) −𝑥(𝑠)| ⩽𝐿 |𝑡−𝑠| |𝑥 (𝑢)| ⩽ 󵄨𝑥(𝑢0)󵄨 +𝐾𝑥 <∞ (32) 𝑥 (37) for 𝑢∈𝑋. The converse implication is also obvious. Thus 𝐶 (𝑋) = 𝐵(𝑋) 𝐶 (𝑋) we can write that 𝜔 ;thatis, 𝜔 contains all for some constant 𝐿𝑥 >0and for all 𝑡, 𝑠 ∈ [𝑎, 𝑏]. This space functions defined and bounded on 𝑋. Further, observe that will be denoted by the symbol Lip[𝑎, 𝑏]. 𝐶 (𝑋) the norm in the space 𝜔 has now the form Our principal aim is to show that the spaces 𝐻𝛼[𝑎, 𝑏] (for 𝛼 ∈ (0, 1] 𝐶[𝑎, 𝑏] 󵄨 󵄨 ) do not form closed subspaces of the space ‖𝑥‖ = 󵄨𝑥(𝑢0)󵄨 + sup |𝑥 (𝑢) −𝑥(V)| . ‖⋅‖ 𝑢,V∈𝑋 (33) with respect to the norm ∞. In order to prove this fact, observe that the space 𝐻𝛼[𝑎, 𝑏] canbenormedinanatural 𝑥∈𝐻[𝑎, 𝑏] It is easy to show that this norm is equivalent to the classical way (cf. Example 2)iffor 𝛼 we put supremum norm ‖⋅‖∞ in the space 𝐵(𝑋). Indeed, we have

‖𝑥‖∞ ⩽ ‖𝑥‖ ⩽3‖𝑥‖∞ (34) |𝑥 (𝑡) −𝑥(𝑠)| ‖𝑥‖ = |𝑥 (𝑎)| + { :𝑡,𝑠∈[𝑎,] 𝑏 ,𝑡=𝑠}.̸ 𝛼 sup |𝑡−𝑠|𝛼 𝑥∈𝐵(𝑋) ||𝑥|| = |𝑥(𝑢)| for any function ,where ∞ sup𝑢∈𝑋 . (38) The examples of the functions 𝑥(𝑢) = 𝑢 −1 and 𝑥(𝑢) =𝑢 in the space 𝐵([0, 2]) with 𝑢0 =0show that the equality signs in (34)areattained. Observe that in view of (36), the above definition can be written in the from 3. Some Properties of the Space of Hölder Functions 𝛼 ‖𝑥‖𝛼 = |𝑥 (𝑎)| +𝐻𝑥 . (39) In order to simplify the considerations in this section, we restrict ourselves to spaces of real functions defined on a fixed interval [𝑎, 𝑏]. In this way we will consider the space 𝐶[𝑎, 𝑏] Obviously, the space 𝐻𝛼[𝑎, 𝑏] with the norm (38)isthe consisting of functions 𝑥:[𝑎,𝑏]→ R,whicharecontinuous Banach space (cf. Section 2). Journal of Function Spaces and Applications 5

Further on let us notice that for an arbitrarily fixed 𝑥∈ which shows that ‖𝑥𝑛‖∞/‖𝑥𝑛‖𝛼 →0as 𝑛→∞.Thusthe 𝐻𝛼[𝑎, 𝑏] and for an arbitrary 𝑡 ∈ [𝑎, 𝑏] we obtain norms ‖⋅‖∞ and ‖⋅‖𝛼 arenotequivalentinthespace𝐻𝛼[0, 1]. Henceweinferthat𝐻𝛼[0, 1] is not closed subspace of the |𝑥 (𝑡)| space 𝐶[0, 1] with respect to the norm ‖⋅‖∞.Obviouslythe [0, 1] ⩽ |𝑥 (𝑡) −𝑥(𝑎)| + |𝑥 (𝑎)| same assertion is true if we replace the interval by an arbitrary interval [𝑎, 𝑏]. 0<𝛼<𝛾⩽1 ⩽ |𝑥 (𝑎)| + sup {|𝑥 (𝑡) −𝑥(𝑎)| :𝑡∈[𝑎,] 𝑏 } In what follows, let us observe that for the following inclusions hold: ⩽ |𝑥 (𝑎)| + sup {|𝑥 (𝑡) −𝑥(𝑠)| :𝑡,𝑠∈[𝑎,] 𝑏 } 𝐻𝛾 [𝑎,] 𝑏 ⊂𝐻𝛼 [𝑎,] 𝑏 ⊂𝐶[𝑎,] 𝑏 . (45) |𝑥 (𝑡) −𝑥(𝑠)| 𝛼 = |𝑥 (𝑎)| + sup { 𝛼 |𝑡−𝑠| |𝑡−𝑠| Particularly, taking into account that for 𝛾=1,wehave

:𝑡,𝑠∈[𝑎,] 𝑏 ,𝑡=𝑠}̸ 𝐻1 [𝑎,] 𝑏 = Lip [𝑎,] 𝑏 , (46) (40)

|𝑥 (𝑡) −𝑥(𝑠)| which yields ⩽ |𝑥 (𝑎)| + { :𝑡,𝑠∈[𝑎,] 𝑏 ,𝑡=𝑠}̸ sup |𝑡−𝑠|𝛼 Lip [𝑎,] 𝑏 ⊂𝐻𝛾 [𝑎,] 𝑏 ⊂𝐻𝛼 [𝑎,] 𝑏 ⊂𝐶[𝑎,] 𝑏 (47) × (𝑏−𝑎)𝛼 for 0<𝛼<𝛾<1. 𝑥∈𝐻[𝑎, 𝑏] 𝛼 |𝑥 (𝑡) −𝑥(𝑠)| Indeed, if 𝛾 ,weobtain ⩽ max {1, (𝑏−𝑎) }{|𝑥 (𝑎)| + sup { 𝛼 |𝑡−𝑠| 𝛾 𝛾 𝛾 𝛼 𝛾−𝛼 |𝑥 (𝑡) −𝑥(𝑠)| ⩽𝐻𝑥|𝑡−𝑠| =𝐻𝑥|𝑡−𝑠| ⋅ |𝑡−𝑠| (48) :𝑡,𝑠∈[𝑎,] 𝑏 ,𝑡=𝑠̸} }. 𝛾−𝛼 𝛾 𝛼 ⩽ (𝑏−𝑎) 𝐻𝑥|𝑡−𝑠| .

Henceweget This shows that 𝑥∈𝐻𝛼[𝑎, 𝑏] and hence we infer that inclusions (47)hold. ‖𝑥‖ ⩽ {1, (𝑏−𝑎)𝛼} ‖𝑥‖ . ∞ max 𝛼 (41) Further, for 𝑥∈𝐻𝛾[𝑎, 𝑏] and for arbitrarily fixed 𝑡, 𝑠 ∈ [𝑎, 𝑏], 𝑡 =𝑠̸, we derive The above inequality means that the norm ‖⋅‖∞ is dominated by the norm ‖⋅‖𝛼. |𝑥 (𝑡) −𝑥(𝑠)| |𝑥 (𝑡) −𝑥(𝑠)| = |𝑡−𝑠|𝛾−𝛼 If the space 𝐻𝛼[𝑎, 𝑏] would be a closed subspace of 𝐶[𝑎, 𝑏] |𝑡−𝑠|𝛼 |𝑡−𝑠|𝛾 with respect to the norm ‖⋅‖∞, then this would mean that (49) 𝐻𝛼[𝑎, 𝑏] (as the linear space) is a Banach space with respect |𝑥 (𝑡) −𝑥(𝑠)| 𝛾−𝛼 ⩽ 𝛾 (𝑏−𝑎) . to the norm ‖⋅‖∞. In other words, the space 𝐻𝛼[𝑎, 𝑏] would |𝑡−𝑠| be a Banach space both with respect to the norm ‖⋅‖∞ and with respect to the norm ‖⋅‖𝛼.Hence,keepinginmind(41) Henceweget andtakingintoaccountthetheoremontheequivalenceof |𝑥 (𝑡) −𝑥(𝑠)| dominated norms [4], we would infer that the norms ‖⋅‖∞ ‖𝑥‖ = |𝑥 (𝑎)| + { :𝑡,𝑠∈[𝑎,] 𝑏 ,𝑡=𝑠}̸ 𝛼 sup 𝑡−𝑠𝛼 and ‖⋅‖𝛼 areequivalentinthespace𝐻𝛼[𝑎, 𝑏]. | | We show that such a conclusion is not true. |𝑥 (𝑡)−𝑥(𝑠)| To this end fix a number 𝛼 ∈ (0, 1] and consider the ⩽ |𝑥 (𝑎)| + { sup |𝑡−𝑠|𝛾 sequence (𝑥𝑛) of real functions defined on the interval [0, 1] by the formula :𝑡,𝑠∈[𝑎,] 𝑏 ,𝑡=𝑠}̸ (𝑏−𝑎)𝛾−𝛼 𝛼 𝛼 1 {𝑛 𝑡 for 𝑡∈[0, ), 𝑛 𝛾−𝛼 𝑥𝑛 (𝑡) = { 1 (42) ⩽ max {1, (𝑏−𝑎) } ‖𝑥‖𝛾 {1 𝑡∈[ ,1], { for 𝑛 (50) for 𝑛 = 1,2,....Observe that (𝑥𝑛) ⊂ 𝐶[0, 1] and ||𝑥𝑛||∞ =1 for 𝑥∈𝐻𝛾[𝑎, 𝑏].Thusthenorm|| ⋅ ||𝛼 is dominated by the for 𝑛 = 1, 2, . . . On the other hand, we have norm || ⋅ ||𝛾. 𝛼 𝛼 We show that the inverse assertion is not true. To this 𝑥𝑛 (𝑡) −𝑥𝑛 (0) 𝑛 𝑡 𝛼 = =𝑛 , 𝑡∈(0, 1] . (43) end, take the function sequence (𝑥𝑛)⊂𝐻𝛾[0, 1],where𝑥𝑛 𝑡𝛼 𝑡𝛼 for is defined by the formula ||𝑥 || =𝑛𝛼 𝑛=1,2,.... This implies that 𝑛 𝛼 ,for Hence we get 1 { 𝛾 𝛾 󵄩 󵄩 {𝑛 𝑡 if 𝑡∈[0, ), 󵄩𝑥 󵄩 1 𝑥 (𝑡) = 𝑛 󵄩 𝑛󵄩∞ = 𝑛 { 1 (51) 󵄩 󵄩 𝑛𝛼 (44) 1 if 𝑡∈[ ,1] 󵄩𝑥𝑛󵄩𝛼 { 𝑛 6 Journal of Function Spaces and Applications for 𝑛 = 1, 2, . ..Itiseasilyseenthat Proof. Atfirst,letusobservethatsince𝐴 is bounded, then there exists a constant 𝑀>0such that for an arbitrary 𝑥∈𝐴 󵄨 󵄨 󵄩 󵄩 󵄨 󵄨 󵄨𝑥 (𝑡) −𝑥 (𝑠)󵄨 the following inequality holds: 󵄩𝑥 󵄩 = 󵄨𝑥 (0)󵄨 + {󵄨 𝑛 𝑛 󵄨 󵄩 𝑛󵄩𝛾 󵄨 𝑛 󵄨 sup |𝑡−𝑠|𝛾 󵄨 󵄨 |𝑥 (𝑢) −𝑥(V)| ‖𝑥‖ = 󵄨𝑥(𝑢 )󵄨 + { 󵄨 0 󵄨 sup 𝜔 (𝑑 (𝑢, V)) :𝑡,𝑠∈[0, 1] ,𝑡=𝑠}̸ (52) (56) :𝑢,V ∈𝑋,𝑢≠V}⩽𝑀, 𝑛𝛾𝑡𝛾 1 = { :𝑡∈(0, ]} = 𝑛𝛾 sup 𝑡𝛾 𝑛 where 𝑢0 is a fixed element of the metric space 𝑋 (cf. (5)). for =1,2,....On the other hand, we have Particularly, we conclude that

𝛾 𝛾 󵄨 󵄨 󵄩 󵄩 𝑛 𝑡 1 󵄨𝑥(𝑢0)󵄨 ⩽𝑀 (57) 󵄩𝑥 󵄩 = { :𝑡∈(0, ]} 󵄩 𝑛󵄩𝛼 sup 𝑡𝛼 𝑛 (53) for an arbitrary 𝑥∈𝐴. Moreover, on the basis of (56)we 𝛾 𝛾−𝛼 1 𝛼 = sup {𝑛 𝑡 :𝑡∈[0, ]} = 𝑛 infer that for any 𝑥∈𝐴and for arbitrary 𝑢, V ∈𝑋,𝑢≠V,the 𝑛 following inequality holds: for 𝑛 = 1, 2, . . . Henceweget |𝑥 (𝑢) −𝑥(V)| ⩽𝑀. (58) 󵄩 󵄩 𝜔 (𝑑 (𝑢, V)) 󵄩𝑥 󵄩 1 󵄩 𝑛󵄩𝛼 = 󵄩 󵄩 𝑛𝛾−𝛼 (54) 󵄩𝑥𝑛󵄩𝛾 The above inequality yields

|𝑥 (𝑢) −𝑥(V)| ⩽𝑀𝜔(𝑑 (𝑢, V)) (59) which yields that ||𝑥𝑛||𝛼/||𝑥𝑛||𝛾 →0as 𝑛→∞.Thisshows that the norm ||⋅||𝛾 is not dominated by the norm ||⋅||𝛼.Thus thesenormsarenotequivalent. for an arbitrary function 𝑥∈𝐴and for all 𝑢, V ∈𝑋. Since lim𝜀→0𝜔(𝜀) = 0, from (59)wededucethatall functions belonging to the set 𝐴 are equicontinuous on the 4. A Sufficient Condition for Relative set 𝑋. Apart from that, from (59) we have that for an arbitrary Compactness in the Space 𝐶𝜔(𝑋) 𝑥∈𝐴the inequality This section is devoted to present a criterion for relative |𝑥 (𝑢) −𝑥(V)| ⩽𝑀𝜔(diam 𝑋) (60) compactness in the space 𝐶𝜔(𝑋) of functions with growths tempered by a given modulus of continuity 𝜔=𝜔(𝜀).More 𝑢, V ∈𝑋 V =𝑢 precisely, we will discuss a sufficient condition for relative holds for all .Puttingintheaboveinequality 0, 𝑥∈𝐴 𝑢∈𝑋 compactness in 𝐶𝜔(𝑋). we infer that for arbitrary and the following It is worthwhile mentioning that as far as we know, there estimate is satisfied: arenoknownconditionsofsuchatype(e.g.,cf.[1–4]). 󵄨 󵄨 󵄨 󵄨 Let us recall that the complete description of the space |𝑥 (𝑢)| − 󵄨𝑥(𝑢0)󵄨 ⩽ 󵄨𝑥 (𝑢) −𝑥(𝑢0)󵄨 ⩽𝑀𝜔(diam 𝑋) . (61) 𝐶𝜔(𝑋) was given in Section 2. This implies that Theorem 4. Assume that 𝜔=𝜔(𝜀)is a given modulus of continuity such that 𝜔(𝜀) →0 as 𝜀→0, and assume 󵄨 󵄨 |𝑥 (𝑢)| ⩽ 󵄨𝑥(𝑢0)󵄨 +𝑀𝜔(diam 𝑋) (62) that 𝑋 is a compact metric space. Let 𝐴 be a bounded subset of the space 𝐶𝜔(𝑋) such that functions belonging to 𝐴 are 𝑥∈𝐴 𝑢∈𝑋 equicontinuous with respect to the modulus of continuity 𝜔; for and .Thismeansthatfunctionsfromtheset 𝐴 that is, the following condition is satisfied: are equibounded. Now, let us consider an arbitrary sequence (𝑥𝑛) in the 𝐴 |𝑥 (𝑢)−𝑥(V)| set . Taking into account the above established facts, we ∀ ∃ ∀ ∀ 𝑢,V∈𝑋 [𝑑 (𝑢, V) ⩽𝛿󳨐⇒ infer that functions of the sequence (𝑥𝑛) are equibounded 𝜀>0 𝛿>0 𝑥∈𝐴 𝜔 (𝑑 (𝑢, V)) 𝑢 ≠V and equicontinuous on the set 𝑋. Hence, in view of Ascoli- Arzela´ criterion we derive that there exists a subsequence of ⩽𝜀]. the sequence (𝑥𝑛) which is uniformly convergent on the set 𝑋 to a function 𝑥=𝑥(𝑢).Toavoidcomplicatednotation,we (55) will denote the mentioned subsequence of the sequence (𝑥𝑛) bythesamesymbol(𝑥𝑛). Observe that the function 𝑥=𝑥(𝑢) 𝐴 𝐶 (𝑋) Then the set isrelativelycompactinthespace 𝜔 . is continuous on the set 𝑋. Journal of Function Spaces and Applications 7

In what follows, we show that 𝑥∈𝐶𝜔(𝑋).Tothisend Next, fix arbitrarily a number 𝜀>0.Let𝛿>0denote a observethatfromthefactthatfunctionsbelongingtothe number corresponding to 𝜀/4 according to the assumption sequence (𝑥𝑛) satisfy inequality (59), we have that about equicontinuity of functions from the set 𝐴 with respect 󵄨 󵄨 to the modulus of continuity 𝜔=𝜔(𝜀). 󵄨𝑥𝑛 (𝑢) −𝑥𝑛 (V)󵄨 ⩽𝑀𝜔(𝑑 (𝑢, V)) (63) 󵄨 󵄨 Now, denote by 𝛽(𝜀) the number defined as follows: for an arbitrary 𝑛∈N and for 𝑢, V ∈𝑋.Sincelim𝑛→∞𝑥𝑛(𝑧) = 𝑥(𝑧) for an arbitrary element 𝑧∈𝑋,thuskeepinginmindthe continuity of the absolute value and other standard facts from 𝜀 𝜀𝜔 (𝛿) 𝛽 (𝜀) = min { , }. (71) mathematical analysis, we deduce from the last inequality 2 4 that |𝑥 (𝑢) −𝑥(V)| ⩽𝑀𝜔(𝑑 (𝑢, V)) (64) Keeping in mind the fact that functions of the sequence (𝑥𝑛) for arbitrary 𝑢, V ∈𝑋. areuniformlyconvergentontheset𝑋 to a function 𝑥,letus Similarly, on the basis of (57), we have choose a natural number 𝑛0 such that 󵄨 󵄨 󵄨𝑥𝑛 (𝑢0)󵄨 ⩽𝑀 (65) 󵄨 󵄨 for 𝑛=1,2....Henceweobtainthat 󵄨𝑥𝑛 (𝑤) −𝑥(𝑤)󵄨 ≤𝛽(𝜀) (72) 󵄨 󵄨 󵄨𝑥(𝑢0)󵄨 ⩽𝑀. (66)

Combining (64)and(66) we derive the following estimate: for all 𝑛⩾𝑛0 and for any 𝑤∈𝑋. ̃2 󵄨 󵄨 |𝑥 (𝑢) −𝑥(V)| Further, let us observe that 𝑑(𝑢, V)>𝛿for (𝑢, V)∈𝑋𝛿. 󵄨𝑥(𝑢 )󵄨 + ⩽2𝑀 󵄨 0 󵄨 𝜔 (𝑑 (𝑢, V)) (67) Henceweinferthat𝜔(𝑑(𝑢, V)) ⩾ 𝜔(𝛿). Thus, for an arbitrary 𝑛⩾𝑛0,inviewof(71)and(72)weget for arbitrary 𝑢, V ∈𝑋,𝑢≠V. This implies that 󵄨 󵄨 |𝑥 (𝑢) −𝑥(V)| 󵄨 󵄨 󵄨𝑥(𝑢0)󵄨 + sup { :𝑢,V ∈𝑋,𝑢≠V}⩽2𝑀 󵄨𝑥 (𝑢) −𝑥(𝑢) −𝑥 (V) +𝑥(V)󵄨 𝜔 (𝑑 (𝑢, V)) {󵄨 𝑛 𝑛 󵄨} sup 𝜔 (𝑑 (𝑢, V)) (68) 𝑢,V ∈𝑋̃2 ( ) 𝛿 which shows that 𝑥∈𝐶𝜔(𝑋). 󵄨 󵄨 󵄨 󵄨 󵄨𝑥𝑛 (𝑢) −𝑥(𝑢)󵄨 + 󵄨𝑥𝑛 (V) −𝑥(V)󵄨 Further on we show that the sequence (𝑥𝑛) is convergent ⩽ { } sup 𝜔 (𝑑 (𝑢, V)) to the function 𝑥 in the sense of the norm of the space 𝐶𝜔(𝑋). (𝑢,V)∈𝑋̃2 𝛿 (73) To this end for convenience, let us denote 󵄨 󵄨 󵄨 󵄨 2 󵄨𝑥 (𝑢) −𝑥(𝑢)󵄨 + 󵄨𝑥 (V) −𝑥(V)󵄨 𝑋 = {(𝑢, V) ∈𝑋×𝑋:𝑢≠V} , ⩽ {󵄨 𝑛 󵄨 󵄨 𝑛 󵄨} 0 sup 𝜔 (𝛿) (𝑢,V)∈𝑋̃2 2 2 𝛿 𝑋𝛿 ={(𝑢, V) ∈𝑋0 :𝑑(𝑢, V) ≤𝛿}, (69) 𝛽 (𝜀) 𝜀 ̃2 2 2 ⩽2 ≤ . 𝑋𝛿 =𝑋0 \𝑋𝛿, 𝜔 (𝛿) 2 where 𝛿>0is a fixed number. 2 2 ̃ 2 ̃ Obviouslywehavethat𝑋 =𝑋 ∪𝑋2 and the sets 𝑋 , 𝑋2 0 𝛿 𝛿 𝛿 𝛿 Next, taking into account the fact that the functions of the are disjoint. sequence (𝑥𝑛) belong to the set 𝐴, in virtue of the choice of Further, for an arbitrarily fixed natural number 𝑛,we the number 𝛿 to the number 𝜀/4,weinferthatforanarbitrary obtain 2 󵄩 󵄩 󵄨 󵄨 pair (𝑢, V)∈𝑋𝛿 and for an arbitrary natural number 𝑛 the 󵄩𝑥 −𝑥󵄩 = 󵄨𝑥 (𝑢 )−𝑥(𝑢 )󵄨 󵄩 𝑛 󵄩 󵄨 𝑛 0 0 󵄨 following inequality is satisfied: 󵄨 󵄨 󵄨𝑥 (𝑢) −𝑥(𝑢) −𝑥 (V) +𝑥(V)󵄨 + {󵄨 𝑛 𝑛 󵄨} sup 󵄨 󵄨 (𝑢,V)∈𝑋2 𝜔 (𝑑 (𝑢, V)) 󵄨 󵄨 0 󵄨𝑥𝑛 (𝑢) −𝑥𝑛 (V)󵄨 𝜀 ⩽ . (74) 󵄨 󵄨 𝜔 (𝑑 (𝑢, V)) 4 = 󵄨𝑥𝑛 (𝑢0)−𝑥(𝑢0)󵄨 󵄨 󵄨 { 󵄨𝑥 (𝑢) −𝑥(𝑢) −𝑥 (V) +𝑥(V)󵄨 + 󵄨 𝑛 𝑛 󵄨 , Henceweobtain max { sup 𝜔 (𝑑 (𝑢, V)) (𝑢,V)∈𝑋̃2 { 𝛿 𝜀 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨} 󵄨𝑥𝑛 (𝑢) −𝑥𝑛 (V)󵄨 ⩽ 𝜔 (𝑑 (𝑢, V)) (75) 󵄨𝑥𝑛 (𝑢) −𝑥(𝑢) −𝑥𝑛 (V) +𝑥(V)󵄨 4 sup } . (𝑢,V)∈𝑋2 𝜔 (𝑑 (𝑢, V)) 𝛿 } (70) for (𝑢, V)∈𝑋such that 𝑑(𝑢, V)⩽𝛿. 8 Journal of Function Spaces and Applications

Letting in the above inequality with 𝑛→∞,weget Henceweinferthat |𝑥 (𝑢) −𝑥(V)| 𝜀 ⩽𝑀 (81) |𝑥 (𝑢) −𝑥(V)| ⩽ 𝜔 (𝑑 (𝑢, V)) 𝜔 (𝑑 (𝑢, V)) 4 (76) 2 for all 𝑢, V ∈𝑋,𝑢≠V. 𝑃>0 for (𝑢, V)∈𝑋such that 𝑑(𝑢, V)⩽𝛿. Consequently, we obtain Further, fix an arbitrary number . According to our assumptions there exists 𝛿0 >0such that 𝜔2(𝑡)/𝜔1(𝑡) ⩽ 𝑃 for 𝑡∈(0,𝛿0].Thisyieldsthat |𝑥 (𝑢) −𝑥(V)| 𝜀 ⩽ (77) 𝜔 (𝑑 (𝑢, V)) 4 𝜔2 (𝑡) ⩽𝑃𝜔1 (𝑡) (82)

for 𝑡∈[0,𝛿0]. 2 for arbitrary (𝑢, V)∈𝑋𝛿. Linking (81)and(82), we deduce that Now, in view of (74)and(77), we derive the following estimates: |𝑥 (𝑢) −𝑥(V)| ⩽𝑃𝑀𝜔1 (𝑑 (𝑢, V)) (83)

󵄨 󵄨 󵄨 󵄨 for all 𝑢, V ∈𝑋such that 𝑑(𝑢, V)⩽𝛿0. 󵄨𝑥𝑛 (𝑢) −𝑥(𝑢) −𝑥𝑛 (V) +𝑥(V)󵄨 sup { } Now, take arbitrary 𝑢, V ∈𝑋such that 𝑑(𝑢, V)⩾𝛿0.Then (𝑢,V)∈𝑋2 𝜔 (𝑑 (𝑢, V)) 𝛿 we get 󵄨 󵄨 󵄨𝑥 (𝑢) −𝑥 (V)󵄨 + |𝑥 (𝑢) −𝑥(V)| 𝜀 ⩽ {󵄨 𝑛 𝑛 󵄨 }⩽ . |𝑥 (𝑢) −𝑥(V)| |𝑥 (𝑢) −𝑥(V)| sup ⩽ (𝑢,V)∈𝑋2 𝜔 (𝑑 (𝑢, V)) 2 𝛿 𝜔1 (𝑑 (𝑢, V)) 𝜔1 (𝛿0) (78) (84) |𝑥 (𝑢) −𝑥(V)| 𝜔 (𝑑 (𝑢, V)) = ⋅ 2 . 𝜔2 (𝑑 (𝑢, V)) 𝜔1 (𝛿0) Combining (70) with inequalities (72), (73), and (78), we ||𝑥 −𝑥||⩽ 𝜀 𝑛∈N,𝑛⩾𝑛 obtain that 𝑛 for 0.Thismeans Hence, in view of (81), we obtain that the sequence (𝑥𝑛) is convergent to the function 𝑥 with respect to the norm of the space 𝐶𝜔(𝑋).Finally,weconclude |𝑥 (𝑢) −𝑥(V)| 𝜔 ( 𝑋) 𝐴 𝐶 (𝑋) ⩽𝑀 2 diam that the set is relatively compact in the space 𝜔 and the 𝜔 (𝑑 (𝑢, V)) 𝜔 (𝛿 ) (85) proof is complete. 1 1 0

Now, based on Theorem 4,weproveamanageableand for all 𝑢, V ∈𝑋such that 𝑑(𝑢, V)⩾𝛿0. handy sufficient condition for relative compactness in the Finally, joining (81), (82), and (85), we deduce that the set 𝐶 (𝑋) 𝐴 𝐶 (𝑋) space 𝜔 . is bounded in the space 𝜔1 . In what follows, let us fix arbitrarily a number 𝜀1 >0. Theorem 5. Assume that 𝜔1, 𝜔2 are moduli of continuity being Keeping in mind the assumption requiring that 𝜔2(𝑡) = continuous at zero and such that 𝜔2(𝜀) = 𝑜(𝜔1(𝜀)) as 𝜀→0, 𝑜(𝜔1(𝑡)) as 𝑡→0,wecanfind𝛿>0such that that is, 𝜔2 (𝑡) ⩽𝜀1 (86) 𝜔1 (𝑡) 𝜔2 (𝜀) lim =0. (79) 𝜀→0𝜔 (𝜀) 1 provided 0<𝑡⩽𝛿. Further, fix an arbitrary number 𝜀>0. Then, according 𝜀 =𝜀/𝑀 𝛿>0 Further, assume that (𝑋, 𝑑) is a compact metric space. Then, if to (86), for the number 1 let us choose .Then, 𝐴 𝐶 (𝑋) 𝐴 for 𝑢, V ∈𝑋such that 𝑢 ≠V and 𝑑(𝑢, V)⩽𝛿we get is a bounded subset of the space 𝜔2 then is relatively compact in the space 𝐶𝜔 (𝑋). 1 𝜔 (𝑑 (𝑢, V)) 𝜀 2 ⩽ . 𝐴 𝐶 (𝑋) 𝜔 (𝑑 (𝑢, V)) 𝑀 (87) Proof. Since is bounded in the space 𝜔2 , then there 1 exists a constant 𝑀>0such that ||𝑥|| ⩽ 𝑀 for any 𝑥∈𝐴,or equivalently Henceweobtain 𝜀 𝜔2 (𝑑 (𝑢, V)) ⩽ 𝜔1 (𝑑 (𝑢, V)) (88) 󵄨 󵄨 |𝑥 (𝑢) −𝑥(V)| 𝑀 󵄨𝑥(𝑢0)󵄨 + sup { :𝑢,V ∈𝑋,𝑢≠V}⩽𝑀. 𝜔2 (𝑑 (𝑢, V)) (80) for 𝑢, V ∈𝑋,𝑢≠V and 𝑑(𝑢, V)⩽𝛿. Journal of Function Spaces and Applications 9

Next, taking 𝑢, V ∈𝑋such that 𝑢 ≠V, 𝑑(𝑢, V)⩽𝛿and described in Example 1) if and only if the function 𝑓 is linear; keeping in mind (81), we derive the following estimate: that is, there exist functions 𝑝, 𝑞 ∈ Lip[𝑎, 𝑏] such that

|𝑥 (𝑢) −𝑥(V)| ⩽𝑀𝜔2 (𝑑 (𝑢, V)) ⩽𝜀𝜔1 (𝑑 (𝑢, V)) . (89) 𝑓 (𝑡, 𝑥) =𝑝(𝑡) +𝑞(𝑡) 𝑥 (93)

Hencewehave for all 𝑡∈[𝑎,𝑏]and 𝑥∈R.Otherresultsinthisdirectioncan |𝑥 (𝑢) −𝑥(V)| be found in [5], for instance. ⩽𝜀 This shows that the investigations on the existence of solu- 𝜔 (𝑑 (𝑢, V)) (90) 1 tions of operator equations in the space 𝐶𝜔(𝑋) consisting of functions with growths tempered by a modulus of continuity for 𝑢, V ∈𝑋such that 𝑢 ≠V and 𝑑(𝑢, V)⩽𝛿. are rather a delicate matter. Finally, in view of Theorem 4 we conclude that the set 𝐴 As we indicate above, the object of the study conducted is relatively compact in the space 𝐶𝜔 (𝑋).Thiscompletesthe 1 in this section is the quadratic integral equation of Fredholm proof. type having the form

Example 6. In order to illustrate the applicability of 𝑏 Theorem 5, let us consider two moduli of continuity of 𝑥 (𝑡) =𝑝(𝑡) +𝑥(𝑡) ∫ 𝑘 (𝑡, 𝜏) 𝑥 (𝜏) 𝑑𝜏, (94) 𝛼 𝛽 𝑎 Holder¨ type having the form 𝜔1(𝜀) = 𝜀 ,𝜔2(𝜀) = 𝜀 ,where 0<𝛼<𝛽⩽1 (cf. Section 3). Then we have where 𝑡 ∈ [𝑎, 𝑏]. In our further considerations we will assume that 𝛽 is a 𝜔2 (𝜀) 𝛽−𝛼 lim = lim𝜀 =0. (91) fixed number in the interval (0, 1]. 𝜀→0𝜔 (𝜀) 𝜀→0 1 Now, we formulate the assumptions under which we will study (94). This shows that the moduli of continuity 𝜔1(𝜀) and 𝜔2(𝜀) satisfy the conditions of Theorem 5. (i) The function 𝑝 = 𝑝(𝑡) belongs to the Holder¨ space 𝐴 Thus, if we assume that a set isboundedinthespace 𝐻𝛽[𝑎, 𝑏]. 𝐶𝜔 ([𝑎, 𝑏]) or 𝐶𝜔 (𝑋),then𝐴 is relatively compact in the 2 𝐶 ([𝑎, 𝑏]) 2 𝐶 (𝑋) (𝑋, 𝑑) (ii) 𝑘 : [𝑎, 𝑏] × [𝑎, 𝑏]→ R is a continuous function space 𝜔1 or 𝜔1 ,respectively,provided is acompactmetricspace. such that it satisfies the Holder¨ condition with the 𝛼 𝛽 0<𝛼<1 exponent 𝛽 with respect to the first variable; that is, In other words, if there exist numbers , with 𝑘 >0 and 𝛼<𝛽⩽1such that 𝐴 is a bounded set in the Holder¨ there exists a constant 𝛽 such that space with the exponent 𝛽, that is, there exists a constant 𝑀> 𝑘 (𝑡, 𝜏) −𝑘(𝑠, 𝜏) ⩽𝑘 𝑡−𝑠𝛽 0 such that | | 𝛽| | (95)

𝛽 |𝑥 (𝑢) −𝑥(V)| ⩽𝑀(𝑑 (𝑢, V)) (92) for all 𝑡, 𝑠, 𝜏 ∈ [𝑎,𝑏]. for any 𝑥∈𝐴and for all 𝑢, V ∈𝑋, then the set 𝐴 is relatively In what follows, based on the above assumptions, we can 𝐾 compact in the Holder¨ space with the exponent 𝛼. define the finite constant by putting 𝑏 𝐾=sup {∫ |𝑘 (𝑡, 𝜏)| 𝑑𝜏 : 𝑡∈ [𝑎,] 𝑏 }. (96) 5. Application to a Quadratic 𝑎 Integral Equation Now, we are prepared to present our last assumption. In this final section we are going to present an application of the results obtained in previous sections to derive an (iii) The following inequality holds: existence result for a quadratic integral equation of Fredholm type. Equations of such a type occur naturally in connection 󵄩 󵄩 𝛽 2 1 󵄩𝑝󵄩𝛽(max {1, (𝑏−𝑎) }) (2𝐾 +𝛽 𝑘 (𝑏−𝑎)) < , (97) with some problems investigated in the theories of radiative 4 transfer, neutron transport, and the kinetic theory of gases where ||𝑝||𝛽 denotes the norm of the function 𝑝 in the [7–11]. Those integral equations are known as Chandrasekhar space 𝐻𝛽[𝑎, 𝑏]. quadratic integral equations (cf. [10]). Let us pay attention to some important facts connected Now, we can formulate our existence result concerning with our considerations. (94). Firstofall,noticethatthestudyoftheexistenceof solutions of functional, differential, and integral equations Theorem 7. Under assumptions (𝑖)–(𝑖𝑖𝑖) (94) has at least one is very complicated in the Lipschitz and Holder¨ function solution belonging to the space 𝐻𝛼[𝑎, 𝑏],where𝛼 is arbitrarily spaces which were considered in Section 3.Tojustifysuchan fixed number such that 0<𝛼<𝛽. opinion, let us recall a result due to Matkowski [6]asserting that the so-called superposition operator generated by the Proof. At the beginning, let us notice that in view of inclu- function 𝑓=𝑓(𝑡,𝑥)maps the space Lip[𝑎, 𝑏] into itself and is sions (47)wehavethatifafunction𝑓 satisfies the Holder¨ Lipschitzian (with respect to the norm in the space Lip[𝑎, 𝑏] condition with the exponent 𝛽,thenitsatisfiesthiscondition 10 Journal of Function Spaces and Applications

with the exponent 𝛼. Thus, from assumptions (i) and (ii) it Hence, in view of the inequalities ||𝑥||∞ ⩽ 𝑘 𝛽 𝛽 follows that there exists a constant 𝛼 such that max{1, (𝑏 − 𝑎) }‖𝑥‖𝛽,𝐻𝑥 ⩽‖𝑥‖𝛽 (cf. (39)and(41)), we derive the following estimate:

𝛼 |𝑘 (𝑡, 𝜏) −𝑘(𝑠, 𝜏)| ⩽𝑘𝛼|𝑡−𝑠| , 󵄨 󵄨 (98) 󵄨𝑝 (𝑡) −𝑝(𝑠)󵄨 ⩽𝐻𝛼|𝑡−𝑠|𝛼 𝛽 𝛽 2 󵄨 󵄨 𝑝 ||𝐹𝑥||𝛽 ⩽ |(𝐹𝑥)(𝑎)|+𝐻𝑝 +𝐾‖𝑥‖∞𝐻𝑥 +𝑘𝛽 (𝑏−𝑎) ‖𝑥‖∞

󵄩 󵄩 2 𝛼 ⩽ 󵄩𝑝󵄩𝛽 +𝐾‖𝑥‖∞ +𝐾‖𝑥‖∞‖𝑥‖𝛽 for all 𝑡, 𝑠, 𝜏 ∈ [𝑎,𝑏],wheretheconstant𝐻𝑝 associated with the function 𝑝 is defined by (36). 𝐹 𝛽 2 2 Now, let us consider the operator defined on the space +𝑘𝛽 (𝑏−𝑎) (max {1, (𝑏−𝑎) }) ‖𝑥‖𝛽 𝐻𝛽[𝑎, 𝑏] by the formula 󵄩 󵄩 𝛽 2 2 ⩽ 󵄩𝑝󵄩𝛽 +2𝐾(max {1, (𝑏−𝑎) }) ‖𝑥‖𝛽 𝑏 (𝐹𝑥)(𝑡) =𝑝(𝑡) +𝑥(𝑡) ∫ 𝑘 (𝑡, 𝜏) 𝑥 (𝜏) 𝑑𝜏, (99) 𝛽 2 2 𝑎 +𝑘𝛽 (𝑏−𝑎) (max {1, (𝑏−𝑎) }) ‖𝑥‖𝛽

󵄩 󵄩 𝛽 2 2 for 𝑡∈[𝑎,𝑏]. ⩽ 󵄩𝑝󵄩𝛽 +(max {1, (𝑏−𝑎) }) (2𝐾 +𝛽 𝑘 (𝑏−𝑎)) ‖𝑥‖𝛽. Further, take an arbitrary function 𝑥∈𝐻𝛽[𝑎, 𝑏].Then,for (101) arbitrarily fixed 𝑡, 𝑠 ∈ [𝑎, 𝑏], in view of assumptions (i) and (ii) we obtain

This shows that the operator 𝐹 transforms the space 𝐻𝛽[𝑎, 𝑏] |(𝐹𝑥)(𝑡) − (𝐹𝑥)(𝑠)| into itself. Moreover, in view of assumption (iii),weinferthat 𝛽 |𝑡−𝑠| 𝐹 transforms into itself the ball 𝐵𝛽 (in the space 𝐻𝛽[𝑎, 𝑏]) 𝜃 𝑟 𝑟 󵄨 󵄨 centered at the zero function and with radius 0,where 0 is 󵄨𝑝 (𝑡) −𝑝(𝑠)󵄨 ⩽ an arbitrary number from the interval [𝑟1,𝑟2],while |𝑡−𝑠|𝛽 󵄨 󵄨 󵄨 𝑏 𝑏 󵄨 󵄨𝑥 (𝑡) ∫ 𝑘 (𝑡, 𝜏) 𝑥 (𝜏) 𝑑𝜏 −𝑥 (𝑠) ∫ 𝑘 (𝑠, 𝜏) 𝑥 (𝜏) 𝑑𝜏󵄨 + 󵄨 𝑎 𝑎 󵄨 𝛽 󵄩 󵄩 |𝑡−𝑠| 1−√1−4𝛾󵄩𝑝󵄩𝛽 𝑟 = , 󵄨 󵄨 1 󵄨 𝑏 𝑏 󵄨 2𝛾 󵄨𝑥 (𝑡) ∫ 𝑘 (𝑡, 𝜏) 𝑥 (𝜏) 𝑑𝜏 −𝑥 (𝑠) ∫ 𝑘 (𝑡, 𝜏) 𝑥 (𝜏) 𝑑𝜏󵄨 ⩽𝐻𝛽 + 󵄨 𝑎 𝑎 󵄨 (102) 𝑝 𝛽 |𝑡−𝑠| 󵄩 󵄩 1+√1−4𝛾󵄩𝑝󵄩 󵄨 󵄨 𝛽 󵄨 𝑏 𝑏 󵄨 𝑟2 = , 󵄨𝑥 (𝑠) ∫ 𝑘 (𝑡, 𝜏) 𝑥 (𝜏) 𝑑𝜏 −𝑥 (𝑠) ∫ 𝑘 (𝑠, 𝜏) 𝑥 (𝜏) 𝑑𝜏󵄨 2𝛾 + 󵄨 𝑎 𝑎 󵄨 |𝑡−𝑠|𝛽

𝑏 𝛽 |𝑥 (𝑡) −𝑥(𝑠)| ⩽𝐻 + ∫ |𝑘 (𝑡, 𝜏)||𝑥 (𝜏)| 𝑑𝜏 𝛽 2 𝑝 𝛽 𝛾=( {1, (𝑏 − 𝑎) }) (2𝐾 + 𝑘 (𝑏 − 𝑎)) |𝑡−𝑠| 𝑎 and max 𝛽 . Now, observe that in view of (47)theball𝐵𝛽 is contained 𝑏 ∫ |𝑘 (𝑡, 𝜏) 𝑥 (𝜏) −𝑘(𝑠, 𝜏)||𝑥 (𝜏)| 𝑑𝜏 in the space 𝐻𝛼[𝑎, 𝑏]. Further, keeping in mind the fact + 𝑥 𝑠 𝑎 | ( )| establishedinExample6,weconcludethattheset𝐵𝛽 is |𝑡−𝑠|𝛽 relatively compact in the space 𝐻𝛼[𝑎, 𝑏]. Moreover, taking 𝑏 𝐵𝛽 𝛽 𝛽 into account estimate (50), we deduce that is also closed ⩽𝐻𝑝 +𝐻𝑥 ∫ |𝑘 (𝑡, 𝜏)|‖𝑥‖∞𝑑𝜏 in the space 𝐻𝛼[𝑎, 𝑏]. 𝑎 Thus, gathering the above obtained facts, we infer that the 𝑏 𝛽 ‖𝑥‖ ∫ 𝑘 |𝑡−𝑠| 𝑑𝜏 set 𝐵𝛽 is a compact and (obviously) convex subset of the space ∞ 𝑎 𝛽 + ‖𝑥‖ 𝐻𝛼[𝑎, 𝑏]. ∞ |𝑡−𝑠|𝛽 In what follows, we will treat the ball 𝐵𝛽 as a subset of the 𝛽 𝛽 2 space 𝐻𝛼[𝑎, 𝑏]. ⩽𝐻𝑝 +𝐻𝑥 ‖𝑥‖∞𝐾+‖𝑥‖∞𝑘𝛽 (𝑏−𝑎) . We now show that the operator 𝐹 is continuous on the (100) space 𝐻𝛼[𝑎, 𝑏]. Journal of Function Spaces and Applications 11 󵄨 󵄨 󵄨[𝑥 (𝑡) −𝑦(𝑡)]−[𝑥(𝑠) −𝑦(𝑠)]󵄨 𝑏 To this end, fix arbitrarily 𝑥∈𝐻𝛼[𝑎, 𝑏] and a number 𝛿> 󵄨 󵄨 ⩽ 𝛼 ‖𝑥‖∞ ∫ |𝑘 (𝑡, 𝜏)| 𝑑𝜏 0. Assume that 𝑦∈𝐻𝛼[𝑎, 𝑏] is an arbitrary function such that |𝑡−𝑠| 𝑎 ||𝑥 − 𝑦|| ⩽𝛿 𝑡, 𝑠 ∈ [𝑎, 𝑏] 𝛼 . Then, for arbitrary we obtain 󵄨 󵄨 +{󵄨[𝑥 (𝑠) −𝑦(𝑠)]−[𝑥(𝑎) −𝑦(𝑎)]󵄨

𝑏 󵄨 󵄨 ∫ |𝑘 (𝑡, 𝜏) −𝑘(𝑠, 𝜏)| 𝑑𝜏 + 󵄨𝑥 (𝑎) −𝑦(𝑎)󵄨} ‖𝑥‖ 𝑎 󵄨 󵄨 󵄨 󵄨 ∞ 𝛼 󵄨[(𝐹𝑥)(𝑡) −(𝐹𝑦)(𝑡)]−[(𝐹𝑥)(𝑠) −(𝐹𝑦)(𝑠)]󵄨 |𝑡−𝑠| 𝛼 󵄨 |𝑡−𝑠| 󵄨 𝑏 󵄨 + 󵄨(𝑦 (𝑡) ∫ 𝑘 (𝑡, 𝜏) [𝑥 (𝜏) −𝑦(𝜏)]𝑑𝜏 󵄨 𝑏 𝑏 󵄨 𝑎 󵄨[𝑥 (𝑡) ∫ 𝑘 (𝑡, 𝜏) 𝑥 (𝜏) 𝑑𝜏 −𝑦 (𝑡) ∫ 𝑘 (𝑡, 𝜏) 𝑦 (𝜏) 𝑑𝜏] 󵄨 𝑎 𝑎 = 󵄨 𝑏 󵄨 󵄨 𝛼 󵄨 󵄨 |𝑡−𝑠| −𝑦 (𝑠) ∫ 𝑘 (𝑡, 𝜏) [𝑥 (𝜏) −𝑦(𝜏)]𝑑𝜏)󵄨 (|𝑡−𝑠|𝛼)−1 󵄨 󵄨 𝑎 󵄨 𝑏 𝑏 󵄨 󵄨 [𝑥 (𝑠) ∫ 𝑘 (𝑠, 𝜏) 𝑥 (𝜏) 𝑑𝜏 −𝑦 (𝑠) ∫ 𝑘 (𝑠, 𝜏) 𝑦 (𝜏) 𝑑𝜏]󵄨 󵄨 𝑏 𝑎 𝑎 󵄨 + 󵄨(𝑦 𝑠 ∫ 𝑘 𝑡, 𝜏 [𝑥 𝜏 −𝑦 𝜏 ]𝑑𝜏 − 󵄨 󵄨 ( ) ( ) ( ) ( ) |𝑡−𝑠|𝛼 󵄨 󵄨 𝑎 󵄨 󵄨 󵄨 𝑏 󵄨 󵄨 󵄨 𝛼 −1 󵄨 𝑏 𝑏 −𝑦 (𝑠) ∫ 𝑘 (𝑠, 𝜏) [𝑥 (𝜏) −𝑦(𝜏)]𝑑𝜏)󵄨 (|𝑡−𝑠| ) 󵄨[𝑥 (𝑡) ∫ 𝑘 (𝑡, 𝜏) 𝑥 (𝜏) 𝑑𝜏 −𝑦 (𝑡) ∫ 𝑘 (𝑡, 𝜏) 𝑥 (𝜏) 𝑑𝜏] 𝑎 󵄨 ⩽ 󵄨 𝑎 𝑎 󵄨 𝛼 󵄨 |𝑡−𝑠| 𝑏 󵄨 󵄩 󵄩 ⩽ 󵄩𝑥−𝑦󵄩𝛼‖𝑥‖∞ sup {∫ |𝑘 (𝑡, 𝜏)| 𝑑𝜏 : 𝑡∈ [𝑎,] 𝑏 } 𝑎 𝑏 𝑏 [𝑦 (𝑡) ∫ 𝑘 (𝑡, 𝜏) 𝑥 (𝜏) 𝑑𝜏 −𝑦 (𝑡) ∫ 𝑘 (𝑡, 𝜏) 𝑦 (𝜏) 𝑑𝜏] 𝑎 𝑎 󵄨 󵄨 + + sup {󵄨[𝑥 (𝑡) −𝑦(𝑡)]−[𝑥(𝑠) −𝑦(𝑠)]󵄨} |𝑡−𝑠|𝛼 𝑡,𝑠∈[𝑎,𝑏]

𝑏 𝑏 𝑏 𝑘 𝑡−𝑠𝛽 [𝑥 (𝑠) ∫ 𝑘 (𝑠, 𝜏) 𝑥 (𝜏) 𝑑𝜏 −𝑦 (𝑠) ∫ 𝑘 (𝑠, 𝜏) 𝑥 (𝜏) 𝑑𝜏] 𝛽| | 𝑎 𝑎 × ‖𝑥‖ ∫ 𝑑𝜏 − ∞ |𝑡−𝑠|𝛼 |𝑡−𝑠|𝛼 𝑎 𝑏 𝛽 𝑏 𝑏 󵄨 󵄨 󵄨 ∫ 𝑘𝛽|𝑡−𝑠| [𝑦 (𝑠) ∫ 𝑘 (𝑠, 𝜏) 𝑥 (𝜏) 𝑑𝜏 −𝑦 (𝑠) ∫ 𝑘 (𝑠, 𝜏) 𝑦 (𝜏) 𝑑𝜏]󵄨 + 󵄨𝑥 (𝑎) −𝑦(𝑎)󵄨 ‖𝑥‖ 𝑎 𝑑𝜏 𝑎 𝑎 󵄨 󵄨 󵄨 ∞ 𝛼 − 󵄨 |𝑡−𝑠| |𝑡−𝑠|𝛼 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨𝑦 (𝑡) −𝑦(𝑠)󵄨 𝑏 󵄨 󵄨 + 󵄨 󵄨 ∫ |𝑘 (𝑡, 𝜏)| 󵄨𝑥 (𝜏) −𝑦(𝜏)󵄨 𝑑𝜏 󵄨 𝛼 󵄨 󵄨 󵄨 𝑏 |𝑡−𝑠| 𝑎 = 󵄨([𝑥 (𝑡) −𝑦(𝑡)] ∫ 𝑘 (𝑡, 𝜏) 𝑥 (𝜏) 𝑑𝜏 󵄨 𝑏 󵄨 𝑎 󵄨 󵄨 |𝑘 (𝑡, 𝜏) −𝑘(𝑠, 𝜏)| 󵄨 󵄨 + 󵄨𝑦 (𝑠)󵄨 ∫ 󵄨𝑥 (𝜏) −𝑦(𝜏)󵄨 𝑑𝜏 󵄨 󵄨 |𝑡−𝑠|𝛼 󵄨 󵄨 𝑏 𝑎 𝛼 −1 +𝑦 (𝑡) ∫ 𝑘 (𝑡, 𝜏) [𝑥 (𝜏) −𝑦(𝜏)]𝑑𝜏)(|𝑡−𝑠| ) 󵄩 󵄩 𝛽−𝛼+1 𝑎 ⩽𝐾‖𝑥‖∞󵄩𝑥−𝑦󵄩𝛼 +𝑘𝛽(𝑏−𝑎) ‖𝑥‖∞ 󵄨 󵄨 𝑏 󵄨[𝑥 (𝑡) −𝑦(𝑡)]−[𝑥(𝑠) −𝑦(𝑠)]󵄨 −([𝑥(𝑠) −𝑦(𝑠)] ∫ 𝑘 (𝑠, 𝜏) 𝑥 (𝜏) 𝑑𝜏 × sup { 𝛼 𝑎 |𝑡−𝑠| 󵄨 𝑏 󵄨 𝛼 𝛼 −1󵄨 × |𝑡−𝑠| :𝑡,𝑠∈[𝑎,] 𝑏 ,𝑡=𝑠}̸ +𝑦 (𝑠) ∫ 𝑘 (𝑠, 𝜏) [𝑥 (𝜏) −𝑦(𝜏)]𝑑𝜏)(|𝑡−𝑠| ) 󵄨 𝑎 󵄨 󵄨 󵄨 𝛽−𝛼+1 󵄨 󵄨 󵄨 𝑏 󵄨 + 󵄨𝑥 (𝑎) −𝑦(𝑎)󵄨 ‖𝑥‖∞𝑘𝛽(𝑏−𝑎) 󵄨[𝑥 (𝑡) −𝑦(𝑡)]−[𝑥(𝑠) −𝑦(𝑠)]󵄨 󵄨∫ 𝑘 (𝑡, 𝜏) 𝑥 (𝜏) 𝑑𝜏󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝑎 󵄨 󵄨 󵄨 𝛼 𝛼 ⩽ 𝛼 𝐻 𝑡−𝑠 𝑏 |𝑡−𝑠| 𝑦 | | + 𝛼 ∫ |𝑘 (𝑡, 𝜏)| |𝑡−𝑠| 𝑎 󵄨 󵄨 󵄨 𝑏 󵄨 󵄨𝑥 (𝑠) −𝑦(𝑠)󵄨 󵄨∫ [𝑘 (𝑡, 𝜏) −𝑘(𝑠, 𝜏)] 𝑥 (𝜏) 𝑑𝜏󵄨 󵄨 󵄨 󵄨 𝑎 󵄨 + 󵄨 󵄨 |𝑡−𝑠|𝛼 ×{sup {󵄨[𝑥 (𝑡)−𝑦(𝑡)]−[𝑥(𝑎)−𝑦(𝑎)]󵄨 𝑡∈[𝑎,𝑏] 󵄨 󵄨 𝑏 󵄨 󵄨 󵄨 + 󵄨(𝑦 (𝑡) ∫ 𝑘 (𝑡, 𝜏) [𝑥 (𝜏) −𝑦(𝜏)]𝑑𝜏 + 󵄨[𝑥 (𝑎) −𝑦(𝑎)]󵄨}}𝑑𝜏 󵄨 𝑎 󵄨 󵄨 󵄨 𝑏 󵄨 󵄩 󵄩 󵄨 𝛼 −1 + 󵄩𝑦󵄩 −𝑦 (𝑠) ∫ 𝑘 (𝑠, 𝜏) [𝑥 (𝜏) −𝑦(𝜏)]𝑑𝜏)󵄨 (|𝑡−𝑠| ) ∞ 𝑎 󵄨 12 Journal of Function Spaces and Applications

󵄩 󵄩 󵄨 󵄨 𝑏 𝑘 𝑡−𝑠 𝛽 𝛼 𝛼󵄩 󵄩 𝛼 󵄨 󵄨 𝛽| | 󵄨 󵄨 +𝐻𝑦 𝐾(𝑏−𝑎) 󵄩𝑥−𝑦󵄩 +𝐻𝑦 𝐾 󵄨𝑥 (𝑎) −𝑦(𝑎)󵄨 × ∫ { {󵄨[𝑥 (𝑡)−𝑦(𝑡)]−[𝑥(𝑎)−𝑦(𝑎)]󵄨 𝛼 |𝑡−𝑠|𝛼 sup 󵄨 󵄨 𝑎 𝑡∈[𝑎,𝑏] 󵄩 󵄩 𝛽+1󵄩 󵄩 +𝑘𝛽󵄩𝑦󵄩∞(𝑏−𝑎) 󵄩𝑥−𝑦󵄩𝛼 󵄨 󵄨 + 󵄨[𝑥 (𝑎) −𝑦(𝑎)]󵄨}}𝑑𝜏 𝛽−𝛼+1󵄩 󵄩 󵄨 󵄨 󵄨 󵄨 +𝑘𝛽(𝑏−𝑎) 󵄩𝑦󵄩∞ 󵄨𝑥 (𝑎) −𝑦(𝑎)󵄨 . 󵄩 󵄩 (103) ⩽𝐾‖𝑥‖∞󵄩𝑥−𝑦󵄩𝛼

𝛽−𝛼+1 +𝑘𝛽(𝑏−𝑎) ‖𝑥‖∞ 󵄨 󵄨 Further, let us observe that we have the following esti- 󵄨[𝑥 (𝑡) −𝑦(𝑡)]−[𝑥(𝑠) −𝑦(𝑠)]󵄨 × {󵄨 󵄨 mates: sup |𝑡−𝑠|𝛼

𝛼 󵄨 󵄨 :𝑡,𝑠∈[𝑎,] 𝑏 ,𝑡=𝑠}̸ (𝑏−𝑎) 󵄨(𝐹𝑥)(𝑎) −(𝐹𝑦)(𝑎)󵄨 󵄨 󵄨 𝑏 𝛽−𝛼+1 󵄨 󵄨 = 󵄨𝑝 𝑎 +𝑥 𝑎 ∫ 𝑘 𝑎, 𝜏 𝑥 𝜏 𝑑𝜏 +𝑘𝛽(𝑏−𝑎) ‖𝑥‖∞ 󵄨𝑥 (𝑎) −𝑦(𝑎)󵄨 󵄨 ( ) ( ) ( ) ( ) 󵄨 𝑎 󵄨 󵄨 +𝐻𝛼 {󵄨[𝑥 (𝑡) −𝑦(𝑡)]−[𝑥(𝑠) −𝑦(𝑠)]󵄨 󵄨 𝑦 sup 󵄨 󵄨 𝑏 󵄨 󵄨 −𝑝 (𝑎) −𝑦(𝑎) ∫ 𝑘 (𝑎, 𝜏) 𝑦 (𝜏) 𝑑𝜏󵄨 𝑏 𝑎 󵄨 :𝑡,𝑠∈[𝑎,] 𝑏 } ∫ |𝑘 (𝑡, 𝜏)| 𝑑𝜏 𝑎 󵄨 𝑏 𝑏 󵄨 = 󵄨𝑥 (𝑎) ∫ 𝑘 (𝑎, 𝜏) 𝑥 (𝜏) 𝑑𝜏 −𝑥 (𝑎) ∫ 𝑘 (𝑎, 𝜏) 𝑦 (𝜏) 𝑑𝜏 𝑏 󵄨 𝛼 󵄨 󵄨 󵄨 𝑎 𝑎 +𝐻𝑦 󵄨𝑥 (𝑎) −𝑦(𝑎)󵄨 ∫ |𝑘 (𝑡, 𝜏)| 𝑑𝜏 𝑎 𝑏 +𝑥(𝑎) ∫ 𝑘 (𝑎, 𝜏) 𝑦 (𝜏) 𝑑𝜏 󵄩 󵄩 𝛽−𝛼+1 +𝑘𝛽󵄩𝑦󵄩∞(𝑏−𝑎) 𝑎 󵄨 󵄨 󵄨 𝑏 󵄨 × sup {󵄨[𝑥 (𝑡)−𝑦(𝑡)]−[𝑥(𝑠)−𝑦(𝑠)]󵄨 󵄨 −𝑦 (𝑎) ∫ 𝑘 (𝑎, 𝜏) 𝑦 (𝜏) 𝑑𝜏󵄨 𝑎 󵄨 :𝑡,𝑠∈[𝑎,] 𝑏 } 󵄨 󵄨 󵄨 𝑏 󵄨 𝛽−𝛼+1󵄩 󵄩 󵄨 󵄨 󵄨 󵄨 +𝑘 (𝑏−𝑎) 󵄩𝑦󵄩 󵄨𝑥 (𝑎) −𝑦(𝑎)󵄨 ⩽ 󵄨𝑥 (𝑎) ∫ 𝑘 (𝑎, 𝜏) (𝑥 (𝜏) −𝑦(𝜏))𝑑𝜏󵄨 𝛽 󵄩 󵄩∞ 󵄨 󵄨 󵄨 𝑎 󵄨 󵄩 󵄩 󵄩 󵄩 ⩽𝐾𝑥 󵄩𝑥−𝑦󵄩 +𝑘 (𝑏−𝑎)𝛽+1 𝑥 󵄩𝑥−𝑦󵄩 󵄨 𝑏 󵄨 ‖ ‖∞󵄩 󵄩𝛼 𝛽 ‖ ‖∞󵄩 󵄩𝛼 󵄨 󵄨 + 󵄨(𝑥 (𝑎) −𝑦(𝑎)) ∫ 𝑘 (𝑎, 𝜏) 𝑦 (𝜏) 𝑑𝜏󵄨 󵄨 󵄨 𝛽−𝛼+1 󵄨 󵄨 󵄨 𝑎 󵄨 +𝑘𝛽(𝑏−𝑎) ‖𝑥‖∞ 󵄨𝑥 (𝑎) −𝑦(𝑎)󵄨 𝑏 󵄨 󵄨 󵄨 󵄨 󵄨[𝑥 (𝑡) −𝑦(𝑡)]−[𝑥(𝑠) −𝑦(𝑠)]󵄨 ⩽ |𝑥 (𝑎)| ∫ |𝑘 (𝑎, 𝜏)| 󵄨𝑥 (𝜏) −𝑦(𝜏)󵄨 𝑑𝜏 +𝐻𝛼𝐾 {󵄨 󵄨 𝑎 𝑦 sup |𝑡−𝑠|𝛼 𝑏 󵄨 󵄨 󵄨 󵄨 + 󵄨𝑥 (𝑎) −𝑦(𝑎)󵄨 ∫ |𝑘 (𝑎, 𝜏)| 󵄨𝑦 (𝜏)󵄨 𝑑𝜏 :𝑡,𝑠∈[𝑎,] 𝑏 ,𝑡=𝑠}̸ (𝑏−𝑎)𝛼 𝑎 󵄩 󵄩 𝑏 ⩽ ‖𝑥‖ 󵄩𝑥−𝑦󵄩 ∫ |𝑘 (𝑎, 𝜏)| 𝑑𝜏 𝛼 󵄨 󵄨 ∞󵄩 󵄩∞ +𝐻𝑦 𝐾 󵄨𝑥 (𝑎) −𝑦(𝑎)󵄨 𝑎 󵄩 󵄩 𝛽−𝛼+1 󵄩 󵄩 󵄩 󵄩 𝑏 +𝑘𝛽󵄩𝑦󵄩 (𝑏−𝑎) 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩∞ + 󵄩𝑥−𝑦󵄩𝛼󵄩𝑦󵄩∞ ∫ |𝑘 (𝑎, 𝜏)| 𝑑𝜏 󵄨 󵄨 𝑎 󵄨[𝑥 (𝑡) −𝑦(𝑡)]−[𝑥(𝑠) −𝑦(𝑠)]󵄨 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 × {󵄨 󵄨 ⩽𝐾(‖𝑥‖ 󵄩𝑥−𝑦󵄩 + 󵄩𝑥−𝑦󵄩 󵄩𝑦󵄩 ). sup |𝑡−𝑠|𝛼 ∞󵄩 󵄩∞ 󵄩 󵄩𝛼󵄩 󵄩∞ (104)

:𝑡,𝑠∈[𝑎,] 𝑏 ,𝑡=𝑠}̸ (𝑏−𝑎)𝛼

󵄩 󵄩 󵄨 󵄨 Moreover, we have that ||𝑦||𝛼 ⩽ ||𝑥||𝛼 +𝛿and from (39)we +𝑘 (𝑏−𝑎)𝛽−𝛼+1󵄩𝑦󵄩 󵄨𝑥 (𝑎) −𝑦(𝑎)󵄨 𝛽 󵄩 󵄩∞ 󵄨 󵄨 derive the following estimate: 󵄩 󵄩 𝛽+1 󵄩 󵄩 ⩽𝐾‖𝑥‖∞󵄩𝑥−𝑦󵄩𝛼 +𝑘𝛽(𝑏−𝑎) ‖𝑥‖∞󵄩𝑥−𝑦󵄩𝛼 󵄨 󵄨 +𝑘 (𝑏−𝑎)𝛽−𝛼+1‖𝑥‖ 󵄨𝑥 (𝑎) −𝑦(𝑎)󵄨 𝛼 󵄩 󵄩 󵄨 󵄨 󵄩 󵄩 𝛽 ∞ 󵄨 󵄨 𝐻𝑦 = 󵄩𝑦󵄩𝛼 − 󵄨𝑦 (𝑎)󵄨 ⩽ 󵄩𝑦󵄩𝛼. (105) Journal of Function Spaces and Applications 13

Combining the above established facts and (41), in view of This shows that functions 𝑝(𝑡) and 𝑘(𝑡, 𝜏) involved in (107) (103)and(104), we obtain following estimates: satisfy assumptions (i) and (ii) of Theorem 7. Further, we can calculate that 󵄩 󵄩 2 󵄩𝑓𝑥 − 𝑓𝑦󵄩 ⩽𝐾( {1, (𝑏−𝑎)𝛼}) (2‖𝑥‖ +𝛿)𝛿 󵄩 󵄩𝛼 max 𝛼 1 𝛽+1 𝐾=sup {∫ |𝑘 (𝑡, 𝜏)| 𝑑𝜏 : 𝑡∈ [0, 1]} +(𝐾+𝑘𝛽(𝑏−𝑎) ) ‖𝑥‖∞ 𝛿 0

𝛽−𝛼+1 𝛼 1 +𝑘 𝑏−𝑎 𝑥 𝛿+𝐾𝑏−𝑎 ( 𝑥 +𝛿)𝛿 √3 2 𝛽( ) ‖ ‖∞ ( ) ‖ ‖𝛼 = sup {∫ 𝑚𝑡 +𝜏𝑑𝜏:𝑡∈[0, 1]} 0 (112) +𝐾(‖𝑥‖𝛼 +𝛿)𝛿 3 3 4 3 4 = sup { [√(𝑚𝑡2 +1) − √(𝑚𝑡2) ]:𝑡∈[0, 1]} 𝛽+1 𝛼 4 +𝑘𝛽(𝑏−𝑎) max {1, (𝑏−𝑎) }(‖𝑥‖𝛼 +𝛿)𝛿 3 𝛽−𝛼+1 𝛼 = [√3 (𝑚+1)4 − √3 𝑚4 ]. +𝑘𝛽(𝑏−𝑎) max {1, (𝑏−𝑎) }(‖𝑥‖𝛼 +𝛿)𝛿. 4 (106) Hencewededucethattheinequalityfromassumption(iii) is This shows that the operator 𝐹 is continuous at the point satisfied provided 𝑥∈𝐻𝛼[𝑎, 𝑏].Since𝑥 was chosen arbitrarily, we deduce that 3 3 4 3 1 𝐹 is continuous on the Holder¨ space 𝐻𝛼[𝑎, 𝑏].Particularly, (√𝑞 + √𝑟)√ { 3 𝑚 + [√(𝑚+1) − √𝑚4 ]} < . 2 4 (113) the operator 𝐹 maps continuously the ball 𝐵𝛽 into itself. Thus, taking into account the fact that the set 𝐵𝛽 is compact in the It is easy to check that the above inequality is satisfied if we space 𝐻𝛼[𝑎, 𝑏] and applying the classical Schauder fixed point put, for example, 𝑞 = 𝑟 = 1/216, 𝑚 =1/128. principle, we complete the proof. Finally, applying Theorem 7,weconcludethat(107)has 𝐻 [0, 1] 0<𝛼<1/2 Now, we illustrate the above obtained result by an exam- at least one solution in the space 𝛼 with , ple. provided inequality (113) is satisfied.

Example 8. Let us consider the following quadratic integral References equation:

1 [1] N. Danford and J. T. Schwartz, Linear Operators I,International 3 𝑥 (𝑡) = √𝑞𝑡 +𝑟 +𝑥(𝑡) ∫ √𝑚𝑡2 +𝜏𝑥 (𝜏) 𝑑𝜏, (107) Publishing, Leyden, The Netherlands, 1963. 0 [2]A.N.KolmogorovandS.V.Fom¯ın, Introductory Real Analysis, Dover Publications, New York, NY, USA, 1975. 𝑡∈[0,1] 𝑞, 𝑟, 𝑚 where and are positive constants. [3] W. Rudin, Functional Analysis,McGraw-Hill,NewYork,NY, Observe that (107)isaparticularcaseof(94)ifweput USA, 2nd edition, 1991. 𝑝(𝑡) = 𝑞𝑡 + 𝑟, 𝑘(𝑡, 𝜏)= √3 𝑚𝑡2 +𝜏 √ . Further, we get [4] J. Wloka, Funktionalanalysis und Anwendungen,Walterde Gruyter, Berlin, Germany, 1971. 󵄨 󵄨 1/2 󵄨𝑝 (𝑡) −𝑝(𝑠)󵄨 ⩽ √𝑞|𝑡−𝑠| . (108) [ 5 ] J. App e l l an d P. P. Z a bre j ko, Nonlinear Superposition Operators, vol. 95 of Cambridge Tracts in Mathematics, Cambridge Univer- Moreover,usingtheinequalityprovedin[12], we obtain sity Press, Cambridge, UK, 1990.

