DOCSLIB.ORG

Bochner's Subordionation and Fractional Caloric

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Bochner's Subordionation and Fractional Caloric BOCHNER'S SUBORDIONATION AND FRACTIONAL CALORIC SMOOTHING IN BESOV AND TRIEBEL{LIZORKIN SPACES VICTORIYA KNOPOVA AND RENE´ L. SCHILLING Abstract. We use Bochner's subordination technique to obtain caloric smooth- ing estimates in Besov- and Triebel{Lizorkin spaces. Our new estimates extend known smoothing results for the Gauß–Weierstraß, Cauchy{Poisson and higher- order generalized Gauß–Weierstraß semigroups. Extensions to other function spaces (homogeneous, hybrid) and more general semigroups are sketched. 1. Introduction f Let (Wt )t≥0 be the f-subordinated Gauß–Weierstraß semigroup; by this we mean the family of operators which is defined through the Fourier transform F(W f u)(ξ) = e−tf(jξj2) Fu(ξ); u 2 S(Rn)(1.1) where the function f : (0; 1) ! (0; 1) is a so-called Bernstein function, see Sec- tion 3. Typical examplesp are f(x) = x (which gives the classical Gauß{Weierstraß semigroup), f(x) = x (which gives the Cauchy{Poisson semigroup) or f(x) = xα, 0 < α < 1 (which leads to the stable semigroups). In this note we prove the caloric f smoothing of (the extension of) (Wt )t≥0 in Besov and Triebel{Lizorkin spaces, see Section 2. \Caloric smoothing" refers to the smoothing effect of the semigroup which can be quantified through inequalities of the following form (f) s+d s s (1.2) Cf;d(t)kWt u j Ap;q k ≤ ku j Ap;qk for all 0 < t ≤ 1 and u 2 Ap;q; where d ≥ 0 is arbitrary, Cf;d(t) is a constant depending only on f and d, and s s Rn s Rn Cf;d(t) ! 0 as t ! 0; Ap;q = Ap;q( ) stands for a Besov space Bp;q( ) or Triebel{ s Rn Lizorkin space Fp;q( ). Results of this type are known for the Gauß–Weierstraß semigroup Wt, i.e. for f(x) = x (see Triebel [15, Theorem 3.35]) and for the generalized Gauß–Weierstraß (m) N semigroup Wt where m 2 ; these operators are also given through the relation (1.1) if we take f(x) = xm, cf. [15, Remark 3.37], but note that for m > 1 this is not a Bernstein function) and recent results by Baaske & Schmeißer [1, Theorem 3.5]. We will use Bochner's subordination technique to prove (1.2) for arbitrary Bern- β stein functions f(x) and arbitrary powers f(x) = x , β > 0. The constant Cf;d(t) is comparable with [f −1(1=t)]−d=2. As an application we generalize the result of Baaske & Schmeißer [1, Theorem 3.5] on the existence and uniqueness of the mild and strong solutions of a nonlinear Cauchy problem with arbitrary (fractional) powers of the Laplacian (−∆)β, β ≥ 1. 2010 Mathematics Subject Classification. Primary: 46E36; 60J35. Secondary: 35K25; 35K55; 60G51. Key words and phrases. Function spaces, caloric smoothing, Gauß–Weierstraß semigroup, subordination. 1 2 V. KNOPOVA AND R.L. SCHILLING Acknowledgement. We thank Hans Triebel from Jena for his encouragement, helpful comments and the possibility to use the preprint of his forthcoming mono- graph [15]. 2. Function spaces n Let us briefly recall some notation. Lp(R ), resp., `q(N0) denote the spaces of pth order integrable functions on Rn, resp., qth order summable sequences indexed n by N0; we admit 0 < p; q ≤ 1. Since we are always working in R , we will usually write Lp instead of Lp(Rn). If p; q < 1 these spaces are quasi-Banach spaces, their (quasi-)norms are denoted by k· j Lpk and k· j `qk, respectively. We write h i ku(k; x) j Lp j `q(N0)k resp. ku(k; x) j `q(N0) j Lpk to indicate that we take first the Lp-norm and then the `q(N0)-norm [resp. first the `q(N0)-norm and then the Lp-norm]. Throughout, we use j; k; m for discrete and x; y; z for continuous variables, so there should be no confusion as to which variable is used for the `q(N0)-norm or Lp-norm. We follow Triebel [15, Definition 1.1 and Remark 1.2 (1.14), (1.15)] for the defini- tion of the scales of Besov- and Triebel{Lizorkin