Real-Variable Theory of Hardy Spaces Associated with Generalized Herz Spaces of Rafeiro and Samko 9789811967870, 9789811967887

Table of contents :
Preface
Abstract
Keywords and Phrases
Contents
1 Generalized Herz Spaces of Rafeiro and Samko
1.1 Matuszewska–Orlicz Indices
1.2 Generalized Herz Spaces
1.3 Convexities
1.4 Absolutely Continuous Quasi-Norms
1.5 Boundedness of Sublinear Operators
1.6 Fefferman–Stein Vector-Valued Inequalities
1.7 Dual and Associate Spaces of Local Generalized Herz Spaces
1.8 Extrapolation Theorems
2 Block Spaces and Their Applications
2.1 Block Spaces
2.2 Duality
2.3 Boundedness of Sublinear Operators
3 Boundedness and Compactness Characterizations of Commutators on Generalized Herz Spaces
3.1 Boundedness Characterizations
3.2 Compactness Characterizations
4 Generalized Herz–Hardy Spaces
4.1 Maximal Function Characterizations
4.2 Relations with Generalized Herz Spaces
4.3 Atomic Characterizations
4.4 Generalized Finite Atomic Herz–Hardy Spaces
4.5 Molecular Characterizations
4.6 Littlewood–Paley Function Characterizations
4.7 Dual Space of Hω,0p,q(Rn)
4.8 Boundedness of Calderón–Zygmund Operators
4.9 Fourier Transform
5 Localized Generalized Herz–Hardy Spaces
5.1 Maximal Function Characterizations
5.2 Relations with Generalized Herz–Hardy Spaces
5.3 Atomic Characterizations
5.4 Molecular Characterizations
5.5 Littlewood–Paley Function Characterizations
5.6 Boundedness of Pseudo-Differential Operators
6 Weak Generalized Herz–Hardy Spaces
6.1 Maximal Function Characterizations
6.2 Relations with Weak Generalized Herz Spaces
6.3 Atomic Characterizations
6.4 Molecular Characterizations
6.5 Littlewood–Paley Function Characterizations
6.6 Boundedness of Calderón–Zygmund Operators
6.7 Real Interpolations
7 Inhomogeneous Generalized Herz Spaces and Inhomogeneous Block Spaces
7.1 Inhomogeneous Generalized Herz Spaces
7.1.1 Convexities
7.1.2 Absolutely Continuous Quasi-Norms
7.1.3 Boundedness of Sublinear Operators and Fefferman–Stein Vector-Valued Inequalities
7.1.4 Dual and Associate Spaces of Inhomogeneous Local Generalized Herz Spaces
7.1.5 Extrapolation Theorems
7.2 Inhomogeneous Block Spaces and Their Applications
7.2.1 Inhomogeneous Block Spaces
7.2.2 Duality Between Inhomogeneous Block Spaces and Global Generalized Herz Spaces
7.2.3 Boundedness of Sublinear Operators
7.3 Boundedness and Compactness Characterizations of Commutators
7.3.1 Boundedness Characterizations
7.3.2 Compactness Characterizations
8 Hardy Spaces Associated with Inhomogeneous Generalized Herz Spaces
8.1 Inhomogeneous Generalized Herz–Hardy Spaces
8.1.1 Maximal Function Characterizations
8.1.2 Relations with Inhomogeneous Generalized Herz Spaces
8.1.3 Atomic Characterizations
8.1.4 Inhomogeneous Generalized Finite Atomic Herz–Hardy Spaces
8.1.5 Molecular Characterizations
8.1.6 Littlewood–Paley Function Characterizations
8.1.7 Dual Space of HKω,0p,q(Rn)
8.1.8 Boundedness of Calderón–Zygmund Operators
8.1.9 Fourier Transform
8.2 Inhomogeneous Localized Generalized Herz–Hardy Spaces
8.2.1 Maximal Function Characterizations
8.2.2 Relations with Inhomogeneous Generalized Herz–Hardy Spaces
8.2.3 Atomic Characterizations
8.2.4 Molecular Characterizations
8.2.5 Littlewood–Paley Function Characterizations
8.2.6 Boundedness of Pseudo-Differential Operators
8.3 Inhomogeneous Weak Generalized Herz–Hardy Spaces
8.3.1 Maximal Function Characterizations
8.3.2 Relations with Inhomogeneous Weak Generalized Herz Spaces
8.3.3 Atomic Characterizations
8.3.4 Molecular Characterizations
8.3.5 Littlewood–Paley Function Characterizations
8.3.6 Boundedness of Calderón–Zygmund Operators
8.3.7 Real Interpolations
Bibliography
Index

Citation preview

Lecture Notes in Mathematics  2320

Yinqin Li Dachun Yang Long Huang

Real-Variable Theory of Hardy Spaces Associated with Generalized Herz Spaces of Rafeiro and Samko

Lecture Notes in Mathematics Volume 2320

Editors-in-Chief Jean-Michel Morel, CMLA, ENS, Cachan, France Bernard Teissier, IMJ-PRG, Paris, France Series Editors Karin Baur, University of Leeds, Leeds, UK Michel Brion, UGA, Grenoble, France Annette Huber, Albert Ludwig University, Freiburg, Germany Davar Khoshnevisan, The University of Utah, Salt Lake City, UT, USA Ioannis Kontoyiannis, University of Cambridge, Cambridge, UK Angela Kunoth, University of Cologne, Cologne, Germany Ariane Mézard, IMJ-PRG, Paris, France Mark Podolskij, University of Luxembourg, Esch-sur-Alzette, Luxembourg Mark Policott, Mathematics Institute, University of Warwick, Coventry, UK Sylvia Serfaty, NYU Courant, New York, NY, USA László Székelyhidi , Institute of Mathematics, Leipzig University, Leipzig, Germany Gabriele Vezzosi, UniFI, Florence, Italy Anna Wienhard, Ruprecht Karl University, Heidelberg, Germany

This series reports on new developments in all areas of mathematics and their applications - quickly, informally and at a high level. Mathematical texts analysing new developments in modelling and numerical simulation are welcome. The type of material considered for publication includes: 1. Research monographs 2. Lectures on a new field or presentations of a new angle in a classical field 3. Summer schools and intensive courses on topics of current research. Texts which are out of print but still in demand may also be considered if they fall within these categories. The timeliness of a manuscript is sometimes more important than its form, which may be preliminary or tentative. Titles from this series are indexed by Scopus, Web of Science, Mathematical Reviews, and zbMATH.

Yinqin Li • Dachun Yang • Long Huang

Real-Variable Theory of Hardy Spaces Associated with Generalized Herz Spaces of Rafeiro and Samko

Yinqin Li Laboratory of Mathematics and Complex Systems (Ministry of Education of China) School of Mathematical Sciences Beijing Normal University Beijing, Beijing, China

Dachun Yang (Corresponding Author) Laboratory of Mathematics and Complex Systems (Ministry of Education of China) School of Mathematical Sciences Beijing Normal University Beijing, Beijing, China

Long Huang School of Mathematics and Information Science Key Laboratory of Mathematics and Interdisciplinary Sciences of the Guangdong Higher Education Institute Guangzhou University Guangzhou, Guangdong, China

This work was supported by National Key Research and Development Program of China (2020YFA0712900), National Natural Science Foundation of China (11971058) (12071197), and Basic and Applied Basic Research Foundation of Guangdong Province (2021A1515110905) ISSN 0075-8434 ISSN 1617-9692 (electronic) Lecture Notes in Mathematics ISBN 978-981-19-6787-0 ISBN 978-981-19-6788-7 (eBook) https://doi.org/10.1007/978-981-19-6788-7 Mathematics Subject Classification: 42B35, 42B30, 42B25, 42B20, 42B10, 46E30, 47B47, 47G30 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

It is well known that Herz spaces certainly play an important role in harmonic analysis and partial differential equations and have been systematically studied and developed so far; see, for instance, [79, 146, 150, 174, 275] for classical Herz spaces, [83, 239–241] for weighted Herz spaces, [132, 196, 202, 235–238, 276] for variable Herz spaces, [61, 68, 69, 93, 179, 262, 263] for Herz-type Hardy spaces, [59, 62, 64, 67, 257–259] for Herz-type Besov spaces, and [57, 60, 212, 256, 260, 261, 264] for Herz-type Triebel–Lizorkin spaces. Observe that the classical Herz space was originally introduced by Herz [104] in 1968 to study the Bernstein theorem on absolutely convergent Fourier transforms, while the research on Herz spaces can be traced back to the work of Beurling [13]. Indeed, in 1964, to study some convolution algebras, Beurling [13] first introduced a special Herz space Ap (Rn ), with p ∈ (1, ∞) [see Remark 7.1.2(iv) for its definition], which is also called the Beurling algebra. After that, great progress has been made on Herz spaces and their applications. For instance, in 1985, Baernstein and Sawyer [9] generalized these Herz spaces and gave many applications in both the embedding and the multiplier theorems for classical Hardy spaces on the n-dimensional Euclidean space; in 1984, to study the Wiener third Tauberian theorem for the n-dimensional Euclidean space, Feichtinger [79] introduced another norm of Ap (Rn ), which is obviously equivalent to the norm defined by Beurling [13]. Moreover, Herz spaces play a crucial role in the convergence and the summability problems of both Fourier transforms and Fourier series. Recall that the study of summability means was originally motivated by the famous convergence problem of Dirichlet integral operators. As one of the deepest results in harmonic analysis, in the celebrated works of both Carleson [28] and Hunt [129], they showed that Dirichlet integral operators converge almost everywhere in one-dimensional case. In recent decades, via replacing Dirichlet integral operators by some other summability means, the summability of Fourier transforms was systematically studied by Butzer and Nessel [19], Trigub and Belinsky [230], and Feichtinger and Weisz [80–82] as well as Weisz [246–253]. Particularly, Weisz [254, 255] established the connection

v

vi

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between the Herz spaces and the summability of both Fourier series and Fourier transforms. Let us give more details on the latter case. To this end, let θ ∈ L1 (Rn ) ∩ C0 (Rn ), where C0 (Rn ) denotes the set of all the continuous functions f on Rn satisfying that lim |f (x)| = 0.

|x|→∞

Recall that, for any given p ∈ [1, 2] and T := (T1 , . . . , Tn ) ∈ (0, ∞)n and for any f ∈ Lp (Rn ), the θ -mean of f is defined by setting, for any x ∈ Rn ,   ξ1 ξn  := θ − ,··· ,− f (ξ )e2πix·ξ dξ, n T T 1 n R 

σTθ f (x)

where fdenotes the Fourier transform of f , i := x · ξ :=

n 

√ −1, and

xj ξj

j =1

for any x := (x1 , · · · , xn ), ξ := (ξ1 , . . . , ξn ) ∈ Rn (see, for instance, [82, (3·1)]). Then, in the recent book [254], Weisz showed that the θ -means of some functions converge to these functions themselves at all their Lebesgue points of these functions under consideration if and only if the Fourier transform of θ belongs to a suitable Herz space. This means that those Herz spaces are the best choice in the study of the summability of Fourier transforms. Furthermore, Herz spaces also prove important in the recent article [207] of Sawano et al. Indeed, in 2017, Sawano et al. [207] introduced both the ball quasi-Banach function space X and the associated Hardy space HX (Rn ) via the grand maximal function. As was pointed out in [207], Sawano et al. used a certain inhomogeneous Herz space to overcome the essential difficulty appearing in the proof of the convergence of the atomic decomposition of HX (Rn ). Also, Herz-type spaces prove useful in the study related to partial differential equations. For instance, Scapellato [210] showed that the variable Herz spaces are the key tools in the study of the regularity of solutions to elliptic equations. Drihem [65, 66] studied semilinear parabolic equations with initial data in Herz spaces or Herz-type Triebel–Lizorkin spaces. In addition, the Fourier–Herz space has been proved to be one of the most suitable spaces to investigate the global stability for fractional Navier–Stokes equations; see, for instance, [31, 41, 159, 190]. For more progress on applications of Herz spaces, we refer the reader to [74, 112, 114, 116, 126, 180, 184, 189, 195, 274]. On the other hand, as a good substitute of Herz spaces, Herz-type Hardy spaces are also useful in many mathematical fields such as harmonic analysis and partial

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vii

differential equations, which have been systematically studied and developed so far; see, for instance, [32, 40, 63, 86, 120, 158, 279]. Recall that the study of Hardy spaces can be traced back to the works of Hardy and Littlewood via tools from complex analysis (see [97, 98, 152]). Then, based on the real-variable methods introduced by Calderón and Zygmund in the 1950s (see [25, 26, 281]), the classical real Hardy space H p (Rn ) was originally initiated by Stein and Weiss [215] and then systematically developed by Fefferman and Stein [77]. Later on, Coifman [44] established the atomic characterization of the Hardy space H p (R); Strömberg [216] showed that an orthonormal wavelet basis is always an unconditional basis of the Hardy space H 1 (Rn ), and this remarkable discovery preluded the wavelet analysis. Moreover, Calderón et al. [21–23] investigated the Cauchy integral on Lipschitz curves using weighted Hardy spaces, and Kenig [143, 144] studied the (weighted) Hardy spaces on Lipschitz domains. For more developments of Hardy spaces, we refer the reader to [105, 106, 148, 160, 186, 187] for the real-variable theory of Hardy spaces as well as [46, 71, 183, 211, 213, 243, 244] for the applications of Hardy spaces to both harmonic analysis and partial differential equations. Furthermore, as a variant of the classical real Hardy spaces, the Hardy spaces associated with the Beurling algebras on the real line were first introduced by Chen and Lau [40] in 1989, in which they studied the dual spaces and the maximal function characterizations of these Herz–Hardy spaces. Later, García-Cuerva [86] in 1989 generalized the results of Chen and Lau [40] to higher-dimensional case, and García-Cuerva and Herrero [87] in 1994 further studied the maximal function, the atomic, and the Littlewood–Paley function characterizations of these Herz–Hardy spaces. Also, since 1990, Lu and Yang made a series of studies on the Hardy spaces associated with the Beurling algebras or with the Herz spaces (see, for instance, the monograph [175] and its references). In particular, in 1992, Lu and Yang [162] first studied both the Littlewood–Paley function and the φ-transform characterizations of the Herz–Hardy space H K2 (Rn ). Moreover, in 1995, Lu and Yang [163] established the atomic and the molecular characterizations of the Herz– Hardy space with general indices and then, in 1997, they further gave various maximal function characterizations of weighted Herz–Hardy spaces in [172]. Meanwhile, Lu and Yang [164] proved that some oscillatory singular integral operators are bounded from Herz–Hardy spaces to Herz spaces and showed that their boundedness fails on Herz–Hardy spaces via a counterexample. Recall that, as was pointed out by Pan [193], the oscillatory singular integral operators may not be bounded from the classical Hardy space H 1 (Rn ) to the Lebesgue space L1 (Rn ). Thus, in some sense, the results of [164] showed that the Herz–Hardy space is a proper substitution of H 1 (Rn ) in the study on oscillatory singular integral operators. In addition, the commutators and the multiplier theorems on the Herz– Hardy spaces were investigated by Lu and Yang, respectively, in [170] and [173]; both the interpolation of generalized Herz spaces and its applications were given in Hernández and Yang [102, 103]; weighted Herz spaces and their applications were considered by Lu and Yang in [166]; embedding theorems for Herz spaces were studied in [167]. Furthermore, Hernández et al. [101] established the φ-transform and the wavelet characterizations for some Herz and Herz-type Hardy spaces by

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means of a local version of the discrete tent spaces at the origin. For more progress on the Herz–Hardy spaces on Rn and their applications, we refer the reader to [111, 119, 134, 165, 169, 171, 175, 198, 245]. Moreover, Herz-type spaces on local fields and Vilenkin groups were also investigated by Fan, Lu, and Yang, respectively, in [168] and [73, 174, 268]. Nowadays, more and more new function spaces continually spring up to meet the increasing requirements arising in harmonic analysis and partial differential equations; see, for instance, [1, 15, 65, 185, 197, 199, 270]. Particularly, extending the classical Herz spaces to some more general settings has also attracted considerable attention recently. For instance, Rafeiro and Samko [197] creatively introduced local and global generalized Herz spaces recently, which are the generalization of classical homogeneous Herz spaces and connect with generalized Morrey type spaces. To be precise, Rafeiro and Samko [197] showed that the scale of these Herz spaces include Morrey type spaces and complementary Morrey type spaces and, as applications, they also obtained the boundedness of a class of sublinear operators on these generalized Herz spaces. Note that Morrey type spaces have been studied in [2, 3, 191, 203, 204]. Moreover, as a generalization of classical variable Herz spaces introduced by Izuki [130], Rafeiro and Samko [199] further extended the generalized Herz spaces to the variable exponent setting. Furthermore, in [199], Rafeiro and Samko showed that the generalized variable exponent Herz spaces coincide with the generalized variable exponent Morrey type spaces, and also established some boundedness of sublinear operators on these spaces. Observe that Herz-type Hardy spaces have found lots of applications in many branches of mathematics and that both the real-variable theory of function spaces and their applications are always one of the central topics of harmonic analysis. Then it is a natural and meaningful topic to introduce and develop the real-variable theory of Hardy spaces associated with local and global generalized Herz spaces of Rafeiro and Samko [197], which is the subject of the present book. To achieve this, we first investigate some basic properties of these generalized Herz spaces and realize that the Hardy spaces associated with generalized Herz spaces completely fall into the framework of Hardy spaces associated with ball quasi-Banach function spaces which were first studied by Sawano et al. [207] as was mentioned above. Indeed, about the real-variable theory of Hardy spaces associated with ball quasiBanach function spaces, Sawano et al. [207] tried to answer the following important issue: Characterize the quasi-Banach space X for which the associated Hardy space HX (Rn ) can be described by various maximal functions, atomic decompositions, molecular decompositions, Littlewood–Paley functions, and so on. To be precise, let X be a ball quasi-Banach function space on Rn (see [207] or Definition 1.2.13 below for its definition). Recall that, if the powered Hardy–Littlewood maximal operators are bounded on both X and its associate space, and if X supports a Fefferman–Stein vector-valued inequality, then Sawano et al. [207] established various real-variable characterizations of the Hardy space HX (Rn ) associated with X, respectively, in terms of atoms, molecules, and the Lusin area function. After the work of Sawano et al. [207], the real-variable theory of function spaces associated with ball quasi-

Preface

ix

Banach function spaces was well developed by Ho in [108, 113] and also by others in [29, 37–39, 122, 233, 242, 266, 277, 278]. This book is devoted to exploring further properties of the local and the global generalized Herz spaces and establishing a complete real-variable theory of Hardy spaces associated with local and global generalized Herz spaces via a totally fresh perspective, which means that we view these generalized Herz spaces as special cases of ball quasi-Banach function spaces. In this perspective, the realvariable theory of Hardy spaces associated with local generalized Herz spaces can be deduced directly from the general framework of the real-variable theory of HX (Rn ) because the local generalized Herz spaces satisfy all the assumptions of the results about Hardy-type spaces associated with ball quasi-Banach function spaces. However, due to the deficiency of the associate space of the global generalized Herz space, the known real-variable characterizations about Hardy-type spaces associated with ball quasi-Banach function spaces are not applicable to Hardy spaces associated with global generalized Herz spaces [see Remark 1.2.19(vi) below for details]. Therefore, the study of the real-variable theory of Hardy spaces associated with global generalized Herz spaces is more difficult. To overcome this obstacle, via replacing the assumptions on associate spaces by some weaker assumptions about the integral representations of quasi-norms of ball quasi-Banach function spaces [see Theorem 4.3.18(ii) below], we first develop some new realvariable characterizations of Hardy-type spaces associated with ball quasi-Banach function spaces, which even improve the known results on Hardy-type spaces associated with ball quasi-Banach function spaces and can be regarded as another way to handle the aforementioned important issue proposed in [207]; moreover, they surely have additional anticipating applications. In particular, applying these improved conclusions, we further obtain a complete real-variable theory of Hardy spaces associated with global generalized Herz spaces. Precisely, in this book, we first give some basic properties of these generalized Herz spaces and obtain the boundedness and the compactness characterizations of commutators on them. Then, based on these local and global generalized Herz spaces, we introduce associated Herz–Hardy spaces, localized Herz–Hardy spaces, and weak Herz–Hardy spaces and develop a complete real-variable theory of these Herz–Hardy spaces, including various maximal function, atomic, molecular as well as various Littlewood–Paley function characterizations. As applications, we establish the boundedness of some important operators arising from harmonic analysis on these Herz–Hardy spaces. Finally, the inhomogeneous Herz–Hardy spaces and their complete real-variable theory are also investigated. We should point out that, with the aforementioned fresh perspective and the aforementioned improved conclusions on the real-variable theory of Hardy spaces associated with ball quasi-Banach function spaces, the exponents in all the obtained conclusions of this book are sharp. Moreover, all of these results in the book are new and have never been published before. To be precise, this book is organized as follows. In Chap. 1, we first recall the concepts of both the function class M(R+ ) and the Matuszewska–Orlicz indices, and the definitions of the local generalized Herz

x

Preface

space K˙ ω,0 (Rn ) as well as the global generalized Herz space K˙ ω (Rn ) introduced by Rafeiro and Samko [197], where p, q ∈ (0, ∞], ω ∈ M(R+ ), and 0 denotes the origin of Rn . In addition, under some reasonable and sharp assumptions, we show that these generalized Herz spaces are special ball quasi-Banach function spaces. Then we give some basic properties about these generalized Herz spaces, which p,q include their convexity, the absolute continuity of the quasi-norm of K˙ ω,0 (Rn ), the boundedness criterion of sublinear operators on them which was essentially obtained by Rafeiro and Samko [197, Theorem 4.3], and Fefferman–Stein vectorvalued inequalities. Furthermore, we find the dual space and the associate space p,q of K˙ ω,0 (Rn ). Finally, we establish the extrapolation theorems of local and global generalized Herz spaces. p,q In Chap. 2, we first introduce the block space B˙ω (Rn ) based on the concepts of (ω, p)-blocks. Then we investigate the properties of block spaces in two aspects. On the one hand, by establishing an equivalent characterization of block spaces via p,q local generalized Herz spaces K˙ ω,0 (Rn ), and borrowing some ideas from the proof p  ,q  of [88, Theorem 6.1], we show that the global generalized Herz space K˙ 1/ω (Rn ) is p,q just the dual space of the block space B˙ω (Rn ). This dual theorem plays an essential role in the study of the real-variable theory of Hardy spaces associated with global generalized Herz spaces in the subsequent chapters. On the other hand, we establish the boundedness of some sublinear operators on block spaces. In particular, the boundedness of powered Hardy–Littlewood maximal operators on block spaces is obtained, which also plays an important role in the subsequent chapters. The main target of Chap. 3 is to study the boundedness and the compactness characterizations of commutators on generalized Herz spaces. Recall that Tao et al. [228] established the boundedness and the compactness characterizations of commutators on ball Banach function spaces. Combining these and the aforementioned fact that the generalized Herz spaces are special ball Banach function spaces, we first show that local generalized Herz spaces satisfy all the assumptions of the results obtained in [228], and then obtain the boundedness and the compactness p,q characterizations of commutators on the local generalized Herz space K˙ ω,0 (Rn ). However, the conclusions obtained in [228] are not applicable to show the boundp,q edness and the compactness characterizations of commutators on K˙ ω (Rn ) due to the deficiency of the associate space of the global generalized Herz space p,q K˙ ω (Rn ) [see Remark 1.2.19(vi) below for details]. Notice that the most important usage of associate spaces in the proof of [228] is that, under the assumption of the boundedness of the Hardy–Littlewood maximal operator on both ball Banach function spaces and their associate spaces, Tao et al. obtained the extrapolation theorem of ball Banach function spaces. Via this extrapolation theorem and some other technical lemmas independent of associate spaces, Tao et al. [228] then established the boundedness and the compactness characterizations of commutators on ball Banach function spaces. Therefore, to overcome the difficulty caused by p,q the deficiency of associate spaces of K˙ ω (Rn ), we establish the new boundedness and the new compactness characterizations of commutators on ball Banach function spaces under the assumption that the extrapolation theorem holds true for ball p,q

p,q

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xi

Banach function spaces instead of the assumption about associate spaces, which improves the corresponding results of [228]. Finally, applying these improved boundedness and compactness characterizations of commutators on ball Banach p,q function spaces and the extrapolation theorem of K˙ ω (Rn ) obtained in Chap. 1, we then obtain the boundedness and the compactness characterizations of commutators on global generalized Herz spaces. Chapter 4 is devoted to introducing the generalized Herz–Hardy space and then establishing its complete real-variable theory. To be precise, we first introduce p,q p,q the generalized Herz–Hardy spaces, H K˙ ω,0 (Rn ) and H K˙ ω (Rn ), associated, p,q respectively, with the local generalized Herz space K˙ ω,0 (Rn ) and the global p,q generalized Herz space K˙ ω (Rn ). Then, using the known real-variable characterizations of Hardy spaces associated with ball quasi-Banach function spaces, we establish various maximal function, atomic, molecular, and Littlewood–Paley p,q function characterizations of the Herz–Hardy space H K˙ ω,0 (Rn ). Moreover, the p,q duality and the Fourier transform properties of H K˙ ω,0 (Rn ) are also obtained based on the corresponding results about Hardy spaces associated with ball quasi-Banach p,q function spaces. However, the study of H K˙ ω (Rn ) is more difficult than that p,q of H K˙ ω,0 (Rn ) due to the deficiency of associate spaces of global generalized Herz spaces. To overcome this obstacle, via replacing the assumptions of the boundedness of powered Hardy–Littlewood maximal operators on associate spaces (see Assumption 1.2.33 below) used in [207, Theorems 3.6, 3,7, and 3.9] by some weaker assumptions about the integral representations of quasi-norms of ball quasi-Banach function spaces as well as some boundedness of powered Hardy– Littlewood maximal operators [see both (ii) and (iii) of Theorem 4.3.18 below], we establish the new atomic and the new molecular characterizations of the Hardy space HX (Rn ) associated with the ball quasi-Banach function space X, which improve the corresponding results obtained by Sawano et al. [207]. Using these improved characterizations and making full use of the obtained duality between block spaces and global generalized Herz spaces in Chap. 2 as well as the construction of the quasi-norm  · K˙ p,q n , we then obtain the maximal function, the (finite) ω (R ) atomic, the molecular, the various Littlewood–Paley function characterizations of p,q H K˙ ω (Rn ) and also give some properties about Fourier transforms of distributions p,q in H K˙ ω (Rn ). Finally, as applications, via first establishing two boundedness criteria of Calderón–Zygmund operators on Hardy spaces associated with ball quasi-Banach function spaces, we obtain the boundedness of Calderón–Zygmund operators on generalized Herz–Hardy spaces. In Chap. 5, we first introduce the localized generalized Herz–Hardy space and then establish its complete real-variable theory. To achieve this, we begin with showing its various maximal function characterizations via the known maximal function characterizations of the local Hardy space hX (Rn ) associated with the ball quasi-Banach function space X. Then, via establishing the new atomic and the new molecular characterizations of hX (Rn ) as well as the boundedness of pseudodifferential operators on hX (Rn ) without using any assumptions about associate spaces (see Theorems 5.3.14, 5.4.11, and 5.6.9 below), we show the atomic and

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the molecular characterizations of localized generalized Herz–Hardy spaces and the boundedness of pseudo-differential operators on localized generalized Herz–Hardy spaces. In addition, to clarify the relation between localized generalized Herz–Hardy spaces and generalized Herz–Hardy spaces, we first establish the relation between hX (Rn ) and the Hardy space HX (Rn ) associated with the ball quasi-Banach function space X. This extends the results obtained by Goldberg [89, Lemma 4] for classical Hardy spaces and also Nakai and Sawano [186, Lemma 9.1] for variable Hardy spaces. Applying this and some auxiliary lemmas about generalized Herz spaces, we then establish the relation between localized generalized Herz–Hardy spaces and generalized Herz–Hardy spaces. As applications, we also establish various Littlewood–Paley function characterizations of hX (Rn ), which, together with the construction of the quasi-norm  · K˙ p,q n , further imply the Littlewood– ω (R ) Paley function characterizations of localized generalized Herz–Hardy spaces. The main target of Chap. 6 is to introduce weak generalized Herz–Hardy spaces and establish their complete real-variable theory. For this purpose, recall that Zhang et al. [278] and Wang et al. [242] investigated the real-variable theory of the weak Hardy space W HX (Rn ) associated with the ball quasi-Banach function space X. Via removing the assumption about associate spaces, we establish the new atomic and the new molecular characterizations of W HX (Rn ) as well as the new real interpolation between the Hardy space HX (Rn ) associated with the ball quasi-Banach function space X and the Lebesgue space L∞ (Rn ), which improve the corresponding results obtained in [278] and [242]. Then, using these improved real-variable characterizations of W HX (Rn ), we obtain various maximal function, atomic, and molecular characterizations of weak generalized Herz–Hardy spaces and also show that the real interpolation spaces between generalized Herz– Hardy spaces and the Lebesgue space L∞ (Rn ) are just the new introduced weak generalized Herz–Hardy spaces. In addition, by establishing a technique lemma about the quasi-norm  · W K˙ p,q n and the Littlewood–Paley function characω (R ) terizations of W HX (Rn ) obtained in [242], we establish various Littlewood–Paley function characterizations of weak generalized Herz–Hardy spaces. Furthermore, we establish two boundedness criteria of Calderón–Zygmund operators from the Hardy space HX (Rn ) to the weak Hardy space W HX (Rn ) and, as a consequence, we finally deduce the boundedness of Calderón–Zygmund operators from generalized Herz–Hardy spaces to weak generalized Herz–Hardy spaces even in the critical case. In Chap. 7, we first introduce the inhomogeneous generalized Herz spaces and then establish their corresponding conclusions obtained in Chaps. 1 through 3. Furthermore, in Chap. 8, based on the inhomogeneous generalized Herz spaces studied in Chap. 7, we introduce the inhomogeneous generalized Herz–Hardy spaces, the inhomogeneous localized generalized Herz–Hardy spaces, and the inhomogeneous weak generalized Herz–Hardy spaces. Then we establish their various real-variable characterizations and also give some applications, which are the corresponding inhomogeneous variants obtained, respectively, in Chaps. 4 through 6.