󵄨 3 3 󵄨 [6] J. Matkowski, “Functional equations and Nemytski˘ıoperators,” |𝑘 (𝑡, 𝜏) −𝑘(𝑠, 𝜏)| = 󵄨√𝑚𝑡2 +𝜏−√𝑚𝑠2 +𝜏󵄨 ⩽ √3 𝑚|𝑡−𝑠|2/3. 󵄨 󵄨 Funkcialaj Ekvaciog,vol.25,no.2,pp.127–132,1982. (109) [7]I.K.Argyros,“Onaclassofquadraticintegralequations with perturbation,” Functiones et Approximatio Commentarii Thus, in our considerations related to Theorem 7 we can take Mathematici,vol.20,pp.51–63,1992. 𝛽=1/2.Thenobviouslywehave [8]L.W.Busbridge,The Mathematics of Radiative Transfer,Cam- bridge University Press, Cambridge, UK, 1960. 2/3 1/2 1/6 |𝑘 (𝑡, 𝜏) −𝑘(𝑠, 𝜏)| ⩽ √3 𝑚|𝑡−𝑠| = √3 𝑚|𝑡−𝑠| |𝑡−𝑠| [9] K. M. Case and P. F. Zweifel, Linear Transport Theory, Addison- Wesley,Reading,Mass,USA,1967. 1/2 ⩽ √3 𝑚|𝑡−𝑠| . [10] S. Chandrasekhar, Radiative Transfer, Oxford University Press, (110) London, UK, 1950. [11] S. Hu, M. Khavanin, and W.Zhuang, “Integral equations arising 3 Henceweinferthat𝑘𝛽 =𝑘1/2 = √𝑚.Apartfromthis,weget in the kinetic theory of gases,” Applicable Analysis,vol.34,no. 3-4, pp. 261–266, 1989. 󵄨 󵄨 󵄩 󵄩 󵄨 󵄨 󵄨𝑝 (𝑡) −𝑝(𝑠)󵄨 [12] R. P. Agarwal, J. Bana´s, K. Bana´s, and D. O’Regan, “Solvability 󵄩𝑝󵄩 = 󵄨𝑝 (0)󵄨 + sup { :𝑡,𝑠∈[0, 1] ,𝑡=𝑠}̸ 1/2 |𝑡−𝑠|1/2 of a quadratic Hammerstein integral equation in the class of functions having limits at infinity,” Journal of Integral Equations = √𝑟+√𝑞. and Applications, vol. 23, no. 2, pp. 157–181, 2011. (111) Hindawi Publishing Corporation Journal of Function Spaces and Applications Volume 2013, Article ID 149659, 8 pages http://dx.doi.org/10.1155/2013/149659

Research Article Boundary Value Problems for a Class of Sequential Integrodifferential Equations of Fractional Order

Bashir Ahmad1 and Juan J. Nieto1,2

1 Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia 2 Departamento de Analisis´ Matematico,´ Facultad de Matematicas,´ Universidad de Santiago de Compostela, 15782 Santiago de Compostela, Spain

Correspondence should be addressed to Juan J. Nieto; [email protected]

Received 16 January 2013; Accepted 13 March 2013

Academic Editor: Jose Luis Sanchez

Copyright © 2013 B. Ahmad and J. J. Nieto. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We investigate the existence of solutions for a sequential integrodifferential equation of fractional order with some boundary conditions. The existence results are established by means of some standard tools of fixed point theory. An illustrative example is also presented.

1. Introduction In this paper, we consider a nonlinear Dirichlet boundary value problem of sequential fractional integrodifferential Nonlinear boundary value problems of fractional differential equations given by equationshavereceivedaconsiderableattentioninthe ( 𝑐 𝛼 +𝑘𝑐 𝛼−1)𝑢(𝑡) =𝑝𝑓(𝑡, 𝑢 (𝑡)) +𝑞𝐼𝛽𝑔 (𝑡, 𝑢 (𝑡)) , last few decades. One can easily find a variety of results 𝐷 𝐷 (1) ranging from theoretical analysis to asymptotic behavior and 0<𝑡<1, numerical methods for fractional equations in the literature on the topic. The interest in the subject has been mainly due to 𝑢 (0) =0, 𝑢(1) =0, (2) the extensive applications of fractional calculus in the mathe- 𝑐 𝛼 matical modeling of several real-world phenomena occurring where 𝐷 denotes the Caputo fractional derivative of order 𝛽 in physical and technical sciences; see, for example, [1–4]. An 𝛼,1<𝛼≤2, 𝐼 (⋅) denotes Riemann-Liouville integral with important feature of a fractional order differential operator, 0<𝛽<1, 𝑓, 𝑔 are given continuous functions, 𝑘 =0̸ ,and distinguishing it from an integer-order differential operator, 𝑝, 𝑞 arerealconstants.Wealsostudythefractionalintegro- is that it is nonlocal in nature. It means that the future differential equation1 ( )subjecttothefollowingboundary stateofadynamicalsystemorprocessbasedonafractional conditions: 󸀠 operator depends on its current state as well as its past states. 𝑢 (0) +𝑘𝑢(0) =𝑎,( 𝑢 1) =𝑏, 𝑎,𝑏∈R, (3) Thus, differential equations of arbitrary order are capable 󸀠 󸀠 of describing memory and hereditary properties of some 𝑢 (0) =𝑎, 𝑢 (0) =𝑢 (1) ,𝑎∈R. (4) important and useful materials and processes. This feature has fascinated many researchers, and they have shifted their focus to fractional order models from the classical integer- 2. Linear Fractional Differential Equations order models. For some recent work on the topic, we refer, For 𝛼 ∈ (1, 2], we consider the following linear fractional for instance, to [5–9]. Recently, in [10], the authors studied differential equation: sequential fractional differential equations with three-point 𝑐 𝛼 𝑐 𝛼−1 boundary conditions. ( 𝐷 +𝑘 𝐷 )𝑢(𝑡) =ℎ(𝑡) , (5) 2 Journal of Function Spaces and Applications

𝑐 𝛼 where 𝐷 denotes the Caputo fractional derivative of order Lemma 2. The unique solution of the problem (5)–(3) is given 𝑐 𝛼 𝑐 −1 𝛼.Rewriting(1)as 𝐷 (𝑢(𝑡) +𝑘 𝐷 𝑢(𝑡)) = ℎ(𝑡),wecan by write its solution as 𝑢 (𝑡) =𝑒𝑘(1−𝑡) 𝑡 −1 1 𝑢 (𝑡) +𝑘 𝑐 𝑢 (𝑡) = ∫ (𝑡−𝑠)𝛼−1ℎ (𝑠) 𝑑𝑠 +𝑐 +𝑐𝑡, 1 𝑠 𝛼−2 𝐷 0 1 (𝑏𝑘) −𝑎 −𝑘(1−𝑠) (𝑠−𝑥) Γ (𝛼) 0 ×[ −∫ 𝑒 (∫ ℎ (𝑥) 𝑑𝑥) 𝑑𝑠] (6) 𝑘 0 0 Γ (𝛼−1) 𝑡 𝑠 (𝑠−𝑥)𝛼−2 𝑎 where 𝑐0,𝑐1 are arbitrary constants. Now, (6) can be expressed + ∫ 𝑒−𝑘(𝑡−𝑠) (∫ ℎ (𝑥) 𝑑𝑥) 𝑑𝑠 + . as 0 0 Γ (𝛼−1) 𝑘 𝑡 𝑡 (14) 1 𝛼−1 𝑢 (𝑡) =−𝑘∫ 𝑢 (𝑠) 𝑑𝑠 + ∫ (𝑡−𝑠) ℎ (𝑠) 𝑑𝑠 Lemma 3. 0 Γ (𝛼) 0 The unique solution of (5) with the boundary (7) conditions (4) is +𝑐 +𝑐𝑡. 0 1 (1 − 𝑒−𝑘𝑡) 𝑢 (𝑡) =− Differentiating7 ( ), we obtain 𝑘(1−𝑒−𝑘)

𝑡 1 𝑠 𝛼−2 󸀠 1 𝛼−2 −𝑘(1−𝑠) (𝑠−𝑥) 𝑢 (𝑡) = −𝑘𝑢 (𝑡) + ∫ (𝑡−𝑠) ℎ (𝑠) 𝑑𝑠1 +𝑐 , (8) ×[𝑘∫ 𝑒 (∫ ℎ (𝑥) 𝑑𝑥) 𝑑𝑠 Γ (𝛼−1) 0 0 0 Γ (𝛼−1) (15) which can alternatively be written as 1 (1−𝑠)𝛼−2 − ∫ ℎ (𝑠) 𝑑𝑠] 0 Γ (𝛼−1) 󸀠 1 𝑡 (𝑢 (𝑡) 𝑒𝑘𝑡) =𝑒𝑘𝑡 ( ∫ (𝑡−𝑠)𝛼−2ℎ (𝑠) 𝑑𝑠 +𝑐 ). 1 (9) 𝛼−2 Γ (𝛼−1) 0 𝑡 𝑠 (𝑠 − 𝑥) + ∫ 𝑒−𝑘(𝑡−𝑠) (∫ ℎ (𝑥) 𝑑𝑥) 𝑑𝑠 +𝑎. 0 0 Γ (𝛼−1) Integrating from 0 to 𝑡,wehave

𝑡 −𝑘𝑡 −𝑘(𝑡−𝑠) 𝛼−1 3. Existence Results for 𝑢 (𝑡) =𝐴𝑒 + ∫ 𝑒 𝐼 ℎ (𝑠) 𝑑𝑠 + 𝐵, (10) 0 the Nonlinear Problems where 𝐴 and 𝐵 are arbitrary constants, and Let P = 𝐶([0, 1], R) denote the Banach space of all continuous functions from [0, 1] into R endowed with the usual 𝑡 𝛼−2 𝛼−1 (𝑡 − 𝑥) norm defined by ‖𝑥‖ = sup{|𝑥(𝑡)|, 𝑡 ∈ [0,. 1]} 𝐼 ℎ (𝑡) =∫ ℎ (𝑥) 𝑑𝑥. (11) 0 Γ (𝛼−1) In view of Lemma 1,wetransformproblem(1)-(2)toan equivalent fixed point problem as Lemma 1. The unique solution of the linear equation (5) 𝑢=V𝑢, subject to the Dirichlet boundary conditions (2) is given by (16) V : P → P −𝑘𝑡 where is defined by (1 − 𝑒 ) 1 𝑠 (𝑠 − 𝑥)𝛼−2 𝑢 (𝑡) = ∫ 𝑒−𝑘(1−𝑠) (∫ ℎ (𝑥) 𝑑𝑥) 𝑑𝑠 (V𝑢)(𝑡) −𝑘 (𝑒 −1) 0 0 Γ (𝛼−1) −𝑘𝑡 (1 − 𝑒 ) 1 𝑠 (𝑠−𝑥)𝛼−2 𝑡 𝑠 (𝑠 − 𝑥)𝛼−2 = ∫ 𝑒−𝑘(1−𝑠) (𝑝 ∫ 𝑓(𝑥,𝑢(𝑥)) 𝑑𝑥 −𝑘(𝑡−𝑠) −𝑘 + ∫ 𝑒 (∫ ℎ (𝑥) 𝑑𝑥) 𝑑𝑠. (𝑒 −1) 0 0 Γ (𝛼−1) 0 0 Γ (𝛼−1) (12) 𝑠 (𝑠−𝑥)𝛼+𝛽−2 +𝑞∫ Γ(𝛼+𝛽−1) Proof. Observe that the general solution of (5)isgivenby 0 (10). Using the given boundary conditions in (10), we find that ×𝑔(𝑥,𝑢(𝑥)) 𝑑𝑥) 𝑑𝑠 1 1 𝑠 (𝑠 − 𝑥)𝛼−2 𝐴=−𝐵= ∫ 𝑒−𝑘(1−𝑠) (∫ ℎ (𝑥) 𝑑𝑥) 𝑑𝑠. −𝑘 (1 − 𝑒 ) 0 0 Γ (𝛼−1) 𝑡 𝑠 𝛼−2 −𝑘(𝑡−𝑠) (𝑠−𝑥) (13) + ∫ 𝑒 (𝑝 ∫ 𝑓 (𝑥,𝑢(𝑥)) 𝑑𝑥 0 0 Γ (𝛼−1) 𝐴 𝐵 Substituting the values of and in (10)yieldsthesolution 𝑠 (𝑠 − 𝑥)𝛼+𝛽−2 (12). This completes the proof. +𝑞∫ 0 Γ(𝛼+𝛽−1) In the next two lemmas, we present the unique solutions of (5) with different kinds of boundary conditions. We do not ×𝑔(𝑥,𝑢(𝑥)) 𝑑𝑥) 𝑑𝑠. provide the proofs for these lemmas as they are similar to that of Lemma 1. (17) Journal of Function Spaces and Applications 3

In a similar manner, we can define a fixed point operator We only present the existence results for the problem (1)- V1 : P → P for the nonlinear problem (1)–(3) as follows: (2). Observe that problem (1)-(2) has solutions if the operator (V 𝑢) (𝑡) equation (16) has fixed points. 1 For computational convenience, we introduce the follow- (𝑏𝑘) −𝑎 1 ing constant: =𝑒𝑘(1−𝑡) [ − ∫ 𝑒−𝑘(1−𝑠) 𝑘 0 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝑠 𝛼−2 󵄨 −𝑘󵄨 󵄨 󵄨 󵄨 󵄨 (𝑠−𝑥) 2 󵄨1−𝑒 󵄨 [󵄨𝑝󵄨 Γ(𝛼+𝛽)+󵄨𝑞󵄨 Γ (𝛼)] ×(𝑝∫ 𝑓 (𝑥,𝑢(𝑥)) 𝑑𝑥 𝑄= . (20) 0 Γ (𝛼−1) |𝑘| Γ(𝛼+𝛽)Γ(𝛼)

𝑠 (𝑠−𝑥)𝛼+𝛽−2 +𝑞∫ 0 Γ(𝛼+𝛽−1) Theorem 4. Assume that 𝑓, 𝑔 : [0, 1] × R → R are continuous functions satisfying the following condition: ×𝑔(𝑥,𝑢(𝑥)) 𝑑𝑥) 𝑑𝑠] 󵄨 󵄨 (𝐴 ) 󵄨𝑓 (𝑡, 𝑢) −𝑓(𝑡, V)󵄨 ≤𝐿 |𝑢−V| , 𝑡 𝑠 (𝑠−𝑥)𝛼−2 1 󵄨 󵄨 1 + ∫ 𝑒−𝑘(𝑡−𝑠) (𝑝 ∫ 𝑓 (𝑥,𝑢(𝑥)) 𝑑𝑥 Γ 𝛼−1 󵄨 󵄨 0 0 ( ) 󵄨𝑔 (𝑡, 𝑢) −𝑔(𝑡, V)󵄨 ≤𝐿2 |𝑢−V| ,∀𝑡∈[0, 1] , (21) 𝑠 (𝑠−𝑥)𝛼+𝛽−2 𝐿 ,𝐿 >0,𝑢,V ∈ R. +𝑞∫ 1 2 0 Γ(𝛼+𝛽−1)

𝑎 Then, the boundary value problem (1)-(2) has a unique solution ×𝑔(𝑥,𝑢(𝑥)) 𝑑𝑥) 𝑑𝑠 + . 𝑘 if 𝐿<1/𝑄,where𝐿=max{𝐿1,𝐿2} and 𝑄 is given by (20). (18) Proof. Let us define 𝑀=max{𝑀1,𝑀2},where𝑀1,𝑀2 |𝑓(𝑡, 0)| =𝑀 , A fixed point operator V2 : P → P for the nonlinear are finite numbers given by sup𝑡∈[0,1] 1 problem (1)–(4) is defined by sup𝑡∈[0,1]|𝑔(𝑡, 0)|2 =𝑀 .Selecting𝑟 ≥ (𝑄𝑀)/(1,we −𝐿𝑄) show that V𝐵𝑟 ⊂𝐵𝑟,where𝐵𝑟 ={𝑢∈P :‖𝑢‖≤𝑟}.For (V 𝑢) (𝑡) 2 𝑢∈𝐵𝑟,wehave (1 − 𝑒−𝑘𝑡) =− −𝑘 󵄨 −𝑘𝑡 󵄨 𝑘(1−𝑒 ) 󵄨1−𝑒 󵄨 ‖(V𝑢)‖ ≤ {󵄨 󵄨 sup 󵄨 1−𝑒−𝑘 󵄨 1 𝑠 (𝑠−𝑥)𝛼−2 𝑡∈[0,1] 󵄨 󵄨 ×[𝑘∫ 𝑒−𝑘(1−𝑠) (𝑝 ∫ Γ (𝛼−1) 1 𝑠 𝛼−2 0 0 −𝑘(1−𝑠) 󵄨 󵄨 (𝑠−𝑥) × ∫ 𝑒 (󵄨𝑝󵄨 ∫ ×𝑓(𝑥,𝑢(𝑥)) 𝑑𝑥 0 0 Γ (𝛼−1) 󵄨 󵄨 × 󵄨𝑓 (𝑥,𝑢(𝑥))󵄨 𝑑𝑥 𝑠 (𝑠−𝑥)𝛼+𝛽−2 󵄨 󵄨 +𝑞∫ Γ(𝛼+𝛽−1) 𝑠 𝛼+𝛽−2 0 󵄨 󵄨 (𝑠−𝑥) + 󵄨𝑞󵄨 ∫ 0 Γ(𝛼+𝛽−1) ×𝑔(𝑥,𝑢(𝑥)) 𝑑𝑥) 𝑑𝑠 󵄨 󵄨 (19) × 󵄨𝑔(𝑥,𝑢(𝑥))󵄨𝑑𝑥)𝑑𝑠 1 (1−𝑠)𝛼−2 −𝑝∫ 𝑓 (𝑠, 𝑢 (𝑠)) 𝑑𝑠 𝑡 0 Γ (𝛼−1) + ∫ 𝑒−𝑘(𝑡−𝑠) 0 1 (1−𝑠)𝛼+𝛽−2 −𝑞 ∫ 𝑔 (𝑠, 𝑢 (𝑠)) 𝑑𝑠] 𝑠 (𝑠−𝑥)𝛼−2 0 Γ(𝛼+𝛽−1) 󵄨 󵄨 ×(󵄨𝑝󵄨 ∫ 0 Γ (𝛼−1) 𝑡 𝑠 𝛼−2 −𝑘(𝑡−𝑠) (𝑠−𝑥) 󵄨 󵄨 + ∫ 𝑒 (𝑝 ∫ 𝑓 (𝑥,𝑢(𝑥)) 𝑑𝑥 ×󵄨𝑓(𝑥,𝑢(𝑥))󵄨𝑑𝑥 0 0 Γ (𝛼−1) 󵄨 󵄨 𝛼+𝛽−2 𝑠 𝛼+𝛽−2 󵄨 󵄨 𝑠 (𝑠−𝑥) (𝑠−𝑥) + 󵄨𝑞󵄨∫ +𝑞∫ 󵄨 󵄨 Γ(𝛼+𝛽−1) 0 Γ(𝛼+𝛽−1) 0

󵄨 󵄨 ×𝑔(𝑥,𝑢(𝑥)) 𝑑𝑥) 𝑑𝑠 +𝑎. ×󵄨𝑔(𝑥,𝑢(𝑥))󵄨𝑑𝑥)𝑑𝑠} 4 Journal of Function Spaces and Applications

󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨1−𝑒−𝑘𝑡 󵄨 1 2 󵄨1−𝑒−𝑘󵄨 [󵄨𝑝󵄨 Γ(𝛼+𝛽)+󵄨𝑞󵄨 Γ (𝛼)] ≤ {󵄨 󵄨 ∫ 𝑒−𝑘(1−𝑠) 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 sup 󵄨 −𝑘 󵄨 ≤(𝐿𝑟) +𝑀 𝑡∈[0,1] 󵄨 1−𝑒 󵄨 0 |𝑘| Γ(𝛼+𝛽)Γ(𝛼)

𝑠 𝛼−2 󵄨 󵄨 (𝑠−𝑥) = (𝐿𝑟) +𝑀 𝑄≤𝑟, ×(󵄨𝑝󵄨 ∫ 0 Γ (𝛼−1) (22) 󵄨 󵄨 ×(󵄨𝑓(𝑥,𝑢(𝑥))−𝑓(𝑥,0)󵄨 which means that V𝐵𝑟 ⊂𝐵𝑟. 󵄨 󵄨 Now, for 𝑢, V ∈ P,weobtain + 󵄨𝑓 (𝑥,0)󵄨 )𝑑𝑥 󵄨 󵄨 ‖V𝑢−VV‖ 𝑠 (𝑠−𝑥)𝛼+𝛽−2 󵄨 󵄨 󵄨 −𝑘𝑡 󵄨 + 󵄨𝑞󵄨 ∫ 󵄨1−𝑒 󵄨 0 Γ(𝛼+𝛽−1) ≤ {󵄨 󵄨 sup 󵄨 1−𝑒−𝑘 󵄨 󵄨 󵄨 𝑡∈[0,1] 󵄨 󵄨 ×(󵄨𝑔 (𝑥,𝑢(𝑥)) −𝑔(𝑥,0)󵄨 󵄨 󵄨 1 𝑠 𝛼−2 −𝑘(1−𝑠) 󵄨 󵄨 (𝑠−𝑥) × ∫ 𝑒 (󵄨𝑝󵄨 ∫ 󵄨 󵄨 0 0 Γ (𝛼−1) + 󵄨𝑔 (𝑥,0)󵄨)𝑑𝑥)𝑑𝑠 󵄨 × 󵄨𝑓 (𝑠, 𝑢 (𝑠)) 𝑡 󵄨 −𝑘(𝑡−𝑠) + ∫ 𝑒 󵄨 0 −𝑓(𝑠, V (𝑠))󵄨 𝑑𝑥

𝑠 𝛼−2 𝑠 𝛼+𝛽−2 󵄨 󵄨 (𝑠−𝑥) 󵄨 󵄨 (𝑠−𝑥) ×(󵄨𝑝󵄨 ∫ + 󵄨𝑞󵄨 ∫ 0 Γ (𝛼−1) 0 Γ(𝛼+𝛽−1) 󵄨 󵄨 󵄨 ×(󵄨𝑓 (𝑥,𝑢(𝑥)) −𝑓(𝑥,0)󵄨 × 󵄨𝑔 (𝑠, 𝑢 (𝑠)) 󵄨 󵄨 + 󵄨𝑓 (𝑥,0)󵄨 )𝑑𝑥 󵄨 󵄨 󵄨 −𝑔(𝑠, V (𝑠))󵄨 𝑑𝑥) 𝑑𝑠 󵄨 󵄨 𝑠 (𝑠−𝑥)𝛼+𝛽−2 + 󵄨𝑞󵄨 ∫ 󵄨 󵄨 Γ(𝛼+𝛽−1) 𝑡 0 + ∫ 𝑒−𝑘(𝑡−𝑠) 󵄨 󵄨 0 ×(󵄨𝑔(𝑥,𝑢(𝑥))−𝑔(𝑥,0)󵄨 󵄨 󵄨 𝑠 (𝑠−𝑥)𝛼−2 ×(󵄨𝑝󵄨 ∫ 󵄨 󵄨 󵄨 󵄨 Γ (𝛼−1) + 󵄨𝑔 (𝑥,0)󵄨 )𝑑𝑥)𝑑𝑠} 0 󵄨 󵄨 × 󵄨𝑓 (𝑠, 𝑢 (𝑠)) −𝑓(𝑠, V (𝑠))󵄨 𝑑𝑥 󵄨 󵄨 󵄨 󵄨 ≤ 󵄨𝑝󵄨 (𝐿1𝑟+𝑀1) 𝑠 𝛼+𝛽−2 󵄨 󵄨 󵄨 󵄨 (𝑠−𝑥) 󵄨1−𝑒−𝑘𝑡 󵄨 + 󵄨𝑞󵄨 ∫ × {󵄨 󵄨 0 Γ(𝛼+𝛽−1) sup 󵄨 −𝑘 󵄨 𝑡∈[0,1] 󵄨 1−𝑒 󵄨 󵄨 󵄨 1 𝑠 (𝑠−𝑥)𝛼−2 × 󵄨𝑔(𝑠, 𝑢(𝑠))−𝑔(𝑠, V(𝑠))󵄨𝑑𝑥)𝑑𝑠} × ∫ 𝑒−𝑘(1−𝑠) (∫ 𝑑𝑥) 𝑑𝑠 0 0 Γ (𝛼−1) 󵄨 󵄨 󵄨 −𝑘󵄨 󵄨 󵄨 󵄨 󵄨 2 󵄨1−𝑒 󵄨 [󵄨𝑝󵄨 Γ(𝛼+𝛽)+󵄨𝑞󵄨 Γ (𝛼)] 𝑡 𝑠 (𝑠−𝑥)𝛼−2 ≤𝐿 ‖𝑢−V‖ + ∫ 𝑒−𝑘(𝑡−𝑠) (∫ 𝑑𝑥)𝑑𝑠} |𝑘| Γ(𝛼+𝛽)Γ(𝛼) 0 0 Γ (𝛼−1) =𝐿𝑄 𝑢−V . 󵄨 󵄨 ‖ ‖ + 󵄨𝑞󵄨 (𝐿 𝑟+𝑀) 󵄨 󵄨 2 2 (23) 󵄨 󵄨 󵄨1−𝑒−𝑘𝑡 󵄨 By the given assumption, 𝐿<1/𝑄, V is a contraction. Thus, × {󵄨 󵄨 sup 󵄨 −𝑘 󵄨 𝑡∈[0,1] 󵄨 1−𝑒 󵄨 the conclusion of the theorem follows by the contraction mapping principle (Banach fixed point theorem). 1 𝑠 (𝑠−𝑥)𝛼+𝛽−2 × ∫ 𝑒−𝑘(1−𝑠) (∫ 𝑑𝑥) 𝑑𝑠 Our next existence result relies on Krasnoselskii’s fixed 0 0 Γ(𝛼+𝛽−1) point theorem. 𝑡 + ∫ 𝑒−𝑘(𝑡−𝑠) Lemma 5 (Krasnoselskii, see [11]). Let 𝑀 be a closed, convex, 0 bounded,andnonemptysubsetofaBanachspace𝑋.Let𝐴, 𝐵 𝐴𝑥+𝐵𝑦 ∈𝑀 𝑥,𝑦∈𝑀 𝑠 (𝑠−𝑥)𝛼+𝛽−2 be the operators such that (i) whenever , ×(∫ 𝑑𝑥) 𝑑𝑠} (ii) 𝐴 is compact, and continuous, and (iii) 𝐵 is a contraction 0 Γ(𝛼+𝛽−1) mapping. Then, there exists 𝑧∈𝑀such that 𝑧=𝐴𝑧+𝐵𝑧. Journal of Function Spaces and Applications 5

Theorem 6. Let 𝑓, 𝑔 : [0, 1] × R → R be continuous Continuities of 𝑓 and 𝑔 imply that the operator 𝜓1 is con- functions satisfying assumption (𝐴1),and tinuous. Also, 𝜓1 is uniformly bounded on 𝐵𝑟 as 󵄨 󵄨 󵄨 󵄨 (𝐴 ) 󵄨𝑓 (𝑡, 𝑢)󵄨 ≤𝜇 (𝑡) , 󵄨𝑔 (𝑡, 𝑢)󵄨 ≤𝜇 (𝑡) , 󵄩 󵄩 2 󵄨 󵄨 1 󵄨 󵄨 2 󵄩𝜓1𝑢󵄩 (24) + 󵄨 󵄨 󵄨 󵄨 󵄩 󵄩 󵄨 󵄨 󵄩 󵄩 ∀ (𝑡, 𝑢) ∈ [0, 1] × R,𝜇1,𝜇2 ∈𝐶([0, 1] , R ). 󵄨 −𝑘󵄨 󵄨 󵄨 󵄩 󵄩 󵄨 󵄨 󵄩 󵄩 󵄨1−𝑒 󵄨 [󵄨𝑝󵄨 󵄩𝜇1󵄩 Γ(𝛼+𝛽)+󵄨𝑞󵄨 󵄩𝜇2󵄩 Γ (𝛼)] (29) ≤ 󵄨 󵄨 . Then, the problem (1)-(2) has at least one solution on [0, 1] |𝑘| Γ(𝛼+𝛽)Γ(𝛼) provided that 𝜓 󵄨 󵄨 󵄨 󵄨 󵄩 󵄩 󵄨 󵄨 󵄩 󵄩 Now, we prove the compactness of the operator 1.Inviewof 󵄨1−𝑒−𝑘󵄨 [󵄨𝑝󵄨 󵄩𝜇 󵄩 Γ(𝛼+𝛽)+󵄨𝑞󵄨 󵄩𝜇 󵄩 Γ 𝛼 ] 󵄨 󵄨 󵄨 󵄨 󵄩 1󵄩 󵄨 󵄨 󵄩 2󵄩 ( ) (𝐴1), we define <1, (25) |𝑘| Γ(𝛼+𝛽)Γ(𝛼) 󵄨 󵄨 󵄨 󵄨 sup 󵄨𝑓 (𝑡, 𝑢)󵄨 = 𝑓, sup 󵄨𝑔 (𝑡, 𝑢)󵄨 = 𝑔. (𝑡,𝑢)∈[0,1]×𝐵 (𝑡,𝑢)∈[0,1]×𝐵 where sup𝑡∈[0,1]|𝜇𝑖(𝑡)| = ‖𝜇𝑖‖, 𝑖 = 1,. 2 𝑟 𝑟 (30) Proof. Let us fix 󵄨 󵄨 Consequently, we have 󵄨 −𝑘󵄨 󵄨 󵄨 󵄩 󵄩 󵄨 󵄨 󵄩 󵄩 2 󵄨1−𝑒 󵄨 [󵄨𝑝󵄨 󵄩𝜇1󵄩 Γ(𝛼+𝛽)+󵄨𝑞󵄨 󵄩𝜇2󵄩 Γ (𝛼)] 𝑟≥ (26) 󵄩 󵄩 |𝑘| Γ(𝛼+𝛽)Γ(𝛼) 󵄩(𝜓1𝑢) 2(𝑡 )−(𝜓1𝑢) 1(𝑡 )󵄩

𝑡 𝑠 𝛼−2 𝐵 ={𝑢∈P :‖𝑢‖≤𝑟} 󵄨 −𝑘𝑡 −𝑘𝑡 󵄨 1 𝑘𝑠 󵄨 󵄨 (𝑠−𝑥) and consider 𝑟 . We define the operators 󵄨 2 1 󵄨 󵄨 󵄨 ≤ 󵄨𝑒 −𝑒 󵄨 ∫ 𝑒 (󵄨𝑝󵄨 𝑓 ∫ 𝑑𝑥 𝜓1 and 𝜓2 on 𝐵𝑟 as 0 0 Γ (𝛼−1) 𝑡 𝑠 (𝑠−𝑥)𝛼−2 𝑠 𝛼+𝛽−2 −𝑘(𝑡−𝑠) 󵄨 󵄨 (𝑠−𝑥) (𝜓1𝑢) (𝑡) = ∫ 𝑒 (𝑝 ∫ 𝑓 (𝑥,𝑢(𝑥)) 𝑑𝑥 + 󵄨𝑞󵄨 𝑔 ∫ 𝑑𝑥) 𝑑𝑠 0 0 Γ (𝛼−1) 0 Γ(𝛼+𝛽−1)

𝑠 𝛼+𝛽−2 𝑡 (𝑠−𝑥) 2 −𝑘(𝑡 −𝑠) +𝑞∫ ≤ ∫ 𝑒 2 0 Γ(𝛼+𝛽−1) 𝑡1

𝑠 𝛼−2 𝑠 𝛼+𝛽−2 󵄨 󵄨 (𝑠−𝑥) 󵄨 󵄨 (𝑠−𝑥) ×𝑔(𝑥,𝑢(𝑥)) 𝑑𝑥) 𝑑𝑠, ×(󵄨𝑝󵄨 𝑓 ∫ 𝑑𝑥+ 󵄨𝑞󵄨 𝑔 ∫ 𝑑𝑥) 𝑑𝑠 0 Γ (𝛼−1) 0 Γ(𝛼+𝛽−1) (31) 𝑡∈[0, 1] ,

−𝑘𝑡 which is independent of 𝑢 and tends to zero as 𝑡2 →𝑡1.Thus, (1 − 𝑒 ) 𝜓 𝐵 (𝜓 𝑢) (𝑡) = 1 is relatively compact on 𝑟.Hence,bytheArzela-Ascoli´ 2 −𝑘 (𝑒 −1) theorem, 𝜓1 is compact on 𝐵𝑟. Thus, all the assumptions of Lemma 5 are satisfied. So, by the conclusion of Lemma 5, 1 𝑠 (𝑠−𝑥)𝛼−2 [0, 1] × ∫ 𝑒−𝑘(1−𝑠) (𝑝 ∫ 𝑓 (𝑥,𝑢(𝑥)) 𝑑𝑥 problem (1)-(2) has at least one solution on . 0 0 Γ (𝛼−1) Now, we show the existence of solutions for the problem 𝑠 (𝑠−𝑥)𝛼+𝛽−2 +𝑞∫ (1)-(2) via Leray-Schauder alternative. 0 Γ(𝛼+𝛽−1) Lemma 7 (nonlinear alternative for single valued maps, see 𝐸 𝐶 ×𝑔(𝑥,𝑢(𝑥)) 𝑑𝑥) 𝑑𝑠, [12]). Let be a Banach space, aclosed,convexsubsetof 𝐸, 𝑈 an open subset of 𝐶,and0∈𝑈.Supposethat𝐹:𝑈→𝐶 is a continuous, compact (that is, 𝐹(𝑈) is a relatively compact 𝑡∈[0, 1] . subset of 𝐶)map.Then,either (27) (i) 𝐹 has a fixed point in 𝑈,or For 𝑢, V ∈𝐵𝑟,wefindthat (ii) there is a 𝑢∈𝜕𝑈(the boundary of 𝑈 in 𝐶)and𝜆∈ 󵄩 󵄩 󵄩𝜓1𝑢+𝜓2V󵄩 (0, 1) with 𝑢 = 𝜆𝐹(𝑢). 󵄨 󵄨 󵄨 −𝑘󵄨 󵄨 󵄨 󵄩 󵄩 󵄨 󵄨 󵄩 󵄩 2 󵄨1−𝑒 󵄨 [󵄨𝑝󵄨 󵄩𝜇1󵄩 Γ(𝛼+𝛽)+󵄨𝑞󵄨 󵄩𝜇2󵄩 Γ (𝛼)] Theorem 8. Let 𝑓, 𝑔 : [0, 1] × R → R be continuous ≤ (28) |𝑘| Γ(𝛼+𝛽)Γ(𝛼) functions and the following assumptions hold. + ≤ 𝑟. (𝐴3) There exist functions 𝜎1,𝜎2 ∈ 𝐶([0, 1], R ),and + + nondecreasing functions 𝜓1,𝜓2 : R → R Thus, 𝜓1𝑢+𝜓2V ∈𝐵𝑟. It follows from assumption such that |𝑓(𝑡, 𝑢)|1 ≤𝜎 (𝑡)𝜓1(‖𝑢‖), |𝑔(𝑡, 𝑢)|≤ (𝐴1) together with (25)that𝜓2 is a contraction mapping. 𝜎2(𝑡)𝜓2(‖𝑢‖),forall(𝑡, 𝑢) ∈ [0, 1]× R. 6 Journal of Function Spaces and Applications

󵄨 󵄨 (𝐴 ) 𝑀>0 󵄨1−𝑒−𝑘𝑡 󵄨 4 There exists a constant such that ≤ 󵄨 󵄨 󵄨 󵄨 󵄨 −𝑘 󵄨 󵄨 −𝑘󵄨 󵄨 󵄨 󵄩 󵄩 󵄨 1−𝑒 󵄨 𝑀×((2󵄨1−𝑒 󵄨 [󵄨𝑝󵄨 𝜓1 (‖𝑢‖) Γ(𝛼+𝛽)󵄩𝜎1󵄩 1 󵄨 󵄨 𝑠 (𝑠−𝑥)𝛼−2 󵄨 󵄨 󵄩 󵄩 × ∫ 𝑒−𝑘(1−𝑠) (󵄨𝑝󵄨 ∫ 𝜎 (𝑥) 𝜓 (‖𝑢‖) 𝑑𝑥 + 󵄨𝑞󵄨 𝜓2 (‖𝑢‖) 󵄩𝜎2󵄩 Γ (𝛼)]) (32) 󵄨 󵄨 1 1 0 0 Γ (𝛼−1) −1 −1 ×(|𝑘| Γ(𝛼+𝛽)Γ(𝛼)) ) >1. 󵄨 󵄨 𝑠 (𝑠−𝑥)𝛼+𝛽−2 + 󵄨𝑞󵄨∫ 𝜎 (𝑥)𝜓 (‖𝑢‖)𝑑𝑥)𝑑𝑠 󵄨 󵄨 Γ(𝛼+𝛽−1) 2 2 Then, the boundary value problem (1)-(2) has at least one 0 [0, 1] solution on . 𝑡 󵄨 󵄨 𝑠 (𝑠−𝑥)𝛼−2 + ∫ 𝑒−𝑘(𝑡−𝑠) (󵄨𝑝󵄨 ∫ 𝜎 (𝑥) 𝜓 (‖𝑢‖) 𝑑𝑥 󵄨 󵄨 Γ (𝛼−1) 1 1 Proof. Consider the operator V : P → P with 𝑢=V𝑢, 0 0 where 𝑠 (𝑠−𝑥)𝛼+𝛽−2 −𝑘𝑡 󵄨 󵄨 (1 − 𝑒 ) + 󵄨𝑞󵄨 ∫ (V𝑢)(𝑡) = 0 Γ(𝛼+𝛽−1) (𝑒−𝑘 −1) ×𝜎 (𝑥) 𝜓 (‖𝑢‖) 𝑑𝑥) 𝑑𝑠 1 𝑠 (𝑠−𝑥)𝛼−2 2 2 × ∫ 𝑒−𝑘(1−𝑠) (𝑝 ∫ Γ 𝛼−1 0 0 ( ) 󵄨 󵄨 󵄩 󵄩 ≤ 󵄨𝑝󵄨 𝜓1 (𝑟) 󵄩𝜎1󵄩 ×𝑓(𝑥,𝑢(𝑥)) 𝑑𝑥 󵄨 −𝑘𝑡 󵄨 1 𝑠 𝛼−2 󵄨1−𝑒 󵄨 −𝑘(1−𝑠) (𝑠−𝑥) ×[󵄨 󵄨 ∫ 𝑒 (∫ 𝑑𝑥) 𝑑𝑠 𝑠 (𝑠−𝑥)𝛼+𝛽−2 󵄨 1−𝑒−𝑘 󵄨 Γ (𝛼−1) +𝑞∫ 󵄨 󵄨 0 0 0 Γ(𝛼+𝛽−1) 𝑡 𝑠 (𝑠−𝑥)𝛼−2 + ∫ 𝑒−𝑘(𝑡−𝑠) (∫ 𝑑𝑥) 𝑑𝑠] ×𝑔(𝑥,𝑢(𝑥)) 𝑑𝑥) 𝑑𝑠 0 0 Γ (𝛼−1) 󵄨 󵄨 󵄨 󵄨 󵄩 󵄩 󵄨1−𝑒−𝑘𝑡 󵄨 𝑡 𝑠 𝛼−2 + 󵄨𝑞󵄨 󵄩𝜎 󵄩 𝜓 (𝑟) [󵄨 󵄨 (𝑠−𝑥) 󵄨 󵄨 󵄩 2󵄩 2 󵄨 −𝑘 󵄨 + ∫ 𝑒−𝑘(𝑡−𝑠) (𝑝 ∫ 𝑓 (𝑥,𝑢(𝑥)) 𝑑𝑥 󵄨 1−𝑒 󵄨 0 0 Γ (𝛼−1) 1 𝑠 𝛼+𝛽−2 −𝑘(1−𝑠) (𝑠−𝑥) 𝑠 (𝑠−𝑥)𝛼+𝛽−2 × ∫ 𝑒 (∫ 𝑑𝑥) 𝑑𝑠 +𝑞∫ 0 0 Γ(𝛼+𝛽−1) 0 Γ(𝛼+𝛽−1) 𝑡 𝑠 (𝑠−𝑥)𝛼+𝛽−2 + ∫ 𝑒−𝑘(𝑡−𝑠) (∫ 𝑑𝑥) 𝑑𝑠] ×𝑔(𝑥,𝑢(𝑥)) 𝑑𝑥) 𝑑𝑠. 0 0 Γ(𝛼+𝛽−1) 󵄨 󵄨 󵄨 −𝑘󵄨 󵄨 󵄨 󵄩 󵄩 (33) ≤(2󵄨1−𝑒 󵄨 [󵄨𝑝󵄨 𝜓1(‖𝑢‖) Γ(𝛼+𝛽)󵄩𝜎1󵄩 V We show that maps bounded sets into bounded sets 󵄨 󵄨 󵄩 󵄩 + 󵄨𝑞󵄨 𝜓2(‖𝑢‖) 󵄩𝜎2󵄩 Γ(𝛼)]) in 𝐶([0, 1], R).Forapositivenumber𝑟,let𝐵𝑟 ={𝑢∈ 󵄨 󵄨 󵄩 󵄩 𝐶([0, 1], R):‖𝑢‖≤𝑟}be a bounded set in 𝐶([0, 1], R).Then, −1 ×(|𝑘| Γ(𝛼+𝛽)Γ(𝛼)) . |(V𝑢)(𝑡)| (34) 󵄨 󵄨 󵄨1−𝑒−𝑘𝑡 󵄨 1 ≤ 󵄨 󵄨 ∫ 𝑒−𝑘(1−𝑠) 󵄨 −𝑘 󵄨 󵄨 1−𝑒 󵄨 0 Consequently, 𝑠 𝛼−2 󵄨 󵄨 (𝑠−𝑥) 󵄨 󵄨 ×(󵄨𝑝󵄨 ∫ 󵄨𝑓 (𝑥,𝑢(𝑥))󵄨 𝑑𝑥 0 Γ (𝛼−1) 󵄨 𝑘󵄨 𝑠 𝛼+𝛽−2 ‖V𝑥‖ ≤(2󵄨1−𝑒 󵄨 󵄨 󵄨 (𝑠−𝑥) 󵄨 󵄨 󵄨 󵄨 + 󵄨𝑞󵄨 ∫ 󵄨𝑔 (𝑥,𝑢(𝑥))󵄨 𝑑𝑥) 𝑑𝑠 0 Γ(𝛼+𝛽−1) 󵄨 󵄨 󵄩 󵄩 ×[󵄨𝑝󵄨 𝜓1(‖𝑢‖) Γ(𝛼 + 𝛽) 󵄩𝜎1󵄩 𝑡 𝑠 𝑠−𝑥 𝛼−2 󵄨 󵄨 󵄩 󵄩 (35) −𝑘(𝑡−𝑠) 󵄨 󵄨 ( ) 󵄨 󵄨 + 󵄨𝑞󵄨 𝜓 𝑢 󵄩𝜎 󵄩 Γ 𝛼 ]) + ∫ 𝑒 (󵄨𝑝󵄨 ∫ 󵄨𝑓 (𝑥,𝑢(𝑥))󵄨 𝑑𝑥 󵄨 󵄨 2(‖ ‖) 󵄩 2󵄩 ( ) 0 0 Γ (𝛼−1) −1 𝑠 𝛼+𝛽−2 ×(|𝑘| Γ(𝛼+𝛽)Γ(𝛼)) . 󵄨 󵄨 (𝑠−𝑥) + 󵄨𝑞󵄨 ∫ 0 Γ(𝛼+𝛽−1)

󵄨 󵄨 Next, we show that V maps bounded sets into equicon- × 󵄨𝑔 (𝑥,𝑢(𝑥))󵄨 𝑑𝑥) 𝑑𝑠 tinuous sets of 𝐶([0, 1], R).Let𝑡1,𝑡2 ∈ [0, 1] with 𝑡1 <𝑡2 and Journal of Function Spaces and Applications 7

𝑢∈𝐵𝑟,where𝐵𝑟 is a bounded set of 𝐶([0, 1], R).Then,we Let 𝑢 be a solution. Then, for 𝑡∈[0,1],andusingthe obtain computations in proving that V is bounded, we have |𝑢 (𝑡)| = |𝜆 (V𝑢)(𝑡)| 󵄨 󵄨 󵄨 −𝑘󵄨 󵄩 󵄩 ≤(2󵄨1−𝑒 󵄨 󵄩(V𝑢) (𝑡2)−(V𝑢) (𝑡1)󵄩 󵄨 󵄨 󵄩 󵄩 ×[󵄨𝑝󵄨 𝜓1 (‖𝑢‖) Γ(𝛼+𝛽)󵄩𝜎1󵄩 󵄨 −𝑘𝑡 −𝑘𝑡 󵄨 (37) 󵄨−𝑒 2 +𝑒 1 󵄨 ≤ 󵄨 󵄨 󵄨 󵄨 󵄩 󵄩 󵄨 −𝑘 󵄨 + 󵄨𝑞󵄨 𝜓 (‖𝑢‖) 󵄩𝜎 󵄩 Γ (𝛼)]) 󵄨 1−𝑒 󵄨 󵄨 󵄨 2 󵄩 2󵄩 −1 1 ×(|𝑘| Γ(𝛼+𝛽)Γ(𝛼)) . −𝑘(1−𝑠) 󵄨 󵄨 󵄩 󵄩 × ∫ 𝑒 ( 󵄨𝑝󵄨 𝜓1 (𝑟) 󵄩𝜎1󵄩 0 Consequently, we have 󵄨 󵄨 󵄨 −𝑘󵄨 󵄨 󵄨 󵄩 󵄩 𝑠 (𝑠−𝑥)𝛼−2 ‖𝑢‖ ×((2󵄨1−𝑒 󵄨 [󵄨𝑝󵄨 𝜓1(‖𝑢‖) Γ(𝛼 + 𝛽) 󵄩𝜎1󵄩 × ∫ 𝑑𝑥 0 Γ (𝛼−1) 󵄨 󵄨 󵄩 󵄩 + 󵄨𝑞󵄨 𝜓2(‖𝑢‖) 󵄩𝜎2󵄩 Γ(𝛼)]) (38) 󵄨 󵄨 󵄩 󵄩 + 󵄨𝑞󵄨 𝜓 (𝑟) 󵄩𝜎 󵄩 −1 󵄨 󵄨 2 󵄩 2󵄩 ×(|𝑘|Γ(𝛼 + 𝛽)Γ(𝛼))−1) ≤1.

𝑠 (𝑠−𝑥)𝛼+𝛽−2 (𝐴 ) 𝑀 ‖𝑢‖ =𝑀̸ × ∫ 𝑑𝑥) 𝑑𝑠 In view of 4 , there exists such that .Letusset 0 Γ(𝛼+𝛽−1) 𝑈={𝑢∈𝐶([0, 1] , R) : ‖𝑢‖ <𝑀} . (39) 󵄨 −𝑘𝑡 −𝑘𝑡 󵄨 󵄨 2 1 󵄨 + 󵄨𝑒 −𝑒 󵄨 Note that the operator V : 𝑈 → 𝐶([0, 1], R) is continuous and completely continuous. From the choice of 𝑈,thereis (36) 𝑡1 no 𝑢∈𝜕𝑈such that 𝑢=𝜆V(𝑢) for some 𝜆 ∈ (0, 1). 𝑘𝑠 󵄨 󵄨 󵄩 󵄩 × ∫ 𝑒 ( 󵄨𝑝󵄨 𝜓1 (𝑟) 󵄩𝜎1󵄩 Consequently, by the nonlinear alternative of Leray-Schauder 0 type (Lemma 7), we deduce that V has a fixed point 𝑢∈𝑈 𝑠 (𝑠−𝑥)𝛼−2 which is a solution of the problem (1)-(2). This completes the × ∫ 𝑑𝑥 proof. 0 Γ (𝛼−1) Example 9. Consider a boundary value problem of integro- 󵄨 󵄨 󵄩 󵄩 + 󵄨𝑞󵄨 𝜓2 (𝑟) 󵄩𝜎2󵄩 differential equations of fractional order given by 𝑐 𝑐 ( 𝐷3/2 +2 𝐷1/2)𝑢 𝑡 𝑠 (𝑠−𝑥)𝛼+𝛽−2 ( ) × ∫ 𝑑𝑥) 𝑑𝑠 0 Γ(𝛼+𝛽−1) 1 = 𝑓 (𝑡, 𝑢 (𝑡)) +𝐼1/2𝑔 (𝑡, 𝑢 (𝑡)) , 0<𝑡<1, (40) 2 𝑡 𝑠 𝛼−2 2 −𝑘(𝑡 −𝑠) 󵄨 󵄨 󵄩 󵄩 (𝑠−𝑥) + ∫ 𝑒 2 (󵄨𝑝󵄨 𝜓 (𝑟) 󵄩𝜎 󵄩 ∫ 𝑑𝑥 𝑢 (0) =0, 𝑢(1) =0, 󵄨 󵄨 1 󵄩 1󵄩 Γ (𝛼−1) 𝑡1 0 where 𝛼=3/2, 𝑘=2, 𝑝=1/2, 𝑞=1, 𝛽=1/2, 𝑓(𝑡, 𝑢) = 󵄨 󵄨 󵄩 󵄩 −1 2 3 + 󵄨𝑞󵄨 𝜓2 (𝑟) 󵄩𝜎2󵄩 (|𝑢|(2+|𝑢|))/(3(1+|𝑢|))+4𝑡, 𝑔(𝑡, 𝑢) = (1/4)tan 𝑢+cos 𝑡+𝑡 + 5. With the given data, it is found that 𝐿1 =2/3,𝐿2 =1/4as 𝑠 (𝑠−𝑥)𝛼+𝛽−2 |𝑓(𝑡, 𝑢)−𝑓(𝑡, V)| ≤ (2/3)|𝑢−V|, |𝑔(𝑡, 𝑢)−𝑔(𝑡, V)| ≤ (1/4)|𝑢−V|, × ∫ 𝑑𝑥) 𝑑𝑠. Γ(𝛼+𝛽−1) and 0 󵄨 󵄨 󵄨 −𝑘󵄨 󵄨 󵄨 󵄨 󵄨 2 󵄨1−𝑒 󵄨 [󵄨𝑝󵄨 Γ(𝛼+𝛽)+󵄨𝑞󵄨 Γ (𝛼)] 𝑄= ≃ 1.3525. (41) |𝑘| Γ(𝛼+𝛽)Γ(𝛼)

Obviously, the right hand side of the previous inequality tends Clearly, 𝐿=max{𝐿1,𝐿2}=2/3and 𝐿<1/𝑄.Thus,all to zero independently of 𝑢∈𝐵𝑟 as 𝑡2 −𝑡1 →0.AsV the assumptions of Theorem 4 are satisfied. Hence, by the satisfies the previous assumptions, therefore it follows by the conclusion of Theorem 4,theproblem(40)hasaunique Arzela-Ascoli´ theorem that V : 𝐶([0, 1], R) → 𝐶([0, 1], R) solution. is completely continuous. The proof will be complete by the application of the Leray- Acknowledgment Schauder nonlinear alternative (Lemma 7)onceweestablish the boundedness of the set of all solutions to equations 𝑢= The authors thank the anonymous referees for their valuable 𝜆V𝑢 for 𝜆 ∈ (0, 1). comments. The research of J. J. Nieto has been partially 8 Journal of Function Spaces and Applications

supported by Ministerio de Economia y Competitividad (Spain), project MTM2010-15314, and co-financed by the European Community fund FEDER.

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Research Article Concerning Asymptotic Behavior for Extremal Polynomials Associated to Nondiagonal Sobolev Norms

Ana Portilla,1 Yamilet Quintana,2 José M. Rodríguez,3 and Eva Tourís4

1 Faculty Mathematics and Computer Science, St. Louis University (Madrid Campus), Avenida del Valle 34, 28003 Madrid, Spain 2 Departamento de Matematicas´ Puras y Aplicadas, Edificio Matematicas´ y Sistemas (MYS), Universidad Simon´ Bol´ıvar, Apartado Postal 89000, Caracas 1080 A, Venezuela 3 Departamento de Matematicas,´ Universidad Carlos III de Madrid, Avenida de la Universidad 30, Leganes,´ 28911 Madrid, Spain 4 Departamento de Matematicas,´ Facultad de Ciencias, Universidad Autonoma´ de Madrid, Campus de Cantoblanco, 28049 Madrid, Spain

Correspondence should be addressed to Yamilet Quintana; [email protected]

Received 28 January 2013; Accepted 8 March 2013

Academic Editor: Jozef´ Bana´s

Copyright © 2013 Ana Portilla et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Let P be the space of polynomials with complex coefficients endowed with a nondiagonal Sobolev norm ‖⋅‖𝑊1,𝑝(𝑉𝜇),wherethe matrix 𝑉 and the measure 𝜇 constitute a 𝑝-admissible pair for 1≤𝑝≤∞. In this paper we establish the zero location and asymptotic behavior of extremal polynomials associated to ‖⋅‖𝑊1,𝑝(𝑉𝜇), stating hypothesis on the matrix 𝑉 rather than on the diagonal matrix appearing in its unitary factorization.

1. Introduction (see Theorem 3 below, which is [3,Theorem8.1]inthecase 𝑁=1). The rest of the above mentioned papers provides In the last decades the asymptotic behavior of Sobolev conditions that ensure the equivalence of norms in Sobolev orthogonal polynomials has been one of the main topics of spaces, and consequently, the boundedness of 𝑀. interest to investigators in the field. In [1]theauthorsobtain Results related to nondiagonal Sobolev norms may be 𝑛 the th root asymptotic of Sobolev orthogonal polynomials found in [5, 6, 14–19]. Particularly, in [5, 6, 15, 18, 19]the when the zeros of these polynomials are contained in a authors establish the asymptotic behavior of orthogonal poly- compact set of the complex plane; however, the boundedness nomials with respect to nondiagonal Sobolev inner products of the zeros of Sobolev orthogonal polynomials is an open andtheauthorsin[5] deal with the asymptotic behavior of problem, but as was stated in [2],itcouldbeobtainedas extremal polynomials with respect to the following nondiag- a consequence of the boundedness of the multiplication onal Sobolev norms. 𝑀𝑓(𝑧) = 𝑧𝑓(𝑧) operator .Thus,findingconditionstoensure Let P be the space of polynomials with complex coef- 𝑀 the boundedness of would provide important information ficients and let 𝜇 be a finite Borel positive measure with about the crucial issue of determining the asymptotic behav- compact support 𝑆(𝜇) consisting of infinitely many points iorofSobolevorthogonalpolynomials(see,e.g.,[3–13]). The in the complex plane; let us consider the diagonal matrix more general result on this topic is [3,Theorem8.1]which Λ:= diag(𝜆𝑗),𝑗=0,...,𝑁,with𝜆𝑗 being positive 𝜇-almost characterizes in terms of equivalent norms in Sobolev spaces everywhere measurable functions, and 𝑈:=(𝑢𝑗𝑘), 0 ≤ 𝑀 the boundedness of for the classical diagonal norm 𝑗,𝑘≤𝑁, a matrix of measurable functions such that the 𝑈(𝑥) = (𝑢 (𝑥)) 0≤𝑗,𝑘≤𝑁 𝜇 1/𝑝 matrix 𝑗𝑘 , is unitary -almost 󵄩 󵄩 𝑁 󵄩 󵄩 𝑉:=𝑈Λ𝑈∗ 𝑈∗ 󵄩𝑞󵄩 := (∑󵄩𝑞(𝑘)󵄩 ) everywhere. If ,where denotes the transpose 󵄩 󵄩𝑊𝑁,𝑝 (𝜇 ,𝜇 ,...,𝜇 ) 󵄩 󵄩 𝑝 (1) 0 1 𝑁 󵄩 󵄩𝐿 (𝜇𝑘) 𝑈 𝑉 𝑘=0 conjugate of (note that then is a positive definite matrix 2 Journal of Function Spaces and Applications

𝜇 1≤𝑝<∞ 𝑝/2 −1 1/(𝑝−1) -almost everywhere), and we define the Sobolev Assume that 1≤𝑝≤2, (𝑐𝑝 𝑑𝜇/𝑑𝑠) ∈𝐿 (𝛾),andthe norm on the space of polynomials P 1,𝑝 𝑝/2 𝑝/2 𝑝/2 1,𝑝 𝑝/2 𝑝/2 norms in 𝑊 ((𝑎𝑝 +𝑐𝑝 )𝜇,𝑝 𝑐 𝜇) and 𝑊 (𝑎𝑝 𝜇,𝑝 𝑐 𝜇) are equivalent on P.Let{𝑞𝑛}𝑛≥0 be a sequence of extremal poly- 󵄩 󵄩 󸀠 (𝑁) 2/𝑝 󵄩𝑞󵄩 𝑁,𝑝 := (∫ [(𝑞,𝑞 ,...,𝑞 )𝑉 nomials with respect to (2). Then the multiplication operator is 󵄩 󵄩𝑊 (𝑉𝜇) 1,𝑝 bounded with the norm 𝑊 (𝑉𝜇) and the zeros of {𝑞𝑛}𝑛≥0 lie 1/𝑝 ∗ 𝑝/2 in the bounded disk {𝑧 : |𝑧| ≤ 2 ‖ 𝑀 ‖}. ×(𝑞,𝑞󸀠,...,𝑞(𝑁)) ] 𝑑𝜇) (2) In this paper we improve Theorem 1 in two directions: on 𝜇 := (∫ [(𝑞,𝑞󸀠,...,𝑞(𝑁))𝑈Λ2/𝑝𝑈∗ the one hand, we enlarge the class of measures considered and, on the other hand, we prove our result for 1≤𝑝<∞ (see Theorem 19). In order to describe the measures we will 1/𝑝 ∗ 𝑝/2 𝑝 ×(𝑞,𝑞󸀠,...,𝑞(𝑁)) ] 𝑑𝜇) . deal with, we introduce the definition of -admissible pairs as follows: given 1≤𝑝<∞, we say that the pair (𝑉,𝜇) is 𝑝- admissible if 𝜇 is a finite Borel measure which can be written In [20, Chapter XIII] certain general conditions imposed as 𝜇=𝜇1 +𝜇2,itssupport𝑆(𝜇) is a compact subset of the on the matrix 𝑉 are requested in order to guarantee the exis- complex plane which contains infinitely many points, and 𝑉 2 tence of an unitary representation with measurable entries. is a positive definite matrix 𝜇-almost everywhere with |𝑏𝑝| ≤ 𝑈 𝜇 If is not the identity matrix -almost everywhere, then (1−𝜀0)𝑎𝑝𝑐𝑝, 𝜇1-almost everywhere for some fixed 0<𝜀0 ≤1; (2) defines a nondiagonal Sobolev norm in which the product the support 𝑆(𝜇2) is contained in a finite union of rectifiable −1 of derivatives of different order appears. We say that 𝑞𝑛(𝑧) = 𝑝/2 1/(𝑝−1) 𝑛 𝑛−1 compact curves 𝛾 with (𝑐𝑝 𝑑𝜇2/𝑑𝑠) ∈𝐿 (𝛾) if 𝛾 =0̸, 𝑧 +𝑎𝑛−1𝑧 +⋅⋅⋅+𝑎1𝑧+𝑎0 is an 𝑛th monic extremal 2/𝑝 𝑎𝑝 𝑏𝑝 polynomial with respect to the norm (2)if 𝑉 := ( ) and 𝑑𝜇2/𝑑𝑠 is the Radon-Nykodim derivative 𝑏𝑝 𝑐𝑝 𝜇 𝛾 󵄩 󵄩 of 2 withrespecttotheEuclideanlengthin . 󵄩𝑞𝑛󵄩𝑊𝑁,𝑝 (𝑉𝜇) We want to make three𝑎 𝑏 remarks about this definition. 𝑉=(2 2 ) Firstofall,since 𝑏 𝑐 is a positive definite matrix 󵄩 󵄩 𝑛 𝑛−1 2 2 = inf {󵄩𝑞󵄩 𝑁,𝑝 :𝑞(𝑧) =𝑧 +𝑏𝑛−1𝑧 2/𝑝 𝑊 (𝑉𝜇) 𝜇-almost everywhere, 𝑉 also has this property and hence 2 |𝑏𝑝| <𝑎𝑝𝑐𝑝, 𝜇-almost everywhere. +⋅⋅⋅+𝑏1𝑧+𝑏0,𝑏𝑗 ∈ C}. 𝑝/2 −1 1/(𝑝−1) (3) In order to obtain (𝑐𝑝 𝑑𝜇2/𝑑𝑠) ∈𝐿 (𝛾) the best choice for 𝜇2 is the restriction of 𝜇 to 𝛾. It is clear that there exists at least an 𝑛th monic extremal Note that the support of 𝜇 is an arbitrary compact set: we 2/𝑝 polynomial. Furthermore, it is unique if 1<𝑝<∞.If𝑝= just require that 𝑆(𝜇2) (the part of 𝑆(𝜇) in which 𝑉 is about 2 2, then the 𝑛th monic extremal polynomial is precisely the to be a degenerated quadratic form, when |𝑏𝑝| is very close 𝑛th monic Sobolev orthogonal polynomial with respect to the to 𝑎𝑝𝑐𝑝) is a union of curves. inner product corresponding to (2). Therefore, with the results on 𝑝-admissible pairs we In [5, Theorem 1] the authors showed that the zeros of the complementandimprovethestudystartedin[22], where the polynomials in {𝑞𝑛}𝑛≥0 areuniformlyboundedinthecomplex case 𝜇=𝜇2 with 1≤𝑝≤2was considered. plane, whenever there exists a constant 𝐶 such that 𝜆𝑗 ≤ Another interesting property which could be studied is 𝐶𝜆𝑘, 𝜇-almost everywhere for 0≤𝑗,𝑘≤𝑁.Thisproperty theasymptoticestimateforthebehaviorofextremalpolyno- made possible to obtain the 𝑛th root asymptotic behavior of mials because, in this setting, there does not exist the usual extremal polynomials (see [5, Theorems 2 and 6]). Although three-term recurrence relation for orthogonal polynomials 2 it is required compact support for 𝜇, this is, certainly, a natural in 𝐿 and this makes it really difficult to find an explicit hypothesis: if 𝑆(𝜇) is not bounded, then we cannot expect to expression for the extremal polynomial of degree 𝑛.Inthis have zeros uniformly bounded, not even in the classical case regard, Theorems 22 and 23 deduce the asymptotic behavior 2 (orthogonal polynomials in 𝐿 ); see [21]. of extremal polynomials as an application of Theorems 18 Taking 𝑁=1, 1≤𝑝≤2and setting up hypothesis and 19.Moreprecisely,weobtainthe𝑛th root and the zero on the matrix 𝑉 (see (4)) rather than on the diagonal matrix counting measure asymptotic both of those polynomials and 𝜆,theauthorsof[22] the following equivalent result to [5, their derivatives to any order. The study of the 𝑛th root Theorem 1]. asymptotic is a classical problem in the theory of orthogonal polynomials; see for instance, [1, 2, 5, 23, 24]. Theorem 1 (see [22, Theorem 4.3]). Let 𝛾 be a finite union of Furthermore, in Theorem 23 we find the following rectifiable compact curves in the complex plane, 𝜇 afiniteBorel asymptotic relation: measure with compact support 𝑆(𝜇) =𝛾, 𝑉 apositivedefinite matrix 𝜇-almost everywhere and (𝑗+1) 𝑑𝜔 (𝑥) 𝑞𝑛 (𝑧) 𝑆(𝜇) lim = ∫ (5) 𝑛→∞ 𝑛𝑞(𝑗) (𝑧) 𝑧−𝑥 𝑎𝑝 𝑏𝑝 𝑛 2/𝑝 𝑉 =( ). (4) 𝑏𝑝 𝑐𝑝 for any 𝑗≥0. Journal of Function Spaces and Applications 3

The main idea of[5, 6, 22] and this paper is to compare In order to bound the zeros of polynomials, one of the nondiagonal and diagonal norms. most successful strategies has certainly been to bound the When it comes to compare nondiagonal and diagonal multiplication operator by the independent variable 𝑀𝑓 ( 𝑧 )= norms, [25] is remarkable, since the authors show that sym- 𝑧𝑓(𝑧),where metric Sobolev bilinear forms, like symmetric matrices, can 󵄩 󵄩 󵄩 󵄩 be rewritten with a diagonal representation; unfortunately, ‖𝑀‖ := sup {󵄩𝑧𝑞(𝑧)󵄩𝑊1,𝑝(𝑉𝜇) : 󵄩𝑞󵄩𝑊1,𝑝(𝑉𝜇) =1}. (8) the entries of these diagonal matrices are real measures, and we cannot use this representation since we need positive Regarding this issue, the following result is known. measures for the Sobolev norms. Finally, we would like to note that the central obstacle in Theorem 2 (see [5,Theorem3]).Let 𝜇 be a finite Borel order to generalize the results given in this paper and [22] measure in C with compact support let and 1≤𝑝<∞.Let to the case of more derivatives is that there are too many {𝑞𝑛}𝑛≥0 be a sequence of extremal polynomials with respect to entries in the matrix 𝑉 and just a few relations to control them (2). Then the zeros of {𝑞𝑛}𝑛≥0 lie in the disk {𝑧 : |𝑧| ≤ 2 ‖ 𝑀 ‖}. (see Lemma 8 and notice that some limits appearing in that Lemma do not provide any new information). In that case we It is also known the following simple characterization of (𝑎 ,𝑏 ,𝑐 ) have just three entries 𝑝 𝑝 𝑝 ,butinthesimplecaseoftwo the boundedness of 𝑀. derivatives (𝑁 = 2) we have Theorem 3 𝜇 𝑎 𝑎 𝑎 (see [3,Theorem8.1]). Let be a finite Borel mea- 11 12 13 sure in C with compact support; 𝛼, 𝛽 nonnegative measurable functions; and 1≤𝑝<∞. Then the multiplication operator is 𝑉:=(𝑎12 𝑎22 𝑎23), (6) 1,𝑝 bounded in 𝑊 (𝛼𝜇, 𝛽𝜇) if and only if the following condition 𝑎13 𝑎23 𝑎33 holds: 1,𝑝 and we would need to control six functions (𝑎11,𝑎12,𝑎13, the norms in 𝑊 ((𝛼 + 𝛽) 𝜇, 𝛽𝜇) 𝑎22,𝑎23,𝑎33);inthegeneralcasewith𝑁 derivatives, we would (9) 1,𝑝 need to control (𝑁 + 1)(𝑁 + 2)/2 functions. and 𝑊 (𝛼𝜇, 𝛽𝜇) are equivalent on P. Theoutlineofthepaperisasfollows.InSection 2 we provide some background and previous results on the It is clear that if there exists a constant 𝐶 such that 𝛽≤ multiplication operator and the location of zeros of extremal 𝐶𝛼 𝜇-almost everywhere, then (9)holds.In[8, 13]some polynomials. We have devoted Section 3 to some technical other very simple conditions implying (9)areshown. lemmas in order to simplify the proof of Theorem 17 about In what follows, we will fix a 𝑝-admissible pair (𝑉,𝜇) with the equivalence of norms; in fact, in these lemmas the hardest 1≤𝑝<∞;then𝑆(𝜇2) is contained in a finite union of part of this proof is collected. In Section 4 we give the proof rectifiable compact curves 𝛾 in the complex plane; each of of that Theorem and in Section 5 we deduce some results on these connected components of 𝛾 is not required to be either asymptotic of extremal polynomials. simple or closed.