spaces. Let Fu denote the Fourier transform of a function u; the extension to the space of tempered distributions 0 Rn 1 1 1 S ( ) is again denoted by F. Fix some φ0 2 C0 such that B(0;1) ≤ φ0 ≤ B(0;3=2) −k −(k+1) P1 and set φk(x) := φ0(2 x) − φ0(2 x). Since k=0 φk(x) = 1, the sequence (φk)k≥0 is a dyadic resolution of unity. By −1 φk(D)u(x) := F (φkFu)(x) we denote the pseudo-differential operator (Fourier multiplier operator) with −J −J n symbol φk. We will also need the dyadic cubes QJ;M = 2 M + 2 (0; 1) , where J 2 Z, M 2 Zn and (0; 1)n is the open unit cube in Rn. Definition 2.1. Let (φk)k≥0 be any dyadic resolution of unity. R s a) Let p 2 (0; 1], q 2 (0; 1] and s 2 . The Besov space Bp;q is the family of all u 2 S0(Rn) such that the following (quasi-)norm is finite: s ks N ku j Bp;qk := k2 φk(D)u(x) j Lp j `q( 0)k: R s b) Let p 2 (0; 1), q 2 (0; 1] and s 2 . The Triebel{Lizorkin space Fp;q is the family of all f 2 S0(Rn) such that the following (quasi-)norm is finite s ks N ku j Fp;qk := k2 φk(D)u(x) j `q( 0) j Lpk: R s c) Let p = 1, q 2 (0; 1) and s 2 . The Triebel{Lizorkin space F1;q is the family of all f 2 S0(Rn) such that the following (quasi-)norm is finite 1 !1=q s Jn=q X ksq q ku j F1;qk := sup 2 2 jφk(D)u(x)j dx : J2N ;M2Zn ˆ 0 QJ;M k=J R s d) Let p = q = 1 and s 2 . The Triebel{Lizorkin space F1;1 is the family of all f 2 S0(Rn) such that the following norm is finite s ku j F1;1k := sup sup sup jφk(D)u(x)j: n J2N0;M2Z x2QJ;M k≥J SUBORDINATION AND FUNCTION SPACES 3 s s R Note that F1;1 = B1;1 for all s 2 and that the norms appearing in Def- s s inition 2.1.d) and 2.1.a) coincide if p = q = 1: ku j F1;1k = ku j B1;1k. Definition 2.1 does not depend on the choice of (φk)k≥0 since different resolutions of unity lead to equivalent (quasi-)norms. Various properties of these spaces as well as their relation to other classical function spaces can be found in Triebel [15], see also [13] and [14]. Consider the heat kernel (Gaussian probability density) related to the Laplace operator on Rn 1 2 (2.1) g (x) := e−|xj =(4t); t > 0; x 2 Rn: t (4πt)n=2 We can use gt(x) to define a convolution operator on the space Bb of bounded Borel functions u : Rn ! R (2.2) Wtu(x) := gt ∗ u(x) = gt(y − x)u(y) dy: ˆRn For positive u ≥ 0 the above integral always exists in [0; 1] and extends Wt to all positive Borel functions. It is not difficult to see that gt+s = gt ∗ gs, i.e. (Wt)t≥0 is a semigroup. The operators are positivity preserving (Wtu ≥ 0 if u ≥ 0) and −tjξj2 conservative (Wt1 ≡ 1). If u 2 S, then F(Wtu)(ξ) = e Fu(ξ). We will need the following simple lemma. We provide the short proof for the readers' convenience. Lemma 2.2. Let (Wt)t≥0 be the Gauß–Weierstraß semigroup. a) Wt : Lp ! Lp, p 2 [1; 1] is a contraction, i.e. kWtu j Lpk ≤ ku j Lpk: n b) Let k(·) be a sequence of positive measurable functions on R such that n ( k(x))k≥0 2 `q(N0) for some q 2 [1; 1] and all x 2 R . Then kWt k(x) j `q(N0)k ≤ Wtk k(·) j `q(N0)k(x): Proof. Part a) follows immediately from Jensen's inequality, see [10, Theorem 13.13], for the probability measure gt(y) dy: p p p kWtu j Lpk = u(x − y)gt(y) dy dx ≤ ju(x − y)j gt(y) dy dx ˆRn ˆRn ˆRn ˆRn p p = ku j Lpk gt(z) dz = ku j Lpk : ˆRn If 0 < p < 1, the inequality is reversed. q0 In order to prove Part b), we fix x and pick a sequence (ak)k2N0 from ` where 0 q q = 1−q . We have hWt k(·); aki = Wth k(·); aki ≤ Wtk k(·) j `qk · kak j `q0 k: In the estimate we use the fact that Wt is linear and positivity preserving, implying that u 7! Wtu is monotone. Taking the supremum over all sequences such that kak j `q0 k = 1 gives kWt k(x) j `qk = sup hWt k(x); aki ≤ Wt (k k(·) j `qk)(x): kakj`q0 k=1 Lemma 2.2 is the key ingredient for our proof that Wt is a contraction in the scales of Besov- and Triebel{Lizorkin spaces. 4 V. KNOPOVA AND R.L. SCHILLING Theorem 2.3. Let (Wt)t≥0 be the Gauß–Weierstraß semigroup and s 2 R.
