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Quantitative Hahn-Banach Theorems and Isometric Extensions for Wavelet and Other Banach Spaces
Axioms 2013, 2, 224-270; doi:10.3390/axioms2020224 OPEN ACCESS axioms ISSN 2075-1680 www.mdpi.com/journal/axioms Article Quantitative Hahn-Banach Theorems and Isometric Extensions for Wavelet and Other Banach Spaces Sergey Ajiev School of Mathematical Sciences, Faculty of Science, University of Technology-Sydney, P.O. Box 123, Broadway, NSW 2007, Australia; E-Mail: [email protected] Received: 4 March 2013; in revised form: 12 May 2013 / Accepted: 14 May 2013 / Published: 23 May 2013 Abstract: We introduce and study Clarkson, Dol’nikov-Pichugov, Jacobi and mutual diameter constants reflecting the geometry of a Banach space and Clarkson, Jacobi and Pichugov classes of Banach spaces and their relations with James, self-Jung, Kottman and Schaffer¨ constants in order to establish quantitative versions of Hahn-Banach separability theorem and to characterise the isometric extendability of Holder-Lipschitz¨ mappings. Abstract results are further applied to the spaces and pairs from the wide classes IG and IG+ and non-commutative Lp-spaces. The intimate relation between the subspaces and quotients of the IG-spaces on one side and various types of anisotropic Besov, Lizorkin-Triebel and Sobolev spaces of functions on open subsets of an Euclidean space defined in terms of differences, local polynomial approximations, wavelet decompositions and other means (as well as the duals and the lp-sums of all these spaces) on the other side, allows us to present the algorithm of extending the main results of the article to the latter spaces and pairs. Special attention is paid to the matter of sharpness. Our approach is quasi-Euclidean in its nature because it relies on the extrapolation of properties of Hilbert spaces and the study of 1-complemented subspaces of the spaces under consideration. -
The Semi-M Property for Normed Riesz Spaces Compositio Mathematica, Tome 34, No 2 (1977), P
COMPOSITIO MATHEMATICA EP DE JONGE The semi-M property for normed Riesz spaces Compositio Mathematica, tome 34, no 2 (1977), p. 147-172 <http://www.numdam.org/item?id=CM_1977__34_2_147_0> © Foundation Compositio Mathematica, 1977, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions géné- rales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infrac- tion pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ COMPOSITIO MATHEMATICA, Vol. 34, Fasc. 2, 1977, pag. 147-172 Noordhoff International Publishing Printed in the Netherlands THE SEMI-M PROPERTY FOR NORMED RIESZ SPACES Ep de Jonge 1. Introduction It is well-known that if (0394, F, IL) is a u-finite measure space and if 1 ~ p 00, then the Banach dual L *p of the Banach space Lp = Lp(0394, IL) can be identified with Lq = Lq(L1, 03BC), where p-1 + q-1 = 1. For p =00 the situation is different; the space Li is a linear subspace of L*, and only in a very trivial situation (the finite-dimensional case) we have Li = Lfi. Restricting ourselves to the real case, the Banach dual L *~ is a (real) Riesz space, i.e., a vector lattice, and Li is now a band in L*. The disjoint complement (i.e., the set of all elements in L* disjoint to all elements in LI) is also a band in L*, called the band of singular linear functionals on Loo. -
Burkholder's Inequalities in Noncommutative Lorentz