2. Background and Previous Results 3. Technical Lemmas In what follows, given 1≤𝑝<∞we define For the sake of clarity and readability, we have opted for 󵄩 󵄩 󵄩𝑓󵄩 proving all the technical lemmas in this section. This makes 󵄩 󵄩𝑊1,𝑝(𝑎𝑝/2𝜇,𝑐𝑝/2𝜇) 𝑝 𝑝 the proof of Theorem 17 much more understandable. 1/𝑝 The following result is well known. 󵄨 󵄨𝑝 󵄨 󵄨𝑝 := (∫ (𝑎𝑝/2󵄨𝑓󵄨 +𝑐𝑝/2󵄨𝑓󸀠󵄨 ) 𝑑𝜇) , 𝑝 󵄨 󵄨 𝑝 󵄨 󵄨 Lemma 4. Let us consider 1≤𝛼<∞.Then 1/𝑝 󵄩 󵄩 󵄨 󵄨2 󵄨 󵄨2 𝑝/2 𝛼 𝛼−1 𝛼 𝛼 󵄩𝑓󵄩 := (∫ (𝑎 󵄨𝑓󵄨 +𝑐 󵄨𝑓󸀠󵄨 ) 𝑑𝜇) , (𝑥+𝑦) ≤2 (𝑥 +𝑦 ) 𝑥, 𝑦 ≥ 0. 󵄩 󵄩𝑊1,𝑝(𝐷𝜇) 𝑝󵄨 󵄨 𝑝󵄨 󵄨 (7) for every (10) 󵄩 󵄩 Lemma 5 (see [22, Lemma 3.1]). Let us consider 0<𝛼≤1. 󵄩𝑓󵄩𝑊1,𝑝(𝑉𝜇) Then 1/𝑝 󵄨 󵄨2 𝑝/2 󵄨 󵄨2 󵄨 󸀠󵄨 󸀠 |𝑦|𝛼 −|𝑥|𝛼 ≤|𝑦−𝑥|𝛼 𝑥, 𝑦 ∈ R := (∫ (𝑎𝑝󵄨𝑓󵄨 +𝑐𝑝󵄨𝑓 󵄨 2R (𝑏𝑝𝑓𝑓 )) 𝑑𝜇) , (1) for every ; 𝛼−1 𝛼 𝛼 𝛼 𝛼 𝛼 (2) 2 (𝑦 +𝑥 ) ≤ (𝑦 + 𝑥) ≤𝑦 +𝑥 for every 𝑥, 𝑦. ≥0 for every polynomial 𝑓. Itisobviouslymucheasiertodealwiththenorms Lemma 6 (see [22,Lemma3.2]). Let {𝑠𝑛}𝑛 and {𝑡𝑛}𝑛 be two ‖⋅‖ 1,𝑝 𝑝/2 𝑝/2 and ‖⋅‖𝑊1,𝑝(𝐷𝜇) than with the one 𝑊 (𝑎𝑝 𝜇,𝑐𝑝 𝜇) sequences of positive numbers. Then ‖⋅‖𝑊1,𝑝(𝑉𝜇). Therefore, one of our main goals is to provide 2𝑠 𝑡 𝑠 weak hypotheses to guarantee the equivalence of these norms 𝑛 𝑛 =1 𝑛 =1. P 𝑛→∞lim 2 2 iif 𝑛→∞lim (11) on the linear space of polynomials (see Section 4). 𝑠𝑛 +𝑡𝑛 𝑡𝑛 4 Journal of Function Spaces and Applications

In what follows 𝑎𝑝,𝑏𝑝,and𝑐𝑝 refer to the coefficients of Since the limit of the product is 1, if we prove that the first, 2/𝑝 𝑛 the fixed matrix 𝑉 . third, and fourth factors tend to 1 as tends to infinity, then the limit of the second factor must also be 1. Definition 7. We say that {𝑓𝑛}𝑛 ⊂ P is an extremal sequence So, our problem is reduced to show 𝑝 𝑛 ‖𝑓‖ ∞ =1 for if, for every , 𝑛 𝐿 (𝜇2) and 󵄨 󵄨𝑝/2 ∫ 󵄨𝑏 𝑓 𝑓󸀠󵄨 𝑑𝜇 󵄨 𝑝 𝑛 𝑛󵄨 2 󵄨 󸀠󵄨𝑝/2 =1, ∫ 󵄨2𝑏 𝑓 𝑓 󵄨 𝑑𝜇 𝑛→∞lim 󵄨 󵄨 𝑝/2 (15) 󵄨 𝑝 𝑛 𝑛󵄨 2 ∫( 𝑎 𝑐 󵄨𝑓 𝑓󸀠󵄨) 𝑑𝜇 lim =1. √ 𝑝 𝑝 󵄨 𝑛 𝑛󵄨 2 𝑛→∞ 󵄨 󵄨2 󵄨 󵄨2 𝑝/2 (12) ∫(𝑎 󵄨𝑓 󵄨 +𝑐 󵄨𝑓󸀠󵄨 ) 𝑑𝜇 𝑝󵄨 𝑛󵄨 𝑝󵄨 𝑛󵄨 2 1/2 𝑝/2 1/2 󵄨 󵄨2 𝑝/2 󵄨 󵄨2 󵄨 󸀠󵄨 2(∫ (𝑎𝑝󵄨𝑓𝑛󵄨 ) 𝑑𝜇2) (∫ (𝑐𝑝󵄨𝑓𝑛󵄨 ) 𝑑𝜇2) Lemma 8. If 1≤𝑝<∞and {𝑓𝑛}𝑛 is an extremal sequence 󵄨 󵄨 for 𝑝,then 𝑛→∞lim 𝑝/2 𝑝/2 󵄨 󵄨2 󵄨 󸀠󵄨2 (16) ∫((𝑎𝑝󵄨𝑓𝑛󵄨 ) +(𝑐𝑝󵄨𝑓𝑛󵄨 ) )𝑑𝜇2 󵄨 󸀠󵄨𝑝/2 ∫ 󵄨𝑏 𝑓 𝑓 󵄨 𝑑𝜇 󵄨 𝑝 𝑛 𝑛󵄨 2 =1, lim 𝑝/2 =1, 𝑛→∞ 󵄨 󸀠󵄨 ∫(√𝑎𝑝𝑐𝑝 󵄨𝑓𝑛𝑓𝑛󵄨) 𝑑𝜇2 𝑝/2 󵄨 󵄨2 𝑝/2 𝑝/2−1 󵄨 󵄨2 󵄨 󸀠󵄨 󵄨 󵄨 𝑝/2 2 ∫((𝑎𝑝󵄨𝑓𝑛󵄨 ) +(𝑐𝑝󵄨𝑓𝑛󵄨 ) )𝑑𝜇2 󵄨 󸀠󵄨 󵄨 󵄨 ∫(√𝑎𝑝𝑐𝑝 󵄨𝑓𝑛𝑓𝑛󵄨) 𝑑𝜇2 =1, (17) 󵄨 󵄨 lim 𝑝/2 lim =1, 𝑛→∞ 󵄨 󵄨2 󵄨 󸀠󵄨2 𝑛→∞ 𝑝/2 1/2 𝑝/2 1/2 ∫(𝑎 󵄨𝑓 󵄨 +𝑐 󵄨𝑓 󵄨 ) 𝑑𝜇 󵄨 󵄨2 󵄨 󸀠󵄨2 𝑝󵄨 𝑛󵄨 𝑝󵄨 𝑛󵄨 2 (∫ (𝑎𝑝󵄨𝑓𝑛󵄨 ) 𝑑𝜇2) (∫ (𝑐𝑝󵄨𝑓𝑛󵄨 ) 𝑑𝜇2) 󵄨 󵄨2 𝑝/2 ∫(𝑎𝑝󵄨𝑓𝑛󵄨 ) 𝑑𝜇2 1/2 𝑝/2 1/2 󵄨 󵄨 󵄨 󵄨2 𝑝/2 󵄨 󸀠󵄨2 lim =1. (18) 2(∫ (𝑎 󵄨𝑓 󵄨 ) 𝑑𝜇 ) (∫ (𝑐 󵄨𝑓 󵄨 ) 𝑑𝜇 ) 𝑛→∞ 2 𝑝/2 𝑝󵄨 𝑛󵄨 2 𝑝󵄨 𝑛󵄨 2 󵄨 󸀠󵄨 ∫(𝑐𝑝󵄨𝑓𝑛󵄨 ) 𝑑𝜇2 lim =1, 𝑛→∞ 󵄨 󵄨2 𝑝/2 󵄨 󵄨2 𝑝/2 ∫((𝑎 󵄨𝑓 󵄨 ) +(𝑐 󵄨𝑓󸀠󵄨 ) )𝑑𝜇 𝑝󵄨 𝑛󵄨 𝑝󵄨 𝑛󵄨 2 Again, we can rewrite the limit in the definition of extremalsequenceasthelimitofthefollowingproduct: 󵄨 󵄨2 𝑝/2 󵄨 󵄨2 𝑝/2 2𝑝/2−1 ∫((𝑎 󵄨𝑓 󵄨 ) +(𝑐 󵄨𝑓󸀠󵄨 ) )𝑑𝜇 𝑝󵄨 𝑛󵄨 𝑝󵄨 𝑛󵄨 2 󵄨 󵄨𝑝/2 ∫ 󵄨𝑏 𝑓 𝑓󸀠󵄨 𝑑𝜇 lim =1, 󵄨 𝑝 𝑛 𝑛󵄨 2 𝑛→∞ 󵄨 󵄨2 󵄨 󵄨2 𝑝/2 󵄨 󵄨 󵄨 󸀠󵄨 𝑛→∞lim 𝑝/2 ∫(𝑎𝑝󵄨𝑓𝑛󵄨 +𝑐𝑝󵄨𝑓𝑛󵄨 ) 𝑑𝜇2 󵄨 󸀠󵄨 ∫(√𝑎𝑝𝑐𝑝 󵄨𝑓𝑛𝑓𝑛󵄨) 𝑑𝜇2 󵄨 󵄨2 𝑝/2 (19) ∫(𝑎 󵄨𝑓 󵄨 ) 𝑑𝜇 󵄨 󵄨 𝑝/2 𝑝󵄨 𝑛󵄨 2 ∫(2 𝑎 𝑐 󵄨𝑓 𝑓󸀠󵄨) 𝑑𝜇 =1. √ 𝑝 𝑝 󵄨 𝑛 𝑛󵄨 2 𝑛→∞lim 𝑝/2 ⋅ =1. 󵄨 󸀠󵄨2 𝑝/2 ∫(𝑐𝑝󵄨𝑓𝑛󵄨 ) 𝑑𝜇2 󵄨 󵄨2 󵄨 󸀠󵄨2 ∫(𝑎𝑝󵄨𝑓𝑛󵄨 +𝑐𝑝󵄨𝑓𝑛󵄨 ) 𝑑𝜇2 (13) The two factors above are nonnegative and less than or Proof. The case 1≤𝑝≤2is a consequence of [22,Lemmas 2 𝑝>2 equal to 1 using, respectively, that |𝑏𝑝| <𝑎𝑝𝑐𝑝 𝜇2-almost 3.5 and 3.6]. We deal now with the case .Firstnotethat 2 2 we can rewrite limit (12)inDefinition 7 as the limit of the everywhere and 2𝑥𝑦 ≤𝑥 +𝑦 .Thus, following product: 󵄨 󵄨𝑝/2 ∫ 󵄨𝑏 𝑓 𝑓󸀠󵄨 𝑑𝜇 󵄨 󵄨𝑝/2 󵄨 𝑝 𝑛 𝑛󵄨 2 󵄨 󸀠󵄨 ∫ 󵄨𝑏𝑝𝑓𝑛𝑓𝑛󵄨 𝑑𝜇2 𝑛→∞lim 󵄨 󵄨 𝑝/2 󵄨 󵄨 ∫( 𝑎 𝑐 󵄨𝑓 𝑓󸀠󵄨) 𝑑𝜇 lim 𝑝/2 √ 𝑝 𝑝 󵄨 𝑛 𝑛󵄨 2 𝑛→∞ 󵄨 󸀠󵄨 ∫(√𝑎𝑝𝑐𝑝 󵄨𝑓𝑛𝑓𝑛󵄨) 𝑑𝜇2 (20) 󵄨 󵄨 𝑝/2 ∫(2 𝑎 𝑐 󵄨𝑓 𝑓󸀠󵄨) 𝑑𝜇 √ 𝑝 𝑝 󵄨 𝑛 𝑛󵄨 2 󵄨 󸀠󵄨 𝑝/2 =1= , ∫(√𝑎𝑝𝑐𝑝 󵄨𝑓𝑛𝑓 󵄨) 𝑑𝜇2 lim 𝑝/2 󵄨 𝑛󵄨 𝑛→∞ 󵄨 󵄨2 󵄨 󸀠󵄨2 ⋅ ∫(𝑎𝑝󵄨𝑓𝑛󵄨 +𝑐𝑝󵄨𝑓𝑛󵄨 ) 𝑑𝜇2 𝑝/2 1/2 𝑝/2 1/2 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨2 󵄨 󸀠󵄨2 (∫ (𝑎𝑝󵄨𝑓𝑛󵄨 ) 𝑑𝜇2) (∫ (𝑐𝑝󵄨𝑓𝑛󵄨 ) 𝑑𝜇2) and (15)holds. 1/2 1/2 Given 𝜀>0,foreach𝑛 let us define the following sets: 󵄨 󵄨2 𝑝/2 󵄨 󵄨2 𝑝/2 2(∫ (𝑎 󵄨𝑓 󵄨 ) 𝑑𝜇 ) (∫ (𝑐 󵄨𝑓󸀠󵄨 ) 𝑑𝜇 ) 𝑝󵄨 𝑛󵄨 2 𝑝󵄨 𝑛󵄨 2 󵄨 󵄨 ⋅ 1 √𝑎𝑝 󵄨𝑓𝑛󵄨 󵄨 󵄨2 𝑝/2 󵄨 󵄨2 𝑝/2 𝐹 := {𝑧 ∈ 𝑆 (𝜇 ): ≤ 󵄨 󵄨 ≤1+𝜀}, ∫((𝑎 󵄨𝑓 󵄨 ) +(𝑐 󵄨𝑓󸀠󵄨 ) )𝑑𝜇 𝑛,𝜀 2 1+𝜀 𝑐 󵄨𝑓󸀠󵄨 𝑝󵄨 𝑛󵄨 𝑝󵄨 𝑛󵄨 2 √ 𝑝 󵄨 𝑛󵄨 (21) 𝑐 𝑝/2 󵄨 󵄨2 𝑝/2 󵄨 󵄨2 𝐹𝑛,𝜀 := 𝑆 (𝜇2)\𝐹𝑛,𝜀. 2𝑝/2−1 ∫((𝑎 󵄨𝑓 󵄨 ) +(𝑐 󵄨𝑓󸀠󵄨 ) )𝑑𝜇 𝑝󵄨 𝑛󵄨 𝑝󵄨 𝑛󵄨 2 ⋅ =1. 𝑝/2 Let us consider the strictly decreasing function 𝐴(𝑡) := 󵄨 󵄨2 󵄨 󸀠󵄨2 2 ∫(𝑎𝑝󵄨𝑓𝑛󵄨 +𝑐𝑝󵄨𝑓𝑛󵄨 ) 𝑑𝜇2 2𝑡/(𝑡 +1)on [1, ∞).If𝑡≥1+𝜀,then𝐴(𝑡) ≤ 𝐴(1 + (14) 𝜀) =:𝜀 𝐶 < 𝐴(1) =.Consequently,if 1 𝑥/𝑦 ≥ 1,then +𝜀 Journal of Function Spaces and Applications 5

2 2 2𝑥𝑦/(𝑥 +𝑦 )=𝐴(𝑥/𝑦)≤𝐶𝜀,andif𝑥/𝑦 ≤ 1/(1 +𝜀),then Since for each 𝑛,wehave 2 2 𝑦/𝑥≥1+𝜀and 2𝑥𝑦/(𝑥 +𝑦 )=𝐴(𝑦/𝑥)≤𝐶𝜀. Therefore, 󵄨 󵄨 𝑝/2 󵄨 󸀠󵄨 ∫ 𝑐 (2√𝑎𝑝𝑐𝑝 󵄨𝑓𝑛𝑓𝑛󵄨) 𝑑𝜇2 𝐹𝑛,𝜀 󵄨 󵄨 󵄨 󸀠󵄨 𝑝/2 ∫ (2 𝑎 𝑐 󵄨𝑓 𝑓 󵄨) 𝑑𝜇 󵄨 󵄨2 󵄨 󵄨2 𝑝/2 𝐹𝑐 √ 𝑝 𝑝 󵄨 𝑛 𝑛󵄨 2 󵄨 󵄨 󵄨 󸀠󵄨 𝑛,𝜀 𝑝/2 ∫ (𝑎𝑝󵄨𝑓𝑛󵄨 +𝑐𝑝󵄨𝑓𝑛󵄨 ) 𝑑𝜇2 ≤𝐶 <1. 𝐹𝑛,𝜀 𝑝/2 𝜀 (22) 󵄨 󵄨2 󵄨 󸀠󵄨2 (25) ∫ 𝑐 (𝑎𝑝󵄨𝑓𝑛󵄨 +𝑐𝑝󵄨𝑓𝑛󵄨 ) 𝑑𝜇2 𝑝/2 𝐹𝑛,𝜀 󵄨 󵄨2 󵄨 󵄨2 ∫ (𝑎 󵄨𝑓 󵄨 +𝑐 󵄨𝑓󸀠󵄨 ) 𝑑𝜇 𝐹𝑐 𝑝󵄨 𝑛󵄨 𝑝󵄨 𝑛󵄨 2 ≤ 𝑛,𝜀 , 𝑝/2 󵄨 󵄨2 󵄨 󸀠󵄨2 Using this fact and (20), we have ∫ (𝑎𝑝󵄨𝑓𝑛󵄨 +𝑐𝑝󵄨𝑓𝑛󵄨 ) 𝑑𝜇2 𝐹𝑛,𝜀

then (24)impliesthat 󵄨 󸀠󵄨 𝑝/2 1= lim ((( (∫ (2 𝑎 󵄨𝑓 𝑓 󵄨) 𝑑𝜇 ) 𝑛→∞ 𝐹 √ 𝑝c𝑝 󵄨 𝑛 𝑛󵄨 2 𝑛,𝜀 󵄨 󵄨 𝑝/2 ∫ (2 𝑎 𝑐 󵄨𝑓 𝑓󸀠󵄨) 𝑑𝜇 𝐹𝑐 √ 𝑝 𝑝 󵄨 𝑛 𝑛󵄨 2 −1 𝑛,𝜀 󵄨 󵄨2 󵄨 󵄨2 𝑝/2 lim =0. (26) ×(∫ (𝑎 󵄨𝑓 󵄨 +𝑐 󵄨𝑓󸀠󵄨 ) 𝑑𝜇 ) ) 𝑛→∞ 󵄨 󵄨2 󵄨 󵄨2 𝑝/2 𝐹𝑐 𝑝󵄨 𝑛󵄨 𝑝󵄨 𝑛󵄨 2 󵄨 󵄨 󵄨 󸀠󵄨 𝑛,𝜀 ∫ (𝑎𝑝󵄨𝑓𝑛󵄨 +𝑐𝑝󵄨𝑓𝑛󵄨 ) 𝑑𝜇2 𝐹𝑛,𝜀

󵄨 󵄨 𝑝/2 󵄨 󸀠󵄨 +((∫ 𝑐 (2√𝑎𝑝c𝑝 󵄨𝑓𝑛𝑓𝑛󵄨) 𝑑𝜇2) On the other hand, using (20)itiseasytodeducethat 𝐹𝑛,𝜀 󵄨 󵄨

𝑝/2 −1 󵄨 󵄨2 󵄨 󸀠󵄨2 ×(∫ (𝑎 󵄨𝑓 󵄨 +𝑐 󵄨𝑓 󵄨 ) 𝑑𝜇 ) )) 𝑝/2 𝐹𝑐 𝑝󵄨 𝑛󵄨 𝑝󵄨 𝑛󵄨 2 󵄨 󸀠󵄨 𝑛,𝜀 lim ((( (∫ (2√𝑎𝑝𝑐𝑝 󵄨𝑓𝑛𝑓 󵄨) 𝑑𝜇2) 𝑛→∞ 𝐹𝑛,𝜀 󵄨 𝑛󵄨

𝑝/2 󵄨 󵄨2 󵄨 󸀠󵄨2 −1 ×(1+((∫ (𝑎 󵄨𝑓 󵄨 +𝑐 󵄨𝑓 󵄨 ) 𝑑𝜇 ) 󵄨 󵄨2 𝑝/2 𝐹 𝑝󵄨 𝑛󵄨 𝑝󵄨 𝑛󵄨 2 󵄨 󵄨2 󵄨 󸀠󵄨 𝑛,𝜀 ×(∫ (𝑎𝑝󵄨𝑓𝑛󵄨 +𝑐𝑝󵄨𝑓 󵄨 ) 𝑑𝜇2) ) 𝐹𝑛,𝜀 󵄨 󵄨 󵄨 𝑛󵄨

−1 𝑝/2 −1 󵄨 󵄨2 󵄨 󸀠󵄨2 ×(∫ (𝑎 󵄨𝑓 󵄨 +𝑐 󵄨𝑓 󵄨 ) 𝑑𝜇) )) ) 󵄨 󸀠󵄨 𝑝/2 𝐹𝑐 𝑝󵄨 𝑛󵄨 𝑝󵄨 𝑛󵄨 󵄨 󵄨 𝑛,𝜀 󵄨 󵄨 +((∫ 𝑐 (2√𝑎𝑝𝑐𝑝 󵄨𝑓𝑛𝑓𝑛󵄨) 𝑑𝜇2) 𝐹𝑛,𝜀 󵄨 󵄨

𝑝/2 −1 𝑝/2 󵄨 󵄨2 󵄨 󸀠󵄨2 󵄨 󵄨2 𝑝/2 ≤(𝐶 + lim inf ((∫ (𝑎 󵄨𝑓 󵄨 +𝑐 󵄨𝑓 󵄨 ) 𝑑𝜇 ) 󵄨 󵄨2 󵄨 󸀠󵄨 𝜀 𝑛→∞ 𝐹 𝑝󵄨 𝑛󵄨 𝑝󵄨 𝑛󵄨 2 ×(∫ (𝑎𝑝󵄨𝑓𝑛󵄨 +𝑐𝑝󵄨𝑓 󵄨 ) 𝑑𝜇2) ) 𝑛,𝜀 𝐹𝑛,𝜀 󵄨 󵄨 󵄨 𝑛󵄨

𝑝/2 −1 󵄨 󵄨2 󵄨 󸀠󵄨2 󵄨 󵄨2 𝑝/2 ×(∫ (𝑎 󵄨𝑓 󵄨 +𝑐 󵄨𝑓 󵄨 ) 𝑑𝜇 ) )) 󵄨 󵄨2 󵄨 󸀠󵄨 𝐹𝑐 𝑝󵄨 𝑛󵄨 𝑝󵄨 𝑛󵄨 2 ×(1+((∫ 𝑐 (𝑎𝑝󵄨𝑓𝑛󵄨 +𝑐𝑝󵄨𝑓𝑛󵄨 ) 𝑑𝜇2) 𝑛,𝜀 𝐹𝑛,𝜀 󵄨 󵄨

−1 󵄨 󵄨2 󵄨 󵄨2 𝑝/2 −1 󵄨 󵄨 󵄨 󸀠󵄨 󵄨 󵄨2 󵄨 󵄨2 𝑝/2 ×(1+lim inf ((∫ (𝑎𝑝󵄨𝑓𝑛󵄨 +𝑐𝑝󵄨𝑓𝑛󵄨 ) 𝑑𝜇2) 󵄨 󵄨 󵄨 󸀠󵄨 𝑛→∞ 𝐹𝑛,𝜀 󵄨 󵄨 ×(∫ (𝑎𝑝󵄨𝑓𝑛󵄨 +𝑐𝑝󵄨𝑓 󵄨 ) 𝑑𝜇2) )) ) 𝐹𝑛,𝜀 󵄨 󵄨 󵄨 𝑛󵄨

−1 −1 󵄨 󵄨2 󵄨 󵄨2 𝑝/2 ×(∫ (𝑎 󵄨𝑓 󵄨 +𝑐 󵄨𝑓󸀠󵄨 ) 𝑑𝜇 ) )) . =1. 𝐹𝑐 𝑝󵄨 𝑛󵄨 𝑝󵄨 𝑛󵄨 2 𝑛,𝜀 (27) (23) Consequently, (24), (26), and (27)give ((∫ (𝑎 |𝑓 |2 +𝑐|𝑓󸀠|2)𝑝/2 If we assume that lim inf 𝑛→∞ 𝐹 𝑝 𝑛 𝑝 𝑛 𝑛,𝜀 󵄨 󸀠󵄨 𝑝/2 −1 ∫ (2 𝑎 𝑐 󵄨𝑓 𝑓 󵄨) 𝑑𝜇 2 󸀠 2 𝑝/2 𝐹 √ 𝑝 𝑝 󵄨 𝑛 𝑛󵄨 2 𝑑𝜇 )(∫ (𝑎 |𝑓 | +𝑐 |𝑓 | ) 𝑑𝜇 ) )=𝑙<∞ 𝑛,𝜀 󵄨 󵄨 2 𝐹𝑐 𝑝 𝑛 𝑝 𝑛 2 ,thenfrom lim =1. 𝑛,𝜀 𝑛→∞ 󵄨 󵄨2 󵄨 󵄨2 𝑝/2 (28) 𝑝/2 󵄨 󵄨 󵄨 󸀠󵄨 1 ≤ ((𝐶 + 𝑙)/(1 𝑙)) < 1 ∫ (𝑎𝑝󵄨𝑓𝑛󵄨 +𝑐𝑝󵄨𝑓𝑛󵄨 ) 𝑑𝜇2 the previous inequality we have 𝜀 + , 𝐹𝑛,𝜀 2 andthisisacontradiction.Hence,lim𝑛→∞((∫ (𝑎𝑝|𝑓𝑛| + 𝐹𝑛,𝜀 −1 󸀠 2 𝑝/2 2 󸀠 2 𝑝/2 Furthermore, since 𝑐𝑝|𝑓𝑛| ) 𝑑𝜇2)(∫ 𝑐 (𝑎𝑝|𝑓𝑛| +𝑐𝑝|𝑓𝑛| ) 𝑑𝜇2) )=∞and 𝐹𝑛,𝜀 consequently, 1 0≤ 𝑝/2 𝑝/2 󵄨 󸀠󵄨2 (1+𝜀) ∫ (𝑐𝑝󵄨𝑓𝑛󵄨 ) 𝑑𝜇2 𝐹𝑛,𝜀 󵄨 󵄨2 󵄨 󵄨2 𝑝/2 ∫ (𝑎 󵄨𝑓 󵄨 +𝑐 󵄨𝑓󸀠󵄨 ) 𝑑𝜇 (29) 𝐹𝑐 𝑝󵄨 𝑛󵄨 𝑝󵄨 𝑛󵄨 2 1 𝑛,𝜀 =0. ≤ , lim 𝑝/2 (24) 𝑝/2 𝑛→∞ 󵄨 󵄨2 󵄨 󸀠󵄨2 󵄨 󸀠󵄨 ∫ (𝑎𝑝󵄨𝑓𝑛󵄨 +𝑐𝑝󵄨𝑓𝑛󵄨 ) 𝑑𝜇2 ∫ (√𝑎𝑝𝑐𝑝 󵄨𝑓𝑛𝑓𝑛󵄨) 𝑑𝜇2 𝐹𝑛,𝜀 𝐹𝑛,𝜀 6 Journal of Function Spaces and Applications we obtain As a consequence of (33)wehave

󵄨 󵄨2 𝑝/2 ∫(𝑎𝑝󵄨𝑓𝑛󵄨 ) 𝑑𝜇2 lim sup 󵄨 󵄨2 𝑝/2 󵄨 󵄨2 󵄨 󵄨2 𝑝/2 𝑛→∞ ∫(𝑐 󵄨𝑓󸀠󵄨 ) 𝑑𝜇 ∫ (𝑎 󵄨𝑓 󵄨 +𝑐 󵄨𝑓󸀠󵄨 ) 𝑑𝜇 𝑝󵄨 𝑛󵄨 2 𝐹𝑐 𝑝󵄨 𝑛󵄨 𝑝󵄨 𝑛󵄨 2 0≤ 𝑛,𝜀 󵄨 󵄨2 𝑝/2 (1+𝜀)𝑝/2 ∫ (𝑐 󵄨𝑓󸀠󵄨 ) 𝑑𝜇 󵄨 󵄨2 𝑝/2 𝐹 𝑝󵄨 𝑛󵄨 2 = lim sup ((( (∫ (𝑎𝑝󵄨𝑓𝑛󵄨 ) 𝑑𝜇2) 𝑛,𝜀 𝑛→∞ 𝐹𝑛,𝜀 (30) 𝑝/2 −1 󵄨 󵄨2 󵄨 󸀠󵄨2 󵄨 󵄨2 𝑝/2 ∫ (𝑎 󵄨𝑓 󵄨 +𝑐 󵄨𝑓 󵄨 ) 𝑑𝜇 ×(∫ (𝑐 󵄨𝑓󸀠󵄨 ) 𝑑𝜇 ) ) 𝐹𝑐 𝑝󵄨 𝑛󵄨 𝑝󵄨 𝑛󵄨 2 𝐹 𝑝󵄨 𝑛󵄨 2 ≤ 𝑛,𝜀 . 𝑛,𝜀 󵄨 󵄨 𝑝/2 ∫ ( 𝑎 𝑐 󵄨𝑓 𝑓󸀠󵄨) 𝑑𝜇 𝐹 √ 𝑝 𝑝 󵄨 𝑛 𝑛󵄨 2 𝑛,𝜀 󵄨 󵄨2 𝑝/2 +((∫ 𝑐 (𝑎𝑝󵄨𝑓𝑛󵄨 ) 𝑑𝜇2) 𝐹𝑛,𝜀

−1 󵄨 󵄨2 𝑝/2 󵄨 󸀠󵄨 ×(∫ (𝑐𝑝󵄨𝑓𝑛󵄨 ) 𝑑𝜇2) )) Therefore, (24), (28), and (30)give 𝐹𝑛,𝜀 󵄨 󵄨

󵄨 󵄨2 𝑝/2 ×(1+((∫ (𝑐 󵄨𝑓󸀠󵄨 ) 𝑑𝜇 ) 𝐹𝑐 𝑝󵄨 𝑛󵄨 2 󵄨 󵄨2 𝑝/2 𝑛,𝜀 󵄨 󵄨2 󵄨 󸀠󵄨 ∫ 𝑐 (𝑎𝑝󵄨𝑓𝑛󵄨 +𝑐𝑝󵄨𝑓𝑛󵄨 ) 𝑑𝜇2 𝐹𝑛,𝜀 󵄨 󵄨 =0. −1 −1 lim 𝑝/2 (31) 󵄨 󵄨2 𝑝/2 𝑛→∞ 󵄨 󸀠󵄨2 ×(∫ (𝑐 󵄨𝑓󸀠󵄨 ) 𝑑𝜇 ) )) ) ∫ (𝑐 󵄨𝑓 󵄨 ) 𝑑𝜇 𝐹 𝑝󵄨 𝑛󵄨 2 𝑝󵄨 𝑛󵄨 2 𝑛,𝜀 𝐹𝑛,𝜀

󵄨 󵄨2 𝑝/2 ∫ (𝑎 󵄨𝑓 󵄨 ) 𝑑𝜇 𝐹 𝑝󵄨 𝑛󵄨 2 = lim sup 𝑛,𝜀 ≤ (1+𝜀)𝑝. 𝑝/2 𝑛→∞ 󵄨 󸀠󵄨2 Similarargumentsallowustoshow ∫ (𝑐𝑝󵄨𝑓𝑛󵄨 ) 𝑑𝜇2 𝐹𝑛,𝜀 (35)

In a similar way we obtain 󵄨 󵄨2 󵄨 󵄨2 𝑝/2 ∫ (𝑎 󵄨𝑓 󵄨 +𝑐 󵄨𝑓󸀠󵄨 ) 𝑑𝜇 𝐹𝑐 𝑝󵄨 𝑛󵄨 𝑝󵄨 𝑛󵄨 2 𝑛,𝜀 󵄨 󵄨2 𝑝/2 lim =0. (32) ∫(𝑎 󵄨𝑓 󵄨 ) 𝑑𝜇 𝑛→∞ 󵄨 󵄨2 𝑝/2 1 𝑝󵄨 𝑛󵄨 2 ∫ (𝑎 󵄨𝑓 󵄨 ) 𝑑𝜇 ≤ . 𝐹 𝑝󵄨 𝑛󵄨 2 𝑝 lim inf 𝑝/2 (36) 𝑛,𝜀 (1+𝜀) 𝑛→∞ 󵄨 󸀠󵄨2 ∫(𝑐𝑝󵄨𝑓𝑛󵄨 ) 𝑑𝜇2

Since these inequalities hold for every 𝜀>0,weconcludethat From (31)and(32)weobtain (18)holds.ApplyingnowLemma 6 we obtain (16). Using Lemma 4,(18), and (34)weobtainthatforevery 𝜀,𝜂>0there exists 𝑁 such that for every 𝑛≥𝑁the following holds: 󵄨 󵄨2 𝑝/2 ∫ (𝑎 󵄨𝑓 󵄨 ) 𝑑𝜇 𝐹𝑐 𝑝󵄨 𝑛󵄨 2 𝑝/2 𝑛,𝜀 󵄨 󵄨2 𝑝/2 󵄨 󵄨2 lim 2𝑝/2−1 ∫((𝑎 󵄨𝑓 󵄨 ) +(𝑐 󵄨𝑓󸀠󵄨 ) )𝑑𝜇 𝑛→∞ 󵄨 󵄨2 𝑝/2 𝑝󵄨 𝑛󵄨 𝑝󵄨 𝑛󵄨 2 ∫ (𝑐 󵄨𝑓󸀠󵄨 ) 𝑑𝜇 𝐹 𝑝󵄨 𝑛󵄨 2 1≤ 𝑛,𝜀 󵄨 󵄨2 󵄨 󵄨2 𝑝/2 ∫(𝑎 󵄨𝑓 󵄨 +𝑐 󵄨𝑓󸀠󵄨 ) 𝑑𝜇 (33) 𝑝󵄨 𝑛󵄨 𝑝󵄨 𝑛󵄨 2 𝑝/2 󵄨 󸀠󵄨2 ∫ (𝑐 󵄨𝑓 󵄨 ) 𝑑𝜇 𝑝/2 𝑝/2 𝐹𝑐 𝑝󵄨 𝑛󵄨 2 𝑝/2−1 󵄨 󵄨2 󵄨 󸀠󵄨2 𝑛,𝜀 2 ∫((𝑎 󵄨𝑓 󵄨 ) +(𝑐 󵄨𝑓 󵄨 ) )𝑑𝜇 =0= lim , 𝑝󵄨 𝑛󵄨 𝑝󵄨 𝑛󵄨 2 𝑛→∞ 󵄨 󵄨2 𝑝/2 ∫ (𝑐 󵄨𝑓󸀠󵄨 ) 𝑑𝜇 ≤ 𝑝󵄨 𝑛󵄨 2 2 2 𝑝/2 𝐹𝑛,𝜀 󵄨 󵄨 󵄨 󸀠󵄨 ∫ (𝑎𝑝󵄨𝑓𝑛󵄨 +𝑐𝑝󵄨𝑓𝑛󵄨 ) 𝑑𝜇2 𝐹𝑛,𝜀 󵄨 󵄨2 𝑝/2 ∫ (𝑎 󵄨𝑓 󵄨 ) 𝑑𝜇 𝐹𝑐 𝑝󵄨 𝑛󵄨 2 𝑝/2 󵄨 󵄨2 𝑝/2 𝑛,𝜀 𝑝/2−1 󵄨 󵄨2 󵄨 󸀠󵄨 lim 2 ∫((𝑎𝑝󵄨𝑓𝑛󵄨 ) +(𝑐𝑝󵄨𝑓 󵄨 ) )𝑑𝜇2 𝑛→∞ 󵄨 󵄨2 𝑝/2 󵄨 󵄨 󵄨 𝑛󵄨 ∫ (𝑎 󵄨𝑓 󵄨 ) 𝑑𝜇 𝐹 𝑝󵄨 𝑛󵄨 2 ≤ 𝑛,𝜀 2 𝑝/2 󵄨 󵄨2 𝑝/2 (1 + (1/(1+𝜀) )) ∫ (𝑎𝑝󵄨𝑓𝑛󵄨 ) 𝑑𝜇2 (34) 𝐹𝑛,𝜀 󵄨 󵄨2 𝑝/2 󵄨 󸀠󵄨 𝑝/2 ∫ 𝑐 (𝑐𝑝󵄨𝑓𝑛󵄨 ) 𝑑𝜇2 𝑝/2 󵄨 󵄨2 𝐹𝑛,𝜀 󵄨 󵄨 2 (1 + 𝜂) ∫ (𝑎 󵄨𝑓 󵄨 ) 𝑑𝜇 =0= lim . 𝑝󵄨 𝑛󵄨 2 𝑛→∞ 󵄨 󵄨2 𝑝/2 ≤ 󵄨 󵄨 𝑝/2 󵄨 󵄨2 𝑝/2 ∫ (𝑎𝑝󵄨𝑓𝑛󵄨 ) 𝑑𝜇2 (1 + (1/ 1+𝜀 2)) ∫ (𝑎 󵄨𝑓 󵄨 ) 𝑑𝜇 𝐹𝑛,𝜀 ( ) 𝑝󵄨 𝑛󵄨 2 𝐹𝑛,𝜀 Journal of Function Spaces and Applications 7

󵄨 󵄨2 𝑝/2 𝑝/2 2 ∫ (𝑎 󵄨𝑓 󵄨 ) 𝑑𝜇 Lemma 13. If 1≤𝑝<∞, {𝑓𝑛}𝑛 is an extremal sequence for 𝑝 2 (1 + 𝜂) 𝐹 𝑝󵄨 𝑛󵄨 2 ≤ ⋅ 𝑛,𝜀 and 𝜀 is small enough, then 𝑝/2 󵄨 󵄨2 𝑝/2 (1 + (1/(1+𝜀)2)) ∫ (𝑎 󵄨𝑓 󵄨 ) 𝑑𝜇 𝐹 𝑝󵄨 𝑛󵄨 2 󵄨 󵄨2 𝑝/2 𝑛,𝜀 ∫ (𝑎 󵄨𝑓 󵄨 ) 𝑑𝜇 𝐴 𝑝󵄨 𝑛󵄨 2 lim 𝜀 =1, 𝑝/2 2 𝑛→∞ 󵄨 󵄨2 𝑝/2 2 (1 + 𝜂) ∫(𝑎 󵄨𝑓 󵄨 ) 𝑑𝜇 = . 𝑝󵄨 𝑛󵄨 2 𝑝/2 (1 + (1/(1+𝜀)2)) (41) 󵄨 󵄨2 𝑝/2 ∫ (𝑐 󵄨𝑓󸀠󵄨 ) 𝑑𝜇 𝐴 𝑝󵄨 𝑛󵄨 2 (37) 𝜀 =1. lim 𝑝/2 𝑛→∞ 󵄨 󸀠󵄨2 ∫(𝑐𝑝󵄨𝑓𝑛󵄨 ) 𝑑𝜇2 Then (17) follows from the previous inequalities, since Proof. If 1≤𝑝≤2, then the result follows from [22,Lemma 𝜀,𝜂>0are arbitrary. 3.11]. For the case 𝑝>2it suffices to follow the proof of[22, This completes the proof. Lemma 3.11] applying Lemmas 8, 10,and12 to conclude the result. Definition 9. For each 0<𝜀<1, we define the sets 𝐴𝜀 and 𝑐 𝐴𝜀 as Lemma 14. If 1≤𝑝<∞and {𝑓𝑛}𝑛 is an extremal sequence 𝑐 for 𝑝,thenforevery𝜀>0small enough with 𝜇2(𝐴𝜀)>0and 󵄨 󵄨 󵄨 󵄨 for every 𝑡 ∈ (0, 1) there exists 𝑁 such that inf𝑧∈𝐴𝑐 |𝑓𝑛(𝑧)| < 𝑡 𝐴𝜀 := {𝑧 ∈ 𝑆2 (𝜇 ):󵄨𝑏𝑝󵄨 > (1−𝜀) √𝑎𝑝𝑐𝑝}, 𝜀 for every 𝑛≥𝑁. (38) 𝐴𝑐 := 𝑆 (𝜇 )\𝐴 . 𝜀 2 𝜀 Proof. If 1≤𝑝≤2, then the result follows from [22,Lemma 3.12]. For the case 𝑝>2it is sufficient to follow the proof of [22, Lemma 3.12] applying Lemma 13 to conclude the result. Lemma 10. If 1≤𝑝<∞and {𝑓𝑛}𝑛 is an extremal sequence for 𝑝 and 𝜀 is small enough, then Definition 15. If 𝑓 is a continuous function on 𝛾, we define 󵄨 󵄨𝑝/2 the oscillation of 𝑓 on 𝛾, and we denote it by osc(𝑓),as ∫ 󵄨𝑏 𝑓 𝑓󸀠󵄨 𝑑𝜇 𝐴 󵄨 𝑝 𝑛 𝑛󵄨 2 󵄨 󵄨 𝜀 =1. (𝑓) := 󵄨𝑓 (𝑧) −𝑓(𝑤)󵄨 . lim 󵄨 󵄨𝑝/2 (39) osc sup 󵄨 󵄨 𝑛→∞ ∫ 󵄨𝑏 𝑓 𝑓󸀠󵄨 𝑑𝜇 𝑧,𝑤∈𝛾 (42) 󵄨 𝑝 𝑛 𝑛󵄨 2 Lemma 16 (see [22, Lemma 3.14]). For 1≤𝑝<∞, 𝛾 (𝑐𝑝/2 𝑑𝜇 /𝑑𝑠)−1 ∈ Remark 11. The statement of the lemma might seem strange, let us assume that is connected and 𝑝 2 1/(𝑝−1) becausewecouldhaveapriori𝜇2(𝐴𝜀)=0;however,the 𝐿 (𝛾),where𝑑𝜇2/𝑑𝑠 is the Radon-Nykodim derivative of existence of the fundamental sequence implies 𝜇2(𝐴𝜀)>0. 𝜇2 with respect to the Euclidean length in 𝛾. (According to one’s notation, if 𝑝=1then 1/(𝑝 − 1) =.) ∞ Then Proof. If 1≤𝑝≤2, then the result follows from [22,Lemma 󵄨 󵄨𝑝 3.8]. For the case 𝑝>2it suffices to follow the proof of[22, ∫ 󵄨𝑓󸀠󵄨 𝑐𝑝/2𝑑𝜇 ≥𝑘⋅ 𝑝 (𝑓) , 󵄨 󵄨 𝑝 2 osc Lemma 3.8] applying Lemma 8 to conclude the result. 𝛾 󵄩 󵄩 󵄩 󵄩 (43) Lemma 12. If 1≤𝑝<∞, {𝑓𝑛}𝑛 is an extremal sequence for 𝑝 1 󵄩 1 󵄩 = 󵄩 󵄩 , 𝜀 with 󵄩 𝑝/2 󵄩 and is small enough, then 𝑘 󵄩(𝑐 (𝑑𝜇 /𝑑𝑠))󵄩 󵄩 𝑝 2 󵄩𝐿1/(𝑝−1)(𝛾)

󵄨 󵄨2 𝑝/2 󵄨 󵄨2 𝑝/2 for every polynomial 𝑓. ∫ ((𝑎 󵄨𝑓 󵄨 ) +(𝑐 󵄨𝑓󸀠󵄨 ) )𝑑𝜇 𝐴 𝑝󵄨 𝑛󵄨 𝑝󵄨 𝑛󵄨 2 𝜀 =1, lim 𝑝/2 𝑝/2 𝑛→∞ 󵄨 󵄨2 󵄨 󸀠󵄨2 4. Equivalent Norms ∫((𝑎𝑝󵄨𝑓𝑛󵄨 ) +(𝑐𝑝󵄨𝑓𝑛󵄨 ) )𝑑𝜇2 (40) Now we prove the announced result about the equivalence of 𝑝/2 󵄨 󵄨2 𝑝/2 norms for 1≤𝑝<∞. 󵄨 󵄨2 󵄨 󸀠󵄨 ∫ 𝑐 ((𝑎𝑝󵄨𝑓𝑛󵄨 ) +(𝑐𝑝󵄨𝑓𝑛󵄨 ) )𝑑𝜇2 𝐴𝜀 󵄨 󵄨 lim =0. Theorem 17. Let one consider 1≤𝑝<∞and (𝑉,𝜇) a 𝑝- 𝑛→∞ 󵄨 󵄨2 𝑝/2 󵄨 󵄨2 𝑝/2 ∫ ((𝑎 󵄨𝑓 󵄨 ) +(𝑐 󵄨𝑓󸀠󵄨 ) )𝑑𝜇 𝑊1,𝑝(𝑎𝑝/2𝜇,𝑝/2 𝑐 𝜇) 𝑊1,𝑝(𝐷𝜇) 𝐴 𝑝󵄨 𝑛󵄨 𝑝󵄨 𝑛󵄨 2 admissible pair. Then the norms 𝑝 𝑝 , , 𝜀 1,𝑝 and 𝑊 (𝑉𝜇) defined as in (3) are equivalent on the space of polynomials P. Proof. If 1≤𝑝≤2, then the result follows from [22,Lemma 3.10]. For the case 𝑝>2it suffices to follow the proof of Proof. The equivalence of the two first norms is straightfor- [22, Lemma 3.10] applying Lemmas 8 and 10 to conclude the ward, by Lemmas 4 and 5. We prove now the equivalence of result. the two last norms. 8 Journal of Function Spaces and Applications

Let us prove that there exists a positive constant 𝐶:= If 1≤𝑝≤2,then[22, Lemma 3.1] (with 𝛼=𝑝/2)gives 𝐶(𝑉,𝜇, 𝑝) such that 󵄨 󵄨2 󵄨 󵄨2 𝑝/2 ∫ (𝑎 󵄨𝑓 󵄨 +𝑐 󵄨𝑓󸀠󵄨 +2R (𝑏 𝑓 𝑓󸀠)) 𝑑𝜇 𝑝󵄨 𝑛󵄨 𝑝󵄨 𝑛󵄨 𝑝 𝑛 𝑛 2 󵄩 󵄩 󵄩 󵄩 𝐶 󵄩𝑓󵄩 1,𝑝 ≤ 󵄩𝑓󵄩 1,𝑝 󵄩 󵄩𝑊 (𝐷𝜇) 󵄩 󵄩𝑊 (𝑉𝜇) 󵄨 󵄨2 󵄨 󵄨2 󵄨 󵄨 𝑝/2 ≥ ∫ (𝑎 󵄨𝑓 󵄨 +𝑐 󵄨𝑓󸀠󵄨 − 󵄨2𝑏 𝑓 𝑓󸀠󵄨) 𝑑𝜇 (44) 𝑝󵄨 𝑛󵄨 𝑝󵄨 𝑛󵄨 󵄨 𝑝 𝑛 𝑛󵄨 2 √ 󵄩 󵄩 ≤ 2 󵄩𝑓󵄩𝑊1,𝑝(𝐷𝜇), for every 𝑓∈P. 󵄨 󵄨2 󵄨 󵄨2 𝑝/2 󵄨 󵄨𝑝/2 ≥ ∫ (𝑎 󵄨𝑓 󵄨 +𝑐 󵄨𝑓󸀠󵄨 ) 𝑑𝜇 − ∫ 󵄨2𝑏 𝑓 𝑓󸀠󵄨 𝑑𝜇 . 𝑝󵄨 𝑛󵄨 𝑝󵄨 𝑛󵄨 2 󵄨 𝑝 𝑛 𝑛󵄨 2

Let us prove first the second inequality ‖𝑓‖𝑊1,𝑝(𝑉𝜇) ≤ (48) √ 2‖𝑓‖ 1,𝑝 . 𝑊 (𝐷𝜇) This right-hand side of the inequality is positive, because |2R(𝑏 𝑓𝑓󸀠)| ≤ |2𝑏 𝑓𝑓 󸀠|≤2𝑎 𝑐 |𝑓𝑓󸀠|≤ 󸀠 2 󸀠 2 Note that 𝑝 𝑝 √ 𝑝 𝑝 |2𝑏𝑝𝑓𝑓 |≤𝑎𝑝|𝑓| +𝑐𝑝|𝑓 | 𝜇2-almost everywhere. This implies 2 󸀠 2 𝑎𝑝|𝑓| +𝑐𝑝|𝑓 | ; therefore, for every polynomial 𝑓 it holds 󵄨 󵄨2 󵄨 󵄨2 𝑝/2 that ((∫ (𝑎 󵄨𝑓 󵄨 +𝑐 󵄨𝑓󸀠󵄨 ) 𝑑𝜇 𝑛→∞lim 𝑝󵄨 𝑛󵄨 𝑝󵄨 𝑛󵄨 2

󵄩 󵄩𝑝 󵄨 󵄨2 󵄨 󵄨2 󵄨 󵄨𝑝/2 󵄩𝑓󵄩 = ∫ (𝑎 󵄨𝑓󵄨 +𝑐 󵄨𝑓󸀠󵄨 − ∫ 󵄨2𝑏 𝑓 𝑓󸀠󵄨 𝑑𝜇 ) 󵄩 󵄩𝑊1,𝑝(𝑉𝜇 ) 𝑝󵄨 󵄨 𝑝󵄨 󵄨 󵄨 𝑝 𝑛 𝑛󵄨 2 (49)

𝑝/2 −1 𝑝/2 󵄨 󵄨2 󵄨 󸀠󵄨2 󸀠 ×(∫ (𝑎 󵄨𝑓 󵄨 +𝑐 󵄨𝑓 󵄨 ) 𝑑𝜇 ) )=0, +2R (𝑏𝑝𝑓𝑓 )) 𝑑𝜇 𝑝󵄨 𝑛󵄨 𝑝󵄨 𝑛󵄨 2 (45) 󵄨 󵄨2 𝑝/2 𝑝/2 󵄨 󵄨2 󵄨 󸀠󵄨 and hence ≤2 ∫ (𝑎𝑝󵄨𝑓󵄨 +𝑐𝑝󵄨𝑓 󵄨 ) 𝑑𝜇 󵄨 󵄨𝑝/2 ∫ 󵄨2𝑏 𝑓 𝑓󸀠󵄨 𝑑𝜇 𝑝/2󵄩 󵄩𝑝 󵄨 𝑝 𝑛 𝑛󵄨 2 =2 󵄩𝑓󵄩 1,𝑝 . =1. 󵄩 󵄩𝑊 (𝐷𝜇) lim 𝑝/2 (50) 𝑛→∞ 󵄨 󵄨2 󵄨 󸀠󵄨2 ∫(𝑎𝑝󵄨𝑓𝑛󵄨 +𝑐𝑝󵄨𝑓𝑛󵄨 ) 𝑑𝜇2 𝐶‖𝑓‖ ≤ In order to prove the first inequality, 𝑊1,𝑝(𝐷𝜇) If 𝑝>2,then ‖𝑓‖𝑊1,𝑝(𝑉𝜇),notethat 2/𝑝 󵄨 󵄨2 𝑝/2 󵄨 󵄨2 󵄨 󸀠󵄨 󸀠 (∫ (𝑎𝑝󵄨𝑓𝑛󵄨 +𝑐𝑝󵄨𝑓𝑛󵄨 +2R (𝑏𝑝𝑓𝑛𝑓𝑛)) 𝑑𝜇2) 󵄩 󵄩𝑝 󵄨 󵄨 󵄩𝑓󵄩 󵄩 󵄩𝑊1,𝑝(𝑉𝜇 ) 1 𝑝/2 2/𝑝 󵄨 󵄨2 󵄨 󸀠󵄨2 ≥(∫ (𝑎 󵄨𝑓 󵄨 +𝑐 󵄨𝑓 󵄨 ) 𝑑𝜇 ) 󵄨 󵄨2 𝑝/2 𝑝󵄨 𝑛󵄨 𝑝󵄨 𝑛󵄨 2 󵄨 󵄨2 󵄨 󸀠󵄨 󸀠 (51) = ∫ (𝑎𝑝󵄨𝑓󵄨 +𝑐𝑝󵄨𝑓 󵄨 +2R (𝑏𝑝𝑓𝑓 )) 𝑑𝜇1 󵄨 󵄨𝑝/2 2/𝑝 −(∫ 󵄨2𝑏 𝑓 𝑓󸀠󵄨 𝑑𝜇 ) 󵄨 𝑝 𝑛 𝑛󵄨 2 󵄨 󵄨2 󵄨 󵄨2 󵄨 󵄨 𝑝/2 ≥ ∫ (𝑎 󵄨𝑓󵄨 +𝑐 󵄨𝑓󸀠󵄨 −2 󵄨𝑏 𝑓𝑓 󸀠󵄨)) 𝑑𝜇 𝑝󵄨 󵄨 𝑝󵄨 󵄨 󵄨 𝑝 󵄨 1 ≥0,

󵄨 󵄨2 󵄨 󵄨 𝑝/2 󸀠 2 󸀠 2 󵄨 󵄨2 󵄨 󸀠󵄨 √ 󵄨 󸀠󵄨 |2𝑏 𝑓𝑓 |≤𝑎|𝑓| +𝑐|𝑓 | 𝜇 ≥ ∫ (𝑎𝑝󵄨𝑓󵄨 +𝑐𝑝󵄨𝑓 󵄨 −2 1−𝜀0√𝑎𝑝𝑐𝑝 󵄨𝑓𝑓 󵄨) 𝑑𝜇1 since 𝑝 𝑝 𝑝 2-almost everywhere. Therefore, 𝑝/2 󵄨 󵄨2 𝑝/2 󵄨 󵄨2 󵄨 󸀠󵄨 2/𝑝 ≥(1−√1−𝜀0) ∫ (𝑎𝑝󵄨𝑓󵄨 +𝑐𝑝󵄨𝑓 󵄨 ) 𝑑𝜇1 󵄨 󵄨2 󵄨 󵄨2 𝑝/2 󵄨 󵄨 (((∫ (𝑎 󵄨𝑓 󵄨 +𝑐 󵄨𝑓󸀠󵄨 ) 𝑑𝜇 ) 𝑛→∞lim 𝑝󵄨 𝑛󵄨 𝑝󵄨 𝑛󵄨 2 𝑝/2󵄩 󵄩𝑝 =(1−√1−𝜀 ) 󵄩𝑓󵄩 . 2/𝑝 0 󵄩 󵄩𝑊1,𝑝(𝐷𝜇 ) 󵄨 󵄨𝑝/2 1 −(∫ 󵄨2𝑏 𝑓 𝑓󸀠󵄨 𝑑𝜇 ) ) 󵄨 𝑝 𝑛 𝑛󵄨 2 (46) 2/𝑝 −1 󵄨 󵄨2 󵄨 󵄨2 𝑝/2 ×((∫ (𝑎 󵄨𝑓 󵄨 +𝑐 󵄨𝑓󸀠󵄨 ) 𝑑𝜇 ) ) )=0, 𝑝󵄨 𝑛󵄨 𝑝󵄨 𝑛󵄨 2 If 𝛾=0(i.e., 𝜇=𝜇1), then we have finished the proof. Assume that 𝛾 =0̸ then we prove 𝐶‖𝑓‖𝑊1,𝑝(𝐷𝜇 ) ≤‖ 2 (52) 𝑓‖ 1,𝑝 𝑊 (𝑉𝜇2), seeking for a contradiction. It is clear that it suffices to prove it when 𝛾 is connected, that is, when 𝛾 is a or, equivalently, rectifiable compact curve. Let us assume that there exists a {𝑓 } ⊂ P 󵄨 󵄨𝑝/2 2/𝑝 sequence 𝑛 𝑛 such that (∫ 󵄨2𝑏 𝑓 𝑓󸀠󵄨 𝑑𝜇 ) 󵄨 𝑝 𝑛 𝑛󵄨 2 =1, 𝑛→∞lim 2/𝑝 (53) 𝑝/2 󵄨 󵄨2 󵄨 󵄨2 𝑝/2 󵄨 󵄨2 󵄨 󸀠󵄨2 󵄨 󵄨 󵄨 󸀠󵄨 ∫(𝑎 󵄨𝑓 󵄨 +𝑐 󵄨𝑓 󵄨 +2R (𝑏 𝑓 𝑓󸀠)) 𝑑𝜇 (∫ (𝑎𝑝󵄨𝑓𝑛󵄨 +𝑐𝑝󵄨𝑓 󵄨 ) 𝑑𝜇2) 𝑝󵄨 𝑛󵄨 𝑝󵄨 𝑛󵄨 𝑝 𝑛 𝑛 2 󵄨 󵄨 󵄨 𝑛󵄨 =0. lim 𝑝/2 (47) 𝑛→∞ 󵄨 󵄨2 󵄨 󸀠󵄨2 ∫(𝑎𝑝󵄨𝑓𝑛󵄨 +𝑐𝑝󵄨𝑓𝑛󵄨 ) 𝑑𝜇2 and (50)alsoholdsfor𝑝>2. Journal of Function Spaces and Applications 9

󸀠 𝑝/2 If 𝑓𝑛 is constant for some 𝑛,then∫|2𝑏𝑝𝑓𝑛𝑓𝑛| 𝑑𝜇2 =0; This latter theorem and Theorem 2 give the following therefore, taking a subsequence if it is necessary, without result. loss of generality we can assume that 𝑓𝑛 is nonconstant and Theorem 19. 1≤𝑝<∞ (𝑉,𝜇) ‖𝑓𝑛‖𝐿∞(𝜇 ) =1for every 𝑛.Then{𝑓𝑛}𝑛 is an extremal sequence Let one consider and a 2 𝑝 {𝑞 } for 𝑝. Applying Lemma 8, -admissible pair such that (59) takes place. Let 𝑛 𝑛≥0 be a sequence of extremal polynomials with respect to (2).Thenthe 󵄨 󵄨2 𝑝/2 {𝑞 } ∫(𝑎𝑝󵄨𝑓𝑛󵄨 ) 𝑑𝜇2 multiplication operator is bounded and the zeros of 𝑛 𝑛≥0 lie =1. {𝑧 : |𝑧| ≤ 2 ‖ 𝑀 ‖} lim 𝑝/2 (54) in the bounded disk . 𝑛→∞ 󵄨 󸀠󵄨2 ∫(𝑐𝑝󵄨𝑓𝑛󵄨 ) 𝑑𝜇2 In general, it is not difficult to check wether or not (59) {𝑧 } ⊂𝑆(𝜇) By Lemma 14, there exists 𝑛 𝑛 2 such that holds.Itisclearthatifthereexistsaconstant𝐶 such that 𝑐𝑝 ≤ |𝑓 (𝑧 )| ≤ 1/2 𝑛≥𝑁 𝑛 𝑛 for every 1.Now,takingintoaccount 𝐶𝑎𝑝 𝜇-almost everywhere, then (59)holds.In[8, 13]some that ‖𝑓𝑛‖ ∞ =1and that 𝛾 is connected, we can apply 𝐿 (𝜇2) other very simple conditions implying (59)areshown. Lemma 16,andthen The following is a direct consequence of Theorem 19. 󵄨 󵄨𝑝 ∫ 󵄨𝑓󸀠󵄨 𝑐𝑝/2𝑑𝜇 ≥𝑘⋅ 𝑝 (𝑓 ) 󵄨 𝑛󵄨 𝑝 2 osc 𝑛 Corollary 20. Let one consider 1≤𝑝<∞and (𝑉,𝜇) a 𝑝- admissible pair. Assume that 𝑐𝑝 ≤𝐶𝑎𝑝, 𝜇-almost everywhere 󵄩 󵄩 󵄨 󵄨 𝑝 𝐶 {𝑞 } ≥𝑘(󵄩𝑓𝑛󵄩 ∞ − 󵄨𝑓𝑛 (𝑧𝑛)󵄨) for some constant .Let 𝑛 𝑛≥0 be a sequence of extremal 󵄩 󵄩𝐿 (𝜇2) 󵄨 󵄨 (55) polynomials with respect to (2). Then the zeros of {𝑞𝑛}𝑛≥0 are 1 𝑝 𝑘 uniformly bounded in the complex plane. ≥𝑘(1− ) = >0 2 2𝑝 Finally, we have the following particular consequence for 𝑛≥𝑁 1/𝑘 = ‖1/(𝑐𝑝/2𝑑𝜇 /𝑑𝑠)‖ for every 1,with 𝑝 2 𝐿1/(𝑝−1)(𝛾). Sobolev orthogonal polynomials. Let us fix 𝜀 small enough. On the one hand, by Lemma 13 Corollary 21. (𝑉,𝜇) 2 it holds that Let be a -admissible pair. Assume that there exists a constant 𝐶 such that 𝑐2 ≤𝐶𝑎2 ,𝜇-almost 󵄨 󵄨2 𝑝/2 󵄨 󵄨2 𝑝/2 ∫ (𝑎𝑝󵄨𝑓𝑛󵄨 ) 𝑑𝜇2 ≤2∫ (𝑎𝑝󵄨𝑓𝑛󵄨 ) 𝑑𝜇2 everywhere. Let {𝑞𝑛}𝑛≥0 be the sequence of Sobolev orthogonal 𝐴 𝜀 polynomials with respect to 𝑉𝜇. Then the zeros of the polyno- {𝑞 } 󵄩 󵄩𝑝 mials in 𝑛 𝑛≥0 are uniformly bounded in the complex plane. ≤2󵄩𝑓 󵄩 ∫ 𝑎𝑝/2𝑑𝜇 󵄩 𝑛󵄩 ∞ 𝑝 2 𝐿 (𝜇2) (56) 𝐴𝜀

𝑝/2 5. Asymptotic of Extremal Polynomials =2∫ 𝑎𝑝 𝑑𝜇2 𝐴 𝜀 We start this section by setting some notation. Let 𝑄𝑛, ‖⋅‖ 2 (𝑆(𝜇)) 𝜔 𝑛 for every 𝑛≥𝑁2 =𝑁2(𝜀). 𝐿 (𝜇),cap ,and 𝑆(𝜇) denote, respectively, the th 2 On the other hand, we have monicorthogonalpolynomialwithrespectto𝐿 (𝜇),theusual 󵄨 󵄨 𝐿2(𝜇) 𝑆(𝜇) 𝜇 (𝐴 )= 𝜇 ({󵄨𝑏 󵄨 > (1−𝜀) √𝑎 𝑐 }) norm in the space , the logarithmic capacity of , lim+ 2 𝜀 lim+ 2 󵄨 𝑝󵄨 𝑝 𝑝 𝜀→0 𝜀→0 and the equilibrium measure of 𝑆(𝜇). Furthermore, in order (57) 󵄨 󵄨 to analyze the asymptotic behavior for extremal polynomials =𝜇 ({󵄨𝑏 󵄨 ≥ √𝑎 𝑐 }) = 0. 2 󵄨 𝑝󵄨 𝑝 𝑝 we will use a special class of measures, “regular measures,” Reg This implies denoted by and defined in [24]. In that work, the authors proved (see Theorem 3.1.1) that, for measures supported on 𝑝/2 𝜇∈Reg lim ∫ 𝑎𝑝 𝑑𝜇2 =0. a compact set of the complex plane, if and only 𝜀→0+ (58) 𝐴𝜀 if 𝛿>0 𝜀 ∫ 𝑎𝑝/2 𝑑𝜇 <𝛿 Given any there exists 1 with 𝐴 𝑝 2 .Hence, 𝜀1 󵄩 󵄩1/𝑛 2 𝑝/2 lim 󵄩𝑄𝑛󵄩𝐿2(𝜇) = cap (𝑆 (𝜇)) . (60) ∫(𝑎𝑝|𝑓𝑛| ) 𝑑𝜇2 <2𝛿for every 𝑛≥𝑁2(𝜀1). Therefore, 𝑛→∞ 2 𝑝/2 lim𝑛→∞ ∫(𝑎𝑝|𝑓𝑛| ) 𝑑𝜇2 =0, which is a contradiction with (54)and(55). Finally, if 𝑧1,𝑧2,...,𝑧𝑛 denote the zeros, repeated according to their multiplicity, of a polynomial 𝑞 whose degree is The following result is a direct consequence of Theorems exactly 𝑛,and𝛿𝑧 is the Dirac measure with mass one at the 3 and 17. 𝑗 point 𝑧𝑗, the expression Theorem 18. Let one consider 1≤𝑝<∞and (𝑉,𝜇) a 𝑝- admissible pair. Then the multiplication operator is bounded in 𝑛 𝑊1,𝑝(𝑉𝜇) 1 ifandonlyifthefollowingconditionholds: ] (𝑞) := ∑𝛿𝑧 𝑛 𝑗 (61) 1,𝑝 𝑝/2 𝑝/2 𝑝/2 𝑗=1 the norms in 𝑊 ((𝑎𝑝 +𝑐𝑝 )𝜇,𝑐𝑝 𝜇) (59) 1,𝑝 𝑝/2 𝑝/2 and 𝑊 (𝑎𝑝 𝜇,𝑝 𝑐 𝜇) are equivalent on P. defines the normalized zero counting measure of 𝑞. 10 Journal of Function Spaces and Applications

We can already state the first result in this section. in (64) for all 𝑧∈C,exceptforasetofcapacityzero,𝑆(𝜔𝑆(𝜇))⊂ {𝑧 : |𝑧| ≤ 2‖𝑀‖} and Theorem 22. 1≤𝑝<∞(𝑉,𝜇) 𝑝 Let one consider , a - (𝑗+1) 𝑑𝜔 (𝑥) 𝑞𝑛 (𝑧) 𝑆(𝜇) admissible pair and {𝑞𝑛}𝑛≥0 the sequence of extremal polyno- = ∫ 𝑛→∞lim (𝑗) 𝑧−𝑥 (66) mials with respect to ‖⋅‖𝑊1,𝑝(𝑉𝜇). Assume that the following 𝑛𝑞𝑛 (𝑧) conditions hold: uniformly on each compact subset of {𝑧 : |𝑧| > 2‖𝑀‖}. 𝑝/2 (i) 𝑎𝑝 𝜇∈Reg; Proof. Note that, in our context, the multiplication 𝑆(𝜇) (ii) is regular with respect to the Dirichlet problem; operator is bounded (see Theorem 3) and the norms of 1,𝑝 𝑝/2 𝑝/2 1,𝑝 (iii) condition (59) takes place. 𝑊 (𝑎𝑝 𝜇,𝑝 𝑐 𝜇) and 𝑊 (𝑉𝜇) definedasin(3)are equivalent (see Theorem 18). This is the crucial fact in the Then, proof of this theorem; once we know this, we just need to 󵄩 (𝑗)󵄩1/𝑛 follow the proof given in [5,Theorem6]pointbypointto 󵄩𝑞 󵄩 = (𝑆 (𝜇)) , 𝑗 ≥0. (62) 𝑛→∞lim 󵄩 𝑛 󵄩𝑆(𝜇) cap conclude the result. 𝑆(𝜇) Furthermore, if the complement of is connected, then Acknowledgments ] (𝑞(𝑗))=𝜔 ,𝑗≥0. 𝑛→∞lim 𝑛 𝑆(𝜇) (63) AnaPortillaandEvaTour´ıs are supported in part by a grantfromMinisteriodeCienciaeInnovacion´ (MTM 2009- in the weak star topology of measures. 12740-C03-01), Spain. Yamilet Quintana is supported in part by the Research Sabbatical Fellowship Program (2011-2012) Proof. Note that, in our context, the hypothesis removed from Universidad SimonBolvar,Venezuela.AnaPortilla,´ with respect to [5,Theorem2]isequivalenttothefollowing JoseM.Rodr´ ´ıguez, and Eva Tour´ıs are supported in part two facts: on the one hand, the multiplication operator is by two grants from Ministerio de Ciencia e Innovacion´ bounded (see Theorem 3),andontheotherhand,thenorms 1,𝑝 𝑝/2 𝑝/2 1,𝑝 (MTM 2009-07800 and MTM 2008-02829-E), Spain. JoseM.´ of 𝑊 (𝑎𝑝 𝜇,𝑝 𝑐 𝜇) and 𝑊 (𝑉𝜇) defined as in (3) are Rodr´ıguez is supported in part by a grant from CONACYT equivalent (see Theorem 18). With this in mind, we just (CONACYT-UAG I0110/62/10 FON.INST.8/10), Mexico.´ This need to follow the proof of [5,Theorem2]toconcludethe word is dedicated to Francisco Marcellan´ Espanol˜ on his 60th result. birthday.