Load more

Recommended publications
  • Preregular Maps Between Banach Lattices
    BULL. AUSTRAL. MATH. SOC. 46A40, 46M05 VOL. II (1974), 231-254. (46BI0, 46BI5) Preregular maps between Banach lattices David A. Birnbaum A continuous linear map from a Banach lattice E into a Banach lattice F is preregular if it is the difference of positive continuous linear maps from E into the bidual F" of F . This paper characterizes Banach lattices B with either of the following properties: (1) for any Banach lattice E , each map in L(E, B) is preregular; (2) for any Banach lattice F , each map in L{B, F) is preregular. It is shown that B satisfies (l) (respectively (2)) if and- only if B' satisfies (2) (respectively (l)). Several order properties of a Banach lattice satisfying (2) are discussed and it is shown that if B satisfies (2) and if B is also an atomic vector lattice then B is isomorphic as a Banach lattice to I (T) for some index set Y . 1. Introduction The following natural question arises in the theory of Banach lattices: Given Banach lattices E and F , is each map in the space L(E, F) of continuous linear maps from E into F the difference of positive (continuous) linear maps? It is known that if F is a C{X) for X an extremally disconnected, compact Hausdorff space X or if E is an A£-space and F has the monotone convergence property then the answer to Received 17 May 1971*. 231 Downloaded from https://www.cambridge.org/core. IP address: 170.106.202.226, on 02 Oct 2021 at 11:23:22, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms.
    [Show full text]
  • Integration of Functions with Values in a Banach Lattice
    INTEGRATION OF FUNCTIONS WITH VALUES IN A BANACH LATTICE G. A. M.JEURNINK INTEGRATION OF FUNCTIONS WITH VALUES IN A BANACH LATTICE PROMOTOR: PROF. DR. А. С. M. VAN ROOIJ INTEGRATION OF FUNCTIONS WITH VALUES IN A BANACH LATTICE PROEFSCHRIFT TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE WISKUNDE EN NATUURWETENSCHAPPEN AAN DE KATHOLIEKE UNIVERSI­ TEIT TE NIJMEGEN, OP GEZAG VAN DE RECTOR MAGNIFICUS PROF DR Ρ G А В WIJDEVELD, VOLGENS BESLUIT VAN HET COI LEGE VAN DEKANEN IN HET OPENBAAR TE VERDEDIGEN OP VRIJDAG 18 JUNI 1982, DES MIDDAGS TE 2 00 ULR PRECIES DOOR GERARDUS ALBERTUS MARIA JEURNINK GEBOREN TE DIEPENVEEN И krips repro meppel 1982 Aan mijn ошіелі Voor hun medewerking aan dit proefschrift ben ik Trees van der Eem-Mijnen en Ciaire Elings-Mesdag zeer dankbaar. CONTENTS INTRODUCTION AND SUMMARY CONVENTIONS AND NOTATIONS CHAPTER I PRELIMINARIES 1 §1 Measurability and integrability of functions with values in a Banach space 1 §2 Banach lattices 13 §3 Summability of sequences in Banach lattices 23 CHAPTER II INTEGRATION 35 51 Integration of functions with values in a Banach lattice 35 §2 Riesz spaces of integrable functions 47 §3 Banach lattice theory for spaces of integrable functions 58 §4 Examples 68 CHAPTER III SPECIAL CLASSES OF OPERATORS AND TENSOR PRODUCTS 75 51 Induced maps between spaces of integrable functions 75 §2 θ-operators 79 5 3 Δ-operators 85 5 4 Tensor products of Banach lattices 92 55 Tensor products of Banach spaces and Banach lattices 103 56 Examples of tensor products 108 CHAPTER IV VECTOR MEASURES 115 §1 Vector measures with values in a Banach lattice 115 §2 Weakly equivalent functions 124 §3 The Radon-Nikodym property 136 54 Weak measurable functions 147 CHAPTER V DANIELL INTEGRATION 153 §1 An extension of the Pettis integral 153 §2 The extension of the integral on S(μ.