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 138, Number 7, July 2010, Pages 2431–2441 S 0002-9939(10)10267-6 Article electronically published on March 24, 2010 BURKHOLDER’S INEQUALITIES IN NONCOMMUTATIVE LORENTZ SPACES YONG JIAO (Communicated by Marius Junge) Abstract. We prove Burkholder’s inequalities in noncommutative Lorentz spaces Lp,q(M), 1 <p<∞, 1 ≤ q<∞, associated with a von Neumann algebra M equipped with a faithful normal tracial state. These estimates generalize the classical inequalities in the commutative case. 1. Introduction Martingale inequalities and sums of independent random variables are important tools in classical harmonic analysis. A fundamental result due to Burkholder [1, 2] can be stated as follows. Given a probability space (Ω, F ,P), let {Fn}n≥1 be a nondecreasing sequence of σ-fields of F such that F = ∨Fn and let En be the conditional expectation operator relative to Fn. Given 2 ≤ p<∞ and an p L -bounded martingale f =(fn)n≥1, we have ∞ ∞ 1/2 1/p 2 p (1.1) fLp ≈ Ek−1(|df k| ) + |df k| . Lp Lp k=1 k=1 The first term on the right is called the conditioned square function of f, while the second is called the p-variation of f. Rosenthal’s inequalities [14] can be regarded as the particular case where the sequence df =(df 1,df2, ...) is a family of independent mean-zero random variables df k = ak. In this case it is easy to reduce Rosenthal’s inequalities to ∞ ∞ ∞ 1/2 1/p ≈ 2 p (1.2) ak Lp ak 2 + ak p . -
Arxiv:1712.01188V2 [Math.AP]
GREEN’S FUNCTION FOR SECOND ORDER ELLIPTIC EQUATIONS WITH SINGULAR LOWER ORDER COEFFICIENTS SEICK KIM AND GEORGIOS SAKELLARIS Abstract. We construct Green’s function for second order elliptic operators of the form Lu = −∇ · (A∇u + bu) + c · ∇u + du in a domain and obtain pointwise bounds, as well as Lorentz space bounds. We assume that the matrix of principal coefficients A is uniformly elliptic and bounded and the lower order coefficients b, c, and d belong to certain Lebesgue classes and satisfy the condition d −∇· b ≥ 0. In particular, we allow the lower order coefficients to be singular. We also obtain the global pointwise bounds for the gradient of Green’s function in the case when the mean oscillations of the coefficients A and b satisfy the Dini conditions and the domain is C1,Dini. 1. Introduction Let Ω be a domain (i.e., an open connected set) in Rn with n ≥ 3. We consider second order elliptic operators in divergence form n n ij i i Lu = − Di(a (x)Dju + b (x)u) + c (x)Diu + d(x)u, iX,j=1 Xi=1 which hereafter shall be abbreviated as Lu = − div(A∇u + bu) + c · ∇u + du. We assume that the principal coefficients A = (aij) are measurable n × n matrices that are bounded and uniformly elliptic; i.e. there is a constant λ> 0 such that n λ|ξ|2 ≤ A(x)ξ · ξ = aij(x)ξiξj, ∀x ∈ Ω, ∀ξ ∈ Rn. (1.1) iX,j=1 We also assume that the lower order coefficients b = (b1,..., bn), c = (c1,..., cn), and d are such that b ∈ Lq(Ω), c ∈ Lr(Ω), d ∈ Ls(Ω) for some q, r ≥ n, s ≥ n/2. -
Fourier Analysis in Function Space Theory
Course No. 401-4463-62L Fourier Analysis in Function Space Theory Dozent: Tristan Rivi`ere Assistant: Alessandro Pigati Contents 1 The Fourier transform of tempered distributions 1 1.1 The Fourier transforms of L1 functions . 1 1.2 The Schwartz Space S(Rn)........................... 4 1.3 Frechet Spaces . 6 1.4 The space of tempered distributions S0(Rn).................. 12 1.5 Convolutions in S0(Rn)............................. 21 2 The Hardy-Littlewood Maximal Function 26 2.1 Definition and elementary properties. 26 2.2 Hardy-Littlewood Lp−theorem for the Maximal Function. 27 2.3 The limiting case p =1. ............................ 30 3 Quasi-normed vector spaces 32 3.1 The Metrizability of quasi-normed vector spaces . 32 3.2 The Lorentz spaces Lp;1 ............................ 36 3.3 Decreasing rearrangement . 37 3.4 The Lorentz spaces Lp;q ............................ 39 3.5 Functional inequalities for Lorentz spaces . 44 3.6 Dyadic characterization of some Lorentz spaces and another proof of Lorentz{ Sobolev embedding (optional) . 49 4 The Lp−theory of Calder´on-Zygmund convolution operators. 52 4.1 Calder´on-Zygmund decompositions. 52 4.2 An application of Calder´on-Zygmund decomposition . 54 4.3 The Marcinkiewicz Interpolation Theorem - The Lp case . 56 4.4 Calderon Zygmund Convolution Operators over Lp ............. 58 4.4.1 A \primitive" formulation . 60 4.4.2 A singular integral type formulation . 64 4.4.3 The case of homogeneous kernels . 69 4.4.4 A multiplier type formulation . 71 4.4.5 Applications: The Lp theory of the Riesz Transform and the Laplace and Bessel Operators . 74 4.4.6 The limiting case p =1........................ -