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xiii

Throughout this book, we always let N := {1, 2, . . .}, Z+ := N ∪ {0}, R+ := (0, ∞), and n Rn+1 + := {(x, t) : x ∈ R , t ∈ (0, ∞)}.

We also use 0 := (0, . . . , 0) to denote the origin of Rn . For any x := (x1 , . . . , xn ) ∈ Rn and θ := (θ1 , . . . , θn ) ∈ (Z+ )n =: Zn+ , let |θ | := θ1 + · · · + θn , x θ := x1θ1 · · · xnθn , and  ∂ γ :=

∂ ∂x1

γ1

 ···

∂ ∂xn

γn .

We always denote by C a positive constant which is independent of the main parameters, but it may vary from line to line. We use C(α,β,··· ) to denote a positive constant depending on the indicated parameters α, β, . . .. The notation f  g means f ≤ Cg and, if f  g  f , then we write f ∼ g. If f ≤ Cg and g = h or g ≤ h, we then write f  g ∼ h or f  g  h, rather than f  g = h or f  g ≤ h. For any s ∈ R, the symbol s denotes the smallest integer not less than s, and the symbol s denotes the largest integer not greater than s. For any set E ⊂ Rn , we denote the set Rn \ E by E  , its characteristic function by 1E , and its n-dimensional Lebesgue measure by |E|. For any q ∈ [1, ∞], we denote by q  its conjugate exponent, that is, 1/q + 1/q  = 1. In addition, we use Sn−1 := {x ∈ Rn : |x| = 1} to denote the unit sphere in Rn and dσ the area measure on Sn−1 . Furthermore, we always use the symbol M (Rn ) to denote the set of all measurable functions on Rn . The symbol Q denotes the set of all cubes with edges parallel to the coordinate axes. Finally, for any cube Q ∈ Q, rQ means a cube with the same center as Q and r times the edge length of Q. This research of both Yinqin Li and Dachun Yang is supported by the National Key Research and Development Program of China (Grant No. 2020YFA0712900) and the National Natural Science Foundation of China (Grant Nos. 11971058 and 12071197). Long Huang is supported by Guangdong Basic and Applied Basic Research Foundation (Grant No. 2021A1515110905). Yinqin Li would like to express his deep gratitude to Dr. Hongchao Jia, Dr. Yangyang Zhang, and Dr. Yirui Zhao for some helpful discussions on the subject of this book. We would also like to thank the four referees of this book for their careful reading and for their many enlightening and useful comments which have definitely improved the presentation and the readability of the book. In particular, we are very grateful to one referee of the aforementioned four referees who gave us both a clear history on the development of the real-variable theory of Hardy spaces (which

xiv

Preface

is now included in this preface) and also several deep insights on Herz spaces and their predual spaces (namely block spaces). In essence, this referee motivated us to improve the related chapters and encouraged us to illustrate both the main definitions and theorems with interesting examples that made the book more readily comprehensible. The improved contents that benefitted from this referee include: Proposition 1.2.3, Corollary 1.2.4, Remark 1.2.5, Theorems 1.2.6 and 1.2.9, Propositions 1.2.10, 1.2.12, 1.2.20, 1.4.6, 1.4.7, and 1.4.8, Theorem 2.1.5, Corollary 2.1.7, Theorems 2.2.4 and 2.2.5, Example 2.2.6, Theorem 2.2.8, Proposition 4.2.4, and Theorems 7.2.3, 7.2.7, 7.2.8, and 7.2.9. Beijing, The People’s Republic of China Beijing, The People’s Republic of China Guangzhou, The People’s Republic of China August 2022

Yinqin Li Dachun Yang Long Huang

Abstract

This book is devoted to exploring properties of generalized Herz spaces and establishing a complete real-variable theory of Hardy spaces associated with local and global generalized Herz spaces via a totally fresh perspective which means that the authors view these generalized Herz spaces as special cases of ball quasi-Banach function spaces. To be precise, in this book, the authors first study some basic properties of generalized Herz spaces and obtain boundedness and compactness characterizations of commutators on them. Then the authors introduce the associated Herz–Hardy spaces, localized Herz–Hardy spaces, and weak Herz–Hardy spaces, and develop a complete real-variable theory of these Herz–Hardy spaces, including their various maximal function, atomic, molecular as well as various Littlewood–Paley function characterizations. As applications, the authors establish the boundedness of some important operators arising from harmonic analysis on these Herz–Hardy spaces. Finally, the inhomogeneous Herz–Hardy spaces and their complete real-variable theory are also investigated. Due to the deficiency of the associate space of the global Herz space, the known real-variable characterizations about Hardy-type spaces associated with ball quasi-Banach function spaces are not applicable to Hardy spaces associated with global generalized Herz spaces which need an improved generalization of the existing one, done by the authors also in this book and having more additional anticipating applications. The authors should also point out that, with the fresh perspective and the improved conclusions on the realvariable theory of Hardy spaces associated with ball quasi-Banach function spaces, the exponents in all the obtained results of this book are sharp. Moreover, all of these results in this book are new and have never been published before.

Keywords and Phrases Generalized Herz space, Ball quasi-Banach function space, Block space, Hardy space, Localized Hardy space, Weak Hardy space, Atom, Molecule, Duality,

xv

xvi

Abstract

Maximal function, Littlewood–Paley function, Hardy–Littlewood maximal operator, Fefferman–Stein vector-valued inequality, Fourier transform, Interpolation, Calderón–Zygmund operator, Commutator, Pseudo-differential operator.

Contents

1

Generalized Herz Spaces of Rafeiro and Samko . . . . .. . . . . . . . . . . . . . . . . . . . 1 1.1 Matuszewska–Orlicz Indices . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2 1.2 Generalized Herz Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 12 1.3 Convexities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 56 1.4 Absolutely Continuous Quasi-Norms .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 58 1.5 Boundedness of Sublinear Operators . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 73 1.6 Fefferman–Stein Vector-Valued Inequalities . . . . . .. . . . . . . . . . . . . . . . . . . . 84 1.7 Dual and Associate Spaces of Local Generalized Herz Spaces . . . . . . . 87 1.8 Extrapolation Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 100

2 Block Spaces and Their Applications . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1 Block Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2 Duality .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3 Boundedness of Sublinear Operators . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

109 109 117 135

3 Boundedness and Compactness Characterizations of Commutators on Generalized Herz Spaces . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 147 3.1 Boundedness Characterizations . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 149 3.2 Compactness Characterizations . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 158 4 Generalized Herz–Hardy Spaces . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1 Maximal Function Characterizations.. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2 Relations with Generalized Herz Spaces. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3 Atomic Characterizations .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.4 Generalized Finite Atomic Herz–Hardy Spaces . . .. . . . . . . . . . . . . . . . . . . . 4.5 Molecular Characterizations .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.6 Littlewood–Paley Function Characterizations . . . . .. . . . . . . . . . . . . . . . . . . . p,q 4.7 Dual Space of H K˙ ω,0 (Rn ) . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.8 Boundedness of Calderón–Zygmund Operators .. .. . . . . . . . . . . . . . . . . . . . 4.9 Fourier Transform .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

169 171 181 189 216 224 244 252 260 290

xvii

xviii

Contents

5 Localized Generalized Herz–Hardy Spaces . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.1 Maximal Function Characterizations.. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2 Relations with Generalized Herz–Hardy Spaces . .. . . . . . . . . . . . . . . . . . . . 5.3 Atomic Characterizations .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.4 Molecular Characterizations .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.5 Littlewood–Paley Function Characterizations . . . . .. . . . . . . . . . . . . . . . . . . . 5.6 Boundedness of Pseudo-Differential Operators . . .. . . . . . . . . . . . . . . . . . . .

301 303 314 323 338 354 380

6 Weak Generalized Herz–Hardy Spaces . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.1 Maximal Function Characterizations.. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.2 Relations with Weak Generalized Herz Spaces. . . .. . . . . . . . . . . . . . . . . . . . 6.3 Atomic Characterizations .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.4 Molecular Characterizations .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.5 Littlewood–Paley Function Characterizations . . . . .. . . . . . . . . . . . . . . . . . . . 6.6 Boundedness of Calderón–Zygmund Operators .. .. . . . . . . . . . . . . . . . . . . . 6.7 Real Interpolations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

399 402 412 416 438 450 459 490

7 Inhomogeneous Generalized Herz Spaces and Inhomogeneous Block Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.1 Inhomogeneous Generalized Herz Spaces . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.1.1 Convexities .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.1.2 Absolutely Continuous Quasi-Norms.. . . . .. . . . . . . . . . . . . . . . . . . . 7.1.3 Boundedness of Sublinear Operators and Fefferman–Stein Vector-Valued Inequalities . . . . . . . . . . . . . . . . . . 7.1.4 Dual and Associate Spaces of Inhomogeneous Local Generalized Herz Spaces . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.1.5 Extrapolation Theorems . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.2 Inhomogeneous Block Spaces and Their Applications.. . . . . . . . . . . . . . . 7.2.1 Inhomogeneous Block Spaces . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.2.2 Duality Between Inhomogeneous Block Spaces and Global Generalized Herz Spaces . . . . . .. . . . . . . . . . . . . . . . . . . . 7.2.3 Boundedness of Sublinear Operators . . . . . .. . . . . . . . . . . . . . . . . . . . 7.3 Boundedness and Compactness Characterizations of Commutators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.3.1 Boundedness Characterizations . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.3.2 Compactness Characterizations . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8 Hardy Spaces Associated with Inhomogeneous Generalized Herz Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.1 Inhomogeneous Generalized Herz–Hardy Spaces.. . . . . . . . . . . . . . . . . . . . 8.1.1 Maximal Function Characterizations . . . . . .. . . . . . . . . . . . . . . . . . . . 8.1.2 Relations with Inhomogeneous Generalized Herz Spaces .. . . 8.1.3 Atomic Characterizations . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.1.4 Inhomogeneous Generalized Finite Atomic Herz–Hardy Spaces . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.1.5 Molecular Characterizations . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

501 501 512 514 518 526 529 532 533 536 541 544 544 548 553 553 554 558 559 567 570

Contents

8.1.6 Littlewood–Paley Function Characterizations .. . . . . . . . . . . . . . . . p,q 8.1.7 Dual Space of H Kω,0 (Rn ) . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.1.8 Boundedness of Calderón–Zygmund Operators . . . . . . . . . . . . . . 8.1.9 Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.2 Inhomogeneous Localized Generalized Herz–Hardy Spaces . . . . . . . . . 8.2.1 Maximal Function Characterizations . . . . . .. . . . . . . . . . . . . . . . . . . . 8.2.2 Relations with Inhomogeneous Generalized Herz–Hardy Spaces . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.2.3 Atomic Characterizations . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.2.4 Molecular Characterizations . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.2.5 Littlewood–Paley Function Characterizations .. . . . . . . . . . . . . . . . 8.2.6 Boundedness of Pseudo-Differential Operators . . . . . . . . . . . . . . . 8.3 Inhomogeneous Weak Generalized Herz–Hardy Spaces .. . . . . . . . . . . . . 8.3.1 Maximal Function Characterizations . . . . . .. . . . . . . . . . . . . . . . . . . . 8.3.2 Relations with Inhomogeneous Weak Generalized Herz Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.3.3 Atomic Characterizations . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.3.4 Molecular Characterizations . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.3.5 Littlewood–Paley Function Characterizations .. . . . . . . . . . . . . . . . 8.3.6 Boundedness of Calderón–Zygmund Operators . . . . . . . . . . . . . . 8.3.7 Real Interpolations .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

xix

573 575 578 580 586 587 591 592 595 599 601 602 603 608 610 612 616 619 628

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 631 Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 643

Chapter 1

Generalized Herz Spaces of Rafeiro and Samko

In this chapter, we first recall the concepts of both the function class M(R+ ) and the Matuszewska–Orlicz indices, and the definitions of the local generalized Herz space p,q K˙ ω,0 (Rn )

as well as the global generalized Herz space K˙ ωp,q (Rn ) creatively introduced by Rafeiro and Samko [197], where p, q ∈ (0, ∞] and ω ∈ M(R+ ). Next, we recall some basic concepts about ball quasi-Banach function spaces introduced by Sawano et al. [207]. Moreover, under some sharp assumptions, we show that these generalized Herz spaces are special ball quasiBanach function spaces. Then we establish some basic properties about these generalized Herz spaces, which include their convexity, the absolutely continuity p,q of the quasi-norm of K˙ ω,0 (Rn ), the boundedness criterion of sublinear operators on them which was essentially obtained by Rafeiro and Samko [197, Theorem 4.3], and Fefferman–Stein vector-valued inequalities. We should point out that the global p,q generalized Herz space K˙ ω (Rn ) may not have the absolutely continuous quasip,q norm. Indeed, using the quasi-norm of K˙ ω (R) and borrowing some ideas from [209, Example 5.1], we construct a special set E and show that its characteristic function 1E belongs to certain global generalized Herz space. But, 1E does not have an absolutely continuous quasi-norm in this global generalized Herz space. Finally, p,q by introducing the local generalized Herz space K˙ ω,ξ (Rn ) for any given ξ ∈ Rn and establishing a dual result of it when p ∈ [1, ∞) and q ∈ (0, ∞), we find the p,q associate space of K˙ ω,0 (Rn ) under some reasonable and sharp assumptions. As an

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 Y. Li et al., Real-Variable Theory of Hardy Spaces Associated with Generalized Herz Spaces of Rafeiro and Samko, Lecture Notes in Mathematics 2320, https://doi.org/10.1007/978-981-19-6788-7_1

1

2

1 Generalized Herz Spaces of Rafeiro and Samko

application, we establish the extrapolation theorem of both K˙ ω,0 (Rn ) and K˙ ω (Rn ) at the end of this chapter. p,q

p,q

1.1 Matuszewska–Orlicz Indices In this section, we first recall the concept of a function class M(R+ ) given in [197] and the concept of Matuszewska–Orlicz indices originally introduced by Matuszewska and Orlicz in [177, 178]. Then we present some fundamental properties related to M(R+ ) and Matuszewska–Orlicz indices, which are widely used throughout this book. To begin with, let ω be a nonnegative function on R+ . Then the function ω is said to be almost increasing (resp., almost decreasing) on R+ if there exists a constant C ∈ [1, ∞) such that, for any t, τ ∈ (0, ∞) satisfying t ≤ τ (resp., t ≥ τ ), ω(t) ≤ Cω(τ ) (see, for instance, [145, p. 30]). Now, we recall the concept of the function class M(R+ ) given in [197, Definition 2.1] as follows. Definition 1.1.1 The function class M(R+ ) is defined to be the set of all the positive functions ω on R+ such that, for any 0 < δ < N < ∞, 0


ln(e + t) − [ln(e

t e+t + t)]2

=

(e + t) ln(e + t) − t (e + t)[ln(e + t)]2

t ln e − t = 0, (e + t)[ln(e + t)]2

which implies that ω(t) is increasing when t ∈ (0, ∞). On the other hand, it is easy to show that ω(t)t −1 is decreasing when t ∈ (0, ∞). Therefore, we conclude that ω ∈ M(R+ ). Moreover, for any t ∈ (0, ∞), we have lim

h→0+

ω(ht) = lim ω(h) h→0+

ht ln(e+ht ) h ln(e+h)

= t lim

h→0+

ln(e + h) = t. ln(e + ht)

From this and Definition 1.1.4, we deduce that ω(ht ) ω(h) )

ln( lim m0 (ω) = sup

t ∈(0,1)

h→0+

ln t

ln t =1 t ∈(0,1) ln t

= sup

and ln( lim M0 (ω) = inf

t ∈(0,1)

h→0+

ln t

ω(ht ) ω(h) )

ln t = 1. t ∈(0,1) ln t

= inf

1.1 Matuszewska–Orlicz Indices

9

In addition, for any t ∈ (0, ∞), we have ht ln(e+ht ) h ln(e+h)

ω(ht) = lim h→∞ ω(h) h→∞ lim

= t lim

1+

h→∞

1+

ln(e + h) h→∞ ln(e + ht)

= t lim

ln( he +1) ln h ln( he +t ) ln h

= t.

This, together with Definition 1.1.4, further implies that ln( lim h→∞

m∞ (ω) = sup

ω(ht ) ω(h) )

t ∈(1,∞)

ln t =1 t ∈(1,∞) ln t

= sup

ln t

and ω(ht ) ) h→∞ ω(h)

ln( lim M∞ (ω) =

inf

t ∈(1,∞)

ln t

=

ln t = 1. t ∈(1,∞) ln t inf

Example 1.1.9 For any given α1 , α2 ∈ R and for any t ∈ (0, ∞), let  ω(t) :=

t α1 (1 − ln t) when t ∈ (0, 1], t α2 (1 + ln t) when t ∈ (1, ∞).

Then, we find that, for any t ∈ (0, 1),

ω(t)t −α1 +1



= (t (1 − ln t)) = − ln t > 0

and, for any t ∈ (1, ∞),

ω(t)t −α2 −1



 =

1 + ln t t



=−

ln t < 0. t2

This further implies that ω(t)t −α1 +1 is increasing when t ∈ (0, 1], and ω(t)t −α2 −1 is decreasing when t ∈ [1, ∞). On the other hand, we can easily know that ω(t)t −α1 is decreasing when t ∈ (0, 1], and ω(t)t −α2 is increasing when t ∈ [1, ∞). Thus, the positive function ω belongs to M(R+ ). In addition, it holds true that, for any h, t ∈ (0, 1), lim

h→0+

  ω(ht) (ht)α1 [1 − ln(ht)] ln t α1 = lim = t lim 1 − = t α1 . ω(h) hα1 (1 − ln h) 1 − ln h h→0+ h→0+

10

1 Generalized Herz Spaces of Rafeiro and Samko

Combining this and Definition 1.1.4, we conclude that ω(ht ) ω(h) )

ln( lim

h→0+

m0 (ω) = sup

ln t

t ∈(0,1)

ln(t α1 ) = α1 t ∈(0,1) ln t

= sup

and ω(ht ) ω(h) )

ln( lim

h→0+

M0 (ω) = inf

t ∈(0,1)

ln t

ln(t α1 ) = α1 . t ∈(0,1) ln t

= inf

Similarly, from Definition 1.1.4, we deduce that m∞ (ω) = M∞ (ω) = α2 . The following equivalent formulae of Matuszewska–Orlicz indices are just [176, Theorem 11.13]. Lemma 1.1.10 Let ω ∈ M(R+ ), and m0 (ω), M0 (ω), m∞ (ω), and M∞ (ω) denote its Matuszewska–Orlicz indices. Then   m0 (ω) = sup α0 ∈ R : ω(t)t −α0 is almost increasing for any t ∈ (0, 1] ,   (1.9) M0 (ω) = inf β0 ∈ R : ω(t)t −β0 is almost decreasing for any t ∈ (0, 1] ,   m∞ (ω) = sup α∞ ∈ R : ω(t)t −α∞ is almost increasing for any t ∈ [1, ∞) , and   M∞ (ω) = inf β∞ ∈ R : ω(t)t −β∞ is almost decreasing for any t ∈ [1, ∞) . Remark 1.1.11 We should point out that the suprema and the infima in Lemma 1.1.10 may not be achieved. Indeed, for any t ∈ (0, ∞), let  ω(t) :=

t (1 − ln t) when t ∈ (0, 1], t (1 + ln t) when t ∈ (1, ∞).

Then, by Example 1.1.9, we conclude that m0 (ω) = 1. However, we have ω( n1 )( n1 )−1 ω( 12 )( 12 )−1

=

1 + ln n →∞ 1 + ln 2

1.1 Matuszewska–Orlicz Indices

11

as n → ∞. This implies that ω(t)t −m0 (ω) is not almost increasing when t ∈ (0, 1]. Thus, in this case, the supremum in (1.9) of the Matuszewska–Orlicz index m0 can not be achieved. Applying Lemma 1.1.10, we immediately obtain the following estimates of the positive function ω ∈ M(R+ ), which were also stated in [197, (25) and (26)]; we omit the details. Lemma 1.1.12 Let ω ∈ M(R+ ), and m0 (ω), M0 (ω), m∞ (ω), and M∞ (ω) denote its Matuszewska–Orlicz indices. Then, for any given ε ∈ (0, ∞), there exists a constant C(ε) ∈ [1, ∞), depending on ε, such that, for any t ∈ (0, 1], −1 M0 (ω)+ε C(ε) t ≤ inf

τ ∈(0,1]

ω(tτ ) ω(tτ ) ≤ sup ≤ C(ε) t m0 (ω)−ε ω(τ ) τ ∈(0,1] ω(τ )

and, for any t ∈ [1, ∞), −1 m∞ (ω)−ε C(ε) t ≤

inf

τ ∈[1,∞)

ω(tτ ) ω(tτ ) ≤ sup ≤ C(ε) t M∞ (ω)+ε . ω(τ ) τ ∈[1,∞) ω(τ )

Remark 1.1.13 We should point out that, in Lemma 1.1.12, the constant C(ε) may not be uniform about ε. Indeed, for any t ∈ (0, ∞), let  ω(t) :=

t (1 − ln t) when t ∈ (0, 1], t (1 + ln t) when t ∈ (1, ∞).

Then, from Example 1.1.9, it follows that m0 (ω) = 1. By this, we conclude that 

 ω(tτ ) m (ω)−ε t, τ ∈(0,1] ω(τ )t 0     ε 1 − ln τ − ln t = sup t ε (1 − ln t) t = sup 1 − ln τ t, τ ∈(0,1] t ∈(0,1] sup

=

eε−1 →∞ ε

as ε → 0+ , which further implies that C(ε) → ∞ as ε → 0+ , where C(ε) is as in Lemma 1.1.12. Here and thereafter, ε → 0+ means ε ∈ (0, ∞) and ε → 0.

12

1 Generalized Herz Spaces of Rafeiro and Samko

1.2 Generalized Herz Spaces The targets of this section are threefold. The first one is to recall the concept of generalized Herz spaces of Rafeiro and Samko given in [197, Definition 2.2], as well as investigate some basic properties of them, which include the mapping property of the geometrical transformation called inversion [see (1.15) below for its definition] and the embedding property or the triviality of some special global generalized Herz spaces. The second one is to recall some concepts related to ball quasi-Banach function spaces. The last one is to show that generalized Herz spaces of Rafeiro and Samko are ball Banach function spaces or ball quasi-Banach function spaces under some reasonable and sharp assumptions. First, we present the concept of generalized Herz spaces which were originally introduced by Rafeiro and Samko in [197, Definition 2.2] under the assumptions p, q ∈ (0, ∞). In what follows, for any x ∈ Rn and r ∈ (0, ∞), let   B(x, r) := y ∈ Rn : |x − y| < r and   B := B(x, r) : x ∈ Rn and r ∈ (0, ∞) .

(1.10)

Moreover, for any ball B := B(x, r) with x ∈ Rn and r ∈ (0, ∞), r is called the p radius of B, which is denoted by r(B). Let p ∈ (0, ∞). We always use L loc (Rn ) p and L loc (Rn \ {0}) to denote the set of all p-order locally integrable functions, n ∞ n respectively, on Rn and Rn \ {0}, and use L∞ loc (R ) and L loc (R \ {0}) to denote the set of all locally essential bounded functions, respectively, on Rn and Rn \ {0}, Definition 1.2.1 Let p, q ∈ (0, ∞] and ω ∈ M(R+ ). p,q (i) The local generalized Herz space K˙ ω,0 (Rn ) is defined to be the set of all the p f ∈ L loc (Rn \ {0}) such that

f K˙ p,q (Rn ) :=

 

ω,0

q  q ω(2k ) f 1B(0,2k )\B(0,2k−1) Lp (Rn )

1 q

(1.11)

k∈Z

is finite. p,q (ii) The global generalized Herz space K˙ ω (Rn ) is defined to be the set of all the p n f ∈ L loc (R ) such that f K˙ p,q n := sup ω (R )

ξ ∈Rn

is finite.

  k∈Z

q  q ω(2 ) f 1B(ξ,2k )\B(ξ,2k−1) Lp (Rn ) k

1 q

(1.12)

1.2 Generalized Herz Spaces

13

Remark 1.2.2 (i) Recall that, in [197, Definition 2.2], Rafeiro and Samko introduced both p,q local generalized Herz space K˙ ω,0 (Rn ) and global generalized Herz space p,q K˙ ω (Rn ) with p, q ∈ (0, ∞) and ω ∈ M(R+ ). In this book, we extend their definitions of generalized Herz spaces to p = ∞ or q = ∞. Moreover, in [197], Rafeiro and Samko also introduced the following continuous versions of both (1.11) and (1.12), respectively, by setting, for any measurable function f on Rn ,  f H p,q (Rn ) := ω,0

q



∞

[ω(t)] 0

|f (y)| dy

q

p

p

t

2 − αp

, k0 +k0

− 2 k0 − 1 + k0 αp ,

] for some k0 ∈ N. In this case, from (1.73) it follows that

2 − αp

2 − αp

− p/s These, together with the assumptions p/s, q/s ∈ (1, ∞), and Theorem 1.2.46, p/s,q/s imply that the local generalized Herz space K˙ ωs ,0 (Rn ) is a BBF space. Combinp,q ing this and Lemma 1.3.1, we further conclude that the Herz space [K˙ ω,0 (Rn )]1/s is a BBF space.

104

1 Generalized Herz Spaces of Rafeiro and Samko

Next, we show (1.134). Indeed, from Lemma 1.3.1, (1.135), and Theorem 1.7.9, we deduce that



 p,q p/s,q/s (p/s) ,(q/s) [K˙ ω,0 (Rn )]1/s = K˙ ωs ,0 (Rn ) = K˙ 1/ωs ,0 (Rn ). (1.136) n In addition, by Lemma 1.1.6 and the assumption s ∈ (0, max{M0 (ω),M ), we ∞ (ω)}+n/p conclude that

  min m0 (1/ωs ), m∞ (1/ωs )



= −s max{M0 (ω), M∞ (ω)} > −s

n n − s p

 =−

n . (p/s)

(1.137)

On the other hand, from Lemma 1.1.6 again and the assumption  r∈

 n ,∞ , min{m0 (ω), m∞ (ω)} + n/p

it follows that   max M0 (1/ωs ), M∞ (1/ωs )



= −s min{m0 (ω), m∞ (ω)} < −s

n n − r p

 =

n n − , (r/s) (p/s)

which, combined with r ∈ (p, ∞), further implies that r  s


0. m∞ ω These, together with (2.38) and Theorems 1.2.6 and 1.2.7, further imply that f B˙ωp,q (Rn ) ∼ h ˙ p ,q 

K1/ω (Rn )

= 0,

which then completes the proof of the necessity and hence Theorem 2.2.5.

 

Moreover, the following special example shows that both Corollary 2.1.7 and Theorem 2.2.5 are sharp. Example 2.2.6 Let p ∈ [1, ∞), q ∈ (0, ∞), α1 , α2 , β1 , β2 ∈ R, and  ωα1 ,α2 ,β1 ,β2 (t) :=

t α1 (1 − ln t)β1 when t ∈ (0, 1], t α2 (1 + ln t)β2 when t ∈ (1, ∞).

Then ωα1 ,α2 ,β1 ,β2 ∈ M(R+ ) and . . m0 ωα1 ,α2 ,β1 ,β2 = M0 ωα1 ,α2 ,β1 ,β2 = α1

(2.39)

. . m∞ ωα1 ,α2 ,β1 ,β2 = M∞ ωα1 ,α2 ,β1 ,β2 = α2 .

(2.40)

and

132

2 Block Spaces and Their Applications

Moreover, (i) if α1 ∈ [ pn , ∞), α2 ∈ (−∞, 0], and β2 ∈ (−∞, 0), then, for any ball B ∈ B, 1B B˙ωp,q

α1 ,α2 ,β1 ,β2

(Rn )

= 0;

p,q (ii) if α1 ∈ (−∞, pn ], α2 ∈ [0, ∞), and β1 , β2 ∈ ( q1 , ∞), then f ∈ B˙ω (Rn ) if n and only if f ∈ M (R ) and

f B˙ωp,q

α1 ,α2 ,β1 ,β2

Here

1 p

+

1 p

(Rn )

= 0.

= 1 and



q :=

⎧ ⎨ ⎩

q when q ∈ [1, ∞), q −1 ∞

when q ∈ (0, 1).

Proof Let all the symbols be the same as in the present example. Similarly to Example 1.1.9, we have both (2.39) and (2.40). We now prove (i). To this end, let B(x0 , r) ∈ B with x0 ∈ Rn and r ∈ (0, ∞) and, in the remainder of this proof, for any x := (x1 , . . . , xn ) ∈ Rn , let   |x|∞ := max |xj | : j = 1, . . . , n . Then, for any j ∈ N ∩ [2 +

ln r ln 2 , ∞),

we choose a ξj ∈ Rn satisfying that

  ξj − x0  ∈ [2j −1 + r, 2j − r]. ∞ For any given j ∈ N ∩ [2 +

ln r ln 2 , ∞)

  y − ξj 

∞

and for any y ∈ B(x0 , r), we have

  ≤ |y − x0 |∞ + x0 − ξj ∞   ≤ |y − x0 | + x0 − ξj ∞ < 2j

and   y − ξj 

∞

  ≥ x0 − ξj ∞ − |y − x0 |∞   ≥ x0 − ξj ∞ − |y − x0 | > 2j −1 ,

2.2 Duality

133

which imply that y ∈ Q(ξj , 2j +1 ) \ Q(ξj , 2j ). By this and the arbitrariness of y, ln r we find that, for any j ∈ N ∩ [2 + ln 2 , ∞), B(x0 , r) ⊂ Q(ξj , 2j +1 ) \ Q(ξj , 2j ). In addition, for any j ∈ N ∩ [2 +

ln r ln 2 , ∞)

(2.41)

(ξ )

and Q ∈ Qj j , let

 −1 1 |Q ∩ B(x0 , r)|− p 1Q∩B(x0,r) . bj,Q := ωα1 ,α2 ,β1 ,β2 (2j −1 ) Then, from Definition 2.1.1, we deduce that, for any j ∈ N ∩ [2 + (ξ ) Qj j ,

ln r ln 2 , ∞)

and

Q ∈ bj,Q is an (ωα1 ,α2 ,β1 ,β2 , p)-block supported in the cube Q. Applying ln r (2.41), we have, for any j ∈ N ∩ [2 + ln 2 , ∞)  

1B(x0 ,r) =

 1 ωα1 ,α2 ,β1 ,β2 (2j −1 ) |Q ∩ B(x0 , r)| p bj,Q .