    [Show full text]
  • Extensions and Liftings of Positive Linear Mappings On
    TRANSACTIONSOF THE AMERICANMATHEMATICAL SOCIETY Volume 211, 1975 EXTENSIONSAND LIFTINGS OF POSITIVE LINEARMAPPINGS ON BANACHLATTICES BY HEINRICH P. LOTZi1) ABSTRACT. Let F be a closed sublattice of a Banach lattice G. We show that any positive linear mapping from F into L (i¿) or C(X) for a Stonian space X has a positive norm preserving extension to G. A dual re- sult for positive norm preserving liftings is also established. These results are applied to obtain extension and lifting theorems for order summable and majorizing linear mappings. We also obtain some partial results concerning positive extensions and liftings of compact linear mappings. The main purpose of this paper is to establish some conditions under which positive linear mappings between Banach lattices have norm preserv- ing positive linear extensions or liftings. These results are then applied to obtain extension and lifting theorems for order summable and majorizing linear mappings and to establish the inductive and projective character of the |cr|- tensor product topology introduced by Jacobs [6]. Finally, we obtain some partial results concerning the problem of finding positive, compact linear extensions and liftings of positive compact linear mappings between Banach lattices. We begin our discussion with a summary of known results concerning the corresponding problems for continuous linear mappings between Banach spaces. Suppose that E, F and G are Banach spaces, that cf> is an isometry of F into G and that T is a continuous linear mapping of F into E. When does T have a norm preserving linear extension T:G —*E, that is, when does there exist a continuous linear mapping T of G into E such that ||T|| = ||T|| and such that the following diagram commutes? If E is the scalar field, the Hahn-Banach theorem asserts that such an exten- sion T of T always exists.
    [Show full text]
  • View This Volume's Front and Back Matter
    http://dx.doi.org/10.1090/gsm/084 Cone s an d Dualit y This page intentionally left blank Cone s an d Dualit y Charalambo s D . Alipranti s RabeeTourk y Graduate Studies in Mathematics Volum e 84 •& Ip^Sn l America n Mathematica l Societ y *0||jjO ? provjcjence i Rhod e Islan d Editorial Board David Cox (Chair) Walter Craig N. V. Ivanov Steven G. Krantz 2000 Mathematics Subject Classification. Primary 46A40, 46B40, 47B60, 47B65; Secondary 06F30, 28A33, 91B28, 91B99. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-84 Library of Congress Cataloging-in-Publication Data Aliprantis, Charalambos D. Cones and duality / Charalambos D. Aliprantis, Rabee Tourky. p. cm. — (Graduate studies in mathematics, ISSN 1065-7339 ; v. 84) Includes bibliographical references and index. ISBN 978-0-8218-4146-4 (alk. paper) 1. Cones (Operator theory). 2. Linear topological spaces, Ordered. I. Tourky, Rabee, 1966- II. Title. QA329 .A45 2007 515'.724—dc22 2007060758 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA.