HEAT FLOW and QUANTITATIVE DIFFERENTIATION Contents 1
HEAT FLOW AND QUANTITATIVE DIFFERENTIATION TUOMAS HYTONEN¨ AND ASSAF NAOR Abstract. For every Banach space (Y; k · kY ) that admits an equivalent uniformly convex norm we prove that there exists c = c(Y ) 2 (0; 1) with the following property. Suppose that n 2 N and that X is an n-dimensional normed space with unit ball BX . Then for every 1-Lipschitz function cn f : BX ! Y and for every " 2 (0; 1=2] there exists a radius r > exp(−1=" ), a point x 2 BX with x + rBX ⊆ BX , and an affine mapping Λ : X ! Y such that kf(y) − Λ(y)kY 6 "r for every y 2 x+rBX . This is an improved bound for a fundamental quantitative differentiation problem that was formulated by Bates, Johnson, Lindenstrauss, Preiss and Schechtman (1999), and consequently it yields a new proof of Bourgain's discretization theorem (1987) for uniformly convex targets. The strategy of our proof is inspired by Bourgain's original approach to the discretization problem, which takes the affine mapping Λ to be the first order Taylor polynomial of a time-t Poisson evolute of an extension of f to all of X and argues that, under appropriate assumptions on f, there must exist a time t 2 (0; 1) at which Λ is (quantitatively) invertible. However, in the present context we desire a more stringent conclusion, namely that Λ well-approximates f on a macroscopically large ball, in which case we show that for our argument to work one cannot use the Poisson semigroup. Nevertheless, our strategy does succeed with the Poisson semigroup replaced by the heat semigroup. -
Regularization and Simulation of Constrained Partial Differential
Regularization and Simulation of Constrained Partial Differential Equations vorgelegt von Diplom-Mathematiker Robert Altmann aus Berlin Von der Fakult¨atII - Mathematik und Naturwissenschaften der Technischen Universit¨atBerlin zur Erlangung des akademischen Grades Doktor der Naturwissenschaften Dr. rer. nat genehmigte Dissertation Promotionsausschuss: Vorsitzende: Prof. Dr. Noemi Kurt Berichter: Prof. Dr. Volker Mehrmann Berichterin: Prof. Dr. Caren Tischendorf Berichter: Prof. Dr. Alexander Ostermann Tag der wissenschaftlichen Aussprache: 29. 05. 2015 Berlin 2015 Contents Zusammenfassung . v Abstract . vii Published Papers. ix 1. Introduction . 1 Part A Preliminaries 5 2. Differential-algebraic Equations (DAEs) . 6 2.1. Index Concepts . 6 2.1.1. Differentiation Index. 7 2.1.2. Further Index Concepts. 8 2.2. High-index DAEs . 8 2.3. Index Reduction Techniques. 9 2.3.1. Index Reduction by Differentiation . 9 2.3.2. Minimal Extension . 9 3. Functional Analytic Tools . 11 3.1. Fundamentals . 11 3.1.1. Dual Operators and Riesz Representation Theorem . 11 3.1.2. Test Functions and Distributions . 12 3.1.3. Sobolev Spaces . 13 3.1.4. Traces . 14 3.1.5. Poincar´eInequality and Negative Norms . 15 3.1.6. Weak Convergence and Compactness . 17 3.2. Bochner Spaces . 17 3.3. Sobolev-Bochner Spaces . 20 3.3.1. Gelfand Triples . 20 3.3.2. Definition and Embeddings . 21 4. Abstract Differential Equations . 23 4.1. Nemytskii Mapping . 23 4.2. Operator ODEs . 24 4.2.1. First-order Equations . 25 4.2.2. Second-order Equations. 26 4.3. Operator DAEs . 27 i 5. Discretization Schemes . 29 5.1. Spatial Discretization . 29 5.1.1. -
Weak-Type Weights and Normable Lorentz Spaces 1