(ξj )

Q∈Qj

Combining this, the fact that {bj,Q }

(ξj )

Q∈Qj

is a sequence of (ωα1 ,α2 ,β1 ,β2 , p)-

blocks, Definition 2.1.3, Remark 2.1.2, and the assumptions α2 ∈ (−∞, 0] and ln r β2 ∈ (−∞, 0), we conclude that, for any j ∈ N ∩ [2 + ln 2 , ∞),   1B(x ,r)  ˙ p,q 0 B

ωα ,α ,β ,β (R 1 2 1 2

≤

⎧ ⎪ ⎪ ⎨   ⎪ ⎪ ⎩Q∈Q(ξj )

n)

ωα1 ,α2 ,β1 ,β2 (2j −1 )

q

q

|Q ∩ B(x0 , r)| p

j

=

⎧ ⎪ ⎪ ⎨  ⎪ ⎪ ⎩Q∈Q(ξj )

⎫1 q ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

q

2(j −1)α2 q [1 + (j − 1) ln 2]β2 q |Q ∩ B(x0 , r)| p

j

1 1  (ξ ) q ≤ 2(j −1)α2 [1 + (j − 1) ln 2]β2 |B(x0 , r)| p Qj j ∼ 2j α2 (1 + j ln 2)β2 → 0 as j → ∞. Therefore, we have   1B(x ,r)  ˙ p,q 0 B

ωα ,α ,β ,β (R 1 2 1 2

which then completes the proof of (i).

n)

= 0,

⎫1 q ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

134

2 Block Spaces and Their Applications

Next, we show (ii). Indeed, using the assumptions α1 ∈ (−∞, pn ], α2 ∈ [0, ∞), and β1 , β2 ∈ ( q1 , ∞), and repeating an argument similar to that used in the proof of Theorem 2.1.5 with ω therein replaced by ωα1 ,α2 ,β1 ,β2 , we find that, for any f ∈ p,q B˙ωα1 ,α2 ,β1 ,β2 (Rn ), f L1 (Rn )+Lp (Rn )  f B˙ωp,q

α1 ,α2 ,β1 ,β2

(Rn ) .

This, together with Definition 2.1.3, further implies that, for any f ∈ M (Rn ), n f B˙ωp,q (Rn ) = 0 if and only if f = 0 almost everywhere in R . This then α1 ,α2 ,β1 ,β2

 

finishes the proof of (ii) and hence Example 2.2.6. Remark 2.2.7 Let p ∈ [1, ∞) and q ∈ (0, ∞).

(i) Example 2.2.6(i) shows that all the assumptions on the indices of ω in Corollary 2.1.7 are sharp. Namely, if ω does not satisfy all the assumptions of Corollary 2.1.7, then  · B˙ωp,q (Rn ) may not be a norm. Indeed, let ωα1 ,α2 ,β1 ,β2 be the same as in Example 2.2.6(i). Then, applying both (2.39) and (2.40), we find that, in this case, ωα1 ,α2 ,β1 ,β2 ∈ M(R+ ), m0 (ωα1 ,α2 ,β1 ,β2 ) ∈ [ pn , ∞), and M∞ (ωα1 ,α2 ,β1 ,β2 ) ∈ (−∞, 0]. These imply that, in this case, ωα1 ,α2 ,β1 ,β2 fails the assumptions of Corollary 2.1.7 and  · B˙ωp,q (Rn ) is not a norm because  · B˙ωp,q f B˙ωp,q

α1 ,α2 ,β1 ,β2

α1 ,α2 ,β1 ,β2

α1 ,α2 ,β1 ,β2

(Rn )

(Rn )

does not satisfy the positive definiteness [namely,

= 0 can not imply f = 0 almost everywhere in Rn ]. Thus,

in this sense, all the assumptions on the indices of ω in Corollary 2.1.7 are sharp. (ii) Example 2.2.6(ii) shows that all the assumptions on the indices of ω in Theorem 2.2.5 are sharp. Precisely, if ω ∈ M(R+ ) fails the assumptions of Theorem 2.2.5, then  · B˙ωp,q (Rn ) may be a true norm. Indeed, let ωα1 ,α2 ,β1 ,β2 be the same as in Example 2.2.6(ii). Then, from both (2.39) and (2.40), we infer that, in this case, ωα1 ,α2 ,β1 ,β2 ∈ M(R+ ), m0 (ωα1 ,α2 ,β1 ,β2 ) ∈ (−∞, pn ], and M∞ (ωα1 ,α2 ,β1 ,β2 ) ∈ [0, ∞). Therefore, in this case, ωα1 ,α2 ,β1 ,β2 does not satisfy the assumptions of Theorem 2.2.5, but ·B˙ωp,q (Rn ) is a true norm. α1 ,α2 ,β1 ,β2

These further imply that, in this sense, all the assumptions on the indices of ω in Theorem 2.2.5 are sharp. In addition, the following conclusion implies that, if p ∈ (0, 1), then the seminorm  · B˙ωp,q (Rn ) is always trivial. Theorem 2.2.8 Let p ∈ (0, 1), q ∈ (0, ∞), and ω ∈ M(R+ ). Then f ∈ B˙ω (Rn ) if and only if f ∈ M (Rn ) and f B˙ωp,q (Rn ) = 0. p,q

Proof Let all the symbols be the same as in the present theorem. We first prove the sufficiency. Indeed, let f ∈ M (Rn ) be such that f B˙ωp,q (Rn ) = 0. Then, by p,q Definition 2.1.3, we have f ∈ B˙ω (Rn ). This finishes the proof of the sufficiency.

2.3 Boundedness of Sublinear Operators

135

p,q Conversely, we deal with the necessity. For this purpose, let f ∈ B˙ω (Rn ). Then, repeating an argument similar to that used in the proof of Theorem 2.2.5 with Theorems 1.2.6 and 1.2.7 used therein replaced by Theorem 2.2.4, we find that there p,q exists a φ ∈ (B˙ω (Rn ))∗ such that

f B˙ωp,q (Rn ) = φ(B˙ωp,q (Rn ))∗ = 0(B˙ωp,q (Rn ))∗ = 0. This then finishes the proof of the necessity and hence Theorem 2.2.8.

 

Remark 2.2.9 Theorems 2.2.5 and 2.2.8 show that, when p ∈ [1, ∞), q ∈ (0, ∞), and ω ∈ M(R+ ) satisfying m0 (ω) ∈ ( pn , ∞) or M∞ (ω) ∈ (−∞, 0), or when p,q p ∈ (0, 1), q ∈ (0, ∞), and ω ∈ M(R+ ), then the block space B˙ω (Rn ) is meaningless. However, it does not affect applications of block spaces in this book. Indeed, in the study of generalized Herz–Hardy spaces in the subsequent chapters, p,q we always use the block space B˙ω (Rn ) under the assumptions p, q ∈ (1, ∞) and ω ∈ M(R+ ) satisfying M0 (ω) ∈ (−∞, pn ) and m∞ (ω) ∈ (0, ∞) (see, for instance, Lemma 4.3.27 below). These block spaces used later do not satisfy the assumptions of both Theorems 2.2.5 and 2.2.8 and, as is proved in Corollary 2.1.7, their seminorms are true norms.

2.3 Boundedness of Sublinear Operators The targets of this section are twofold. The first one is to establish a boundedp,q ness criterion of sublinear operators on the block space B˙ω (Rn ) under some reasonable and sharp assumptions. To achieve this, we show the lattice property of block spaces (see Lemma 2.3.2 below). Using this lattice property and the characterization of block spaces via local generalized Herz spaces obtained in the last section, we conclude the boundedness of sublinear operators. As an application, we give the boundedness of powered Hardy–Littlewood maximal operators on block spaces, which plays an important role in the subsequent chapters. The second target is to investigate the boundedness of Calderón–Zygmund operators on block p,q spaces. Indeed, by embedding B˙ω (Rn ) into certain weighted Lebesgue space (see Lemma 2.3.8 below), the boundedness of Calderón–Zygmund operators can be concluded directly by that of sublinear operators just proved. To begin with, we present the boundedness criterion of sublinear operators on p,q B˙ω (Rn ) as follows. Theorem 2.3.1 Let p ∈ (1, ∞), q ∈ (0, ∞), and ω ∈ M(R+ ) satisfy −

n n < m0 (ω) ≤ M0 (ω) <  p p

136

2 Block Spaces and Their Applications

and −

= 1. Let T be a bounded sublinear operator on Lp (Rn ) satisfying p,q that there exists a positive constant C such that, for any f ∈ K˙ ω,0 (Rn ) and x ∈ / n supp (f ) := {x ∈ R : f (x) = 0}, where

1 p

+

n n < m∞ (ω) ≤ M∞ (ω) <  , p p

1 p

 |T (f )(x)| ≤ C

Rn

|f (y)| dy |x − y|n

and, for any {fj }j ∈N ⊂ M (Rn ) and almost every x ∈ Rn ,  ⎛ ⎞        T ⎝ ⎠ fj (x) ≤ |T (fj )(x)|.    j ∈N j ∈N

(2.42)

p,q If T is well defined on B˙ω (Rn ), then there exists a positive constant C such that, p,q n ˙ for any f ∈ Bω (R ),

T (f )B˙ωp,q (Rn ) ≤ Cf B˙ωp,q (Rn ) . In order to show Theorem 2.3.1, we require the following lattice property of block spaces. Lemma 2.3.2 Let p, q ∈ (0, ∞) and ω ∈ M(R+ ). Then a measurable function f p,q on Rn belongs to the block space B˙ω (Rn ) if and only if there exists a measurable p,q n ˙ function g ∈ Bω (R ) such that |f | ≤ g almost everywhere in Rn . Moreover, for these f and g, f B˙ωp,q (Rn ) ≤ gB˙ωp,q (Rn ) . Proof Let p, q ∈ (0, ∞) and ω ∈ M(R+ ). We first show the necessity. To this end, p,q let f ∈ B˙ω (Rn ). Then, by Definition 2.1.3, we find that f =

  l∈N k∈Z Q∈Q(ξl ) k

λξl ,k,Q bξl ,k,Q

2.3 Boundedness of Sublinear Operators

137

almost everywhere in Rn and ⎡ ⎤1 q 

 ⎢  q ⎥ λξl ,k,Q ⎦ ∈ f B˙ωp,q (Rn ) , f B˙ωp,q (Rn ) + 1 , ⎣ l∈N

k∈Z Q∈Q(ξl ) k

where {ξl }l∈N ⊂ Rn , {λξl ,k,Q } (ξ )

(ξ )

l∈N, k∈Z, Q∈Qk l

⊂ [0, ∞), and, for any l ∈ N, k ∈ Z,

and Q ∈ Qk l , bξl ,k,Q is an (ω, p)-block supported in the cube Q. Let g :=

 

λξl ,k,Q |bξl ,k,Q |.

l∈N k∈Z Q∈Q(ξl ) k (ξ )

Obviously, |f | ≤ g and, for any l ∈ N, k ∈ Z, and Q ∈ Qk l , the function |bξl ,k,Q | is also an (ω, p)-block supported in Q. This further implies that

gB˙ωp,q (Rn )

⎡ ⎤1 q  ⎢  q ⎥ ≤ λξl ,k,Q ⎦ < f B˙ωp,q (Rn ) + 1 < ∞. ⎣ l∈N

k∈Z Q∈Q(ξl ) k

p,q Thus, g ∈ B˙ω (Rn ) and hence we finish the proof of the necessity. p,q Now, we show the sufficiency. To achieve this, let f ∈ M (Rn ) and g ∈ B˙ω (Rn ) n be such that |f | ≤ g almost everywhere in R . Assume that

g=

 

λξl ,k,Q bξl ,k,Q

l∈N k∈Z Q∈Q(ξl ) k

almost everywhere in Rn , where {ξl }l∈N ⊂ Rn , {λξl ,k,Q } (ξ )

(ξ )

l∈N, k∈Z, Q∈Qk l

⊂ [0, ∞),

and, for any l ∈ N, k ∈ Z, and Q ∈ Qk l , bξl ,k,Q is an (ω, p)-block supported in the cube Q. Then we have 1{y∈Rn : g(y)=0} =

 

1 λξl ,k,Q bξl ,k,Q 1{y∈Rn : g(y)=0}, g (ξ )

l∈N k∈Z Q∈Q l k

which, together with the assumption |f | ≤ g almost everywhere in Rn , implies that f = f 1{y∈Rn: g(y)=0} =

 

f λξl ,k,Q bξl ,k,Q 1{y∈Rn : g(y)=0}. g (ξ )

l∈N k∈Z Q∈Q l k

138

2 Block Spaces and Their Applications (ξ )

Now, for any l ∈ N, k ∈ Z, and Q ∈ Qk l , let bξl ,k,Q :=

f bξ ,k,Q 1{y∈Rn : g(y)=0}. g l (ξ )

Obviously, for any l ∈ N, k ∈ Z, and Q ∈ Qk l , supp (bξl ,k,Q ) ⊂ Q. In addition, from the assumption |f | ≤ g almost everywhere in Rn , it follows that, for any (ξ ) l ∈ N, k ∈ Z, and Q ∈ Qk l ,      bξl ,k,Q 

Lp (Rn )

−1    1 ≤ bξl ,k,Q Lp (Rn ) ≤ ω |Q| n , (ξ )

which further implies that, for any l ∈ N, k ∈ Z, and Q ∈ Qk l , the function bξl ,k,Q is an (ω, p)-block supported in Q, and hence

f B˙ωp,q (Rn )

⎡ ⎤1 q  ⎢  q ⎥ ≤ λξl ,k,Q ⎦ . ⎣ l∈N

k∈Z Q∈Q(ξl ) k

This, combined with the choice of {λξl ,k,Q } p,q B˙ω (Rn ), further implies that

(ξ )

l∈N, k∈N, Q∈Qk l

, and the assumption g ∈

f B˙ωp,q (Rn ) ≤ gB˙ωp,q (Rn ) < ∞, which completes the proof of the sufficiency and hence Lemma 2.3.2.

 

Remark 2.3.3 Let p, q ∈ (0, ∞) and ω ∈ M(R+ ). We point out that a measurable p,q function f on Rn belongs to the block space B˙ω (Rn ) if and only if the function p,q n |f | belongs to B˙ω (R ). Indeed, the sufficiency of this conclusion is deduced from p,q Lemma 2.3.2 directly. On the other hand, let f ∈ B˙ω (Rn ) and g be as in the proof of the necessity of Lemma 2.3.2. Then, from the proof of the necessity of p,q Lemma 2.3.2, it follows that g ∈ B˙ω (Rn ). Applying this, the fact that |f | ≤ g, p,q and Lemma 2.3.2 again, we conclude that |f | ∈ B˙ω (Rn ), which completes the proof of the necessity of the above claim. Moreover, repeating an argument similar p,q to that used in the proof of Lemma 2.3.2, we easily find that, for any f ∈ B˙ω (Rn ), f B˙ωp,q (Rn ) =  |f | B˙ωp,q (Rn ) . Via the equivalent characterization of block spaces established in Lemma 2.2.2(ii), we obtain the following boundedness of sublinear operators on block spaces.

2.3 Boundedness of Sublinear Operators

139

Proposition 2.3.4 Let p, q ∈ (0, ∞), ω ∈ M(R+ ), and T be a bounded sublinear p,q p,q operator on K˙ ω,0 (Rn ) such that (2.42) holds true. If T is well defined on B˙ω (Rn ), p,q n then there exists a positive constant C such that, for any f ∈ B˙ω (R ), T (f )B˙ωp,q (Rn ) ≤ Cf B˙ωp,q (Rn ) . p,q Proof Let p, q ∈ (0, ∞), ω ∈ M(R+ ), and f ∈ B˙ω (Rn ). Then, from Lemma 2.2.2(ii) and Remark 1.7.2, it follows that, for any k ∈ N, there exist 4 (k) sequences {ξl }l∈N ⊂ Rn and {fξ (k) }l∈N ⊂ M (Rn ) such that both f = l∈N fξ (k) l

almost everywhere in Rn and

l



   (k)  fξ (k) · + ξl  ˙ p,q

Kω,0

l

l∈N

(Rn )

1  f B˙ωp,q (Rn ) + . k

(2.43)

Moreover, by (2.42), we find that, for any k ∈ N and almost every x ∈ Rn , |T (f )(x)| ≤

    T fξ (k) (x) .

(2.44)

l

l∈N

p,q Applying the boundedness of the operator T on K˙ ω,0 (Rn ), we conclude that, for any k, l ∈ N,





p,q T fξ (k) · + ξl(k) ∈ K˙ ω,0 (Rn ) l

and 

  (k)   fξ (k) · + ξl 





  (k)  T fξ (k) · + ξl  ˙ p,q

Kω,0 (Rn )

l

˙ p,q (Rn ) K ω,0

l

,

where the implicit positive constant is independent of k, l, and f . This, combined with Lemma 2.3.2, Lemma 2.2.2(ii), (2.44), and (2.43), further implies that, for any k ∈ N, T (f )B˙ωp,q (Rn )

  

   ≤ T fξ (k)    l

B˙ωp,q (Rn )

l∈N

  

   ∼ T fξ (k)    p,q l l∈N



   (k)   T fξ (k) · + ξl  ˙ p,q l

l∈N





   (k)  fξ (k) · + ξl  ˙ p,q l∈N

l

1  f B˙ωp,q (Rn ) + , k

B˙ω (Rn )

Kω,0 (Rn )

Kω,0 (Rn )

(2.45)

140

where 

2 Block Spaces and Their Applications

4

 l∈N T (fξ (k) )B˙ωp,q (Rn )

4

is defined as in Lemma 2.2.2(ii) with f therein

l

replaced by l∈N T (fξ (k) ), and the implicit positive constants are independent of l both f and k. Letting k → ∞ in (2.45), we obtain T (f )B˙ωp,q (Rn )  f B˙ωp,q (Rn ) , where the implicit positive constant is independent of f . This finishes the proof of Proposition 2.3.4.   We now show Theorem 2.3.1. Proof of Theorem 2.3.1 Let all the symbols be as in the present theorem. Then, combining Theorem 1.5.1 and Proposition 2.3.4, we find that there exists a positive p,q constant C such that, for any f ∈ B˙ω (Rn ), T (f )B˙ωp,q (Rn ) ≤ Cf B˙ωp,q (Rn ) .  

This finishes the proof of Theorem 2.3.1.

As a consequence, we now give the boundedness of powered Hardy–Littlewood maximal operators on block spaces as follows. Corollary 2.3.5 Let p ∈ (1, ∞), q ∈ (0, ∞), r ∈ [1, p), and ω ∈ M(R+ ) satisfy   1 n 1 − − < m0 (ω) ≤ M0 (ω) < n p r p and   1 n 1 − . − < m∞ (ω) ≤ M∞ (ω) < n p r p p,q Then there exists a positive constant C such that, for any f ∈ B˙ω (Rn ),

   (r)  M (f ) ˙ p,q

Bω (Rn )

≤ Cf B˙ωp,q (Rn ) ,

where M(r) is as in (1.55) with θ therein replaced by r. Proof Let all the symbols be as in the present corollary. We first show that the operator M(r) satisfies (2.42). Indeed, applying the Minkowski inequality of Lr (Rn ), we know that, for any {fj }j ∈N ⊂ M (Rn ), any x ∈ Rn , and any ball B ∈ B containing x, r  ⎤ 1r  1      r  r     ⎦ ⎣  f (y) dy ≤ (y) dy , f j j   B j ∈N B  j ∈N ⎡

2.3 Boundedness of Sublinear Operators

141

which implies that ⎡ ⎣ 1 |B|

 r ⎤ 1r  1    1   r   r     ⎦ fj (y) dy fj (y) dy ≤  |B| B B j ∈N  j ∈N  ≤ M(r)(fj )(x). j ∈N

This further implies that, for any {fj }j ∈N ⊂ M (Rn ) and x ∈ Rn , ⎛ M(r) ⎝



⎞ fj ⎠ (x) ≤

j ∈N



M(r) (fj )(x).

j ∈N

Thus, the powered Hardy–Littlewood maximal operator M(r) is a sublinear operator which satisfies (2.42). Next, we prove that M(r) is bounded on the local generalized Herz space p,q K˙ ω,0 (Rn ). Indeed, from Lemma 1.3.1, it follows that 

p,q K˙ ω,0 (Rn )

1/r

p/r,q/r = K˙ ωr ,0 (Rn ).

In addition, applying Lemma 1.1.6 and p/r ∈ (1, ∞), we conclude that   n min m0 (ωr ), m∞ (ωr ) = r min{m0 (ω), m∞ (ω)} > − p/r and   max M0 (ωr ), M∞ (ωr ) = r max{M0 (ω), M∞ (ω)}   n n n − = . λ}

υ(y) dy

< ∞.

Note that Chang et al. [29, Lemma 4.7] gave an embedding lemma about ball quasi-Banach function spaces. However, due to the deficiency of the Fatou property [namely, whether Definition 1.2.13(iii) holds true for block spaces is unknown], the p,q block space B˙ω (Rn ) may not be a ball quasi-Banach function space. Fortunately, we find that the proof of [29, Lemma 4.7] is still valid for Lemma 2.3.8 below; we omit the details. Lemma 2.3.8 Let p, q, and ω be as in Theorem 2.3.6. Then there exists an ε ∈ (0, 1) and a positive constant C such that, for any f ∈ M (Rn ), f L1υ (Rn ) ≤ Cf B˙ωp,q (Rn ) ,

2.3 Boundedness of Sublinear Operators

143

where υ := [M(1B(0,1))]ε with M being the Hardy–Littlewood maximal operator as in (1.54). In order to prove Theorem 2.3.6, we also require the following property of A1 (Rn )-weights, which is just [70, Theorem 7.7(1)]. Lemma 2.3.9 Let f ∈ L1loc (Rn ) satisfy M(f ) < ∞ almost everywhere in Rn and let δ ∈ [0, 1). Then υ := [M(f )]δ ∈ A1 (Rn ). The following conclusion shows the weak boundedness of Calderón–Zygmund operators on weighted Lebesgue spaces, which is just [70, Theorem 7.12]. Lemma 2.3.10 Let υ ∈ A1 (Rn ), d ∈ Z+ , and T be a d-order Calderón–Zygmund operator as in Definition 1.5.8. Then there exists a positive constant C such that, for any f ∈ L1υ (Rn ) and λ ∈ (0, ∞),  {x∈Rn : |T (f )|>λ}

υ(y) dy ≤

C f L1υ (Rn ) λ

and hence T (f )W L1υ (Rn ) ≤ Cf L1υ (Rn ) . To prove Theorem 2.3.6, we also need the following technical estimate about the Hardy–Littlewood maximal operator of characteristic functions of balls (see, for instance, [90, Example 2.1.8]). Lemma 2.3.11 Let r ∈ (0, ∞) and M be the Hardy–Littlewood maximal operator as in (1.54). Then, for any x ∈ Rn , . 6n r n rn (x) ≤ ≤ M 1 . B(0,r) (r + |x|)n (r + |x|)n Via above preparations, we now show Theorem 2.3.6. Proof of Theorem 2.3.6 Let all the symbols be as in the present theorem and f ∈ p,q B˙ω (Rn ). Then, from Lemma 2.3.8, we deduce that f ∈ L1υ (Rn ), where υ := [M(1B(0,1))]ε with an ε ∈ (0, 1). Applying Lemma 2.3.9, we find that υ ∈ A1 (Rn ). This, together with the fact that f ∈ L1υ (Rn ) and Lemma 2.3.10, further implies p,q that T (f ) ∈ W L1υ (Rn ), and hence T is well defined on B˙ω (Rn ). In addition, using Lemma 2.2.2(ii), we conclude that there exists a sequence n {ξl }l∈N ⊂ Rn and 4a sequence {fξl }l∈N ⊂ M (R )n such that, for any l ∈ N, fξl ∈ p,q n ˙ Kω,ξl (R ), f = l∈N fξl almost everywhere in R and   fξ  ˙ p,q n < ∞. l K (R ) l∈N

ω,ξl

(2.46)

144

2 Block Spaces and Their Applications

We next prove that, for almost every x ∈ Rn , |T (f )(x)| ≤

 - .  T fξ (x) . l

(2.47)

l∈N

To achieve this, for any N ∈ N, let f (N) :=

N 

fξl .

l=1

Then, from (2.46), it follows that, for any N ∈ N, N    fξ  ˙ p,q l K

ω,ξl (R

l=1

n)

≤

  fξ  ˙ p,q l K

ω,ξl (R

l∈N

n)

< ∞,

p,q which, combined with Lemma 2.2.2(ii) again, implies that f (N) ∈ B˙ω (Rn ). By this, Lemma 2.3.8, the linearity of T , Lemmas 2.3.10 and 2.2.2(ii), and (2.46), we conclude that, for any N ∈ N,

f − f (N) ∈ L1υ (Rn ) and, for any λ ∈ (0, ∞), 

 {x∈Rn : |T (f )(x)−T (f (N) )(x)|>λ}

. ε M 1B(0,1) (y) dy



  1 1     f − f (N)  1 n  f − f (N)  ˙ p,q n Lυ (R ) Bω (R ) λ λ



∞  1   fξ  ˙ p,q n → 0 l K ω,ξl (R ) λ

(2.48)

l=N+1

as N → ∞. Now, we claim that there exists a subsequence {f (Nj ) }j ∈N ⊂ {f (N) }N∈N such that T (f (Nj ) ) → T (f ) almost everywhere in Rn as j → ∞. Indeed, applying Lemma 2.3.11 with r := 1, we find that, for any x ∈ Rn , 

. ε M 1B(0,1) (x) ∼ (1 + |x|)−εn .

2.3 Boundedness of Sublinear Operators

145

By this and (2.48), we conclude that, for any λ ∈ (0, ∞),    x ∈ B(0, 1) :  



 !    T (f )(x) − T f (N) (x) > λ  

{x∈Rn : |T (f )(x)−T (f (N) )(x)|>λ}

. ε M 1B(0,1) (y) dy → 0

as N → ∞ and, for any k ∈ N,   !

      x ∈ B(0, 2k ) \ B(0, 2k−1 ) : T (f )(x) − T f (N) (x) > λ    . ε  2kεn M 1B(0,1) (y) dy → 0 {x∈Rn : |T (f )(x)−T (f (N) )(x)|>λ}

as N → ∞. These further imply that the sequence {T (f (N) )}N∈N converges in measure to T (f ) both in B(0, 1) and B(0, 2k ) \ B(0, 2k−1 ) for any k ∈ N. From this, the Riesz theorem, and a diagonalization argument, we deduce that there exists a subsequence {f (Nj ) }j ∈N ⊂ {f (N) }N∈N such that T (f (Nj ) ) → T (f ) almost everywhere in Rn as j → ∞, which completes the proof of the above claim. On the other hand, using the linearity of T , we conclude that, for any j ∈ N,     Nj Nj 

    - .   - . .   T fξ  ≤ T fξ  . T fξl (x) ≤ T f (Nj )  =  l l  l=1  l=1 l∈N

(2.49)

By the above claim and letting j → ∞ in (2.49), we further find that, for almost every x ∈ Rn , |T (f )(x)| ≤

 - .  T fξ (x) . l l∈N

This finishes the proof of (2.47). Combining this, Corollary 1.5.10, and an argument similar to that used in the proof of (2.45), we conclude that T (f )B˙ωp,q (Rn )  f B˙ωp,q (Rn ) , where the implicit positive constant is independent of f , which completes the proof of Theorem 2.3.6.  