    [Show full text]
  • Preregular Maps Between Banach Lattices
    BULL. AUSTRAL. MATH. SOC. 46A40, 46M05 VOL. II (1974), 231-254. (46BI0, 46BI5) Preregular maps between Banach lattices David A. Birnbaum A continuous linear map from a Banach lattice E into a Banach lattice F is preregular if it is the difference of positive continuous linear maps from E into the bidual F" of F . This paper characterizes Banach lattices B with either of the following properties: (1) for any Banach lattice E , each map in L(E, B) is preregular; (2) for any Banach lattice F , each map in L{B, F) is preregular. It is shown that B satisfies (l) (respectively (2)) if and- only if B' satisfies (2) (respectively (l)). Several order properties of a Banach lattice satisfying (2) are discussed and it is shown that if B satisfies (2) and if B is also an atomic vector lattice then B is isomorphic as a Banach lattice to I (T) for some index set Y . 1. Introduction The following natural question arises in the theory of Banach lattices: Given Banach lattices E and F , is each map in the space L(E, F) of continuous linear maps from E into F the difference of positive (continuous) linear maps? It is known that if F is a C{X) for X an extremally disconnected, compact Hausdorff space X or if E is an A£-space and F has the monotone convergence property then the answer to Received 17 May 1971*. 231 Downloaded from https://www.cambridge.org/core. IP address: 170.106.202.8, on 27 Sep 2021 at 07:26:14, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms.
    [Show full text]
  • The Problems of Employment in Mathematical Sciences 718 Some Super-Classics of Mathematics 723 News Items An:D Announcements 722, 726, 730, 738, 742
    OF THE AMERICAN MATHEMATICAL SOCIETY Edited by Everett Pitcher and Gordon L. Walker CONTENTS MEETINGS Calendar of Meetings o o o o o o o o o o o o o o o o o o o o Inside Front Cover Program for the August Meeting in University Park, Pennsylvania 690 Abstracts for the Meeting: Pages 752-792 PRELIMINARY ANNOUNCEMENTS OF MEETINGS o o o o o o o o o o 715 THE PROBLEMS OF EMPLOYMENT IN MATHEMATICAL SCIENCES 718 SOME SUPER-CLASSICS OF MATHEMATICS 723 NEWS ITEMS AN:D ANNOUNCEMENTS 722, 726, 730, 738, 742 MEMORANDA TO MEMBERS • o o o 0 0 0 0 0 0 0 0 0 0 727 Contributing Members Grants for Scientific Research Mathematical Sciences Employment Register Annual Salary Survey Change of Address??????? LETTERS TO THE EDITOR 731 SPECIAL MEETINGS INFORMATION CENTER 733 PERSONAL ITEMS o o o o 736 NEW AMS PUBLICATIONS 739 BACKLOG OF MATHEMATICS RESEARCH JOURNALS 743 VISITING MATHEMATICIANS o o o o o o o 744 ABSTRACTS OF CO~TRIBUTED PAPERS 751 ABSTRACTS PRESENTED TO THE SOCIETY 793 ERRATA TO ABSTRACTS o 792, 842 INDEX TO ADVERTISERS o 848 The Seventy-Sixth Summer Meeting Pennsylvania State University University Park, Pennsylvania August 31-September 3, 1971 The seventy-sixth summer meeting matics. of the American Mathematical Society By invitation of the Committee to will be held at The Pennsylvania State Select Hour Speakers for Annual and University, University Park, Pennsylva­ Summer Meetings, there will be five in­ nia, from Tuesday, August 31, 1971, vited hour addresses at the meeting. Pro­ through Friday, September 3, 1971.
    [Show full text]
  • NARROW ORTHOGONALLY ADDITIVE OPERATORS 3 Means That Y Is a Fragment of X
    NARROW ORTHOGONALLY ADDITIVE OPERATORS MARAT PLIEV AND MIKHAIL POPOV Abstract. We extend the notion of narrow operators to nonlinear maps on vector lattices. The main objects are orthogonally additive operators and, in particular, abstract Uryson operators. Most of the re- sults extend known theorems obtained by O. Maslyuchenko, V. Mykhay- lyuk and the second named author published in Positivity 13 (2009), pp. 459–495, for linear operators. For instance, we prove that every orthogonally additive laterally-to-norm continuous C-compact operator from an atomless Dedekind complete vector lattice to a Banach space lc is narrow. Another result asserts that the set Uon(E, F ) of all order narrow laterally continuous abstract Uryson operators is a band in the vector lattice of all laterally continuous abstract Uryson operators from an atomless vector lattice E with the principal projection property to a Dedekind complete vector lattice F . The band generated by the dis- jointness preserving laterally continuous abstract Uryson operators is lc the orthogonal complement to Un (E, F ). 1. Introduction 1.1. About the paper. Narrow operators were introduced and studied in 1990 by Plichko and the second named author [14] as a generalization of compact operators defined on symmetric function spaces. Since that, narrow operators were defined on much more general domain spaces generalizing the previous cases, like K¨othe function spaces [3], vector lattices [10] and lattice normed spaces [16]. Now it is a subject of an intensive study (see recent monograph [18]). Some properties of AM-compact operators are generalized to narrow op- erators, but not all of them.