In the following theorem, we use 𝑔Ω(𝑧; ∞) to denote the Green’s function for Ω with logarithmic singularity at ∞, References where Ω is the unbounded component of the complement [1] G. Lopez´ Lagomasino and H. Pijeira Cabrera, “Zero location of 𝑆(𝜇). Notice that, if 𝑆(𝜇) is regular with respect to the 𝑔 (𝑧; ∞) and nth root asymptotics of Sobolev orthogonal polynomials,” Dirichlet problem, then Ω is continuous up to the Journal of Approximation Theory,vol.99,no.1,pp.30–43,1999. boundary and it can be extended continuously to all C,with C \Ω [2] G. Lopez´ Lagomasino, H. Pijeira Cabrera, and I. Perez´ value zero on . Izquierdo, “Sobolev orthogonal polynomials in the complex plane,” Journal of Computational and Applied Mathematics,vol. Theorem 23. 1≤𝑝<∞(𝑉,𝜇) Let one consider , 127, no. 1-2, pp. 219–230, 2001, Numerical analysis 2000, Vol. V, a 𝑝-admissible pair and {𝑞𝑛}𝑛≥0 the sequence of extremal Quadrature and orthogonal polynomials. polynomials with respect to ‖⋅‖𝑊1,𝑝(𝑉𝜇). Assume that the [3]V.Alvarez,D.Pestana,J.M.Rodr´ıguez, and E. Romera, following conditions hold: “Weighted Sobolev spaces on curves,” Journal of Approximation 𝑝/2 Theory,vol.119,no.1,pp.41–85,2002. (i) 𝑎𝑝 𝜇∈Reg; [4] E. Colorado, D. Pestana, J. M. Rodrguez, and E. Romera, (ii) 𝑆(𝜇) is regular with respect to the Dirichlet problem; “Muckenhoupt inequalitywith three measures and Sobolev orthogonal polynomials”. (iii) condition (59) takes place. [5] G. L. Lagomasino, I. Perez´ Izquierdo, and H. Pijeira Cabrera, 𝑗≥0 “Asymptotic of extremal polynomials in the complex plane,” Then, for each , Journal of Approximation Theory,vol.137,no.2,pp.226–237, 󵄨 (𝑗) 󵄨1/𝑛 𝑔 (𝑧;∞) 2005. 󵄨 󵄨 Ω lim sup󵄨𝑞𝑛 (𝑧)󵄨 ≤ cap (𝑆 (𝜇)) 𝑒 , (64) [6] A. Portilla, J. M. Rodr´ıguez, and E. Tour´ıs, “The multiplica- 𝑛→∞ 󵄨 󵄨 tion operator, zero location and asymptotic for non-diagonal uniformly on compact subsets of C.Furthermore,foreach𝑗≥ Sobolev norms,” Acta Applicandae Mathematicae, vol. 111, no. 2,pp.205–218,2010. 0, [7] J.M.Rodr´ıguez, “The multiplication operator in Sobolev spaces 󵄨 (𝑗) 󵄨1/𝑛 𝑔 (𝑧;∞) with respect to measures,” Journal of Approximation Theory,vol. 󵄨𝑞 (𝑧)󵄨 = (𝑆 (𝜇)) 𝑒 Ω , 𝑛→∞lim 󵄨 𝑛 󵄨 cap (65) 109, no. 2, pp. 157–197, 2001. [8] J. M. Rodr´ıguez, “A simple characterization of weighted uniformly on each compact subset of {𝑧 : |𝑧| > 2‖𝑀‖}. ∩Ω Sobolev spaces with bounded multiplication operator,” Journal Finally, if the complement of 𝑆(𝜇) is connected, one has equality of Approximation Theory,vol.153,no.1,pp.53–72,2008. Journal of Function Spaces and Applications 11

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Research Article Functions of Bounded 𝜅𝜑-Variation in the Sense of Riesz-Korenblum

Mariela Castillo,1 Sergio Rivas,2 María Sanoja,1 and Iván Zea1

1 Escuela de Matematica,´ Universidad Central de Venezuela, Los Chaguaramos, Caracas 1050, Venezuela 2 Area´ de Matematica,´ Universidad Nacional Abierta, San Bernandino, Caracas 1010, Venezuela

Correspondence should be addressed to Mar´ıa Sanoja; sanoja [email protected]

Received 4 December 2012; Revised 6 February 2013; Accepted 7 February 2013

Academic Editor: Jozef´ Bana´s

Copyright © 2013 Mariela Castillo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We present the space of functions of bounded 𝜅𝜑-variation in the sense of Riesz-Korenblum, denoted by 𝜅BV 𝜑[a,b], which is a combination of the notions of bounded 𝜑-variation in the sense of Riesz and bounded 𝜅-variation in the sense of Korenblum. Moreover, we prove that the space generated by this class of functions is a Banach space with a given norm and we prove that the uniformly bounded composition operator satisfies Matkowski’s weak condition.

1. Introduction of two 𝜅-decreasing functions. In 1986, S. K. Kim and J. Kim [9]andPark[10], in 2010, introduced the notion of functions The concept of functions of bounded variation has been well of 𝜅𝜙-bounded variation on compact interval [𝑎, 𝑏] ⊂ R known since Jordan [1] gave the complete characterization which is a combination of concepts of bounded 𝜅-variation of functions of a bounded variation as the difference of and bounded 𝜙-variation in the sense of Schramm [11], and two increasing functions in 1881. This class of functions in 2011 Aziz et al. [12] showed that the space of bounded 𝜅- immediatelyprovedtobeimportantinconnectionwith variation satisfies Matkowski’s weak condition. the rectification of curves and with Dirichlet’s theorem on Recently in [13] Castillo et al. introduce the notion of the convergence of Fourier series. Functions of a bounded bounded 𝜅-variation in the sense of Riesz-Korenblum, which variation exhibit many interesting properties that make them is a combination of the notions of bounded 𝑝-variation in a suitable class of functions in a variety of contexts with wide the sense of Riesz and bounded 𝜅-variation in the sense of applications in pure and applied mathematics (see [2–4]). Korenblum. Riesz [5] in 1910 generalized the notion of Jordan and The purpose of this paper is twofold. First, to intro- introduced the concept of bounded 𝑝-variation (1<𝑝<∞) duce the concept of bounded 𝜅𝜑-variation in the sense of andshowedthat,for1<𝑝<∞, this class coincides with the Riesz-Korenblum, which is a combination of the notions of class of functions absolutely continuous with the derivative bounded 𝜑-variation in the sense of Riesz and bounded 𝜅- in the space 𝐿𝑝. On the other hand, this notion of bounded variationinthesenseofKorenblum.Weprovesomeproper- 𝑝-variation was generalized by Medvedev [6]in1953who ties of this class of functions and its relation with the functions introduced the concept of bounded 𝜑-variation in the sense of bounded 𝜅-variation and bounded 𝜑-variation in the sense of Riesz and also showed a Riesz’s lemma for this class of of Riesz. Second we prove that the space generated by this functions. class of functions is a Banach space with a given norm and Korenblum [7] in 1975 introduced the notion of bounded that the uniformly bounded composition operator satisfies 𝜅-variation. This concept differs from others due tothe Matkowski’s weak condition in this space. The Matkowski fact that it introduces a distortion function 𝜅 that measures property has been studied by several authors (see [14–16]), intervals in the domain of the function and not in the range. andforMatkowski’sweakproperty,seealso[3, 17–21]. In In 1985, Cyphert and Kelingos [8]showedthatafunction𝑢 [22–24] Matkowski, Merentes, and others authors have been is of bounded 𝜅-variation if it can be written as the difference studying a weaker condition on the composition operator 2 Journal of Function Spaces and Applications such as uniformly bounded and uniformly continuous com- The notion of bounded variation due to Jordan position operators. (Definition)wasgeneralizedbyMedvedev(see[ 1 6]) as follows.

2. Preliminaries Definition 4. Let 𝜑 be a 𝜑-function and 𝑢:[𝑎,𝑏]→ R be a 𝜋:𝑎=𝑡 <𝑡 <⋅⋅⋅<𝑡 =𝑏 In this section we present some definitions and preliminary function. For each partition 0 1 𝑛 of [𝑎, 𝑏] results related to the notion of functions of bounded 𝜅𝜑- the interval , we define variation in the sense of Riesz-Korenblum. 𝑅 𝑅 𝑉𝜑 (𝑢) =𝑉𝜑 (𝑢; [𝑎,] 𝑏 ) 𝑢:[𝑎,𝑏]→ R Definition 1. Let be a function. For each 󵄨 󵄨 𝜋:𝑎=𝑡 <𝑡 <⋅⋅⋅<𝑡 =𝑏 [𝑎, 𝑏] 𝑛 󵄨𝑢(𝑡)−𝑢(𝑡 )󵄨 󵄨 󵄨 (4) partition 0 1 𝑛 of the interval , := ∑𝜑(󵄨 𝑖 𝑖−1 󵄨) 󵄨𝑡 −𝑡 󵄨 , sup 󵄨 󵄨 󵄨 𝑖 𝑖−1󵄨 we define 𝜋 𝑖=1 󵄨𝑡𝑖 −𝑡𝑖−1󵄨 𝑛 󵄨 󵄨 𝑉 (𝑢; [𝑎,] 𝑏 ) := sup∑ 󵄨𝑢(𝑡𝑖)−𝑢(𝑡𝑖−1)󵄨 , (1) where the supremum is taken over all partitions 𝜋 of the 𝜋 𝑖=1 𝑅 interval [𝑎, 𝑏].If𝑉𝜑 (𝑢; [𝑎, 𝑏]) <∞,wesaythat𝑢 has 𝜑 where the supremum is taken over all partitions 𝜋 of the bounded -variation in the sense of Riesz. We denote by 𝑉𝑅[𝑎, 𝑏] 𝜑 interval [𝑎, 𝑏].If𝑉(𝑢; [𝑎, 𝑏]),wesaythat <∞ 𝑢 has 𝜑 the class of all functions of bounded -variation in bounded variation. We denote by 𝐵𝑉[𝑎, 𝑏] the collection of the sense of Riesz on [𝑎, 𝑏]. all functions of bounded variation on [𝑎, 𝑏]. Nowwegivesomepropertiesoftheclassoffunctions 𝑅 Now, we will give some well-known properties of the 𝑉𝜑 [𝑎, 𝑏]. space of functions 𝐵𝑉[𝑎,. 𝑏] 𝑅 (1) If 𝑢∈ Lip [𝑎, 𝑏],then𝑢∈𝑉[𝑎, 𝑏];thatis,Lip[𝑎, (1) If the function 𝑢 is monotone, then 𝑉(𝑢; [𝑎, 𝑏])= 𝜑 𝑏] ⊂ 𝑉𝑅[𝑎, 𝑏] |𝑢(𝑏) − 𝑢(𝑎)|. 𝜑 . (2) If 𝑢∈𝐵𝑉[𝑎,𝑏],then𝑢 is bounded on [𝑎, 𝑏]. 𝑅 (2) If 𝜑 is convex then 𝑉𝜑 [𝑎, 𝑏] ⊂ 𝐵𝑉[𝑎, 𝑏] and if (3) A function 𝑢 hasboundedvariationinaninterval 𝑅 lim𝑡→∞(𝜑(𝑡)/𝑡) = 𝛾<∞,then𝑉𝜑 [𝑎, 𝑏] = 𝐵𝑉[𝑎,. 𝑏] [𝑎, 𝑏] if and only if it can be decomposed as a 𝑅 difference of increasing functions. (3) If 𝜑 satisfies the condition ∞1 and 𝑉𝜑 (𝑢; [𝑎, 𝑏]) <∞, (4) Every function of bounded variation has left- and then 𝑢 is bounded on [𝑎, 𝑏]. right-hand limits at each point of its domain. 𝑅 (4) 𝑉𝜑 [𝑎, 𝑏] is a symmetric convex set. (5) If 𝑢∈ Lip [𝑎, 𝑏],then𝑢∈𝐵𝑉[𝑎,𝑏]that is, Lip [𝑎, 𝑏] ⊂ 𝐵𝑉[𝑎, 𝑏] 𝑅 . (5) If 𝜑 is convex, then 𝜅𝑉𝜑 [𝑎, 𝑏] is a vector space if and (6) 𝐵𝑉[𝑎, 𝑏] is a Banach space endowed with the norm only if 𝜑 satisfies the condition Δ 2(∞).

𝜑 𝑅𝑉𝜑[𝑎, 𝑏] = {𝑢 : [𝑎, 𝑏]→ ‖𝑢‖𝐵𝑉 = |𝑢 (𝑎)| +𝑉(𝑢; [𝑎,] 𝑏 ) ,𝑢∈𝐵𝑉[𝑎,] 𝑏 . (6) If is convex, then the set (2) 𝑅 R :∃𝜆>0,𝑉𝜑 (𝜆𝑢) < ∞} is a Banach space endowed In 1937, Young (see [25]) introduced the definition of 𝜑- with the norm function as follows. 𝑢 ||𝑢||𝜑 = |𝑢 (𝑎)| + inf {𝜀>0:𝑉𝜑 ( )≤1}. (5) Definition 2. Afunction𝜑 : [0, ∞) → [0, ∞) is said to be a 𝜀 𝜑-function if it satisfies the following properties. 𝜑 𝜑 (a) 𝜑 is continuous on [0, ∞). (7) Let be a convex -function such that it satisfies the condition ∞1.Afunction𝑢 has bounded 𝜑-variation 𝜑(𝑡) =0 𝑡=0 (b) if and only if . in an interval [𝑎, 𝑏] if and only if 𝑢 is absolutely 𝑏 (c) 𝜑 is strictly increasing. [𝑎, 𝑏] ∫ 𝜑(|𝑢󸀠(𝑡)|)𝑑𝑡 <∞ continuous in and 𝑎 .Also, (d) lim𝑡→∞𝜑(𝑡) =∞. 𝑏 󵄨 󵄨 ∞ Δ 𝜑 𝜑 𝑉𝑅 (𝑢; [𝑎,] 𝑏 ) =∫ 𝜑 (󵄨𝑢󸀠 (𝑡)󵄨) 𝑑𝑡. Definition 3 (conditions 1 and 2). Let be a convex - 𝜑 󵄨 󵄨 (6) function, then 𝑎

(a) 𝜑 satisfies the condition ∞1 if lim𝑡→∞(𝜑(𝑡)/𝑡) =∞, Other generalization of the notion of bounded variation (b) 𝜑 satisfies the condition Δ 2(∞) if there is 𝐶>0, 𝑥0 > was introduced by Korenblum. Korenblum employed a func- 0 such that tion 𝜅 : [0, 1] → [0, 1] called 𝜅-function. This function 𝜅 can be viewed as a rescaling of lengths of subintervals of [𝑎, 𝑏] 𝜑 (2𝑡) ≤𝐶𝜑(𝑡) ,𝑡≥𝑥. 0 (3) such that the length of [𝑎, 𝑏] is 1 if 𝜅(1) = 1. Journal of Function Spaces and Applications 3

Definition 5. Afunction𝜅 : [0, 1] → [0, 1] is said to be a Definition 7. Let 𝜑 be a 𝜑-function, 𝜅∈K,and𝑢:[𝑎,𝑏]→ 𝜅-function if it satisfies the following properties: R be a function. For each partition 𝜋:𝑎=𝑡0 <𝑡1 <⋅⋅⋅< 𝑡𝑛 =𝑏of the interval [𝑎, 𝑏], we define (a) 𝜅 is continuous with 𝜅(0) = 0 and 𝜅(1) = 1, 𝜅 𝑅 𝑅 (b) is concave (down), increasing, and 𝜅𝑉𝜑 (𝑢) =𝜅𝑉𝜑 (𝑢; [𝑎,] 𝑏 )

(c) lim𝑡→0+ (𝜅(𝑡)/𝑡) =∞. 𝑛 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 ∑𝑖=1 𝜑(󵄨𝑢(𝑡𝑖)−𝑢(𝑡𝑖−1)󵄨 / 󵄨𝑡𝑖 −𝑡𝑖−1󵄨) 󵄨𝑡𝑖 −𝑡𝑖−1󵄨 := sup 𝑛 , Thesetofall𝜅-functions will be denoted by K.Notethat, 𝜋 ∑𝑖=1 𝜅 ((𝑡𝑖 −𝑡𝑖−1)/(𝑏−𝑎)) every 𝜅-function 𝜅 is subadditive; that is, (11)

𝜅(𝑡 +𝑡 )≤𝜅(𝑡)+𝜅(𝑡 ), 𝑡 ,𝑡 ∈ [0, 1] . 1 2 1 2 1 2 (7) where the supremum is taken over all partitions 𝜋 of the 𝑅 interval [𝑎, 𝑏].If𝜅𝑉 (𝑢; [𝑎, 𝑏]) <∞,wesaythat𝑢 has 𝜋:𝑎=𝑡 <𝑡 < ⋅⋅⋅ < 𝑡 =𝑏 𝜑 Then, for all partition 0 1 𝑛 of bounded 𝜅𝜑-variation in the sense of Riesz-Korenblum. We [𝑎, 𝑏],wehave 𝑅 will denote by 𝜅𝑉𝜑 [𝑎, 𝑏] the class of all functions of bounded 𝑛 𝑛 𝜅𝜑-variation in the sense of Riesz-Korenblum on [𝑎, 𝑏]. 𝑡𝑖 −𝑡𝑖−1 𝑡𝑖 −𝑡𝑖−1 1=𝜅(1) =𝜅(∑ )≤∑𝜅( ). (8) 𝑏−𝑎 𝑏−𝑎 𝑅 𝑖=1 𝑖=1 Remark 8. Note that the class 𝜅𝑉𝜑 [𝑎, 𝑏] is not empty since for an affine function 𝑢:[𝑎,𝑏]→ R is defined by 𝑢(𝑡) := Korenblum (see [7]) introduces the definition of bounded 𝑑𝑡 + 𝑒, 𝑡 ∈ [𝑎, 𝑏],where𝑑, 𝑒 are fixed real numbers. For a 𝜅 -variation as follows. given partition 𝜋:𝑎=𝑡0 <𝑡1 <⋅⋅⋅<𝑡𝑛 =𝑏of [𝑎, 𝑏] we have 𝑢 [𝑎, 𝑏] 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 Definition 6. Arealfunction on is said to be of ∑𝑛 𝜑(󵄨𝑢(𝑡)−𝑢(𝑡 )󵄨 / 󵄨𝑡 −𝑡 󵄨) 󵄨𝑡 −𝑡 󵄨 𝜅 𝑖=1 󵄨 𝑖 𝑖−1 󵄨 󵄨 𝑖 𝑖−1󵄨 󵄨 𝑖 𝑖−1󵄨 bounded -variation, if 𝑛 ∑𝑖=1 𝜅 ((𝑡𝑖 −𝑡𝑖−1)/(𝑏−𝑎)) 𝜅𝑉 (𝑢) =𝜅𝑉(𝑢; [𝑎,] 𝑏 ) 𝜑 (|𝑑|)(𝑏−𝑎) (12) = 𝑛 . 𝑛 󵄨 󵄨 󵄨 󵄨 ∑𝑖=1 𝜅 ((𝑡𝑖 −𝑡𝑖−1)/(𝑏−𝑎)) ∑𝑖=1 󵄨𝑢(𝑡𝑖)−𝑢(𝑡𝑖−1)󵄨 (9) := sup 𝑛 <∞, 𝜋 ∑𝑖=1 𝜅 ((𝑡𝑖 −𝑡𝑖−1)/(𝑏−𝑎)) Taking the supremum over all partitions 𝜋 of the interval 𝜋 [𝑎, 𝑏], the greater value of the right side of the above where the supremum is taken over all partitions of the 𝜋:𝑎=𝑡 <𝑡 =𝑏 interval [𝑎, 𝑏]. We denote by 𝜅𝐵𝑉[𝑎, 𝑏] the collection of all expression is obtain for the partition 0 1 functions of bounded 𝜅-variation on [𝑎, 𝑏]. andinthiscaseweget 𝜅𝐵𝑉[𝑎, 𝑏] 𝜑 (|𝑑|)(𝑏−𝑎) 𝜑 (|𝑑|)(𝑏−𝑎) Next, some properties of the space are exposed = =𝜑(|𝑑|)(𝑏−𝑎) . (see [8]). 𝜅((𝑡1 −𝑡0)/(𝑏−𝑎)) 𝜅 (1) (13) (1) If the function 𝑢 is monotone, then 𝜅𝑉(𝑢; [𝑎, 𝑏])= |𝑢(𝑏) − 𝑢(𝑎)|. Therefore, (2) If 𝑢 ∈ 𝜅𝐵𝑉[𝑎,,then 𝑏] 𝑢 is bounded on [𝑎, 𝑏]. 𝑅 (3) If 𝑢∈𝐵𝑉[𝑎,𝑏],then𝑢 ∈ 𝜅𝐵𝑉[𝑎,;thatis, 𝑏] 𝐵𝑉[𝑎, 𝑏]⊂ 𝜅𝑉𝜑 (𝑢; [𝑎,] 𝑏 ) =𝜑(|𝑑|)(𝑏−𝑎) . (14) 𝜅𝐵𝑉[𝑎,. 𝑏] (4) A function 𝑢 has bounded 𝜅-variation in an interval [𝑎, 𝑏] In the following proposition, we prove two important if and only if it can be decomposed as a 𝜅𝑅𝑉 [𝑎, 𝑏] difference of 𝜅-decreasing functions. properties of the space 𝜑 . (5) Every function of bounded 𝜅-variation has left- and Proposition 9. Let 𝜑 be a convex 𝜑-function, then right-hand limits at each point of its domain. 𝜅𝑉𝑅[𝑎, 𝑏] ⊂ 𝜅𝐵𝑉[𝑎, 𝑏] (6) 𝜅𝐵𝑉[𝑎, 𝑏] is a Banach space endowed with the norm (a) 𝜑 . (𝜑(𝑡)/𝑡) = 𝛾<∞ 𝜅𝑉𝑅[𝑎, 𝑏] = 𝜅𝐵𝑉[𝑎, ||𝑢||𝜅 = |𝑢 (𝑎)| +𝜅𝑉(𝑢; [𝑎,] 𝑏 ) , 𝑢 ∈ 𝜅𝐵𝑉 [𝑎,] 𝑏 . (10) (b) If lim𝑡→∞ ,then 𝜑 𝑏].

𝑅 3. Main Results Proof. (a) Let 𝑢∈𝜅𝑉𝜑 [𝑎, 𝑏], 𝜋 :0 𝑎=𝑡 <𝑡1 < ⋅⋅⋅ < 𝑡𝑛 =𝑏, [𝑎, 𝑏] In this section we present the principal results of this paper. be a partition of the interval and Next, we introduce the definition of function of bounded 𝜅𝜑- 󵄨 󵄨 󵄨𝑢(𝑡)−𝑢(𝑡 )󵄨 variationinthesenseofRiesz-Korenblumforthefunction 𝜎:={𝑖=1,...,𝑛: 󵄨 𝑖 𝑖−1 󵄨 ≤1}. 𝑢:[𝑎,𝑏]→ R 󵄨 󵄨 (15) . 󵄨𝑡𝑖 −𝑡𝑖−1󵄨 4 Journal of Function Spaces and Applications

Since 𝜑 is a convex 𝜑-function and 𝜑(0) = 0,wehave Now, we will show part (b). If 𝑡 1 𝜑 (𝑡) 𝜑 (1) =𝜑( )≤ 𝜑 (𝑡) ,𝑡≥1, (16) 0< lim =𝛾<∞, (22) 𝑡 𝑡 𝑡→∞ 𝑡 so 𝑡 ≤ 𝜑(𝑡)/𝜑(1). then there exist 𝑥0 >0and 𝑐>0,suchthat Hence, for 𝑖∉𝜎we get 𝜑 (𝑡) ≤ 𝑐𝑡, 𝑡0 ≥𝑥 . (23) 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨𝑢(𝑡𝑖)−𝑢(𝑡𝑖−1)󵄨 𝜑(󵄨𝑢(𝑡𝑖)−𝑢(𝑡𝑖−1)󵄨 / 󵄨𝑡𝑖 −𝑡𝑖−1󵄨) 󵄨 󵄨 ≤ (17) Let us consider the partition 𝜋:𝑎=𝑡0 <𝑡1 <⋅⋅⋅<𝑡𝑛 =𝑏 󵄨𝑡 −𝑡 󵄨 𝜑 (1) 󵄨 𝑖 𝑖−1󵄨 of the interval [𝑎, 𝑏], 𝑢 ∈ 𝜅𝐵𝑉[𝑎,,and 𝑏] 󵄨 󵄨 multiplied by |𝑡𝑖 −𝑡𝑖−1| andapplyingthesumonbothsidesof 󵄨𝑢(𝑡)−𝑢(𝑡 )󵄨 𝛼 := {𝑖=1,...,𝑛: 󵄨 𝑖 𝑖−1 󵄨 >𝑥} , the above inequality, we have 𝑥0 󵄨 󵄨 0 (24) 󵄨𝑡𝑖 −𝑡𝑖−1󵄨 󵄨 󵄨 𝑛 󵄨𝑢(𝑡)−𝑢(𝑡 )󵄨 󵄨 󵄨 ∑󵄨 𝑖 𝑖−1 󵄨 󵄨𝑡 −𝑡 󵄨 then 󵄨 󵄨 󵄨 𝑖 𝑖−1󵄨 󵄨𝑡𝑖 −𝑡𝑖−1󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝑖∉𝜎 󵄨 󵄨 ∑𝑛 𝜑(󵄨𝑢(𝑡)−𝑢(𝑡 )󵄨 / 󵄨𝑡 −𝑡 󵄨) 󵄨𝑡 −𝑡 󵄨 (18) 𝑖=1 󵄨 𝑖 𝑖−1 󵄨 󵄨 𝑖 𝑖−1󵄨 󵄨 𝑖 𝑖−1󵄨 𝑛 󵄨 󵄨 𝑛 1 󵄨𝑢(𝑡)−𝑢(𝑡 )󵄨 󵄨 󵄨 ∑ 𝜅((𝑡𝑖 −𝑡𝑖−1)/(𝑏−𝑎)) ≤ ∑𝜑(󵄨 𝑖 𝑖−1 󵄨) 󵄨𝑡 −𝑡 󵄨 , 𝑖=1 󵄨 󵄨 󵄨 𝑖 𝑖−1󵄨 𝜑 (1) 𝑖∉𝜎 󵄨𝑡𝑖 −𝑡𝑖−1󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 ∑𝑖∈𝛼 𝜑(󵄨𝑢(𝑡𝑖)−𝑢(𝑡𝑖−1)󵄨 / 󵄨𝑡𝑖 −𝑡𝑖−1󵄨) 󵄨𝑡𝑖 −𝑡−1󵄨 𝑥0 󵄨 󵄨 󵄨 󵄨 󵄨 i 󵄨 = 𝑛 then ∑𝑖=1 𝜅 ((𝑡𝑖 −𝑡𝑖−1)/(𝑏−𝑎)) 󵄨 󵄨 ∑𝑛 󵄨𝑢(𝑡)−𝑢(𝑡 )󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝑖=1 󵄨 𝑖 𝑖−1 󵄨 ∑𝑖∉𝛼 𝜑(󵄨𝑢(𝑡𝑖)−𝑢(𝑡𝑖−1)󵄨 / 󵄨𝑡𝑖 −𝑡𝑖−1󵄨) 󵄨𝑡𝑖 −𝑡𝑖−1󵄨 𝑛 𝑥0 + 𝑛 ∑𝑖=1 𝜅 ((𝑡𝑖 −𝑡𝑖−1)/(𝑏−𝑎)) ∑𝑖=1 𝜅((𝑡𝑖 −𝑡𝑖−1)/(𝑏−𝑎)) 𝑛 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 ∑𝑖=1 (󵄨𝑢(𝑡𝑖)−𝑢(𝑡𝑖−1)󵄨 / 󵄨𝑡𝑖 −𝑡𝑖−1󵄨) 󵄨𝑡𝑖 −𝑡𝑖−1󵄨 󵄨 󵄨 = ∑𝑖∈𝛼 𝑐 󵄨𝑢(𝑡𝑖)−𝑢(𝑡𝑖−1)󵄨 ∑𝑛 𝜅 ((𝑡 −𝑡 )/(𝑏−𝑎)) ≤ 𝑥0 𝑖=1 𝑖 𝑖−1 ∑𝑛 𝜅((𝑡 −𝑡 )/(𝑏−𝑎)) 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝑖=1 𝑖 𝑖−1 ∑ (󵄨𝑢(𝑡)−𝑢(𝑡 )󵄨 / 󵄨𝑡 −𝑡 󵄨) 󵄨𝑡 −𝑡 󵄨 𝑖∈𝜎 󵄨 𝑖 𝑖−1 󵄨 󵄨 𝑖 𝑖−1󵄨 󵄨 𝑖 𝑖−1󵄨 󵄨 󵄨 = 𝑛 ∑𝑖∉𝛼 𝜑(𝑥0) 󵄨𝑡𝑖 −𝑡𝑖−1󵄨 𝑥0 󵄨 󵄨 ∑𝑖=1 𝜅 ((𝑡𝑖 −𝑡𝑖−1)/(𝑏−𝑎)) + , ∑𝑛 𝜅((𝑡 −𝑡 )/(𝑏−𝑎)) 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝑖=1 𝑖 𝑖−1 ∑𝑖∉𝜎 (󵄨𝑢(𝑡𝑖)−𝑢(𝑡𝑖−1)󵄨 / 󵄨𝑡𝑖 −𝑡𝑖−1󵄨) 󵄨𝑡𝑖 −𝑡𝑖−1󵄨 (25) + 𝑛 ∑ 𝜅 ((𝑡𝑖 −𝑡𝑖−1)/(𝑏−𝑎)) 𝑖=1 thus, 󵄨 󵄨 ∑ 󵄨𝑡 −𝑡 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝑖∈𝜎 󵄨 𝑖 𝑖−1󵄨 ∑𝑛 𝜑(󵄨𝑢(𝑡)−𝑢(𝑡 )󵄨 / 󵄨𝑡 −𝑡 󵄨) 󵄨𝑡 −𝑡 󵄨 ≤ 𝑛 𝑖=1 󵄨 𝑖 𝑖−1 󵄨 󵄨 𝑖 𝑖−1󵄨 󵄨 𝑖 𝑖−1󵄨 ∑𝑖=1 𝜅((𝑡𝑖 −𝑡𝑖−1)/(𝑏−𝑎)) 𝑛 ∑𝑖=1 𝜅 ((𝑡𝑖 −𝑡𝑖−1)/(𝑏−𝑎)) 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 1 ∑𝑖∉𝜎 𝜑(󵄨𝑢(𝑡𝑖)−𝑢(𝑡𝑖−1)󵄨 / 󵄨𝑡𝑖 −𝑡𝑖−1󵄨) 󵄨𝑡𝑖 −𝑡𝑖−1󵄨 󵄨 󵄨 + . 𝑐∑𝑖∈𝛼 󵄨𝑢(𝑡𝑖)−𝑢(𝑡𝑖−1)󵄨 𝑛 𝑥0 󵄨 󵄨 𝜑 (1) ∑𝑖=1 𝜅 ((𝑡𝑖 −𝑡𝑖−1)/(𝑏−𝑎)) ≤ 𝑛 ∑𝑖=1 𝜅 ((𝑡𝑖 −𝑡𝑖−1)/(𝑏−𝑎)) (𝑏−𝑎) 1 𝑅 󵄨 󵄨 ≤ 𝑛 + 𝜅𝑉𝜑 (𝑢) 𝜑(𝑥 ) ∑ 󵄨𝑡 −𝑡 󵄨 (26) ∑ 𝜅((𝑡 −𝑡 )/(𝑏−𝑎)) 𝜑 (1) 0 𝑖∉𝛼𝑥 󵄨 𝑖 𝑖−1󵄨 𝑖=1 𝑖 𝑖−1 + 0 ∑𝑛 𝜅 ((𝑡 −𝑡 )/(𝑏−𝑎)) 1 𝑖=1 𝑖 𝑖−1 ≤ (𝑏−𝑎) + 𝜅𝑉𝑅 (𝑢) . 𝜑 (1) 𝜑 (𝑏−𝑎) ≤𝑐⋅𝜅𝑉(𝑢) +𝜑(𝑥0) 𝑛 (19) ∑𝑖=1 𝜅((𝑡𝑖 −𝑡𝑖−1)/(𝑏−𝑎))

Then ≤𝑐⋅𝜅𝑉(𝑢) +𝜑(𝑥0) (𝑏−𝑎) . 𝑛 󵄨 󵄨 ∑ 󵄨𝑢(𝑡)−𝑢(𝑡 )󵄨 1 𝜋 𝑖=1 󵄨 𝑖 𝑖−1 󵄨 ≤ (𝑏−𝑎) + 𝜅𝑉𝑅 (𝑢) . Then by considering the supremum over all partitions 𝑛 𝜑 [𝑎, 𝑏] ∑𝑖=1 𝜅 ((𝑡𝑖 −𝑡𝑖−1)/(𝑏−𝑎)) 𝜑 (1) of the interval of the left side, we get (20) 𝑅 𝜅𝑉𝜑 (𝑢) ≤𝑐⋅𝜅𝑉(𝑢) +𝜑(𝑥0) (𝑏−𝑎) , (27) Considering the supremum over all partitions 𝜋 of the interval [𝑎, 𝑏] in the above expression, we get that is, 𝑅 1 𝜅𝐵𝑉 [𝑎,] 𝑏 ⊂𝜅𝑉 [𝑎,] 𝑏 . (28) 𝜅𝑉 (𝑢) ≤𝑏−𝑎+ 𝜅𝑉𝑅 (𝑢) , 𝜑 𝜑 (1) 𝜑 (21) 𝑅 Therefore, from part (a) and (28)wehave𝜅𝑉𝜑 [𝑎, 𝑏] = 𝑅 𝜅𝐵𝑉[𝑎, 𝑏] therefore 𝜅𝑉𝜑 [𝑎, 𝑏] ⊂ 𝜅𝐵𝑉[𝑎,. 𝑏] . Journal of Function Spaces and Applications 5

𝑅 𝑅 Proposition 10. Let Lip [𝑎, 𝑏] be the Banach space of all (b) If 𝑢 is constant, then 𝜅𝑉𝜑 (𝑢) =.Nowif 0 𝜅𝑉𝜑 (𝑢) =, 0 𝑅 Lipschitz functions 𝑢:[𝑎,𝑏]→ R.ThenLip[𝑎, 𝑏] 𝜑⊂ 𝜅𝑉 [𝑎, we get that 𝑏]. 𝜑 (|𝑢 (𝑡) −𝑢(𝑎)| / |𝑡−𝑎|) |𝑡−𝑎| 0=𝜅𝑉𝑅 (𝑢) ≥ , 𝜑 𝜅 ((𝑡−𝑎) / (𝑏−𝑎)) +𝜅((𝑏−𝑡) / (𝑏−𝑎)) Proof. Let 𝜋:𝑎=𝑡0 <𝑡1 < ⋅⋅⋅ < 𝑡𝑛 =𝑏be a partition of the interval [𝑎, 𝑏] and 𝑢∈ Lip [𝑎, 𝑏], then there exists 𝑐∈R (33) 𝑥, 𝑦 ∈ [𝑎, 𝑏] |𝑢(𝑥)−𝑢(𝑦)|≤ 𝑐|𝑥−𝑦| such that for any we have . for some 𝑡 ∈ (𝑎, 𝑏].FromDefinitions2 and 5,wehave Hence, 𝜑 (|𝑢 (𝑡) −𝑢(𝑎)| / |𝑡−𝑎|) |𝑡−𝑎| 𝑛 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 0≥ ≥0, ∑𝑖=1 𝜑(󵄨𝑢(𝑡𝑖)−𝑢(𝑡𝑖−1)󵄨 / 󵄨𝑡𝑖 −𝑡𝑖−1󵄨) 󵄨𝑡𝑖 −𝑡𝑖−1󵄨 (34) 𝑛 𝜅 ((𝑡−𝑎) / (𝑏−𝑎)) +𝜅((𝑏−𝑡) / (𝑏−𝑎)) ∑𝑖=1 𝜅((𝑡𝑖 −𝑡𝑖−1)/(𝑏−𝑎)) 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 hence ∑𝑛 𝜑(𝑐󵄨𝑡 −𝑡 󵄨 / 󵄨𝑡 −𝑡 󵄨) 󵄨𝑡 −𝑡 󵄨 ≤ 𝑖=1 󵄨 𝑖 𝑖−1󵄨 󵄨 𝑖 𝑖−1󵄨 󵄨 𝑖 𝑖−1󵄨 |𝑢 (𝑡) −𝑢(𝑎)| 𝑛 𝜑( ) |𝑡−𝑎| =0, 𝑡∈(𝑎,] 𝑏 . (35) ∑𝑖=1 𝜅 ((𝑡𝑖 −𝑡𝑖−1)/(𝑏−𝑎)) |𝑡−𝑎| 𝑛 󵄨 󵄨 𝜑(0) = 0 𝑡=0 ∑𝑖=1 𝜑 (𝑐) 󵄨𝑡𝑖 −𝑡𝑖−1󵄨 (29) Since if and only if ,weget = 𝑛 ∑𝑖=1 𝜅((𝑡𝑖 −𝑡𝑖−1)/(𝑏−𝑎)) |𝑢 (𝑡) −𝑢(𝑎)| =0, (36) 𝜑 (𝑐)(𝑏−𝑎) |𝑡−𝑎| = ∑𝑛 𝜅((𝑡 −𝑡 )/(𝑏−𝑎)) therefore, 𝑢(𝑡) = 𝑢(𝑎) 𝑡 ∈[𝑎,𝑏].So,𝑢 is constant. 𝑖=1 𝑖 𝑖−1 𝑅 (c) Suppose that 𝑢∈𝜅𝑉𝜑 [𝑎, 𝑏] is unbounded on [𝑎, 𝑏]. ≤𝜑(𝑐)(𝑏−𝑎) , Then, there exists a sequence {𝑡𝑛}𝑛≥1, 𝑡𝑛 ∈ [𝑎, 𝑏],forall 𝑛≥1 𝑛→∞|𝑢(𝑡𝑛)| = ∞ {𝑡𝑛 }𝑛 ≥1 considering the supremum over all partitions 𝜋 of the interval ,suchthatlim .Let 𝑘 𝑘 be a {𝑡𝑛}𝑛 1 {𝑡𝑛 }𝑛 ≥1 [𝑎, 𝑏],weget subsequence of ≥ such that 𝑘 𝑘 converges to point 𝑥∈[𝑎,𝑏] {𝑢(𝑡 )} {𝑢(𝑡 )} 𝑅 .Then, 𝑛𝑘 𝑛𝑘≥1 is a subsequence of 𝑛 𝑛≥1. 𝜅𝑉𝜑 (𝑢) ≤𝜑(𝑐)(𝑏−𝑎) <∞. (30) |𝑢(𝑡 )| = ∞ So, lim𝑘→∞ 𝑛𝑘 . [𝑎, 𝑏] ⊂ 𝜅𝑉𝑅[𝑎, 𝑏] Therefore, Lip 𝜑 . Case 1.Supposethat𝑥 =𝑎̸.Since 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 The class of functions of a bounded 𝜅𝜑-variation has 𝜑(󵄨𝑢(𝑡 )−𝑢(𝑎)󵄨 / 󵄨𝑡 −𝑎󵄨) 󵄨𝑡 −𝑎󵄨 󵄨 𝑛𝑘 󵄨 󵄨 𝑛𝑘 󵄨 󵄨 𝑛𝑘 󵄨 many interesting properties as the following proposition 𝜅 ((𝑡 −𝑎)/(𝑏−𝑎))+𝜅((𝑏−𝑡 )/(𝑏−𝑎)) showes. 𝑛𝑘 𝑛𝑘 (37) 𝑅 Proposition 11. Let 𝜑 be a 𝜑-function, 𝜅∈K and 𝑢:[𝑎, ≤𝜅𝑉𝜑 (𝑢) <∞, 𝑏] → R be a function, then for all 𝑛𝑘 ≥1and since 𝜑 is continuous, we have 𝜅𝑉𝑅(⋅) : 𝜅𝑉𝑅[𝑎, 𝑏] → R (a) The function 𝜑 𝜑 is an even func- 󵄨 󵄨 𝑅 𝑅 𝜑(lim𝑘→∞ (󵄨𝑢(𝑡𝑛 )−𝑢(𝑎)󵄨 / |𝑥−𝑎|)) |𝑥−𝑎| tion, that is, 𝜅𝑉𝜑 (𝑢) = 𝜑𝜅𝑉 (−𝑢). 󵄨 𝑘 󵄨 𝑅 𝜅 ((𝑥−𝑎) / (𝑏−𝑎)) +𝜅((𝑏−𝑥) / (𝑏−𝑎)) (b) 𝜅𝑉𝜑 (𝑢) = 0 if and only if 𝑢 is constant. 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝑅 𝜑(󵄨𝑢(𝑡 )−𝑢(𝑎)󵄨 / 󵄨𝑡 −𝑎󵄨) 󵄨𝑡 −𝑎󵄨 (c) If 𝜅𝑉 (𝑢) < ∞ then 𝑢 is bounded on [𝑎, 𝑏]. 󵄨 𝑛𝑘 󵄨 󵄨 𝑛𝑘 󵄨 󵄨 𝑛𝑘 󵄨 𝜑 = lim 𝑘→∞ 𝑅 𝜅((𝑡𝑛 −𝑎)/(𝑏−𝑎))+𝜅((𝑏−𝑡𝑛 )/(𝑏−𝑎)) (d) 𝜑 is convex if and only if the function 𝜅𝑉𝜑 (⋅) : 𝑘 𝑘 𝑅 𝜅𝑉𝜑 [𝑎, 𝑏] → [0, ∞) defined by 𝑅 ≤𝜅𝑉𝜑 (𝑢) . 𝑅 𝑅 𝑅 𝜅𝑉𝜑 (𝑢) := 𝜅𝑉𝜑 (𝑢; [𝑎,] 𝑏 ) ,𝑢∈𝜅𝑉𝜑 [𝑎,] 𝑏 (31) (38)

|𝑢(𝑡𝑛 )−𝑢(𝑎)| 𝑘→∞ is convex. On the other hand, 𝑘 tends to infinity as . Then, since 𝜑(𝑡) →∞ as 𝑡→∞,weget 󵄨 󵄨 Proof. (a) From Definition,wehave 4 𝜑(󵄨𝑢(𝑡 )−𝑢(𝑎)󵄨 / |𝑥−𝑎|) |𝑥−𝑎| 󵄨 𝑛𝑘 󵄨 𝜅𝑉𝑅 (𝑢) lim =∞, (39) 𝜑 𝑘→∞ 𝜅 ((𝑥−𝑎) / (𝑏−𝑎)) +𝜅((𝑏−𝑥) / (𝑏−𝑎)) 𝑛 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝑅 ∑𝑖=1 𝜑(󵄨𝑢(𝑡𝑖)−𝑢(𝑡𝑖−1)󵄨 / 󵄨𝑡𝑖 −𝑡𝑖−1󵄨) 󵄨𝑡𝑖 −𝑡𝑖−1󵄨 then 𝜅𝑉𝜑 (𝑢) =, ∞ which is a contradiction. = sup 𝑛 𝜋 ∑ 𝜅 ((𝑡 −𝑡 )/(𝑏−𝑎)) 𝑖=1 𝑖 𝑖−1 𝑥=𝑎 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 Case 2.Supposethat .Then,since ∑𝑛 𝜑(󵄨−𝑢 (𝑡 )−(−𝑢(𝑡 ))󵄨 / 󵄨𝑡 −𝑡 󵄨) 󵄨𝑡 −𝑡 󵄨 𝑖=1 󵄨 𝑖 𝑖−1 󵄨 󵄨 𝑖 𝑖−1󵄨 󵄨 𝑖 𝑖−1󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 = 𝑛 𝜑(󵄨𝑢 (𝑏) −𝑢(𝑡 )󵄨 / 󵄨𝑏−𝑡 󵄨) 󵄨𝑏−𝑡 󵄨 sup 󵄨 𝑛𝑘 󵄨 󵄨 𝑛𝑘 󵄨 󵄨 𝑛𝑘 󵄨 𝜋 ∑𝑖=1 𝜅 ((𝑡𝑖 −𝑡𝑖−1)/(𝑏−𝑎)) 𝜅((𝑡 −𝑎)/(𝑏−𝑎))+𝜅((𝑏−𝑡 )/(𝑏−𝑎)) 𝑅 𝑛𝑘 𝑛𝑘 (40) =𝜅𝑉𝜑 (−𝑢) . 𝑅 (32) ≤𝜅𝑉𝜑 (𝑢) <∞, 6 Journal of Function Spaces and Applications

𝑛 󵄨 󵄨 for all 𝑛𝑘 ≥1, and since 𝜑 is continuous, we have 󵄨(𝛼𝑢 +𝛽V)(𝑡𝑖)−(𝛼𝑢+𝛽V)(𝑡𝑖−1)󵄨 = sup (∑ 𝜑( 󵄨 󵄨 ) 𝜋 𝑖=1 󵄨𝑡𝑖 −𝑡𝑖−1󵄨 󵄨 󵄨 𝜑( 󵄨𝑢 (𝑏) −𝑢(𝑡 )󵄨 / |𝑏−𝑥|) |𝑏−𝑥| −1 lim𝑘→∞ 󵄨 𝑛𝑘 󵄨 𝑛 󵄨 󵄨 𝑡 −𝑡 𝜅 ((𝑥−𝑎) / (𝑏−𝑎)) +𝜅((𝑏−𝑥) / (𝑏−𝑎)) × 󵄨𝑡 −𝑡 󵄨 )(∑ 𝜅( 𝑖 𝑖−1 )) 󵄨 𝑖 𝑖−1󵄨 𝑏−𝑎 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝑖=1 𝜑(󵄨𝑢 (𝑏) −𝑢(𝑡 )󵄨 / 󵄨𝑏−𝑡 󵄨) 󵄨𝑏−𝑡 󵄨 󵄨 𝑛𝑘 󵄨 󵄨 𝑛𝑘 󵄨 󵄨 𝑛𝑘 󵄨 𝑅 = lim =𝜅𝑉𝜑 (𝛼𝑢 +𝛽V). 𝑘→∞ 𝜅((𝑡𝑛 −𝑎)/(𝑏−𝑎)) + 𝜅 ((𝑏𝑛 −𝑡 )/(𝑏−𝑎)) 𝑘 𝑘 (44) 𝑅 ≤𝜅𝑉𝜑 (𝑢) . (41) 𝑅 𝑅 𝑅 So 𝜅𝑉𝜑 (𝛼𝑢V +𝛽 )≤𝛼𝜅𝑉𝜑 (𝑢) + 𝛽𝜅𝑉𝜑 (V) for all 𝛼, 𝛽 ∈ [0, 1] 𝑅 such that 𝛼+𝛽=1, and therefore 𝜅𝑉𝜑 (⋅) is convex. 𝜑(𝑡) →∞ 𝑡→∞ |𝑢(𝑏)−𝑢(𝑡 )| = ∞ Since as and lim𝑘→∞ 𝑛𝑘 , In order to prove the order direction, let us consider the we get following functions. 𝑢:[𝑎,𝑏]→ R defined by 𝑢(𝑡) := 𝑥𝑡 and V : [𝑎, 𝑏] → R defined by V(𝑡) := 𝑦𝑡, 𝑡 ∈ [𝑎, 𝑏]. Therefore, we 󵄨 󵄨 𝜑(󵄨𝑢 (𝑏) −𝑢(𝑡 )󵄨 / |𝑏−𝑥|) |𝑏−𝑥| have 󵄨 𝑛𝑘 󵄨 lim =∞, (42) 𝑘→∞ 𝜅 ((𝑥−𝑎) / (𝑏−𝑎)) +𝜅((𝑏−𝑥) / (𝑏−𝑎)) 𝑅 𝑅 𝜅𝑉𝜑 (𝑢) =𝜑(𝑥)(𝑏−𝑎) ,𝜅𝑉𝜑 (V) =𝜑(𝑦)(𝑏−𝑎) , (45) 𝑅 𝑅 so 𝜅𝑉𝜑 =∞which is a contradiction. In both cases a 𝜅𝑉𝜑 (𝛼𝑢V +𝛽 )=𝜑(𝛼𝑥+𝛽𝑦)(𝑏−𝑎) . contradiction is reached. Hence 𝑢 is bounded. (d) Let 𝑢, V : [𝑎, 𝑏] → R and 𝛼, 𝛽 ∈ [0, 1] such that 𝜅𝑉𝑅(⋅) 𝛼+𝛽=1.Supposethat𝜑 is convex, then Since 𝜑 is convex, we get

𝛼𝜅𝑉𝑅 (𝑢) +𝛽𝜅𝑉𝑅 (V) 𝑅 𝜑 𝜑 𝜑(𝛼𝑥+𝛽𝑦)(𝑏−𝑎) =𝜅𝑉𝜑 (𝛼𝑢 +𝛽V) 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 ∑𝑛 𝜑(󵄨𝑢(𝑡)−𝑢(𝑡 )󵄨 / 󵄨𝑡 −𝑡 󵄨) 󵄨𝑡 −𝑡 󵄨 𝑅 𝑅 𝑖=1 󵄨 𝑖 𝑖−1 󵄨 󵄨 𝑖 𝑖−1󵄨 󵄨 𝑖 𝑖−1󵄨 ≤𝛼𝜅𝑉𝜑 (𝑢) +𝛽𝜅𝑉𝜑 (V) (46) =𝛼sup 𝑛 𝜋 ∑𝑖=1 𝜅((𝑡𝑖 −𝑡𝑖−1)/(𝑏−𝑎)) =𝛼𝜑(𝑥)(𝑏−𝑎) +𝛽𝜑(𝑦)(𝑏−𝑎) , 𝑛 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 ∑𝑖=1 𝜑(󵄨V (𝑡𝑖)−V (𝑡𝑖−1)󵄨 / 󵄨𝑡𝑖 −𝑡𝑖−1󵄨) 󵄨𝑡𝑖 −𝑡𝑖−1󵄨 +𝛽sup 𝑛 𝜋 ∑ 𝜅 ((𝑡 −𝑡 )/(𝑏−𝑎)) 𝑖=1 𝑖 𝑖−1 which implies that 󵄨 󵄨 𝑛 󵄨𝑢(𝑡)−𝑢(𝑡 )󵄨 󵄨 󵄨 ≥ (∑ [𝛼𝜑 (󵄨 𝑖 𝑖−1 󵄨) 󵄨𝑡 −𝑡 󵄨 sup 󵄨 󵄨 󵄨 𝑖 𝑖−1󵄨 𝜋 𝑖=1 󵄨𝑡𝑖 −𝑡𝑖−1󵄨 𝜑 (𝛼𝑥+𝛽𝑦) ≤𝛼𝜑(𝑥) +𝛽𝜑(𝑦) . (47) 󵄨 󵄨 󵄨V (𝑡 )−V (𝑡 )󵄨 󵄨 󵄨 +𝛽𝜑 (󵄨 𝑖 𝑖−1 󵄨) 󵄨𝑡 −𝑡 󵄨]) 󵄨 󵄨 󵄨 𝑖 𝑖−1󵄨 Therefore, 𝜑 is convex, and the proof of the theorem is 󵄨𝑡𝑖 −𝑡𝑖−1󵄨 completed. −1 𝑛 𝑡 −𝑡 ×(∑ 𝜅( 𝑖 𝑖−1 )) , Remark 12. The part (c) of Proposition 11 is a consequence of 𝑏−𝑎 𝑖=1 the part (a) of Proposition 9 if the 𝜑-function is convex. (43) Proposition 13. Let 𝜑 be a 𝜑-function, 𝜅∈K,and𝑢, V : [𝑎, 𝑏] → R be functions, then where the supremum is taken over all partitions 𝜋 of the interval [𝑎, 𝑏]. 𝑅 𝑅 𝑅 Since 𝜑 is convex, we have 𝜅𝑉𝜑 (𝛼𝑢 +𝛽V)≤𝜅𝑉𝜑 (𝑢) +𝜅𝑉𝜑 (V) (48) 𝛼, 𝛽 ∈ [0, 1] ,𝛼+𝛽=1. 𝑅 𝑅 𝛼𝜅𝑉𝜑 (𝑢) +𝛽𝜅𝑉𝜑 (V)

𝑛 󵄨 󵄨 𝛼, 𝛽 ∈ [0, 1] 𝛼+𝛽=1 𝑥, 𝑦 ∈ [0, ∞) 󵄨𝛼(𝑢(𝑡𝑖)−𝑢(𝑡𝑖−1)) + 𝛽 (V (𝑡𝑖)−V (𝑡𝑖−1))󵄨 Proof. Let such that and . ≥ sup (∑ 𝜑( 󵄨 󵄨 ) Since 𝜑 is nondecreasing and nonnegative and 𝛼𝑥 + 𝛽𝑦 is one 𝜋 󵄨𝑡𝑖 −𝑡𝑖−1󵄨 𝑖=1 of the segment joining point 𝑥 with 𝑦,thenwehave −1 󵄨 󵄨 𝑛 𝑡 −𝑡 × 󵄨𝑡 −𝑡 󵄨 )(∑ 𝜅( 𝑖 𝑖−1 )) 󵄨 𝑖 𝑖−1󵄨 𝜑 (𝛼𝑥+𝛽𝑦) ≤ max {𝜑 (𝑥) ,𝜑(𝑦)} ≤𝜑(𝑥) +𝜑(𝑦) . (49) 𝑖=1 𝑏−𝑎 Journal of Function Spaces and Applications 7

From the inequality above, we deduce that Also, if 𝑥, 𝑦 ∈𝐴 and 𝛼>0, 𝛽>0,then 𝑅 𝑅 𝜅𝑉𝜑 (𝑢) +𝜅𝑉𝜑 (V) 𝛼 𝛽 𝛼𝑥+𝛽𝑦=(𝛼+𝛽)( 𝑥+ 𝑦) ∈ (𝛼 + 𝛽) 𝐴. (54) 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝛼+𝛽 𝛼+𝛽 ∑𝑛 𝜑(󵄨𝑢(𝑡)−𝑢(𝑡 )󵄨 / 󵄨𝑡 −𝑡 󵄨) 󵄨𝑡 −𝑡 󵄨 = 𝑖=1 󵄨 𝑖 𝑖−1 󵄨 󵄨 𝑖 𝑖−1󵄨 󵄨 𝑖 𝑖−1󵄨 sup 𝑛 ⋃ 𝜆𝐴 𝜋 ∑𝑖=1 𝜅((𝑡𝑖 −𝑡𝑖−1)/(𝑏−𝑎)) Therefore, 𝜆>0 is a vector space. ⋃ 𝜆𝐴={𝑥∈𝑋:∃𝜆>0,𝜆𝑥∈ 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 Let us now prove that 𝜆>0 ∑𝑛 𝜑(󵄨V (𝑡 )−V (𝑡 )󵄨 / 󵄨𝑡 −𝑡 󵄨) 󵄨𝑡 −𝑡 󵄨 𝐴} 𝑧=𝜆𝑥,𝑥∈𝐴,𝜆>0 1/𝑧 = 𝑥 ∈𝐴 + 𝑖=1 󵄨 𝑖 𝑖−1 󵄨 󵄨 𝑖 𝑖−1󵄨 󵄨 𝑖 𝑖−1󵄨 .Indeed,if ,then . sup 𝑛 𝜆𝑥 ∈𝐴 𝜆>0 𝑥 = (1/𝜆)(𝜆𝑥) ∈ 𝜋 ∑𝑖=1 𝜅 ((𝑡𝑖 −𝑡𝑖−1)/(𝑏−𝑎)) Conversely, if ,forsome ,then (1/𝜆)𝐴. 𝑛 󵄨 󵄨 󵄨𝛼(𝑢(𝑡𝑖)−𝑢(𝑡𝑖−1)) + 𝛽 (V (𝑡𝑖)−V (𝑡𝑖−1))󵄨 𝑅 ≥ sup (∑ 𝜑( 󵄨 󵄨 ) As a consequence of Lemma 15 and since 𝜅𝑉𝜑 is a convex 𝜋 󵄨𝑡 −𝑡 󵄨 𝑖=1 󵄨 𝑖 𝑖−1󵄨 and symmetric set, we have the following corollary. −1 󵄨 󵄨 𝑛 𝑡 −𝑡 Corollary 16. 𝜑 𝜑 𝜅∈K × 󵄨𝑡 −𝑡 󵄨 )(∑ 𝜅( 𝑖 𝑖−1 )) , Let be a -function and .Then,the 󵄨 𝑖 𝑖−1󵄨 𝜅𝑉𝑅[𝑎, 𝑏] 𝑖=1 𝑏−𝑎 vector space generated by the class 𝜑 is equal to the (50) following thus 𝑅 𝜅𝑅𝑉𝜑 [𝑎,] 𝑏 ={𝑢:[𝑎,] 𝑏 → R :∃𝜆>0,𝜅𝑉𝜑 (𝜆𝑢) <∞}. 𝑅 𝑅 𝜅𝑉𝜑 (𝑢) +𝜅𝑉𝜑 (V) (55) 𝑛 󵄨 󵄨 Theorem 17. 𝜑 𝜑 𝜅∈K 󵄨(𝛼𝑢V +𝛽 )(𝑡𝑖)−(𝛼𝑢+𝛽V)(𝑡𝑖−1)󵄨 Let be a -function and ,then = sup (∑ 𝜑( 󵄨 󵄨 ) 𝜋 󵄨𝑡 −𝑡 󵄨 𝑖=1 󵄨 𝑖 𝑖−1󵄨 𝑅𝑉𝜑 [𝑎,] 𝑏 ⊂𝜅𝑅𝑉𝜑 [𝑎,] 𝑏 ⊂𝜅𝐵𝑉[𝑎,] 𝑏 . (56) (51) 𝑛 −1 󵄨 󵄨 𝑡 −𝑡 Proof. First we prove that 𝑅𝑉𝜑[𝑎, 𝑏] ⊂𝜑 𝜅𝑅𝑉 [𝑎, 𝑏].Let𝑢∈ × 󵄨𝑡 −𝑡 󵄨 )(∑ 𝜅( 𝑖 𝑖−1 )) 󵄨 𝑖 𝑖−1󵄨 𝑅𝑉 [𝑎, 𝑏] 𝜆>0 𝑉𝑅(𝜆𝑢) <∞ 𝑖=1 𝑏−𝑎 𝜑 , then there exists such that 𝜑 and 𝜋:𝑎=𝑡0 <𝑡1 <⋅⋅⋅<𝑡𝑛 =𝑏is a partition of the interval 𝑅 =𝜅𝑉𝜑 (𝛼𝑢 +𝛽V). [𝑎, 𝑏],thenfrominequality(8)wehave 𝑅 𝑅 𝑅 󵄨 󵄨 Therefore, 𝜅𝑉 (𝛼𝑢V +𝛽 )≤𝜅𝑉(𝑢) + 𝜅𝑉 (V) for all 𝛼, 𝛽 ∈ 𝑛 󵄨𝜆𝑢 (𝑡 )−𝜆𝑢(𝑡 )󵄨 󵄨 󵄨 𝜑 𝜑 𝜑 ∑𝜑(󵄨 𝑖 𝑖−1 󵄨) 󵄨𝑡 −𝑡 󵄨 [0, 1] 𝛼+𝛽=1 󵄨 󵄨 󵄨 𝑖 𝑖−1󵄨 such that . 𝑖=1 󵄨𝑡𝑖 −𝑡𝑖−1󵄨 𝜅𝑉𝑅[𝑎, 𝑏] 𝑅 Remark 14. From Propositions 11 and 13,wehave 𝜑 ≤𝑉𝜑 (𝜆𝑢) to be convex and symmetric. 𝑛 (57) 𝑅 𝑡𝑖 −𝑡𝑖−1 The following lemma allows us to give a characterization =𝑉𝜑 (𝜆𝑢) 𝜅(∑ ) 𝑖=1 𝑏−𝑎 of the space 𝜅𝑅𝑉𝜑[𝑎, 𝑏]. 𝑛 𝑡 −𝑡 Lemma 15. Let 𝑋 be a vector space and 𝐴⊂𝑋anonempty ≤𝑉𝑅 (𝜆𝑢) ∑𝜅( 𝑖 𝑖−1 ). 𝜑 𝑏−𝑎 convex and symmetric set. Then one has the following. 𝑖=1 0∈𝐴 (a) . Thus, 𝐴 (b) The vector space associated by is iqual to: 𝑛 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 ∑𝑖=1 𝜑(󵄨𝜆𝑢𝑖 (𝑡 )−𝜆𝑢(𝑡𝑖−1)󵄨 / 󵄨𝑡𝑖 −𝑡𝑖−1󵄨) 󵄨𝑡𝑖 −𝑡𝑖−1󵄨 𝑅 ⟨𝐴⟩ = {𝑥∈𝑋:∃𝜆>0,𝜆𝑥∈𝐴} = ⋃𝜆𝐴. 𝑛 ≤𝑉𝜑 (𝜆𝑢) . (52) ∑𝑖=1 𝜅 ((𝑡𝑖 −𝑡𝑖−1)/(𝑏−𝑎)) 𝜆>0 (58) Proof. (a) For all 𝑥∈𝐴,wehavethat−𝑥 ∈ 𝐴 0 = (1/2)𝑥 + (1/2)(−𝑥) ∈. 𝐴 Then, considering the supremum of the left side, we get (b) By definition, 𝐴⊂⋃𝜆>0 𝜆𝐴 ⊂ ⟨𝐴⟩. In order to show 𝑅 𝑅 𝜅𝑉𝜑 (𝜆𝑢) ≤𝑉𝜑 (𝜆𝑢) , (59) the other inclusion, we have to prove that ⋃𝜆>0 𝜆𝐴 is a vector space. Indeed, if 𝛼∈R,weget therefore, 𝑢∈𝜅𝑅𝑉𝜑[𝑎, 𝑏] and 𝑅𝑉𝜑[𝑎, 𝑏] ⊂𝜑 𝜅𝑅𝑉 [𝑎, 𝑏]. 𝛼 (𝜆𝑥) ∈ ⋃𝜆𝐴, 𝛼 >0, On the other hand, by part (a) of Proposition 9 we get that 𝜆>0 𝜅𝑅𝑉𝜑[𝑎, 𝑏] ⊂ 𝜅𝐵𝑉[𝑎, 𝑏]. 𝛼 (𝜆𝑥) =0∈⋃𝜆𝐴, 𝛼 =0, Theorem 18. 𝜑 𝜑 (53) Let be a convex -function, then 𝜆>0 𝑅 Λ={𝑢∈𝜅𝑅𝑉𝜑 [𝑎,] 𝑏 :𝜅𝑉𝜑 (𝑢) ≤1} (60) 𝛼 (𝜆𝑥) = (−𝛼𝜆)(−𝑥) ∈ ⋃𝜆𝐴, 𝛼 <0. 𝜆>0 is a symmetric convex absorbent subset of 𝜅𝑅𝑉𝜑[𝑎, 𝑏]. 8 Journal of Function Spaces and Applications

Proof. First we show the convexity. Let 𝛼, 𝛽 ∈ [0, 1] such that (c) 𝛼+𝛽=1and 𝑢, V ∈Λ.Then,byProposition 11 we get ||𝑢+V||𝜅𝜑 = |(𝑢+V)(𝑎)| +𝜇Λ (𝑢+V) 𝑅 𝑅 𝑅 𝜅𝑉𝜑 (𝛼𝑢 +𝛽V)≤𝛼𝜅𝑉𝜑 (𝑢) +𝛽𝜅𝑉𝜑 (V) ≤𝛼⋅1+𝛽⋅1=1, = |(𝑢)(𝑎) + V (𝑎)| +𝜇Λ (𝑢+V) (61) (68) ≤ |𝑢 (𝑎)| + |V (𝑎)| +𝜇Λ (𝑢) +𝜇Λ (V) thus 𝛼𝑢V +𝛽 ∈Λ. Now let 𝑢∈Λand 0<|𝜆|≤1.If0<𝜆≤1by = ||𝑢||𝜅𝜑 + ||V||𝜅𝜑. Proposition 11,wehave ||𝑢 + V|| ≤ ||𝑢|| +||V|| 𝑅 𝑅 Thus, 𝜅𝜑 𝜅𝜑 𝜅𝜑. 𝜅𝑉𝜑 (𝜆𝑢) ≤𝜆𝜅𝑉𝜑 (𝑢) ≤𝜆⋅1≤1. (62) (d) Let us now prove that ||𝑢||𝜅𝜑 =0if and only if 𝑢=0. Indeed, suppose that ||𝑢||𝜅𝜑 =0,thatis, For the case −1≤𝜆<0, by the symmetric and convexity 𝜅𝑉𝑅(⋅) of the functional 𝜑 given in Proposition 11 we get that |𝑢 (𝑎)| +𝜇Λ (𝑢) =0, (69) 𝜅𝑉𝑅 (𝜆𝑢) =𝜅𝑉𝑅 (−𝜆 (−𝑢)) ≤ −𝜆𝜅𝑉𝑅 (−𝑢) = |𝜆| 𝜅𝑉𝑅 (𝑢) . 𝜑 𝜑 𝜑 𝜑 then |𝑢(𝑎)| =0 and 0=𝜇Λ(𝑢) = inf{𝜆 > 0 : 𝑅 (63) 𝜅𝑉𝜑 (𝑢/𝜆) ≤. 1} This implies that for each positive integer 𝑛, there exists 𝜆𝑛 such that 1/𝜆𝑛 >𝜆𝑛 >0and Hence, we have shown that Λ is balance. Now we will 𝑅 𝑅 𝜅𝑉𝜑 (𝑢/𝜆𝑛)≤1.Asthefunction𝜅𝑉𝜑 (⋅) is convex, we show that Λ is absorbent. Let 𝑢∈𝜅𝑅𝑉𝜑[𝑎, 𝑏] then there exist 𝜅𝑉𝑅(𝑢) ≤ 𝜆 𝑛→∞ 𝛼>0 𝜅𝑉𝑅(𝛼𝑢) <∞ 𝜅𝑉𝑅(𝛼𝑢) ≤1 𝛼𝑢 ∈Λ have 𝜑 𝑛. Taking the limit as ,weget such that 𝜑 .If 𝜑 then . 𝑅 𝑅 𝜅𝑉𝜑 (𝑢) =. 0 Moreover, by part (b) of Proposition 11 On the other hand, if 𝜅𝑉𝜑 (𝛼𝑢) >1 we have we have that 𝑢 is constant; that is, 𝑢(𝑡) = 𝑢(𝑎), 𝑡∈ [𝑎, 𝑏] 𝑢=0 𝛼𝑢 1 , therefore . 𝜅𝑉𝑅 ( )≤ 𝜅𝑉𝑅 (𝛼𝑢) =1, 𝜑 𝑅 𝑅 𝜑 (64) Now, suppose that 𝑢=0.Then|𝑢(𝑎)| =0 and 𝜇Λ(𝑢) = 𝜅𝑉𝜑 (𝛼𝑢) 𝜅𝑉𝜑 (𝛼𝑢) 0.Hence||𝑢||𝜅𝜑 =0. Therefore (𝜅𝑅𝑉𝜑[𝑎, 𝑏], || ⋅𝜅𝜑 || ) is 𝑅 anormedspace. in this case, (𝛼/𝜅𝑉𝜑 (𝛼𝑢))𝑢.Hence, ∈Λ Λ is absorbent.

Remark 19. As a consequence of Theorem 18,theMinkowski Lemma 21. 𝑢∈𝜅𝑅𝑉𝑅[𝑎, 𝑏] 𝜆>0 𝜇 (𝑢) ≤ 𝜆 functional associated to the set Λ defines a seminorm on Let 𝜑 .If ,then Λ if 𝜅𝑅𝑉 [𝑎, 𝑏] 𝑅 𝜑 and is defined by and only if 𝜅𝑉𝜑 (𝑢/𝜆; [𝑎, 𝑏]). ≤1 𝑢 𝜇 (𝑢) = {𝜆 > 0 : 𝜅𝑉𝑅 ( )≤1}. Proof Λ inf 𝜑 𝜆 (65) 𝑅 Case 1.Let𝑢∈𝜅𝑅𝑉𝜑 [𝑎, 𝑏] and 𝜇Λ(𝑢) <.Then,bythe 𝜆 0<𝑘<𝜆 𝜅𝑉𝑅(𝑢/𝑘) ≤1 Theorem 20. Let 𝜑 be a convex 𝜑-function. Then, (𝜅𝑅𝑉𝜑[𝑎, infimum property, there is such that 𝜑 . 𝑏], || ⋅ || ) || ⋅ || :𝜅𝑅𝑉[𝑎, 𝑏] → R 𝑅 𝜅𝜑 ,wherethefunctional 𝜅𝜑 𝜑 , Hence, by the convexity of 𝜅𝑉𝜑 (⋅), defined by 𝑅 𝑢 𝑅 𝑢 𝑘 𝑘 𝑅 𝑢 𝑘 𝑢 = 𝑢 𝑎 +𝜇 𝑢 ,𝑢∈𝜅𝑅𝑉𝑎, 𝑏 , 𝜅𝑉 ( )=𝜅𝑉 ( )≤ 𝜅𝑉 ( )≤ ≤1. || ||𝜅𝜑 | ( )| Λ ( ) 𝜑 [ ] (66) 𝜑 𝜆 𝜑 𝑘 𝜆 𝜆 𝜑 𝑘 𝜆 (70) is a normed space. 𝑅 Case 2.Let𝑢∈𝜅𝑅𝑉𝜑 [𝑎, 𝑏] and 𝜇Λ(𝑢) =.Then,bythe 𝜆 infimum property, there exists a sequence {𝜆𝑛}𝑛∈N such that Proof. Let 𝑢, V ∈𝜅𝑅𝑉𝜑[𝑎, 𝑏], 𝛼∈R.Then,wehavethefol- lowing. 𝑢 𝜆 >𝜆, 𝜅𝑉𝑅 ( )≤1, 𝑛∈N, 𝜆 =𝜆. 𝑛 𝜑 𝑘 𝑛→∞lim 𝑛 (71) (a) ||𝑢||𝜅𝜑 ≥0since |𝑢(𝑎)| ≥0 and 𝜇Λ(𝑢) ≥. 0 (b) Since 𝑢/𝜆𝑛 pointwise converges to 𝑢/𝜆 on [𝑎, 𝑏] as 𝑛→ 𝑅 ∞,bythelowersemicontinuityof𝜅𝑉𝜑 (⋅),weobtainthat ||𝛼𝑢||𝜅𝜑 = |𝛼𝑢 (𝑎)| + |𝛼| 𝜇Λ (𝑢) 𝑅 𝑢 𝑅 𝑢 = |𝛼||𝑢 (𝑎)| + |𝛼| 𝜇 (𝑢) 𝜅𝑉 ( )≤ 𝜅𝑉 ( )≤1. Λ 𝜑 𝜆 𝑛→∞lim 𝜑 𝜆 (72) (67) 𝑛 = |𝛼| (|𝑢 (𝑎)| +𝜇 (𝑢)) Λ As, by the definition of the infimum, the converse is

= |𝛼|||𝑢||𝜅𝜑. obvious, this completes the proof.