Chapter 3

Boundedness and Compactness Characterizations of Commutators on Generalized Herz Spaces

In this chapter, we investigate the boundedness and the compactness characterizations of commutators on local and global generalized Herz spaces. Recall that the commutator plays an important role in various branch of mathematics, such as harmonic analysis (see, for instance, [7, 8, 30, 34, 95, 188, 208]) and partial differential equations (see, for instance, [33, 35, 225]). In 1976, Coifman et al. [47] first obtained a boundedness characterization of commutators on the Lebesgue space Lp (Rn ) with p ∈ (1, ∞). To be precise, let b ∈ L1loc (Rn ) and T be a singular integral operator with homogeneous kernel . Coifman et al. [47] proved that, for any given p ∈ (1, ∞) and any given b ∈ BMO (Rn ), the commutator [b, T ] is bounded on Lp (Rn ) and also that, if, for any given b ∈ L1loc (Rn ) and any Riesz transform Rj with j ∈ {1, . . . , n}, [b, Rj ] is bounded on Lp (Rn ), then b ∈ BMO (Rn ). Furthermore, in 1978, Uchiyama [231] showed that, for any given p ∈ (1, ∞), the commutator [b, T ] is bounded on Lp (Rn ) if and only if b ∈ BMO(Rn ). Later on, as extensions of the results in Lebesgue spaces, such boundedness characterizations were established on various function spaces (see [56] for Morrey spaces, [161] for weighted Lebesgue spaces, and [141] for variable Lebesgue spaces). On the other hand, the compactness characterizations of commutators were also studied. In 1978, Uchiyama [231] first showed that, for any given p ∈ (1, ∞), the commutator [b, T ] is compact on Lp (Rn ) if and only if b ∈ CMO(Rn ). After that, this compactness characterization was extended to Morrey spaces (see [36]) and to weighted Lebesgue spaces (see [43, 96]). Moreover, very recently, Tao et al. [228] studied the boundedness and the compactness characterizations of commutators on ball Banach function spaces. The main target of this chapter is to study the boundedness and the compactness characterizations of commutators on generalized Herz spaces. As was proved in Chap. 1, the generalized Herz spaces are special ball Banach function spaces under some reasonable and sharp assumptions of exponents. Thus, we can obtain the boundedness and the compactness characterizations of commutators on generalized © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 Y. Li et al., Real-Variable Theory of Hardy Spaces Associated with Generalized Herz Spaces of Rafeiro and Samko, Lecture Notes in Mathematics 2320, https://doi.org/10.1007/978-981-19-6788-7_3

147

148

3 Boundedness and Compactness Characterizations of Commutators. . .

Herz spaces via proving that generalized Herz spaces satisfy all the assumptions of the results obtained in [228]. This approach is feasible for the boundedness and the compactness characterizations of commutators on local generalized Herz spaces. p,q However, since the associate space of the global generalized Herz space K˙ ω (Rn ) is still unknown, we can not establish the boundedness and the compactness p,q characterizations of commutators on K˙ ω (Rn ) by the aforementioned method. To overcome this difficulty, we replace the assumptions of conclusions in [228] about associate spaces by other assumptions about ball Banach function spaces which p,q are more convenient to check for K˙ ω (Rn ). In particular, we point out that the p,q n ˙ extrapolation theorem of Kω (R ) established in Chap. 1 is a key tool in the proof of the boundedness and the compactness characterizations of commutators p,q on K˙ ω (Rn ). We now recall some basic concepts. Let  be a Lipschitz function on the unit sphere of Rn , which is homogeneous of degree zero and has mean value zero, namely, for any x, y ∈ Sn−1 and μ ∈ (0, ∞), |(x) − (y)| ≤ |x − y|,

(3.1)

(μx) = (x),

(3.2)

and  Sn−1

(x) dσ (x) = 0,

(3.3)

here and thereafter, Sn−1 := {x ∈ Rn : |x| = 1} denotes the unit sphere of Rn and dσ the area measure on Sn−1 . Furthermore, we state the following L∞ -Dini condition (see, for instance, [70, p. 93]). Definition 3.0.12 A function  ∈ L∞ (Sn−1 ) is said to satisfy the L∞ -Dini condition if 

1 0

ω∞ (τ ) dτ < ∞, τ

where ω∞ is defined by setting, for any τ ∈ (0, 1), ω∞ (τ ) :=

sup {x, y∈Sn−1 : |x−y| rB , which implies that B(x, t) ⊂ [B(xB , rB )] and hence contradicts to the assumption B(xB , rB ) ∩ B(x, t) = ∅. Thus, we obtain |x − xB | < t + rB and finish the proof of the above claim. Combining this and the assumption x ∈ (2B) , we further conclude that t > |x − xB | − rB ≥

|x − xB | . 2

From this, Definition 4.3.4(iii), and the Taylor remainder theorem, we deduce that, for any y ∈ B = B(xB , rB ), there exists a ty ∈ (0, 1) such that |a ∗ φt (x)|  ⎤  ⎡      ∂ γ φt (x − xB ) ⎥  ⎢  γ⎥  ⎢ (y − xB ) ⎦ dy  a(y) ⎣φt (x − y) − = n γ !   R γ ∈Zn +   |γ |≤d

4.3 Atomic Characterizations

203

 ⎤  ⎡       ⎥ ⎢  ∂ γ φt (x − ty y − (1 − ty )xB ) γ⎥  ⎢ (y − xB ) ⎦ dy  = a(y) ⎣ γ !   B(xB ,rB ) γ ∈Zn +   |γ |=d+1   t −n−d−1 |a(y)||y − xB |d+1 dy B(xB ,rB )



|x

rBd+1 aL1 (Rn ) . − xB |n+d+1

(4.22)

Next, we claim that rB ⊂ [B(xB , rB ) ∩ B(x, |x − xB |)] , B t0 x + (1 − t0 )xB , 2 where t0 :=

rB 2|x−xB | .

Indeed, for any y ∈ B(t0 x + (1 − t0 )xB , r2B ), we have

|y − xB | ≤ |y − t0 x − (1 − t0 )xB | + t0 |x − xB |
− p/s and - . M∞ ωs = sM∞ (ω) < 0. These, combined with the assumptions ps , qs ∈ (1, ∞) and Lemma 4.3.27 with p, q, and ω therein replaced, respectively, by ps , qs , and ωs , further imply that, for any f ∈ M (Rn ), f K˙ p/s,q/s (Rn ) ωs

 ∼ sup fgL1 (Rn ) : g ˙ (p/s) ,(q/s) B1/ωs

 (Rn )

=1

(4.41)

with the positive equivalence constants independent of f . On the other hand, by p/s,q/s p,q Theorem 1.3.2, we find that K˙ ωs (Rn ) = [K˙ ω (Rn )]1/s . Combining this and (4.41), we conclude that, for any f ∈ M (Rn ), f [K˙ p,q n 1/s ω (R )]

 ∼ sup fgL1 (Rn ) : g ˙ (p/s) ,(q/s) B1/ωs

 (Rn )

=1 . 

(4.42) 

p,q (p/s) ,(q/s) Therefore, K˙ ω (Rn ) satisfies Theorem 4.3.18(ii) with Y := B˙1/ωs (Rn ). p,q Finally, we show that Theorem 4.3.18(iii) is satisfied for K˙ ω (Rn ). Indeed, from 1 n Lemma 4.3.23, it follows that, for any f ∈ Lloc (R ),

   ((r/s))  (f ) ˙ (p/s) ,(q/s) M B1/ωs

(Rn )

 f  ˙ (p/s) ,(q/s) B1/ωs

(Rn )

.

(4.43)

This implies that Theorem 4.3.18(iii) holds true for K˙ ω (Rn ) with p,q





(p/s) ,(q/s) Y := B˙1/ωs (Rn ).

Moreover, combining (4.40), (4.42), and (4.43), we further find that the Herz space p,q K˙ ω (Rn ) under consideration satisfies all the assumptions of Theorem 4.3.18 and then complete the proof of Theorem 4.3.16.   As an application of this atomic characterization, we give the atomic characterip,q zation of the generalized Hardy–Morrey space H M ω (Rn ). First, we introduce the p,q n following (M ω (R ), r, d)-atoms.

4.3 Atomic Characterizations

215

Definition 4.3.28 Let p, q, and ω be as in Definition 4.3.12, r ∈ [1, ∞], and d ∈ p,q Z+ . Then a measurable function a on Rn is called an (M ω (Rn ), r, d)-atom if there exists a ball B ∈ B such that (i) supp (a) := {x ∈ Rn : a(x) = 0} ⊂ B; 1/r (ii) aLr (Rn ) ≤ 1B |B|p,q ; M ω (Rn )

(iii) for any α ∈ Zn+ with |α| ≤ d,  Rn

a(x)x α dx = 0.

By Theorem 4.3.16 and Remarks 1.2.2(iv) and 4.0.20(ii), we immediately obtain p,q the atomic characterization of the generalized Hardy–Morrey space H M ω (Rn ) as follows; we omit the details. Corollary 4.3.29 Let p, q, ω, r, d, and s be as in Corollary 4.3.13. The generalized p,q,r,d,s atomic Hardy–Morrey space H M ω (Rn ), associated with the global generalp,q n ized Morrey space M ω (R ), is defined to be the set of all the f ∈ S  (Rn ) such that p,q there exists a sequence {aj }j ∈N of (M ω (Rn ), r, d)-atoms supported, respectively, in the balls {Bj }j ∈N ⊂ B, and a sequence {λj }j ∈N ⊂ [0, ∞) such that f =



λj aj

j ∈N

in S  (Rn ) and ⎧ ⎫1    # $s ⎨ ⎬s  λj   1 Bj   ⎩  ⎭ 1Bj M p,q n ω (R )  j ∈N 

< ∞. p,q

M ω (Rn )

p,q,r,d,s

Moreover, for any f ∈ H M ω

f H M p,q,r,d,s (Rn ) ω

(Rn ),

⎧⎧ ⎫1    # $s ⎪ ⎨⎨ ⎬s  λj   := inf  1 Bj  p,q ⎪ ⎭  ⎩  1  n B ⎩ j ∈N j M ω (R ) 

⎫ ⎪ ⎬ p,q

M ω (Rn )

⎪ ⎭

where the infimum is taken over all the decompositions of f as above. Then n p,q,r,d,s H M p,q (Rn ) ω (R ) = H M ω

with equivalent quasi-norms.

,

216

4 Generalized Herz–Hardy Spaces

4.4 Generalized Finite Atomic Herz–Hardy Spaces In this section, we first introduce the generalized finite atomic Herz–Hardy spaces, which are equipped with a finite atomic quasi-norm, and we then prove that, on these generalized finite atomic Herz–Hardy spaces, this finite atomic quasi-norm is equivalent to the quasi-norm of the corresponding generalized Herz–Hardy spaces. We begin with the following concept of finite atomic Hardy spaces associated p,q with K˙ ω,0 (Rn ). Definition 4.4.1 Let p, q ∈ (0, ∞), ω ∈ M(R+ ) with m0 (ω) ∈ (− pn , ∞) and m∞ (ω) ∈ (− pn , ∞), r ∈ (max{1, p, min{m0 (ω),mn∞ (ω)}+n/p }, ∞],   s ∈ 0, min 1, p, q,

n max{M0 (ω), M∞ (ω)} + n/p

 ,

and d ≥ n(1/s − 1) be a fixed integer. Then the generalized finite atomic Herz– p,q,r,d,s p,q Hardy space H K˙ ω,0,fin (Rn ), associated with K˙ ω,0 (Rn ), is defined to be the set p,q of all finite linear combinations of (K˙ ω,0 (Rn ), r, d)-atoms. Moreover, for any f ∈ p,q,r,d,s H K˙ (Rn ), ω,0,fin

f H K˙ p,q,r,d,s (Rn ) ω,0,fin

⎧ ⎪ ⎪ ⎨

⎧ ⎫1   N #  $s ⎨ ⎬s  λj   := inf  1 Bj  p,q  ⎩  ⎭ ⎪ 1  n B ˙ ⎪ j K (R )  ˙ p,q ⎩  j =1 ω,0

Kω,0 (Rn )

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

,

where the infimum is taken over all finite linear combinations of f , namely, N ∈ N, f =

N 

λj aj ,

j =1 N n ˙ {λj }N j =1 ⊂ [0, ∞), and {aj }j =1 being (Kω,0 (R ), r, d)-atoms supported, respecN tively, in the balls {Bj }j =1 ⊂ B. p,q

Based on the known finite atomic characterization of the Hardy space HX (Rn ) associated with the ball quasi-Banach function space X (see [266, Theorem 1.10] or Lemma 4.4.3 below), we have the following equivalence between the quasi-norms p,q,r,d,s  · H K˙ p,q (Rn ) and  · H K˙ p,q,r,d,s (Rn ) on H K˙ ω,0,fin (Rn ). Throughout this book, ω,0

ω,0,fin

C(Rn ) is defined to be the set of all continuous functions on Rn .

4.4 Generalized Finite Atomic Herz–Hardy Spaces

217

Theorem 4.4.2 Let p, q, ω, d, s, and r be as in Definition 4.4.1. (i) If   r ∈ max 1, p,

  n ,∞ , min{m0 (ω), m∞ (ω)} + n/p

then  · H K˙ p,q,r,d,s (Rn ) and  · H K˙ p,q (Rn ) are equivalent quasi-norms on the ω,0

ω,0,fin

p,q,r,d,s generalized finite atomic Herz–Hardy space H K˙ ω,0,fin (Rn ). (ii) If r = ∞, then ·H K˙ p,q,∞,d,s (Rn ) and ·H K˙ p,q (Rn ) are equivalent quasi-norms ω,0

ω,0,fin

p,q,∞,d,s on H K˙ ω,0,fin (Rn ) ∩ C(Rn ).

To prove this theorem, recall that, on Hardy spaces associated with ball quasiBanach function spaces, Yan et al. [266, Theorem 1.10] introduced the following finite atomic Hardy spaces, which are equipped with a finite atomic quasi-norm, and then showed that, on these finite atomic Hardy spaces, this finite atomic quasinorm is equivalent to the quasi-norm of the corresponding Hardy spaces. Lemma 4.4.3 Let X, r, d, and s be as in Lemma 4.3.6. Then the finite atomic X,r,d,s Hardy space Hfin (Rn ), associated with X, is defined to be the set of all finite X,r,d,s linear combinations of (X, r, d)-atoms. Moreover, for any f ∈ Hfin (Rn ),

f H X,r,d,s (Rn ) fin

⎧ ⎫1   N 

s ⎨ ⎬s  λj   := inf  1 Bj   ⎪ ⎩  ⎭ 1  B X ⎩  j =1 j  ⎧ ⎪ ⎨

⎫ ⎪ ⎬ ⎪ ⎭

,

(4.44)

X

where the infimum is taken over all finite linear combinations of f , namely, N ∈ N, f =

N 

λj aj ,

j =1 N {λj }N j =1 ⊂ [0, ∞), and {aj }j =1 being (X, r, d)-atoms supported, respectively, in N the balls {Bj }j =1 ⊂ B. Then

(i) if r ∈ (1, ∞),  · H X,r,d,s (Rn ) and  · HX (Rn ) are equivalent quasi-norms on fin

X,r,d,s Hfin (Rn ); (ii) if r = ∞,  · H X,∞,d,s (Rn ) and  · HX (Rn ) are equivalent quasi-norms on fin

X,∞,d,s Hfin (Rn ) ∩ C(Rn ).

218

4 Generalized Herz–Hardy Spaces

Remark 4.4.4 X,r,d,s (i) We point out that the finite atomic Hardy space Hfin (Rn ) may not be equal to the Hardy space HX (Rn ). Indeed, let X := L1 (R) and, for any x ∈ R, let

f (x) :=

x . |x|(1 + x 2 )

Then HX (Rn ) goes back to the Hardy space H 1 (R) and, as was mentioned in [213, p. 178, Subsection 6.2], f ∈ H 1 (R). But, f does not have compact support and hence f does not belong to any finite atomic Hardy space associated with L1 (R). However, as was showed in [207, Remark 3.12], when the ball quasi-Banach function space X has an absolutely continuous quasiX,r,d,s norm, then Hfin (Rn ) is dense in HX (Rn ). (ii) To simplify the presentation, in what follows, if a Hardy space has properties similar to those in Lemma 4.4.3 (or Theorem 4.4.2), we simply call that this Hardy space has the finite atomic characterization. Remark 4.4.5 We should point out that Lemma 4.4.3 has a wide range of applications. Here we present several function spaces to which Lemma 4.4.3 can be applied. (i) Let p := (p1 , . . . , pn ) ∈ (0, ∞)n , 5  d≥ n

1 −1 min{p1 , . . . , pn }

6

be a fixed nonnegative integer, r ∈ (max{1, p1 , . . . , pn }, ∞], and s ∈ (0, min{1, p1 , . . . , pn }). Then, in this case, by both Remarks 1.2.31(iii) and 1.2.34(iii), we easily conclude that the mixed-norm Lebesgue space Lp (Rn ) satisfies all the assumptions of Lemma 4.4.3. This implies that Lemma 4.4.3 with X therein replaced by Lp (Rn ) holds true. This result is just the finite atomic characterization of mixed-norm Hardy spaces showed in [123, Theorem 5.9] (see also [127, Theorem 4.5]). (ii) Let 0 < q ≤ p < ∞, 5  6 1 d≥ n −1 q be a fixed nonnegative integer, r ∈ (max{1, p}, ∞], and s ∈ (0, min{1, q}). Then, in this case, from both Remarks 1.2.31(iv) and 1.2.34(iv), we can p easily deduce that the Morrey space Mq (Rn ) satisfies all the assumptions of p Lemma 4.4.3. This then implies that Lemma 4.4.3 with X := Mq (Rn ) holds

4.4 Generalized Finite Atomic Herz–Hardy Spaces

219

true. To the best of our knowledge, this finite atomic characterization of Hardy– Morrey spaces is totally new. (iii) Let p(·) ∈ C log (Rn ) satisfy 0 < p− ≤ p+ < ∞, where p− and p+ are defined, respectively, in (1.59) and (1.60). Let 5  6 1 d≥ n −1 p− be a fixed nonnegative integer, r ∈ (max{1, p+ }, ∞], and s := min{1, p− }. Then, in this case, using both Remarks 1.2.31(v) and 1.2.34(v), we can easily find that the variable Lebesgue space Lp(·) (Rn ) satisfies all the assumptions of Lemma 4.4.3. Thus, Lemma 4.4.3 with X := Lp(·)(Rn ) holds true. This result is just the finite atomic characterization of variable Hardy spaces established in [156, Theorem 5.4]. Lemma 4.4.3 implies that, to show Theorem 4.4.2, we only need to prove that, p,q under the assumptions of Theorem 4.4.2, K˙ ω,0 (Rn ) satisfies all the assumptions of Lemma 4.4.3. Applying this idea and some technical lemmas obtained in the last section, we now show Theorem 4.4.2. Proof of Theorem 4.4.2 Let p, q, ω, d, s, and r be as in the present theorem. Then, using the assumption m0 (ω) ∈ (− pn , ∞) and Theorem 1.2.42, we conclude that the p,q local generalized Herz space K˙ ω,0 (Rn ) is a BQBF space. Thus, in order to finish the proof of the present theorem, it suffices to show that all the assumptions of p,q Lemma 4.4.3 hold true for K˙ ω,0 (Rn ). First, let θ ∈ (0, s) be such that 6   5  6 5  1 1 1 −1 ≤n −1 < n −1 + 1. n s θ s This implies that d ≥ n(1/s − 1) = n(1/θ − 1). We now prove that K˙ ω,0 (Rn ) satisfies Assumption 1.2.29 for the above θ and s. Indeed, from Lemma 4.3.11, it follows that, for any {fj }j ∈N ⊂ L1loc (Rn ), p,q

⎧ ⎫1/s    ⎨  s ⎬    (θ) M (fj )   ⎩ ⎭   j ∈N  ˙ p,q

Kω,0 (Rn )

⎛ ⎞1/s       ⎝ s⎠   |fj |     j ∈N  ˙ p,q

,

Kω,0 (Rn )

which implies that K˙ ω,0 (Rn ) satisfies Assumption 1.2.29 for the above θ and s. p,q On the other hand, by Lemma 1.8.6, we find that [K˙ ω,0 (Rn )]1/s is a BBF space and, for any f ∈ L1loc (Rn ), p,q

   ((r/s))  (f ) M

˙ (Rn )]1/s ) ([K ω,0 p,q

 f ([K˙ p,q (Rn )]1/s ) . ω,0

220

4 Generalized Herz–Hardy Spaces

This further implies that, under the assumptions of the present theorem, K˙ ω,0 (Rn ) satisfies all the assumptions of Lemma 4.4.3, which completes the proof of Theorem 4.4.2.   p,q

Then, as an application of Theorem 4.4.2, we have the following finite atomic characterization of generalized Hardy–Morrey spaces, which is a simple corollary of Theorem 4.4.2 and Remarks 1.2.2(iv) and 4.0.20(ii); we omit the details. Corollary 4.4.6 Let p, q, ω, r, d, and s be as in Corollary 4.3.13. Then the p,q,r,d,s generalized finite atomic Hardy–Morrey space H M ω,0,fin (Rn ), associated with p,q the local generalized Morrey space M ω,0 (Rn ), is defined to be the set of all p,q finite linear combinations of (M ω,0 (Rn ), r, d)-atoms. Moreover, for any f ∈ p,q,r,d,s

H M ω,0,fin (Rn ),

f H M p,q,r,d,s (Rn ) ω,0,fin

⎫ ⎪ ⎪ ⎬

⎧ ⎪ ⎪ ⎨

⎧ ⎫1   N #  $s ⎨ ⎬s  λj   := inf  1 Bj  p,q  ⎩  ⎪ ⎭ 1  Bj M (Rn ) ⎪ ⎩  j =1  ω,0

p,q M ω,0 (Rn )

⎪ ⎪ ⎭

,

where the infimum is taken over all finite linear combinations of f , namely, N ∈ N, f =

N 

λj aj ,

j =1 p,q

N n {λj }N j =1 ⊂ [0, ∞), and {aj }j =1 being (M ω,0 (R ), r, d)-atoms supported, respecN tively, in the balls {Bj }j =1 ⊂ B. Then

(i) if  r∈

 n ,∞ , min{m0 (ω), m∞ (ω)} + n/p

 · H M p,q,r,d,s (Rn ) and  · H M p,q (Rn ) are equivalent quasi-norms on the ω,0

ω,0,fin

p,q,r,d,s

generalized finite atomic Hardy–Morrey space H M ω,0,fin (Rn ); (ii) if r = ∞,  · H M p,q,∞,d,s (Rn ) and  · H M p,q (Rn ) are equivalent quasi-norms on ω,0,fin

p,q,∞,d,s

H M ω,0,fin

ω,0

(Rn ) ∩ C(Rn ).

Next, we are devoted to introducing and investigating the finite atomic Hardy p,q space associated with the global generalized Herz space K˙ ω (Rn ). For this purpose, we first introduce the following generalized finite atomic Herz–Hardy space.

4.4 Generalized Finite Atomic Herz–Hardy Spaces

221

Definition 4.4.7 Let p, q ∈ (0, ∞), ω ∈ M(R+ ) with m0 (ω) ∈ (− pn , ∞) and −

n < m∞ (ω) ≤ M∞ (ω) < 0, p

∈

(max{1, p, min{m0 (ω),mn∞ (ω)}+n/p }, ∞], s ∈ (0, min{1, p, q, and d ≥ n(1/s − 1) be a fixed integer. Then p,q,r,d,s the generalized finite atomic Herz–Hardy space H K˙ ω,fin (Rn ), associated p,q with K˙ ω (Rn ), is defined to be the set of all finite linear combinations of p,q p,q,r,d,s (K˙ ω (Rn ), r, d)-atoms. Moreover, for any f ∈ H K˙ (Rn ), r

n max{M0 (ω),M∞ (ω)}+n/p }),

ω,fin

f H K˙ p,q,r,d,s (Rn ) ω,fin

⎧ ⎫1   N #  $s ⎨ ⎬s  λj   := inf  1 Bj   ⎩  ⎪ p,q ⎭ 1  Bj K ˙ ω (Rn ) ⎩  j =1  ˙ p,q ⎧ ⎪ ⎨

Kω (Rn )

⎫ ⎪ ⎬ ⎪ ⎭

,

where the infimum is taken over all finite linear combinations of f , namely, N ∈ N, f =

N 

λj aj ,

j =1 N n ˙ {λj }N j =1 ⊂ [0, ∞), and {aj }j =1 being (Kω (R ), r, d)-atoms supported, respectively, in the balls {Bj }N j =1 ⊂ B. p,q

Then the following conclusion implies that the quasi-norms of both the generalp,q ized Herz–Hardy space H K˙ ω (Rn ) and the generalized finite atomic Herz–Hardy p,q,r,d,s p,q,r,d,s space H K˙ ω,fin (Rn ) are equivalent on H K˙ ω,fin (Rn ). Theorem 4.4.8 Let p, q, ω, d, s, and r be as in Definition 4.4.7. (i) If   r ∈ max 1, p,

  n ,∞ , min{m0 (ω), m∞ (ω)} + n/p

then  · H K˙ p,q,r,d,s (Rn ) and  · H K˙ p,q n are equivalent quasi-norms on the ω (R ) ω,fin

p,q,r,d,s generalized finite atomic Herz–Hardy space H K˙ ω,fin (Rn ). (ii) If r = ∞, then ·H K˙ p,q,∞,d,s (Rn ) and ·H K˙ p,q n are equivalent quasi-norms ω (R ) ω,fin

p,q,∞,d,s (Rn ) ∩ C(Rn ). on H K˙ ω,fin

To show Theorem 4.4.8, note that the associate space of the global generalized p,q Herz space K˙ ω (Rn ) is still unknown. This means that we can not prove Theorem 4.4.8 via applying Lemma 4.4.3 directly. Fortunately, by the proof of [266, Theorem 1.10], we find that, if the atomic characterization of HX (Rn ) holds true,

222

4 Generalized Herz–Hardy Spaces

Lemma 4.4.3 still holds true even when there is not any assumption about the associate space of X. Namely, we have the following finite atomic characterization of HX (Rn ). Lemma 4.4.9 Let X be a ball quasi-Banach function space satisfy Assumption 1.2.29 with some θ, s ∈ (0, 1], r ∈ (1, ∞], and d ≥ n(1/θ − 1) be a fixed integer such that HX (Rn ) = H X,r,d,s (Rn ) with equivalent quasi-norms, where the atomic Hardy space H X,r,d,s (Rn ) is defined as in Definition 4.3.17. Then (i) if r ∈ (1, ∞),  · H X,r,d,s (Rn ) and  · HX (Rn ) are equivalent quasi-norms on fin

X,r,d,s Hfin (Rn ); (ii) if r = ∞,  · H X,∞,d,s (Rn ) and  · HX (Rn ) are equivalent quasi-norms on fin

X,∞,d,s Hfin (Rn ) ∩ C(Rn ). X,r,d,s Proof Let all the symbols be as in the present lemma and f ∈ Hfin (Rn ). Then, applying Lemma 4.1.4(ii), (4.44), (4.19), and the assumption that

HX (Rn ) = H X,r,d,s (Rn ) with equivalent quasi-norms, we find that f HX (Rn ) ∼ f H X,r,d,s (Rn )  f H X,r,d,s (Rn ) .

(4.45)

fin

Conversely, using Assumption 1.2.29 and repeating the proof of [266, Theorem 1.10] with q therein replaced by r, we conclude that, if r ∈ (1, ∞), then f H X,r,d,s (Rn )  f HX (Rn ) fin

and, if r = ∞ and f ∈ C(Rn ), then f H X,∞,d,s (Rn )  f HX (Rn ) . fin

These, combined with (4.45), further imply that both (i) and (ii) of the present lemma hold true. Thus, the proof of Lemma 4.4.9 is completed.   Via this finite atomic characterization of HX (Rn ), we now show Theorem 4.4.8. Proof of Theorem 4.4.8 Let p, q, ω, d, s, and r be as in the present theorem. Then, applying the assumptions m0 (ω) ∈ (− pn , ∞) and M∞ (ω) ∈ (−∞, 0), and p,q Theorem 1.2.44, we find that the global generalized Herz space K˙ ω (Rn ) is a BQBF

4.4 Generalized Finite Atomic Herz–Hardy Spaces

223

space. Therefore, to complete the proof of the present theorem, we only need to p,q show that all the assumptions of Lemma 4.4.9 are satisfied for K˙ ω (Rn ). First, we prove that there exists a θ ∈ (0, 1] such that d ≥ n(1/θ − 1) p,q and, for this θ and the s same as in the present theorem, K˙ ω (Rn ) satisfies 1 n Assumption 1.2.29, namely, for any {fj }j ∈N ⊂ L loc (R ), ⎧ ⎫1/s    ⎨  s ⎬    M(θ) (fj )   ⎩ ⎭   j ∈N  ˙ p,q

Kω,0 (Rn )

⎛ ⎞1/s       ⎝   |fj |s ⎠     j ∈N  ˙ p,q

.

(4.46)

Kω,0 (Rn )

Indeed, from the definition of n(1/s − 1), it follows that 6   5  6 5  1 1 1 −1 ≤n −1 < n −1 + 1. n s s s Thus, we can choose a θ ∈ (0, s) such that 6   5  6 5  1 1 1 −1 ≤n −1 < n −1 + 1, n s θ s which further implies that d ≥ n(1/s − 1) = n(1/θ − 1). On the other hand, by Lemma 4.3.25, we conclude that (4.46) holds true. This implies that Assumption 1.2.29 holds true for this θ and s. Next, using Theorem 4.3.16, we find that H K˙ ωp,q (Rn ) = H K˙ ωp,q,r,d,s (Rn ) p,q with equivalent quasi-norms. This further implies that K˙ ω (Rn ) satisfies all the assumptions of Lemma 4.4.9, which completes the proof of Theorem 4.4.8.  

Combining Theorem 4.4.8 and Remarks 1.2.2(iv) and 4.0.20(ii), we immediately obtain the following finite atomic characterization of the generalized Hardy–Morrey p,q space H M ω (Rn ); we omit the details. Corollary 4.4.10 Let p, q, ω, r, d, and s be as in Corollary 4.3.13. Then the p,q,r,d,s generalized finite atomic Hardy–Morrey space H M ω,fin (Rn ), associated with p,q n the global generalized Morrey space M ω (R ), is defined to be the set of all p,q finite linear combinations of (M ω (Rn ), r, d)-atoms. Moreover, for any f ∈ p,q,r,d,s H M ω,fin (Rn ),

f H M p,q,r,d,s (Rn ) ω,fin

⎧ ⎫1   N #  $s ⎨ ⎬s  λj   := inf  1 Bj  ⎪ ⎭ ⎩  1Bj M p,q n ⎩  ω (R )  j =1  ⎧ ⎪ ⎨

⎫ ⎪ ⎬ p,q

M ω (Rn )

⎪ ⎭

,

224

4 Generalized Herz–Hardy Spaces

where the infimum is taken over all finite linear combinations of f , namely, N ∈ N, f =

N 

λj aj ,

j =1 p,q

N n {λj }N j =1 ⊂ [0, ∞), and {aj }j =1 being (M ω (R ), r, d)-atoms supported, respecN tively, in the balls {Bj }j =1 ⊂ B. Then

(i) if  r∈

 n ,∞ , min{m0 (ω), m∞ (ω)} + n/p

then  · H M p,q,r,d,s (Rn ) and  · H M p,q n are equivalent quasi-norms on the ω (R ) ω,fin

p,q,r,d,s

generalized finite atomic Hardy–Morrey space H M ω,fin (Rn ); (ii) if r = ∞, then ·H M p,q,∞,d,s (Rn ) and ·H M p,q n are equivalent quasi-norms ω (R ) ω,fin

p,q,∞,d,s

on H M ω,fin

(Rn ) ∩ C(Rn ).