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 124, Number 3, March 1996 WEAK-TYPE WEIGHTS AND NORMABLE LORENTZ SPACES MAR´IA J. CARRO, ALEJANDRO GARC´IA DEL AMO, AND JAVIER SORIA (Communicated by J. Marshall Ash) Abstract. We show that the Lorentz space Λ1(w) is a Banach space if and only if the Hardy-Littlewood maximal operator M satisfies a certain weak-type estimate. We also consider the case of general measures. Finally, we study some properties of several indices associated to these spaces. 1. Introduction We are going to study weighted Lorentz spaces of functions defined in Rn as follows (for standard notation we refer to [BS] and [GR]): If u is a weight in Rn, + w is a weight in R , fu∗ denotes the decreasing rearrangement of f with respect to the measure u(x) dx and 0 <p< , we define ∞ 1/p ∞ p p p Λu(w)= f; f Λu(w) = fu∗(t) w(t)dt < . k k 0 ∞ Z If u 1, we will only write Λp(w). Classical examples are obtained by choosing ≡ (p/q) 1 p q,p w(t)=t − .InthiscaseΛ(w)=L . A classical result of G.G. Lorentz (see [Lo]) shows that Λ1(w) is a norm, if and only if, w is a decreasing function. The problem of findingk·k conditions on w so that Λp(w) is a Banach space (that is, there exists a norm equivalent to Λp(w)) was solved, for p>1, by E. Sawyer ([Sa]). This condition is that the Hardy-Littlewoodk·k maximal operator is bounded on Λp(w). -
HARDY-TYPE OPERATORS in LORENTZ-TYPE SPACES DEFINED on MEASURE SPACES1 Qinxiu Sun∗, Xiao Yu∗∗ and Hongliang Li∗∗∗
Indian J. Pure Appl. Math., 51(3): 1105-1132, September 2020 °c Indian National Science Academy DOI: 10.1007/s13226-020-0453-1 HARDY-TYPE OPERATORS IN LORENTZ-TYPE SPACES DEFINED ON MEASURE SPACES1 Qinxiu Sun¤, Xiao Yu¤¤ and Hongliang Li¤¤¤ ¤Department of Mathematics, Zhejiang University of Science and Technology, Hangzhou 310023, China ¤¤Department of Mathematics, Shangrao Normal University, Shangrao 334001, China ¤¤¤Department of Mathematics, Zhejiang International Studies University, Hangzhou 310012, China e-mail: [email protected] (Received 2 June 2018; accepted 12 June 2019) Weight criteria for the boundedness and compactness of generalized Hardy-type operators Z T f(x) = u1(x) f(y)u2(y)v0(y) d¹(y); x 2 X; (0.1) fÁ(y)·Ã(x)g in Orlicz-Lorentz spaces defined on measure spaces is investigated where the functions Á; Ã; u1; u2; v0 are positive measurable functions. Some sufficient conditions of boundedness of G0 G1 G0 G1;1 T :Λv0 (w0) ! Λv1 (w1) and T :Λv0 (w0) ! Λv1 (w1) are obtained on Orlicz-Lorentz spaces. Furthermore, we achieve sufficient and necessary conditions for T to be bounded and p0 p1;q1 compact from a weighted Lorentz space Λv0 (w0) to another Λv1 (w1). It is notable that the function spaces concerned here are quasi-Banach spaces instead of Banach spaces. Key words : Hardy operator; Orlicz-Lorentz spaces; weighted Lorentz spaces; boundedness; compactness. 2010 Mathematics Subject Classification : 46E30, 46B42. 1. INTRODUCTION R x For the Hardy operator S defined by Sf(x) = 0 f(t)dt, the weighted Lebesgue-norm inequalities have been characterized by many authors (e.g. -
Generalized Lorentz Spaces and Köthe Duality
Generalized Lorentz spaces and Köthe duality Anna Kaminska´ and Yves Raynaud University of Memphis, Paris University VI Conference on Non Linear Functional Analysis, Universitat Politecnica de Valencia, Spain, 17-20 October 2017 Anna Kaminska´ and Yves Raynaud Generalized Lorentz spaces and Köthe duality µ measure on the the measure space (Ω; A; µ) L0(Ω) = L0(Ω; A; µ) , µ-measurable real valued functions on Ω 0 0 L+(Ω) non-negative functions from L (Ω). L1 = L1(Ω), kf k1, L1 = L1(Ω), kf k1 L1 + L1(Ω), kf kL1+L1(Ω) = inffkgk1 + khk1 : f = g + hg < 1 L1 \ L1(Ω), kf kL1\L1(Ω) = maxfkf k1; kf k1g < 1. A Banach function space E over (Ω; A), is a complete vector space 0 E ⊂ L (Ω) equipped with a norm k · kE such that if 0 ≤ f ≤ g, where 0 g 2 E and f 2 L (Ω), then f 2 E and kf kE ≤ kgkE . The space E satisfies the Fatou property whenever for any 0 f 2 L (Ω), fn 2 E such that fn " f a.e. and sup kfnkE < 1 it follows that f 2 E and kfnkE " kf kE . µ distribution of f with respect to µ, df (s) = µfjf j > sg, s ≥ 0, and its ∗,µ µ decreasing rearrangement f (t) = inffs > 0 : dg (s) ≤ tg, 0 < t < µ(Ω). f ; g are equimeasurable (with respect to the measures µ and ν) if µ ν ∗,µ ∗,ν df (s) = dg (s), s ≥ 0; equivalently, f = g . Anna Kaminska´ and Yves Raynaud Generalized Lorentz spaces and Köthe duality A Banach function space E is symmetric space (with respect to µ) whenever kf kE = kgkE for every µ-equimeasurable functions f ; g 2 E. -