In the next theorem, we prove that (𝜅𝑅𝑉𝜑[𝑎, 𝑏], || ⋅𝜅𝜑 || ) is Therefore, ||𝛼𝑢||𝜅𝜑 = |𝛼|||𝑢||𝜅𝜑. a Banach space. Journal of Function Spaces and Applications 9

Theorem 22. Let 𝜑 be a convex 𝜑-function such that 𝜑 satisfies We defined on [𝑎, 𝑏] the function 𝑢(𝑡) := lim𝑛→∞𝑢𝑛(𝑡). the condition ∞1, then the space (𝜅𝑅𝑉𝜑[𝑎, 𝑏], || ⋅𝜅𝜑 || ) is a We claim that 𝑢∈𝜅𝑅𝑉𝑝[𝑎, 𝑏]. In fact, let 𝜋:𝑎=𝑡1 < Banach space. 𝑡2 <⋅⋅⋅<𝑡𝑛 =𝑏be a partition of the interval [𝑎, 𝑏].Then,for 𝑛≥𝑁and 𝜀>0one has {𝑢 } (𝜅𝑅𝑉 [𝑎, 𝑏], Proof. Let 𝑛 𝑛∈N be a Cauchy sequence in 𝜑 𝑢 −𝑢 || ⋅ || ) 0<𝜀<1 𝑁∈N 𝜅𝑉𝑅 ( 𝑛 ) 𝜅𝜑 ,thengiventhat ,thereis ,such 𝜑 𝜀 that for 𝑛, 𝑚 ≥𝑁 we have 󵄨 󵄨 󵄨󵄨 󵄨󵄨 𝑛 󵄨(𝑢 −𝑢)(𝑡)−(𝑢 −𝑢)(𝑡 )󵄨 󵄨󵄨𝑢 −𝑢 󵄨󵄨 <𝜀, 𝑛,𝑚≥𝑁, 󵄨 𝑛 𝑖 𝑛 𝑖−1 󵄨 󵄨󵄨 𝑛 𝑚󵄨󵄨𝜅𝜑 (73) = sup (∑ 𝜑( 󵄨 󵄨 ) 𝜋 𝑖=1 𝜀 󵄨𝑡𝑖 −𝑡𝑖−1󵄨 that is, −1 𝑛 𝑡 −𝑡 󵄨 󵄨 󵄨 󵄨 𝑖 𝑖−1 󵄨(𝑢𝑛 −𝑢𝑚) (𝑎)󵄨 +𝜇Λ (𝑢𝑛 −𝑢𝑚) <𝜀, 𝑛,𝑚≥𝑁. (74) × 󵄨𝑡𝑖 −𝑡𝑖−1󵄨 )(∑ 𝜅( )) 𝑖=1 𝑏−𝑎 Then, 𝑛 󵄨 󵄨 󵄨 󵄨 = sup (∑ 𝜑(󵄨(𝑢𝑛 − lim 𝑢𝑚)(𝑡𝑖) 󵄨𝑢𝑛 (𝑎) −𝑢𝑚 (𝑎)󵄨 <𝜀,Λ 𝜇 (𝑢𝑛 −𝑢𝑚)<𝜀 𝑛,𝑚≥𝑁, 󵄨 𝑚→∞ 󵄨 󵄨 (75) 𝜋 𝑖=1

󵄨 󵄨 󵄨 −1 by Lemma 21 and the last inequality, we have −(𝑢 − 𝑢 )(𝑡 )󵄨 (𝜀 󵄨𝑡 −𝑡 󵄨) ) 𝑛 𝑚→∞lim 𝑚 𝑖−1 󵄨 󵄨 𝑖 𝑖−1󵄨 𝑅 𝑢𝑛 −𝑢𝑚 𝜅𝑉𝜑 ( ; [𝑎,] 𝑏 )<1. (76) −1 𝜀 󵄨 󵄨 𝑛 𝑡 −𝑡 × 󵄨𝑡 −𝑡 󵄨 )(∑ 𝜅( 𝑖 𝑖−1 )) 󵄨 𝑖 𝑖−1󵄨 𝑏−𝑎 Then, for a partition 𝜋:𝑎≤𝑥<𝑦≤𝑏we get that 𝑖=1 󵄨 󵄨 𝑛 󵄨 󵄨 󵄨(𝑢 −𝑢 )(𝑦)−(𝑢 −𝑢 ) (𝑥)󵄨 󵄨 󵄨 󵄨(𝑢𝑛 −𝑢𝑚)(𝑡𝑖)−(𝑢𝑛 −𝑢𝑚)(𝑡𝑖−1)󵄨 𝜑(󵄨 𝑛 𝑚 𝑛 𝑚 󵄨) 󵄨𝑦−𝑥󵄨 = (∑ 𝜑(󵄨 󵄨) 󵄨 󵄨 󵄨 󵄨 sup𝑚→∞lim 󵄨 󵄨 𝜀 󵄨𝑦−𝑥󵄨 𝜋 𝑖=1 𝜀 󵄨𝑡𝑖 −𝑡𝑖−1󵄨 −1 𝑥−𝑎 𝑦−𝑥 𝑏−𝑦 𝑛 −1 ×(𝜅( )+𝜅( )+𝜅( )) (77) 󵄨 󵄨 𝑡𝑖 −𝑡𝑖−1 𝑏−𝑎 𝑏−𝑎 𝑏−𝑎 × 󵄨𝑡𝑖 −𝑡𝑖−1󵄨 )(∑ 𝜅( )) . 𝑖=1 𝑏−𝑎 𝑢 −𝑢 ≤𝜅𝑉𝑅 ( 𝑛 𝑚 ; [𝑎,] 𝑏 )<1. (82) 𝜑 𝜀 Since for 𝑛, 𝑚, ≥𝑁 Hence, we have 𝑅 𝑢𝑛 −𝑢𝑚 󵄨 󵄨 1≥𝜅𝑉𝜑 ( ) 󵄨(𝑢 −𝑢 )(𝑦)−(𝑢 −𝑢 ) (𝑥)󵄨 󵄨 󵄨 𝜀 𝜑(󵄨 𝑛 𝑚 𝑛 𝑚 󵄨) 󵄨𝑦−𝑥󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝜀 󵄨𝑦−𝑥󵄨 𝑛 󵄨(𝑢 −𝑢 )(𝑡)−(𝑢 −𝑢 )(𝑡 )󵄨 󵄨 𝑛 𝑚 𝑖 𝑛 𝑚 𝑖−1 󵄨 󵄨 󵄨 ≥(∑𝜑( 󵄨 󵄨 ) 󵄨𝑡𝑖 −𝑡𝑖−1󵄨) 𝑥−𝑎 𝑦−𝑥 𝑏−𝑦 𝜀 󵄨𝑡 −𝑡 󵄨 <𝜅( )+𝜅( )+𝜅( ) (78) 𝑖=1 󵄨 𝑖 𝑖−1󵄨 𝑏−𝑎 𝑏−𝑎 𝑏−𝑎 −1 𝑛 𝑡 −𝑡 ≤3⋅𝜅(1) =3, ×(∑𝜅( 𝑖 𝑖−1 )) , 𝑖=1 𝑏−𝑎 therefore (83) 󵄨 󵄨 3 󵄨 󵄨 then 󵄨(𝑢 −𝑢 )(𝑦)−(𝑢 −𝑢 ) (𝑥)󵄨 <𝜑−1 ( ) 󵄨𝑦−𝑥󵄨 𝜀. 󵄨 𝑛 𝑚 𝑛 𝑚 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨𝑦−𝑥󵄨 𝑛 󵄨(𝑢 −𝑢 )(𝑡)−(𝑢 −𝑢 )(𝑡 )󵄨 (∑ 𝜑(󵄨 𝑛 𝑚 𝑖 𝑛 𝑚 𝑖−1 󵄨) (79) 𝑚→∞lim 󵄨 󵄨 𝑖=1 𝜀 󵄨𝑡𝑖 −𝑡𝑖−1󵄨 −1 Since 𝜑 satisfies the condition ∞1 and 𝜑 is continuous, 𝑛 −1 󵄨 󵄨 𝑡𝑖 −𝑡𝑖−1 (84) we obtain that × 󵄨𝑡𝑖 −𝑡𝑖−1󵄨 )(∑ 𝜅( )) 𝑖=1 𝑏−𝑎 −1 3 󵄨 󵄨 𝜑 (󵄨 󵄨) 󵄨𝑦−𝑥󵄨 ,𝑥,𝑦∈[𝑎,] 𝑏 ,𝑥=𝑦,̸ (80) 󵄨𝑦−𝑥󵄨 ≤1, 𝑛≥𝑁. is bounded. Let 𝑀>0be a upper bound of (80), then Thus, 󵄨 󵄨 𝑅 𝑢𝑛 −𝑢 󵄨(𝑢 −𝑢 )(𝑦)−(𝑢 −𝑢 ) (𝑥)󵄨 <𝑀𝜀. 𝜅𝑉 ( )≤1. 󵄨 𝑛 𝑚 𝑛 𝑚 󵄨 (81) 𝜑 𝜀 (85)

As a consequence, the sequence {𝑢𝑛}𝑛∈N is a uniformly Therefore, (𝑢𝑛 − 𝑢)/𝜀 ∈𝜑 𝜅𝑅𝑉 [𝑎, 𝑏] for all 𝑛≥𝑁.As𝑢𝑛 ∈ Cauchy sequence, on the interval [𝑎, 𝑏], and by the complete- 𝜅𝑅𝑉𝜑[𝑎, 𝑏] for all 𝑛≥𝑁,and𝜅𝑅𝑉𝜑[𝑎, 𝑏] is a vector space, ness of R lim𝑛→∞𝑢𝑛(𝑡) exist for all 𝑡 ∈ [𝑎, 𝑏]. then 𝑢∈𝜅𝑅𝑉𝜑[𝑎, 𝑏]. 10 Journal of Function Spaces and Applications

Finally, let us prove that {𝑢𝑛}𝑛≥1 converge in norm to 𝑢. Theorem 28. Let 𝜑 be a 𝜑-function, 𝜅∈K,andℎ:[𝑎,𝑏]× Let 𝜀>0be arbitrary, then R → R be a function continuous with respect to the second variable. Suppose that the composition operator 𝐻 generated 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 by ℎ maps 𝜅𝑅𝑉𝜑[𝑎, 𝑏] into itself and satisfies the following 󵄨󵄨𝑢𝑛 −𝑢󵄨󵄨 = 󵄨󵄨𝑢𝑛 − lim 𝑢𝑚󵄨󵄨 𝜅𝜑 󵄨󵄨 𝑚→∞ 󵄨󵄨𝜅𝜑 inequality: (86) 󵄨󵄨 󵄨󵄨 ||𝐻 (𝑢) −𝐻(V)||𝜅𝜑 ≤𝛾(||𝑢−V||𝜅𝜑), 𝑢,V ∈𝜅𝑅𝑉𝜑 [𝑎,] 𝑏 = lim 󵄨󵄨𝑢𝑛 −𝑢𝑚󵄨󵄨𝜅𝜑 <𝜀, ∀𝑛≥𝑁. 𝑚→∞ (90)

Therefore, the sequence {𝑢𝑛}𝑛≥1 converges to 𝑢 in the for some function 𝛾 : [0, ∞) → [0, ∞).Then,thereexist || ⋅ || 𝜅𝑅𝑉 [𝑎, 𝑏] − norm 𝜅𝜑 and thus 𝜑 is a Banach space. functions 𝛼, 𝛽 ∈𝜑 𝜅𝑅𝑉 [𝑎, 𝑏] such that

ℎ− (𝑡, 𝑥) =𝛼(𝑡) 𝑥+𝛽(𝑡) ,𝑡∈[𝑎,] 𝑏 ,𝑥∈R, 3.1. Uniformly Bounded Composition Operator. In this sec- (91) tion, we present the other main result of this paper; namely, − where ℎ (⋅, 𝑥) : (𝑎, 𝑏]→ R is the left regularization of ℎ(⋅, 𝑥) we show that any uniformly bounded composition operator for all 𝑥∈R. that maps the space the 𝜅𝑅𝑉𝜑[𝑎, 𝑏] into itself necessarily satisfies the so-called Matkowski’s weak condition. Proof. By hypothesis, for 𝑥∈R fixed the constant function 𝑢(𝑡) = 𝑥, 𝑡 ∈ [𝑎, 𝑏] belongs to 𝜅𝑅𝑉𝜑[𝑎, 𝑏].Since𝐻 maps Definition 23. For a given function ℎ:[𝑎,𝑏]×R → R,the [𝑎,𝑏] [𝑎,𝑏] 𝜅𝑅𝑉𝜑[𝑎, 𝑏] intoitself,wehavethatthefunction(𝐻𝑢)(𝑡) = composition operator 𝐻:R → R generated by ℎ is ℎ(𝑡, 𝑢(𝑡)) = ℎ(𝑡,𝜑 𝑥)∈𝜅𝑅𝑉 [𝑎, 𝑏].ByLemma 26, the left defined by − − regularization ℎ (⋅, 𝑥) ∈ 𝜅𝑅𝑉𝜑 for every 𝑥∈R. [𝑎,𝑏] From inequality (90) and the definition of the norm ||⋅||𝜅𝜑 𝐻𝑢 (𝑡) =ℎ(𝑡, 𝑢 (𝑡)) ,𝑢∈R ,𝑡∈[𝑎,] 𝑏 . (87) we obtain for 𝑢1,𝑢2 ∈𝜅𝑅𝑉𝜑[𝑎, 𝑏] that 󵄨󵄨 󵄨󵄨 [𝑎,𝑏] 𝜇 (𝐻 (𝑢 )−𝐻(𝑢 )) ≤ 󵄨󵄨𝐻(𝑢 )−𝐻(𝑢 )󵄨󵄨 Here R denotes the family of all functions 𝑢:[𝑎,𝑏]→ R. Λ 1 2 󵄨󵄨 1 2 󵄨󵄨𝜅𝜑 (92) 𝜅𝑅𝑉 [𝑎, 𝑏] ⊂ 𝜅𝐵𝑉[𝑎, 𝑏] 󵄨󵄨 󵄨󵄨 Remark 24. Since 𝜑 ,thenevery ≤𝛾(󵄨󵄨𝑢1 −𝑢2󵄨󵄨𝜅𝜑). function of bounded 𝜅𝜑-variation in the sense of Riesz- Korenblum has left- and right-hand limits at each point of From the inequality (92)andLemma 21,if𝛾(||𝑢1 − its domain (see [8]). 𝑢2||𝜅𝜑)>0,then

Now, we will give the definition of left regularization ofa 𝐻(𝑢 )−𝐻(𝑢 ) 𝜅𝑉𝑅 ( 1 2 )≤1. function. 𝜑 󵄨󵄨 󵄨󵄨 (93) 𝛾(󵄨󵄨𝑢1 −𝑢2󵄨󵄨𝜅𝜑) Definition 25. Let 𝑢∈𝜅𝑅𝑉𝜑[𝑎, 𝑏], one defined its left regular- − On the other hand, if 𝑎≤𝑟<𝑠≤𝑏,thenfrom ization 𝑢 : (𝑎, 𝑏] → R of mapping 𝑢 by the following: 𝑅 the definitions of the operator 𝐻,thefunctional𝜅𝑉𝜑 ,and inequality (93), we have lim 𝑢 (𝑠) ,𝑡∈(𝑎,] 𝑏 , 𝑢− (𝑡) := {𝑠→𝑡− 󵄨 󵄨 (88) 󵄨𝐻(𝑢 ) (𝑠) −𝐻(𝑢 ) (𝑠) −𝐻(𝑢 ) (𝑟) +𝐻(𝑢 ) (𝑟)󵄨 𝑢 (𝑎) ,𝑡=𝑎. 𝜑(󵄨 1 2 1 2 󵄨) 󵄨󵄨 󵄨󵄨 𝛾(󵄨󵄨𝑢1 −𝑢2󵄨󵄨𝜅𝜑) |𝑠−𝑟| − We will denote by 𝜅𝑅𝑉𝜑 [𝑎, 𝑏] the subset in 𝜅𝑅𝑉𝜑[𝑎, 𝑏] 𝑟−𝑎 𝑠−𝑟 𝑏−𝑠 −1 which consists of those functions that are left continuous on × |𝑠−𝑟| (𝜅 ( )+𝜅( )+𝜅( )) (𝑎, 𝑏]. 𝑏−𝑎 𝑏−𝑎 𝑏−𝑎 Lemma 26. 𝑢∈𝜅𝑅𝑉[𝑎, 𝑏] 𝑢− ∈𝜅𝑅𝑉−[𝑎, 𝑏] 𝐻(𝑢 )−𝐻(𝑢 ) If 𝜑 ,then 𝜑 . ≤𝜅𝑉𝑅 ( 1 2 )≤1, 𝜑 󵄨󵄨 󵄨󵄨 𝛾(󵄨󵄨𝑢1 −𝑢2󵄨󵄨 ) Thus, if a function 𝑢 has bounded 𝜅𝜑-variation in the 𝜅𝜑 sense of Riesz-Korenblum, then its left regularization is a left (94) continuous function. whence Definition 27 (see [24, Definition 1]). Let X and Y be two 󵄨 󵄨 󵄨ℎ(𝑠,𝑢1 (𝑠))−ℎ(𝑠,𝑢2 (𝑠))−ℎ(𝑟,𝑢1 (𝑟))+ℎ(𝑟,𝑢2 (𝑟))󵄨 metric (or normed) spaces. One says that a mapping 𝐻: 𝜑( 󵄨󵄨 󵄨󵄨 ) 𝛾(󵄨󵄨𝑢 −𝑢 󵄨󵄨 ) |𝑠−𝑟| X → Y is uniformly bounded if, for any 𝑡>0, there exists a 󵄨󵄨 1 2󵄨󵄨𝜅𝜑 nonnegative real number 𝛾(𝑡) such that for any nonempty set 𝑟−𝑎 𝑠−𝑟 𝑏−𝑠 −1 𝐵⊂X we have × |𝑠−𝑟| (𝜅 ( )+𝜅( )+𝜅( )) ≤1. 𝑏−𝑎 𝑏−𝑎 𝑏−𝑎 diam 𝐵≤𝑡󳨐⇒diam 𝐻 (𝐵) ≤𝛾(𝑡) . (89) (95) Journal of Function Spaces and Applications 11

− − Let 𝑥1,𝑥2 ∈ R,𝑥1 <𝑥2 andconsiderthefunctions𝑢1, Since ℎ (⋅, 𝑥) ∈ 𝜅𝑅𝑉𝜑 [𝑎, 𝑏] for all 𝑥∈R in particular for 𝑢2 : [𝑎, 𝑏] → R defined by 𝑥=0, 𝑥 −𝑥 𝑥 +𝑥 − 𝑢 (𝑡) := 1 2 (𝑡−𝑟) + 𝑘 2 ,𝑡∈[𝑎,] 𝑏 ,𝑘=1,2 𝛽 (𝑡) := ℎ (𝑡,) 0 𝑡∈[𝑎,] 𝑏 , (106) 𝑘 2 (𝑠−𝑟) 2 − (96) therefore 𝛽∈𝜅𝑅𝑉𝜑 [𝑎, 𝑏]. On the other hand, considering 𝑥=1we obtain that then 𝑢𝑘 ∈𝜅𝑅𝑉𝜑[𝑎, 𝑏], 𝑘 =.From( 1,2 96), we have − 𝛼 (𝑡) =ℎ (𝑡, 𝑥) −𝛽(𝑡) ,𝑡∈[𝑎,] 𝑏 , (107) 𝑥1 −𝑥2 𝑢1 −𝑢2 = , (97) 2 as 𝜅𝑅𝑉𝜑[𝑎, 𝑏] is a vector space which implies that 𝛼∈ − 𝜅𝑅𝑉 [𝑎, 𝑏]. therefore, 𝜑 󵄨𝑥 −𝑥 󵄨 󵄨󵄨 󵄨󵄨 󵄨 1 2 󵄨 Theorem 29. Let 𝜑 be a function and 𝜅∈K.Supposethata 󵄨󵄨𝑢1 −𝑢2󵄨󵄨 = 󵄨 󵄨 . (98) 𝜅𝜑 󵄨 2 󵄨 function ℎ:[𝑎,𝑏]×R → R is continuous with respect to the second variable. If the composition operator 𝐻 generated by ℎ Moreover, maps the space 𝜅𝑅𝑉𝜑[𝑎, 𝑏] into itself and is uniformly bounded, 𝑥 +𝑥 𝛼, 𝛽 ∈− 𝜅𝑅𝑉 [𝑎, 𝑏] 𝑢 (𝑠) =𝑥,𝑢(𝑟) = 1 2 =𝑢 (𝑠) ,𝑢(𝑟) =𝑥. then there exist the functions 𝜑 ,suchthat 1 1 1 2 2 2 2 − (99) ℎ (𝑡, 𝑥) =𝛼(𝑡) 𝑥+𝛽(𝑡) ,𝑡∈[𝑎,] 𝑏 ,𝑥∈R. (108)

From (95)weget Proof. Take any 𝑡≥0and 𝑢, V ∈𝜅𝑅𝑉𝜑[𝑎, 𝑏] such that 󵄨 󵄨 󵄨 𝑥1 +𝑥2 𝑥1 +𝑥2 󵄨 𝜑(󵄨ℎ(𝑠,𝑥 )−ℎ(𝑠, )−ℎ(𝑟, )+ℎ(𝑟,𝑥 )󵄨 ||𝑢−V||𝜅𝜑 ≤𝑡. 󵄨 1 2 2 2 󵄨 (109) {𝑢, V}≤𝑡 𝐻 󵄨 󵄨 −1 3 Since diam , by the uniform boundedness of , ×(𝛾(󵄨(𝑥 −𝑥 )/2󵄨) |𝑠−𝑟|) )≤ , 󵄨 1 2 󵄨 |𝑠−𝑟| we have (100) diam 𝐻 ({𝑢, V}) ≤𝛾(𝑡) , (110) then that is, 󵄨 󵄨 󵄨 𝑥1 +𝑥2 𝑥1 +𝑥2 󵄨 󵄨ℎ(𝑠,𝑥 )−ℎ(𝑠, )−ℎ(𝑟, )+ℎ(𝑟,𝑥 )󵄨 ||𝐻 (𝑢) −𝐻(V)|| = 𝐻 ({𝑢, V}) ≤𝛾(||𝑢−V|| ), 󵄨 1 2 2 2 󵄨 𝜅𝜑 diam 𝜅𝜑 󵄨 󵄨 (111) −1 3 󵄨𝑥1 −𝑥2 󵄨 ≤𝜑 ( )𝛾(󵄨 󵄨) |𝑠−𝑟| . |𝑠−𝑟| 󵄨 2 󵄨 therefore, by the Theorem 28 we have (101) − ℎ (𝑡, 𝑥) =𝛼(𝑡) 𝑥+𝛽(𝑡) ,𝑡∈[𝑎,] 𝑏 ,𝑥∈R. (112) Since 𝜑 satisfies the condition ∞1,wehave

−1 1 𝜑 ( )𝑡=0, (102) 𝑡→0lim Remark 30. Observe that similar results hold for the right 𝑡 + regularization ℎ of ℎ defined by ℎ− 𝑟 𝑠 then from the definition of and letting tend to in (101), + ℎ (𝑡, 𝑥) := lim ℎ (𝑠, 𝑥) 𝑡∈[𝑎,) 𝑏 . we obtain 𝑠→𝑡+ (113) 󵄨 󵄨 󵄨 − − 𝑥1 +𝑥2 − 󵄨 󵄨ℎ (𝑠,1 𝑥 )−2ℎ (𝑠, )+ℎ (𝑠,2 𝑥 )󵄨 ≤0, (103) 󵄨 2 󵄨 Acknowledgments therefore Thanks are due to the referee for the useful suggestion and 𝑥 +𝑥 ℎ− (𝑠, 𝑥 )+ℎ− (𝑠, 𝑥 ) comments to improve this paper. This research has been ℎ− (𝑠, 1 2 )= 1 2 , 2 2 partly supported by the Central Bank of Venezuela. The (104) authorswanttothanksthelibrarystaffofB.C.Vforcompiling 𝑠∈(𝑎,] 𝑏 ,𝑥1,𝑥2 ∈ R the references. whichprovesthatforeveryfixed𝑠 ∈ (𝑎, 𝑏] the function − ℎ (𝑠, ⋅) satisfies the Jensen functional equation in R (see [26, References ℎ page 315]). The continuity of with respect to the second [1] C. Jordan, “Sur la serie´ de Fourier,” Comptes Rendus De variable implies that for every 𝑡∈[𝑎,𝑏]there exist 𝛼(𝑡), 𝛽(𝑡) ∈ L’Ac a d emie´ Des Sciences, vol. 2, pp. 228–230, 1881. R such that [2] D. Bugajewski, “On BV-solutions of some nonlinear integral ℎ− (𝑡, 𝑥) =𝛼(𝑡) 𝑥+𝛽(𝑡) ,𝑡∈[𝑎,] 𝑏 ,𝑥∈R. equations,” Integral Equations and Operator Theory,vol.46,no. (105) 4, pp. 387–398, 2003. 12 Journal of Function Spaces and Applications

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Research Article Homogeneous Triebel-Lizorkin Spaces on Stratified Lie Groups

Guorong Hu

Graduate School of Mathematical Sciences, The Universityf o Tokyo, 3-8-1 Komaba, Mekuro-ku, Tokyo 153-8914, Japan

Correspondence should be addressed to Guorong Hu; [email protected]

Received 7 January 2013; Accepted 3 March 2013

Academic Editor: Jozef´ Bana´s

Copyright © 2013 Guorong Hu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Homogeneous Triebel-Lizorkin spaces with full range of parameters are introduced on stratified Lie groups in terms of Littlewood- Paley-type decomposition. It is shown that the scale of these spaces is independent of the choice of Littlewood-Paley-type decomposition and the sub-Laplacian used for the construction of the decomposition. Some basic properties of these spaces are given. As the main result of this paper, boundedness of a class of singular integral operators on these function spaces is obtained.

1. Introduction function spaces via Littlewood-Paley-type decomposition. We find that a helpful way to treat the case that either the In recent years there were several efforts of extending Besov integrability parameter 𝑝 or the summability parameter 𝑞 and Triebel-Lizorkin spaces from Euclidean spaces to other is less than 1 is to take the Peetre type maximal function domains and non-isotropic settings. In particular, Han et al. into consideration. With the help of the almost orthogonality [1] developed a theory of these function spaces on spaces estimate on stratified Lie groups (see Lemma 2), we show of homogenous type with the additional reverse doubling that our definition of homogeneous Triebel-Lizorkin spaces property. That setting is quite general and includes for is independent of the choice of the Littlewood-Paley-type example Lie groups of polynomial growth. However, the high decomposition and the sub-Laplacian used for the construc- level of generality imposes restrictions on the possible values tion of the decomposition. Thus, these function spaces reflect of the parameters of the function spaces. of properties of the group, not of the sub-Laplacian used for For the purpose of studying subelliptic regularity, Folland the construction of the decomposition. Singular integral theory is a powerful tool for the study [2] introduced fractional Sobolev spaces and Lipschitz spaces 𝑝 on stratified Lie groups. Later, Folland and Stein [3]estab- of partial differential equations. The 𝐿 -boundedness of con- lished the theory of Hardy spaces on general homogeneous volution operators with homogeneous distribution kernels groups. Besov spaces on stratified Lie groups were first on Lie groups endowed with suitable homogeneous structure introduced by Saka [4], by means of the heat semigroup was proved by Knapp and Stein [6](for𝑝=2)andKoranyi´ associated to the sub-Laplacian. Recently, Fuhr¨ and Mayeli and Vagi´ [7](for1<𝑝<∞). In Section 4 of this paper, [5] introduced homogeneous Besov spaces on stratified Lie we prove the boundedness on homogeneous Triebel-Lizorkin groups in terms of Littlewood-Paley-type decomposition and spaces of a class of convolution type singular integral oper- established wavelet characterization of them. However, the ators on stratified Lie groups, which includes convolution integrability parameter 𝑝 and the summability parameter 𝑞 operators with homogeneous distribution kernels. of the function spaces studied in both [4, 5] are restricted to This paper is organized as follows. After reviewing be no less than 1. Moreover, systematic treatment of Triebel- some basic notions concerning stratified Lie groups and Lizorkin spaces on stratified Lie groups can not be found in their associated sub-Laplaicans in Section 2,inSection 3 we ̇𝛼 the literature, to our best knowledge. introduce homogeneous Triebel-Lizorkin spaces 𝐹𝑝,𝑞(𝐺) on Thepurposeofthispaperistointroduceandstudyhomo- stratified Lie groups, and give some basic properties of them. 𝐹̇𝛼 (𝐺) geneousTriebel-Lizorkinspaceswithfullrangeofparameters In Section 4 we show the 𝑝,𝑞 -boundedness of a class on stratified Lie groups. Motivated by5 [ ], we define these of convolution singular integral operators. Throughout this 2 Journal of Function Spaces and Applications

paper the letter 𝐶 will denote a positive constant which is is a basis of 𝑉1. We denote by 𝑌1,...,𝑌𝑛 the corresponding independent of the main variables involved but whose value basis for right-invariant vector fields, that is, may differ from line to line. The notation 𝑎≲𝑏or 𝑏≳𝑎for some variable quantities 𝑎 and 𝑏 means that 𝑎≤𝐶𝑏for some 𝑑 𝑌𝑗𝑓 (𝑥) = 𝑓(exp (𝑡𝑋𝑗)𝑥)|𝑡=0. (4) constant 𝐶>0; 𝑎∼𝑏stands for 𝑎≲𝑏≲𝑎. We agree that the 𝑑𝑡 set N of natural numbers contains 0. 𝑛 𝐼 𝑖1 𝑖𝑛 If 𝐼=(𝑖1,...,𝑖𝑛)∈N is a multi-index we set 𝑋 =𝑋1 ⋅⋅⋅𝑋𝑛 𝐼 𝑖 𝑖 𝑌 =𝑌1 ⋅⋅⋅𝑌𝑛 2. Preliminaries and 1 𝑛 .Moreover,weset 𝑛 𝑛 In this section we briefly review the basic notions concerning |𝐼| = ∑𝑖𝑘,𝑑(𝐼) = ∑𝑑𝑘𝑖𝑘, (5) stratified Lie groups and their associated sub-Laplacians. For 𝑘=1 𝑘=1 more details we refer the reader to the monograph by Folland and Stein [3]. A Lie group 𝐺 is called a stratified Lie group where the integers 𝑑1 ≤ ⋅⋅⋅ ≤ 𝑑𝑛 are given according to that if it is connected and simply connected, and its Lie algebra g 𝑋 ∈𝑉 𝑋𝐼 𝑌𝐼 𝑘 𝑑𝑘 .Then (resp., ) is a left-invariant (resp., right- may be decomposed as a direct sum g =𝑉1 ⊕⋅⋅⋅⊕𝑉𝑚,with invariant) differential operator, homogeneous of degree 𝑑(𝐼), [𝑉 ,𝑉]=𝑉 1≤𝑘≤𝑚−1 [𝑉 ,𝑉 ]=0 1 𝑘 𝑘+1 for and 1 𝑚 .Such with respect to the dilations 𝛿𝑡, 𝑡>0. agroup𝐺 is clearly nilpotent, and thus it may be identified A complex-valued function 𝑃 on 𝐺 is called a polynomial g : g →𝐺 with (as a manifold) via the exponential map exp . on 𝐺 if 𝑃∘exp is a polynomial on g.Let𝜉1,...,𝜉𝑛 be the basis R𝑛 Examples of stratified Lie groups include Euclidean spaces for the linear forms on g dual to the basis 𝑋1,...,𝑋𝑛 for g, H𝑛 −1 and the Heisenberg group . and set 𝜂𝑗 =𝜉𝑗 ∘ exp . From our definition of polynomials g {𝛿 : The algebra is equipped with a family of dilations 𝑡 on 𝐺, 𝜂1,...,𝜂𝑛 are generators of the algebra of polynomials 𝑡>0}which are the algebra automorphisms defined by on 𝐺.Thus,everypolynomialon𝐺 canbewrittenuniquely as 𝑚 𝑚 𝛿 (∑𝑋 )=∑𝑡𝑗𝑋 (𝑋 ∈𝑉). 𝑃=∑𝑎 𝜂𝐼,𝑎∈ C, 𝑡 𝑗 𝑗 𝑗 𝑗 (1) 𝐼 𝐼 (6) 𝑗=1 𝑗=1 𝐼

Under our identification of 𝐺 with g, 𝛿𝑡 may also be viewed whereallbutfinitelymanyofthecoefficientsvanish,and 𝐼 𝑖1 𝑖𝑛 as a map 𝐺→𝐺. We generally write 𝑡𝑥 instead of 𝛿𝑡(𝑥),for 𝜂 =𝜂⋅⋅⋅𝜂 .Apolynomialofthetype(6)iscalled of 𝑥∈𝐺. We shall denote by homogeneous degree 𝐿,where𝐿∈N,if𝑑(𝐼) ≤𝐿 holds for all multi-indices 𝐼 with 𝑎𝐼 =0̸ .WeletP denote the space 𝑚 𝐺 P Δ=∑𝑗[ (𝑉 )] of all polynomials on , and let 𝐿 denote the space of dim 𝑗 (2) 𝐺 𝐿 P 𝑗=1 polynomials on of homogeneous degree .Notethat 𝐿 is invariant under left and right translations (see3 [ ,Proposition 𝑓:𝐺 → C the homogeneous dimension of 𝐺. 1.25]). A function is said to have vanishing 𝐿 A homogeneous norm onGisacontinuousfunction moments of order ,if 𝑥 󳨃→|𝑥|from 𝐺 to [0, ∞) smooth away from 0 (the group −1 0 |𝑥 |=|𝑥| ∀𝑃 ∈ P𝐿 : ∫ 𝑓 (𝑥) 𝑃 (𝑥) 𝑑𝑥=0, (7) identity), vanishing only at ,andsatisfying and 𝐺 |𝑡𝑥| = 𝑡|𝑥| for all 𝑥∈𝐺and 𝑡>0.Homogeneousnormson𝐺 always exist and any two of them are equivalent. We assume with the absolute convergence of the integral. 𝐺 is provided with a fixed homogeneous norm. It satisfies a The Schwartz class on 𝐺 is defined by triangle inequality: there exists a constant 𝛾≥1such that |𝑥𝑦| ≤ 𝛾(|𝑥|+|𝑦|) for all 𝑥, 𝑦.If ∈𝐺 𝑥∈𝐺and 𝑟>0we define 𝜕|𝐼|𝑓 S (𝐺) ={𝑓∈𝐶∞ (𝐺) :𝑃 ∈𝐿∞ (𝐺) , −1 𝑖 𝑖 the ball of radius 𝑟 about 𝑥 by 𝐵(𝑥, 𝑟) = {𝑦 ∈ 𝐺:|𝑦 𝑥| < 𝜕𝜂 1 ⋅⋅⋅𝜕𝜂𝑛 𝑟}.TheLebesguemeasureong induces a bi-invariant Haar (8) measure on 𝐺.Asdonein[3], we fix the normalization of ∀𝐼 ∈ N𝑛,∀𝑃∈P}; Haar measure by requiring that the measure of 𝐵(0, 1) be 1. We shall denote the measure of any measurable 𝐸⊂𝐺by |𝐸|. Δ Clearly we have |𝛿𝑡(𝐸)| = 𝑡 |𝐸|. All integrals on 𝐺 are with that is, 𝑓∈S(𝐺) if and only if 𝑓∘exp ∈ S(g).Asisindicated respect to (the normalization of) Haar measure. Convolution in [3,p.35],S(𝐺) is a Frechet´ space and several different is defined by choicesoffamiliesofnormsinducethesametopologyon S(𝐺).Inthispaper,forourpurposeweusethefamily −1 −1 𝑓∗𝑔(𝑥) = ∫ 𝑓(𝑦)𝑔(𝑦 𝑥) 𝑑𝑦 = ∫ 𝑓(𝑥𝑦 )𝑔(𝑦)𝑑𝑦. {‖ ⋅ ‖(𝐿,𝑁) :𝐿,𝑁∈N} of norms given by 󵄩 󵄩 󵄨 󵄨 󵄨 󵄨 (3) 󵄩𝜙󵄩 = (1+|𝑥|)𝑁 (󵄨𝑋𝐼𝜙 (𝑥)󵄨 + 󵄨𝑋𝐼𝜙̃(𝑥)󵄨). 󵄩 󵄩(𝐿,𝑁) sup 󵄨 󵄨 󵄨 󵄨 𝑑(𝐼)≤𝐿,𝑥∈𝐺 (9) We consider g as the Lie algebra of all left-invariant vector fields on 𝐺,andfixabasis𝑋1,...,𝑋𝑛 of g,obtainedasaunion Here and in what follows, we use the notation convention ̃ −1 of bases of the 𝑉𝑗.Inparticular,𝑋1,...,𝑋],with] = dim(𝑉1), 𝜙(𝑥) = 𝜙(𝑥 ) for any function 𝜙:𝐺 → C.Thedualspace Journal of Function Spaces and Applications 3

󸀠 S (𝐺) of S(𝐺) is the space of tempered distributions on 𝐺.If Lemma 2. Let 𝐿, 𝑁 ∈ N with 𝑁≥𝐿+Δ+2. Suppose both 󸀠 𝑓∈S (𝐺) and 𝜙∈S(𝐺) we shall denote the evaluation of 𝑓 𝜙, 𝜓 ∈ S(𝐺) have vanishing moments of order 𝐿. Then there on 𝜙 by ⟨𝑓,𝜙⟩. exists a constant 𝐶>0such that for all 𝑗, ℓ ∈ Z and all 𝑥∈𝐺, Using the above conventions for the choice of the basis 𝑋 ,...,𝑋 ] = (𝑉 ) 󵄨 󵄨 1 𝑛,and dim 1 ,the sub-Laplacian is defined by 󵄨(𝐷2𝑗 𝜙) ∗ (𝐷2ℓ 𝜓) (𝑥)󵄨 ] 2 󵄨 󵄨 L =−∑𝑗=1 𝑋𝑗 . When restricted to smooth functions with (𝑗∧ℓ)Δ compact support, L is essentially self-adjoint. Its closure has 󵄩 󵄩 󵄩 󵄩 −|𝑗−ℓ|𝐿 2 (14) 2 2 ≤𝐶󵄩𝜙󵄩(𝐿+1,𝑁)󵄩𝜓󵄩(𝐿+1,𝑁)2 , domain D ={𝑢∈𝐿(𝐺) : L𝑢∈𝐿(𝐺)},whereL𝑢 is taken (1 + 2𝑗∧ℓ |𝑥|)𝑁 in the sense of distributions. We denote this extension still by L L the symbol . By the spectral theorem, admits a spectral where 𝑗∧ℓ:=min{𝑗, ℓ}. resolution ∞ Proof. Using dilations and the facts (𝐷2𝑗 𝜙) ∗ (𝐷2ℓ 𝜓)(𝑥) = L = ∫ 𝜆𝑑𝐸(𝜆) , −1 (10) (𝐷 ℓ 𝜓)̃ ∗ (𝐷 𝑗 𝜙)(𝑥̃ ) ‖𝜙‖ =‖𝜙‖̃ 0 2 2 and (𝐿,𝑁) (𝐿,𝑁) (see (9)), we may assume ℓ≥𝑗=0. To proceed we follow the idea in 𝑑𝐸(𝜆) 𝑚 where is the projection measure. If is a bounded the proof of [11, Lemma B.1]. Let 𝐷1 ={𝑦:|𝑦|<1}, [0, ∞) −1 Borel measurable function on ,theoperator 𝐷2 ={𝑦:|𝑦|≥1and |𝑥𝑦 | ≤ |𝑥|/2𝛾} and 𝐷3 ={𝑦: ∞ −1 |𝑦| ≥ 1 and |𝑥𝑦 | > |𝑥|/2𝛾}.Let𝑦 󳨃→𝑃𝑥,𝜙(𝑦) be the left 𝑚 (L) = ∫ 𝑚 (𝜆) 𝑑𝐸 (𝜆) (11) 𝜙 𝑥 𝐿 0 Taylor polynomial of at of homogeneous degree (see [3, pp. 26-27]). Then using vanishing moments of 𝜓 2 is bounded on 𝐿 (𝐺), and commutes with left translations. 󵄨 󵄨 Thus, by the Schwartz kernel theorem, there exists a tempered 󵄨𝜙∗(𝐷2ℓ 𝜓) (𝑥)󵄨 distribution 𝑀 on 𝐺 such that 󵄨 󵄨 󵄨 −1 −1 󵄨 󵄨 󵄨 ≤ ∫ 󵄨𝜙(𝑥𝑦 )−𝑃𝑥,𝜙 (𝑦 )󵄨 󵄨(𝐷2ℓ 𝜓) (𝑦)󵄨 𝑑𝑦 𝑚 (L) 𝑓 = 𝑓 ∗ 𝑀, ∀𝑓∈ S (𝐺) . (12) 󵄨 󵄨 (15) 𝜆=0 Note that the point may be neglected in the spectral ≡ ∫ + ∫ + ∫ . {0} resolution, since the projection measure of is zero (see 𝐷1 𝐷2 𝐷3 [8,p.76]).Consequentlyweshouldregard𝑚 as functions on R+ ≡(0,∞) [0, ∞) rather than on . For 𝑦∈𝐷1, the stratified Taylor formula (cf. [3,Corollary S(R+) R+ Let denote the space of restrictions to of 1.44]) yields that, with 𝑏 asuitablepositiveconstant, functions in S(R).AnimportantfactprovedbyHulanicki 󵄨 󵄨 [9]isasinthefollowinglemma. 󵄨 −1 −1 󵄨 󵄨𝜙(𝑥𝑦 )−𝑃𝑥,𝜙 (𝑦 )󵄨 + Lemma 1. If 𝑚∈S(R ) then the distribution kernel 𝑀 of 󵄨 󵄨𝐿+1 󵄨 󵄨 ≲ 󵄨𝑦󵄨 󵄨(𝑋𝐼𝜙) (𝑥𝑧)󵄨 𝑚(L) is in S(𝐺). 󵄨 󵄨 sup 󵄨 󵄨 |𝑧|≤𝑏𝐿+1|𝑦| 𝑑(𝐼)=𝐿+1 Moreover, from the proof of [10,Corollary1]weseethat + 𝑚 S(R ) 󵄨 󵄨𝐿+1 󵄨 󵄨𝐿+1 if is a function in which vanishes identically near 󵄩 󵄩 󵄨𝑦󵄨 󵄩 󵄩 󵄨𝑦󵄨 𝑀 ≲ 󵄩𝜙󵄩 󵄨 󵄨 ≲ 󵄩𝜙󵄩 󵄨 󵄨 , the origin, then is a Schwartz function with all moments 󵄩 󵄩(𝐿+1,𝑁) sup 𝑁 󵄩 󵄩(𝐿+1,𝑁) 𝑁 𝐿+1 1+ 𝑥𝑧 1+ 𝑥 vanishing. |𝑧|≤𝑏 ( | |) ( | |) In the sequel, if not other specified, we will generally use (16) + Greak alphabets with hats to denote functions in S(R ),and 𝐿+1 𝐿+1 𝐿+1 use Greek alphabets without hats to denote the associated since 2𝑏 (1 + |𝑥𝑧|) ≥ 2𝑏 +|𝑥𝑧|≥2𝑏 + |𝑥|/𝛾 − |𝑧| ≥ 𝜙∈̂ S(R+) 𝐿+1 𝐿+1 distribution kernels; for example, for we shall 𝑏 + |𝑥|/𝛾 ≳ (1 + |𝑥|)/𝛾 if |𝑧| ≤ 𝑏 .Hencewehave ̂ denote by 𝜙 the distribution kernel of the operator 𝜙(L), L 󵄨 󵄨𝐿+1 where is a sub-Laplacian fixed in the context. 󵄩 󵄩 󵄩 󵄩 2ℓΔ 󵄨𝑦󵄨 ∫ ≲󵄩𝜙󵄩 󵄩𝜓󵄩 ∫ 󵄨 󵄨 𝑑𝑦 󵄩 󵄩(𝐿+1,𝑁)󵄩 󵄩(0,𝑁) 𝑁 ℓ 󵄨 󵄨 𝑁 𝐷1 (1+|𝑥|) 𝐷1 (1 + 2 󵄨𝑦󵄨) 3. Homogeneous Triebel-Lizorkin Spaces on 󵄨 󵄨 󵄨 󵄨𝐿+1 Stratified Lie Groups −ℓ(𝐿+1) 󵄨2ℓ𝑦󵄨 󵄩 󵄩 󵄩 󵄩 2 󵄨 󵄨 ℓ 1 = 󵄩𝜙󵄩 󵄩𝜓󵄩 ∫ 𝑑(2 𝑦) ℎ 𝐺 𝑡>0 𝐿 󵄩 󵄩(𝐿+1,𝑁)󵄩 󵄩(0,𝑁) 𝑁 ℓ 󵄨 󵄨 𝑁 For any function on and , we define the - (1+|𝑥|) 𝐺 (1 + 2 󵄨𝑦󵄨) normalized dilation of ℎ by 2−ℓ(𝐿+1) Δ 󵄩 󵄩 󵄩 󵄩 𝐷 ℎ (𝑥) =𝑡 ℎ (𝑡𝑥) . ≲ 󵄩𝜙󵄩 󵄩𝜓󵄩 , 𝑡 (13) (𝐿+1,𝑁) (0,𝑁) (1+|𝑥|)𝑁 Before we introduce the homogeneous Triebel-Lizorkin (17) spaces on stratified Lie groups, we prove the following basic estimate, which is a generalization of [11,LemmaB.1]andwill where for the last inequality we used [3,Corollary1.17]and be frequently used throughout this paper. that 𝑁−𝐿−1≥Δ+1. 4 Journal of Function Spaces and Applications

−1 2−ℓ(𝑁−Δ) For 𝑦∈𝐷2,wehave|𝑦| ≥ |𝑥|/𝛾−|𝑥𝑦 | ≥ |𝑥|/𝛾−|𝑥|/2𝛾 = 󵄩 󵄩 󵄩 󵄩 −1 ≲ 󵄩𝜙󵄩 󵄩𝜓󵄩 |𝑥|/2𝛾. On the other hand, |𝑦| ≤ 𝛾(|𝑥| +|𝑥𝑦 |) ≤ 𝛾(|𝑥| + (𝐿,𝑁) (0,𝑁) (1+|𝑥|)𝑁 |𝑥|/2𝛾) = (𝛾 + (1/2))|𝑥|.Thus,wehave 1 ×[∫ 𝑑𝑦 󵄨 󵄨 𝑁 𝐺 (1 + 󵄨𝑦󵄨) ℓ 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 2 (1 + 󵄨𝑦󵄨) 2ℓ (1+|𝑥|) |𝑥|𝐿 1+2ℓ 󵄨𝑦󵄨 ≥2ℓ 󵄨𝑦󵄨 ≥ 󵄨 󵄨 ≥ , + ∫ 𝑑𝑦] 󵄨 󵄨 󵄨 󵄨 𝑁 2 4𝛾 (1+|𝑥|) |𝑦|≤𝛾(|𝑥|+|𝑥|/2𝛾) (18) 1 𝐿 −ℓ(𝑁−Δ) 𝐿+Δ 󵄨 󵄨𝐿 𝐿 󵄩 󵄩 󵄩 󵄩 2 |𝑥| 󵄨𝑦󵄨 ≤(𝛾+ ) |𝑥| . ≲ 󵄩𝜙󵄩 󵄩𝜓󵄩 [1 + ] 2 (𝐿,𝑁) (0,𝑁) (1+|𝑥|)𝑁 (1+|𝑥|)𝑁

−ℓ𝐿 󵄩 󵄩 󵄩 󵄩 2 ≲ 󵄩𝜙󵄩 󵄩𝜓󵄩 , (𝐿,𝑁) (0,𝑁) (1+|𝑥|)𝑁 Also, we note that by [12, Proposition 20.3.14] the left Taylor (20) polynomial 𝑃𝑥,𝜙 is of the form Δ whereweusedthat|{𝑦 : |𝑦| ≤ 𝛾(|𝑥| + |𝑥|/2𝛾)}| ∼|𝑥| and 𝑁−Δ>𝐿. −1 For 𝑦∈𝐷3 we have |𝑥𝑦 | > |𝑥|/2𝛾,and,hence 𝑃𝑥,𝜙 (𝑦) 󵄩 󵄩 󵄩 󵄩 𝐿 ℎ ∫ ≲ 󵄩𝜙󵄩 󵄩𝜓󵄩 𝜂𝑖 (𝑦)...𝜂𝑖 (𝑦) 󵄩 󵄩(𝐿,𝑁)󵄩 󵄩(0,𝑁) 1 𝑘 𝐷 =𝜙(𝑥) + ∑ ∑ ∑ 𝑋𝑖 ⋅⋅⋅𝑋𝑖 𝜙 (𝑥) , 3 𝑘! 1 𝑘 ℎ=1 𝑘=1 1≤𝑖1,...,𝑖𝑘≤𝑛 󵄨 󵄨ℎ 𝑑 +⋅⋅⋅+𝑑 =ℎ 1 󵄨𝑦󵄨 𝑖1 𝑖𝑘 × ∫ [ + ∑ 󵄨 󵄨 ] 󵄨 󵄨 𝑁 𝑁 (19) 𝐷 󵄨 −1󵄨 (1+|𝑥|) 3 (1 + 󵄨𝑥𝑦 󵄨) 0≤ℎ≤𝐿 2ℓΔ × 𝑑𝑦 󵄨 ℓ 󵄨 𝑁 (1 + 󵄨2 𝑦󵄨) where the integers 𝑑1 ≤ ⋅⋅⋅ ≤ 𝑑𝑛 are given according to that 𝑋𝑗 ∈𝑉𝑑 . From these remarks, it follows that ℓΔ 󵄨 󵄨𝐿 𝑗 󵄩 󵄩 󵄩 󵄩 2 󵄨𝑦󵄨 ≲ 󵄩𝜙󵄩 󵄩𝜓󵄩 ∫ 󵄨 󵄨 𝑑𝑦 󵄩 󵄩(𝐿,𝑁)󵄩 󵄩(0,𝑁) 𝑁 ℓ 󵄨 󵄨 𝑁 (1+|𝑥|) |𝑦|≥1 (1 + 2 󵄨𝑦󵄨) 󵄨 󵄨𝐿 −ℓ𝐿 󵄨2ℓ𝑦󵄨 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 2 󵄨 󵄨 ℓ ∫ ≲ 󵄩𝜙󵄩 󵄩𝜓󵄩 ≤ 󵄩𝜙󵄩 󵄩𝜓󵄩 ∫ 󵄨 󵄨 𝑑(2 𝑦) 󵄩 󵄩(𝐿,𝑁)󵄩 󵄩(0,𝑁) 󵄩 󵄩(𝐿,𝑁)󵄩 󵄩(0,𝑁) 𝑁 󵄨 󵄨 𝑁 𝐷 (1+|𝑥|) 𝐺 ℓ 󵄨 󵄨 2 (1 + 2 󵄨𝑦󵄨) 󵄨 󵄨ℎ 1 󵄨𝑦󵄨 󵄩 󵄩 󵄩 󵄩 2−ℓ𝐿 × ∫ [ + ∑ 󵄨 󵄨 ] ≲ 󵄩𝜙󵄩 󵄩𝜓󵄩 , 󵄨 󵄨 𝑁 𝑁 󵄩 󵄩(𝐿,𝑁)󵄩 󵄩(0,𝑁) 𝑁 𝐷 󵄨 −1󵄨 (1+|𝑥|) (1+|𝑥|) 2 (1 + 󵄨𝑥𝑦 󵄨) 0≤ℎ≤𝐿 (21) 2ℓΔ × 𝑑𝑦 󵄨 󵄨 𝑁 (1 + 󵄨2ℓ𝑦󵄨) where for the last inequality we used [3,Corollary1.17]and 󵄨 󵄨 𝑁−𝐿>Δ+1. 󵄩 󵄩 󵄩 󵄩 1 |𝑥|𝐿 Combining the above estimates, we arrive at ≲ 󵄩𝜙󵄩 󵄩𝜓󵄩 ∫ [ + ] 󵄩 󵄩(𝐿,𝑁)󵄩 󵄩(0,𝑁) 󵄨 󵄨 𝑁 𝑁 𝐷 󵄨 −1󵄨 (1+|𝑥|) 1 2 (1 + 󵄨𝑥𝑦 󵄨) 󵄨 󵄨 󵄩 󵄩 󵄩 󵄩 −ℓ𝐿 󵄨𝜙∗(𝐷2ℓ 𝜓) (𝑥)󵄨 ≤ 󵄩𝜙󵄩 󵄩𝜓󵄩 2 . (𝐿+1,𝑁) (𝐿+1,𝑁) (1+|𝑥|)𝑁 2ℓΔ × 𝑑𝑦 (22) [2ℓ (1+|𝑥|)]𝑁 This is exactly what we need. 󵄩 󵄩 󵄩 󵄩 2−ℓ(𝑁−Δ) = 󵄩𝜙󵄩 󵄩𝜓󵄩 Z(𝐺) 󵄩 󵄩(𝐿,𝑁)󵄩 󵄩(0,𝑁) 𝑁 Let denote the space of Schwartz functions with all (1+|𝑥|) moments vanishing. We then consider Z(𝐺) as a subspace of S(𝐺) 1 ,includingthetopology.Itisshownin[5, Lemma 3.3] ×[∫ 𝑑𝑦 that Z(𝐺) is a closed subspace of S(𝐺),andthetopologydual 󵄨 󵄨 𝑁 󸀠 𝐺 (1 + 󵄨𝑥𝑦−1󵄨) Z (𝐺) of Z(𝐺) can be canonically identified with the factor 󵄨 󵄨 󸀠 space S (𝐺)/P. |𝑥|𝐿 + ∫ 𝑑𝑦] We now have the following Calderon´ type reproducing 𝑁 (1+|𝑥|) |𝑥𝑦−1|≤|𝑥|/2𝛾 formula. Journal of Function Spaces and Applications 5

̂ Lemma 3. Suppose L is a sub-Laplacian on 𝐺,and𝜙∈ Definition 4. Let 𝛼∈R, 0<𝑝<∞and 0<𝑞≤∞.LetL S(R+) ̂ ̇𝛼 ̂ is a function with compact support, vanishing identi- be a sub-Laplacian on 𝐺 and 𝜙∈A. We define 𝐹𝑝,𝑞(L, 𝜙) as 󸀠 cally near the origin, and satisfying the space of all 𝑓∈S (𝐺)/P such that ̂ −2𝑗 + 󵄩 󵄩 ∑𝜙(2 𝜆) = 1, ∀𝜆 ∈ R . 󵄩 1/𝑞󵄩 (23) 󵄩 𝑞 󵄩 𝑗∈Z 󵄩 󵄩 󵄩 𝑗𝛼 󵄨 󵄨 󵄩 󵄩𝑓󵄩𝐹̇𝛼 (L,𝜙)̂ ≡ 󵄩(∑(2 󵄨𝑓∗(𝐷2𝑗 𝜙)󵄨) ) 󵄩 <∞, (29) 𝑝,𝑞 󵄩 󵄩 󵄩 𝑗∈Z 󵄩 𝑝 Then for all 𝑔∈Z(𝐺),itholdsthat 𝐿 with the usual modification for 𝑞=∞. 𝑔= lim ∑ 𝑔∗(𝐷2𝑗 𝜙) , 𝑘→∞ (24) |𝑗|≤𝑘 We then introduce the Peetre type maximal functions: 󸀠 ̂ + Given 𝑓∈S (𝐺), 𝜙∈S(R ), L asub-Laplacian,and with convergence in Z(𝐺). Duality entails that, for all 𝑢∈ 𝑎, 𝑡, >0 we define S󸀠(𝐺)/P , 󵄨 󵄨 󵄨𝑓∗(𝐷−1 𝜙) (𝑥)󵄨 𝑀∗ (𝑓, L, 𝜙)̂ (𝑥) = 󵄨 𝑡 󵄨 𝑎,𝑡 sup −1 󵄨 −1 󵄨 𝑎 𝑢= lim ∑ 𝑢∗(𝐷2𝑗 𝜙) , 𝑦∈𝐺 (1 + 𝑡 󵄨𝑦 𝑥󵄨) 𝑘→∞ (25) 󵄨 󵄨 |𝑗|≤𝑘 󵄨 󵄨 (30) ∗∗ 󵄨𝑋𝑘 [𝑓 ∗ (𝐷𝑡−1 𝜙)] (𝑥)󵄨 󸀠 𝑀 (𝑓, L, 𝜙)̂ (𝑥) = . S (𝐺)/P 𝑎,𝑡 sup −1 󵄨 −1 󵄨 𝑎 and the convergence is in . 𝑦∈𝐺 (1 + 𝑡 󵄨𝑦 𝑥󵄨) 1≤𝑘≤] Proof. Firstnotethatthe2-homogeneity of L implies that ̂ −2𝑗 Lemma 5. L 𝜙∈̂ A the distribution kernel of 𝜙(2 L) coincides with 𝐷2𝑗 𝜙.Let Suppose is a sub-Laplacian and .Then 𝑛 𝑎>0 𝐶>0 𝐼∈N and 𝑁∈N be arbitrarily chosen. Then take 𝐿∈N for every there is a constant such that for all 𝑓∈S󸀠(𝐺)/P 𝑗∈Z 𝑥∈𝐺 such that 𝐿≥𝑁+𝑑(𝐼)+1.Sinceboth𝑔 and 𝜙 are Schwartz ,all ,andall , functions with all moments vanishing, it follows by Lemma 2 ∗∗ ̂ 𝑗 ∗ ̂ 𝑀 −𝑗 (𝑓, L, 𝜙) (𝑥) ≤𝐶2𝑀 −𝑗 (𝑓, L, 𝜙) (𝑥) . (31) that 𝑎,2 𝑎,2 𝑁 󵄨 𝐼 󵄨 Proof. Because of (28)itispossibletofindafunc- (1+|𝑥|) 󵄨𝑋 [𝑔∗(𝐷2𝑗 𝜙)] (𝑥)󵄨 + −2 2 󵄨 󵄨 tion 𝜓∈̂ S(R ) supported in [2 ,2 ] such that 󵄨 󵄨 ̂ −2𝑗 −2𝑗 + ̂ 𝑑(𝐼)𝑗 𝑁 󵄨 𝐼 󵄨 ∑ 𝜙(2 𝜆)𝜓(2̂ 𝜆) = 1 for 𝜆∈R .Set𝜁(𝜆) = =2 (1+|𝑥|) 󵄨𝑔∗[𝐷2𝑗 (𝑋 𝜙)] (𝑥)󵄨 𝑗∈Z 1 ̂ −2𝑗 ̂ −2𝑗 + ̂ ̂ ̂ ∑𝑗=−1 𝜙(2 𝜆)𝜓(2 𝜆), 𝜆∈R .Then𝜙(𝜆) = 𝜙(𝜆)𝜁(𝜆) for 𝑑(𝐼)𝑗 −|𝑗|𝐿 (𝑗∧0)Δ (𝑗∧0) −(𝐿+Δ+2) 𝑁 + ≤𝐶2 2 2 (1 + 2 |𝑥|) (1+|𝑥|) all 𝜆∈R .Consequently,for𝑗∈Z and 1≤𝑘≤], 󵄨 󵄨 −𝑁 󵄨𝑋 [𝑓 ∗ (𝐷 𝑗 𝜙)] (𝑦)󵄨 ≤𝐶2𝑑(𝐼)𝑗2−|𝑗|𝐿2(𝑗∧0)Δ(1 + 2(𝑗∧0) |𝑥|) (1+|𝑥|)𝑁 󵄨 𝑘 2 󵄨 󵄨 󵄨 = 󵄨𝑋𝑘 [𝑓 ∗ (𝐷2𝑗 𝜙) ∗ (𝐷2𝑗 𝜁)] (𝑦)󵄨 ≤𝐶2−|𝑗|(𝐿−𝑁−𝑑(𝐼)), 󵄨 󵄨 (26) 󵄨 󵄨 󵄨 −1 󵄨 ≤ ∫ 󵄨𝑓∗(𝐷2𝑗 𝜙) (𝑧)󵄨 󵄨[𝑋𝑘 (𝐷2𝑗 𝜁)] (𝑧 𝑦)󵄨 𝑑𝑧 𝐶 where the constant is a suitable multiple of 󵄨 󵄨 ‖𝑔‖ ‖𝑋𝐼𝜙‖ 𝑗(Δ+1) 󵄨 󵄨 󵄨 𝑗 −1 󵄨 (𝐿+1,𝐿+Δ+2) (𝐿+1,𝐿+Δ+2). This implies that =2 ∫ 󵄨𝑓∗(𝐷2𝑗 𝜙) (𝑧)󵄨 󵄨(𝑋𝑘𝜁) (2 (𝑧 𝑦))󵄨 𝑑𝑧 𝑁 𝐼 󵄨 󵄨 (32) ∑𝑗∈Z(1 + |𝑥|) |𝑋 [𝑔 ∗ (𝐷2𝑗 𝜙)](𝑥)| converges uniformly 𝑥 𝐼∈N𝑛 𝑁∈N 𝑗 ∗ ̂ in ,forevery and every .Consequentlythere ≲2𝑀𝑎,2−𝑗 (𝑓, L, 𝜙) (𝑥) exists ℎ∈S(𝐺) such that ∑|𝑗|≤𝑘 𝑔∗(𝐷2𝑗 𝜙) converges in the 󵄨 󵄨 𝑎 󵄨 󵄨 −𝑎−(Δ+1) topology of S(𝐺) to ℎ,as𝑘→∞. On the other hand, by 𝑗Δ 𝑗 󵄨 −1 󵄨 𝑗 󵄨 −1 󵄨 × ∫ 2 (1 + 2 󵄨𝑧 𝑥󵄨) (1 + 2 󵄨𝑧 𝑦󵄨) 𝑑𝑧 (23) and the spectral theorem (cf. [13, Theorem VII.2]), 󵄨 󵄨 𝑎 −2𝑗 𝑗 ∗ ̂ 𝑗 󵄨 −1 󵄨 ̂ ≲2𝑀 −𝑗 (𝑓, L, 𝜙) (𝑥) (1 + 2 󵄨𝑦 𝑥󵄨) , 𝑔=∑𝜙(2 L)𝑔= ∑𝑔∗(𝐷2𝑗 𝜙) 𝑎,2 󵄨 󵄨 𝑗∈Z 𝑗∈Z (27) where for the last inequality we used [3,Corollary1.17]and 𝑗 −1 𝑎 𝑗 −1 −𝑎 𝑗 −1 𝑎 2 (1+2 |𝑧 𝑥|) (1+2 |𝑧 𝑦|) ≲(1+2|𝑦 𝑥|) holds in 𝐿 -norm. Therefore, ℎ=𝑔, which completes the that . Dividing (1 + 2𝑗|𝑦−1𝑥|)𝑎 proof. both sides of the above estimate by ,andthen taking the supremum over 𝑦∈𝐺and 1≤𝑘≤],weobtain ̂ + Let A denote the class of all functions 𝜙 in S(R ) the desired estimate. satisfying ̂ Lemma 6. Suppose L is a sub-Laplacian, 𝜙∈A,and𝑟>0. ̂ −2 2 󸀠 supp 𝜙⊂[2 ,2 ], Then there exists a constant 𝐶 such that for all 𝑓∈S (𝐺)/P, all 𝑗∈Z,andall𝑥∈𝐺, 󵄨 󵄨 3 2 5 2 (28) 󵄨 ̂ 󵄨 1/𝑟 󵄨𝜙 (𝜆)󵄨 ≥𝐶>0 for 𝜆∈[( ) ,( ) ]. ∗ ̂ 󵄨 󵄨𝑟 󵄨 󵄨 5 3 𝑀𝑎,2−𝑗 (𝑓, L, 𝜙) (𝑥) ≤ 𝐶 [𝑀󵄨 ( 𝑓∗(𝐷2𝑗 𝜙)󵄨 ) (𝑥)] , (33) 6 Journal of Function Spaces and Applications

𝑎 where 𝑎=Δ/𝑟,and𝑀 is the Hardy-Littlewood maximal that 𝐶𝜀(1 + 𝑏𝜀) <1/2), and taking the supremum over 𝑦0 ∈ operator on 𝐺. 𝐺, we get the desired estimate.