4.5 Molecular Characterizations The target of this section is to establish the molecular characterization of generalized Herz–Hardy spaces. Indeed, we first show the molecular characterization of the genp,q eralized Herz–Hardy space H K˙ ω,0 (Rn ) via the known molecular characterization of the Hardy space HX (Rn ) associated with the ball quasi-Banach function space X (see Lemma 4.5.5 below). However, we should point out that, due to the deficiency p,q of the associate space of K˙ ω (Rn ), the molecular characterization of the generalized p,q Herz–Hardy space H K˙ ω (Rn ) can not be obtained by applying Lemma 4.5.5 [see Remark 1.2.19(vi) for the details]. To overcome this obstacle, we first establish an improved molecular characterization of HX (Rn ) (see Theorem 4.5.11 below) via borrowing some ideas from the proof of [207, Theorem 3.9] and get the rid of associate spaces. Then, by this molecular characterization, we obtain the molecular p,q characterization of H K˙ ω (Rn ). p,q We first use molecules to characterize the Hardy space H K˙ ω,0 (Rn ) associated p,q with the local generalized Herz space K˙ ω,0 (Rn ). To this end, we now introduce p,q the following concept of (K˙ ω,0 (Rn ), r, d, τ )-molecules. In what follows, for any j ∈ N and B ∈ B, let



Sj (B) := 2j B \ 2j −1 B and S0 (B) := B.

4.5 Molecular Characterizations

225

Definition 4.5.1 Let p, q ∈ (0, ∞), ω ∈ M(R+ ) with m0 (ω) ∈ (− pn , ∞), τ ∈ (0, ∞), r ∈ [1, ∞], and d ∈ Z+ . Then a measurable function m on Rn is called a p,q (K˙ ω,0 (Rn ), r, d, τ )-molecule centered at a ball B ∈ B if (i) for any j ∈ Z+ ,   m1S (B)  r n ≤ 2−τj j L (R )

|B|1/r ; 1B K˙ p,q (Rn ) ω,0

(ii) for any α ∈ Zn+ with |α| ≤ d,  Rn

m(x)x α dx = 0.

Then we establish the molecular characterization of the generalized Herz–Hardy p,q space H K˙ ω,0 (Rn ) as follows. Theorem 4.5.2 Let p, q ∈ (0, ∞), ω ∈ M(R+ ) with m0 (ω) ∈ (− pn , ∞) and m∞ (ω) ∈ (− pn , ∞),   s ∈ 0, min 1, p, q,

n max{M0 (ω), M∞ (ω)} + n/p

 ,

d ≥ n(1/s − 1) be a fixed integer,   r ∈ max 1, p,

  n ,∞ , min{m0 (ω), m∞ (ω)} + n/p

p,q and τ ∈ (0, ∞) with τ > n(1/s − 1/r). Then f ∈ H K˙ ω,0 (Rn ) if and only if p,q f ∈ S  (Rn ) and there exists a sequence {mj }j ∈N of (K˙ ω,0 (Rn ), r, d, τ )-molecules centered, respectively, at the balls {Bj }j ∈N ⊂ B and a sequence {λj }j ∈N ⊂ [0, ∞) 4 such that f = j ∈N λj mj in S  (Rn ) and

⎧ ⎫1    # $s  ⎨ ⎬s  λj   1 Bj   ⎭  ⎩ 1Bj K˙ p,q (Rn )  j ∈N  ˙ p,q ω,0

Kω,0 (Rn )

< ∞.

226

4 Generalized Herz–Hardy Spaces

Moreover, there exist two positive constants C1 and C2 such that, for any f ∈ p,q H K˙ ω,0 (Rn ),

C1 f H K˙ p,q (Rn ) ω,0

⎧ ⎫1  ⎧  ⎪ # $s ⎪ ⎨⎨ ⎬s  λi   ≤ inf  1 Bj   ⎩  ⎪ p,q ⎭ 1Bj K˙ (Rn ) ⎪  ˙ p,q ⎩ j ∈N ω,0

Kω,0 (Rn )

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

≤ C2 f H K˙ p,q (Rn ) , ω,0

where the infimum is taken over all the decompositions of f as above. To show this theorem, we need the molecular characterization of Hardy spaces associated with ball quasi-Banach function spaces. First, we present the definition of (X, r, d, τ )-molecules introduced in [207, Definition 3.8]. Definition 4.5.3 Let X be a ball quasi-Banach function space, r ∈ [1, ∞], d ∈ Z+ , and τ ∈ (0, ∞). Then a measurable function m on Rn is called an (X, r, d, τ )molecule centered at a ball B ∈ B if (i) for any j ∈ Z+ , 1/r   m1S (B)  r n ≤ 2−τj |B| ; j L (R ) 1B X

(ii) for any α ∈ Zn+ with |α| ≤ d,  Rn

m(x)x α dx = 0.

Remark 4.5.4 Let X, r, d, and τ be as in Definition 4.5.3. Then it is easy to see that, for any (X, r, d)-atom a supported in the ball B ∈ B, a is also an (X, r, d, τ )molecule centered at B. Via these molecules, Sawano et al. [207, Theorem 3.9] established the molecular characterization of the Hardy space HX (Rn ), which plays a key role in the proof p,q of the molecular characterization of H K˙ ω,0 (Rn ). Namely, the following conclusion holds true. Lemma 4.5.5 Let X, r, d, θ , and s be as in Lemma 4.3.6 and τ ∈ (0, ∞) with τ > n(1/θ − 1/r). Then f ∈ HX (Rn ) if and only if f ∈ S  (Rn ) and there exists a sequence {mj }j ∈N of (X, r, d, τ )-molecules centered, respectively, at the balls {Bj }j ∈N ⊂ B and a sequence {λj }j ∈N ⊂ [0, ∞) such that f =

 j ∈N

λj mj

4.5 Molecular Characterizations

227

in S  (Rn ) and ⎡ ⎤1  

s s    λj ⎣  1Bj ⎦  < ∞.    1Bj X  j ∈N  X

Moreover, for any f ∈ HX (Rn ),

f HX (Rn )

⎧⎡ ⎤1  

s s ⎪  ⎨  λi ⎣  ∼ inf  1 Bj ⎦    ⎪ 1  Bj X ⎩ j ∈N 

⎫ ⎪ ⎬ ⎪ ⎭ X

with the positive equivalence constants independent of f , where the infimum is taken over all the decompositions of f as above. Remark 4.5.6 We should point out that Lemma 4.5.5 has a wide range of applications. Here we present several function spaces to which Lemma 4.5.5 can be applied. (i) Let p := (p1 , . . . , pn ) ∈ (0, ∞)n , 5  d≥ n

1 −1 min{p1 , . . . , pn }

6

be a fixed nonnegative integer, r ∈ (max{1, p1 , . . . , pn }, ∞], s ∈ (0, min{1, p1 , . . . , pn }), and   τ∈ n

  1 1 − ,∞ . min{1, p1 , . . . , pn } r

Then, in this case, from both Remarks 1.2.31(iii) and 1.2.34(iii), we easily infer that the mixed-norm Lebesgue space Lp (Rn ) satisfies all the assumptions of Lemma 4.5.5. Therefore, Lemma 4.5.5 with X := Lp (Rn ) holds true. This result goes back to the molecular characterization of mixed-norm Hardy spaces established in [155, Theorem 1]. (ii) Let 0 < q ≤ p < ∞, 6 5  1 −1 d≥ n q

228

4 Generalized Herz–Hardy Spaces

be a fixed nonnegative integer, r ∈ (max{1, p}, ∞], s ∈ (0, min{1, q}), and   τ∈ n

  1 1 − ,∞ . min{1, q} r

Then, in this case, combining both Remarks 1.2.31(iv) and 1.2.34(iv), we p can easily find that the Morrey space Mq (Rn ) satisfies all the assumptions p of Lemma 4.5.5. This then implies that Lemma 4.5.5 with X := Mq (Rn ) holds true. Recall that the molecular characterization of Hardy–Morrey spaces was also given in [107, Theorem 5.10] via replacing Definition 4.5.3(i) by some pointwise estimates (namely, [107, Definition 5.3]) which is stronger than Definition 4.5.3(i). Therefore, this aforementioned result extends [107, Theorem 5.10]. (iii) Let p(·) ∈ C log (Rn ) satisfy 0 < p− ≤ p+ < ∞, where p− and p+ are defined, respectively, in (1.59) and (1.60). Let 5  6 1 d≥ n −1 p− be a fixed nonnegative integer, r ∈ (max{1, p+ }, ∞], s := min{1, p− }, and   τ∈ n

  1 1 − ,∞ . min{1, p− } r

Then, in this case, using both Remarks 1.2.31(v) and 1.2.34(v), we easily conclude that the variable Lebesgue space Lp(·)(Rn ) satisfies all the assumptions of Lemma 4.5.5. Thus, Lemma 4.5.5 with X := Lp(·) (Rn ) holds true. This result coincides with the molecular characterization of variable Hardy spaces given in [154, Theorem 3.1]. Applying Lemma 4.5.5, we next show Theorem 4.5.2. Proof of Theorem 4.5.2 Let p, q, ω, r, d, s, and τ be as in the present theorem. Then, from the assumption m0 (ω) ∈ (− pn , ∞) and Theorem 1.2.42, it follows that p,q the local generalized Herz space K˙ ω,0 (Rn ) is a BQBF space. Therefore, to complete the proof of the present theorem, we only need to show that all the assumptions of p,q Lemma 4.5.5 are satisfied for K˙ ω,0 (Rn ). First, let θ ∈ (0, s) satisfy 6   5  6 5  1 1 1 −1 ≤n −1 < n −1 +1 n s θ s and  τ >n

 1 1 − . θ r

4.5 Molecular Characterizations

229

p,q This implies that d ≥ n(1/s − 1) = n(1/θ − 1). We now show that K˙ ω,0 (Rn ) satisfies Assumption 1.2.29 for the above θ and s. Indeed, applying Lemma 4.3.11, we find that, for any {fj }j ∈N ⊂ L1loc (Rn ),

⎧ ⎫1/s    ⎨  s ⎬    (θ) M (fj )   ⎩ ⎭   j ∈N  ˙ p,q

Kω,0 (Rn )

⎛ ⎞1/s       ⎝ s⎠   |fj |     j ∈N  ˙ p,q

,

(4.47)

Kω,0 (Rn )

p,q which implies that Assumption 1.2.29 holds true for K˙ ω,0 (Rn ) with the above θ and s. p,q On the other hand, by Lemma 1.8.6, we conclude that [K˙ ω,0 (Rn )]1/s is a BBF space and, for any f ∈ L1loc (Rn ),

   ((r/s))  (f ) M

˙ (Rn )]1/s ) ([K ω,0 p,q

 f ([K˙ p,q (Rn )]1/s ) . ω,0

This, together with (4.47), further implies that, under the assumptions of the present p,q theorem, K˙ ω,0 (Rn ) satisfies all the assumptions of Lemma 4.5.5 and hence finishes the proof of Theorem 4.5.2.   As an application of Theorem 4.5.2, we now consider the molecular characterp,q ization of the generalized Hardy–Morrey space H M ω,0 (Rn ). For this purpose, we p,q first introduce (M ω,0 (Rn ), r, d, τ )-molecules as follows. Definition 4.5.7 Let p, q ∈ [1, ∞), ω ∈ M(R+ ) with M∞ (ω) ∈ (−∞, 0) and −

n < m0 (ω) ≤ M0 (ω) < 0, p

τ ∈ (0, ∞), r ∈ [1, ∞], and d ∈ Z+ . Then a measurable function m on Rn is called p,q an (M ω,0 (Rn ), r, d, τ )-molecule centered at a ball B ∈ B if (i) for any j ∈ Z+ ,  m1S

 

j (B) Lr (Rn )

≤ 2−τj

|B|1/r ; 1B M p,q (Rn ) ω,0

(ii) for any α ∈ Zn+ with |α| ≤ d,

/ Rn

m(x)x α dx = 0. p,q

Then we have the following molecular characterization of H M ω,0 (Rn ), which can be deduced directly from Theorem 4.5.2 and Remarks 1.2.2(iv) and 4.0.20(ii); we omit the details. Corollary 4.5.8 Let p, q, ω, r, d, and s be as in Corollary 4.3.13 and τ ∈ (0, ∞) p,q with τ > n(1/s − 1/r). Then f ∈ H M ω,0 (Rn ) if and only if f ∈ S  (Rn )

230

4 Generalized Herz–Hardy Spaces p,q

and there exists a sequence {mj }j ∈N of (M ω,0 (Rn ), r, d, τ )-molecules centered, respectively, at the balls {Bj }j ∈N ⊂ B and a sequence {λj }j ∈N ⊂ [0, ∞) such that f =



λj mj

j ∈N

in S  (Rn ) and ⎧ ⎫1    # $s ⎨ ⎬s  λj   1 Bj   ⎩ ⎭  1Bj M p,q (Rn )  j ∈N  ω,0

< ∞.

p,q M ω,0 (Rn )

p,q

Moreover, for any f ∈ H M ω,0 (Rn ),

f H M p,q (Rn ) ω,0

⎧ ⎫1  ⎧  ⎪ # $s ⎪ ⎨⎨ ⎬s  λi   ∼ inf  1 Bj  p,q  ⎩  ⎪ ⎭ 1  Bj M (Rn ) ⎪  ⎩ j ∈N ω,0

⎫ ⎪ ⎪ ⎬ p,q

M ω,0 (Rn )

⎪ ⎪ ⎭

with the positive equivalence constants independent of f , where the infimum is taken over all the decompositions of f as above. Next, we establish the molecular characterization of the generalized Herz–Hardy p,q space H K˙ ω (Rn ). To achieve this, we first introduce the following concept of p,q (K˙ ω (Rn ), r, d, τ )-molecules. Definition 4.5.9 Let p, q ∈ (0, ∞), ω ∈ M(R+ ) with m0 (ω) ∈ (− pn , ∞) and M∞ (ω) ∈ (−∞, 0), τ ∈ (0, ∞), r ∈ [1, ∞], and d ∈ Z+ . Then a measurable p,q function m on Rn is called a (K˙ ω (Rn ), r, d, τ )-molecule centered at a ball B ∈ B if (i) for any j ∈ Z+ ,   m1S (B)  r n ≤ 2−j τ j L (R )

|B|1/r ; 1B K˙ p,q n ω (R )

(ii) for any α ∈ Zn+ with |α| ≤ d,  Rn

m(x)x α dx = 0.

Via these molecules, we characterize the generalized Herz–Hardy space p,q p,q H K˙ ω (Rn ) associated with the global generalized Herz space K˙ ω (Rn ) as follows.

4.5 Molecular Characterizations

231

Theorem 4.5.10 Let p, q ∈ (0, ∞), ω ∈ M(R+ ) with m0 (ω) ∈ (− pn , ∞) and −

n < m∞ (ω) ≤ M∞ (ω) < 0, p

(max{1, p, min{m0 (ω),mn∞ (ω)}+n/p }, ∞], s ∈ (0, min{1, p, q, n 1 }), d ≥ n( − 1) be a fixed integer, and τ ∈ (n( 1s − 1r ), ∞). max{M0 (ω),M∞ (ω)}+n/p s p,q Then f ∈ H K˙ ω (Rn ) if and only if f ∈ S  (Rn ) and there exists {λj }j ∈N ⊂ [0, ∞) p,q and a sequence {mj }j ∈N of (K˙ ω (Rn ), r, d, τ )-molecules centered, respectively, r

∈

at the balls {Bj }j ∈N ⊂ B such that

f =



λj mj

j ∈N

in S  (Rn ) and ⎧ ⎫1    # $s  ⎨ ⎬s  λj   1 Bj   ⎩  ⎭ 1Bj K˙ p,q n ω (R )  j ∈N  ˙ p,q

< ∞.

Kω (Rn )

Moreover, there exist two positive constants C1 and C2 such that, for any f ∈ p,q H K˙ ω (Rn ),

C1 f H K˙ p,q n ω (R )

⎧⎧ ⎫1    # $s ⎪ ⎬s  ⎨⎨ λj   ≤ inf  1 Bj  p,q   ⎩ ⎭ ⎪ 1  n Bj K ˙ ω (R ) ⎩ j ∈N  ˙ p,q

Kω (Rn )

⎫ ⎪ ⎬ ⎪ ⎭

≤ C2 f H K˙ p,q n , ω (R ) where the infimum is taken over all the decompositions of f as above. Due to the deficiency of associate spaces, we can not show Theorem 4.5.10 via using the known molecular characterization of the Hardy space HX (Rn ) associated with the ball quasi-Banach function space X (see Lemma 4.5.5 above) directly [see Remark 1.2.19(vi) for the details]. To overcome this obstacle, we first give the following molecular characterization of HX (Rn ) when there is no assumption about the associate space X . Theorem 4.5.11 Let X, r, d, s, and θ be as in Theorem 4.3.18 and τ ∈ (0, ∞) with τ > n( θ1 − 1r ). Then f ∈ HX (Rn ) if and only if f ∈ S  (Rn ) and there exists {λj }j ∈N ⊂ [0, ∞) and a sequence {mj }j ∈N of (X, r, d, τ )-molecules centered,

232

4 Generalized Herz–Hardy Spaces

respectively, at the balls {Bj }j ∈N ⊂ B such that f =



(4.48)

λj mj

j ∈N

in S  (Rn ) and ⎡ ⎤1  

s s    λj ⎣  1Bj ⎦  < ∞.    1Bj X  j ∈N 

(4.49)

X

Moreover, there exist two positive constants C1 and C2 such that, for any f ∈ HX (Rn ),

C1 f HX (Rn )

⎧⎡ ⎤1  

s s ⎪  ⎨  λi ⎣  ≤ inf  1 Bj ⎦    ⎪ 1  Bj X ⎩ j ∈N 

⎫ ⎪ ⎬ ⎪ ⎭

≤ C2 f HX (Rn ) ,

(4.50)

X

where the infimum is taken over all the decompositions of f as above. Remark 4.5.12 We should point out that Theorem 4.5.11 is an improved version of the known molecular characterization obtained in [207, Theorem 3.9]. Indeed, if Y ≡ (X1/s ) in Theorem 4.5.11, then this theorem goes back to [207, Theorem 3.9]. In order to show the above theorem, we require the following pointwise estimate on the radial maximal function of (X, r, d, τ )-molecules. Lemma 4.5.13 Let X be a ball quasi-Banach function space, r ∈ (1, ∞], θ ∈ (0, 1], d ≥ n(1/θ − 1) be a fixed integer, and τ ∈ (n( θ1 − 1r ), ∞). Assume that φ ∈ S(Rn ) satisfies supp (φ) ⊂ B(0, 1). Then there exists a positive constant C such that, for any (X, r, d, τ )-molecule m centered at the ball B ∈ B, ⎡ M(m, φ) ≤ C ⎣M(m)14B +

∞ 

. M m1(2j+1 B)\(2j−2 B) 1(2j B)\(2j−1 B)

j =3

 1 (θ) + M (1B ) , 1B X where M is the radial maximal function defined as in Definition 4.1.1(i). Proof Let all the symbols be as in the present lemma and B := B(x0 , r0 ) with x0 ∈ Rn and r0 ∈ (0, ∞). Then, from Lemma 4.3.21 with  and f therein replaced,

4.5 Molecular Characterizations

233

respectively, by φL∞ (Rn ) 1B(0,1) and m, we deduce that, for any x ∈ 4B, M(m, φ)(x) = sup |m ∗ φt (x)|  M(m)(x), t ∈(0,∞)

(4.51)

which is the desired estimate of M(m, φ)(x) when x ∈ 4B. Next, we estimate M(m, φ)(x) for any x ∈ Sj (B) with j ∈ N ∩ [3, ∞). To this end, let j ∈ N ∩ [3, ∞), x ∈ Sj (B), t ∈ (0, ∞), and 6 5  1 −1 . dθ := n θ Then, by Definition 4.5.3(ii), we find that |m ∗ φt (x)|      φt (x − y)m(y) dy  = Rn

  ⎡ ⎤        x−· γ  1  ⎢ ⎥ ∂ (φ( t ))(x0 ) γ⎥  ⎢φ x − y − =  n (y − x m(y) dy ) 0 ⎦  ⎣ t γ!  t Rn  n γ ∈Z+   |γ |≤d     1  x−y ≤ n  φ t t 2j−2 B −



    ⎥  (y − x0 )γ ⎥ m(y) dy  ⎦ γ!         · · ·  +  · · ·  j−2 j−2  ⎤

∂ γ (φ( x−· t ))(x0 )

γ ∈Zn + |γ |≤dθ

  + 

θ

(2j+1 B)\(2

B)

(2

B)

         x−· γ    ∂ (φ( t ))(x0 ) γ φ x − y − ≤ (y − x ) 0  |m(y)| dy  t γ! 2j−2 B   γ ∈Zn +   |γ |≤dθ      + φt (x − y)m(y) dy  (2j+1 B)\(2j−2 B)

1 + n t



(2j+1 B)

     φ x − y  |m(y)| dy    t

234

4 Generalized Herz–Hardy Spaces

     γ    ∂ (φ( x−· ))(x ) 1 0 t γ  (y − x0 )  |m(y)| dy + n t (2j−2 B)  n γ!  +  |γγ ∈Z  |≤d 

θ

=: A1 + A2 + A3 + A4 .

(4.52)

We next estimate A1 , A2 , A3 , and A4 , respectively. First, we deal with A1 . Indeed, from the Taylor remainder theorem and the Tonelli theorem, it follows that, for any y ∈ 2j −2 B, there exists a ty ∈ (0, 1) such that         j −2 x−· γ   ∂ (φ( t ))(ty + (1 − ty )x0 ) 1 γ  A1 = − x ) (y 0   n t Sk (B)  γ!  ∈Zn k=0 +  |γγ|=d  +1 θ

× |m(y)| dy 

j −2   k=0

    γ  ∂ φ x − ty − (1 − ty )x0  |y − x0 |dθ +1 |m(y)| dy   t Sk (B)

 × 

γ ∈Zn + |γ |=dθ +1

1 t n+dθ +1

j −2   k=0 Sk (B)

|y − x0 |dθ +1 |m(y)| dy. |x − ty − (1 − ty )x0 |n+dθ +1

Observe that, for any y ∈ 2j −2 B, we have |y − x0 | < 2j −2 r0 ≤

1 |x − x0 |, 2

which, combined with the assumption ty ∈ (0, 1), further implies that   x − ty − (1 − ty )x0  ≥ |x − x0 | − |y − y0 | > 1 |x − x0 |. 2 Using this and the assumption n(1/θ − 1) < dθ + 1, we conclude that A1 



j −2   k=0 Sk (B)

|y − x0 |dθ +1 |m(y)| dy |x − x0 |n+dθ +1

j −2  

|y − x0 | θ −n

k=0 Sk (B)

n

|x − x0 |

n θ

dy 

n j −2   (2k r0 ) θ −n  m1S (B)  1 n . n k L (R ) θ k=0 |x − x0 |

(4.53)

4.5 Molecular Characterizations

235

Notice that, for any k ∈ Z+ , by the Hölder inequality and Definition 4.5.3(i), we find that     m1S (B)  1 n ≤ |Sk (B)|1− 1r m1S (B)  r n k k L (R ) L (R ) 1

1

≤ |Sk (B)|1− r 2−kτ

rn n |B| r ∼ 2−k(τ −n+ r ) 0 . 1B X 1B X

(4.54)

From this, (4.53), and the assumption τ ∈ (n( θ1 − 1r ), ∞), we deduce that n

n

j −2  r0θ r0θ 1 1 −k(τ − θn + nr ) A1  2 ∼ n n , 1B X |x − x0 | θ 1B X |x − x0 | θ k=0

(4.55)

which is the desired estimate of A1 . Next, we estimate A2 . Indeed, applying Lemma 4.3.21 with f m1(2j+1 B)\(2j−2 B) and  := φL∞ (Rn ) 1B(0,1), we conclude that    . A2 =  m1(2j+1 B)\(2j−2 B) ∗ φ(x)  M m1(2j+1 B)\(2j−2 B) (x).

:=

(4.56)

This finishes the estimate of A2 . We now deal with A3 . Notice that, for any y ∈ (2j +1 B) , we have |y − x0 | ≥ 2j +1 r0 > 2|x − x0 |, which further implies that |x − y| ≥ |y − x0 | − |x − x0 | > |x − x0 |. From this, the Tonelli theorem, and (4.54), it then follows that A3 

∞   k=j +2 Sk (B)



∞  k=j +2

∞    1 1 m1S (B)  1 n |m(y)| dy  k L (R ) n n |x − y| |x − x0 | k=j +2

2−k(τ + r ) n

1 (2k r0 )n . 1B X |x − x0 |n

In addition, for any k ∈ N ∩ [j + 2, ∞), we have |x − x0 | < 2j r0 ≤ 2k−2 r0 ,

236

4 Generalized Herz–Hardy Spaces

which implies that and τ ∈

(n( θ1

−

1 r ), ∞),

∞ 

A3 

2k r0 |x−x0 |

2

∈ [4, ∞). This, together with the assumptions θ ∈ (0, 1]

further implies that

−k(τ + nr )

k=j +2

1 1B X



2k r0 |x − x0 |

 nθ

n

n

∞  r0θ r0θ n n 1 1 ∼ 2−k(τ − θ + r ) ∼ n n , 1B X |x − x0 | θ 1B X |x − x0 | θ k=j +2

(4.57)

which is the desired estimate of A3 . Finally, we turn to estimate A4 . Indeed, for any y ∈ (2j −2 B) , it holds true that |y − x0 | ≥ 2j −2 r0 > which further implies that

4|y−x0 | |x−x0 |

1 |x − x0 |, 4

∈ (1, ∞). Applying this, the assumption dθ ≤

n(1/θ − 1), (4.54), and the assumption τ ∈ (n( θ1 − 1r ), ∞), we conclude that A4 

∞   k=j −1



t n+|γ |

∞    k=j −1



γ ∈Zn + |γ |≤dθ

1

γ ∈Zn + |γ |≤dθ

∞  

Sk (B)

 Sk (B)

     γ ∂ φ x − x0  |y − x0 ||γ | |m(y)| dy   t

|y − x0 ||γ | dy |x − x0 |n+|γ | n

|y − x0 | θ −n n

|x − x0 | θ

k=j −1 Sk (B)

|m(y)| dy

n ∞   (2k r0 ) θ −n  m1S (B)  1 n  n k L (R ) θ k=j −1 |x − x0 | n

n

∞  r0θ r0θ 1 1 −k(τ − θn + nr )  2 ∼ n n , 1B X |x − x0 | θ 1B X |x − x0 | θ k=j −1

which is the desired estimate of A4 . Combining this, (4.52), (4.55), (4.56), (4.57), and an argument similar to that used in the estimation of (4.23) with rB and xB therein replaced, respectively, by r0 and x0 , we find that, for any x ∈ Sj (B) with

4.5 Molecular Characterizations

237

j ∈ N ∩ [3, ∞) and t ∈ (0, ∞), . |m ∗ φt (x)|  M m1(2j+1 B)\(2j−2 B) (x) + .  M m1(2j+1 B)\(2j−2 B) (x) +

n

r0θ 1 n 1B X |x − x0 | θ 1 M(θ) (1B ) (x), 1B X

which, together with the arbitrariness of t and (4.51), further implies that M(m, φ)  M(m)14B +

∞ 

. M m1(2j+1 B)\(2j−2 B) 1(2j B)\(2j−1 B)

j =3

+

1 M(θ) (1B ) . 1B X  

This then finishes the proof of Lemma 4.5.13. We now show Theorem 4.5.11.

Proof of Theorem 4.5.11 Let all the symbols be as in the present theorem. We first prove the necessity. Indeed, let f ∈ HX (Rn ). Then, applying Theorem 4.3.18, we find that, under the assumptions of the present theorem, HX (Rn ) = H X,r,d,s (Rn ) with equivalent quasi-norms. This implies that f ∈ H X,r,d,s (Rn ). Therefore, there exists a sequence {λj }j ∈N ⊂ [0, ∞) and {aj }j ∈N of (X, r, d)-atoms supported, respectively, in the balls {Bj }j ∈N ⊂ B such that f =



(4.58)

λj aj

j ∈N

in S  (Rn ) and ⎡ ⎤1  

s s    λj ⎣  ⎦ 1Bj  < ∞.    1Bj X  j ∈N 

(4.59)

X

Moreover, by Remark 4.5.4, we conclude that, for any j ∈ N, aj is a (X, r, d, τ )molecule centered at the ball Bj . This, combined with (4.58) and (4.59), then finishes the proof of the necessity. In addition, from the choice of {λj }j ∈N , Definition 4.3.17, and Theorem 4.3.18, we deduce that ⎧⎡ ⎤1  

s s ⎪  ⎨  λj ⎣  inf  1 Bj ⎦  ⎪   1  Bj X ⎩ j ∈N 

⎫ ⎪ ⎬ ⎪ ⎭ X

≤ f H X,r,d,s (Rn ) ∼ f HX (Rn ) ,

(4.60)

238

4 Generalized Herz–Hardy Spaces

where the infimum is taken over all the sequences {λj }j ∈N ⊂ [0, ∞) and {mj }j ∈N of (X, r, d, τ )-molecules centered, respectively, at the balls {Bj }j ∈N ⊂ B such that (4.48) and (4.49) hold true. sufficiency. To this end, let f ∈ S  (Rn ) satisfy f = 4 Next, we show the n j ∈N λj mj in S (R ), where {λj }j ∈N ⊂ [0, ∞) and {mj }j ∈N is a sequence of (X, r, d, τ )-molecules centered, respectively, at the balls {Bj }j ∈N ⊂ B such that ⎡ ⎤1  

s s    λj ⎣  ⎦ 1Bj  < ∞.    1Bj X  j ∈N 

(4.61)

X

To show the sufficiency, we only need to prove that f ∈ HX (Rn ). / To achieve this, we choose a φ ∈ S(Rn ) satisfying that supp (φ) ⊂ B(0, 1) and Rn φ(x) dx = 0. Then, by Lemma 4.1.4, we find that, to prove f ∈ HX (Rn ), it suffices to show 4 M(f, φ)X < ∞. Indeed, applying both the assumption that f = j ∈N λj mj in S  (Rn ) and an argument similar to that used in the proof of (4.25) with aj therein replaced by mj for any j ∈ N, we conclude that M(f, φ) ≤



λj M(mj , φ).

j ∈N

From this, the assumption d ≥ n(1/θ − 1), and Lemma 4.5.13 with m replaced by mj for any j ∈ N, we deduce that        M(f, φ)X   λ M(m )1 j j 4Bj    j ∈N X     ∞

    + λ M m 1 1 k+1 k−2 k k−1 j j (2 Bj )\(2 Bj ) (2 Bj )\(2 Bj )   j ∈N k=3  X      λj  + M(θ) (1Bj )   j ∈N 1Bj X  X

=: III1 + III2 + III3 .