Arxiv:Math/9912098V1 [Math.CA] 13 Dec 1999 Otelrnzspace Lorentz the to H Ouinkresspotdo Uvsi H Ln [0,[7) Hs O These [27])
SHARP LORENTZ SPACE ESTIMATES FOR ROUGH OPERATORS Andreas Seeger and Terence Tao Abstract. We demonstrate the (H1,L1,2) or (Lp,Lp,2) mapping properties of several rough operators. In all cases these estimates are sharp in the sense that the Lorentz exponent 2 cannot be replaced by any lower number. 1. Introduction In this paper we consider the endpoint behaviour on Hardy spaces of two classes of operators, namely singular integral operators with rough homogeneous kernels [4] and singular integral operators with con- volution kernels supported on curves in the plane ([20], [27]). These operators fall outside the Calder´on- Zygmund theory; however weak type (L1,L1,∞) or (H1,L1,∞) inequalities have been established in the previous literature ([7], [9], [16] [18], [25], [29]) We shall show that the target space L1,∞ can be improved to the Lorentz space L1,2, possibly at the cost of moving to a stronger type of Hardy space (e.g. product H1). Examples of Christ [8], [17] show that these types of results are optimal in the sense that one cannot replace L1,2 by L1,q for any q< 2. The space L1,2 arises naturally as the interpolation space halfway between L1,∞ and L1. As a gross caricature of how this space arises, suppose that we have a collection of functions fi which are uniformly bounded in L1, and whose maximal function sup f is in weak L1, and we wish to estimate the quantity i | i| γifi L1,2 Xi 2 for some l co-efficients γi. If the fi are sufficiently orthogonal, we may hope to control this quantity by the square function 2 1/2 (1.1) γifi . -
Upper Estimates for Gâteaux Differentiability of Bump Functions in Orlicz-Lorentz Spaces
Volume 8 (2007), Issue 4, Article 113, 8 pp. UPPER ESTIMATES FOR GÂTEAUX DIFFERENTIABILITY OF BUMP FUNCTIONS IN ORLICZ-LORENTZ SPACES B. ZLATANOV DEPARTMENT OF MATHEMATICS AND INFORMATICS PLOVDIV UNIVERSITY, 24 “TZAR ASSEN” STR. PLOVDIV, 4000 BULGARIA [email protected] Received 29 January, 2007; accepted 17 November, 2007 Communicated by C.P. Niculescu ABSTRACT. Upper estimates for the order of Gâteaux smoothness of bump functions in Orlicz– Lorentz spaces d(w, M, Γ), Γ uncountable, are obtained. The best possible order of Gâteaux differentiability in the class of all equivalent norms in d(w, M, Γ) is found. Key words and phrases: Orlicz function, Orlicz–Lorentz space, Smooth bump functions. 2000 Mathematics Subject Classification. 46B25, 40E30. 1. INTRODUCTION The existence of higher order Fréchet smooth norms and bump functions and its impact on the geometrical properties of a Banach space have been subject to many investigations begin- ning with the classical result for Lp–spaces in [1] and [6]. An extensive study and bibliography may be found in [2]. As any negative result on the existence of Gâteaux smooth bump func- tions immediately applies to the problem of existence of Fréchet smooth bump functions and norms, the question arises of estimating the best possible order of Gâteaux smoothness of bump functions in a given Banach space. A variational technique (the Ekeland variational principle) was applied in [2] to show that in `1(Γ), Γ uncountable, there is no continuous Gâteaux dif- ferentiable bump function. Following the same idea and using Stegall’s variational principle, an extension of this result to Banach spaces with uncountable unconditional basis was given in [4] and to Banach spaces with uncountable symmetric basis in [9].