∞ (1) (2) Proof. Let 𝑔∈𝐶 (𝐺) and 𝑦0 ∈𝐺.Thestratifiedmeanvalue Theorem 7. Suppose L , L are any two sub-Laplacians 𝛿>0 ̂(1) ̂(2) theorem (cf. [3, Theorem 1.41]) gives that for every and on 𝐺,and𝜙 , 𝜙 are any two functions in A.Then,for𝛼∈ 𝑦∈𝐺 |𝑦−1𝑦 |<𝛿 every with 0 , R, 0<𝑝<∞and 0<𝑞≤∞,wehavethe(quasi-)norm 󵄨 󵄨 equivalence 󵄨𝑔(𝑦)−𝑔(𝑦0)󵄨 󵄨 −1 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄩 󵄩 󵄩 󵄩 󸀠 ≤𝐶󵄨𝑦0 𝑦󵄨 sup 󵄨(𝑋𝑘𝑔) 0(𝑦 𝑧)󵄨 󵄩 󵄩 󵄩 󵄩 󵄨 󵄨 󵄨 󵄨 󵄩𝑓󵄩 ̇𝛼 (1) ̂(1) ∼ 󵄩𝑓󵄩 ̇𝛼 (2) ̂(2) ,𝑓∈S (𝐺) /P. −1 󵄩 󵄩𝐹𝑝,𝑞(L ,𝜙 ) 󵄩 󵄩𝐹𝑝,𝑞(L ,𝜙 ) (37) |𝑧|≤𝑏|𝑦0 𝑦| 1≤𝑘≤] (34) 󵄨 󵄨 ≤𝐶𝛿sup 󵄨(𝑋𝑘𝑔) 0(𝑦 𝑧)󵄨 , Proof. First we note that if 𝑎>Δ/min{𝑝, 𝑞} then we have |𝑧|≤𝑏𝛿 1≤𝑘≤] 󵄩 󵄩 𝑏 󵄩 1/𝑞󵄩 where isasuitablepositiveconstant.Hencewehave 󵄩 󵄨 󵄨 𝑞 󵄩 󵄩(∑(2𝑗𝛼 󵄨𝑀∗ (𝑓, L(𝑖), 𝜙̂(𝑖))󵄨) ) 󵄩 󵄨 󵄨 󵄨 󵄨 󵄩 󵄨 𝑎,2−𝑗 󵄨 󵄩 󵄨𝑔(𝑦 )󵄨 ≤𝐶𝛿 󵄨𝑋 𝑔(𝑦)󵄨 󵄩 𝑗∈Z 󵄩 󵄨 0 󵄨 sup 󵄨 𝑘 󵄨 󵄩 󵄩𝐿𝑝 𝑦−1𝑦 ≤𝑏𝛿 | 0| (38) 1≤𝑘≤] 󵄩 󵄩 󵄩 1/𝑞󵄩 (35) 󵄩 󵄨 󵄨 𝑞 󵄩 󵄩 𝑗𝛼 󵄨 (𝑖) 󵄨 󵄩 1/𝑟 ∼ 󵄩(∑(2 󵄨𝑓∗(𝐷2𝑗 𝜙 )󵄨) ) 󵄩 −𝑎 󵄨 󵄨𝑟 󵄩 󵄨 󵄨 󵄩 +𝛿 (∫ 󵄨𝑔(𝑦)󵄨 𝑑𝑦) . 󵄩 𝑗∈Z 󵄩 𝑝 −1 󵄩 󵄩𝐿 |𝑦 𝑦0|≤𝛿

𝑗 −1 𝑎 Putting 𝑔=𝑓∗(𝐷2𝑗 𝜙), dividing both sides by (1+2 |𝑦 𝑥|) , (𝑖) 0 for 𝑖=1,2,where𝜙 denotes the distribution kernel of and using Lemma 5,wehave ̂(𝑖) (𝑖) 𝜙 (L ).Indeed,thedirection“≳”of(38)isobvious,andthe 󵄨 󵄨 󵄨𝑓∗(𝐷2𝑗 𝜙) 0(𝑦 )󵄨 other direction follows by Lemma 6 and the Fefferman-Stein 𝑗 󵄨 −1 󵄨 𝑎 vector-valued maximal inequality on spaces of homogeneous (1 + 2 󵄨𝑦0 𝑥󵄨) type (see, e.g., [14]). Thus, to prove (37), it suffices to show 󵄨 󵄨 𝑎 󵄨 󵄨 𝑗 󵄨 −1 󵄨 that 󵄨𝑋 [𝑓 ∗ (𝐷 𝑗 𝜙)] (𝑦)󵄨 (1 + 2 󵄨𝑦 𝑥󵄨) ≤𝐶𝛿 󵄨 𝑘 2 󵄨 󵄨 󵄨 sup 𝑗 󵄨 −1 󵄨 𝑎 𝑗 󵄨 −1 󵄨 𝑎 −1 (1 + 2 󵄨𝑦 𝑥󵄨) (1 + 2 󵄨𝑦 𝑥󵄨) |𝑦 𝑦0|≤𝑏𝛿 󵄨 󵄨 󵄨 0 󵄨 󵄩 󵄩 󵄩 1/𝑞󵄩 1≤𝑘≤] 󵄩 󵄨 󵄨 𝑞 󵄩 󵄩 𝑗𝛼 󵄨 ∗ (1) ̂(1) 󵄨 󵄩 󵄩(∑(2 󵄨𝑀𝑎,2−𝑗 (𝑓, L , 𝜙 )󵄨) ) 󵄩 −𝑎 1/𝑟 󵄩 󵄨 󵄨 󵄩 𝛿 󵄩 𝑗∈Z 󵄩 󵄨 󵄨𝑟 󵄩 󵄩𝐿𝑝 + 󵄨 󵄨 𝑎 (∫ 󵄨𝑓∗(𝐷2𝑗 𝜙) (𝑦)󵄨 𝑑𝑦) 𝑗 󵄨 −1 󵄨 −1 (39) (1 + 2 󵄨𝑦 𝑥󵄨) |𝑦 𝑦0|≤𝛿 󵄩 󵄩 󵄨 0 󵄨 󵄩 1/𝑞󵄩 󵄩 𝑗𝛼 󵄨 ∗ (2) (2) 󵄨 𝑞 󵄩 𝑗 󵄨 −1 󵄨 𝑎 ∗∗ 󵄩 󵄨 ̂ 󵄨 󵄩 ≤𝐶𝛿 (1 + 2 󵄨𝑦 𝑦 󵄨) 𝑀 (𝑓, L, 𝜙)̂ (𝑥) ∼ 󵄩(∑(2 󵄨𝑀𝑎,2−𝑗 (𝑓, L , 𝜙 )󵄨) ) 󵄩 . sup 󵄨 0󵄨 𝑎,2−𝑗 󵄩 󵄨 󵄨 󵄩 −1 󵄩 𝑗∈Z 󵄩 𝑝 |𝑦 𝑦0|≤𝑏𝛿 󵄩 󵄩𝐿 𝛿−𝑎 + 󵄨 󵄨 𝑎 ̂(1) + (1 + 2𝑗 󵄨𝑦−1𝑥󵄨) To this end, let 𝜓 be a function in S(R ) with support 󵄨 0 󵄨 −2 2 ̂(1) −2𝑗 (1) −2𝑗 in [2 ,2 ] such that ∑𝑗∈Z 𝜙 (2 𝜆)𝜓̂ (2 𝜆) = 1 for 𝜆∈ 1/𝑟 + 󸀠 󵄨 󵄨𝑟 R .For𝑓∈S (𝐺)/P,byLemma 3 we have ×(∫ 󵄨𝑓∗(𝐷2𝑗 𝜙) (𝑦)󵄨 𝑑𝑦) −1 −1 |𝑦 𝑥|≤𝛾(𝛿+|𝑦0 𝑥|)

𝑗 𝑎 𝑗 ∗ (1) (1) ≤𝐶𝛿(1+2𝑏𝛿) 2 𝑀 (𝑓, L, 𝜙)̂ 𝑥 𝑓=∑𝑓∗(𝐷2𝑗 𝜙 )∗(𝐷2𝑗 𝜓 ) 𝑎,2−𝑗 ( ) (40) 𝑗∈Z 󵄨 󵄨 𝑎 𝛿−𝑎𝛾𝑎(𝛿 + 󵄨𝑦−1𝑥󵄨) 󵄨 0 󵄨 󵄨 󵄨𝑟 1/𝑟 + [𝑀󵄨 ( 𝑓∗(𝐷 𝑗 𝜙)󵄨 ) (𝑥)] 𝑗 󵄨 −1 󵄨 𝑎 󵄨 2 󵄨 󸀠 (1) (1 + 2 󵄨𝑦0 𝑥󵄨) with convergence in S (𝐺)/P.Here𝜓 is the distribution (1) (1) (2) kernel of 𝜓̂ (L ). Hence, since 𝜙 ∈ Z(𝐺),wehavethe ≤𝐶𝜀(1+𝑏𝜀)𝑎𝑀∗ (𝑓, L, 𝜙)̂ (𝑥) 𝑎,2−𝑗 pointwise representation 𝑎 −1 𝑎 󵄨 󵄨𝑟 1/𝑟 +𝛾 (1 + 𝜀 ) [𝑀󵄨 ( 𝑓∗(𝐷2𝑗 𝜙)󵄨 ) (𝑥)] , (2) (1) (1) (36) 𝑓∗(𝐷2ℓ 𝜙 )(𝑦)= ∑𝑓∗(𝐷2𝑗 𝜙 )∗(𝐷2𝑗 𝜓 ) 𝑗∈Z (41) 𝛿=2−𝑗𝜀 (1+2𝑗𝜀−1|𝑦−1𝑥|)/(1+ wherewehaveset andusedthat 0 (2) 𝑗 −1 −1 ∗(𝐷2ℓ 𝜙 )(𝑦), ∀𝑦∈𝐺. 2 |𝑦0 𝑥|) ≲ 1 + 𝜀 . Finally, taking 𝜀 sufficiently small (such Journal of Function Spaces and Applications 7

̇𝛼 ̂ It follows that Remark 8. From Theorem 7 we see that the space 𝐹𝑝,𝑞(L, 𝜙) 󵄨 (2) 󵄨 L 𝜙̂ 󵄨𝑓∗(𝐷 ℓ 𝜙 )(𝑦)󵄨 is actually independent of the choice of and .Thus,in 󵄨 2 󵄨 ̂ what follows we don’t specify the choice of L and 𝜙 and write 󵄨 󵄨 𝐹̇𝛼 (𝐺) 𝐹̇𝛼 (L, 𝜙)̂ 󵄨 (1) 󵄨 𝑝,𝑞 instead of 𝑝,𝑞 . Henceforth we shall fix any sub- ≤ ∑ ∫ 󵄨𝑓∗(𝐷2𝑗 𝜙 ) (𝑧)󵄨 𝑗∈Z Laplacian L. Moreover, for the sake of briefness, we will write 𝑀∗ (𝑓, 𝜙)(𝑥)̂ 𝑀∗ (𝑓, L, 𝜙)(𝑥)̂ 󵄨 󵄨 𝑎,𝑡 instead of 𝑎,𝑡 . 󵄨 (1) (2) −1 󵄨 × 󵄨(𝐷2𝑗 𝜓 )∗(𝐷2ℓ 𝜙 )(𝑧 𝑦)󵄨 𝑑𝑧 Proposition 9. For 𝛼∈R, 0<𝑝<∞and 0<𝑞≤∞, ∗ (1) ̂(1) (42) Z(𝐺) 󳨅→ 𝐹̇𝛼 (𝐺) 󳨅→ ≤ ∑𝑀𝑎,2−𝑗 (𝑓, L , 𝜙 )(𝑦) one has the continuous inclusion maps 𝑝,𝑞 󸀠 𝑗∈Z S (𝐺)/P. 𝑎 󵄨 󵄨 𝑗 󵄨 (1) (2) 󵄨 𝑔∈Z(𝐺) 𝜙∈̂ A 𝑁 ≥ (Δ+ 1)/𝑝 × ∫ (1 + 2 |𝑧|) 󵄨(𝐷2𝑗 𝜓 )∗(𝐷2ℓ 𝜙 ) (𝑧)󵄨 𝑑𝑧 Proof. Let and .Choose and 𝐿≥𝑁+|𝛼|+1.Sinceboth𝑔 and 𝜙 are Schwartz functions ∗ (1) ̂(1) with all moments vanishing, it follows by Lemma 2 that ≡ ∑𝑀𝑎,2−𝑗 (𝑓, L , 𝜙 )(𝑦)𝐼𝑗,ℓ. 𝑗∈Z 󵄨 󵄨 󵄨𝑔∗(𝐷2𝑗 𝜙) (𝑥)󵄨 (1) (2) 󵄨 󵄨 Since both 𝜓 and 𝜙 are Schwartz functions with all 󵄩 󵄩 −|𝑗|𝐿 (𝑗∧0)Δ 𝑗∧0 −(𝐿+Δ+2) moments vanishing, we can use Lemma 2 to estimate that, ≤𝐶󵄩𝑔󵄩(𝐿+1,𝐿+Δ+2)2 2 (1 + 2 |𝑥|) with 𝐿 sufficiently large, 󵄩 󵄩 −𝑁 ≤𝐶󵄩𝑔󵄩 2−|𝑗|𝐿2(𝑗∧0)Δ(1 + 2𝑗∧0 |𝑥|) 𝑗 𝑎 −|𝑗−ℓ|𝐿 (𝑗∧ℓ)Δ 󵄩 󵄩(𝐿+1,𝐿+Δ+2) 𝐼𝑗,ℓ ≤𝐶∫ (1 + 2 |𝑧|) 2 2 󵄩 󵄩 −|𝑗|(𝐿−𝑁) −𝑁 ≤𝐶󵄩𝑔󵄩(𝐿+1,𝐿+Δ+2)2 (1+|𝑥|) −(𝐿+Δ+2) ×(1+2𝑗∧ℓ |𝑧|) 𝑑𝑧 (46)

𝑗 𝑎 −|𝑗−ℓ|𝐿 (𝑗∧ℓ)Δ ≤𝐶∫ (1 + 2 |𝑧|) 2 2 for all 𝑗∈Z and all 𝑥∈𝐺,wheretheconstant𝐶 is a suitable (43) multiple of ‖𝜙‖(𝐿+1,𝐿+Δ+2). This together with3 [ ,Corollary −(𝑎+Δ+1) ×(1+2𝑗∧ℓ |𝑧|) 𝑑𝑧 1.17] give that

−(Δ+1) 1/𝑞 ≤𝐶2−|𝑗−ℓ|(𝐿−𝑎) ∫ 2(𝑗∧ℓ)Δ (1 + 2𝑗∧ℓ |𝑧|) 𝑑𝑧 󵄩 󵄩 󵄩 󵄩 −|𝑗|(𝐿−𝑁−|𝛼|)𝑞 󵄩𝑔󵄩 ̇𝛼 ≲ 󵄩𝑔󵄩 (∑2 ) 󵄩 󵄩𝐹𝑝,𝑞(𝐺) 󵄩 󵄩(𝐿+1,𝐿+Δ+2) 𝑗∈Z (47) ≤𝐶2−|𝑗−ℓ|(𝐿−𝑎). 󵄩 󵄩 ≲ 󵄩𝑔󵄩(𝐿+1,𝐿+Δ+2), Here, the constant 𝐶 is a suitable multiple of (1) (2) ‖𝜓 ‖(𝐿+1,𝐿+Δ+2)‖𝜙 ‖(𝐿+1,𝐿+Δ+2). On the other hand we Z(𝐺) 󳨅→ 𝐹̇𝑠 (𝐺) observe that which implies that 𝑝,𝑞 continuously. 𝛼 Now we show the other embedding. Let 𝑓∈𝐹̇ (𝐺) 𝑀∗ (𝑓, L(1), 𝜙̂(1))(𝑦) 𝑝,𝑞 𝑎,2−𝑗 ̂ ̂ andtakeany𝜓∈Z(𝐺).Let𝜙∈A, and then let 𝜁∈ + −2 2 ∗ (1) ̂(1) 𝑗 󵄨 −1 󵄨 𝑎 S(R ) be a function with support in [2 ,2 ] such that ≤𝑀 −𝑗 (𝑓, L , 𝜙 ) (𝑥) (1 + 2 󵄨𝑦 𝑥󵄨) 𝑎,2 󵄨 󵄨 ∑ 𝜙(2̂ −2𝑗𝜆)𝜁(2̂ −2𝑗𝜆) = 1 𝜆∈R+ (44) 𝑗∈Z for .ThenbyLemma 3 ∗ (1) ̂(1) ≲𝑀𝑎,2−𝑗 (𝑓, L , 𝜙 ) (𝑥) we have

ℓ 󵄨 −1 󵄨 𝑎 (𝑗−ℓ)𝑎 󵄨 󵄨 ×(1+2 󵄨𝑦 𝑥󵄨) max (1, 2 ). 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨⟨𝑓,𝜓⟩󵄨 = 󵄨⟨∑𝑓∗(𝐷 𝑗 𝜙) ∗ (𝐷 𝑗 𝜁) , 𝜓⟩󵄨 󵄨 󵄨 󵄨 2 2 󵄨 Putting these estimates into (42), multiplying both sides by 󵄨 𝑗∈Z 󵄨 2ℓ𝛼 (1 + 2ℓ|𝑦−1𝑥|)𝑎 , dividing both sides by and then taking 󵄨 ̃ 󵄨 ≤ ∑ 󵄨⟨𝑓 ∗ (𝐷 𝑗 𝜙) , 𝜓 ∗ (𝐷 𝑗 𝜁)⟩󵄨 the supremum over 𝑦∈𝐺,weobtain 󵄨 2 2 󵄨 (48) 𝑗∈Z 2ℓ𝛼𝑀∗ (𝑓, L(2), 𝜙̂(2)) 𝑥 󵄩 󵄩 𝑎,2−ℓ ( ) 󵄩 󵄩 󵄩 ̃ 󵄩 ≤ ∑󵄩𝑓∗(𝐷 𝑗 𝜙)󵄩 󵄩𝜓∗(𝐷 𝑗 𝜁)󵄩 . 󵄩 2 󵄩𝐿∞ 󵄩 2 󵄩𝐿1 −|𝑗−ℓ|(𝐿−2𝑎−|𝛼|) 𝑗𝛼 ∗ (1) ̂(1) (45) 𝑗∈Z ≤ ∑2 2 𝑀𝑎,2−𝑗 (𝑓, L , 𝜙 ) (𝑥) . 𝑗∈Z To proceed we claim that In view of [15,Lemma2],taking𝐿>2𝑎+|𝛼|in the above ≲ inequality yields the direction “ ”of(39). By symmetricity, 󵄩 󵄩 𝑗Δ/𝑝 󵄩 󵄩 󵄩𝑓∗(𝐷 𝑗 𝜙)󵄩 ≤𝐶2 󵄩𝑓∗(𝐷 𝑗 𝜙)󵄩 (39)holds,andtheproofiscomplete. 󵄩 2 󵄩𝐿∞ 󵄩 2 󵄩𝐿𝑝 (49) 8 Journal of Function Spaces and Applications for all 0<𝑝<∞. Assuming the claim for a moment, it Sinceallthenecessarytoolsaredevelopedintheabove follows from (48)that arguments, the following proposition can be proved in the same manner as its Euclidean counterpart; see, for example, 󵄨 󵄨 𝑗𝛼󵄩 󵄩 the proof of [16, Theorem 2.3.3]. 󵄨⟨𝑓,𝜓⟩󵄨 ≲(sup 2 󵄩𝑓∗(𝐷2𝑗 𝜙)󵄩𝐿𝑝 ) 𝑗∈Z (50) Proposition 10. For 𝛼∈R, 0<𝑝<∞and 0<𝑞≤∞, 𝑗[(Δ/𝑝)−𝛼]󵄩 ̃ 󵄩 ̇𝛼 × ∑2 󵄩𝜓∗(𝐷 𝑗 𝜁)󵄩 . 𝐹 (𝐺) 󵄩 2 󵄩𝐿1 𝑝,𝑞 is a quasi-Banach space. 𝑗∈Z Let us introduce a class of functions. We say that 𝑓∈ It is easy to see that ̂ + R(𝐺), if there exists 𝜙∈S(R ) whose support is compact 𝑗𝛼󵄩 󵄩 󵄩 󵄩 and which vanishes identically near the origin, and 𝑔∈S(𝐺), sup2 󵄩𝑓∗(𝐷2𝑗 𝜙)󵄩 𝑝 ≲ 󵄩𝑓󵄩 ̇𝛼 . 𝐿 𝐹𝑝,𝑞(𝐺) (51) ̂ 𝑗∈Z such that 𝑓=𝜙(L)𝑔.ClearlyR(𝐺) ⊂ Z(𝐺).

To estimate the sum in (50), we note that if we choose 𝑁≥ Lemma 11. Let 𝛼∈R and 0<𝑝,𝑞<∞.ThenR(𝐺) is dense Δ+1 𝐿 ≥ 𝑁 + (Δ/𝑝) + |𝛼|+1 ̇𝛼 ̇𝛼 and then similarly to (46)we in 𝐹𝑝,𝑞(𝐺).Inparticular,Z(𝐺) is dense in 𝐹𝑝,𝑞(𝐺). have 󵄨 󵄨 ̇𝛼 󵄨 ̃ 󵄨 󵄩 󵄩 −|𝑗|(𝐿−𝑁) −𝑁 Proof. Take any 𝑢∈𝐹𝑝,𝑞(𝐺) and any 𝜀>0.Intheappendix 󵄨𝜓∗(𝐷 𝑗 𝜁) (𝑥)󵄨 ≤𝐶󵄩𝜓󵄩 2 (1+|𝑥|) , 󵄨 2 󵄨 󵄩 󵄩(𝐿+1,𝐿+Δ+2) ̇𝛼 we show that 𝐹𝑝,𝑞(𝐺) admits smooth atomic decomposition. (52) ∞ By the smooth atomic decomposition, we see that 𝐶0 (𝐺) ∩ ̃ ̇𝛼 ̇𝛼 where the constant 𝐶 is a suitable multiple of ‖𝜁‖(𝐿+1,𝐿+Δ+2). 𝐹𝑝,𝑞(𝐺) is dense in 𝐹𝑝,𝑞(𝐺),for𝛼∈R and 0<𝑝,𝑞<∞.Thus, ∞ ̇𝛼 From this it follows that 𝑔∈𝐶 (𝐺)∩𝐹 (𝐺) ‖𝑔−𝑢‖ ̇𝛼 <𝜀/2 we can find 0 𝑝,𝑞 such that 𝐹𝑝,𝑞(𝐺) . 𝑗(Δ/𝑝−𝛼)󵄩 ̃ 󵄩 ∑2 󵄩𝜓∗(𝐷 𝑗 𝜁)󵄩 On the other hand, the argument in Step 5 of the proof of 󵄩 2 󵄩𝐿1 𝑗∈Z [16, Theorem 2.3.3] shows that there exists a sufficiently large 𝑁 𝑁∈N ‖𝑔 − ∑ 𝑔∗(𝐷 𝑗 𝜙)‖ ̇𝛼 <𝜀/2 󵄩 󵄩 such that 𝑗=−𝑁 2 𝐹𝑝,𝑞(𝐺) .Nowwe ≲ ∑2−|𝑗|[𝐿−𝑁−(Δ/𝑝)−|𝛼|]󵄩𝜓󵄩 󵄩 󵄩(𝐿+1,𝐿+Δ+2) (53) put 𝑗∈Z 󵄩 󵄩 𝑁 𝑁 ≲ 󵄩𝜓󵄩 . ̂ −2𝑗 󵄩 󵄩(𝐿+1,𝐿+Δ+2) ℎ= ∑ 𝜙(2 L)𝑔= ∑ 𝑔∗(𝐷2𝑗 𝜙) . (57) 𝑗=−𝑁 𝑗=−𝑁 Therefore, 󵄨 󵄨 󵄩 󵄩 󵄩 󵄩 ℎ∈R(𝐺) 󵄨⟨𝑓, 𝜓⟩󵄨 ≲ 󵄩𝑓󵄩 󵄩𝜓󵄩 . Then ,andwehave 󵄨 󵄨 󵄩 󵄩𝐹̇𝛼 (𝐺)󵄩 󵄩(𝐿+1,𝐿+Δ+2) (54) 𝑝,𝑞 󵄩 󵄩 󵄩 󵄩 ‖ℎ−𝑢‖ 𝛼 ≲ ℎ−𝑔 + 𝑔−𝑢 <𝜀. 𝐹̇(𝐺) 󵄩 󵄩𝐹̇𝛼 (𝐺) 󵄩 󵄩𝐹̇𝛼 (𝐺) (58) ̇𝛼 󸀠 𝑝,𝑞 𝑝,𝑞 𝑝,𝑞 This implies that 𝐹𝑝,𝑞(𝐺) 󳨅→ S (𝐺)/P continuously. We are left with showing the claim. Indeed, if 𝑟>0is fixed This proves the claimed statement. then by Lemma 6 we have, for all 𝑥∈𝐺, ̇𝛼 We next consider lifting property of 𝐹𝑝,𝑞(𝐺).For𝜎∈R, 󵄨 󵄨 𝜎 󵄨𝑓∗(𝐷 𝑗 𝜙) (𝑥)󵄨 L 󵄨 󵄨 󵄨 2 󵄨 the power is naturally given by 󵄨𝑓∗(𝐷 𝑗 𝜙) (𝑥)󵄨 ∼ 󵄨 2 󵄨 inf 󵄨 󵄨 Δ/𝑟 𝑦∈𝐵(𝑥,2−𝑗) 𝑗 󵄨 −1 󵄨 ∞ (1 + 2 󵄨𝑥 𝑦󵄨) 𝜎 𝜎 𝜎 2 L 𝑓=∫ 𝜆 𝑑𝐸 (𝜆) 𝑓, 𝑓 ∈ Dom (L )⊂𝐿 (𝐺) . (59) ∗ ̂ 0 ≤ inf 𝑀Δ/𝑟,2−𝑗 (𝑓, 𝜙) (𝑦) 𝑦∈𝐵(𝑥,2−𝑗) Remark 12. By [17,Theorem13.24],wehaveR(𝐺) ⊂ (L𝜎) 𝜎∈R (L𝜎)∩ 󵄨 󵄨𝑟 1/𝑟 Dom for all .Asa consequence,Dom 󵄨 󵄨 ̇𝛼 ̇𝛼 ≲ inf [𝑀 (󵄨𝑓∗(𝐷2𝑗 𝜙)󵄨 )(𝑦)] . 𝐹 (𝐺) is dense in 𝐹 (𝐺),forall𝛼, 𝜎 ∈ R and 0<𝑝,𝑞<∞. 𝑦∈𝐵(𝑥,2−𝑗) 𝑝,𝑞 𝑝,𝑞 𝐹̇𝛼 (𝐺) (55) We now have the lifting property of 𝑝,𝑞 . Taking 𝑟<𝑝in the above estimate and using Hardy- Theorem 13. Let 𝛼, 𝜎 ∈ R and 0<𝑝, 𝑞<∞. Littlewood maximal inequality, we have 𝜎 𝜎 (i) The operator L ,initiallydefinedonDom(L )∩ 𝛼 𝛼 󵄨 󵄨𝑝 𝐹̇ (𝐺), extends to a continuous operator from 𝐹̇ (𝐺) 󵄨𝑓∗(𝐷2𝑗 𝜙) (𝑥)󵄨 𝑝,𝑞 𝑝,𝑞 ̇𝛼−2𝜎 to 𝐹𝑝,𝑞 (𝐺). 1 󵄨 󵄨𝑟 𝑝/𝑟 ≲ ∫ [𝑀 (󵄨𝑓∗(𝐷 𝑗 𝜙)󵄨 )(𝑦)] 𝑑𝑦 𝛼 𝛼−2𝜎 󵄨 −𝑗 󵄨 󵄨 2 󵄨 𝑇 : 𝐹̇ (𝐺) → 𝐹̇ (𝐺) 󵄨𝐵(𝑥,2 )󵄨 𝐵(𝑥,2−𝑗) (ii) Let 𝜎 𝑝,𝑞 𝑝,𝑞 denote the continuous 󵄨 󵄨 𝜎 extension of L .Then𝑇𝜎 is an isomorphism, and 𝑗Δ 󵄨 󵄨𝑝 ̇𝛼 󵄨 󵄨 ‖𝑇𝜎𝑓‖𝐹̇𝛼−2𝜎(𝐺) is an equivalent quasi-norm of 𝐹𝑝,𝑞(𝐺). ≤2 ∫ 󵄨𝑓∗(𝐷2𝑗 𝜙) (𝑦)󵄨 𝑑𝑦. 𝑝,𝑞 (56) ̂ ̂ 𝜎 Proof. (i) Let 𝜙∈A.Set𝜁(𝜆) =𝜆 and 𝜓(𝜆)̂ = ̂ ̂ ̂ 𝜎 Since 𝑥 is arbitrary, the claim follows. 𝜙(𝜆)𝜁(𝜆) = 𝜙(𝜆)𝜆 .Clearly𝜓̂ is also in A,and Journal of Function Spaces and Applications 9

2𝑗𝜎 −2𝑗 ̂ −2𝑗 ̂ 2 𝜓(2̂ 𝜆) = 𝜙(2 𝜆)𝜁(𝜆).By[17,Theorem13.24],wehave Our next goal is to show the Lusin and Littlewood-Paley ̂ −2𝑗 ̂ ̂ −2𝑗 ̂ 𝐹̇𝛼 (𝐺) 𝛼∈R 0<𝑞<∞ 𝜙(2 L)∘𝜁(L)⊂(𝜙(2 ⋅)𝜁(⋅))(L), and moreover function characterizations of 𝑝,𝑞 .If , , 𝜆>0, 𝑏∈R,and𝑢(𝑥, 𝑘) is a function on 𝐺×Z, we define ̂ −2𝑗 ̂ Dom (𝜙(2 L)∘𝜁 (L)) ̂ ̂ −2𝑗 ̂ = Dom (𝜁 (L))∩Dom ((𝜙(2 ⋅) 𝜁 (⋅)) (L)) (60) ∞ 1/𝑞 𝑞 𝑔𝛼 (𝑢)(𝑥) =( ∑ (2𝑘𝛼 |𝑢 (𝑥,) 𝑘 |) ) , 𝜎 2 𝜎 𝑞 = Dom (L )∩𝐿 (𝐺) = Dom (L ). 𝑘=−∞ 𝑓∈ (L𝜎) 22𝑗𝜎𝜓(2̂ −2𝑗L)𝑓 = ∞ Hence, for every Dom ,wehave 𝛼 𝑘𝛼 󵄨 󵄨 𝑞 ̂ −2𝑗 𝜎 𝐺𝜆,𝑞 (𝑢)(𝑥) =[ ∑ ∫ (2 󵄨𝑢(𝑦,𝑘)󵄨) 𝜙(2 L)L 𝑓. 𝐺 𝜎 ̇𝑠 𝑘=−∞ Now let 𝑓∈Dom(L )∩𝐹𝑝,𝑞(𝐺). By the above remarks, 2𝑗𝜎 𝜎 1/𝑞 2 [𝑓 ∗ (𝐷 𝑗 𝜓)] =L ( 𝑓) ∗ (𝐷 𝑗 𝜙) we have 2 2 .Itfollowsthat 𝑘 󵄨 −1 󵄨 −𝜆𝑞 𝑘Δ (62) ×(1 + 2 󵄨𝑦 𝑥󵄨) 2 𝑑𝑦] , 󵄩 𝜎 󵄩 󵄨 󵄨 󵄩L 𝑓󵄩 ̇𝛼−2𝜎 󵄩 󵄩𝐹𝑝,𝑞 (𝐺) 𝛼 󵄩 1/𝑞󵄩 𝑆𝑏,𝑞 (𝑢)(𝑥) 󵄩 󵄩 󵄩 𝑗(𝛼−2𝜎) 󵄨 𝜎 󵄨 𝑞 󵄩 ∼ 󵄩(∑(2 󵄨(L 𝑓) ∗ (𝐷 𝑗 𝜙)󵄨) ) 󵄩 󵄩 󵄨 2 󵄨 󵄩 ∞ 1/𝑞 󵄩 󵄩 𝑞 󵄩 𝑗∈Z 󵄩 𝑝 𝑘𝛼 󵄨 󵄨 (𝑘−𝑏)Δ 𝐿 =( ∑ ∫ (2 󵄨𝑢(𝑦,𝑘)󵄨) 2 𝑑𝑦) . (61) |𝑦−1𝑥|≤2𝑏−𝑘 󵄩 1/𝑞󵄩 𝑘=−∞ 󵄩 󵄩 󵄩 𝑗𝛼 󵄨 󵄨 𝑞 󵄩 = 󵄩(∑(2 󵄨𝑓∗(𝐷 𝑗 𝜓)󵄨) ) 󵄩 󵄩 󵄨 2 󵄨 󵄩 󵄩 𝑗∈Z 󵄩 󵄩 󵄩𝐿𝑝 ̇𝛼 The following proposition shows that the spaces 𝐹𝑝,𝑞(𝐺) are 󵄩 󵄩 ∼ 󵄩𝑓󵄩 ̇𝛼 . 󵄩 󵄩𝐹𝑝,𝑞(𝐺) characterized by Lusin and Littlewood-Paley functions. 𝜎 ̇𝛼 ̇𝛼 Proposition 14. 𝛼∈R 0<𝑝𝑞<∞𝑏∈R 𝜆> Since Dom(L )∩𝐹𝑝,𝑞(𝐺) is dense in 𝐹𝑝,𝑞(𝐺) (see Remark 12), Let , , , and 𝜎 𝜎 ̇𝛼 ̇𝛼 {Δ/𝑝, Δ/𝑞} 𝑓∈S󸀠(𝐺)/P 𝜙∈̂ A 𝑢(𝑥, 𝑘) = the mapping L : Dom(L )∩𝐹𝑝,𝑞(𝐺) 󳨃→ 𝐹𝑝,𝑞(𝐺) extends to max .Let and .Put ̇𝛼 ̇𝛼−2𝜎 𝑓∗(𝐷2𝑘 𝜙)(𝑥),for𝑥∈𝐺and 𝑘∈Z.Thenonehas acontinuousoperatorfrom𝐹𝑝,𝑞(𝐺) to 𝐹𝑝,𝑞 (𝐺). We denote ̇𝛼 ̇𝛼−2𝜎 this extension by 𝑇𝜎 : 𝐹𝑝,𝑞(𝐺) 󳨃→ 𝐹𝑝,𝑞 (𝐺). ̇𝛼 󵄩 𝛼 󵄩 󵄩 𝛼 󵄩 󵄩 𝛼 󵄩 (ii) Let us first show that the mapping 𝑇𝜎 : 𝐹𝑝,𝑞(𝐺) → 󵄩𝑔 (𝑢)󵄩 ∼ 󵄩𝑆 (𝑢)󵄩 ∼ 󵄩𝐺 (𝑢)󵄩 . 󵄩 𝑞 󵄩 𝑝 󵄩 𝑏,𝑞 󵄩 𝑝 󵄩 𝜆,𝑞 󵄩 𝑝 (63) ̇𝛼−2𝜎 ̇𝛼 󵄩 󵄩𝐿 󵄩 󵄩𝐿 󵄩 󵄩𝐿 𝐹𝑝,𝑞 (𝐺) is injective. Indeed, assume 𝑓∈𝐹𝑝,𝑞(𝐺) such that ̇𝛼−2𝜎 𝑇𝜎𝑓 is the zero element of 𝐹𝑝,𝑞 (𝐺).ByRemark 12 we can 𝜎 ̇𝛼 𝜆> {Δ/𝑝, Δ/𝑞} find a sequence 𝑓ℓ in Dom(L )∩𝐹𝑝,𝑞(𝐺) which converges Proof. Step 1. Show that if max then ̇𝛼 𝜎 𝜎 in 𝐹𝑝,𝑞(𝐺) to 𝑓. Then applying (i) to L yields that L 𝑓ℓ ̇𝑠−2𝜎 𝜎 converges in 𝐹 (𝐺) to the zero element. Since L 𝑓ℓ ∈ 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 𝑝,𝑞 󵄩𝑔𝛼 (𝑢)󵄩 ≲ 󵄩𝐺𝛼 (𝑢)󵄩 ≲ 󵄩𝑆𝛼 (𝑢)󵄩 . −𝜎 ̇𝛼−2𝜎 −𝜎 𝜎 󵄩 𝑞 󵄩𝐿𝑝 󵄩 𝜆,𝑞 󵄩𝐿𝑝 󵄩 𝑏,𝑞 󵄩𝐿𝑝 (64) Dom(L )∩𝐹𝑝,𝑞 (𝐺) and 𝑓ℓ = L (L 𝑓ℓ), applying (i) −𝜎 ̇𝛼 to the operator L we see that 𝑓ℓ converges in 𝐹𝑝,𝑞(𝐺) to the 𝑓 𝐹̇𝛼 (𝐺) zero element. Therefore, is the zero element in 𝑝,𝑞 .This Indeed, the proof of the first inequality in64 ( )isessentially 𝑇 proves that 𝜎 is injective. the same as that of [18, Theorem 2.3]. To see the second ̇𝛼 ̇𝛼−2𝜎 Next we show that 𝑇𝜎 : 𝐹𝑝,𝑞(𝐺) → 𝐹𝑝,𝑞 (𝐺) is surjective. inequality in (64),oneonlyneedstoexaminetheproofsof ̇𝛼−2𝜎 Indeed, given 𝑓∈𝐹 (𝐺),welet𝑓ℓ be a sequence in Theorems 1, 2 and 4 in [19, Chapter 4] and observe that, 𝑝,𝑞 𝑢 (L−𝜎)∩𝐹̇𝛼−2𝜎(𝐺) 𝐹̇𝛼−2𝜎(𝐺) 𝑓 although the function considered in [19] is defined on the Dom 𝑝,𝑞 which converges in 𝑝,𝑞 to . R𝑛+1 = R𝑛 ×(0,∞) L−𝜎𝑓 𝐹̇𝛼 (𝐺) half space + , the arguments there can also Thenfrom(i)weseethat ℓ converges in 𝑝,𝑞 .Denote be adapted to functions 𝑢 which are defined on 𝐺×Z. this limit by 𝑔.Weclaimthat𝑇𝜎𝑔=𝑓. Indeed, since 𝑓ℓ = 𝜎 −𝜎 𝛼 𝛼 L (L 𝑓ℓ), it follows from (i) that 𝑓ℓ converges to 𝑇𝜎𝑔 in Step 2. We prove that ‖𝑆𝑏,𝑞(𝑢)‖𝐿𝑝 ≲‖𝑔𝑞 (𝑢)‖𝐿𝑝 .Firstnotethat, ̇𝛼 ̇𝛼−2𝜎 𝐹𝑝,𝑞(𝐺).Hence𝑇𝜎𝑔=𝑓in 𝐹𝑝,𝑞 (𝐺).Thisprovesthat𝑇𝜎 is by an argument similar to the proofs of [19,Theorems1and 𝛼 𝛼 surjective. 2], it follows that ‖𝑆𝑏,𝑞(𝑢)‖𝐿𝑝 ∼‖𝑆0,𝑞(𝑢)‖ for every fixed 𝑏∈R. The above arguments also show that both 𝑇−𝜎 ∘𝑇𝜎 and ̇𝛼 Hence, in view of (38), it is enough to show that 𝑇𝜎 ∘𝑇−𝜎 are identity operators on 𝐹𝑝,𝑞(𝐺).Furthermore,by an easy density argument we see that (61)holdsforall𝑓∈ ̇𝛼 𝜎 𝐹𝑝,𝑞(𝐺),providedthatL in (61)isreplacedby𝑇𝜎.Thus,𝑇𝜎 : 󵄩 1/𝑞󵄩 󵄩 󵄩 𝐹̇𝛼 (𝐺) → 𝐹̇𝛼−2𝜎(𝐺) ‖𝑇 𝑓‖ 󵄩 𝛼 󵄩 󵄩 𝑗𝛼 󵄨 ∗ ̂ 󵄨 𝑞 󵄩 𝑝,𝑞 𝑝,𝑞 is an isomorphism, and 𝜎 𝐹̇𝛼−2𝜎(𝐺) is 󵄩𝑆 (𝑢)󵄩 ≲ 󵄩(∑(2 󵄨𝑀 −𝑗 (𝑓, 𝜙)󵄨) ) 󵄩 . 𝑝,𝑞 󵄩 0,𝑞 󵄩𝐿𝑝 󵄩 󵄨 𝜆,2 󵄨 󵄩 (65) 𝛼 󵄩 󵄩 ̇ 󵄩 𝑗∈Z 󵄩 𝑝 an equivalent quasi-norm of 𝐹𝑝,𝑞(𝐺). 󵄩 󵄩𝐿 10 Journal of Function Spaces and Applications

But this is a consequence of the following elementary esti- Remark 17. Using [3, Proposition 1.29], it is easy to verify that mate: (67) is equivalent to the following condition: 𝑞 󵄨 󵄨 𝑗𝛼 󵄨 󵄨 𝑗Δ 󵄨 𝐼 󵄨 −𝑄−𝑑(𝐼) ∫ (2 󵄨𝑓∗(𝐷2𝑗 𝜙) (𝑦)󵄨) 2 𝑑𝑦 󵄨𝑌 𝐾 (𝑥)󵄨 ≤𝐶𝐼|𝑥| , for |𝐼| ≤𝑟,𝑥=0.̸ (70) |𝑦−1𝑥|<2−𝑗 󵄨 󵄨

𝑗𝛼 󵄨 󵄨 𝑞 Examples of such kernels include the class of distributions (2 󵄨𝑓∗(𝐷 𝑗 𝜙) (𝑦)󵄨) 󵄨 2 󵄨 which are homogeneous of degree −Δ (see Folland and Stein ≲ sup ∞ 𝑗 󵄨 −1 󵄨 𝜆𝑞 𝑦∈𝐵 𝑥,2−𝑗 (1 + 2 󵄨𝑦 𝑥󵄨) [3,p.11]fordefinition)andagreewith𝐶 functions away ( ) 󵄨 󵄨 󸀠 (66) from 0.Indeed,assume𝐾∈S (𝐺) is such a distribution, then 𝑗𝛼 󵄨 󵄨 𝑞 it is easy to verify that 𝐾 satisfies the regularity condition (i) in (2 󵄨𝑓∗(𝐷2𝑗 𝜙) (𝑦)󵄨) ≤ sup 󵄨 󵄨 𝜆𝑞 Definition 16; moreover, from [3,Proposition6.13]weseethat 𝑦∈𝐺 (1 + 2𝑗 󵄨𝑦−1𝑥󵄨) 𝐾 ∫ 𝐾(𝑥)𝑑𝑥 = 󵄨 󵄨 is a principle value distribution such that 𝜀<|𝑥|<𝐿 𝑞 0 for all 0<𝜀<𝐿<∞. Hence, for every normalized bump =[2𝑗𝛼𝑀∗ (𝑓, 𝜙)̂ (𝑥)] . 𝜆,2−𝑗 function 𝜙, by the homogeneity of 𝐾 we have 󵄨 󵄨 The proof of Proposition 14 is thus complete. 󵄨 𝑅 󵄨 󵄨 󵄨 󵄨⟨𝐾, 𝜙 ⟩󵄨 = 󵄨⟨𝐾, 𝜙⟩󵄨 Corollary 15. 𝐺 𝐹̇0 (𝐺) = 󵄨 󵄨 Let be a stratified Lie group. Then 𝑝,2 󵄨 󵄨 𝑝 = 󵄨lim ∫ 𝐾 (𝑥) [𝜙 (𝑥) −𝜙(0)]𝑑𝑥󵄨 𝐻 (𝐺) with equivalent (quasi-)norms, for 0<𝑝<∞.Here 󵄨𝜀→0 󵄨 𝑝 󵄨 𝜀<|𝑥|<2 󵄨 (71) 𝐻 (𝐺) are Hardy spaces on 𝐺. 󵄨 󵄨 ≤ ∫ |𝐾 (𝑥)| 󵄨𝜙 (𝑥) −𝜙(0)󵄨 𝑑𝑥. Proof. In [3,Chapter7],FollandandSteinprovedthechar- |𝑥|<2 𝑝 acterization of Hardy spaces 𝐻 (𝐺) by continuous version Lusin function, for 0<𝑝<∞. Note that the arguments Using stratified mean value theorem (cf.3 [ , Theorem 1.41]) in [3,Chapter7]arestillvalidifwereplacethecontinuous and (67)–(69), it is easy to verify that the last integral version Lusin function by discrete version one defined above; converges absolutely and is bounded by a constant inde- see also [20] for a treatment of discrete version Lusin funcion. pendent of 𝜙 and 𝑅.Hence𝐾 satisfies the condition (ii) in This fact together with Proposition 14 yield the identification Definition 16. ̇0 𝑝 of 𝐹𝑝,2(𝐺) with 𝐻 (𝐺) for 0<𝑝<∞. Now we state the main result of this section. Theorem 18. Let 𝛼∈R, 0<𝑝,𝑞<∞,andlet𝑟 be a positive 4. Convolution Singular Integral Operators on 𝑟>Δ/ {𝑝, 𝑞} + |𝛼| +2 𝑇 ̇𝛼 integer such that min .Suppose is 𝐹𝑝,𝑞(𝐺) a singular integral operator of order 𝑟.Then𝑇 extends to a ̇𝛼 bounded operator on 𝐹𝑝,𝑞(𝐺). In this section we study boundedness of convolution singular 󸀠 integral operators on homogeneous Triebel-Lizorkin spaces If 𝐾∈S (𝐺) and 𝑡>0, we define 𝐷𝑡𝐾 as the tempered −1 on stratified Lie groups. Motivated by21 [ ,Section5.3in distribution given by ⟨𝐷𝑡𝐾, 𝜙⟩ = ⟨𝐾, 𝜙(𝑡 ⋅)⟩ (∀𝜙 ∈ S(𝐺)). Chapter XIII], we introduce a class of singular convolution For the proof of Theorem 18, we will need the following kernels as follows. lemma, in which 𝑏 isthepositiveconstantasin[3,Corollary 1.44]. Definition 16. Let 𝑟 be a positive integer. A kernel of order 𝑟 󸀠 is a distribution 𝐾∈S (𝐺) with the following properties: 𝑟 Lemma 19. Let 𝑟 be a positive integer. Suppose 𝐾 is a kernel of (i) 𝐾 coincides with a 𝐶 function 𝐾(𝑥) away from the 𝑟 𝜙 𝐵(0, 1/4𝛾2𝑏𝑟) 0 order ,and is a smooth function supported in group identity and enjoys the regularity condition: and having vanishing moments of order 𝑟−1.Then,thereexists 𝑛 󵄨 𝐼 󵄨 −𝑄−𝑑(𝐼) aconstant𝐶>0such that for all 𝑗∈Z,all𝐼∈N with |𝐼| ≤, 𝑟 󵄨𝑋 𝐾 (𝑥)󵄨 ≤𝐶𝐼|𝑥| , for |𝐼| ≤𝑟,𝑥=0.̸ (67) 󵄨 󵄨 and all 𝑥∈𝐺 (ii) 𝐾 satisfies the cancellation condition: For all normal- 󵄨 𝐼 󵄨 󵄨 𝐼 󵄨 max {󵄨𝑋 [𝜙 ∗ (𝐷2𝑗 𝐾)] (𝑥)󵄨 , 󵄨𝑌 [𝜙 ∗ (𝐷2𝑗 𝐾)] (𝑥)󵄨} ized bump function 𝜙 and all 𝑅>0,wehave 󵄨 󵄨 󵄨 󵄨 (72) 󵄨 󵄨 −Δ−𝑟 󵄨 𝑅 󵄨 ≤𝐶(1+|𝑥|) . 󵄨⟨𝐾, 𝜙 ⟩󵄨 ≤𝐶, (68)

𝑅 Moreover, 𝜙∗(𝐷2𝑗 𝐾) have vanishing moments of the same where 𝜙 (𝑥) = 𝜙(𝑅𝑥),and𝐶 is a constant independent of order as 𝜙. 𝜙 and 𝑅. Here, by a normalized bump function we mean a 𝜙 𝐵(0, 1) function supported in and satisfying Proof. Recall that the convolution of 𝜙∈S(𝐺) with 𝐾∈ 󵄨 󵄨 S󸀠(𝐺) 𝜙 ∗ 𝐾(𝑥) := ⟨𝐾,(𝑥𝜙)∼⟩ 𝑥𝜙 󵄨𝑋𝐼𝜙 (𝑥)󵄨 ≤1, ∀|𝐼| ≤𝑁,∀𝑥∈𝐺, is defined by ,where is 󵄨 󵄨 (69) 𝑥 ̃ the function given by 𝜙(𝑧) = 𝜙(𝑥𝑧),andasbefore𝑓(𝑥) := −1 for some fixed positive integer 𝑁. 𝑓(𝑥 ) for any function 𝑓:𝐺 → C.From[3,p.38]we ∞ The convolution operator 𝑇 with kernel of order 𝑟 is called see that 𝜙∗(𝐷2𝑗 𝐾) are 𝐶 functions, 𝑗∈Z.Weclaimthat 𝑥 ∼ a singular integral operator of order 𝑟. for every 𝑥 with |𝑥| ≤ 1/2𝛾,thefunction𝑧 󳨃→(𝜙) (𝑧) Journal of Function Spaces and Applications 11

is a normalized bump function multiplied with a constant Observe that 𝐷2𝑗 𝐾 satisfies (67)withthebound𝐶 indepen- 𝑟−|𝐼| independent of 𝑥. Indeed, using the quasi-triangle inequality dent of 𝑗∈Z. Also note that for 𝑦∈supp 𝜙 and |𝑧| ≤ 𝑏 |𝑦| satisfied by the homogeneous norm it is easy to verify that we have 𝑧𝑥 ∈ 𝐺 \ {0}.Thus,forall𝐽 with 𝑑(𝐽) = 𝑟 − |𝐼| and all 𝑥 ∼ 𝑟−|𝐼| the function 𝑧 󳨃→( 𝜙) (𝑧) is supported in 𝐵(0, 1);moreover, 𝑧 with |𝑧| ≤ 𝑏 |𝑦|,byusing(73)and(67)(with𝐾 replaced |𝑥| ≤ 1/2𝛾 since and since by 𝐷2𝑗 𝐾)wehave 𝑌𝐼 = ∑ 𝑃 𝑋𝐽, 𝐼,𝐽 󵄨 󵄨 |𝐽|≤|𝐼| (73) 󵄨 𝐽 𝐼 󵄨 󵄨𝑌 𝑋 (𝐷 𝑗 𝐾) (𝑧𝑥)󵄨 𝑑(𝐽)≥𝑑(𝐼) 󵄨 2 󵄨

󵄨 󸀠 󵄨 󵄨 󵄨 󵄨 𝐽 +𝐼 󵄨 𝑃 𝑑(𝐽)−𝑑(𝐼) ≤ ∑ 󵄨𝑃 󸀠 (𝑧𝑥)󵄨 󵄨𝑋 (𝐷 𝑗 𝐾) (𝑧𝑥)󵄨 where 𝐼,𝐽 are polynomials of homogeneous degree 󵄨 𝐽,𝐽 󵄨 󵄨 2 󵄨 (see [3,Proposition1.29]),wehave |𝐽󸀠|≤|𝐽| 𝑑(𝐽󸀠)≥𝑑(𝐽) 󵄨 𝐼 𝑥 ∼ 󵄨 󵄨𝑋 [( 𝜙) ] (𝑧)󵄨 (78) 󵄨 󵄨 󸀠 󸀠 ≲ ∑ |𝑧𝑥|𝑑(𝐽 )−𝑑(𝐽)|𝑧𝑥|−Δ−𝑑(𝐽 +𝐼) 󵄨 𝐼 𝑥 −1 󵄨 = 󵄨𝑌 ( 𝜙) (𝑧 )󵄨 󸀠 󵄨 󵄨 |𝐽 |≤|𝐽| 𝑑(𝐽󸀠)≥𝑑(𝐽) 󵄨 −1 󵄨 󵄨 𝐽 𝑥 −1 󵄨 ≲ ∑ 󵄨𝑃𝐼,𝐽 (𝑧 )󵄨 󵄨𝑋 ( 𝜙) (𝑧 )󵄨 󵄨 󵄨 󵄨 󵄨 (74) −Δ−𝑟+|𝐼|−𝑑(𝐼) |𝐽|≤|𝐼| ≲ |𝑧𝑥| . 𝑑(𝐽)≥𝑑(𝐼)

󵄨 −1 󵄨 󵄨 𝐽 −1 󵄨 = ∑ 󵄨𝑃 (𝑧 )󵄨 󵄨(𝑋 𝜙) (𝑥𝑧 )󵄨 ≤𝐶. 󵄨 𝐼,𝐽 󵄨 󵄨 󵄨 𝐼 Here, for the second inequality we also used the observation |𝐽|≤|𝐼| 󸀠 󸀠 𝑑(𝐽)≥𝑑(𝐼) that when 𝑑(𝐽) = 𝑟 − |𝐼| and |𝐽 |≤|𝐽|,wehave|𝐽 +𝐼|≤ |𝐽+𝐼|≤𝑑(𝐽)+|𝐼|=𝑟−|𝐼|+|𝐼|=𝑟. Inserting (78)into(77) 𝐶 𝐼 𝑥 Here 𝐼 is a constant depending on but not on .Hence we obtain the claim is true. The above argument also shows that, for 𝑛 𝑥 𝐼 ∼ every 𝑥 with |𝑥| ≤ 1/2𝛾 and for every 𝐼∈N , [ (𝑌 𝜙)] is 󵄨 󵄨 󸀠 󵄨 𝐼 −1 ̃ −1 󵄨 𝐶 󵄨𝑋 (𝐷2𝑗 𝐾) (𝑦 𝑥) − 𝑃𝑥,𝑋𝐼(𝐷 𝐾) (𝑦 )󵄨 a normalized bump function multiplied with a constant 𝐼 󵄨 2𝑗 󵄨 independent of 𝑥. Thus, by the condition (ii) in Definition 16, 𝑛 󵄨 󵄨𝑟−|𝐼| −Δ−𝑟+|𝐼|−𝑑(𝐼) (79) there exits a constant 𝐶>0such that for all 𝑗∈Z,all𝐼∈N ≤𝐶󵄨𝑦󵄨 sup |𝑧𝑥| . with |𝐼| ≤,andall 𝑟 𝑥 with |𝑥| ≤ 1/2𝛾 |𝑧|≤𝑏𝑟−|𝐼||𝑦| 󵄨 󵄨 󵄨 𝐼 󵄨 󵄨𝑌 [𝜙 ∗ (𝐷2𝑗 𝐾)] (𝑥)󵄨 𝑟−|𝐼| 󵄨 󵄨 Notice that for |𝑥| ≥ 1/2𝛾, 𝑦∈supp 𝜙 and |𝑧| ≤ 𝑏 |𝑦|,we 󵄨 𝐼 󵄨 = 󵄨(𝑌 𝜙) ∗ (𝐷2𝑗 𝐾)] (𝑥)󵄨 have |𝑧𝑥| ∼ .Thus,byusingthevanishingmomentsof|𝑥| 𝜙 󵄨 ∼ 󵄨 (75) and (79), we have 󵄨 𝑥 𝐼 󵄨 = 󵄨⟨𝐷2𝑗 𝐾, [ (𝑌 𝜙)] ⟩󵄨 󵄨 󵄨 󵄨 𝑥 𝐼 ∼ −𝑗 󵄨 󵄨 𝐼 󵄨 󵄨 󵄨 󵄨𝑋 [𝜙 ∗ (𝐷 𝑗 𝐾)] (𝑥)󵄨 = 󵄨⟨𝐾, [ (𝑌 𝜙)] (2 ⋅)⟩󵄨 ≤𝐶. 󵄨 2 󵄨 󵄨 󵄨 󵄨 𝐼 −1 󵄨 From this and [3,Proposition1.29],wealsogetthat,forall = 󵄨∫ 𝜙(𝑦)𝑋 (𝐷 𝑗 𝐾) (𝑦 𝑥) 𝑑𝑦󵄨 𝑛 󵄨 2 󵄨 𝑗∈Z,all𝐼∈N with |𝐼| ≤,andall 𝑟 𝑥 with |𝑥| ≤ 1/2𝛾 󵄨 󵄨 󵄨 𝐼 󵄨 󵄨 󵄨 󵄨 𝐼 −1 ̃ −1 󵄨 󵄨𝑋 [𝜙 ∗ (𝐷 𝑗 𝐾)] (𝑥)󵄨 ≤ ∫ 󵄨𝜙(𝑦)󵄨 󵄨𝑋 (𝐷2𝑗 𝐾) (𝑦 𝑥) − 𝑃𝑥,𝑋𝐼(𝐷 𝐾) (𝑦 )󵄨 𝑑𝑦 󵄨 2 󵄨 󵄨 󵄨 󵄨 2𝑗 󵄨

󸀠 󵄨 󸀠 󵄨 𝑑(𝐼 )−𝑑(𝐼) 󵄨 𝐼 󵄨 ≤𝐶 ∑ |𝑥| 󵄨𝑌 [𝜙 ∗ (𝐷 𝑗 𝐾)] (𝑥)󵄨 ≤𝐶. −Δ−𝑟+|𝐼|−𝑑(𝐼)󵄨 󵄨𝑟−|𝐼| 󵄨 󵄨 󵄨 2 󵄨 (76) ≤𝐶 ∫ |𝑧𝑥| 󵄨𝑦󵄨 󵄨𝜙(𝑦)󵄨 𝑑𝑦 󸀠 sup 󵄨 󵄨 󵄨 󵄨 |𝐼 |≤|𝐼| |𝑧|≤𝑏𝑟−|𝐼||𝑦| 𝑑(𝐼󸀠)≥𝑑(𝐼) 󵄨 󵄨𝑟−|𝐼| 󵄨 󵄨 𝑛 ≤𝐶|𝑥|−Δ−𝑟+|𝐼| −𝑑(𝐼) ∫ 󵄨𝑦󵄨 󵄨𝜙(𝑦)󵄨 𝑑𝑦 Let now |𝑥| > 1/2𝛾.Let𝑦∈supp 𝜙.Let𝐼∈N with 󵄨 󵄨 󵄨 󵄨 ̃ |𝐼| ≤, 𝑟 and denote by 𝑃𝑥,𝑌𝐼(𝐷 𝐾) the right Taylor polynomial 2𝑗 −Δ−𝑟+|𝐼| −𝑑(𝐼) 𝐼 ≤𝐶|𝑥| . of 𝑌 (𝐷2𝑗 𝐾) at 𝑥 of homogeneous degree 𝑟−|𝐼|−1(see [3,pp. 26-27]). Then by the right-invariant version of [3,Corollary (80) 1.44], we have 󵄨 𝐼 −1 −1 󵄨 𝑛 󵄨𝑋 (𝐷 𝑗 𝐾) (𝑦 𝑥) − 𝑃̃ 𝐼 (𝑦 )󵄨 Combining (76)and(80), we see that, for all 𝑗∈Z,all𝐼∈N 󵄨 2 𝑥,𝑋 (𝐷 𝑗 𝐾) 󵄨 2 with |𝐼| <,andall 𝑟 𝑥∈𝐺 󵄨 󵄨 󵄨 󵄨𝑟−|𝐼| 󵄨 𝐽 𝐼 󵄨 (77) ≤𝐶󵄨𝑦󵄨 sup 󵄨𝑌 𝑋 (𝐷2𝐽 𝐾) (𝑧𝑥)󵄨 . 𝑧 ≤𝑏𝑟−|𝐼| 𝑦 󵄨 󵄨 | | | | 󵄨 𝐼 󵄨 −Δ−𝑟+|𝐼|−𝑑(𝐼) 𝑑(𝐽)=𝑟−|𝐼| 󵄨𝑋 [𝜙∗(𝐷2𝑗 𝐾)] (𝑥)󵄨 ≤𝐶(1+|𝑥|) . (81) 12 Journal of Function Spaces and Applications

𝑛 ̂ From this and (73), we also get that, for all 𝑗∈Z,all𝐼∈N with convergence in Z(𝐺).Let𝜙∈A,andlet𝐾 be the with |𝐼| <,andall 𝑟 𝑥∈𝐺 convolution kernel of the operaotor 𝑇.Thenwehavethe 󵄨 󵄨 󵄨 𝐼 󵄨 representation 󵄨𝑌 [𝜙 ∗ (𝐷2𝑗 𝐾)] (𝑥)󵄨 𝑓∗𝐾∗(𝐷 ℓ 𝜙) 󸀠 󵄨 2 󵄨 󵄨 𝐼 󵄨 ≤ ∑ 󵄨𝑃 󸀠 (𝑥)󵄨 𝑋 [𝜙 ∗ (𝐷 𝑗 𝐾)] (𝑥)󵄨 󵄨 𝐼,𝐼 󵄨 2 󵄨 |𝐼󸀠|≤|𝐼| = ∑𝑓∗(𝐷2𝑗 𝜓) ∗ (𝐷2𝑗 𝜁)∗𝐾∗(𝐷2ℓ 𝜙) 𝑑(𝐼󸀠)≥𝑑(𝐼) 𝑗∈Z (85) (82) 󸀠 󸀠 󸀠 ≤𝐶 ∑ |𝑥|𝑑(𝐼 )−𝑑(𝐼)(1+|𝑥|)−Δ−𝑟+|𝐼 |−𝑑(𝐼 ) = ∑𝑓∗(𝐷2𝑗 𝜓) ∗2 𝐷 𝑗 [𝜁 ∗ (𝐷2−𝑗 𝐾)] ∗ 2(𝐷 ℓ 𝜙) , 𝑗∈Z |𝐼󸀠|≤|𝐼| 𝑑(𝐼󸀠)≥𝑑(𝐼) 󸀠 which holds pointwise and also in the sense of S (𝐺).Since 𝜁∗(𝐷 −𝑗 𝐾) ≤𝐶(1+|𝑥|)−Δ−𝑟+|𝐼|−𝑑(𝐼). (by Lemma 19) 2 satisfies the decay condition72 ( ) (with the bound 𝐶 independent of 𝑗∈Z)andhasvanishing Since |𝐼| ≤ 𝑑(𝐼),(81)alongwith(82)yield(72). moments of the same order as 𝜁,fromtheproofofLemma 2 It is straightforward to verify that 𝜙∗(𝐷2𝑗 𝐾) have we see that vanishing moments of the same order as 𝜙. The proof of 󵄨 󵄨 󵄨𝐷2𝑗 [𝜁 ∗ (𝐷2−𝑗 𝐾)] ∗ 2(𝐷 ℓ 𝜙) (𝑦)󵄨 Lemma 19 is therefore complete. 2(𝑗∧ℓ)Δ (86) The proof of Theorem 18 also relies on the existence ≤𝐶2−|𝑗−ℓ|(𝑟−2) . 𝑗∧ℓ 󵄨 󵄨 𝑟+Δ of smooth functions with compact support and having (1 + 2 󵄨𝑦󵄨) arbitrarily high order vanishing moments. This together with85 ( )givesthat Lemma 20. Given any nonnegative integer 𝐿 and any positive 󵄨 󵄨 󵄨𝑓∗𝐾∗(𝐷 ℓ 𝜙) (𝑥)󵄨 ̂ + 󵄨 2 󵄨 number 𝛿, there exists a function 𝜁∈S(R ) with the following properties: 󵄨 󵄨 ≤ ∑ ∫ 󵄨𝑓∗(𝐷2𝑗 𝜓) (𝑧)󵄨 ̂ −2(𝑘0+1) −2(𝑘0−1) 𝑗∈Z (i) |𝜁(𝜆)| ≥ 𝐶 >0 for 𝜆∈[2 ,2 ],with𝑘0 some (large) positive integer; 󵄨 −1 󵄨 × 󵄨𝐷 𝑗 [𝜁 ∗ (𝐷 −𝑗 𝐾)] ∗ (𝐷 ℓ 𝜙) (𝑧 𝑥)󵄨 𝑑𝑧 (ii) 𝜁 is a Schwartz function on 𝐺 having vanishing 󵄨 2 2 2 󵄨 moments of order 𝐿; (𝑗∧ℓ)Δ 󵄨 󵄨 −|𝑗−ℓ|(𝑟−2) 2 󵄨𝑓∗(𝐷2𝑗 𝜓) (𝑧)󵄨 𝜁 ⊂ 𝐵(0, 𝛿) ≲ ∑2 ∫ 𝑑𝑧 (87) (iii) supp . 𝑗∧ℓ 󵄨 −1 󵄨 𝑟+Δ 𝑗∈Z (1 + 2 󵄨𝑧 𝑥󵄨) Proof. From the appendix of [22] we see that there exists 󵄨 󵄨 ̂ + ̂ −|𝑗−ℓ|(𝑟−2−𝑎) 󵄨𝑓∗(𝐷2𝑗 𝜓) (𝑧)󵄨 𝜃∈S(R ) such that 𝜃(0) = 1 and 𝜃 has compact support. ≲ ∑2 [ ] sup 𝑗 󵄨 −1 󵄨 𝑎 ̂ −2 𝑘 ̂ −2 + 𝑧∈𝐺 (1 + 2 󵄨𝑧 𝑥󵄨) Now let us define 𝜁(𝜆) = (𝑡 𝜆) 𝜃(𝑡 𝜆), 𝜆∈R .Here𝑡>0 𝑗∈Z 󵄨 󵄨 𝑘 𝜁(𝑥) =𝐷 (L𝑘𝜃)(𝑥) = and is a nonnegative integer. Then 𝑡 (𝑗∧ℓ)Δ Δ 𝑘 2 𝑡 (L 𝜃)(𝑡𝑥).Hence,ifwetake𝑡, 𝑘 sufficiently large, then × ∫ 𝑑𝑧, ̂ 𝑗∧ℓ 󵄨 −1 󵄨 𝑟+Δ−𝑎 (ii) and (iii) follow immediately. Moreover, since 𝜃(0) =, 1 (1 + 2 󵄨𝑧 𝑥󵄨) it is easy to see that (i) is also satisfied, provided that 𝑘0 is 𝑗∧ℓ −1 −𝑎 whereforthelastinequalityweusedthat(1 + 2 |𝑧 𝑥|) ≤ sufficiently large. |𝑗−ℓ|𝑎 𝑗 −1 −𝑎 2 (1 + 2 |𝑧 𝑥|) . By the hypothesis we can choose 𝑎 WearenowreadytoproveTheorem 18. such that 𝑎>Δ/min{𝑝, 𝑞} and 𝑟−2−𝑎−|𝛼|>. 0 From [3, Corollary 1.17] we see that the last integral converges ̂ + Proof of Theorem 1. Choose a function 𝜁∈S(R ) which absolutely. Consequently, we obtain satisfies conditions (i)–(iii) in Lemma 20 with 𝐿=𝑟−1and 2 𝑟 ℓ𝛼 󵄨 󵄨 𝛿 = 1/4𝛾 𝑏 . The condition (i) guarantees the existence ofa 2 󵄨𝑓∗𝐾∗(𝐷2ℓ 𝜙) (𝑥)󵄨 + function 𝜓∈̂ S(R ) with the following properties: −|𝑗−ℓ|(𝑟−2−𝑎−|𝛼|) 𝑗𝛼 ∗ ̂ (88) ≲ ∑2 2 𝑀𝑎,2−𝑗 (𝑓, 𝜓) (𝑥) . −2(𝑘0+1) −2(𝑘0−1) supp 𝜓⊂[2̂ ,2 ], 𝑗∈Z

󵄨 󵄨 −2(𝑘 +1) −2(𝑘 −1) 󵄨𝜓̂ (𝜆)󵄨 >0 𝜆∈(2 0 ,2 0 ), Hence, it follows by [15,Lemma2]that 󵄨 󵄨 for (83) 󵄩 1/𝑞󵄩 −2𝑗 ̂ −2𝑗 + 󵄩 𝑞 󵄩 ∑𝜓(2̂ 𝜆) 𝜁(2 𝜆) = 1, ∀𝜆 ∈ R . 󵄩 ℓ𝛼 󵄨 󵄨 󵄩 󵄩(∑(2 󵄨𝑓∗𝐾∗(𝐷2ℓ 𝜙) (𝑥)󵄨) ) 󵄩 𝑗∈Z 󵄩 󵄩 󵄩 ℓ∈Z 󵄩𝐿𝑝 −2𝑘0 (89) Note that 𝜓(2̂ ⋅) ∈ A.For𝑓∈Z(𝐺),byLemma 3 we have 󵄩 1/𝑞󵄩 󵄩 󵄩 󵄩 𝑗𝛼 󵄨 ∗ 󵄨 𝑞 󵄩 𝑓=∑𝑓∗(𝐷 𝑗 𝜓) ∗ (𝐷 𝑗 𝜁) ≲ 󵄩(∑(2 󵄨𝑀 −𝑗 (𝑓, 𝜓)̂ 󵄨) ) 󵄩 . 2 2 󵄩 󵄨 𝑎,2 󵄨 󵄩 (84) 󵄩 𝑗∈Z 󵄩 𝑗∈Z 󵄩 󵄩𝐿𝑝 Journal of Function Spaces and Applications 13

] ‖𝑓 ∗ 𝐾‖ ̇𝛼 ≲‖𝑓‖ ̇𝛼 ‖𝑓‖ ̇𝛼 ≲ ∑ This together with38 ( ) imply that 𝐹𝑝,𝑞(𝐺) 𝐹𝑝,𝑞(𝐺) Inserting this into (93), we obtain 𝐹𝑝,𝑞(𝐺) 𝑗=1 ̇𝛼 ̇𝛼 𝑓∈Z(𝐺) Z(𝐺) 𝐹 (𝐺) 𝑓 󳨃→𝑓∗𝐾 ‖𝑋 𝑓‖ ̇𝛼−1 𝑓∈R(𝐺) R(𝐺) 𝐹 (𝐺) for all .Since is dense in 𝑝,𝑞 , 𝑗 𝐹𝑝,𝑞 (𝐺) for all .Since is dense in 𝑝,𝑞 , 𝐹̇𝛼 (𝐺) ̇𝛼 extends to an bounded operator on 𝑝,𝑞 . This completes thelatterinequalityalsoholdsforall𝑓∈𝐹𝑝,𝑞(𝐺).This the proof of Theorem 18. completes the proof.