(4.62)

We first estimate III1 . Indeed, applying Definition 4.5.3(i) and an argument similar to that used in the estimation of II1 in the proof of Theorem 4.3.18 with

4.5 Molecular Characterizations

239

M(aj )12Bj therein replaced by M(mj )14Bj for any j ∈ N, we find that ⎡ ⎤1  

s s    λj  ⎣ III1   1 Bj ⎦  .   1Bj X   j ∈N

(4.63)

X

This is the desired estimate of III1 . Next, we deal with III2 . Indeed, from the Lr (Rn ) boundedness of M and Definition 4.5.3(i), it follows that, for any j ∈ N and k ∈ N ∩ [3, ∞),  

  M mj 1(2k+1Bj )\(2k−2 Bj ) 1(2k Bj )\(2k−1Bj ) 

Lr (Rn )

     mj 1(2k+1Bj )\(2k−2 Bj ) 

Lr (Rn )

k+1      mj 1(2l Bj )\(2l−1 Bj ) 



Lr (Rn )

l=k−1

$ # 1/r n 12k Bj X |B | |2k Bj |1/r j  2−kτ ∼ 2−k(τ + r ) , 1Bj X 1Bj X 12k Bj X where the implicit positive constants are independent of both j and k. This implies that there exists a positive constant C such that, for any j ∈ N and k ∈ N ∩ [3, ∞),  

  M mj 1(2k+1Bj )\(2k−2 Bj ) 1(2k Bj )\(2k−1Bj )  # n

≤ C 2−k(τ + r )

12k Bj X

$

1Bj X

Lr (Rn )

|2k Bj |1/r . 12k Bj X

(4.64)

For any j ∈ N and k ∈ N ∩ [3, ∞), let μj,k := C2−k(τ + r ) n

12k Bj X 1Bj X

and aj,k := C −1 2k(τ + r ) n

1Bj X 12k Bj X

M mj 1(2k+1 Bj )\(2k−2Bj ) 1(2k Bj )\(2k−1Bj ) .

Then, by (4.64), we conclude that, for any j ∈ N and k ∈ N ∩ [3, ∞), supp (aj,k ) ⊂ 2k Bj and k 1/r   aj,k  r n ≤ |2 Bj | . L (R ) 2k Bj X

240

4 Generalized Herz–Hardy Spaces

These, together with the definitions of both μj,k and aj,k , and an argument similar to that used in the estimation of II1 in the proof of Theorem 4.3.18 with {λj }j ∈N and {M(aj )12Bj }j ∈N therein replaced, respectively, by   λj μj,k j ∈N,k∈N∩[3,∞) and ! aj,k 12k Bj

j ∈N,k∈N∩[3,∞)

,

further imply that     ∞     III2 ∼  λ μ a j j,k j,k    j ∈N k=3 X ⎡ ⎤1  

s s ∞    λj μj,k  ⎣  1 2 k Bj ⎦    12k Bj X   j ∈N k=3 X ⎡ ⎤1  

s ∞ s     n λj ⎣  ∼ 2−ks(τ + r ) 12k Bj ⎦  .   1Bj X k=3  j ∈N 

(4.65)

X

In order to complete the estimation of III2 , we now estimate the characteristic function 12k Bj with j ∈ N and k ∈ N ∩ [3, ∞). Indeed, from (1.55), we deduce that, for any j ∈ N, k ∈ N ∩ [3, ∞), and x ∈ 2k Bj , 

M

(θ)

-

.

1Bj (x)

s

 ≥  ∼

1 k |2 Bj | |Bj | |2k Bj |



 2 k Bj

s

θ

1Bj (y)

∼ 2−

nks θ

θ

s

θ

dy

,

which implies that 1 2 k Bj  2

nks θ



- .s M(θ) 1Bj .

Combining this and the assumption τ ∈ (n( θ1 − 1r ), ∞), we further conclude that ∞  k=3

n

2−ks(τ + r ) 12k Bj 

∞  k=3

 - .s  (θ) - .s n n 1 Bj 2−ks(τ − θ + r ) M(θ) 1Bj ∼ M .

4.5 Molecular Characterizations

241

Using this, (4.65), and Definition 4.3.17(i) with fj therein replaced by for any j ∈ N, we find that ⎧   #

$s ⎫ 1s  ⎨  ⎬ λj   M(θ) III2   1 Bj  ⎩ ⎭  1Bj X  j ∈N  X ⎡  1 ⎤ 

s s    λj ⎣   1 Bj ⎦  ,   1Bj X  j ∈N 

λj 1Bj X 1Bj

(4.66)

X

which completes the estimation of III2 . Finally, for the term III3 , from (4.30), we infer that ⎡ ⎤1  

s s    λj ⎣  III3   1 Bj ⎦  .   1Bj X  j ∈N 

(4.67)

X

This is the desired estimate of III3 . Thus, combining (4.62), (4.63), (4.66), (4.67), and (4.61), we conclude that ⎡ ⎤1  

s s    λj ⎣  M(f, φ)X   1Bj ⎦  < ∞,   1Bj X  j ∈N 

(4.68)

X

which further implies that f ∈ HX (Rn ) and hence completes the proof of the sufficiency. Moreover, from Lemma 4.1.4, (4.68), and the choice of {λj }j ∈N , it follows that f HX (Rn ) ∼ M(f, φ)X ⎧⎡ ⎤1  

s s ⎪ ⎨   λj ⎣   inf  1 Bj ⎦  ⎪   1  Bj X ⎩ j ∈N 

⎫ ⎪ ⎬ ⎪ ⎭

,

X

where the infimum is taken over all the sequences {λj }j ∈N ⊂ [0, ∞) and {mj }j ∈N of (X, r, d, τ )-molecules centered, respectively, at the balls {Bj }j ∈N ⊂ B such that (4.48) and (4.49) hold true. This, together with (4.60), implies that (4.50) holds true, which completes the proof of Theorem 4.5.11.   Via the above molecular characterization of HX (Rn ), we next prove Theorem 4.5.10.

242

4 Generalized Herz–Hardy Spaces

Proof of Theorem 4.5.10 Let p, q, ω, r, d, s, and τ be as in the present theorem. Then, combining the assumptions, m0 (ω) ∈ (− pn , ∞) and M∞ (ω) ∈ (−∞, 0), and p,q Theorem 1.2.44, we find that the global generalized Herz space K˙ ω (Rn ) under consideration is a BQBF space. Therefore, in order to show the present theorem, we only need to prove that all the assumptions of Theorem 4.5.11 are satisfied. Indeed, let θ ∈ (0, s) be such that 6   5  6 5  1 1 1 −1 ≤n −1 < n −1 +1 n s θ s and  1 1 − . τ >n θ r 

Thus, we have d ≥ n(1/s − 1) = n(1/θ − 1). Then, from (4.40), (4.42), and (4.43), it follows that the following three statements hold true: (i) for any {fj }j ∈N ⊂ L1loc (Rn ), ⎧ ⎫1/s    ⎨  s ⎬    M(θ) (fj )   ⎩ ⎭   j ∈N  ˙ p,q

Kω (Rn )

⎛ ⎞1/s       ⎝   |fj |s ⎠     j ∈N  ˙ p,q

;

Kω (Rn )

(ii) for any f ∈ M (Rn ),  f [K˙ p,q ∼ sup fgL1 (Rn ) : g ˙(p/s) ,(q/s) n 1/s ω (R )] B1/ωs

(Rn )

 =1 ;

(iii) for any f ∈ L1loc (Rn ),     ((r/s)) (f ) ˙ (p/s) ,(q/s) M B1/ωs

(Rn )

 f  ˙ (p/s) ,(q/s) B1/ωs

(Rn )

.

These further imply that, under the assumptions of the present theorem, K˙ ω (Rn ) satisfies all the assumptions of Theorem 4.5.11, which completes the proof of Theorem 4.5.10.   p,q

Similarly, we now characterize the generalized Hardy–Morrey space p,q H M ω (Rn ) via molecules. To this end, we first introduce the following p,q (M ω (Rn ), r, d, τ )-molecules.

4.5 Molecular Characterizations

243

Definition 4.5.14 Let p, q, ω, r, d, and τ be as in Definition 4.5.7. Then a p,q measurable function m on Rn is called an (M ω (Rn ), r, d, τ )-molecule centered at a ball B ∈ B if (i) for any j ∈ Z+ ,  m1S

 

j (B) Lr (Rn )

≤ 2−τj

|B|1/r ; 1B M p,q n ω (R )

(ii) for any α ∈ Zn+ with |α| ≤ d,  Rn

m(x)x α dx = 0.

Then we immediately obtain the following molecular characterization of p,q the generalized Hardy–Morrey space H M ω (Rn ) via Theorem 4.5.10 and Remarks 1.2.2(iv) and 4.0.20(ii); we omit the details. Corollary 4.5.15 Let p, q, ω, r, d, and s be as in Corollary 4.3.13 and τ ∈ (0, ∞) p,q with τ > n(1/s − 1/r). Then f ∈ H M ω (Rn ) if and only if f ∈ S  (Rn ) p,q and there exists a sequence {mj }j ∈N of (M ω (Rn ), r, d, τ )-molecules centered, respectively, at the balls {Bj }j ∈N ⊂ B and a sequence {λj }j ∈N ⊂ [0, ∞) such that f =



λj mj

j ∈N

in S  (Rn ) and ⎧ ⎫1    # $s ⎬s  ⎨ λj   1 Bj   p,q ⎭ ⎩  1Bj M ω (Rn )  j ∈N 

< ∞. p,q

M ω (Rn )

p,q

Moreover, for any f ∈ H M ω (Rn ),

f H M p,q n ω (R )

⎧⎧ ⎫1    # $s ⎪ ⎨⎨ ⎬s  λi   ∼ inf  1 Bj  p,q ⎪ ⎭   ⎩ 1  n B ⎩ j ∈N j M ω (R ) 

⎫ ⎪ ⎬ p,q

M ω (Rn )

⎪ ⎭

with the positive equivalence constants independent of f , where the infimum is taken over all the decompositions of f as above.

244

4 Generalized Herz–Hardy Spaces

4.6 Littlewood–Paley Function Characterizations In this section, we establish various Littlewood–Paley function characterizations of generalized Herz–Hardy spaces. To be precise, we characterize generalized Herz– Hardy spaces via the Lusin area function, the Littlewood–Paley g-function, and the Littlewood–Paley gλ∗ -function. Throughout this book, for any ϕ ∈ S(Rn ), F ϕ or  ϕ always denotes its Fourier transform, namely, for any x ∈ Rn ,  F ϕ(x) :=  ϕ (x) :=

Rn

ϕ(ξ )e−2πix·ξ dξ,

√ 4n where i := −1 and x · ξ := j =1 xj ξj for any x := (x1 , . . . , xn ), ξ := (ξ1 , . . . , ξn ) ∈ Rn . Moreover, for any f ∈ S  (Rn ), its Fourier transform, also denoted by F f or f, is defined by setting, for any φ ∈ S(Rn ), ; < ; < . F f, φ := f, φ := f, φ Then we state the definitions of the Lusin area function, the Littlewood–Paley gfunction, and the Littlewood–Paley gλ∗ -function as follows (see, for instance, [29, Definitions 4.1 and 4.2]). Definition 4.6.1 Let ϕ ∈ S(Rn ) satisfy  ϕ (0) = 0 and, for any ξ ∈ Rn \ {0}, there exists a t ∈ (0, ∞) such that  ϕ (tξ ) = 0. Then, for any f ∈ S  (Rn ), the Lusin area function S(f ) and the Littlewood–Paley gλ∗ -function gλ∗ (f ) with λ ∈ (0, ∞) are defined, respectively, by setting, for any x ∈ Rn ,  S(f )(x) :=

|f ∗ ϕt (y)|2 (x)

dy dt t n+1

1 2

and gλ∗ (f )(x)

#



∞

:= 0

Rn

t t + |x − y|

λn

dy dt |f ∗ ϕt (y)| n+1 t 2

$1 2

,

here and thereafter, for any x ∈ Rn , (x) := {(y, t) ∈ Rn+1 + : |y − x| < t}. Definition 4.6.2 Let ϕ ∈ S(Rn ) satisfy  ϕ (0) = 0 and, for any x ∈ Rn \ {0}, there j exists a j ∈ Z such that  ϕ (2 x) = 0. Then, for any f ∈ S  (Rn ), the Littlewood–

4.6 Littlewood–Paley Function Characterizations

245

Paley g-function g(f ) is defined by setting, for any x ∈ Rn , 

∞

g(f )(x) :=

|f ∗ ϕt (x)|2

0

dt t

1 2

.

Recall that f ∈ S  (Rn ) is said to vanish weakly at infinity if, for any φ ∈ S(Rn ), f ∗ φt → 0 in S  (Rn ) as t → ∞ with φt (·) = t −n φ(·/t). Now, we establish various Littlewood–Paley function characterizations of the generalized Herz–Hardy spaces p,q p,q H K˙ ω,0 (Rn ) and H K˙ ω (Rn ) as follows. To begin with, we give the Littlewood– p,q Paley function characterizations of H K˙ ω,0 (Rn ). Theorem 4.6.3 Let p, q ∈ (0, ∞), ω ∈ M(R+ ) satisfy m0 (ω) ∈ (− pn , ∞) and m∞ (ω) ∈ (− pn , ∞),  s0 := min 1, p, q,

 n , max{M0 (ω), M∞ (ω)} + n/p

and λ ∈ (max{1, 2/s0 }, ∞). Then the following four statements are mutually equivalent: (i) (ii) (iii) (iv)

f f f f

p,q ∈ H K˙ ω,0 (Rn ); p,q ∈ S  (Rn ), f vanishes weakly at infinity, and S(f ) ∈ K˙ ω,0 (Rn ); p,q ∈ S  (Rn ), f vanishes weakly at infinity, and g(f ) ∈ K˙ ω,0 (Rn ); p,q  n ∗ ∈ S (R ), f vanishes weakly at infinity, and gλ (f ) ∈ K˙ ω,0 (Rn ).

Moreover, for any f ∈ H K˙ ω,0 (Rn ), p,q

  f H K˙ p,q (Rn ) ∼ S(f )K˙ p,q (Rn ) ∼ g(f )K˙ p,q (Rn ) ∼ gλ∗ (f )K˙ p,q (Rn ) , ω,0

ω,0

ω,0

ω,0

where the positive equivalence constants are independent of f . To show Theorem 4.6.3, recall that Chang et al. [29, Theorems 4.9, 4.11, and 4.13] investigated the Lusin area function, the g-function, and the gλ∗ -function characterizations of the Hardy space HX (Rn ) as follows, which is vital in the proof p,q of the Littlewood–Paley function characterizations of H K˙ ω,0 (Rn ). Lemma 4.6.4 Let s ∈ (0, 1], θ ∈ (0, s), λ ∈ (max{1, 2s }, ∞), and X be a ball quasi-Banach function space. Assume that Assumption 1.2.29 holds true for both X and Xs/2 with θ and s as above and X satisfies Assumption 1.2.33 with s as above. Then the following four statements are mutually equivalent: (i) (ii) (iii) (iv)

f f f f

∈ HX (Rn ); ∈ S  (Rn ), f vanishes weakly at infinity, and S(f ) ∈ X; ∈ S  (Rn ), f vanishes weakly at infinity, and g(f ) ∈ X; ∈ S  (Rn ), f vanishes weakly at infinity, and gλ∗ (f ) ∈ X.

246

4 Generalized Herz–Hardy Spaces

Moreover, for any f ∈ HX (Rn ),   f HX (Rn ) ∼ S(f )X ∼ g(f )X ∼ gλ∗ (f )X with the positive equivalence constants independent of f . Remark 4.6.5 We point out that Lemma 4.6.4 has a wide range of applications. Here we give several function spaces to which Lemma 4.6.4 can be applied (see also [29, Section 5]). (i) Let p ∈ (0, ∞) and λ ∈ (max{1, p2 }, ∞). Then, in this case, by both Remarks 1.2.31(i) and 1.2.34(i), we can easily conclude that the Lebesgue space Lp (Rn ) satisfies all the assumptions of Lemma 4.6.4. Therefore, Lemma 4.6.4 with X := Lp (Rn ) holds true. This result is the known Littlewood–Paley function characterizations of the Hardy space H p (Rn ) (see, for instance, [85, Chapter 7]). (ii) Let p ∈ (0, ∞), υ ∈ A∞ (Rn ), and λ ∈ (max{1, 2qpυ }, ∞), where qυ is the same as in (1.58). Then, in this case, as was pointed out in [29, Subsection p 5.4], the weighted Lebesgue space Lυ (Rn ) satisfies all the assumptions of p Lemma 4.6.4. This then implies that Lemma 4.6.4 with X := Lυ (Rn ) holds true (see, for instance, [29, Theorems 5.16 and 5.18]). (iii) Let p := (p1 , . . . , pn ) ∈ (0, ∞)n and   λ ∈ max 1,

  2 ,∞ . min{p1 , . . . , pn }

Then, in this case, as was mentioned in [29, Subsection 5.2], the mixed-norm Lebesgue space Lp (Rn ) satisfies all the assumptions of Lemma 4.6.4. Thus, Lemma 4.6.4 with X := Lp (Rn ) holds true. These Littlewood–Paley function characterizations of mixed-norm Hardy spaces were also established in [29, Theorems 5.8 and 5.10] (see also [123, 127]). (iv) Let 0 < q ≤ p < ∞ and λ ∈ (max{1, q2 }, ∞). Then, in this case, as was p pointed out in [29, Subsection 5.1], the Morrey space Mq (Rn ) satisfies all the p assumptions of Lemma 4.6.4. Therefore, Lemma 4.6.4 with X := Mq (Rn ) holds true (see, for instance, [29, Theorems 5.3 and 5.5]). (v) Let p(·) ∈ C log (Rn ) satisfy 0 < p− ≤ p+ < ∞, and λ ∈ (max{1, p2− }, ∞), where p− and p+ are defined, respectively, in (1.59) and (1.60). Then, in this case, as was mentioned in [29, Subsection 5.3], the variable Lebesgue space Lp(·) (Rn ) satisfies all the assumptions of Lemma 4.6.4. This further implies that Lemma 4.6.4 with X := Lp(·) (Rn ) holds true (see, for instance, [29, Theorems 5.12 and 5.14]). Via this lemma, we now show Theorem 4.6.3.

4.6 Littlewood–Paley Function Characterizations

247

Proof of Theorem 4.6.3 Let all the symbols be as in the present theorem. Then, combining the assumption m0 (ω) ∈ (− pn , ∞) and Theorem 1.2.42, we find that p,q the local generalized Herz space K˙ ω,0 (Rn ) is a BQBF space. This implies that, to complete the proof of the present theorem, we only need to show that, under p,q the assumptions of the present theorem, K˙ ω,0 (Rn ) satisfies all the assumptions of Lemma 4.6.4. 2 For this purpose, let s ∈ ( λ2 , s0 ) and θ ∈ (0, min{s, s2 }). We now prove that both p,q p,q K˙ ω,0 (Rn ) and [K˙ ω,0 (Rn )]s/2 satisfy Assumption 1.2.29 with these θ and s. Indeed, applying Lemma 4.3.11, we conclude that, for any {fj }j ∈N ⊂ L1loc (Rn ), ⎧ ⎫1/s    ⎨  s ⎬    (θ) M (fj )   ⎩ ⎭   j ∈N  ˙ p,q

Kω,0 (Rn )

⎛ ⎞1/s       ⎝ s⎠   |fj |     j ∈N  ˙ p,q

.

Kω,0 (Rn )

This implies that, for the above θ and s, Assumption 1.2.29 holds true for K˙ ω,0 (Rn ). p,q

2

On the other hand, from the assumptions θ ∈ (0, min{s, s2 }) and   s ∈ 0, min p, we deduce that

s θ

n max{M0 (ω), M∞ (ω)} + n/p

 ,

∈ (1, ∞) and

 min p,

n max{M0 (ω), M∞ (ω)} + n/p

 >s>

2θ . s

This, together with Lemma 4.3.11 with K˙ ω,0 (Rn ) therein replaced by [K˙ ω,0 (Rn )]s/2, further implies that, for any {fj }j ∈N ⊂ L1loc (Rn ), p,q

⎧ ⎫1/s    ⎨  s ⎬    M(θ) (fj )   ⎭  ⎩  j ∈N 

˙ p,q (Rn )]s/2 [K ω,0

p,q

⎛ ⎞1/s       ⎝   |fj |s ⎠     j ∈N 

,

˙ p,q (Rn )]s/2 [K ω,0

p,q which completes the proof that Assumption 1.2.29 holds true for both K˙ ω,0 (Rn ) and p,q [K˙ ω,0 (Rn )]s/2. In addition, notice that λ ∈ (max{1, 2s }, ∞) and we can choose an

  r ∈ max 1, p,

  n ,∞ . min{m0 (ω), m∞ (ω)} + n/p

248

4 Generalized Herz–Hardy Spaces

Then, by Lemma 1.8.6, we find that [K˙ ω,0 (Rn )]1/s is a BBF space and, for any f ∈ L1loc (Rn ), p,q

   ((r/s))  (f ) M

˙ (Rn )]1/s ) ([K ω,0 p,q

 f ([K˙ p,q (Rn )]1/s ) , ω,0

which further implies that all the assumptions of Lemma 4.6.4 are satisfied for p,q   K˙ ω,0 (Rn ). This finishes the proof of Theorem 4.6.3. Remark 4.6.6 We point out that, in Theorem 4.6.3, when p ∈ (1, ∞), q ∈ [1, ∞), and ω(t) := t α for any t ∈ (0, ∞) and for any given α ∈ (− pn , pn ), from Remark 4.0.18 and Theorem 4.2.1, it follows that the generalized Herz–Hardy space p,q α,q H K˙ ω,0 (Rn ) coincides with the classical homogeneous Herz space K˙ p (Rn ) and hence the conclusion obtained in this theorem goes back to [175, Theorem 1.1.1]. Using Theorem 4.6.3 and Remarks 1.2.2(iv) and 4.0.20(ii), we immediately obtain the following Littlewood–Paley function characterizations of the generalized p,q Hardy–Morrey space H M ω,0 (Rn ); we omit the details. Corollary 4.6.7 Let p, q ∈ [1, ∞), ω ∈ M(R+ ) satisfy −

n < m0 (ω) ≤ M0 (ω) < 0 p

and −

n < m∞ (ω) ≤ M∞ (ω) < 0, p

and λ ∈ (2, ∞). Then the following four statements are mutually equivalent: (i) (ii) (iii) (iv)

f f f f

p,q

∈ H M ω,0 (Rn ); p,q ∈ S  (Rn ), f vanishes weakly at infinity, and S(f ) ∈ M ω,0 (Rn ); p,q  n ∈ S (R ), f vanishes weakly at infinity, and g(f ) ∈ M ω,0 (Rn ); p,q ∈ S  (Rn ), f vanishes weakly at infinity, and gλ∗ (f ) ∈ M ω,0 (Rn ). p,q

Moreover, for any f ∈ H M ω,0 (Rn ),   f H M p,q (Rn ) ∼ S(f )M p,q (Rn ) ∼ g(f )M p,q (Rn ) ∼ gλ∗ (f )M p,q (Rn ) , ω,0

ω,0

ω,0

ω,0

where the positive equivalence constants are independent of f . On the other hand, we now show the following Littlewood–Paley function p,q characterizations of the generalized Herz–Hardy space H K˙ ω (Rn ).

4.6 Littlewood–Paley Function Characterizations

249

Theorem 4.6.8 Let p, q ∈ (0, ∞), ω ∈ M(R+ ) satisfy m0 (ω) ∈ (− pn , ∞) and −

n < m∞ (ω) ≤ M∞ (ω) < 0, p

 s0 := min 1, p, q,

 n , max{M0 (ω), M∞ (ω)} + n/p

and λ ∈ (max{1, 2/s0 }, ∞). Then the following four statements are mutually equivalent: (i) (ii) (iii) (iv)

f f f f

∈ H K˙ ω (Rn ); p,q ∈ S  (Rn ), f vanishes weakly at infinity, and S(f ) ∈ K˙ ω (Rn ); p,q  n ∈ S (R ), f vanishes weakly at infinity, and g(f ) ∈ K˙ ω (Rn ); p,q ∈ S  (Rn ), f vanishes weakly at infinity, and gλ∗ (f ) ∈ K˙ ω (Rn ). p,q

p,q Moreover, for any f ∈ H K˙ ω (Rn ),

 ∗    ˙ p,q n , f H K˙ p,q n ∼ S(f )K n ∼ g(f )K n ∼ gλ (f ) K ˙ p,q ˙ p,q (R ) ω (R ) ω (R ) ω (R ) ω

where the positive equivalence constants are independent of f . To prove Theorem 4.6.8, we first give some symbols. Recall that, for any given ξ ∈ Rn , the translation operator τξ is defined by setting, for any f ∈ M (Rn ) and x ∈ Rn , τξ (f )(x) := f (x − ξ ).

(4.69)

Furthermore, the translation operator τξ of distributions with ξ ∈ Rn is defined by setting, for any f ∈ S  (Rn ) and φ ∈ S(Rn ), ;

< ; < τξ (f ), φ := f, τ−ξ (φ) = f, φ(· + ξ ) .

The following technical lemma establishes the relations among translations, convolutions, and various Littlewood–Paley functions, which plays a key role in the proof of Theorem 4.6.8. Lemma 4.6.9 Let f ∈ S  (Rn ), φ ∈ S(Rn ), and ξ ∈ Rn . Then (i) [τξ (f )] ∗ φ = τξ (f ∗ φ); (ii) M(τξ (f ), φ) = τξ (M(f, φ)), where the radial maximal function M is defined as in Definition 4.1.1(i); (iii) A(τξ (f )) = τξ (A(f )), where A ∈ {S, g, gλ∗ } with λ ∈ (0, ∞); (iv) if f vanishes weakly at infinity, then τξ (f ) vanishes weakly at infinity.

250

4 Generalized Herz–Hardy Spaces

Proof Let all the symbols be as in the present lemma. We first prove (i). Indeed, for any x ∈ Rn , we have 

 ; < τξ (f ) ∗ φ(x) = τξ (f ), φ(x − ·) = f, φ (x − (· + ξ )) = f, φ(x − ξ − ·) = (f ∗ φ) (x − ξ ) = τξ (f ∗ φ)(x),

which implies that (i) holds true. Next, we show (ii). Applying (i) with φ therein replaced by φt for any t ∈ (0, ∞), we find that     .  M τξ (f ), φ = sup  τξ (f ) ∗ φt  = sup τξ (f ∗ φt ) t ∈(0,∞)



t ∈(0,∞)



sup {|f ∗ φt |} = τξ (M(f, φ)) .

= τξ

t ∈(0,∞)

This finishes the proof of (ii). We then prove (iii). First, let A := S. Then, from Definition 4.6.1 and (i) with φ = ϕt for any t ∈ (0, ∞), it follows that, for any x ∈ Rn ,  . 2 S τξ (f ) (x) =

 

    τξ (f ) ∗ ϕt (y)2 dy dt t n+1 (x) |(f ∗ ϕt )(y − ξ )|2

= 

(x)

|f ∗ ϕt (y)|2

= (x−ξ )

dy dt t n+1

 2 dy dt = τξ (S(f ))(x) . n+1 t

Thus, S(τξ (f )) = τξ (S(f )) holds true. Similarly, we can obtain g(τξ (f )) = τξ (g(f )) and gλ∗ (τξ (f )) = τξ (gλ∗ (f )) and hence (iii) holds true. Finally, we prove (iv). Indeed, for any ψ, using (i) with φ therein replaced by ψt for any t ∈ (0, ∞), and the assumption that f vanishes weakly at infinity, we find that, for any η ∈ S(Rn ), 

 Rn

 τξ (f ) ∗ ψt (y)η(y) dy =

 

=

Rn

Rn

(f ∗ ψt ) (y − ξ )η(y) dy (f ∗ ψt ) (y)η(y + ξ ) dy → 0

as t → ∞. This further implies that, for any ψ ∈ S(Rn ), [τξ (f )] ∗ ψt → 0 in S  (Rn ) as t → ∞, which completes the proof of (iv) and hence Lemma 4.6.9.  