    [Show full text]
  • Arxiv:1908.06786V2 [Math.PR]
    BOCHNER’S SUBORDINATION AND FRACTIONAL CALORIC SMOOTHING IN BESOV AND TRIEBEL–LIZORKIN SPACES VICTORIYA KNOPOVA AND RENE´ L. SCHILLING Abstract. We use Bochner’s subordination technique to obtain caloric smooth- ing estimates in Besov- and Triebel–Lizorkin spaces. Our new estimates extend known smoothing results for the Gauß–Weierstraß, Cauchy–Poisson and higher- order generalized Gauß–Weierstraß semigroups. Extensions to other function spaces (homogeneous, hybrid) and more general semigroups are sketched. 1. Introduction f Let (Wt )t≥0 be the f-subordinated Gauß–Weierstraß semigroup; by this we mean the family of operators which is defined through the Fourier transform 2 (1.1) (W f u)(ξ)= e−tf(|ξ| ) u(ξ), u S(Rn), F t F ∈ where the function f : (0, ) (0, ) is a so-called Bernstein function, see Sec- tion 3. Typical examples are∞ f→(x) =∞x (which gives the classical Gauß–Weierstraß semigroup), f(x)= √x (which gives the Cauchy–Poisson semigroup) or f(x)= xα, 0 <α< 1 (which leads to the stable semigroups). In this note we prove the caloric f smoothing of (the extension of) (Wt )t≥0 in Besov and Triebel–Lizorkin spaces, see Section 2. “Caloric smoothing” refers to the smoothing effect of the semigroup which can be quantified through inequalities of the following form (1.2) C (t) W f u As+d u As for all 0 < t 1 and u As , f,d k t | p,q k≤k | p,qk ≤ ∈ p,q where d 0 is arbitrary, Cf,d(t) is a constant depending only on f and d, and ≥ s s Rn s Rn Cf,d(t) 0 as t 0; the symbol Ap,q = Ap,q( ) stands for a Besov space Bp,q( ) → → s Rn or a Triebel–Lizorkin space Fp,q( ).
    [Show full text]
  • Kernel Mean Embedding of Probability Measures and Its Applications to Functional Data Analysis
    Kernel Mean Embedding of Probability Measures and its Applications to Functional Data Analysis Saeed Hayati Kenji Fukumizu [email protected] [email protected] Afshin Parvardeh [email protected] November 5, 2020 Abstract This study intends to introduce kernel mean embedding of probability measures over infinite-dimensional separable Hilbert spaces induced by functional response statistical models. The embedded function represents the concentration of probability measures in small open neighborhoods, which identifies a pseudo-likelihood and fosters a rich framework for sta- tistical inference. Utilizing Maximum Mean Discrepancy, we devise new tests in functional response models. The performance of new derived tests is evaluated against competitors in three major problems in functional data analysis including function-on-scalar regression, functional one-way ANOVA, and equality of covariance operators. 1 Introduction Functional response models are among the major problems in the context of Functional Data Analysis. A fundamental issue in dealing with functional response statistical models arises due to the lack of practical frameworks on characterizing probability measure on function spaces. This is mainly a con- sequence of the tremendous gap on how we present probability measures in arXiv:2011.02315v1 [math.ST] 4 Nov 2020 finite-dimensional and infinite-dimensional spaces. A useful property of finite-dimensional spaces is the existence of a locally fi- nite, strictly positive, and translation invariant measure like Lebesgue or count- ing measure, which makes it easy to take advantage of probability measures directly in the statistical inference. Fitting a statistical model, and estimat- ing parameters, hypothesis testing, deriving confidence regions and developing goodness of fit indices, all can be applied by integrating distribution or condi- tional distribution of response variables as a presumption into statistical proce- dures.
    [Show full text]