Corollary 21. Let 𝛼∈R, 0<𝑝,𝑞<∞,andlet𝑘 be a nonnegative integer. Then Appendix 󵄩 󵄩 󵄩 󵄩 󵄩𝑓󵄩 ∼ ∑ 󵄩𝑋𝐼𝑓󵄩 . 󵄩 󵄩𝐹̇𝛼 (𝐺) 󵄩 󵄩 ̇𝛼−𝑘 𝛼 𝑝,𝑞 󵄩 󵄩𝐹𝑝,𝑞 (𝐺) (90) ̇ 𝑑(𝐼)=𝑘 Smooth Atomic Decomposition of 𝐹𝑝,𝑞(𝐺) Proof. Note that by the Poincare-Birkhoff-Witt´ theorem (cf. 𝐼 In this appendix we show that homogeneous Triebel-Lizorkin [23, I.2.7]), the operators 𝑋 form a basis of the algebra of the spaces on stratified Lie groups admit smooth atomic decom- left-invariant differential operators on 𝐺. By this fact and the position. We follow the proof of [24, Theorem 6.6.3] with stratification of 𝐺,itsufficestoshowthat necessary modifications. 󵄩 󵄩 ] 󵄩 󵄩 Equipped with Haar measure and the quasi-distance 󵄩𝑓󵄩 ∼ ∑󵄩𝑋 𝑓󵄩 . 󵄩 󵄩𝐹̇𝛼 (𝐺) 󵄩 𝑗 󵄩 ̇𝛼−1 (91) defined by the homogeneous norm, the group 𝐺 is a space of 𝑝,𝑞 󵄩 󵄩𝐹𝑝,𝑞 (𝐺) 𝑗=1 homogeneous type in the sense of Coifman and Weiss [25]. To this end, we first note that when restricted to Schwartz On such type of spaces, Christ [26]constructedadyadicgrid −1/2 functions, 𝑋𝑗L are convolution operators with distribu- analogous to that of the Euclidean space as follows. −Δ tion kernels homogeneous of degree and coincide with Lemma A.1. 𝐺 smooth functions in 𝐺\{0}. This follows from the fact that the Let be a stratified Lie group. There exists a −1/2 {𝑄𝑘 :𝑘∈Z,𝛼 ∈ A } 𝐺 operator L is a convolution operator whose distribution collection 𝛼 𝑘 of open subsets of ,where A 𝛿 ∈ (0, 1) kernel is homogeneous of degree −Δ+ 1 and coincides with a 𝑘 is some (possibly finite) index set, and constants 𝐴 ,𝐴 >0 smooth function in 𝐺\{0}(see [2, Proposition 3.17]). Hence, and 1 2 such that −1/2 by Theorem 18, 𝑋𝑗L extend to bounded operators on 𝑘 𝑘 𝑘 (i) |𝐺 \ ⋃𝛼∈A 𝑄 |=0for each fixed 𝑘 and 𝑄 ∩𝑄 =0 𝐹̇𝛼 (𝐺) 𝑘 𝛼 𝛼 𝛽 𝑝,𝑞 . From this fact and the lifting property (Theorem 7), if 𝛼 =𝛽̸ ; we deduce that ℓ 𝑘 ℓ 󵄩 󵄩 󵄩 󵄩 (ii) for any 𝛼, 𝛽, 𝑘,ℓ with ℓ≥𝑘,either𝑄 ⊂𝑄 or 𝑄 ∩ 󵄩𝑋 𝑓󵄩 = 󵄩(𝑋 L−1/2) L1/2𝑓󵄩 𝛽 𝛼 𝛽 󵄩 𝑗 󵄩𝐹̇𝛼−1(𝐺) 󵄩 𝑗 󵄩𝐹̇𝛼−1(𝐺) 𝑘 𝑝,𝑞 𝑝,𝑞 𝑄𝛼 =0; 󵄩 󵄩 (92) 󵄩 1/2 󵄩 󵄩 󵄩 (iii) for each (𝑘, 𝛼) and ℓ<𝑘, there exists a unique 𝛽 such ≲ 󵄩L 𝑓󵄩 𝛼−1 ∼ 󵄩𝑓󵄩 ̇𝛼 . 󵄩 󵄩𝐹̇(𝐺) 󵄩 󵄩𝐹𝑝,𝑞(𝐺) 𝑘 ℓ 𝑝,𝑞 that 𝑄𝛼 ⊂𝑄𝛽; ] Hence ∑𝑗=1 ‖𝑋𝑗𝑓‖𝐹̇𝛼−1(𝐺) ≲‖𝑓‖𝐹̇𝛼 (𝐺).Toseetheconverse,we 𝑘 𝑘 𝑘 −1 𝑝,𝑞 𝑝,𝑞 (iv) diam(𝑄𝛼)≤𝐴1𝛿 ,wherediam(𝑄𝛼):=sup{|𝑥 𝑦| : need to use [2, Lemma 4.12], which asserts that there exists 𝑥, 𝑦𝑘 ∈𝑄 } 𝐾 ,...,𝐾 𝛼 ; tempered distributions 1 ] homogeneous of degree 𝑘 𝑘 𝑘 𝑘 −Δ+ 1 and coinciding with smooth functions in 𝐺\{0}such (v) each 𝑄𝛼 contains some ball 𝐵(𝑧𝛼,𝐴2𝛿 ),where𝑧𝛼 ∈𝐺. ] that 𝑓=∑ (𝑋𝑗𝑓) ∗𝑗 𝐾 for all 𝑓∈S(𝐺).Bythisresultand 𝑗=1 𝑘 −1/2 𝑄 Theorem 7,wehave,atleastfor𝑓∈R(𝐺) (⊂ Dom(L )), The set 𝛼 can be thought of as a dyadic cube with 𝛿𝑘 𝑧𝑘 D 󵄩 󵄩 󵄩 󵄩 diameter roughly and centered at 𝛼. We denote by the 󵄩𝑓󵄩 = 󵄩L (L−1/2𝑓)󵄩 𝑘 󵄩 󵄩𝐹̇𝛼 (𝐺) 󵄩 󵄩 ̇𝛼−1 𝐺 𝑘∈Z D ={𝑄 : 𝑝,𝑞 󵄩 󵄩𝐹𝑝,𝑞 (𝐺) family of all dyadic cubes on .For ,weset 𝑘 𝛼 𝛼∈A𝑘},sothatD =⋃ D𝑘.Foranydyadiccube𝑄∈D, 󵄩 󵄩 𝑘∈Z = 󵄩L (𝑓 ∗1 𝑅 )󵄩𝐹̇𝛼−1(𝐺) we denote by 𝑧𝑄 the “center” of 𝑄 and by 𝑘𝑄 theuniqueinteger 𝑝,𝑞 𝑘 𝑄∈D 󵄩 󵄩 such that 𝑘. 󵄩 ] 󵄩 𝛿= 󵄩 󵄩 Without loss of generality, in what follows we assume = 󵄩L (∑ (𝑋 𝑓) ∗ 𝐾 ∗𝑅 )󵄩 (93) 1/2 2𝑗 𝛿−𝑗 󵄩 𝑗 𝑗 1 󵄩 . Otherwise we need to replace in Definition 4 by , 󵄩 𝑗=1 󵄩 ̇𝛼−1 𝐹𝑝,𝑞 (𝐺) andalsomakesomeothernecessarychanges;see[27,pp.96– 󵄩 󵄩 98] for more details. 󵄩 ] 󵄩 󵄩 󵄩 = 󵄩∑ (𝑋𝑗𝑓) ∗ L (𝐾𝑗 ∗𝑅1)󵄩 , 𝑄 𝐿 󵄩 󵄩 Definition A.2. Let be a dyadic cube and let be a 󵄩𝑗=1 󵄩𝐹̇𝛼−1(𝐺) 𝑝,𝑞 nonnegative integer. A smooth function 𝑎𝑄 on 𝐺 is called a −1/2 smooth L-atom for 𝑄 if it satisfies where 𝑅1 is the convolution kernel of the operator L .As −𝑘𝑄 is indicated in [2,p.190],L(𝐾𝑗 ∗𝑅1) are distributions homo- (i) 𝑎𝑄 is supported in 𝐵(𝑧𝑄,(𝛾(1+𝐴1)/𝐴2)2 ); −Δ geneous of degree and coincide with smooth functions ∫𝑎 (𝑥)𝑃(𝑥)𝑑𝑥 =0 𝑃∈P away from 0.ThusitfollowsbyTheorem 18 that (ii) 𝑄 for all 𝐿; 󵄩 󵄩 󵄩 󵄩 |𝑋𝐼𝑎 (𝑥)| ≤ |𝑄|−(𝑑(𝐼)/Δ)−(1/2) 𝐼 󵄩(𝑋 𝑓) ∗ L (𝐾 ∗𝑅 )󵄩 ≲ 󵄩𝑋 𝑓󵄩 . (iii) 𝑄 for all multi-indices 󵄩 𝑗 𝑗 1 󵄩𝐹̇𝛼−1(𝐺) 󵄩 𝑗 󵄩𝐹̇𝛼−1(𝐺) (94) 𝑝,𝑞 𝑝,𝑞 with |𝐼| ≤ 𝐿. +1 14 Journal of Function Spaces and Applications

Definition A.3. Let 𝛼∈R and 0<𝑝,𝑞<∞.Thesequence Thus, it follows from Lemma 2 that 𝛼̇ space 𝑓𝑝,𝑞 consists of all sequences {𝑠𝑄}𝑄∈D such that the function 󵄨 󸀠 󵄨 󵄨 󸀠󸀠 󵄨 1/𝑞 󵄨𝑎 ∗(𝐷 𝑗 𝜙) (𝑥)󵄨 = 󵄨(𝐷 ℓ 𝑎 )∗(𝐷 𝑗 𝜙) (𝑥)󵄨 󵄨 𝑄 2 󵄨 󵄨 2 𝑄 2 󵄨 𝛼,𝑞 −(𝛼/Δ) −(1/2) 󵄨 󵄨 𝑞 𝑔 ({𝑠𝑄} )=(∑ (|𝑄| 󵄨𝑠𝑄󵄨 𝜒𝑄) ) (A.1) 𝑄 2−(𝑗∧ℓ)Δ (A.9) 𝑄∈D ≲2−ℓΔ/22−|𝑗−ℓ|𝐿 , 󵄨 󵄨 𝑁 𝑝 1+(2𝑗∧ℓ 󵄨𝑧−1𝑥󵄨) is in 𝐿 (𝐺).Forsuchasequence𝑠={𝑠𝑄}𝑄 we set 󵄨 𝑄 󵄨 󵄩 𝛼,𝑞 󵄩 ‖𝑠‖ 𝛼̇= 󵄩𝑔 (𝑠)󵄩 𝑝 . 𝑓𝑝,𝑞 󵄩 󵄩𝐿 (A.2) ̂ The smooth atomic decomposition of homogeneous where 𝜙∈A,and𝑁 can be taken to be arbitrarily large. Triebel-Lizorkin spaces on stratified Lie groups can be stated Consequently, as follows.

Theorem A.4. Let 𝛼∈R, 0<𝑝,𝑞<∞,andlet𝐿 be a 󵄨 󵄨 󵄨𝑎 ∗(𝐷 𝑗 𝜙) (𝑥)󵄨 nonnegative integer satisfying 𝐿>[Δmax(1, 1/𝑝, 1/𝑞) − Δ− 󵄨 𝑄 2 󵄨 𝛼] + 1 𝐶 󵄨 󵄨 . Then there is a constant Δ,𝑝,𝑞,𝛼 such that for every 󵄨 󸀠 −1 󵄨 = 󵄨𝑎𝑄 ∗(𝐷2𝑗 𝜙) 𝑄(𝑧 𝑥)󵄨 sequence of smooth 𝐿-atoms {𝑎𝑄}𝑄∈D and every sequence of 󵄨 󵄨 {𝑠 } (A.10) complex scalars 𝑄 𝑄∈D one has 2−(𝑗∧ℓ)Δ 󵄩 󵄩 ≲2−ℓΔ/22−|𝑗−ℓ|𝐿 . 󵄩 󵄩 󵄨 󵄨 𝑁 󵄩 󵄩 󵄩 󵄩 1+(2𝑗∧ℓ 󵄨𝑧−1𝑥󵄨) 󵄩 ∑ 𝑠 𝑎 󵄩 ≤𝐶 󵄩{𝑠 } 󵄩 . 󵄨 𝑄 󵄨 󵄩 𝑄 𝑄󵄩 Δ,𝑝,𝑞,𝛼󵄩 𝑄 𝑄∈D󵄩 𝛼̇ (A.3) 󵄩 󵄩 𝑓𝑝,𝑞 󵄩𝑄∈D 󵄩 ̇𝛼 󵄩 󵄩𝐹𝑝,𝑞(𝐺) 𝐶󸀠 Conversely, there is a constant Δ,𝑝,𝑞,𝛼 such that given any Note that if 𝑟 ∈ (0, 1] and 𝑁>Δ/𝑟then we have ̇𝛼 distribution 𝑓∈𝐹𝑝,𝑞(𝐺) and any 𝐿≥0, there exists a sequence of smooth 𝐿-atoms {𝑎𝑄}𝑄∈D such that 󵄨 󵄨 󵄨𝑠𝑄󵄨 𝑓= ∑ 𝑠𝑄𝑎𝑄, ∑ (A.4) 󵄨 −1 󵄨 𝑁 𝑄∈D 𝑄∈D (1 + 2𝑗∧ℓ 󵄨𝑧 𝑥󵄨) ℓ 󵄨 𝑄 󵄨 󸀠 where the sum converges in S (𝐺)/P and moreover 1/𝑟 󵄩 󵄩 󸀠 󵄩 󵄩 󵄨 󵄨𝑟 󵄩{𝑠 } 󵄩 ≤𝐶 󵄩𝑓󵄩 . 󵄨𝑠𝑄󵄨 󵄩 𝑄 𝑄󵄩 𝛼̇ Δ,𝑝,𝑞,𝛼󵄩 󵄩𝐹̇𝛼 (𝐺) (A.5) 𝑓𝑝,𝑞 𝑝,𝑞 ≤(∑ ) 󵄨 −1 󵄨 𝑁𝑟 𝑄∈D (1 + 2𝑗∧ℓ 󵄨𝑧 𝑥󵄨) ℓ 󵄨 𝑄 󵄨 Proof. Let 𝑄∈Dℓ for some ℓ∈Z,andlet𝑎𝑄 be a smooth 𝐿 𝑄 𝑎󸀠 (𝑥) = 𝑎 (𝑧 𝑥) 𝑎󸀠 -atom for .Weset 𝑄 𝑄 𝑄 .Then 𝑄 is supported 1/𝑟 󵄨 󵄨𝑟 −ℓ ℓ 󵄨𝑠 󵄨 𝑄 −1𝜒 𝑧 in 𝐵(0, (𝛾(1 +𝐴1)/𝐴2)2 );moreover(since1+2 |𝑦| : 1 for 󵄨 𝑄󵄨 | | 𝑄 ( ) 󸀠 =(∫ ∑ 𝑑𝑧) 𝑦∈supp 𝑎 ), 󵄨 −1 󵄨 𝑁𝑟 𝑄 𝐺 𝑄∈D (1 + 2𝑗∧ℓ 󵄨𝑧 𝑥󵄨) ℓ 󵄨 𝑄 󵄨 ℓ𝑑(𝐼)+ℓΔ 󵄨 𝐼 󸀠 󵄨 −ℓΔ/2 2 󵄨𝑋 𝑎 (𝑦)󵄨 ≲2 ,∀|𝐼| ≤𝐿+1, 󵄨 𝑄 󵄨 ℓ 󵄨 󵄨 𝑁 (A.6) 󵄨 󵄨𝑟 (1 + 2 󵄨𝑦󵄨) { 󵄨𝑠𝑄󵄨 𝜒𝑄 (𝑧) ≲2ℓΔ/𝑟 ∫ ( ∑ 󵄨 󵄨 )𝑑𝑧 { 𝑁𝑟 󸀠󸀠 −1 −(𝑗∧ℓ) 𝑗∧ℓ 󵄨 −1 󵄨 𝑁 𝑎 (𝑥) = |𝑥 𝑧|≤(2𝛾𝐴1)2 𝑄∈D (1 + 2 󵄨𝑧 𝑥󵄨) where canbechosentobearbitrarilylarge.Set 𝑄 { ℓ 󵄨 𝑄 󵄨 −ℓΔ 󸀠 −ℓ 󸀠 󸀠󸀠 2 𝑎𝑄(2 𝑥),thatis,𝑎𝑄 =𝐷2ℓ 𝑎𝑄.Thentheaboveinequality canberewrittenas ∞ 󵄨 󵄨 1 + ∑ ∫ 󵄨 𝐼 󸀠󸀠 󵄨 −ℓΔ/2 (2𝛾𝐴 )2𝑘2−(𝑗∧ℓ)≤|𝑥−1𝑧|≤(2𝛾𝐴 )2𝑘+12−(𝑗∧ℓ) 󵄨𝑋 𝑎 (𝑦)󵄨 ≲2 ,∀|𝐼| ≤𝐿+1. 𝑘=0 1 1 󵄨 𝑄 󵄨 󵄨 󵄨 𝑁 (A.7) (1 + 󵄨𝑦󵄨) 1/𝑟 𝑎󸀠󸀠 ⊂𝐵(0,(𝛾(1+ 󵄨 󵄨𝑟 Using this estimate, (73), and that supp 𝑄 󵄨𝑠𝑄󵄨 𝜒𝑄 (𝑧) } 𝐴 ))/𝐴 ) ×( ∑ )𝑑𝑧} 1 2 , we also deduce that 󵄨 −1 󵄨 𝑁𝑟 𝑄∈D (1 + 2𝑗∧ℓ 󵄨𝑧 𝑥󵄨) 󵄨 ∼ 󵄨 󵄨 󵄨 ℓ 󵄨 𝑄 󵄨 } 󵄨𝑋𝐼(𝑎󸀠󸀠) (𝑦) 󵄨 = 󵄨(𝑌𝐼𝑎󸀠󸀠)(𝑦−1)󵄨 󵄨 𝑄 󵄨 󵄨 𝑄 󵄨

󵄨 −1󵄨𝑑(𝐽)−𝑑(𝐼) 󵄨 𝐽 󸀠󸀠 −1 󵄨 { ≲ ∑ 󵄨𝑦 󵄨 󵄨(𝑋 𝑎 )(𝑦 )󵄨 ℓΔ/𝑟 󵄨 󵄨𝑟 󵄨 󵄨 󵄨 𝑄 󵄨 ≲2 ∫ −1 −(𝑗∧ℓ) ( ∑ 󵄨𝑠𝑄󵄨 𝜒𝑄 (𝑧))𝑑𝑧 { |𝑥 𝑧|≤(2𝛾𝐴1)2 |𝐽|≤|𝐼| 𝑄∈D 𝑑(𝐽)≥𝑑(𝐼) (A.8) { ℓ

−ℓΔ/2 1 ∞ ≲2 ,∀|𝐼| ≤𝐿+1. 𝑘 −𝑁𝑟 󵄨 󵄨 𝑁 + ∑(1 + 𝐴 2 ) (1 + 󵄨𝑦󵄨) 1 󵄨 󵄨 𝑘=0 Journal of Function Spaces and Applications 15

−2𝑘 𝛼 ̂ 0 ̇ × ∫ 𝜓(2 ⋅) ∈ A.Given𝑓∈𝐹𝑝,𝑞(𝐺),itfollowsfromLemma 3 (2𝛾𝐴 )2𝑘2−(𝑗∧ℓ)≤|𝑥−1𝑧|≤(2𝛾𝐴 )2𝑘+12−(𝑗∧ℓ) 1 1 that

1/𝑟 𝑓=∑𝑓∗(𝐷 𝑗 𝜓) ∗ (𝐷 𝑗 𝜁) } 2 2 (A.12) 󵄨 󵄨𝑟 𝑗∈Z ×( ∑ 󵄨𝑠𝑄󵄨 𝜒𝑄 (𝑧))𝑑𝑧} 𝑄∈D ℓ } 󸀠 with the convergence in S (𝐺)/P. Let us decompose 𝑓 as

−(𝑗∧ℓ)Δ ℓΔ/𝑟 { 2 −1 ≲2 ∫ 𝑓=∑ ∑ ∫ 𝑓∗(𝐷2𝑗 𝜓) (𝑦)2 (𝐷 𝑗 𝜁) (𝑦 𝑥) 𝑑𝑦 {󵄨 −(𝑗∧ℓ) 󵄨 󵄨𝐵(𝑥,2𝛾𝐴 2 )󵄨 𝐵(𝑥,2𝛾𝐴 2−(𝑗∧ℓ)) 𝑗∈Z 𝑄∈D 𝑄 {󵄨 1 󵄨 1 𝑗 (A.13) = ∑ ∑ 𝑠𝑄𝑎𝑄, 𝑗∈Z 𝑄∈D 󵄨 󵄨𝑟 𝑗 ×( ∑ 󵄨𝑠𝑄󵄨 𝜒𝑄 (𝑧))𝑑𝑧 𝑄∈Dℓ wherewehaveset,for𝑄∈D𝑗, ∞ (𝑘+1)Δ −(𝑗∧ℓ)Δ 𝑘 −𝑁𝑟 2 2 1/2 󵄨 󵄨 󵄩 𝐼 󵄩 + ∑(1 + 𝐴 2 ) 𝑠 ≡ |𝑄| 󵄨𝑓∗(𝐷 𝑗 𝜓) (𝑦)󵄨 󵄩𝑋 𝜁󵄩 , 1 󵄨 𝑘+1 −(𝑗∧ℓ) 󵄨 𝑄 sup 󵄨 2 󵄨 sup 󵄩 󵄩𝐿1 𝑘=0 󵄨𝐵(𝑥,2𝛾𝐴12 2 )󵄨 𝑦∈𝑄 |𝐼|≤𝐿+1 (A.14) 1 −1 𝑎𝑄 (𝑥) ≡ ∫ 𝑓∗(𝐷2𝑗 𝜓) (𝑦)2 (𝐷 𝑗 𝜁) (𝑦 𝑥) 𝑑𝑦. × ∫ 𝑠𝑄 𝑄 𝑘+1 −(𝑗∧ℓ) 𝐵(𝑥,2𝛾𝐴12 2 )

𝑎𝑄 1/𝑟 It is straightforward to verify that is supported in 𝐵(𝑧 ,(𝛾(1 + 𝐴 )/𝐴 )2−𝑗) 𝑎 󵄨 󵄨𝑟 } 𝑄 1 2 ,andthat 𝑄 has vanishing ×(∑ 󵄨𝑠 󵄨 𝜒 (𝑧))𝑑𝑧 󵄨 𝑄󵄨 𝑄 } moments of the same order as 𝜁.Moreover,for𝑄∈D𝑗 we 𝑄∈D ℓ } have

Δ𝑗 𝑑(𝐼)𝑗 1/𝑟 󵄨 󵄨 2 2 󵄨 󵄨 ∞ 󵄨 𝐼 󵄨 󵄨 𝐼 𝑗 −1 󵄨 ℓΔ/𝑟 −(𝑗∧ℓ)Δ 𝑘 −𝑁𝑟 (𝑘+1)Δ −(𝑗∧ℓ)Δ 󵄨𝑋 𝑎𝑄󵄨 ≤ ∫ 󵄨𝑓∗(𝐷2𝑗 𝜓) (𝑦) (𝑋 𝜁) (2 (𝑦 𝑥))󵄨 𝑑𝑦 ≤2 {2 + ∑(1 + 𝐴12 ) 2 2 } 𝑠𝑄 𝑄 𝑘=0 𝑑(𝐼)𝑗 2 󵄩 𝐼 󵄩 󵄨 󵄨 ≤ 󵄩(𝑋 𝜁)󵄩 󵄨𝑓∗(𝐷 𝑗 𝜓) (𝑦)󵄨 1/𝑟 𝑠 󵄩 󵄩𝐿1 sup 󵄨 2 󵄨 󵄨 󵄨𝑟 𝑄 𝑦∈𝑄 × {𝑀 ( ∑ 󵄨𝑠𝑄󵄨 𝜒𝑄) (𝑥)} 𝑄∈D ℓ ≤ |𝑄|−(𝑑(𝐼)/Δ)−(1/2)

1/𝑟 (A.15) { 󵄨 󵄨𝑟 } ≲2ℓΔ/𝑟2−(𝑗∧ℓ)Δ/𝑟 𝑀( ∑ 󵄨𝑠 󵄨 𝜒 ) (𝑥) { 󵄨 𝑄󵄨 𝑄 } for all |𝐼| ≤ 𝐿+1.Hencethefunction𝑎𝑄 asmooth𝐿-atom for 𝑄∈D { ℓ } 𝑄.Chooseany𝜆>Δ/min{𝑝, 𝑞}.Wenotethat

𝑞 1/𝑟 ∑ ( 𝑄 −(𝛼/Δ)−(1/2)𝑠 𝜒 𝑥 ) { } | | 𝑄 𝑄 ( ) max(ℓ−𝑗,0)Δ/𝑟 󵄨 󵄨𝑟 𝑄∈D =2 {𝑀( ∑ 󵄨𝑠𝑄󵄨 𝜒𝑄) (𝑥)} . 𝑗 𝑄∈D { ℓ } 𝑞 𝑗𝛼 󵄨 󵄨 (A.11) ∼ ∑ (2 sup 󵄨𝑓∗(𝐷2𝑗 𝜓) (𝑦)󵄨 𝜒𝑄 (𝑥)) 𝑄∈D 𝑦∈𝑄 −1 𝑗 (A.16) Here we used the fact that for 𝑧∈𝑄and |𝑥 𝑧| ≥ 𝑘 −(𝑗∧ℓ) 𝑗∧ℓ −1 𝑁𝑟 −𝑘 𝑁𝑟 𝑗𝛼 󵄨 󵄨 𝑞 (2𝛾𝐴1)2 2 one has (1 + 2 |𝑧 𝑥|) ≥(1+𝐴12 ) , 󵄨 󵄨 𝑄 ≲ sup [2 󵄨𝑓∗(𝐷2𝑗 𝜓) (𝑦)󵄨] −𝑗 which can be easily verified by using Lemma 19 (iv) and the 𝑦∈𝐵(𝑥,𝐴12 ) quasi-triangle inequality satisfied by the homogeneous norm. 𝑗𝛼 ∗ ̂ 𝑞 The above estimate and (A.10), along with the argument in ≲[2 𝑀𝜆,2−𝑗 (𝑓, 𝜓) (𝑥)] . [24, pp. 80-81], yield (A.3). Now we show the converse statement of the theorem. ̂ + ̂ We thus obtain, using (38), By Lemma 20, there exists 𝜁∈S(R ) such that |𝜁(𝜆)| ≥ −2(𝑘 +1) −2(𝑘 −1) 󵄩 󵄩 𝐶>0for 𝜆∈[2 0 ,2 0 ] for some (large) 󵄩 1/𝑞󵄩 󵄩 󵄩 󵄩 󵄨 󵄨𝑞 󵄩 𝑘 𝜁 𝐵(0, 1) 󵄩 󵄩 󵄩 󵄨 𝑗𝛼 ∗ ̂ 󵄨 󵄩 󵄩 󵄩 positive integer 0,andthat is supported in and has 󵄩{𝑠𝑄} 󵄩 ≲ 󵄩(∑󵄨2 𝑀𝜆,2−𝑗 (𝑓, 𝜓)󵄨 ) 󵄩 ≲ 󵄩𝑓󵄩 ̇𝛼 . 󵄩 𝑄󵄩𝑓𝛼̇ 󵄩 󵄨 󵄨 󵄩 󵄩 󵄩𝐹𝑝,𝑞(𝐺) 𝐿 𝑝,𝑞 󵄩 𝑗∈Z 󵄩 vanishing moments of order .Thenitispossibletofind 󵄩 󵄩𝐿𝑝 𝜓∈̂ S(R+) 𝜓⊂ afunction with the properties that supp (A.17) −2(𝑘 +1) −2(𝑘 −1) −2(𝑘 +1) −2(𝑘 −1) [2 0 ,2 0 ], |𝜓(𝜆)|̂ >0 for 𝜆∈(2 0 ,2 0 ), −2𝑗 ̂ −2𝑗 + and ∑𝑗∈Z 𝜓(2̂ 𝜆)𝜁(2 𝜆) = 1 for 𝜆∈R .Notethat This proves (A.5). 16 Journal of Function Spaces and Applications

Acknowledgments [18] L. Paiv¨ arinta,¨ “Equivalent quasinorms and Fourier multipliers 𝑠 in the Triebel spaces 𝐹𝑝,𝑞,” Mathematische Nachrichten,vol.106, The author would like to thank Professor Hitoshi Arai for his pp. 101–108, 1982. patient guidance and constant support. He is also grateful to [19] J.-O. Stromberg¨ and A. Torchinsky, Weighted Hardy Spaces,vol. Professor Yoshihiro Sawano for his valuable comments. 1381 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1989. References [20] M. Bownik, B. Li, D. Yang, and Y. Zhou, “Weighted anisotropic product Hardy spaces and boundedness of sublinear operators,” [1] Y. Han, D. Muller,¨ and D. Yang, “A theory of Besov and Triebel- Mathematische Nachrichten,vol.283,no.3,pp.392–442,2010. Lizorkin spaces on metric measure spaces modeled on Carnot- [21] E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthog- Caratheodory´ spaces,” Abstract and Applied Analysis,vol.2008, onality, and Oscillatory Integrals,vol.43ofPrinceton Mathemat- ArticleID893409,250pages,2008. ical Series, Princeton University Press, Princeton, NJ, USA, 1993. [2] G. B. Folland, “Subelliptic estimates and function spaces on [22] L. Grafakos and X. Li, “Bilinear operators on homogeneous nilpotent Lie groups,” Arkiv for¨ Matematik,vol.13,no.2,pp.161– groups,” Journal of Operator Theory,vol.44,no.1,pp.63–90, 207, 1975. 2000. [3]G.B.FollandandE.M.Stein,Hardy Spaces on Homogeneous [23] N. Bourbaki, Groupes et Algebres` de Lie,(Elements´ de Math., Groups,vol.28ofMathematical Notes, Princeton University Fasc. 26 et 37), Chapter I–III, Hermann, Paris, France, 1960 and Press, Princeton, NJ, USA, 1982. 1972. [4] K. Saka, “Besov spaces and Sobolev spaces on a nilpotent Lie [24] L. Grafakos, Modern Fourier Analysis,vol.250ofGraduate Texts group,” The Tohokuˆ Mathematical Journal,vol.31,no.4,pp.383– in Mathematics, Springer, New York, NY, USA, 2nd edition, 437, 1979. 2009. [5] H. Fuhr¨ and A. Mayeli, “Homogeneous Besov spaces on [25] R. R. Coifman and G. Weiss, “Extensions of Hardy spaces and stratified Lie groups and their wavelet characterization,” Journal their use in analysis,” Bulletin of the American Mathematical of Function Spaces and Applications,vol.2012,ArticleID523586, Society,vol.83,no.4,pp.569–645,1977. 41 pages, 2012. [26] M. Christ, “A 𝑇(𝑏) theorem with remarks on analytic capacity [6] A. W. Knapp and E. M. Stein, “Intertwining operators for and the Cauchy integral,” Colloquium Mathematicum,vol.60- semisimple groups,” Annals of Mathematics,vol.93,pp.489– 61,no.2,pp.601–628,1990. 578, 1971. [27] Y. S. Han and E. T. Sawyer, “Littlewood-Paley theory on [7] A. Koranyi´ and S. Vagi,´ “Singular integrals on homogeneous spaces of homogeneous type and the classical function spaces,” spaces and some problems of classical analysis,” Annali della Memoirs of the American Mathematical Society,vol.110,no.530, Scuola Normale Superiore di Pisa,vol.25,pp.575–648,1971. pp. 1–126, 1994. 𝑝 [8] M. Christ, “𝐿 bounds for spectral multipliers on nilpotent groups,” Transactions of the American Mathematical Society,vol. 328, no. 1, pp. 73–81, 1991. [9] A. Hulanicki, “A functional calculus for Rockland operators on nilpotent Lie groups,” Studia Mathematica,vol.78,no.3,pp. 253–266, 1984. [10] D. Geller and A. Mayeli, “Continuous wavelets and frames on stratified Lie groups. I,” The Journal of Fourier Analysis and Applications,vol.12,no.5,pp.543–579,2006. [11] M. Frazier and B. Jawerth, “A discrete transform and decompositions of distribution spaces,” Journal of Functional Analysis, vol.93,no.1,pp.34–170,1990. [12] A. Bonfiglioli, E. Lanconelli, and F. Uguzzoni, Stratified Lie Groups and Potential Theory for Their Sub-Laplacians, Springer Monographs in Mathematics, Springer, Berlin, Germany, 2007. [13]M.ReedandB.Simon,Methods of Modern Mathematical Physics. I: Functional Analysis, Academic Press, New York, NY, USA, 2nd edition, 1980. [14] L. Grafakos, L. Liu, and D. Yang, “Vector-valued singular integrals and maximal functions on spaces of homogeneous type,” Mathematica Scandinavica,vol.104,no.2,pp.296–310, 2009. [15]V.S.Rychkov,“OnatheoremofBui,Paluszynski,´ and Taible- son,” ProceedingsoftheSteklovInstituteofMathematics,vol.227, pp.280–292,1999. [16] H. Triebel, TheoryofFunctionSpaces,vol.78ofMonographs in Mathematics,Birkhauser,¨ Basel, Switzerland, 1983. [17] W. Rudin, Functional Analysis, International Series in Pure and Applied Mathematics, McGraw-Hill, New York, NY, USA, 2nd edition, 1991. Hindawi Publishing Corporation Journal of Function Spaces and Applications Volume 2013, Article ID 865835, 12 pages http://dx.doi.org/10.1155/2013/865835

Research Article A Note on Weighted Besov-Type and Triebel-Lizorkin-Type Spaces

Canqin Tang

Department of Mathematical, Dalian Maritime University, Dalian, Liaoning 116026, China

Correspondence should be addressed to Canqin Tang; [email protected]

Received 16 November 2012; Revised 1 February 2013; Accepted 1 February 2013

Academic Editor: Jozef´ Bana´s

Copyright © 2013 Canqin Tang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

̇𝑠,𝜏 𝑛 Let 𝑠, 𝜏 ∈ R, 𝑞 ∈ (0, ∞],and𝜔 be in the class 𝐴∞ of Muckenhoupt. We introduce the weighted Besov-type spaces 𝐵𝑝,𝑞,𝜔(R ) and ̇𝑠,𝜏 𝑛 weighted Triebel-Lizorkin-type spaces 𝐹𝑝,𝑞,𝜔(R ) for 𝑝∈(0,∞)and then establish the 𝜑-transform characterizations of these new spaces in the sense of Frazier and Jawerth.

𝑓(𝜉)̂ ≡∫ 𝑓(𝑥)𝑒−𝑖𝑥⋅𝜉 𝑑𝑥 1. Introduction where R𝑛 .Throughoutthepaper,forall 𝑛 𝑗𝑛 𝑗 𝑗∈Z and 𝑥∈R ,weput𝜑𝑗(𝑥) ≡ 2 𝜑(2 𝑥). Function spaces have been a central topic in modern analysis As in [16], we set and are now of increasing applications in areas such as harmonic analysis and partial differential equations. 𝑛 𝑛 𝛾 Since Besov spaces and Triebel-Lizorkin spaces were S∞ (R )≡ {𝜑∈S (R ):∫ 𝜑 (𝑥) 𝑥 𝑑𝑥 =0 introduced in [1–3],thesespacesbecamethefocusofmany R𝑛 (5) scholars. Bui studied weighted Besov and Triebel-Lizorkin 𝑛 spaces in [4, 5]. In 2010, Yuan et al. gave a unified treatment for all multi-indices 𝛾∈(N ∪ {0}) }. of Morrey, Campanato, Besov, Lizorkin, and Triebel spaces in [6]. A series of research results on these topics can be found S󸀠 (R𝑛) S (R𝑛) in [6–15].TheydevelopatheoryofspacesofBesov-Triebel- Let ∞ be the topological dual of ∞ ,namely,the S (R𝑛) LizorkintypebuiltonMorreyspaces.Basedontheirwork, set of all continuous linear functionals on ∞ .Endowed S󸀠 (R𝑛) ∗ S󸀠 (R𝑛) this note will generalize these spaces to the weighted cases. ∞ with the weak -topology, then ∞ is complete; 𝑛 𝑛 Let S(R ) be the space of all Schwartz functions on R . see [17]. 𝑛 𝐴 =∪ 𝐴 𝜔∈𝐴 Let 𝜑 and 𝜓 be functions on R satisfying Let ∞ 𝑝≥1 𝑝.If ∞, then there exist independent positive constants 𝐶𝑖 and 𝛼𝑖, 𝑖=1,2,suchthat

𝛼 𝛼 |𝐸| 1 𝜔 (𝐸) |𝐸| 2 𝑛 𝐶 ( ) ≤( )≤𝐶( ) 𝜑, 𝜓 ∈ S (R ) , (1) 1 |𝑄| 𝜔 (𝑄) 2 |𝑄| (6) 1 󵄨 󵄨 𝜑,̂ 𝜓⊂{𝜉∈̂ R𝑛 : ≤ 󵄨𝜉󵄨 ≤2}, supp 2 󵄨 󵄨 (2) for each cube 𝑄 and each subcube 𝐸⊃𝑄. Before stating our theorems on the weighted Triebel- 󵄨 󵄨 󵄨 󵄨 3 󵄨 󵄨 5 󵄨𝜑̂ (𝜉)󵄨 , 󵄨𝜓̂ (𝜉)󵄨 ≥𝐶>0 if ≤ 󵄨𝜉󵄨 ≤ , (3) Lizorkin-type spaces, we first give the definition of these 5 3 spaces. ∑𝜑(2̂ 𝑗𝜉)𝜓(2̂ 𝑗𝜉) = 1 𝜉 =0,̸ if (4) Definition 1. Let 𝜏, 𝑠 ∈ R, 𝑞 ∈ (0, ∞], 𝜔∈𝐴∞ and let 𝜑 be a 𝑗∈Z Schwartz function satisfying (1)through(3). 2 Journal of Function Spaces and Applications

(i) Let 𝑝 ∈ (0, ∞]. The weighted Besov-type space Theorem 3. Let 𝜔∈𝐴∞, 𝑝, 𝑞 ∈ (0, ∞], 𝜀 ∈ ((𝑛(𝛼2 − ̇𝑠,𝜏 𝑛 󸀠 𝑛 𝐵𝑝,𝑞,𝜔(R ) is defined to be the set of all 𝑓∈S∞(R ) 𝛼1))/𝑝, ∞), 𝑠∈R,and𝜏∈[0,(𝛼1/𝑝) + (𝜀/2𝑛)).Thenall𝜀- 𝑠,𝜏̇ 𝑛 ‖𝑓‖ ̇𝑠,𝜏 𝑛 <∞ almost diagonal operators are bounded on 𝑎𝑝,𝑞,𝜔(R ). such that 𝐵𝑝,𝑞,𝜔(R ) ,where 󵄩 󵄩 𝛼 =𝛼 =1 𝜀 ∈ (0, ∞) 𝜏∈ 󵄩𝑓󵄩𝐵̇𝑠,𝜏 (R𝑛) Remark 4. Let 1 2 ,then and 𝑝,𝑞,𝜔 [0, (1/𝑝) + (𝜀/2𝑛)).Thenall𝜀-almost diagonal operators are 𝑠,𝜏 𝑛 1 bounded on 𝑎𝑝,𝑞̇ (R ), which is just the results established by ≡ sup 𝜏 𝑃 |𝑃| Yuan et al.; see [6, Theorem 3.1] and [10,Theorem4.1]. dyadic (7) 1/𝑞 ∞ This paper is organized as follows. In Section 2,we { 󵄨 󵄨 𝑝 𝑞/𝑝} 𝑗𝑠 󵄨 󵄨 establish the 𝜑-transform characterizations of the spaces × { ∑ [∫ (2 󵄨𝜑𝑗 ∗𝑓(𝑥)󵄨) 𝜔 (𝑥) 𝑑𝑥] } ̇𝑠,𝜏 𝑛 𝑃 𝐴 (R ). And the boundedness of almost diagonal oper- 𝑗=𝑗𝑃 𝑝,𝑞,𝜔 { } ̇𝑠,𝜏 𝑛 ators on 𝐴𝑝,𝑞,𝜔(R ) is considered in Section 3. 𝑝=∞ 𝑞= with suitable modifications made when or At the end of this section, we make some conventions ∞ . on notation.Throughoutthepaper,𝐶 denotes unspecified (ii) Let 𝑝 ∈ (0, ∞). The weighted Triebel-Lizorkin-type positive constants, possibly different at each occurrence; the ̇𝑠,𝜏 𝑛 𝑋≲𝑌 𝐶 space 𝐹𝑝,𝑞,𝜔(R ) is defined to be the set of all 𝑓∈ symbol means that there exists a positive constant 󸀠 𝑛 𝑋≤𝐶𝑌 𝑋∼𝑌 𝐶−1𝑌≤𝑋≤𝐶𝑌 S (R ) ‖𝑓‖ 𝑠,𝜏 <∞ such that ,and means . ∞ such that 𝐹̇(R𝑛) ,where 𝑛 𝑛 𝑝,𝑞,𝜔 For any 𝜑∈S(R ),weset𝜑(𝑥)̃ ≡ 𝜑(−𝑥) for all 𝑥∈R .For 󵄩 󵄩 𝑛 󵄩 󵄩 𝑘=(𝑘1,...,𝑘𝑛)∈Z and 𝑗∈Z, 𝑄𝑗𝑘 denotes the dyadic cube 󵄩𝑓󵄩 ̇𝑠,𝜏 𝑛 𝐹𝑝,𝑞,𝜔(R ) 𝑗 𝑄𝑗𝑘 ≡{(𝑥1,...,𝑥𝑛):𝑘𝑖 ≤2𝑥𝑖 <𝑘𝑖 +1for 𝑖 = 1,...,𝑛}and 1 −𝑗 ≡ Q ≡{𝑄𝑗𝑘}𝑗,𝑘. We denote by 𝑥𝑄 the lower left-corner 2 𝑘 of sup |𝑃|𝜏 𝑃 dyadic 𝑄=𝑄𝑗𝑘.Throughoutthepaper,whendyadiccube𝑄 appears (8) ∑ {⋅} 𝑄 1/𝑝 as an index, such as 𝑄 and 𝑄,itisunderstoodthat runs 𝑝/𝑞 R𝑛 𝑄 { ∞ } over all dyadic cubes in .Foreachcube ,wedenoteitsside 𝑗𝑠 󵄨 󵄨 𝑞 × ∫ [ ∑ (2 󵄨𝜑 ∗𝑓(𝑥)󵄨) ] 𝜔 (𝑥) 𝑑𝑥 length by 𝑙(𝑄) and its center by 𝑐𝑄,andfor𝑟>0, we denote { 󵄨 𝑗 󵄨 } 𝑃 𝑗=𝑗 by 𝑟𝑄 the cube concentric with 𝑄 having the side length 𝑟𝑙(𝑄). { [ 𝑃 ] } 𝑛 Let 𝐸 be a set of R .Denoteby𝜒𝐸 its characteristic function ∘ with suitable modifications made when 𝑞=∞, and 𝐸 its interior.Also,setN ≡ {1, 2, . . .} and Z+ ≡ N ∪{0}. where 𝑙(𝑃) is the side length of dyadic cube 𝑃, 𝑗𝑃 ≡ − 𝑙(𝑃) log2 , and the supremum is taken over all dyadic 𝜑 𝐴̇𝑠,𝜏 (R𝑛) cubes 𝑃. 2. -Transform Characterizations of 𝑝,𝑞,𝜔

Obviously, letting 𝜏=0,theabovespacesareweighted In this section, we establish the 𝜑-transform characterizations ̇𝑠 𝑛 𝐴̇𝑠,𝜏 (R𝑛) Besov space 𝐵𝑝,𝑞,𝜔(R ) and weighted Triebel-Lizorkin space of the spaces 𝑝,𝑞,𝜔 .Tothisend,weintroducetheir ̇𝑠 𝑛 ̇𝑠,𝜏 𝑛 corresponding sequence spaces as follows. 𝐹𝑝,𝑞,𝜔(R ) in [5, 18]. If 𝑝 = 𝑞 ∈ (0, ∞),then𝐵𝑝,𝑞,𝜔(R )= 𝑠,𝜏 𝑛 𝑠,𝜏 𝑛 𝐹̇ (R ). For simplicity, in what follows, we use 𝐴̇ (R ) 𝑝,𝑞,𝜔 𝑝,𝑞,𝜔 Definition 5. Let 𝜏, 𝑠 ∈ R, 𝑞 ∈ (0, ∞],and𝜔∈𝐴∞. ̇𝑠,𝜏 𝑛 ̇𝑠,𝜏 𝑛 ̇𝑠,𝜏 𝑛 to denote either 𝐵𝑝,𝑞,𝜔(R ) or 𝐹𝑝,𝑞,𝜔(R ).If 𝐴𝑝,𝑞,𝜔(R ) means 𝐹̇𝑠,𝜏 (R𝑛) 𝑝=∞ 𝑝,𝑞,𝜔 , then the case is excluded. Below are the ̇𝑠,𝜏 𝑛 (i) Let 𝑝∈(0,∞]. The sequence space 𝑏 (R ) is main results of the paper. 𝑝,𝑞,𝜔 {𝑡 } ⊂ C Let 𝜑 and 𝜓 satisfy (1)through(4). Recall that the 𝜑- defined to be the set of all sequences 𝑄 𝑄 such that ‖𝑡‖𝑏̇𝑠,𝜏 (R𝑛) <∞,where transform 𝑆𝜑 isdefinedtobethemaptakingeach𝑓∈ 𝑝,𝑞,𝜔 󸀠 𝑛 S∞(R ) to the sequence 𝑆𝜑𝑓={(𝑆𝜑𝑓)𝑄}𝑄,where(𝑆𝜑𝑓)𝑄 ≡ ⟨𝑓,𝜑𝑄⟩ for all dyadic cubes 𝑄;theinverse𝜑-transform 𝑇𝜓 ‖𝑡‖𝑏̇𝑠,𝜏 (R𝑛) is defined to be the map taking a sequence 𝑡={𝑡𝑄}𝑄 to 𝑝,𝑞,𝜔 𝑇 𝑡=∑𝑡 𝜓 𝜓 𝑄 𝑄 𝑄;see,forexample,[19, 20]. The following 1 𝜑 theorem is about the -transform characterizations of the ≡ sup 𝜏 ̇𝑠,𝜏 𝑛 𝑃 |𝑃| spaces 𝐴𝑝,𝑞,𝜔(R ). dyadic 1/𝑞 𝑝 𝑞/𝑝 Theorem 2. Let 𝑠∈R, 𝜏∈[0,∞), 𝑝, 𝑞 ∈ (0, ∞],and𝜑 and 𝜓 { ∞ } ̇𝑠,𝜏 𝑛 𝑗𝑠𝑞 󵄨 󵄨 satisfy (1) through (4).Thentheoperators𝑆𝜑 : 𝐴 (R )→ × ∑2 [∫ ( ∑ 󵄨𝑡 󵄨 𝜒̃ (𝑥)) 𝜔 (𝑥) 𝑑𝑥] 𝑝,𝑞,𝜔 { 󵄨 𝑄󵄨 𝑄 } 𝑠,𝜏 𝑛 𝑠,𝜏 𝑛 𝑠,𝜏 𝑛 𝑃 −𝑗 𝑎̇ (R ) 𝑇 : 𝑎̇ (R )→𝐴̇ (R ) 𝑗=𝑗𝑃 𝑙(𝑄)=2 𝑝,𝑞,𝜔 and 𝜓 𝑝,𝑞,𝜔 𝑝,𝑞,𝜔 are bounded. { [ ] } ̇𝑠,𝜏 𝑛 Furthermore, 𝑇𝜓 ∘𝑆𝜑 is the identity on 𝐴𝑝,𝑞,𝜔(R ). (9)

Next is the result of an 𝜀-almost diagonal operator 𝑠,𝜏 𝑛 −1/2 (defined in Section 3)on𝑎𝑝,𝑞,𝜔̇ (R ). and 𝜒̃𝑄 ≡|𝑄| 𝜒𝑄. Journal of Function Spaces and Applications 3

𝑠,𝜏̇ 𝑛 (ii) Let 𝑝 ∈ (0, ∞). The sequence space 𝑓𝑝,𝑞,𝜔(R ) is Lemma 8. Suppose 0<𝑎≤𝑟<∞and 𝜆>𝑛𝑟𝛼2/𝑎.Let 𝜇, V ∈ Z 𝜇≤V 𝑥∈𝑄 𝑙(𝑄)− =2 V defined to be the set of all sequences {𝑡𝑄}𝑄 ⊂ C such and .Thenforeach and ,we ‖𝑡‖ 𝑠,𝜏̇ 𝑛 <∞ have that 𝑓𝑝,𝑞,𝜔(R ) ,where 1/𝑟 𝑟 ‖𝑡‖ 𝑠,𝜏̇ 𝑛 󵄨 󵄨 𝑓𝑝,𝑞,𝜔(R ) 󵄨𝑡 󵄨 ( ∑ 󵄨 𝑅󵄨 ) 󵄨 󵄨 𝜆 1 −𝜇 −1 󵄨 󵄨 𝑙(𝑅)=2 (1 + 𝑙(𝑅) 󵄨𝑥𝑅 −𝑥𝑄󵄨) ≡ sup 𝜏 𝑃 |𝑃| (14) dyadic 1/𝑎 𝑎 𝑝/𝑞 1/𝑝 [ 󵄨 󵄨 ] ≤𝐶 𝑀𝜔 ( ∑ 󵄨𝑡𝑅󵄨 𝜒𝑘) (𝑥) . { −𝑠/𝑛 󵄨 󵄨 𝑞 } 󵄨 󵄨 𝑙(𝑅)=2−𝜇 × {∫ [ ∑ (|𝑄| 󵄨𝑡𝑄󵄨 𝜒̃𝑄 (𝑥)) ] 𝜔 (𝑥) 𝑑𝑥} . [ ] { 𝑃 𝑄⊂𝑃 } (10) Proof. Assume 𝑥𝑄 =0.Let 󵄨 󵄨 Obviously, we have 󵄨𝑥 󵄨 𝐴 ={𝑅 :𝑙(𝑅) =2−𝜇, 󵄨 𝑅󵄨 ≤1}, 0 dyadic 𝑙 (𝑅) ‖𝑡‖𝑏̇𝑠,𝜏 (R𝑛) 𝑝,𝑞,𝜔 󵄨 󵄨 (15) 󵄨𝑥 󵄨 1 −𝜇 𝑘−1 󵄨 𝑅󵄨 𝑘 = 𝐴𝑘 ={𝑅dyadic :𝑙(𝑅) =2 ,2 < ≤2} sup |𝑃|𝜏 𝑙 (𝑅) 𝑃 dyadic

𝑞/𝑝 1/𝑞 for 𝑘=1,2,3,.... { } 𝐵⊃∪̃ 𝑅 { ∞ [ ] } Denote 𝑅∈𝐴𝑘 ,then [ −𝑠/𝑛−1/2 1/𝑝 󵄨 󵄨 𝑝] × ∑ ∑ (|𝑄| 𝜔(𝑄) 󵄨𝑡𝑄󵄨) . { [ 󵄨 󵄨 ] } 󵄨 󵄨𝑟 {𝑗=𝑗 −𝑗 } 󵄨 󵄨 𝑃 𝑙(𝑄)=2 󵄨𝑡𝑅󵄨 𝑄⊂𝑃 ∑ { [ ] } 󵄨 󵄨 𝜆 (1 + 󵄨𝑥 󵄨 /𝑙 (𝑅)) (11) 𝑅∈𝐴𝑘 󵄨 𝑅󵄨 ̇𝑠,𝜏 𝑛 −𝑘𝜆 󵄨 󵄨𝑟 In a similar manner to consider 𝐴𝑝,𝑞,𝜔(R ), we use ≤𝐶2 ∑ 󵄨𝑡𝑅󵄨 𝑠,𝜏 𝑛 ̇𝑠,𝜏 𝑛 𝑠,𝜏̇ 𝑛 𝑅∈𝐴𝑘 𝑎𝑝,𝑞,𝜔̇ (R ) to denote either 𝑏𝑝,𝑞,𝜔(R ) or 𝑓𝑝,𝑞,𝜔(R ). If 𝑠,𝜏 𝑛 𝑠,𝜏̇ 𝑛 𝑟/𝑎 𝑎𝑝,𝑞,𝜔̇ (R ) means 𝑓𝑝,𝑞(R ), then the case 𝑝=∞is excluded. −𝑘𝜆 󵄨 󵄨𝑎 To prove Theorem 2, we need some technical lemmas. ≤𝐶2 ( ∑ 󵄨𝑡𝑅󵄨 ) The next lemma is a special case of[21, Lemma 2.11]. 𝑅∈𝐴𝑘

󵄨 󵄨𝑎 𝑟/𝑎 Lemma 6. Let 𝛿∈R, 𝜔∈𝐴∞. Then there exist positive 󵄨 󵄨 (16) −𝑘𝜆 󵄨𝑡𝑅󵄨 constants 𝐿0 and 𝐶 such that for all 𝑗∈Z, ≤𝐶2 (∫ ∑ 𝜒𝑅𝜔 (𝑥) 𝑑𝑥) ̃ 𝜔 𝑅 𝐵 𝑅∈𝐴 ( ) 𝛿 𝑘 𝜔(𝑄) 𝑛(2|𝛿|+1)|𝑗| ∑ ≤𝐶2 . 𝑟/𝑎 𝐿 𝑟𝛼 /𝑎 󵄨 󵄨𝑛 0 (12) ̃ 2 𝑄∈Q,𝑙(𝑄)=2−𝑗 (1 + 󵄨𝑥 󵄨 / {1, |𝑄|}) |𝐵| 󵄨 󵄨𝑎 󵄨 𝑄󵄨 max ≤𝐶2−𝑘𝜆( ) [𝑀 ( ∑ 󵄨𝑡 󵄨 𝜒 )] 𝑅 𝜔 󵄨 𝑅󵄨 𝑅 | | 𝑅∈𝐴 For all 𝑠∈R, 𝜏∈[0,∞), 𝑝 ∈ (0, ∞),and𝑞 ∈ (0, ∞], [ 𝑘 ] ̇𝑠,𝜏 𝑛 𝑠,𝜏̇ 𝑛 ̇𝑠,𝜏 𝑛 we see that 𝑏𝑝,min{𝑝,𝑞},𝜔(R )⊂𝑓𝑝,𝑞,𝜔(R )⊂𝑏𝑝,max{𝑝,𝑞},𝜔(R ) 𝑟/𝑎 𝑎 by Minkowski’s inequality. The following conclusions is easily −𝑘(𝜆−𝑛𝑟𝛼2/𝑎)[ 󵄨 󵄨 ] ≤𝐶2 𝑀𝜔 ( ∑ 󵄨𝑡𝑅󵄨 𝜒𝑅) . verifiedsimilarlytotheproofoflemma2.7in[6]. Here we 𝑅∈𝐴 [ 𝑘 ] omit the proof. 𝑘 𝑟 Lemma 7. Let 𝑠∈R, 𝜏∈[0,∞), 𝑝,𝑞∈(0,∞],and𝜓∈ Summing on and taking the th roots yield the result. S (R𝑛) 𝜔∈𝐴 𝑡∈𝑎𝑠,𝜏̇ (R𝑛) 𝑇 𝑡≡∑ 𝑡 𝜓 ∞ , ∞.Thenforall 𝑝,𝑞,𝜔 , 𝜓 𝑄 𝑄 𝑄 Lemma 9. 𝜏=0 󸀠 𝑛 𝑠,𝜏 𝑛 󸀠 𝑛 Let ,then converges in S∞(R );moreover,𝑇𝜓 : 𝑎𝑝,𝑞,𝜔̇ (R )→S∞(R ) 󵄩 ∗ 󵄩 is continuous. 󵄩𝑡 󵄩 ∼ ‖𝑡‖ 𝑠,0 , 󵄩 𝑝∧𝑞,𝜆󵄩 ̇𝑠,0 𝑛 𝑏̇(R𝑛) 󵄩 󵄩𝑏𝑝,𝑞,𝜔(R ) 𝑝,𝑞,𝜔 𝑡={𝑡} 𝑟 ∈ (0, ∞] 𝜆∈ (17) For a sequence 𝑄 𝑄, ,andafixed 󵄩 ∗ 󵄩 󵄩𝑡 󵄩 ∼ ‖𝑡‖ 𝑠,0 . (0, ∞) 󵄩 𝑝∧𝑞,𝜆󵄩 𝑠,0̇ 𝑛 𝑓̇(R𝑛) ,set 󵄩 󵄩𝑓𝑝,𝑞,𝜔(R ) 𝑝,𝑞,𝜔 1/𝑟 󵄨 󵄨𝑟 󵄨𝑡 󵄨 ∗ ∗ 󵄨 𝑅󵄨 Proof. For all 𝑄,wehave|𝑡𝑄|≤(𝑡𝑝∧𝑞,𝜆)𝑄.Thenitisobvious (𝑡𝑟,𝜆) ≡ ( ∑ ) (13) 𝑄 −1 󵄨 󵄨 𝜆 that {𝑅:𝑙(𝑅)=𝑙(𝑄)} (1 + 𝑙(𝑅) 󵄨𝑥𝑅 −𝑥𝑄󵄨) 󵄩 ∗ 󵄩 ∗ ∗ ‖𝑡‖ 𝑠,0 ≤𝐶󵄩𝑡 󵄩 . 𝑡 ≡{(𝑡 ) } 𝑓̇(R𝑛) 󵄩 𝑝∧𝑞,𝜆󵄩 𝑠,0̇ 𝑛 and 𝑟,𝜆 𝑟,𝜆 𝑄 𝑄. We have the following estimates. 𝑝,𝑞,𝜔 󵄩 󵄩𝑓𝑝,𝑞,𝜔(R ) (18) 4 Journal of Function Spaces and Applications

Let 𝜀=−1+(𝜆/𝑛𝛼2)>0,choose𝑎 = (𝑝∧𝑞)/(1+𝜀/2),then Applying the fact of Lemma 9 that for each sequence ∗ ∗ 𝑝/𝑎, 𝑞/𝑎 >1 𝑡={𝑡 } ‖𝑡 ‖ 𝑠,0 ∼‖𝑡‖̇𝑠,0 𝑛 ‖𝑡 ‖ 𝑠,0 ∼ . Using weighted Fefferman-Stein inequality, we 𝑄 𝑄, 𝑝∧𝑞,𝜆 𝑏̇(R𝑛) 𝑏𝑝,𝑞,𝜔(R ) and 𝑝∧𝑞,𝜆 𝑓̇(R𝑛) have 𝑝,𝑞,𝜔 𝑝,𝑞,𝜔 ‖𝑡‖ 𝑠,0̇ 𝑛 𝑓𝑝,𝑞,𝜔(R ),wethenhave

󵄩 󵄩 󵄩𝑡∗ 󵄩 1 󵄩 𝑝∧𝑞,𝜆󵄩𝑓𝑠,0̇(R𝑛) 𝐼 ≡ 𝑝,𝑞,𝜔 𝑃 |𝑃|𝜏 󵄩 󵄩 󵄩 1/𝑞󵄩 󵄩 󵄨 󵄨 𝑞 󵄩 𝑞/𝑝 1/𝑞 = 󵄩[∑(|𝑄|−𝑠/𝑛 󵄨(𝑡∗ ) 󵄨 𝜒𝑄)̃ ] 󵄩 󵄩 󵄨 𝑝∧𝑞,𝜆 𝑄󵄨 󵄩 { } 󵄩 󵄨 󵄨 󵄩 { ∞ 𝑝 } 󵄩 𝑄 󵄩 𝑝 { [ −𝑠/𝑛−1/2 1/𝑝 ∗ ] } 𝐿𝜔 × ∑ [ ∑ [|𝑄| 𝜔(𝑄) (𝑟 ) ] ] { [ 𝑝∧𝑞,𝜆 𝑄 ] } { −𝑗 } 󵄩 1/𝑝∧𝑞 𝑗=𝑗𝑃 𝑙(𝑄)=2 󵄩 󵄨 󵄨𝑝∧𝑞 󵄩[ 󵄨𝑡 󵄨 { [ 𝑄⊂𝑃 ] } ≤󵄩[∑(|𝑄|−𝑠/𝑛( ∑ 󵄨 𝑅󵄨 ) 󵄩 −1󵄨 󵄨𝜆 󵄩 𝑄 {𝑅:𝑙(𝑅)=𝑙(𝑄)} (1+𝑙(𝑅) 󵄨𝑥 −𝑥 󵄨 ) 1 󵄩 ∗ 󵄩 1 󵄩 󵄨 𝑅 𝑄󵄨 ≤ 󵄩𝑟 󵄩 ≲ ‖𝑟‖ 𝑠,0 ≲ ‖𝑡‖ 𝑠,𝜏 󵄩[ 𝜏 󵄩 𝑝∧𝑞,𝜆󵄩 ̇𝑠,0 𝑛 𝜏 𝑏̇(R𝑛) 𝑏̇(R𝑛) |𝑃| 󵄩 󵄩𝑏𝑝,𝑞,𝜔(R ) |𝑃| 𝑝,𝑞,𝜔 𝑝,𝑞,𝜔 𝑞 1/𝑞󵄩 󵄩 (22) 󵄩 ] 󵄩 × 𝜒𝑄)̃ ] 󵄩 󵄩 󵄩 and similarly, ] 󵄩 𝑝 󵄩𝐿𝜔

1/𝑎 󵄩 𝑎 𝑞/𝑎 𝑎/𝑞󵄩 𝑝/𝑞 󵄩 󵄩 𝑞 󵄩 󵄩 1 { −𝑠/𝑛−1/2 ∗ 󵄩 −𝑠/𝑛 󵄨 󵄨 󵄩 ̃ 󵄩[ 󵄨 󵄨 ] 󵄩 𝐼𝑃 ≡ ∫ [ ∑ [|𝑄| (𝑟 ) 𝜒𝑄 (𝑥)] ] ≤𝐶󵄩 ∑ (𝑀𝜔 ( ∑ |𝑅| 󵄨𝑡𝑅󵄨 𝜒̃𝑅) ) 󵄩 𝑃 𝜏 { 𝑝∧𝑞,𝜆 𝑄 󵄩 󵄩 | | 𝑃 𝑄⊂𝑃 󵄩 𝜈∈Z 𝑙(𝑅)=2−𝜈 󵄩 { 󵄩[ ] 󵄩 𝑝/𝑎 󵄩 󵄩𝐿𝜔 1/𝑝 󵄩 𝑞 1/𝑞󵄩 } (23) 󵄩 󵄩 󵄩 −𝑠/𝑛 󵄨 󵄨 󵄩 ×𝜔(𝑥) 𝑑𝑥} ≤𝐶󵄩[ ∑ ( ∑ |𝑅| 󵄨𝑡 󵄨 𝜒̃ ) ] 󵄩 󵄩 󵄨 𝑅󵄨 𝑅 󵄩 } 󵄩 𝜈∈Z −𝜈 󵄩 󵄩[ 𝑙(𝑅)=2 ] 󵄩 𝑝 󵄩 󵄩𝐿𝜔 ≲ ‖𝑡‖ 𝑠,𝜏̇ 𝑛 . 𝑓𝑝,𝑞,𝜔(R ) = ‖𝑡‖ 𝑠,0̇ 𝑛 . 𝑓𝑝,𝑞,𝜔(R ) (19) On the other hand, let 𝑄⊂𝑃be a dyadic cube with 𝑙(𝑄) = −𝑖 ̃ 2 𝑙(𝑃) for some 𝑖∈Z+.Suppose𝑄 is any dyadic cube with ̃ −𝑖 ̃ ∗ 𝑙(𝑄) = 𝑙(𝑄) =2 𝑙(𝑃) and 𝑄 ⊂ 𝑃 + 𝑘𝑙(𝑃) ⊆3𝑃̸ for some ‖𝑡 ‖ ∼‖𝑡‖𝑠,0 Therefore, 𝑝∧𝑞,𝜆 𝑠,0̇ 𝑛 𝑓̇(R𝑛).Similarly,wecan 𝑛 𝑓𝑝,𝑞,𝜔(R ) 𝑝,𝑞,𝜔 𝑘∈Z ,where𝑃 + 𝑘𝑙(𝑃) ≡ {𝑥 + 𝑘𝑙(𝑃).Then :𝑥∈𝑃} |𝑘| ≥ 2 ∗ −1 𝑖 ‖𝑡 ‖ ∼‖𝑡‖𝑠,0 1+𝑙(𝑄)̃ |𝑥 −𝑥 |∼2|𝑘| verify that 𝑝∧𝑞,𝜆 ̇𝑠,0 𝑛 𝑏̇(R𝑛). and 𝑄 𝑄̃ .For 𝑏𝑝,𝑞,𝜔(R ) 𝑝,𝑞,𝜔 Lemma 10. 𝜔∈𝐴 𝑠∈R 𝜏∈[0,∞)𝑝, 𝑞 ∈ (0, ∞] Let ∞, , , , 𝜔 (𝑄) 𝜔 (𝑄) 𝜔 (𝑃) 𝜔 (𝑃+𝑘𝑙(𝑃)) and 𝜆 ∈ (𝑛2 +𝑛𝛼 ,∞). Then there exists a constant 𝐶∈[1,∞) = 𝑡∈𝑎𝑠,𝜏̇ (R𝑛) 𝜔(𝑄)̃ 𝜔 (𝑃) 𝜔 (𝑃+𝑘𝑙(𝑃)) 𝜔(𝑄)̃ such that for all 𝑝,𝑞,𝜔 , (24) 𝑛𝛼 𝑖𝑛(𝛼 −𝛼 ) ≤𝐶𝑘 2 2 2 1 , 󵄩 󵄩 󵄩 ∗ 󵄩 ‖𝑡‖𝑎𝑠,𝜏̇(R𝑛) ≤ 󵄩𝑡 󵄩 ≤𝐶‖𝑡‖𝑎𝑠,𝜏̇(R𝑛). 𝑝,𝑞,𝜔 󵄩 min{𝑝,𝑞},𝜆󵄩𝑎𝑠,𝜏̇(R𝑛) 𝑝,𝑞,𝜔 (20) 𝑝,𝑞,𝜔 we have

1 ∗ 𝐽𝑃 ≡ 𝜏 Proof. Notice that |𝑡𝑄|≤(𝑡𝑟,𝜆)𝑄 holds for all dyadic cubes |𝑃| 𝑄 ‖𝑡‖ 𝑠,𝜏 𝑛 ≤ . This observation immediately implies that 𝑎𝑝,𝑞,𝜔̇(R ) ∗ ‖𝑡 ‖ 𝑝∧𝑞= {𝑝, 𝑞} { ∞ 𝑝∧𝑞,𝜆 𝑎𝑠,𝜏̇(R𝑛),where min . { [ 𝑝,𝑞,𝜔 × ∑ [ ∑ [|𝑄|−𝑠/𝑛−1/2𝜔(𝑄)1/𝑝 To see the converse, fix a dyadic cube 𝑃.Let𝑟𝑄 ≡𝑡𝑄 if { [ { −𝑖 𝑄⊂3𝑃 𝑟 ≡0 𝑢 ≡𝑡 −𝑟 𝑖=0 𝑙(𝑄)=2 𝑙(𝑃) and 𝑄 otherwise, and let 𝑄 𝑄 𝑄.Set { [ 𝑄⊂𝑃 𝑟≡{𝑟𝑄}𝑄 and 𝑢≡{𝑢𝑄}𝑄. Then for all dyadic cubes 𝑄,we have 𝑞/𝑝 1/𝑞 } 𝑝] } ∗ ] ×(𝑢𝑝∧𝑞,𝜆) ] ] } 𝑝∧𝑞 𝑝∧𝑞 𝑝∧𝑞 𝑄 } (𝑡∗ ) =(𝑟∗ ) +(𝑢∗ ) . 𝑝∧𝑞,𝜆 𝑄 𝑝∧𝑞,𝜆 𝑄 𝑝∧𝑞,𝜆 𝑄 (21) ] } Journal of Function Spaces and Applications 5

1 1/𝑞 ≲ { } |𝑃|𝜏 { ∞ } 𝑖𝑛(𝛼2−𝛼1)−𝜆 𝑛𝛼2−𝜆 ≲ ‖𝑡‖𝑏̇𝑠,𝜏 (R𝑛) ∑2 ( ∑ |𝑘| ) 𝑝,𝑞,𝜔 { } {𝑖=0 𝑘∈Z𝑛 } { |𝑘|≥2 { ∞ { } { 𝑖𝑛(𝛼 −𝛼 )𝑞/𝑝−𝑖𝜆𝑞/(𝑝∧𝑞) × ∑2 2 1 { ≲ ‖𝑡‖𝑏̇𝑠,𝜏 (R𝑛). {𝑖=0 𝑝,𝑞,𝜔 { (27)

‖𝑡∗ ‖ ≲ (𝐼 +𝐽)≲ Therefore, by (21), min{𝑝,𝑞},𝜆 ̇𝑠,𝜏 𝑛 sup𝑃∈Q 𝑃 𝑃 [ 𝑏𝑝,𝑞,𝜔(R ) [ 𝑛𝛼 −𝜆 2 ‖𝑡‖ ̇𝑠,𝜏 𝑛 × [ ∑ |𝑘| 𝑏𝑝,𝑞,𝜔(R ). [ 𝑛 𝑘∈Z𝑛 𝑖∈Z 𝑘∈Z |𝑘|≥2 To complete the proof, for any +, +,anddyadic [ cube 𝑃,set 󵄨 󵄨−𝑠/𝑛−1/2 1/𝑝 󵄨̃󵄨 ̃ −𝑖 × ∑ (󵄨𝑄󵄨 𝜔(𝑄) 𝐴 (𝑖, 𝑘,) 𝑃 ≡{𝑄∈̃ Q :𝑙(𝑄)̃ = 2 𝑙 (𝑃) , 𝑙(𝑄̃)=2−𝑖𝑙(𝑃) (28) 𝑄⊂𝑃+𝑘𝑙̃ (𝑃) 𝑄⊂𝑃+𝑘𝑙̃ (𝑃) , 𝑄∩̃ (3𝑃) =0}.