4.6 Littlewood–Paley Function Characterizations

251

Now, we show Theorem 4.6.8 via Lemma 4.6.9. Proof of Theorem 4.6.8/ Let p, q, ω, and λ be as in the present theorem, and let φ ∈ S(Rn ) satisfy that Rn φ(x) dx = 0. We now first show that (i) implies (ii). To p,q this end, let f ∈ H K˙ ω (Rn ). Then, applying this and Theorem 4.1.8, we find that p,q M(f, φ) ∈ K˙ ω (Rn ). This, together with Lemma 4.6.9(ii), further implies that, for any ξ ∈ Rn ,    . M τξ (f ), φ  ˙ p,q n = τξ (M(f, φ)) ˙ p,q n K (R ) K (R ) ω,0

ω,0

≤ M(f, φ)K˙ p,q n ∼ f H K n < ∞. ˙ p,q ω (R ) ω (R ) p,q Therefore, for any ξ ∈ Rn , M(τξ (f ), φ) ∈ K˙ ω,0 (Rn ). From this and Theorem 4.1.2, p,q we deduce that, for any ξ ∈ Rn , τξ (f ) ∈ H K˙ ω,0 (Rn ). Combining this and Theorem 4.6.3, we conclude that, for any ξ ∈ Rn , τξ (f ) vanishes weakly at infinity, p,q S(τξ (f )) ∈ K˙ ω,0 (Rn ), and

   . S τξ (f )  ˙ p,q n ∼ τξ (f ) ˙ p,q n . K (R ) H K (R ) ω,0

(4.70)

ω,0

In particular, letting ξ := 0, we have f vanishes weakly at infinity. Then, from Lemma 4.6.9(iii), (4.70), Theorem 4.1.2, and Lemma 4.6.9(ii), we further deduce that, for any ξ ∈ Rn ,    . τξ (S(f )) ˙ p,q n = S τξ (f )  ˙ p,q n Kω,0 (R ) Kω,0 (R )    . ∼ τξ (f )H K˙ p,q (Rn ) ∼ M τξ (f ), φ K˙ p,q (Rn ) ω,0 ω,0   ∼ τξ (M(f, φ))K˙ p,q (Rn ) . ω,0

By this, the definition of  · K˙ p,q n , and Theorem 4.1.8, we find that ω (R ) S(f )K˙ p,q n ∼ M(f, φ)K n ∼ f H K n < ∞, ˙ p,q ˙ p,q ω (R ) ω (R ) ω (R )

(4.71)

p,q which further implies that S(f ) ∈ K˙ ω (Rn ), and hence (i) implies (ii). Conversely, we show that (ii) implies (i), namely, assume f ∈ S  (Rn ) vanishes p,q p,q weakly at infinity and S(f ) ∈ K˙ ω (Rn ), we need to prove that f ∈ H K˙ ω (Rn ). Indeed, from Lemma 4.6.9(iii), we deduce that, for any ξ ∈ Rn ,

 . S τξ (f )  ˙ p,q

Kω,0 (Rn )

  = τξ (S(f ))K˙ p,q (Rn ) ≤ S(f )K˙ p,q n < ∞, ω (R )

(4.72)

ω,0

which implies that S(τξ (f )) ∈ K˙ ω,0 (Rn ). On the other hand, by Lemma 4.6.9(iv), we conclude that, for any ξ ∈ Rn , τξ (f ) vanishes weakly at infinity. Combining this, p,q the fact that S(τξ (f )) ∈ K˙ ω,0 (Rn ) for any ξ ∈ Rn , and Theorem 4.6.3, we further p,q conclude that, for any ξ ∈ Rn , τξ (f ) ∈ H K˙ ω,0 (Rn ). Then, applying Theorem 4.1.8, p,q

252

4 Generalized Herz–Hardy Spaces

Lemma 4.6.9(ii), Theorems 4.1.2 and 4.6.3, and (4.72), we find that   τξ (M(f, φ)) ˙ p,q n f H K˙ p,q n ∼ sup K (R ) ω (R ) ξ ∈Rn

ω,0

   . ∼ sup M τξ (f ), φ K˙ p,q (Rn ) ∼ sup τξ (f )H K˙ p,q (Rn ) ξ ∈Rn

ω,0

ξ ∈Rn

ω,0

 . ∼ sup S τξ (f ) K˙ p,q (Rn )  S(f )K˙ p,q n < ∞, ω (R ) ξ ∈Rn

ω,0

p,q which further implies that f ∈ H K˙ ω (Rn ), and hence (ii) implies (i). Moreover, p,q from (4.71), it follows that, for any f ∈ H K˙ ω (Rn ),

f H K˙ p,q n ∼ S(f )K n . ˙ p,q ω (R ) ω (R ) Similarly, we can obtain (i) is equivalent to both (iii) and (iv) and, for any f ∈ p,q H K˙ ω (Rn ), f H K˙ p,q n ∼ A(f )K n , ˙ p,q ω (R ) ω (R ) where A ∈ {g, gλ∗ }. This then finishes the proof of Theorem 4.6.8.

 

As an application, we then characterize the generalized Hardy–Morrey space p,q H M ω (Rn ) via the Lusin area function, the g-function, and the gλ∗ -function. Namely, the following conclusion holds true, which is a direct corollary of Theorem 4.6.8 and Remarks 1.2.2(iv) and 4.0.20(ii); we omit the details. Corollary 4.6.10 Let p, q, ω, and λ be as in Corollary 4.6.7. Then the following four statements are mutually equivalent: (i) (ii) (iii) (iv)

f f f f

p,q

∈ H M ω (Rn ); p,q ∈ S  (Rn ), f vanishes weakly at infinity, and S(f ) ∈ M ω (Rn ); p,q  n ∈ S (R ), f vanishes weakly at infinity, and g(f ) ∈ M ω (Rn ); p,q ∈ S  (Rn ), f vanishes weakly at infinity, and gλ∗ (f ) ∈ M ω (Rn ). p,q

Moreover, for any f ∈ H M ω (Rn ),  ∗    p,q n f H M p,q n ∼ S(f )M p,q (Rn ) ∼ g(f )M p,q (Rn ) ∼ gλ (f ) M (R ) ω (R ) ω ω ω

with the positive equivalence constants independent of f .

˙ p,q (Rn ) 4.7 Dual Space of H K ω,0 In this section, we investigate the dual space of the generalized Herz–Hardy space p,q H K˙ ω,0 (Rn ). Throughout this book, for any d ∈ Z+ , the symbol Pd (Rn ) denotes the set of all polynomials on Rn with degree not greater than d. Moreover, for any

p,q 4.7 Dual Space of H K˙ ω,0 (Rn )

253

ball B ∈ B and any g ∈ L1loc (Rn ), PBd g denotes the minimizing polynomial of g with degree not greater than d, which means that PBd g is the unique polynomial f ∈ Pd (Rn ) such that, for any h ∈ Pd (Rn ),  [g(x) − f (x)]h(x) dx = 0. B

Now, we introduce the following Campanato-type function spaces associated with local generalized Herz spaces, which were originally introduced in [277, Definition 3.2] for any given general ball quasi-Banach function space X. Definition 4.7.1 Let p, q, s ∈ (0, ∞), r ∈ [1, ∞), d ∈ Z+ , and ω ∈ M(R+ ) p,q,r,d,s with m0 (ω) ∈ (− pn , ∞). Then the Campanato-type function space L˙ω,0 (Rn ), p,q associated with the local generalized Herz space K˙ ω,0 (Rn ), is defined to be the set of all the f ∈ Lrloc (Rn ) such that f L˙ p,q,r,d,s (Rn ) ω,0

 # $s  1s  −1     m λi := sup  1 Bi    n  ˙ p,q  i=1 1Bi K˙ p,q ω,0 (R ) ×

Kω,0 (Rn )

⎧ m ⎨  j =1

λj |Bj | ⎩ 1Bj K˙ p,q (Rn ) ω,0

#

1 |Bj |

 Bj

r    f (x) − PBdj f (x) dx

$1 ⎫ r⎬ ⎭

m is finite, where the supremum is taken over all m ∈ N, {Bj }m j =1 ⊂ B, and {λj }j =1 ⊂ 4m [0, ∞) with j =1 λj = 0.

Remark 4.7.2 By Definition 4.7.1, we can easily show that Pd (Rn ) ⊂ p,q,r,d,s p,q,r,d,s (Rn ) and, for any f ∈ L˙ ω,0 (Rn ), f L˙ p,q,r,d,s (Rn ) = 0 if and only if L˙ ω,0 ω,0

p,q,r,d,s f ∈ Pd (Rn ). Therefore, in what follows, we always identify f ∈ L˙ω,0 (Rn ) n with {f + P : P ∈ Pd (R )}.

Applying [277, Remark 3.3(iii) and Proposition 3.4] with X therein replaced p,q by K˙ ω,0 (Rn ), we immediately obtain the following equivalent characterizations of these Campanato-type function spaces; we omit the details.

254

4 Generalized Herz–Hardy Spaces

Proposition 4.7.3 Let p, q, ω, r, d, and s be as in Definition 4.7.1. Then the following three statements are equivalent: p,q,r,d,s (i) f ∈ L˙ ω,0 (Rn ); r n (ii) f ∈ L loc (R ) and

f ˙ p,q,r,d,s Lω,0

(Rn )

 $s  1s   m # −1    λ i := sup inf  1 Bi    p,q  i=1 1Bi K˙ ω,0 (Rn )  ˙ p,q ×

Kω,0 (Rn )

⎧ m ⎨  j =1

λj |Bj | ⎩ 1Bj K˙ p,q (Rn )

#

ω,0

1 |Bj |

 |f (x) − P (x)|r dx

$1 ⎫ r⎬

Bj

⎭

is finite, where the supremum is the same as in Definition 4.7.1 and the infimum is taken over all P ∈ Pd (Rn ); (iii) f ∈ Lrloc (Rn ) and = ˙ p,q,r,d,s n f L (R ) ω,0

 # $s  1s  −1     λ i := sup  1 Bi    n  ˙ p,q  i∈N 1Bi K˙ p,q ω,0 (R ) ×

Kω,0 (Rn )

⎧ ⎨ j ∈N

λj |Bj | ⎩ 1Bj K˙ p,q (Rn ) ω,0

#

1 |Bj |

 Bj

 r   f (x) − PBdj f (x) dx

$1 ⎫ r⎬ ⎭

is finite, where the supremum is taken over all {Bj }j ∈N ⊂ B and {λj }j ∈N ⊂ [0, ∞) satisfying  # $s  1s       λ i  1 Bi    p,q 1  Bi K ˙ (Rn )  i∈N  ˙ p,q ω,0

Kω,0

∈ (0, ∞). (Rn )

Moreover, there exist two positive constants C1 and C2 such that, for any f ∈ Lrloc (Rn ), C1 f L˙ p,q,r,d,s (Rn ) ≤ f ˙ p,q,r,d,s ω,0

Lω,0

(Rn )

≤ C2 f L˙ p,q,r,d,s (Rn ) ω,0

p,q 4.7 Dual Space of H K˙ ω,0 (Rn )

255

and = ˙ p,q,r,d,s n = f  ˙ p,q,r,d,s n . f L (R ) L (R ) ω,0

ω,0

Via the known dual theorem of the Hardy space HX (Rn ) associated with the ball quasi-Banach function space X, we now show that the dual space of the p,q generalized Herz–Hardy space H K˙ ω,0 (Rn ) is just the Campanato-type function  p,q,r ,d,s space L˙ (Rn ). Namely, we have the following theorem. ω,0

Theorem 4.7.4 Let p, q ∈ (0, ∞), ω ∈ M(R+ ) with m0 (ω) ∈ (− pn , ∞) and m∞ (ω) ∈ (− pn , ∞),  p− := min 1, p,

 n , max{M0 (ω), M∞ (ω)} + n/p

d ≥ n(1/p− − 1) be a fixed integer, s ∈ (0, min{p− , q}), and   r ∈ max 1, p,

  n ,∞ . min{m0 (ω), m∞ (ω)} + n/p



p,q,r ,d,s p,q Then L˙ ω,0 (Rn ) is the dual space of H K˙ ω,0 (Rn ) in the following sense: p,q,r  ,d,s

(i) Let g ∈ L˙ ω,0

(Rn ). Then the linear functional  Lg : f → Lg (f ) :=

Rn

(4.73)

f (x)g(x) dx,

p,q,r,d,s initially defined for any f ∈ H K˙ ω,0,fin (Rn ), has a bounded extension to the p,q generalized Herz–Hardy space H K˙ ω,0 (Rn ). p,q (ii) Conversely, any continuous linear functional on H K˙ ω,0 (Rn ) arises as in (4.73)  p,q,r ,d,s with a unique g ∈ L˙ (Rn ). ω,0

Moreover, there exist two positive constants C1 and C2 such that, for any g ∈ p,q,r  ,d,s L˙ ω,0 (Rn ), C1 g ˙ p,q,r  ,d,s Lω,0

(Rn )

≤ Lg (H K˙ p,q (Rn ))∗ ≤ C2 g ˙ p,q,r  ,d,s ω,0

Lω,0

p,q p,q where (H K˙ ω,0 (Rn ))∗ denotes the dual space of H K˙ ω,0 (Rn ).

(Rn )

,

256

4 Generalized Herz–Hardy Spaces

To show this theorem, we first recall the definition of the ball Campanato-type function space LX,r,d,s (Rn ) associated with the general ball quasi-Banach function space X as follows, which is just [277, Definition 3.2]. Definition 4.7.5 Let X be a ball quasi-Banach function space, r ∈ [1, ∞), s ∈ (0, ∞), and d ∈ Z+ . Then the ball Campanato-type function space LX,r,d,s (Rn ), associated with X, is defined to be the set of all the f ∈ Lrloc (Rn ) such that f LX,r,d,s (Rn ) # −1 $1     s s   m λ i := sup  1 Bi      i=1 1Bi X X ⎧ $1 ⎫ #   m ⎨ r⎬ r  λj |Bj | 1   × f (x) − PBdj f (x) dx ⎩ 1Bj X |Bj | Bj ⎭ j =1

m is finite, where the supremum is taken over all m ∈ N, {Bj }m j =1 ⊂ B, and {λj }j =1 ⊂ 4m [0, ∞) with j =1 λj = 0.

Notice that Zhang et al. [277, Theorem 3.14] established the following duality between the Hardy space HX (Rn ) and the ball Campanato-type function space LX,r  ,d,s (Rn ), which is an essential tool for us to show Theorem 4.7.4. Lemma 4.7.6 Let X be a ball quasi-Banach function space satisfy: (i) there exists a p− ∈ (0, ∞) such that, for any given θ ∈ (0, p− ) and u ∈ (1, ∞), and for any {fj }j ∈N ⊂ L1loc (Rn ), ⎧ ⎫1   u ⎨  u ⎬    M(fj )   ⎭  ⎩  j ∈N 

X 1/θ

⎧ ⎫1    ⎬u   ⎨   u  |fj |  ⎭  ⎩  j ∈N 

,

X 1/θ

where the implicit positive constant is independent of {fj }j ∈N ; (ii) for the above p− , there exists an s0 ∈ (0, min{1, p− }) and an r0 ∈ (s0 , ∞) such that X1/s0 is a ball Banach function space and, for any f ∈ L1loc (Rn ),    ((r0/s0 ) )  (f ) M

(X 1/s0 )

 f (X1/s0 ) ,

where the implicit positive constant is independent of f ; (iii) X has an absolutely continuous quasi-norm.

p,q 4.7 Dual Space of H K˙ ω,0 (Rn )

257

Assume d ≥ n(1/ min{1, p− } − 1), s ∈ (0, s0 ], and r ∈ (max{1, r0 }, ∞]. Then the dual space of HX (Rn ), denoted by (HX (Rn ))∗ , is LX,r  ,d,s (Rn ) in the following sense: (a) Let g ∈ LX,r  ,d,s (Rn ). Then the linear functional  Lg : f → Lg (f ) :=

Rn

f (x)g(x) dx,

(4.74)

X,r,d,s initially defined for any f ∈ Hfin (Rn ), has a bounded extension to HX (Rn ). (b) Conversely, any continuous linear functional on HX (Rn ) arises as in (4.74) with a unique g ∈ LX,r  ,d,s (Rn ).

Moreover, gLX,r  ,d,s (Rn ) ∼ Lg (HX (Rn ))∗ , where the positive equivalence constants are independent of g. Remark 4.7.7 We point out that the dual theorem, Lemma 4.7.6, has a wide range of applications. Here we present two function spaces to which Lemma 4.7.6 can be applied (see also [277, Section 6]). (i) Let p := (p1 , . . . , pn ) ∈ (0, ∞)n , 5  d≥ n

1 −1 min{p1 , . . . , pn }

6

be a fixed nonnegative integer, s ∈ (0, min{1, p1 , . . . , pn }), and r ∈ (max{1, p1 , . . . , pn }, ∞]. Then, in this case, as was pointed out in [277, Subsection 6.1], the mixednorm Lebesgue space Lp (Rn ) satisfies all the assumptions of Lemma 4.7.6. Therefore, Lemma 4.7.6 with X := Lp (Rn ) holds true. This result coincides with [277, Theorem 6.2] (see also [128]). Moreover, the dual theorem of the mixed-norm Hardy space H p (Rn ), with p ∈ (0, 1]n , was first established in [124, Theorem 3.10] (see also [127]), which is a special case of the aforementioned result. (ii) Let p(·) ∈ C log (Rn ) satisfy 0 < p− ≤ p+ < ∞, where p− and p+ are defined, respectively, in (1.59) and (1.60). Let 5  6 1 d≥ n −1 p− be a fixed nonnegative integer, s ∈ (0, min{1, p− }), and r ∈ (max{1, p+ }, ∞]. Then, in this case, as was mentioned in [277, Subsection 6.2], the variable Lebesgue space Lp(·) (Rn ) satisfies all the assumptions of Lemma 4.7.6. Thus, Lemma 4.7.6 with X := Lp(·) (Rn ) holds true. This dual result is just [277, Theorem 6.7] (see also [125]) and extends [186, Theorem 7.5]. Indeed, when

258

4 Generalized Herz–Hardy Spaces

0 < p− ≤ p+ ≤ 1, then, in this case, the aforementioned result coincides with [186, Theorem 7.5]. With the help of the above lemma, we then prove Theorem 4.7.4. Proof of Theorem 4.7.4 Let all the symbols be as in the present theorem. Then, from the assumption m0 (ω) ∈ (− pn , ∞) and Theorem 1.2.42, we deduce that the p,q local generalized Herz space K˙ ω,0 (Rn ) is a BQBF space. This implies that, to finish p,q the proof of the present theorem, we only need to show that K˙ ω,0 (Rn ) satisfies (i), (ii), and (iii) of Lemma 4.7.6. p,q First, we prove that Lemma 4.7.6(i) holds true for K˙ ω,0 (Rn ). Indeed, for any given θ ∈ (0, p− ) and u ∈ (1, ∞), by Lemma 4.3.10 with r := θ , we conclude that, for any {fj }j ∈N ⊂ L1loc (Rn ), ⎧ ⎫1   u ⎨  u ⎬    M(fj )   ⎭  ⎩  j ∈N 

˙ (Rn )]1/θ [K ω,0 p,q

⎛ ⎞1   u    ⎝ u⎠   |fj |     j ∈N 

,

˙ (Rn )]1/θ [K ω,0 p,q

which implies that Lemma 4.7.6(i) holds true for K˙ ω,0 (Rn ). p,q Next, we show that K˙ ω,0 (Rn ) satisfies Lemma 4.7.6(ii). To this end, let s0 ∈ (s, min{p− , q}) and p,q

  r0 ∈ max 1, p,

  n ,r . min{m0 (ω), m∞ (ω)} + n/p

Then, from Lemma 1.8.6 with s and r therein replaced, respectively, by s0 and r0 , it p,q follows that [K˙ ω,0 (Rn )]1/s0 is a BBF space and, for any f ∈ L1loc (Rn ),    ((r0/s0 ) )  (f ) M

˙ p,q (Rn )]1/s0 ) ([K ω,0

 f ([K˙ p,q (Rn )]1/s0 ) . ω,0

p,q This implies that, under the assumptions of the present theorem, K˙ ω,0 (Rn ) satisfies Lemma 4.7.6(ii). Finally, applying Theorem 1.4.1, we find that the local generalized Herz space p,q K˙ ω,0 (Rn ) has an absolutely continuous quasi-norm. Therefore, all the assumptions p,q of Lemma 4.7.6 hold true for K˙ ω,0 (Rn ). Thus, the proof of Theorem 4.7.4 is then completed.  

Remark 4.7.8 We point out that the absolutely continuous quasi-norm of the ball quasi-Banach function space X plays a key role in the dual theorem of the associated Hardy space HX (Rn ). However, by Example 1.4.4, we find that the global generalized Herz space does not have an absolutely continuous quasi-norm. Thus, p,q the dual space of the generalized Herz–Hardy space H K˙ ω (Rn ) is still unknown.

p,q 4.7 Dual Space of H K˙ ω,0 (Rn )

259

From the above dual theorem, we immediately deduce the following equivalence p,q,r,d,s of the Campanato-type function space L˙ ω,0 (Rn ); we omit the details. Corollary 4.7.9 Let p, q, ω, p− , d, and s be as in Theorem 4.7.4,  p+ := max 1, p,

 n , min{m0 (ω), m∞ (ω)} + n/p

 ), d := n(1/p − 1), and s ∈ (0, min{p , q}). Then r ∈ [1, p+ 0 − 0 − p,q,r,d,s p,q,1,d ,s (Rn ) = L˙ ω,0 0 0 (Rn ) L˙ ω,0

with equivalent quasi-norms. As an application of Theorem 4.7.4, we next investigate the dual space of p,q the generalized Hardy–Morrey space H M ω,0 (Rn ) via introducing the following p,q,r,d,s Campanato-type function space L˙ω,0 (Rn ) associated with the local generalized Morrey space. Definition 4.7.10 Let p, q, r ∈ [1, ∞), s ∈ (0, ∞), d ∈ Z+ , and ω ∈ M(R+ ) satisfy M∞ (ω) ∈ (−∞, 0) and −

n < m0 (ω) ≤ M0 (ω) < 0. p

p,q,r,d,s Then the Campanato-type function space L˙ω,0 (Rn ), associated with the local p,q n generalized Morrey space M ω,0 (R ), is defined to be the set of all the f ∈ Lrloc (Rn ) such that

f L˙ p,q,r,d,s (Rn ) ω,0

 $s  1s   m # −1    λ i := sup  1 Bi    n  i=1 1Bi M p,q  p,q ω,0 (R ) ×

M ω,0 (Rn )

⎧ m ⎨  j =1

λj |Bj | ⎩ 1Bj M p,q (Rn ) ω,0

#

1 |Bj |

 Bj

 r   f (x) − PBdj f (x) dx

$1 ⎫ r⎬ ⎭

m is finite, where the supremum is taken over all m ∈ N, {Bj }m j =1 ⊂ B, and {λj }j =1 ⊂ 4m [0, ∞) with j =1 λj = 0.

Using Theorem 4.7.4 and Remarks 1.2.2(iv) and 4.0.20(ii), we immediately obtain the following conclusion, which shows that the dual space of the generalized p,q Hardy–Morrey space H M ω,0 (Rn ) is just the Campanato-type function space  p,q,r ,d,s (Rn ); we omit the details. L˙ ω,0

260

4 Generalized Herz–Hardy Spaces

Corollary 4.7.11 Let p, q ∈ [1, ∞), ω ∈ M(R+ ) with −

n < m0 (ω) ≤ M0 (ω) < 0 p

and −

n < m∞ (ω) ≤ M∞ (ω) < 0, p

d ∈ Z+ , s ∈ (0, 1), and  r∈

 n ,∞ . min{m0 (ω), m∞ (ω)} + n/p



p,q,r ,d,s p,q Then L˙ω,0 (Rn ) is the dual space of H M ω,0 (Rn ) in the following sense: p,q,r  ,d,s

(i) Let g ∈ L˙ω,0

(Rn ). Then the linear functional  Lg : f → Lg (f ) :=

Rn

f (x)g(x) dx,

(4.75)

p,q,r,d,s

initially defined for any f ∈ H M ω,0,fin (Rn ), has a bounded extension to the p,q generalized Hardy–Morrey space H M ω,0 (Rn ). p,q (ii) Conversely, any continuous linear functional L ∈ (H M ω,0 (Rn ))∗ arises as in  p,q,r ,d,s (4.75) with a unique g ∈ L˙ (Rn ). ω,0

Moreover, there exist two positive constants C1 and C2 such that, for any g ∈ p,q,r  ,d,s L˙ (Rn ), ω,0

C1 g

p,q,r  ,d,s L˙ω,0 (Rn )

≤ Lg (H M p,q (Rn ))∗ ≤ C2 g

, p,q,r  ,d,s L˙ω,0 (Rn )

ω,0

where (H M ω,0 (Rn ))∗ denotes the dual space of H M ω,0 (Rn ). p,q

p,q

4.8 Boundedness of Calderón–Zygmund Operators The main target of this section is to investigate the boundedness of Calderón– Zygmund operators on generalized Herz–Hardy spaces. To this end, we first establish two general boundedness criteria of Calderón–Zygmund operators on Hardy spaces associated with ball quasi-Banach function spaces (see Proposition 4.8.12 and Theorem 4.8.17 below) under some reasonable assumptions. Via these results and the facts that both local and global generalized Herz spaces are

4.8 Boundedness of Calderón–Zygmund Operators

261

ball quasi-Banach function spaces, we then obtain the boundedness of Calderón– Zygmund operators on generalized Herz–Hardy spaces. Let d ∈ Z+ and T be a d-order Calderón–Zygmund operator defined as in Definition 1.5.8. Recall the well-known assumption on T that, for any γ ∈ Zn+ with |γ | ≤ d, T ∗ (x γ ) = 0, namely, for any a ∈ /L2 (Rn ) having compact support and satisfying that, for any γ ∈ Zn+ with |γ | ≤ d, Rn a(x)x γ dx = 0, it holds true that  T (a)(x)x γ dx = 0 Rn

(see, for instance, [160, p. 119]). Definition 4.8.1 Let d ∈ Z+ . A d-order Calderón–Zygmund operator T is said to have the vanishing moments up to order d if, for any γ ∈ Zn+ with |γ | ≤ d, T ∗ (x γ ) = 0. We should point out that the assumption that the d-order Calderón–Zygmund operator T has the vanishing moments up to order d is reasonable. Indeed, this assumption holds true automatically when T is a Calderón–Zygmund operator with kernel K(x, y) := K1 (x − y) for a locally integrable function K1 on Rn \ {0}. To be precise, we have the following interesting proposition about convolutional type Calderón–Zygmund operators, which might be well known. But, we do not find its detailed proof in the literature. Thus, for the convenience of the reader, we present its detailed proof as follows. Proposition 4.8.2 Let d ∈ Z+ , K ∈ S  (Rn ), and the operator T be defined by setting, for any f ∈ S(Rn ), T (f ) := K ∗ f . Assume that the following three statements hold true:  ∈ L∞ (Rn ); (i) K (ii) K coincides with a function belonging to C d (Rn \ {0}) in the sense that, for any given a ∈ L2 (Rn ) with compact support and for any x ∈ / supp (a) := {x ∈ Rn : a(x) = 0},  T (a)(x) = K(x − y)a(y) dy; Rn

(iii) there exists a positive constant C and a δ ∈ (0, 1] such that, for any γ ∈ Zn+ with |γ | ≤ d and for any x ∈ Rn \ {0},   γ ∂ K(x) ≤

C |x|n+|γ |

(4.76)

262

4 Generalized Herz–Hardy Spaces

and, for any γ ∈ Zn+ with |γ | = d and x, y ∈ Rn with |x| > 2|y|,   γ ∂ K(x − y) − ∂ γ K(x) ≤ C

|y|δ . |x|n+d+δ

(4.77)

Then T has a vanishing moments up to order d. To show Proposition 4.8.2, we need some preliminary lemmas. Recall that the following conclusion gives the Lp (Rn ) boundedness of convolutional type Calderón–Zygmund operators, which is just [70, Theorem 5.1]. Lemma 4.8.3 Let p ∈ (1, ∞) and T be as in Proposition 4.8.2. Then T is well defined on Lp (Rn ) and there exists a positive constant C such that, for any f ∈ Lp (Rn ), T (f )Lp (Rn ) ≤ Cf Lp (Rn ) . Via borrowing some ideas from the proof of [213, p. 117, Lemma], we next establish the following technical estimates about the kernel K, which plays an important role in the proof of Proposition 4.8.2. Lemma 4.8.4 Let d, K, and δ be as in Proposition 4.8.2. For any t ∈ (0, ∞), let K (t ) := K ∗ φt , where φ ∈ S(Rn ) with supp (φ) ⊂ B(0, 1). Then there exists a positive constant C such that (i) for any t ∈ (0, ∞), γ ∈ Zn+ with |γ | ≤ d, and x ∈ Rn \ {0},    γ (t )  ∂ K (x) ≤

C ; |x|n+|γ |

(ii) for any t ∈ (0, ∞), γ ∈ Zn+ with |γ | = d, and x, y ∈ Rn with |x| > 4|y|,    γ (t )  ∂ K (x − y) − ∂ γ K (t ) (x) ≤ C

|y|δ . |x|n+d+δ

Proof Let all the symbols be as in the present lemma and t ∈ (0, ∞). We first prove (i). Indeed, notice that, for any x ∈ Rn ,  K (t ) (x) =

Rn

 ) e2πix·ξ K(ξ φ (tξ ) dξ.

From this, we further deduce that, for any γ ∈ Zn+ with |γ | ≤ d and for any x ∈ Rn ,  ∂ γ K (t )(x) =

Rn

 ) φ(tξ ) dξ. (2πiξ )γ e2πix·ξ K(ξ

(4.78)

4.8 Boundedness of Calderón–Zygmund Operators

263

 ∈ L∞ (Rn ), further implies that, for any which, together with the assumption K n x ∈ R with 0 < |x| < 2t,     γ (t )  ∂ K (x)  ∼

Rn

  φ(tξ ) dξ ∼ |ξ |γ 

1 t n+|γ |



1 |x|n+|γ |



1 t n+|γ |

Rn

   φ(ξ ) dξ (4.79)

,

where the implicit positive constants are independent of t. On the other hand, applying the both assumptions Proposition 4.8.2(ii) and supp (φt ) ⊂ B(0, t), we find that, for any x ∈ Rn with |x| ≥ 2t,  K (t )(x) =

Rn

K(x − y)φt (y) dy.