𝑞/(𝑝∧𝑞) 1/𝑞 } ̃ −1 𝑖 } Recall that 1+𝑙(𝑄) |𝑥𝑄 −𝑥𝑄̃|∼2|𝑘| for any 𝑄⊂𝑃and 󵄨 󵄨 𝑝∧𝑞] } 󵄨 󵄨 ] 𝑄̃ ∈ 𝐴(𝑖, 𝑘,𝑃) 𝑑∈[0,1] {𝑎 } ⊂ C × 󵄨𝑡𝑄̃󵄨 ) ] } . ,andthatforall and 𝑗 𝑗 , 󵄨 󵄨 ] } } ] } 𝑑 󵄨 󵄨 󵄨 󵄨𝑑 󵄨 󵄨 󵄨 󵄨 (25) (∑ 󵄨𝑎𝑗󵄨) ≤ ∑󵄨𝑎𝑗󵄨 . (29) 𝑗 𝑗

When 𝑝≤𝑞,by𝜆>𝑛+𝑛𝛼2,wehave Similarly to the proof of Lemma A.2 (see [18,RemarkA.3]), 𝑥∈𝑃 𝑎 ∈ (0, 𝑝 ∧𝑞] 𝐽 ≲ ‖𝑡‖ ̇𝑠,𝜏 𝑛 we obtain that for all and , 𝑃 𝑏𝑝,𝑞,𝜔(R )

1/𝑞 󵄨 󵄨−𝑠/𝑛−1/2 󵄨 󵄨 𝑝∧𝑞 𝑞/𝑝 󵄨̃󵄨 󵄨 󵄨 (󵄨𝑄󵄨 󵄨𝑡𝑄̃󵄨) { ∞ } 󵄨 󵄨 󵄨 󵄨 { 𝑖𝑛(𝛼 −𝛼 )𝑞/𝑝−𝑖𝜆𝑞/𝑝 𝑛𝛼 −𝜆 } ∑ × ∑2 2 1 ( ∑ |𝑘| 2 ) (26) −1 󵄨 󵄨 𝜆 { } ̃ ̃ 󵄨 󵄨 { } 𝑄∈𝐴(𝑖,𝑘,𝑃) (1 + 𝑙(𝑄) 󵄨𝑥 −𝑥̃󵄨) {𝑖=0 𝑘∈Z𝑛 } 󵄨 𝑄 𝑄󵄨 { |𝑘|≥2 } −𝜆 ≲(2𝑖 |𝑘|) ≲ ‖𝑡‖ ̇𝑠,𝜏 𝑛 . 𝑏𝑝,𝑞,𝜔(R ) 𝑎 (𝑝∧𝑞)/𝑎 󵄨̃󵄨−𝑠/𝑛−1/2 󵄨 󵄨 (󵄨𝑄󵄨 󵄨𝑡̃󵄨) When 𝑝>𝑞,byHolder’s¨ inequality and 𝜆>𝑛+𝑛𝛼2,we [ 󵄨 󵄨 󵄨 𝑄󵄨 ] × ∫ ∑ 𝜒 𝜔 (𝑥) 𝑑𝑥 obtain [ 𝑄̃ ] 𝑃+𝑘𝑙(𝑃) 𝜔(𝑄)̃ [ ] 1 𝐽 ≲ −𝑖(𝜆−𝑛𝛼 (𝑝∧𝑞)/𝑎) −𝜆 𝑃 |𝑃|𝜏 ≲2 2 |𝑘|

{ { ∞ [ 𝑎 { 𝑖𝑛(𝛼 −𝛼 )𝑞/𝑝−𝑖𝜆 󵄨 󵄨−𝑠/𝑛 󵄨 󵄨 2 1 [ 󵄨̃󵄨 󵄨 󵄨 × {∑2 × [𝑀 ( ∑ (󵄨𝑄󵄨 󵄨𝑡̃󵄨 𝜒̃̃) ) { [ 𝜔 󵄨 󵄨 󵄨 𝑄󵄨 𝑄 {𝑖=0 𝑙(𝑄̃)=2−𝑖𝑙(𝑃) { [ 𝑄⊂𝑃+𝑘𝑙̃ (𝑃)

(𝑝∧𝑞)/𝑎 [ [ 𝑛𝛼 −𝜆 ] × [ ∑ |𝑘| 2 ] [ × (𝑥+𝑘𝑙(𝑃)) ] , 𝑘∈Z𝑛 ] |𝑘|≥2 [ ] 1/𝑞 (30) } 𝑞]} 󵄨̃󵄨−𝑠/𝑛−1/2 ̃ 1/𝑝 󵄨 󵄨 ] 𝑀 × ∑ (󵄨𝑄󵄨 𝜔(𝑄) 󵄨𝑡̃󵄨) ] where herein and in what follows, 𝜔 denotes the Hardy- 󵄨 󵄨 󵄨 𝑄󵄨 ]} 𝑛 −𝑖 } 𝑙(𝑄̃)=2 𝑙(𝑃) } Littlewood maximal function on R .Let𝑎≡(2𝑛(𝑝∧ 𝑄⊂𝑃+𝑘𝑙̃ (𝑃) ]} 𝑞)𝛼2)/(𝜆 + 𝑛𝛼2).Then𝑎 ∈ (0, 𝑝 ∧𝑞). Applying Minkowski’s 6 Journal of Function Spaces and Applications

𝑓∈𝐴̇𝑠,𝜏 (R𝑛) 𝐶−1‖ (𝑓)‖ ≤ inequality, Fefferman-Stein’s weighted vector-valued inequal- such that for all 𝑝,𝑞,𝜔 , inf 𝛾 𝑠,𝜏 𝑛 𝑎𝑝,𝑞,𝜔̇(R ) ity, and Holder’s¨ inequality, we have ‖𝑓‖ 𝑠,𝜏 ≤‖ (𝑓)‖ 𝑠,𝜏 ≤𝐶‖ (𝑓)‖ 𝐴̇ (R𝑛) sup 𝑎̇(R𝑛) inf 𝛾 𝑠,𝜏 𝑛 . 𝑝,𝑞,𝜔 𝑝,𝑞,𝜔 𝑎𝑝,𝑞,𝜔̇(R ) 1 𝐽̃ ≡ 𝑃 |𝑃|𝜏 Proof of Theorem 2. With Lemmas 7, 10,and11,theargument for Theorem 2 follows from the method pioneered by Frazier 𝑝/𝑞 1/𝑝 { 𝑞 } and Jawerth (see [18, pages 50-51]); see also the proof of [10, × ∫ [ ∑ [|𝑄|−(𝑠/𝑛)−(1/2)(𝑢∗ ) 𝜒 (𝑥)] ] 𝑑𝑥 Theorem 3.1]. We omit the details. { 𝑝∧𝑞,𝜆 𝑄 𝑄 } 𝑃 𝑄⊂𝑃 { } From Theorem 2,weimmediatelydeducethefollowing 1 conclusion. ≲ |𝑃|𝜏 Corollary 12. With all the notations as in Definition,then 1 ̇𝑠,𝜏 𝑛 the spaces 𝐴𝑝,𝑞,𝜔(R ) areindependentofthechoiceof𝜑. { { [ { [ ∞ [ −𝑖(𝜆−𝑛𝛼 (𝑝∧𝑞)/𝑎) −𝜆 𝑠,𝜏 𝑛 × ∫ [∑ ( ∑ 2 2 |𝑘| 𝐴̇ (R ) { [ 3. Almost Diagonal Operators on 𝑝,𝑞,𝜔 { 𝑃 [𝑖=0 𝑘∈Z𝑛 { |𝑘|≥2 As an application of Theorem 2, we study boundedness of { [ ̇𝑠,𝜏 𝑛 operators in 𝐴𝑝,𝑞,𝜔(R ) by first considering their bounded- 𝑠,𝜏 𝑛 ness in corresponding 𝑎̇ (R ).Inthissection,weshow [ 𝑝,𝑞,𝜔 [ 󵄨 󵄨−𝑠/𝑛 󵄨 󵄨 𝑎 𝑎𝑠,𝜏̇ (R𝑛) ×[𝑀 ( ∑ (󵄨𝑄̃󵄨 󵄨𝑡 󵄨 𝜒̃ ) ) that almost diagonal operators are bounded on 𝑝,𝑞,𝜔 for [ 𝜔 󵄨 󵄨 󵄨 𝑄̃󵄨 𝑄̃ ̃ −𝑖 appropriate indices, which generalize the classical results on 𝑙(𝑄)=2 𝑙(𝑃) ̇𝑠,0 𝑛 𝑠,0̇ 𝑛 [ 𝑄⊂𝑃+𝑘𝑙̃ (𝑃) 𝑏𝑝,𝑞,𝜔(R ) and 𝑓𝑝,𝑞,𝜔(R );see[16, 18].

𝑞/(𝑝∧𝑞) 𝑝/𝑞 𝑠∈R 𝑝, 𝑞 ∈ (0, ∞] 𝐽≡𝑛/ {1, 𝑝} (𝑝∧𝑞)/𝑎 Definition 13. Let , , min ] 𝑏̇𝑠,𝜏 (R𝑛) 𝐽≡𝑛/ {1, 𝑝, 𝑞} ] for the space 𝑝,𝑞,𝜔 , min for the space ] ] 𝑠,𝜏 𝑛 ] ] 𝑓̇ (R ) 𝜀 ∈ (0, ∞) 𝐴 × (𝑥+𝑘𝑙(𝑃)) ] ) ] 𝑝,𝑞,𝜔 ,and .Anoperator associated with ] ] {𝑎 } 𝑡={𝑡 } ⊂ C ] amatrix 𝑄𝑃 𝑄,𝑃,namely,forallsequences 𝑄 𝑄 , ] 𝐴𝑡 ≡ {(𝐴𝑡)𝑄}𝑄 ≡{∑𝑃 𝑎𝑄𝑃𝑡𝑃}𝑄, is called 𝜀-almost diagonal on 𝑠,𝜏 𝑛 ] 𝑎𝑝,𝑞,𝜔̇ (R ) if the matrix {𝑎𝑄𝑃}𝑄,𝑃 satisfies 1/𝑝 󵄨 󵄨 󵄨𝑎 󵄨 } 󵄨 𝑄𝑃󵄨 } sup <∞, (32) } 𝑄,𝑃 𝜔𝑄𝑃 (𝜀) ×𝑑𝑥 } } } where } 𝑠 󵄨 󵄨 −𝐽−𝜀 𝑙 (𝑄) 󵄨𝑥𝑄 −𝑥𝑃󵄨 𝜔𝑄𝑃 (𝜀) ≡( ) (1 + ) ≲ ‖𝑡‖ 𝑠,𝜏̇ 𝑛 . 𝑙 (𝑃) max (𝑙 (𝑃) ,𝑙(𝑄)) 𝑓𝑝,𝑞,𝜔(R ) (31) 𝑙 (𝑄) (𝑛+𝜀)/2 𝑙 (𝑃) (𝑛+𝜀)/2+𝐽−𝑛 × min [( ) ,( ) ]. ‖𝑡∗ ‖ ≲ (𝐼̃ + 𝑙 (𝑃) 𝑙 (𝑄) Therefore, by (21) again, min{𝑝,𝑞},𝜆 𝑠,𝜏̇ 𝑛 sup𝑃∈Q 𝑃 𝑓𝑝,𝑞,𝜔(R ) ̃ (33) 𝐽 ) ≲ ‖𝑡‖ 𝑠,𝜏̇ 𝑛 𝑃 𝑓𝑝,𝑞,𝜔(R ), which completes the proof of Lemma 10. We remark that an 𝜀-almost diagonal operator is also an almost diagonal operator introduced by Frazier and Jawerth Let 𝜑 satisfy (1)through(3). Since 𝜑(𝑥)̃ ≡ 𝜑(−𝑥) also in [18]. In [18, Section 9], Frazier and Jawerth showed satisfy (1)through(3), we may take 𝜑̃ in place of 𝜑 in the that certain appropriate Calderon-Zygmund´ operators and ̇𝑠,𝜏 𝑛 󸀠 𝑛 definition of 𝐵𝑝,𝑞,𝜔(R ).Forany𝑓∈S∞(R ) and 𝑄∈Q certain classes of Fourier multiplier operators correspond to −𝑗 𝜑 with 𝑙(𝑄) =2 , define the sequence sup(𝑓) ≡sup { 𝑄(𝑓)}𝑄 almost diagonal matrices, and hence, the -transform simul- 1/2 taneously “almost diagonalizes” these operators. Moreover, by setting sup (𝑓) ≡ |𝑄| sup |𝜑̃𝑗 ∗ 𝑓(𝑦)|,andfor 𝑄 𝑦∈𝑄 Yang and Yuan proved that all almost diagonal operators are any 𝛾∈Z+,thesequenceinf𝛾(𝑓) ≡inf { 𝑄,𝛾 (𝑓)}𝑄 by 𝑠,𝜏̇ 𝑛 ̇𝑠,𝜏 𝑛 bounded on 𝑓 (R ) and 𝑏 (R );see[9, 10]. These results (𝑓) ≡ |𝑄|1/2 { |𝜑̃ ∗ 𝑓(𝑦)| : 𝑙(𝑄)̃ = 𝑝,𝑞 𝑝,𝑞 setting inf𝑄,𝛾 max inf𝑦∈𝑄̃ 𝑗 can be generalized into the weighted Besov- and Triebel- −𝛾 ̃ 2 𝑙(𝑄), 𝑄⊂𝑄}.Choosing𝜆>𝑛+𝑛𝛼2 as in the proof of [6, Lizorkin-type spaces. We turn to prove it. Lemma 2.9], we have the following estimates. 𝑠,𝜏 𝑛 Proof of Theorem 3. Let 𝑡={𝑡𝑄}𝑄 ∈ 𝑎𝑝,𝑞,𝜔̇ (R ) and let Lemma 11. Let 𝑠∈R, 𝜏∈[0,∞), 𝑝, 𝑞 ∈ (0, ∞],and𝛾∈Z+ 𝐴 be an 𝜀-almost diagonal operator associated with the be sufficiently large. Then there exists a constant 𝐶∈[1,∞) matrix {𝑎𝑄𝑅}𝑄,𝑅 and 𝜀 ∈ (𝑛(𝛼1 −𝛼2)/𝑝, ∞).Withoutloss Journal of Function Spaces and Applications 7

1/2+𝜏 −1/𝑝 𝑚−1 𝑠=0 |𝑅| 𝜔(𝑅) ‖𝑡‖ ̇0,𝜏 𝑛 𝑅∈Q 2 𝑙(𝑅) ≤ of generality, we may assume that . Indeed, if the 𝑏𝑝,𝑞,𝜔(R ) for all .Since 𝑠=0 ̃𝑡 ≡ 𝑙(𝑅)−𝑠𝑡 𝐵 𝑚 conclusion holds for ,let 𝑅 𝑅 and let be |𝑥𝑄 −𝑥𝑅|<2 𝑙(𝑅), 𝑙(𝑄) ≤ 𝑙(𝑅), the operator associated with the matrix {𝑏𝑄𝑅}𝑄,𝑅,where𝑏𝑄𝑅 ≡ 𝑠 (𝑙(𝑅)/𝑙(𝑄)) 𝑎 𝑄, 𝑅 ∈ Q ‖𝐴𝑡‖ 𝑠,𝜏 𝑛 = 𝑄𝑅 for all .Thenwehave 𝑎𝑝,𝑞,𝜔̇(R ) ̃ ̃ ‖𝐵𝑡‖ 0,𝜏 𝑛 ≲‖𝑡‖ 0,𝜏 𝑛 ∼‖𝑡‖𝑠,𝜏 𝑛 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝑎𝑝,𝑞,𝜔̇(R ) 𝑎𝑝,𝑞,𝜔̇(R ) 𝑎𝑝,𝑞,𝜔̇(R ),whichdeducesthe 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝑚 󵄨𝑥−𝑥𝑅󵄨 ≤ 󵄨𝑥−𝑥𝑄󵄨 + 󵄨𝑥𝑄 −𝑥𝑅󵄨 ≤ √𝑛𝑙 (𝑄) +2 𝑙 (𝑅) desired conclusion. 𝑏̇0,𝜏 (R𝑛) 𝑚 𝑚 We now consider the space 𝑝,𝑞,𝜔 in the case ≤(2 + √𝑛) 𝑙 (𝑅) ≲𝐶2 𝑙 (𝑅) (36) min(𝑝, 𝑞).Forall >1 𝑄∈Q,wewrite𝐴≡𝐴0 + 𝑚+𝑗−𝑖 𝐴1 with (𝐴0𝑡)𝑄 ≡∑{𝑅: 𝑙(𝑅)≥𝑙(𝑄)} 𝑎𝑄𝑅𝑡𝑅 and (𝐴1𝑡)𝑄 ≡ ≤𝐶2 𝑙 (𝑄) . ∑{𝑅:𝑙(𝑅)<𝑙(𝑄)} 𝑎𝑄𝑅𝑡𝑅.ByDefinition 13,weseethatforall𝑄∈Q,

(𝑛+𝜀)/2 𝑄=2̃ 𝑚+𝑗−𝑖𝑄 𝑝≤1𝜏 < (𝜀/2𝑛) +(𝛼 /𝑝) 𝜀> 󵄨 󵄨 𝑙 (𝑄) Let .If , 1 , 󵄨(𝐴 𝑡) 󵄨 ≲ ∑ ( ) (𝑛(𝛼 −𝛼 )/𝑝) 󵄨 0 𝑄󵄨 𝑙 (𝑅) 2 1 ,then {𝑅: 𝑙(𝑅)≥𝑙(𝑄)} 󵄨 󵄨 (34) 󵄨𝑡 󵄨 × 󵄨 𝑅󵄨 , 󵄨 󵄨 𝑛+𝜀 1 (1 + 𝑙(𝑅)−1 󵄨𝑥 −𝑥 󵄨) B ≡ 󵄨 𝑄 𝑅󵄨 |𝑃|𝜏

{ ∞ 𝑗 −1 ∞ 𝜀/2 { 𝑃 𝑙 (𝑄) and therefore × ∑ [∫ ( ∑ ∑ ∑ ∑ ( ) { 𝑙 (𝑅) 𝑗=𝑗 𝑃 −𝑗 𝑖=−∞ 𝑚=0 𝑅∈ 𝑈 𝑃 [ 𝑙(𝑄)=2 𝑚,𝑖 󵄩 󵄩 { 󵄩𝐴0𝑡󵄩 ̇0,𝜏 𝑛 󵄩 󵄩𝑏𝑝,𝑞,𝜔(R ) 𝑝 1 × |𝑅|𝜏𝜔(𝑅)−1/𝑝2−𝑚(𝑛+𝜀)𝜒 (𝑥) ) ≲ 𝑄 sup |𝑃|𝜏 𝑃 dyadic 𝑞/𝑝 1/𝑞 } { ∞ 𝑙 (𝑄) (𝑛+𝜀)/2 ×𝜔(𝑥) 𝑑𝑥] × ∑ [∫ ( ∑ ∑ ( ) } { 𝑙 (𝑅) 𝑗=𝑗 𝑃 −𝑗 {𝑅: 𝑙(𝑄)≤𝑙(𝑅)≤𝑙(𝑃)} } { 𝑃 [ 𝑙(𝑄)=2 { ∞ 𝑗 −1 ∞ 𝜀𝑝/2 󵄨 󵄨 𝑝 { 𝑃 𝑙 (𝑄) 󵄨 󵄨 𝑗𝑃𝑛𝜏 [ 󵄨𝑡𝑅󵄨 𝜒̃𝑄 (𝑥) ≤2 ∑ ∫ ∑ ∑ ∑ ∑ ( ) × ) { 𝑙 (𝑅) 𝑛+𝜀 𝑗=𝑗 𝑃 −𝑗 𝑖=−∞ 𝑚=0 𝑅∈ 𝑈 −1 󵄨 󵄨 𝑃 [ 𝑙(𝑄)=2 𝑚,𝑖 (1 + 𝑙(𝑅) 󵄨𝑥𝑄 −𝑥𝑅󵄨) { −1 𝜏𝑝 −(𝑛+𝜀)𝑚𝑝 𝑞/𝑝 1/𝑞 ×𝜔(𝑅) |𝑅| 2 } ] ×𝜔(𝑥) 𝑑𝑥 } 𝑞/𝑝 1/𝑞 } } ] } ×𝜒𝑝 (𝑥) 𝜔 (𝑥) 𝑑𝑥] 𝑄 } 1 ] + } sup |𝑃|𝜏 𝑃 dyadic { ∞ 𝑗𝑃−1 ∞ 𝜀𝑝/2 𝑗 𝑛𝜏 { 𝑙 (𝑄) 𝑝 ≤2𝑃 ∑ [ ∑ ∑ ∑ ∑ ( ) { ∞ { 𝑙 (𝑅) 𝑗=𝑗 −𝑗 𝑖=−∞ 𝑚=0 𝑅∈ 𝑈 × ∑ [∫ ( ∑ ∑ ⋅⋅⋅) 𝑃 [𝑙(𝑄)=2 𝑚,𝑖 { { 𝑗=𝑗 𝑃 −𝑗 𝑃 [ 𝑙(𝑄)=2 {𝑅: 𝑙(𝑅)>𝑙(𝑃)} { 𝑞/𝑝 1/𝑞 𝜔 (𝑄) } 𝑞/𝑝 1/𝑞 ×𝑙(𝑅)𝑛𝜏𝑝2−(𝑛+𝜀)𝑚𝑝 ] } 𝜔 (𝑅) } ×𝜔(𝑥) 𝑑𝑥] ≡𝐼 +𝐼. ] } 1 2 } ] } { ∞ 𝑗𝑃−1 ∞ 𝜀𝑝/2 (35) 𝑗 𝑛𝜏 { 𝑙 (𝑄) ≤2𝑃 ∑ [ ∑ ∑ ∑ ∑ ( ) { 𝑙 (𝑅) 𝑗=𝑗 −𝑗 𝑖=−∞ 𝑚=0 𝑅∈ 𝑈 { 𝑃 [l(𝑄)=2 𝑚,𝑖 𝑖∈Z 𝑚∈N 𝑈 ≡{𝑅∈Q : 𝑙(𝑅)−𝑖 =2 For all and ,set 0,𝑖 1/𝑞 −𝑖 𝑞/𝑝 and |𝑥𝑄 −𝑥𝑅| < 𝑙(𝑅)} and 𝑈𝑚,𝑖 ≡{𝑅∈Q : 𝑙(𝑅) =2 and 𝜔(𝑄)̃ } 𝑛𝜏𝑝 −(𝑛+𝜀)𝑚𝑝 𝜔 (𝑄) ] 2𝑚−1𝑙(𝑅) ≤ |𝑥 −𝑥 |<2𝑚𝑙(𝑅)} 𝑈 ≲2𝑚𝑛 ×𝑙(𝑅) 2 } 𝑄 𝑅 .Thenwehave# 𝑚,𝑖 , 𝜔 (𝑅) 𝜔(𝑄)̃ } where #𝑈𝑚,𝑖 denotes the cardinality of 𝑈𝑚,𝑖.Noticethat|𝑡𝑅|≤ ] } 8 Journal of Function Spaces and Applications

{ ∞ 𝑗𝑃−1 ∞ Thus, by 0≤𝜏<(𝛼1/𝑝) + (𝜀/2𝑛), 𝑗 𝑛𝜏 { (𝑖𝜀𝑝/2)−𝑖𝑛𝜏𝑝+𝑖𝑛𝛼 −(𝑗𝜀𝑝/2)−𝑗𝑛𝛼 ≤𝐶2𝑃 ∑ [ ∑ ∑ 2 1 2 1 { 𝑗=𝑗 𝑖=−∞ 𝑚=0 { 𝑃 [ 1 𝑞/𝑝 1/𝑞 𝐼2 ≲ ‖𝑡‖𝑏̇0,𝜏 (R𝑛) sup 𝜏 } 𝑝,𝑞,𝜔 |𝑃| 𝑚𝑛(𝛼 −𝛼 ) −𝑚𝜀𝑝) } 𝑃 dyadic ×2 2 1 2 ] ≤𝐶, } ] { ∞ 𝑗 −1 ∞ (𝑛+𝜀)/2 } { 𝑃 𝑙 (𝑄) × ∑ [∫ ( ∑ ∑ ∑ ∑ ( ) (37) { 𝑙 (𝑅) 𝑗=𝑗 𝑃 −𝑗 𝑖=−∞ 𝑚=0 𝑅∈ 𝑈 { 𝑃 [ 𝑙(𝑄)=2 𝑚,𝑖

1/2+𝜏 −1/𝑝 𝑝 |𝑅| 𝜔(𝑅) 𝜒̃𝑄 (𝑥) ̃ 𝑚𝑛𝛼2 ̃ × ) here we use 𝜔(𝑄)/𝜔(𝑅) ≲2 and 𝜔(𝑄)/𝜔(𝑄) ≲ −1 󵄨 󵄨 𝑛+𝜀 −𝑛(𝑚+𝑗−𝑖)𝛼 (1 + 𝑙(𝑅) 󵄨𝑥 −𝑥 󵄨) 2 1 . 󵄨 𝑄 𝑅󵄨 𝑝>1 If , using Minkowski’s inequality, we have 𝑞/𝑝 1/𝑞 } ×𝜔(𝑥) 𝑑𝑥] } ] } 𝑗 −1 { 𝑃 ∞ 𝜀/2 [ 𝑙 (𝑄) 1 {∫ ∑ ∑ ∑ ∑ ( ) 𝑙 (𝑅) ≲ ‖𝑡‖𝑏̇0,𝜏 (R𝑛) sup 𝜏 𝑃 𝑙(𝑄)=2−𝑗 𝑖=−∞ 𝑚=0 𝑅∈ 𝑈 𝑝,𝑞,𝜔 |𝑃| { [ 𝑚,𝑖 𝑃 dyadic

𝑝 1/𝑝 { ∞ 𝑗 −1 ∞ 𝜀/2 } { 𝑃 𝑙 (𝑄) × |𝑅|𝜏𝜔(𝑅)−1/𝑝2−𝑚(𝑛+𝜀)𝜒 (𝑥) ] 𝜔 (𝑥) 𝑑𝑥 × ∑ [∫ ( ∑ ∑ ∑ ∑ ( ) 𝑄 } { 𝑙 (𝑅) 𝑗=𝑗 𝑃 −𝑗 𝑖=−∞ 𝑚=0 𝑅∈ 𝑈 ] } { 𝑃 [ 𝑙(𝑄)=2 𝑚,𝑖 𝑗 −1 𝑃 ∞ 𝑙 (𝑄) 𝜀/2 𝑝 ≤ ∑ ∑ ∑ ∑ [ ] × |𝑅|𝜏𝜔(𝑅)−1/𝑝2−𝑚(𝑛+𝜀)𝜒 (𝑥) ) −𝑗 𝑙 (𝑅) 𝑄 𝑙(𝑄)=2 𝑖=−∞ 𝑚=0 𝑅∈𝑚,𝑖 𝑈

𝜔 (𝑄) 1/𝑝 ×[ ] |𝑅|𝜏2−𝑚(𝑛+𝜀) 𝑞/𝑝 1/𝑞 } 𝜔 (𝑅) ] ×𝜔(𝑥) 𝑑𝑥 } 𝑗 −1 } 𝑃 ∞ 𝑙 (𝑄) 𝜀/2 ] ≤ ∑ ∑ ∑ ∑ [ ] } 𝑙 (𝑅) −𝑗 𝑖=−∞ 𝑚=0 𝑅∈ 𝑈 𝑙(𝑄)=2 𝑚,𝑖 ≲ ‖𝑡‖ ̇0,𝜏 𝑛 . 𝑏𝑝,𝑞 𝜔(R ) 1/𝑝 1/𝑝 (40) 𝜔(𝑄)̃ 𝜔 (𝑄) ×[ ] [ ] |𝑅|𝜏2−𝑚(𝑛+𝜀). 𝜔 (𝑅) 𝜔(𝑄)̃ (38) For 𝐼1,let𝑟 and 𝑢 bethesameasintheproofofLemma 10. We see that

Therefore, 1 𝐼 ≲ 1 sup |𝑃|𝜏 𝑃 dyadic 𝑗 𝑛𝜏 B ≤𝐶2𝑃 { ∞ 𝑗 × ∑ [∫ ( ∑ ∑ 2(𝑖−𝑗)(𝑛+𝜀)/2 𝑗 −1 { ∞ 𝑃 ∞ 𝑃 −𝑗 { 𝑗=𝑗𝑃 𝑙(𝑄)=2 𝑖=𝑗𝑃 [ 𝑖((𝜀/2)+(𝑛𝛼1/𝑝)−𝑛𝜏) [ × { ∑ ∫ ∑ ∑ ∑ ∑ 2 { 𝑗=𝑗 𝑃 −𝑗 𝑖=−∞ 𝑚=0 𝑅∈ 𝑈 { 𝑃 [ 𝑙(𝑄)=2 𝑚,𝑖 󵄨 󵄨 𝑝 󵄨𝑟𝑅󵄨 𝜒̃𝑄 (𝑥) 𝑞 1/𝑞 × ∑ ) −1 󵄨 󵄨 𝑛+𝜀 } 𝑙(𝑅)=2−𝑖 (1 + 𝑙(𝑅) 󵄨𝑥𝑄 −𝑥𝑅󵄨) 𝑗(−(𝜀/2)−(𝑛𝛼1/𝑝)) −𝑚(𝑛+𝜀) 𝑛𝑚(𝛼2−𝛼1)/𝑝] ×2 2 2 } ] } 𝑞/𝑝 1/𝑞 } 1 ≤𝐶. ×𝑑𝑥] + } sup |𝑃|𝜏 𝑃 dyadic (39) ] } Journal of Function Spaces and Applications 9

|𝑘| ≥ 2 1+𝑙(𝑅)−1|𝑥 −𝑥 | ∼ |𝑘|𝑙(𝑃)/𝑙(𝑅) { and 𝑄 𝑅 . Therefore, by { ∞ 𝑗 × ∑ [∫ ( ∑ ∑ 2(𝑖−𝑗)(𝑛+𝜀)/2 Holder’s¨ inequality, { {𝑗=𝑗 𝑃 −𝑗 𝑖=𝑗 { 𝑃 [ 𝑙(𝑄)=2 𝑃 1 𝐽 ≲ { 2 sup 𝜏+1+𝜀/𝑛 𝑃 |𝑃| 󵄨 󵄨 𝑝 dyadic 󵄨𝑢 󵄨 𝜒̃ (𝑥) × ∑ 󵄨 𝑅󵄨 𝑄 ) −1 󵄨 󵄨 𝑛+𝜀 { ∞ 𝑗 −𝑖 (1 + 𝑙(𝑅) 󵄨𝑥 −𝑥 󵄨) { [ 𝑙(𝑅)=2 󵄨 𝑄 𝑅󵄨 × ∑ 2−𝑗𝑞𝜀/2 [∫ ( ∑ ∑ 2−𝑖(𝑛+𝜀)/2 ∑ |𝑘|−𝑛−𝜀 { [ {𝑗=𝑗 𝑃 −𝑗 𝑖=𝑗 𝑘∈Z𝑛 𝑞/𝑝 1/𝑞 𝑃 𝑙(𝑄)=2 𝑃 } { [ |𝑘|≥2 ×𝑑𝑥] ≡𝐽 +𝐽. } 1 2 𝑝 ] } 󵄨 󵄨 × ∑ 󵄨𝑡 󵄨 𝜒 (𝑥)) (41) 󵄨 𝑅󵄨 𝑄 𝑙(𝑅)=2−𝑖 𝑅⊂𝑃+𝑘𝑙(𝑃)

𝑞/𝑝 1/𝑞 Applying Lemma 8 with 𝑎=1,forall𝑥∈𝑄,wehave } ] } ] ×𝜔(𝑥) 𝑑𝑥] } } 󵄨 󵄨 󵄨𝑟 󵄨 󵄨 󵄨 ] } ∑ 󵄨 𝑅󵄨 ≲𝑀 ( ∑ 󵄨𝑟 󵄨 𝜒 ) (𝑥) . −1 󵄨 󵄨 𝑛+𝜀 𝜔 󵄨 𝑅󵄨 𝑅 1 −𝑖 (1 + 𝑙(𝑅) 󵄨𝑥 −𝑥 󵄨) −𝑖 𝑙(𝑅)=2 󵄨 𝑄 𝑅󵄨 𝑙(𝑅)=2 ≲ ‖𝑡‖ ̇0,𝜏 𝑛 𝑏𝑝,𝑞,𝜔(R ) sup 𝜀/𝑛 (42) 𝑃 dyadic |𝑃| 𝑞 1/𝑞 { } { ∞ [ 𝑗 ] } 𝐿𝑝 (R𝑛) −𝑗𝑞𝜀/2[ −𝑖𝜀/2 −𝑛−𝜀] Hence Holder’s¨ inequality and the 𝜔 boundedness for × { ∑ 2 [ ∑ 2 ∑ |𝑘| ] } { 𝑛 } 𝑝 ∈ (1, ∞] of the Hardy-Littlewood maximal operator yield 𝑗=𝑗𝑃 𝑖=𝑗𝑃 𝑘∈Z { [ |𝑘|≥2 ] }

≲ ‖𝑡‖𝑏̇0,𝜏 (R𝑛). 1 𝑝,𝑞,𝜔 𝐽 ≲ (44) 1 sup |𝑃|𝜏 𝑃 dyadic Hence, { 𝑝 { ∞ 𝑗 󵄩 󵄩 [ ((𝑖−𝑗)𝜀)/2 󵄨 󵄨 󵄩𝐴0𝑡󵄩 ̇0,𝜏 𝑛 ≲ ‖𝑡‖ ̇0,𝜏 𝑛 . × ∑ ∫ ( ∑ 2 𝑀 ( ∑ 󵄨𝑟 󵄨 𝜒̃ ) (𝑥)) 󵄩 󵄩𝑏𝑝,𝑞,𝜔(R ) 𝑏𝑝,𝑞,𝜔(R ) (45) { 𝜔 󵄨 𝑅󵄨 𝑅 {𝑗=𝑗 𝑃 𝑖=𝑗 −𝑖 { 𝑃 [ 𝑃 𝑙(𝑅)=2 { The similar computations to 𝐼1 yield

𝑞/𝑝 1/𝑞 󵄩 󵄩 } 󵄩𝐴 𝑡󵄩 ≲ ‖𝑡‖ 0,𝜏 . } 󵄩 1 󵄩𝑏̇0,𝜏 (R𝑛) 𝑏̇(R𝑛) (46) ×𝜔(𝑥) 𝑑𝑥] 𝑝,𝑞,𝜔 𝑝,𝑞,𝜔 } ] 𝑠,𝜏̇ 𝑛 } For Triebel-Lizorkin-type spaces 𝑓𝑝,𝑞,𝜔(R ),wealsohave 1 ≲ sup 𝜏 󵄩 󵄩 |𝑃| 󵄩𝐴0𝑡󵄩 0,𝜏̇ 𝑛 𝑃 dyadic 󵄩 󵄩𝑓𝑝,𝑞,𝜔(R )

1/𝑞 1 𝑝 𝑞/𝑝 { ∞ } ≲ { } sup |𝑃|𝜏 [ 󵄨 󵄨 ] 𝑃 dyadic × { ∑ ∫ ( ∑ 󵄨𝑡𝑅󵄨 𝜒̃𝑅) (𝑥)𝜔(𝑥)𝑑𝑥 } { 3𝑃 } 𝑖=𝑗 𝑙(𝑅)=2−𝑖 { 𝑃[ ] } 𝑙 (𝑄) (𝑛+𝜀)/2 ×{∫ [ ∑ ( ∑ ( ) 𝑙 (𝑅) ≲ ‖𝑡‖ ̇0,𝜏 𝑛 , 𝑃 𝑄⊂𝑃 {𝑅: 𝑙(𝑄)≤𝑙(𝑅)≤𝑙(𝑃)} 𝑏𝑝,𝑞,𝜔(R ) (43) 󵄨 󵄨 𝑞 𝑝/𝑞 󵄨𝑡 󵄨 𝜒̃ (𝑥) × 󵄨 𝑅󵄨 𝑄 ) ] −1 󵄨 󵄨 𝑛+𝜀 (1 + 𝑙(𝑅) 󵄨𝑥𝑄 −𝑥𝑅󵄨) ] where the last inequality follows from Minkowski’s inequality if 𝑞>𝑝or (29)if𝑞≤𝑝. 1/𝑝 To estimate 𝐽2,wenoticethatif𝑅 ∩ (3𝑃),then =0 𝑅⊂ ×𝜔(𝑥) 𝑑𝑥} 𝑛 𝑃 + 𝑘𝑙(𝑃) and (𝑃 + 𝑘𝑙(𝑃)) ∩ (3𝑃) =0 for some 𝑘∈Z with 10 Journal of Function Spaces and Applications

1 { ∞ + sup 𝜏 { 𝑃 |𝑃| × ∫ [ ∑ ∑ dyadic { 𝑃 𝑗=𝑗 −𝑗 [ 𝑃 𝑙(𝑄)=2 𝑝/𝑞 1/𝑝 { { 𝑞 } × ∫ [ ∑ ( ∑ ⋅⋅⋅) ] 𝜔 (𝑥) 𝑑𝑥 𝑗 { } (𝑖−𝑗)(𝑛+𝜀)/2 { 𝑃 𝑄⊂𝑃 {𝑅: 𝑙(𝑅)>𝑙(𝑃)} } ×(∑ 2 𝑖=𝑗 ̃ ̃ 𝑃 ≡ 𝐼1 + 𝐼2. 𝑞 𝑝/𝑞 (47) |𝑢 |𝜒̃ (𝑥) × ∑ 𝑅 𝑄 ) ] −1 󵄨 󵄨 𝑛+𝜀 𝑙(𝑅)=2−𝑖 (1 + 𝑙(𝑅) 󵄨𝑥𝑄−𝑥𝑅󵄨) ] Similarly to the estimate for 𝐼2, applying the fact that |𝑡𝑅|≤ 1/2+𝜏 −1/𝑝 1/𝑝 |𝑅| 𝜔(𝑅) ‖𝑡‖ ̇0,𝜏 𝑛 0≤𝜏<(𝛼1/𝑝) + (𝜀/2𝑛) } 𝑏𝑝,𝑞,𝜔(R ) and ,we } ×𝜔(𝑥) 𝑑𝑥 ≡ 𝐽̃ + 𝐽̃ . obtain } 1 2 } 1 ̃ (49) 𝐼2 ≲ ‖𝑡‖𝑓0,𝜏̇(R𝑛) sup 𝜏 𝑝,𝑞,𝜔 |𝑃| 𝑃 dyadic

{ ∞ Similarly to the estimate of 𝐽1,by[18,RemarkA.3], Holder’s¨ [ × {∫ ∑ ∑ inequality and the Fefferman-Stein weighted vector-valued 𝑃 𝑗=𝑗 −𝑗 { [ 𝑃 𝑙(𝑄)=2 inequality, we obtain

𝑗 −1 𝑃 ∞ 𝑙 (𝑄) (𝑛+𝜀)/2 ×( ∑ ∑ ∑ ( ) ̃ 1 𝑖=−∞ 𝑚=0 𝑅∈ 𝑈 𝑙 (𝑅) 𝐽1 ≲ sup 𝜏 𝑚,𝑖 |𝑃| 𝑃 dyadic 𝑞 𝑝/𝑞 |𝑅|1/2+𝜏𝜔(𝑅)−1/𝑝𝜒̃ (𝑥) 𝑄 ] { ∞ 𝑗 × 𝑛+𝜀 ) [ (𝑖−𝑗)𝜀/2 −1 󵄨 󵄨 × ∫ ( ∑ ∑ 2 𝑀𝐻𝐿 (1 + 𝑙(𝑅) 󵄨𝑥𝑄 −𝑥𝑅󵄨) { 󵄨 󵄨 ] 𝑃 𝑗=𝑗 𝑖=𝑗 { 𝑃 [ 𝑃 1/𝑝 ×𝜔(𝑥) 𝑑𝑥} 𝑞 𝑝/𝑞 󵄨 󵄨 ] ×( ∑ 󵄨𝑟𝑅󵄨 𝜒̃𝑅) (𝑥) ) 𝑙(𝑅)=2−𝑖 ] ≲ ‖𝑡‖ 0,𝜏̇ 𝑛 . 𝑓𝑝,𝑞,𝜔(R ) 1/𝑝 (48) } ×𝜔(𝑥) 𝑑𝑥 } ̃ To estimate 𝐼1,noticethat } 1 ≲ 1 sup |𝑃|𝜏 ̃ 𝑃 dyadic 𝐼1 ≲ sup 𝜏 𝑃 |𝑃| dyadic 1/𝑝 𝑞 𝑝/𝑞 { ∞ } { 󵄨 󵄨 } { ∞ × ∫ [ ∑ ( ∑ 󵄨𝑡 󵄨 𝜒̃ (𝑥)) ] 𝜔(𝑥)𝑑𝑥 { { 󵄨 𝑅󵄨 𝑅 } × ∫ [ ∑ ∑ 3𝑃 𝑖=𝑗 −𝑖 { [ 𝑃 𝑙(𝑅)=2 ] 𝑃 𝑗=𝑗 −𝑗 { } { [ 𝑃 𝑙(𝑄)=2 ≲ ‖𝑡‖𝑓0,𝜏̇(R𝑛). 𝑗 𝑝,𝑞,𝜔 ×(∑ 2(𝑖−𝑗)(𝑛+𝜀)/2 (50)

𝑖=𝑗𝑃 󵄨 󵄨 𝑞 𝑝/𝑞 𝐽̃ 𝐽 󵄨𝑟 󵄨 𝜒̃ (𝑥) For 2, similarly to the estimate for 2,byMinkowski’s × ∑ 󵄨 𝑅󵄨 𝑄 ) ] inequality and the Fefferman-Stein weighted vector-valued −1 󵄨 󵄨 𝑛+𝜀 𝑙(𝑅)=2−𝑖 (1 + 𝑙(𝑅) 󵄨𝑥𝑄 −𝑥𝑅󵄨) ] maximal inequality, we obtain 1/𝑝 } 1 1 𝐽̃ ≲ ∑ |𝑘|−(𝑛+𝜀) ×𝜔(𝑥) 𝑑𝑥} + sup 𝜏 2 sup 𝜏+𝜀/𝑛 } |𝑃| 𝑛 𝑃 |𝑃| 𝑃 dyadic 𝑘∈Z dyadic } |𝑘|≥2 Journal of Function Spaces and Applications 11

{ { ∞ [ 𝑗 References [ (𝑖−𝑗)𝜀/2 −𝑖𝜀 × {∫ ( ∑ [ ∑ 2 2 { 𝑃 [1] H. Triebel, Theory of Function Spaces,Birkhauser,¨ Basel, 𝑗=𝑗𝑃 𝑖=𝑗𝑃 { [ Switzerland, 1983. [2] H. Triebel, Theoryoffunctionspaces.II,Birkhauser,¨ Basel, Switzerland, 1992. 󵄨 󵄨 ×𝑀 ( ∑ |𝑅|−1/2 󵄨𝑡 󵄨 𝜒 ) [3] H. Triebel, Theory of function spaces. III,Birkhauser,¨ Basel, HL 󵄨 𝑅󵄨 𝑅 Switzerland, 2006. 𝑙(𝑅)=2−𝑖 𝑅⊂𝑃+𝑘𝑙(𝑃) [4] H.-Q. Bui, “Characterizations of weighted Besov and Triebel- Lizorkin spaces via temperatures,” Journal of Functional Analy- 𝑞 𝑝/𝑞 sis,vol.55,no.1,pp.39–62,1984. ] [5] H.-Q. Bui, M. Paluszynski,´ and M. H. Taibleson, “A maximal ] × (𝑥+𝑘𝑙(𝑃)) ] ) function characterization of weighted Besov-Lipschitz and Triebel-Lizorkin spaces,” Studia Mathematica,vol.119,no.3,pp. ] 219–248, 1996. 1/𝑝 [6] W. Yuan, W. Sickel, and D. Yang, Morrey and Campanato } } meet Besov, Lizorkin and Triebel,vol.2005ofLecture Notes in Mathematics, Springer, Berlin, Germany, 2010. ×𝜔(𝑥) 𝑑𝑥} ≲ ‖𝑡‖𝑓0,𝜏̇(R𝑛). } 𝑝,𝑞,𝜔 [7]Y.Liang,Y.Sawano,T.Ullrich,D.Yang,andW.Yuan,“New } characterizations of Besov-Triebel-Lizorkin-Hausdorff spaces (51) including coorbits and wavelets,” The Journal of Fourier Analysis and Applications, vol. 18, no. 5, pp. 1067–1111, 2012. [8]Y.Sawano,D.Yang,andW.Yuan,“NewapplicationsofBesov- type and Triebel-Lizorkin-type spaces,” Journal of Mathematical ‖𝐴 𝑡‖ 0,𝜏̇ 𝑛 ≲‖𝑡‖ 0,𝜏̇ 𝑛 Hence 0 𝑓𝑝,𝑞 (R ) 𝑓𝑝,𝑞,𝜔(R ). Analysis and Applications,vol.363,no.1,pp.73–85,2010. ̃ Some similar estimates to 𝐼1 also yield that [9] D. Yang and W. Yuan, “A new class of function spaces connect- ing Triebel-Lizorkin spaces and 𝑄 spaces,” Journal of Functional ‖𝐴1𝑡‖𝑓0,𝜏̇(R𝑛) ≲‖𝑡‖𝑓0,𝜏̇(R𝑛).Thus,weobtainthedesired 𝑝,𝑞,𝜔 𝑝,𝑞,𝜔 Analysis,vol.255,no.10,pp.2760–2809,2008. conclusion for the case min{𝑝, 𝑞} >1. {𝑝, 𝑞} ≤1 [10] D. Yang and W. Yuan, “New Besov-type spaces and Triebel- The case min is a simple consequence of the 𝑄 {𝑝, 𝑞} >1 𝛿 ∈ (0, 𝑝 ∧𝑞) Lizorkin-type spaces including spaces,” Mathematische case min .Indeed,choosinga ,then Zeitschrift,vol.265,no.2,pp.451–480,2010. 𝑝/𝛿 >1 𝑞/𝛿 >1 𝐴̃ and .Let be an operator associated with [11] D. Yang and W. Yuan, “Characterizations of Besov-type and {𝑎̃ } ≡ {|𝑎 |𝛿(𝑙(𝑄)/𝑙(𝑃))𝑛/2−𝛿𝑛/2} 𝐴̃ the matrix 𝑄𝑃 𝑄,𝑃 𝑄𝑃 𝑄,𝑃.Then Triebel-Lizorkin-type spaces via maximal functions and local is a 𝜀̃-almost diagonal operator with 𝑠=0and 𝜀=𝛿𝜀̃ . means,” Nonlinear Analysis: Theory, Methods & Applications, ̃𝑡 ≡ {𝑙(𝑄)𝑛/2−𝛿𝑛/2|𝑡 |𝛿} ‖̃𝑡‖1/𝛿 = vol. 73, no. 12, pp. 3805–3820, 2010. Define 𝑄 𝑄.Then 𝑎0,𝜏𝛿̇ (R𝑛) 𝑝/𝛿,𝑞/𝛿,𝜔 [12] D. Yang and W. Yuan, “Relations among besov-type spaces, ‖𝑡‖ 0,𝜏 𝑛 𝛿<1 ‖𝐴𝑡‖ 0,𝜏 𝑛 ≲ 𝑎𝑝,𝑞,𝜔̇(R ).Since ,wehave 𝑎𝑝,𝑞,𝜔̇(R ) Triebel-Lizorkin-type spaces and generalized Carleson measure ‖𝐴̃̃𝑡‖1/𝛿 {𝑝, 𝑞} >1 spaces,” Applicable Analysis, 2011. 𝑎0,𝜏𝛿̇ (R𝑛). Applying the conclusions for min 𝑝/𝛿,𝑞/𝛿,𝜔 [13] W. Yuan, Y. Sawano, and D. Yang, “Decompositions of Besov- ̃ 1/𝛿 1/𝛿 ‖𝐴𝑡‖ 0,𝜏 𝑛 ≲‖𝐴̃𝑡‖ ≲‖̃𝑡‖ ∼ Hausdorff and Triebel-Lizorkin-Hausdorff spaces and their yields 𝑎𝑝,𝑞,𝜔̇(R ) 𝑎0,𝜏𝛿̇ (R𝑛) 𝑎0,𝜏𝛿̇ (R𝑛) 𝑝/𝛿,𝑞/𝛿,𝜔 𝑝/𝛿,𝑞/𝛿,𝜔 applications,” Journal of Mathematical Analysis and Applica- ‖𝑡‖ 0,𝜏 𝑛 𝑎𝑝,𝑞,𝜔̇(R ), which completes the proof of Theorem 3. tions,vol.369,no.2,pp.736–757,2010. [14] D. Yang and W.Yuan, “Dual properties of Triebel-Lizorkin-type ̇𝑠,0 𝑛 ̇𝑠 𝑛 Remark 14. Recall that 𝑏𝑝,𝑞,𝜔(R )=𝑏𝑝,𝑞,𝜔(R ) and spaces and their applications,” Zeitschriftur f¨ Analysis und ihre 𝑠,0̇ 𝑛 𝑠̇ 𝑛 Anwendungen,vol.30,no.1,pp.29–58,2011. 𝑓𝑝,𝑞,𝜔(R )=𝑓𝑝,𝑞,𝜔(R ).Thus,if𝜏=0,byTheorem 3,we [15] D. Yang, W. Yuan, and C. Zhuo, “Fourier multipliers on see that for all 𝜀 ∈ (0, ∞),all𝜀-almost diagonal operators 𝑏̇𝑠 (R𝑛) 𝑓𝑠̇ (R𝑛) Triebel-Lizorkin-type spaces,” Journal of Function Spaces and are bounded on 𝑝,𝑞,𝜔 and 𝑝,𝑞,𝜔 ,whichisjustthe Applications, vol. 2012, Article ID 431016, 37 pages, 2012. classical results established by Frazier and Jawerth; see [18, [16] M. Frazier, B. Jawerth, and G. Weiss, Littlewood-Paley Theory Theorem 3.3] and [16,Theorem6.20]. and the Study of Function Spaces, CBMS-AMS Regional Con- ference, Auburn University, Washington, DC, USA, 1989. [17] I. M. Gel’fand and G. E. Shilov, Generalized Functions, Academic Acknowledgments Press, London, UK, 1968. [18] M. Frazier and B. Jawerth, “A discrete transform and decom- The author cordially thanks Prof. D. C. Yang and Dr. W. Yuan positions of distribution spaces,” Journal of Functional Analysis, for their valuable comments. This work was supported by vol. 93, no. 1, pp. 34–170, 1990. National Natural Science Foundation (Grand no. 11021043) of [19] M. Frazier and B. Jawerth, “Decomposition of Besov spaces,” China and the Fundamental Research Funds for the Central Indiana University Mathematics Journal,vol.34,no.4,pp.777– Universities. Also, we thank the referee for his many valuable 799, 1985. remarks which greatly improved the presentation of this [20] M. Frazier and B. Jawerth, “The 𝜑-transform and applications paper. to distribution spaces,”in Function Spaces and Applications,vol. 12 Journal of Function Spaces and Applications

1302 of Lecture Notes in Mathematics, pp. 223–246, Springer, Berlin, Germany, 1988. [21] M. Bownik, “Anisotropic Triebel-Lizorkin spaces with doubling measures,” The Journal of Geometric Analysis,vol.17,no.3,pp. 387–424, 2007. Hindawi Publishing Corporation Journal of Function Spaces and Applications Volume 2013, Article ID 275702, 3 pages http://dx.doi.org/10.1155/2013/275702

Research Article The Space of Continuous Periodic Functions Is a Set of First Category in 𝐴𝑃(𝑋)

Zhe-Ming Zheng,1 Hui-Sheng Ding,1 and Gaston M. N’Guérékata2

1 College of Mathematics and Information Science, Jiangxi Normal University, Nanchang, Jiangxi 330022, China 2 Department of Mathematics, Morgan State University, 1700 E. Cold Spring Lane, Baltimore, MD 21251, USA

Correspondence should be addressed to Hui-Sheng Ding; [email protected]

Received 17 January 2013; Accepted 12 February 2013

Academic Editor: Jozef´ Bana´s

Copyright © 2013 Zhe-Ming Zheng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We prove that the space of continuous periodic functions is a set of first category in the space of almost periodic functions, and we also show that the space of almost periodic functions is a set of first category in the space of almost automorphic functions.

1. Introduction Definition 1 (see [4]). A function 𝑓∈𝐶(R,𝑋)is called almost periodic if, for every 𝜀>0, there exists 𝑙(𝜀) >0 such that every Since the last century, the study on almost periodic type func- interval of length 𝑙(𝜀) contains a number 𝜏 with the property tions and their applications to evolution equations has been that of great interest for many mathematicians. There is a large literature on this topic. Several books are especially devoted 󵄩 󵄩 to almost periodic type functions and their applications to sup 󵄩𝑓 (𝑡+𝜏) −𝑓(𝑡)󵄩 <𝜀. 𝑡∈R (1) differential equations and dynamical systems. For example, let us indicate the books of Amerio and Prouse [1], Bezandry and Diagana [2], Bohr [3], Corduneanu [4], Diagana [5], Fink We denote the collection of all such functions by 𝐴𝑃(𝑋). [6], Levitan and Zhikov [7], N’Guer´ ekata´ [8, 9], Pankov [10], Shen and Yi [11], Zaidman [12], and Zhang [13]. Recall that 𝐴𝑃(𝑋) is a Banach space under the supremum Although almost periodic functions have a very wide norm. rangeofapplicationsnow,itseemsthatgivinganexample of almost periodic (not periodic) functions is more difficult Definition 2. Afunction𝑓∈𝐶(R,𝑋) is called periodic if than giving an example of periodic functions. Also, there is there exists 𝑙>0such that a similar problem for almost automorphic functions. In this paper, we aim to compare the “amount” of almost periodic 𝑓 (𝑡+𝑙) =𝑓(𝑡) ,∀𝑡∈R. functions (not periodic) with the “amount” of continuous (2) periodic functions, and we also discuss the related problems for almost automorphic functions. Here, 𝑙 is called a period of 𝑓. We denote the collection of all such functions by 𝑃(𝑋).For𝑓 ∈ 𝑃(𝑋),wecall𝑙0 the 2. Main Results fundamental period if 𝑙0 is the smallest period of 𝑓. Throughout the rest of this paper, we denote by R the set of Remark 3. Similar to the proof in [4,page1],itisnotdifficult real numbers, by 𝑋 aBanachspace,andby𝐶(R,𝑋)the set of to show that if 𝑓 ∈ 𝑃(𝑋) is not constant, and then 𝑓 has the all continuous functions 𝑓:R →𝑋. fundamental period. 2 Journal of Function Spaces and Applications

Definition 4 (see [8]). A function 𝑓∈𝐶(R,𝑋) is called where 𝑘 is a fixed positive integer. Letting 𝜀=min{𝜀𝑙 },and 1≤𝑖≤𝑘 𝑖 almost automorphic if, for every real sequence (𝑠𝑚),there 󵄩 󵄩 exists a subsequence (𝑠𝑛) such that 󵄩 󵄩 𝑁 (𝑓, 𝜀) := {𝑔∈𝐴𝑃(𝑋) : 󵄩𝑔−𝑓󵄩𝐴𝑃(𝑋) <𝜀} , (12) 𝑔 (𝑡) = lim 𝑓(𝑡+𝑠𝑛) (3) 𝑛→∞ for every 𝑔∈𝑁(𝑓,𝜀),weclaimthat𝑔∉𝑃𝑛.Infact,forevery 𝑙∈[𝑛,𝑛+1] 𝑖∈{1,...,𝑘} is well defined for each 𝑡∈R and , there exists such that

𝑔 (𝑡−𝑠 ) =𝑓(𝑡) 𝑙∈(𝑙𝑖 −𝛿𝑙 ,𝑙𝑖 +𝛿𝑙 ). (13) 𝑛→∞lim 𝑛 (4) 𝑖 𝑖 for each 𝑡∈R.Denoteby𝐴𝐴(𝑋) the set of all such functions. Then, by9 ( ), we have 󵄩 󵄩 󵄩𝑓(𝑡𝑙 +𝑙)−𝑓(𝑡𝑙 )󵄩 ≥3𝜀𝑙 ≥3𝜀, (14) Recall that there exists an almost automorphic function 󵄩 𝑖 𝑖 󵄩 𝑖 which is not almost periodic, for instance, the following which yields that function: 󵄩 󵄩 1 󵄩𝑔(𝑡 +𝑙)−𝑔(𝑡 )󵄩 󵄩 𝑙𝑖 𝑙𝑖 󵄩 𝑓 (𝑡) = sin ,𝑡∈R. (5) 2+cos 𝑡+cos √2𝑡 󵄩 󵄩 󵄩 󵄩 ≥ 󵄩𝑓(𝑡 +𝑙)−𝑓(𝑡 )󵄩 − 󵄩𝑓(𝑡 +𝑙)−𝑔(𝑡 +𝑙)󵄩 󵄩 𝑙𝑖 𝑙𝑖 󵄩 󵄩 𝑙𝑖 𝑙𝑖 󵄩 Before the proof of our main results, we need to recall the 󵄩 󵄩 (15) notion about the first category. − 󵄩𝑓(𝑡 )−𝑔(𝑡 )󵄩 󵄩 𝑙𝑖 𝑙𝑖 󵄩 Definition 5 (see [14]). Let 𝑆 beatopologicalspace.Aset ≥3𝜀−𝜀−𝜀=𝜀>0, 𝐸⊂𝑆is said to be nowhere dense if its closure has an empty interior.Thesetsofthefirstcategoryin𝑆 are those that are where ‖𝑔 − 𝑓‖𝐴𝑃(R) <𝜀wasused.So,weknowthat𝑁(𝑓,𝜀) ⊂ countable unions of nowhere dense sets. Any subset of S that 𝐴𝑃(𝑋)𝑛 \𝑃 ,whichmeansthat𝑃𝑛 is a closed subset of 𝐴𝑃(𝑋). is not of the first category is said to be of the second category in 𝑆. Step 2.Every𝑃𝑛 has an empty interior. It suffices to prove that, for every 𝑓∈𝑃𝑛 and 𝛿>0, Theorem 6. 𝑃(𝑋) is a set of first category in 𝐴𝑃(𝑋). 𝑁(𝑓,𝛿) ⋂(𝐴𝑃(𝑋)𝑛 \𝑃 ) =0̸ . Now let 𝑓∈𝑃𝑛 and 𝛿>0.In thefollowing,wediscusstwocases. Proof. For 𝑛 = 1, 2, ., . we denote 𝑓 𝑃 ={𝑓∈𝐶(R,𝑋) : 𝑙∈[𝑛, 𝑛] +1 Case I. is constant. 𝑛 there exists We denote (6) such that 𝑓 (𝑡+𝑙) =𝑓(𝑡) ∀𝑡 ∈ R}. cos 𝑡+cos (√2𝑡) 𝑓𝛿 (𝑡) = ⋅𝛿⋅𝑥0 +𝑓(𝑡) ,𝑡∈R, (16) Then, it is easy to see that 3

∞ where 𝑥0 ∈𝑋is some constant with ‖𝑥0‖=1.Then𝑓𝛿 ∈ 𝑃 (𝑋) = ⋃𝑃𝑛. (7) 𝑁(𝑓,𝛿) and 𝑓𝛿 ∉𝑃𝑛 since 𝑓𝛿 is not periodic. 𝑛=1 𝑓 We divide the remaining proof into two steps. Case II. is not constant. By Remark 3, 𝑓 has a fundamental period 𝑙0. We denote Step 1.Every𝑃𝑛 is a closed subset of 𝐴𝑃(𝑋). 𝑡 𝛿 𝑓 (𝑡) =𝑓(𝑡) +𝑓( ) ⋅ ,𝑡∈R, Let 𝑓 ∈ 𝐴𝑃(𝑋)𝑛 \𝑃 . Then, for every 𝑙∈[𝑛,𝑛+1],there 𝛿 (17) 𝜋 𝑀𝑓 exists 𝑡𝑙 ∈ R such that 𝑓(𝑡𝑙 +𝑙)=𝑓(𝑡̸ 𝑙).Denote 1 󵄩 󵄩 where 𝑀𝑓 = sup𝑡∈R‖𝑓(𝑡)‖.Obviously,𝑓𝛿 ∈ 𝑁(𝑓,𝛿).Also,we 𝜀 := 󵄩𝑓(𝑡 +𝑙)−𝑓(𝑡)󵄩 >0, 𝑙∈[𝑛, 𝑛] +1 . (8) 𝑓 ∉𝑃 𝑙 4 󵄩 𝑙 𝑙 󵄩 claim that 𝛿 𝑛. In fact, if this is not true, then there exists 𝑇∈[𝑛,𝑛+1]such that In addition, due to the continuity of 𝑓,forevery𝑙∈[𝑛,𝑛+1], 𝑓 𝑡+𝑇 =𝑓 𝑡 ,𝑡∈R, there exists 𝛿𝑙 >0such that 𝛿 ( ) 𝛿 ( ) (18) 󵄩 󵄩 󵄩𝑓 (𝑡𝑙 +𝑠) −𝑓(𝑡𝑙)󵄩 ≥3𝜀𝑙,∀𝑠∈(𝑙−𝛿𝑙,𝑙+𝛿𝑙) . (9) that is, 𝑡+𝑇 𝛿 𝑡 𝛿 Obviously, we have 𝑓 (𝑡+𝑇) +𝑓( )⋅ =𝑓(𝑡)+𝑓( )⋅ ,𝑡∈R. 𝜋 𝑀 𝜋 𝑀 [𝑛, 𝑛] +1 ⊂ ⋃ (𝑙−𝛿,𝑙+𝛿). 𝑓 𝑓 𝑙 𝑙 (10) 𝑙∈[𝑛,𝑛+1] (19) Let Then, by the Heine-Borel theorem, there exists 𝑙1,...,𝑙𝑘 ∈ [𝑛, 𝑛 + 1] such that 𝐹1 (𝑡) =𝑓(𝑡+𝑇) −𝑓(𝑡) , 𝑘 𝛿 𝑡 𝑡+𝑇 (20) [𝑛, 𝑛] +1 ⊂ ⋃ (𝑙 −𝛿 ,𝑙 +𝛿 ), 𝐹 (𝑡) = [𝑓 ( )−𝑓( )] , 𝑡 ∈ R. 𝑖 𝑙𝑖 𝑖 𝑙𝑖 (11) 2 𝑖=1 𝑀𝑓 𝜋 𝜋 Journal of Function Spaces and Applications 3

Then 𝐹1(𝑡) ≡2 𝐹 (𝑡).If𝐹1(𝑡) ≡2 𝐹 (𝑡) ≡,where 𝐶 𝐶 is a fixed Acknowledgments constant, then H.-S. Ding acknowledges support from the NSF of China 𝑓 (𝑡+𝑇) =𝑓(𝑡) +𝐶, 𝑡∈R, (21) (11101192), the Chinese Ministry of Education (211090), the NSF of Jiangxi Province (20114BAB211002), and the Jiangxi which yields Provincial Education Department (GJJ12173). 𝑓 (𝑘𝑇) −𝑓(0) 𝐶= 󳨀→ 0 , 𝑘󳨀→ ∞, (22) References 𝑘 [1] L. Amerio and G. Prouse, Almost-Periodic Functions and Func- 𝑓 since is bounded. Thus, we have tional Equations, The University Series in Higher Mathematics, Van Nostrand Reinhold, New York, NY, USA, 1971. 𝑡 𝑡+𝑇 𝑓 (𝑡+𝑇) =𝑓(𝑡) ,𝑓()=𝑓( ), 𝑡∈R. [2]P.H.BezandryandT.Diagana,Almost Periodic Stochastic 𝜋 𝜋 Processes, Springer, New York, NY, USA, 2011. (23) [3] H. Bohr, Almost Periodic Functions, Chelsea Publishing, New York, NY, USA, 1947. Noting that 𝑙0 is the fundamental period of 𝑓 and 𝜋𝑙0 is 𝑓(⋅/𝜋) [4] C. Corduneanu, Almost Periodic Functions, Chelsea Publishing, the fundamental period of , there exist two positive New York, NY, USA, 2nd edition, 1989. integers 𝑝 and 𝑞 such that [5] T. Diagana, Pseudo Almost Periodic Functions in Banach Spaces, Nova Science, New York, BY, USA, 2007. 𝑝𝑙0 =𝑇=𝑞𝜋𝑙0, (24) [6] A. M. Fink, Almost Periodic Differential Equations,vol.377of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1974. that is, 𝜋=𝑝/𝑞, which is a contradiction. If 𝐹1 =𝐹2 is not [7]B.M.LevitanandV.V.Zhikov,Almost Periodic Functions and constant, then, by Remark 3, we can assume that 𝑇0 is the Differential Equations, Cambridge University Press, Cambridge, fundamental period of 𝐹1 and 𝐹2.Notingthat𝑙0 is a period UK, 1982. of 𝐹1 and 𝜋𝑙0 is a period of 𝐹2,similartotheaboveproof, wecanalsoshowthat𝜋 is a rational number, which is a [8]G.M.N’Guer´ ekata,´ Almost Automorphic and Almost Periodic Functions in Abstract Spaces, Kluwer Academic, New York, NY, contradiction. USA, 2001. In conclusion, 𝑃(𝑋) is countable unions of closed subsets with empty interior. So 𝑃(𝑋) is a set of first category. [9] G. M. N’Guer´ ekata,´ Topics in Almost Automorphy,Springer, New York, NY, USA, 2005. Remark 7. Since 𝐴𝑃(𝑋) is a set of second category, it follows [10] A. A. Pankov, Bounded and Almost Periodic Solutions of Nonlin- from Theorem 6 that 𝐴𝑃(𝑋) \ 𝑃(𝑋) is a set of second earOperatorDifferentialEquations,vol.55ofMathematics and its Applications (Soviet Series), Kluwer Academic, Dordrecht, category, which means that, to some extent, the “amount” of The Netherlands, 1990. almost periodic functions (not periodic) is far more than the “amount” of continuous periodic functions. [11] W.X. Shen and Y.F.Yi, Almost Automorphic and Almost Periodic Dynamics in Skew-Product Semiflows, no. 647 of Memoirs of the American Mathematical Society, 1998. Theorem 8. 𝐴𝑃(𝑋) is a set of first category in 𝐴𝐴(𝑋). [12] S. Zaidman, Almost-Periodic Functions in Abstract Spaces,vol. Proof. Firstly, 𝐴𝑃(𝑋) is a closed subset of 𝐴𝐴(𝑋). Secondly, 126 of Research Notes in Mathematics,Pitman,Boston,Mass, 𝐴𝑃(𝑋) has an empty interior in 𝐴𝐴(𝑋). In fact, letting USA, 1985. [13] C. Zhang, Almost Periodic Type Functions and Ergodicity, 1 Kluwer Academic, Dordrecht, The Netherlands, 2003. 𝜑 (𝑡) = ,𝑡∈R, sin (25) 2+cos 𝑡+cos (√2𝑡) [14] W. Rudin, Functional Analysis, International Series in Pure and Applied Mathematics, McGraw-Hill, New York, NY, USA, 2nd edition, 1991. for every 𝑓 ∈ 𝐴𝑃(𝑋) and 𝛿>0,wehave𝑓𝛿 ∉ 𝐴𝑃(𝑋) and 𝑓𝛿 ∈ 𝑁(𝑓,𝛿),where

𝛿𝑥0 󵄩 󵄩 𝑓𝛿 (𝑡) =𝑓(𝑡) + 𝜑 (𝑡) ,𝑡∈R,𝑀𝜑 = sup 󵄩𝜑 (𝑡)󵄩 , 𝑀𝜑 𝑡∈R (26) and 𝑥0 ∈𝑋is some constant with ‖𝑥0‖=1.Thiscompletes the proof.

Remark 9. By Theorem 8, 𝐴𝐴(𝑋) \ 𝐴𝑃(𝑋) is a set of second category in 𝐴𝐴(𝑋), which means that, to some extent, the “amount” of almost automorphic functions (not almost periodic) is far more than the “amount” of almost periodic functions.