This implies that, for any γ ∈ Zn+ with |γ | ≤ d and for any x ∈ Rn with |x| ≥ 2t,  ∂ K (x) = γ

(t )

Rn

∂ γ K(x − y)φt (y) dy



∂ γ K(x − y)φt (y) dy.

=

(4.80)

B(0,t )

In addition, for any x ∈ Rn with |x| ≥ 2t and for any y ∈ B(0, t), we have |x − y| ≥ |x| − |y| >

1 |x|. 2

Combining this, (4.80), and (4.76) with x therein replaced by x − y, we further conclude that, for any γ ∈ Zn+ with |γ | ≤ d and for any x ∈ Rn with |x| ≥ 2t,    γ (t )  ∂ K (x) 

1 |x − y|n+|γ |

 Rn

|φ(y)| dy 

1 |x|n+|γ |

,

(4.81)

where the implicit positive constants are independent of t. Moreover, from (4.79) and (4.81), it follows that (i) holds true. Next, we show (ii). To this end, fix x, y ∈ Rn with |x| > 4|y|. We now prove (ii) by considering the following two cases on x.

264

4 Generalized Herz–Hardy Spaces

Case (1) x ∈ Rn with |x| < 4t. In this case, we have |y| < 14 |x| < t. Then,  ∈ L∞ (Rn ) using (4.78), the Lagrange mean value theorem, and the assumptions K n and δ ∈ (0, 1], we find that there exists a y1 ∈ R such that, for any γ ∈ Zn+ with |γ | = d,     γ (t ) ∂ K (x − y) − ∂ γ K (t )(x)        ) =  φ(tξ ) dξ  (2πiξ )γ e2πi(x−y)·ξ − e2πix·ξ K(ξ Rn

  = 

(2πiξ ) (2πiξ ) · ye γ

Rn



 |y| 

Rn

2πi(x−y1 )·ξ

  |ξ |d+1  φ(tξ ) dξ ∼

|y|

    K(ξ )φ (tξ ) dξ  

t n+d+1

Rn

   φ(ξ ) dξ

|y|δ 1 |y|δ  , δ t |x|n+d+δ

t n+d

(4.82)

where the implicit positive constants are independent of t. This finishes the proof of (ii) under the assumption |x| < 4t. Case (2) |x| ≥ 4t. In this case, we first claim that |x − y| > t. Indeed, when |y| < t, we have |x − y| ≥ |x| − |y| > 3t > t. On the other hand, if |y| ≥ t, from the assumption |x| > 4|y|, we deduce that |x − y| ≥ |x| − |y| > 3|y| > t. Therefore, the above claim holds true. Using this claim and the assumptions Proposition 4.8.2(ii), |x| ≥ 4t, and supp (φt ) ⊂ B(0, t), we conclude that  K (x) = (t )

|z| n, we further conclude that  Rn

M (T (a), φ) (x)|x||γ | dx 

=

M (T (a), φ) (x)|x| B(x0 ,4r0 )

 (x0 + 4r0 )

|γ |

|γ |

 dx +



[B(x0 ,4r0 )]

···

M (T (a), φ) (x) dx B(x0 ,4r0 )





  |x0 ||γ | 1 + aL2 (Rn ) dx + n+d+δ |x − x0 |n+d−|γ |+δ [B(x0 ,4r0 )] |x − x0 |      1 1 dx  aL2 (Rn ) 1 + + n+d+δ |x − x0 |n+d−|γ |+δ [B(x0 ,4r0 )] |x − x0 | < ∞. This implies that, for any γ ∈ Zn+ with |γ | ≤ d, M(T (a), φ)| · ||γ | ∈ L1 (Rn ) and hence finishes the proof of Lemma 4.8.5.   Recall that the Hardy space H 1 (Rn ) is defined as in Definition 4.2.2 with X := The following atomic decomposition of H 1 (Rn ) can be deduced from the proof of [160, Chapter 2, Proposition 3.3] immediately, which is an essential tool in the proof of Proposition 4.8.2; we omit the details. L1 (Rn ).

Lemma 4.8.6 Let N ∈ N, d ∈ Z+ , and f ∈ H 1 (Rn ) ∩ L2 (Rn ). Then there exists {λi,j }i∈Z, j ∈N ⊂ [0, ∞) and a sequence {ai,j }i∈Z, j ∈N of (L1 (Rn ), ∞, d)-atoms supported, respectively, in the balls {Bi,j }i∈Z, j ∈N ⊂ B such that f =



λi,j ai,j

i∈Z j ∈N

almost everywhere in Rn and there exists a positive constant C such that, for any i ∈ Z and j ∈ N, the following three statements hold true: i (i) λ 0i,j |ai,j | ≤ C2 ; (ii) 4j ∈N Bi,j = i := {x ∈ Rn : MN (f )(x) > 2i } with MN as in (4.3); (iii) j ∈N 1Bi,j ≤ C.

268

4 Generalized Herz–Hardy Spaces

In order to show Proposition 4.8.2, we also require some auxiliary conclusions about weighted function spaces. The following one about powered weights is a part of [90, Example 7.1.7]. Lemma 4.8.7 Let a ∈ R and p ∈ (1, ∞). Then | · |a ∈ Ap (Rn ) if and only if −n < a < n(p − 1). Moreover, the following conclusion gives the strong type inequality characterization of Ap (Rn )-weights with p ∈ (1, ∞), which can be found in [70, Theorem 7.3]. Lemma 4.8.8 Let p ∈ (1, ∞) and υ ∈ A∞ (Rn ). Then the Hardy–Littlewood p maximal operator M is bounded on Lυ (Rn ) if and only if υ ∈ Ap (Rn ). Via the above two lemmas, we show the following technical conclusion about grand maximal functions and radial maximal functions, which is vital in the proof of Proposition 4.8.2. Lemma 4.8.9 Let d ∈ Z+ , N ∈ N ∩ [n + d + 1, ∞), f ∈ S  (Rn ), and φ ∈ S(Rn ) with supp (φ) ⊂ B(0, 1) and  Rn

φ(x) dx = 0.

Assume that M(f, φ)| · |d ∈ L1 (Rn ), then MN (f )| · |d ∈ L1 (Rn ) and     MN (f )| · |d 

L1 (Rn )

    ∼ M(f, φ)| · |d 

L1 (Rn )

with the positive equivalence constants independent of f . n Proof Let all the symbols be as in the present lemma and r ∈ (0, n+d ). Then we 1 have d < n( r − 1). This, together with Lemma 4.8.7 with a := d and p := 1r , implies that | · |d ∈ A1/r (Rn ). Applying this and Lemma 4.8.8 with p := 1r and υ := | · |d , we conclude that, for any g ∈ L1loc (Rn ),

M(g)L1/r (Rn )  gL1/r (Rn ) . |·|d

(4.89)

|·|d

1/r

In addition, observe that L|·|d (Rn ) = [L1|·|d (Rn )]1/r . Thus, from this, (4.89), and Lemma 4.1.4 with X := L1|·|d (Rn ), it follows that     MN (f )| · |d 

L1 (Rn )

    ∼ M(f, φ)| · |d 

L1 (Rn )

,

where the positive equivalence constants are independent of f . This finishes the proof of Lemma 4.8.9.  

4.8 Boundedness of Calderón–Zygmund Operators

269

Based on above preparations, we now prove Proposition 4.8.2. Proof of Proposition 4.8.2 Let all the symbols be as in the present proposition and a ∈ L2 (Rn ), with compact support, satisfying that, for any γ ∈ Zn+ with |γ | ≤ d,  Rn

a(x)x γ dx = 0.

Then, from Lemma 4.8.3 with p = 2, we deduce that T (a) ∈ L2 (Rn ). In addition, let φ ∈ S(Rn ) with supp (φ) ⊂ B(0, 1) and  Rn

φ(x) dx = 0.

By Lemma 4.8.5 with γ = 0, we find that M(T (a), φ) ∈ L1 (Rn ) and hence T (a) ∈ H 1 (Rn ). Thus, we have T (a) ∈ H 1 (Rn ) ∩ L2 (Rn ). Choose an N ∈ N ∩ [n + d + 1, ∞). Then, applying Lemma 4.8.6 with f therein replaced by T (a), we conclude that there exists {λi,j }i∈Z, j ∈N ⊂ [0, ∞) and a sequence {ai,j }i∈Z, j ∈N of (L1 (Rn ), ∞, d)-atoms such that T (a) =



λi,j ai,j

(4.90)

i∈Z j ∈N

almost everywhere in Rn and (i), (ii), and (iii) of Lemma 4.8.6 hold true. Assume E ⊂ Rn such that |E| = 0 and, for any x ∈ Rn \ E, T (a)(x) =



λi,j ai,j (x).

i∈Z j ∈N

Now, we show that 

  λi,j ai,j   MN (T (a))

i∈Z, j ∈N

almost everywhere in Rn by considering the following two cases on x. 0 Case (1) x ∈ ( i∈Z i ) \ E. Here and thereafter, for any i ∈ Z, i is defined as in Lemma 4.8.6(ii) with f replaced by T (a). In this case, we have MN (T (a))(x) = 0. In addition, for any i ∈ Z and j ∈ N, by the assumption supp (ai,j ) ⊂ Bi,j and Lemma 4.8.6(ii), we conclude that ai,j (x) = 0. This, together with MN (T (a))(x) = 0, further implies that  i∈Z, j ∈N

  λi,j ai,j (x) = 0 = MN (T (a)) (x).

(4.91)

270

4 Generalized Herz–Hardy Spaces

Thus, 

  λi,j ai,j   MN (T (a))

i∈Z, j ∈N

0 almost everywhere in ( i∈Z i ) . Case (2) x ∈ (i0 \ i0 +1 ) \ E for some i0 ∈ Z. In this case, for any i ∈ N ∩ [i0 + 1, ∞) and j ∈ N, we have ai,j (x) = 0. Applying this, both (i) and (iii) of Lemma 4.8.6, the assumption supp (ai,j ) ⊂ Bi,j for any i ∈ Z and j ∈ N, and the definition of i , we find that, for any x ∈ (i0 \ i0 +1 ) \ E, 

i0       λi,j ai,j (x) = λi,j ai,j (x)

i∈Z, j ∈N

i=−∞ j ∈N



i0  i=−∞

2i



1Bi,j 

j ∈N

i0 

2i ∼ 2i0

i=−∞

∼ MN (T (a)) (x),

(4.92)

which further implies that 

  λi,j ai,j   MN (T (a))

i∈Z, j ∈N

almost everywhere in i0 \ i0 +1 . Combining (4.91) and (4.92), we then conclude that    λi,j ai,j   MN (T (a)) i∈Z, j ∈N

almost everywhere in Rn . Therefore, for any γ ∈ Zn+ with |γ | ≤ d, from both Lemmas 4.8.5 and 4.8.9 with d therein replaced by |γ |, we deduce that  i∈Z, j ∈N

  λi,j ai,j  | · ||γ |  MN (T (a))| · ||γ | ∈ L1 (Rn ).

4.8 Boundedness of Calderón–Zygmund Operators

271

Using this, (4.90), the dominated convergence theorem, and the assumption that {ai,j }i∈Z, j ∈N is a sequence of (L1 (Rn ), ∞, d)-atoms, we further find that, for any γ ∈ Zn+ with |γ | ≤ d,  Rn

T (a)(x)x γ dx =

 n i∈Z j ∈N R

ai,j (x)x γ dx = 0,  

which completes the proof of Proposition 4.8.2.

Under the reasonable assumption that the Calderón–Zygmund operator T has vanishing moments, we then have the boundedness of Calderón–Zygmund operators p,q on the generalized Herz–Hardy space H K˙ ω,0 (Rn ) as follows. n Theorem 4.8.10 Let d ∈ Z+ , δ ∈ (0, 1], p, q ∈ ( n+d+δ , ∞), K be a d-order standard kernel defined as in Definition 1.5.7, T a d-order Calderón–Zygmund operator with kernel K having the vanishing moments up to order d, and ω ∈ M(R+ ) with

−

n n < m0 (ω) ≤ M0 (ω) < n − + d + δ p p

and −

n n < m∞ (ω) ≤ M∞ (ω) < n − + d + δ. p p

Then T has a unique extension on H K˙ ω,0 (Rn ) and there exists a positive constant p,q C such that, for any f ∈ H K˙ ω,0 (Rn ), p,q

T (f )H K˙ p,q (Rn ) ≤ Cf H K˙ p,q (Rn ) . ω,0

ω,0

Remark 4.8.11 We should point out that, in Theorem 4.8.10, when d = 0, p ∈ (1, ∞), and ω(t) := t α for any t ∈ (0, ∞) and for any given α ∈ [n(1 − p1 ), n(1 − 1 p ) + δ),

then Theorem 4.8.10 goes back to [157, Theorem 1].

To show Theorem 4.8.10, we first establish the boundedness of Calderón– Zygmund operators on Hardy spaces associated with ball quasi-Banach function spaces as follows. Proposition 4.8.12 Let X be a ball quasi-Banach function space satisfying Assumption 1.2.29 for some 0 < θ < s ≤ 1. Assume that X1/s is a ball Banach function space and there exists an r0 ∈ (1, ∞) and a positive constant C such that, for any f ∈ L1loc (Rn ),    ((r0/s) )  (f ) M

(X 1/s )

≤ C f (X1/s ) .

(4.93)

272

4 Generalized Herz–Hardy Spaces

Let d ∈ Z+ , δ ∈ (0, 1], K be a d-order standard kernel defined as in Definition 1.5.7, and T a d-order Calderón–Zygmund operator with kernel K having the n n vanishing moments up to order d. If θ ∈ ( n+d+δ , n+d ] and X has an absolutely continuous quasi-norm, then T has a unique extension on HX (Rn ) and there exists a positive constant C such that, for any f ∈ HX (Rn ), T (f )HX (Rn ) ≤ Cf HX (Rn ) . Remark 4.8.13 We point out that Proposition 4.8.12 has a wide range of applications. Here we present several function spaces to which Proposition 4.8.12 can be applied. In what follows, let d and δ be the same as in Proposition 4.8.12. n (i) Let p ∈ ( n+d+δ , ∞). Then, in this case, combining Remarks 1.2.27, 1.2.31(i), and 1.2.34(i), we can easily conclude that the Lebesgue space Lp (Rn ) satisfies all the assumptions of Proposition 4.8.12. This then implies that Proposition 4.8.12 with X := Lp (Rn ) holds true. If further assume that p ∈ (0, 1] and T is a convolutional type operators defined as in Proposition 4.8.2, then, in this case, the aforementioned result is just [213, p. 115, Theorem 4]. nqυ (ii) Let υ ∈ A∞ (Rn ), qυ be the same as in (1.58), and p ∈ ( n+d+δ , ∞). Then, in this case, from Remarks 1.2.27, 1.2.31(ii), and 1.2.34(ii), we can easily p deduce that the weighted Lebesgue space Lυ (Rn ) satisfies all the assumptions p of Proposition 4.8.12. Therefore, Proposition 4.8.12 with X := Lυ (Rn ) holds true. If further assume that p ∈ (0, 1] and d := 0 in Proposition 4.8.12, then, in this case, the aforementioned result goes back to [194, Theorem 3]. n (iii) Let p := (p1 , . . . , pn ) ∈ ( n+d+δ , ∞)n . Then, in this case, by Remarks 1.2.27, 1.2.31(iii), and 1.2.34(iii), we can easily find that the mixed-norm Lebesgue space Lp (Rn ) satisfies all the assumptions of Proposition 4.8.12. Thus, Proposition 4.8.12 with X := Lp (Rn ) holds true. If further assume that K is a convolutional type operator defined as in Proposition 4.8.2, and d := 0, then, in this case, the aforementioned result was established in [123, Theorem 6.4] (see also [127, 155]). n (iv) Let p(·) ∈ C log (Rn ) satisfy n+d+δ < p− ≤ p+ < ∞, where p− and p+ are defined, respectively, in (1.59) and (1.60). Then, in this case, from Remarks 1.2.27, 1.2.31(v), and 1.2.34(v), we can easily infer that the variable Lebesgue space Lp(·)(Rn ) satisfies all the assumptions of Proposition 4.8.12. This then implies that Proposition 4.8.12 with X := Lp(·) (Rn ) holds true. If further assume that p+ ∈ (0, 1] and d := 0 in Proposition 4.8.12, then, in this case, the aforementioned result was also given in [280, Theorem 6.6].

To show Proposition 4.8.12, we need the following auxiliary lemma which established the relations among Calderón–Zygmund operators, atoms, and molecules. Lemma 4.8.14 Let X be a ball quasi-Banach function space, r ∈ [2, ∞), and d ∈ Z+ . Assume that K is a d-order standard kernel defined as in Definition 1.5.7 with some δ ∈ (0, 1], and T a d-order Calderón–Zygmund operator with kernel K having the vanishing moments up to order d. Then, for any (X, r, d)-

4.8 Boundedness of Calderón–Zygmund Operators

273

atom a supported in the ball B ∈ B, T (a) is a harmless constant multiple of an (X, r, d, d + δ + rn )-molecule centered at B. Proof Let all the symbols be as in the present lemma and a an (X, r, d)-atom supported in the ball B := B(x0 , r0 ) with x0 ∈ Rn and r0 ∈ (0, ∞). Then, combining Definitions 4.3.4(iii) and 4.8.1, we find that, for any γ ∈ Zn+ with |γ | ≤ d,  Rn

T (a)(x)x γ dx = 0.

This implies that T (a) satisfies Definition 4.5.3(ii). Next, we show that Definition 4.5.3(i) holds true for a harmless constant multiple of T (a). Indeed, from Lemma 1.5.9 with p := r and Definition 4.3.4(ii), it follows that T (a)1B Lr (Rn ) ≤ T (a)Lr (Rn )  aLr (Rn ) 

|B|1/r 1B X

(4.94)

and   T (a)1(2B)\B  r n L (R ) n |B|1/r |B|1/r ∼ 2−(d+δ+ r  ) . 1B X 1B X

≤ T (a)Lr (Rn )  aLr (Rn ) 

(4.95)

These are the desired estimates of T (a)1B Lr (Rn ) and T (a)1(2B)\B Lr (Rn ) , respectively. In addition, recall that, for any j ∈ N, Sj (B) := (2j B) \ (2j −1 B). We next estimate T (a)1Sj+1 (B) Lr (Rn ) . Indeed, by Definition 1.5.8(ii), we find that, for any j ∈ N and x ∈ Sj +1 (B),  T (a)(x) =

Rn

K(x, y)a(y).

This, together with Definition 4.3.4(iii) and the Taylor remainder theorem, further implies that, for any j ∈ N and y ∈ B, there exists a ty ∈ [0, 1] such that   T (a)1S (B)  r n j+1 L (R ) # = Sj+1 (B)

$1  r r    K(x, y)a(y) dy  dx   B

274

4 Generalized Herz–Hardy Spaces

r  ⎡ ⎫1 ⎤ r   ⎪   ⎪ γ ⎬  ∂(2) K(x, x0 )   ⎢ ⎥  ⎢K(x, y) − a(y) dy  dx (y − x0 )γ ⎥ =  ⎣ ⎦ ⎪ ⎪ γ!  ⎪ ⎪ γ ∈Zn ⎩ Sj+1 (B)  B ⎭ +  |γ |≤d ⎧   ⎪  ⎪ γ γ  ⎨  ∂(2) K(x, ty y + (1 − ty )x0 ) − ∂(2) K(x, x0 )   =  ⎪ γ! ⎪ ⎩ Sj+1 (B)  B γ ∈Zn+ |γ |=d ⎧ ⎪ ⎪ ⎨

r  1r  × (y − x0 )γ a(y) dy  dx #

 

|y − x0 | a(y) d

B

Sj+1 (B)

⎤1 r     γ r ⎥  γ × ∂(2) K(x, ty y + (1 − ty )x0 ) − ∂(2) K(x, x0) dx ⎥ ⎦ dy.

(4.96)

γ ∈Zn + |γ |=d

On the other hand, for any j ∈ N, x ∈ Sj +1 (B), and y ∈ B, we have |x − x0 | ≥ 2j r0 > 2|y − x0 | ≥ 2|ty (y − x0 )|. Using this, (4.96), (1.104) with y and z therein replaced, respectively, by x0 and ty y + (1 − ty )x0 for any y ∈ B, the Hölder inequality, and Definition 4.3.4(ii), we conclude that, for any j ∈ N,   T (a)1S (B)  r n j+1 L (R ) #

$1 r δr |y − x | 0  r0d |a(y)| dx dy (n+d+δ)r B Sj+1 (B) |x − x0 |  1  |a(y)| dy  r0d+δ (2j r0 )−n+d+δ Sj +1 (B) r 

B

 2−j (d+δ+ r  ) aLr (Rn )  2−j (d+δ+ r  ) n

n

∼ 2−(j +1)(d+δ+ r  ) n

|B|1/r , 1B X

|B|1/r 1B X (4.97)

4.8 Boundedness of Calderón–Zygmund Operators

275

which is the desired estimate of T (a)1Sj+1 (B) Lr (Rn ) with j ∈ N. Applying (4.94), (4.95), and (4.97), we find that there exists a positive constant C, independent of a, such that, for any j ∈ Z+ , n

CT (a)1Sj (B) Lr (Rn ) ≤ 2−j (d+δ+ r  )

|B|1/r . 1B X

This further implies that CT (a) satisfies Definition 4.5.3(ii) and hence CT (a) is an (X, r, d, d + δ + rn )-molecule centered at B, which then completes the proof of Lemma 4.8.14.   Via Lemma 4.8.14, we next show Proposition 4.8.12. Proof of Proposition 4.8.12 Let all the symbols be as in the present proposition and r := max{2, r0 }. Then, from the Hölder inequality, it follows that, for any f ∈ L1loc (Rn ), x ∈ Rn , and B ∈ B satisfying x ∈ B, 

1 |B|







|f (y)|(r/s) dy

1 (r/s)

 ≤

B



1 |B|

≤M





|f (y)|(r0/s) dy

1 (r0 /s)

B

((r0 /s) )

(f )(x),

which further implies that 



M((r/s) ) (f )(x) ≤ M((r0 /s) ) (f )(x). Using this, (4.93), the assumption that X1/s is a BBF space, Remark 1.2.18, and Definition 1.2.13(ii), we find that, for any f ∈ L1loc (Rn ),    ((r/s))  (f ) M

(X 1/s )

     ≤ M((r0/s) ) (f )

(X 1/s )

 f (X1/s ) .

(4.98)

n n On the other hand, by the assumptions θ ∈ ( n+d+δ , n+d ] and δ ∈ (0, 1], we conclude that     1 1 −1 −δ ≥n −1 −1 d>n θ θ

and  d≤n

 1 −1 . θ

276

4 Generalized Herz–Hardy Spaces

These imply that d = n(1/θ −1). From this, (4.98), the assumption that X satisfies Assumption 1.2.29 for the above θ and s, and Lemma 4.4.3, we deduce that, for any X,r,d,s g ∈ Hfin (Rn ), gH X,r,d,s (Rn ) ∼ gHX (Rn )

(4.99)

fin

with the positive equivalence constants independent of g. X,r,d,s m Now, let f ∈ Hfin (Rn ), m ∈ N, {λj }m j =1 ⊂ [0, ∞), and {aj }j =1 of (X, r, d)atoms supported, respectively, in the balls {Bj }m j =1 ⊂ B such that f =

m 

λj aj .

j =1

This, combined with the linearity of T , implies that T (f ) =

m 

(4.100)

T (aj ).

j =1

On the other hand, from Lemma 4.8.14, it follows that, for any j ∈ {1, . . . , m}, T (aj ) is a harmless constant multiple of an (X, r, d, d + δ + rn )-molecule centered n at Bj . Moreover, by the assumption θ > n+d+δ , we find that   n 1 1 − . d +δ+  >n r θ r Combining this, the assumption that X satisfies Assumption 1.2.29 for the above θ and s, (4.98), (4.100), the fact that, for any j ∈ {1, . . . , m}, T (aj ) is a harmless constant multiple of an (X, r, d, d + δ + rn )-molecule centered at Bj , and Lemma 4.5.5, we further conclude that T (f ) ∈ HX (Rn ) and

T (f )HX (Rn )

⎡ ⎤1   m

s s    λj ⎣   1 Bj ⎦  .   1Bj X  j =1  X

This, together with the choice of {λj }m j =1 , (4.44), and (4.99), implies that T (f )HX (Rn )  f H X,r,d,s (Rn ) ∼ f HX (Rn ) . fin

X,r,d,s Therefore, T is bounded on the finite atomic Hardy space Hfin (Rn ). Finally, from the assumption that X has an absolutely continuous quasi-norm and X,r,d,s [207, Remark 3.12], we deduce that the finite atomic Hardy space Hfin (Rn ) is n dense in the Hardy space HX (R ). Thus, by a standard density argument, we find

4.8 Boundedness of Calderón–Zygmund Operators

277

that T has a unique extension on HX (Rn ) and, for any f ∈ HX (Rn ), T (f )HX (Rn )  f HX (Rn ) ,  

which completes the proof of Proposition 4.8.12.

Via the above boundedness of Calderón–Zygmund operators on the Hardy space HX (Rn ), we next prove Theorem 4.8.10. Proof of Theorem 4.8.10 Let all the symbols be as in the present theorem. Then, from the assumption m0 (ω) ∈ (− pn , ∞) and Theorem 1.2.42, it follows that p,q the local generalized Herz space K˙ ω,0 (Rn ) is a BQBF space. We now show Theorem 4.8.10 via proving that all the assumptions of Proposition 4.8.12 hold true p,q for K˙ ω,0 (Rn ). First, we show that there exist θ, s ∈ (0, 1] such that Assumption 1.2.29 holds p,q true for K˙ ω,0 (Rn ). Indeed, by the assumption max{M0 (ω), M∞ (ω)} ∈ (− pn , n − n p + d + δ), we conclude that n ∈ max{M0 (ω), M∞ (ω)} + n/p



 n ,∞ . n+d +δ

n This, combined with the assumptions p, q ∈ ( n+d+δ , ∞), further implies that

 min 1, p, q,

n max{M0 (ω), M∞ (ω)} + n/p



 ∈

 n ,∞ . n+d +δ

Therefore, we can choose an    n n , min 1, p, q, s∈ n+d +δ max{M0 (ω), M∞ (ω)} + n/p and a  θ∈

  n n , min s, . n+d +δ n+d

Then, applying Lemma 4.3.11, we find that, for any {fj }j ∈N ⊂ L1loc (Rn ), ⎧ ⎫1/s    ⎨  s ⎬    M(θ) (fj )   ⎭  ⎩  j ∈N  ˙ p,q

Kω,0 (Rn )

⎛ ⎞1/s       ⎝   |fj |s ⎠     j ∈N  ˙ p,q

.

(4.101)

Kω,0 (Rn )

p,q This implies that, for the above θ and s, K˙ ω,0 (Rn ) satisfies Assumption 1.2.29.

278

4 Generalized Herz–Hardy Spaces

Next, from Lemma 1.8.6, we deduce that the Herz space [K˙ ω,0 (Rn )]1/s is a BBF space. In addition, let p,q

  r0 ∈ max 1, p,

  n ,∞ . min{m0 (ω), m∞ (ω)} + n/p

We now show that, for this r0 and the above s, (4.13) holds true. Indeed, using Lemma 1.8.6 with r := r0 , we conclude that, for any f ∈ L1loc (Rn ),     ((r0/s) ) (f ) M

˙ p,q (Rn )]1/s ) ([K ω,0

 f ([K˙ p,q (Rn )]1/s ) . ω,0

(4.102)

Finally, from Theorem 1.4.1, it follows that the local generalized Herz space p,q K˙ ω,0 (Rn ) has an absolutely continuous quasi-norm. Combining this, (4.101), the p,q fact that [K˙ ω,0 (Rn )]1/s is a BBF space, and (4.102), we find that all the assumptions p,q of Lemma 4.8.12 hold true for K˙ ω,0 (Rn ) under consideration and then complete the proof of Theorem 4.8.10.   As an application, we have the following boundedness of Calderón–Zygmund p,q operators on the generalized Hardy–Morrey space H M ω,0 (Rn ), which can be deduced from Theorem 4.8.10 and Remarks 1.2.2(iv) and 4.0.20(ii) immediately; we omit the details. Corollary 4.8.15 Let d ∈ Z+ , p, q ∈ [1, ∞), K be a d-order standard kernel defined as in Definition 1.5.7 with some δ ∈ (0, 1], T a d-order Calderón–Zygmund operator with kernel K having the vanishing moments up to order d, and ω ∈ M(R+ ) with −

n < m0 (ω) ≤ M0 (ω) < 0 p

and −

n < m∞ (ω) ≤ M∞ (ω) < 0. p p,q

Then T has a unique extension on H M ω,0 (Rn ) and there exists a positive constant p,q C such that, for any f ∈ H M ω,0 (Rn ), T (f )H M p,q (Rn ) ≤ Cf H M p,q (Rn ) . ω,0

ω,0

The remainder of this section is devoted to showing the boundedness of p,q Calderón–Zygmund operators on the generalized Herz–Hardy space H K˙ ω (Rn ). Precisely, we turn to prove the following theorem.

4.8 Boundedness of Calderón–Zygmund Operators

279

n Theorem 4.8.16 Let d ∈ Z+ , δ ∈ (0, 1], p, q ∈ ( n+d+δ , ∞), K be a d-order standard kernel with defined as in Definition 1.5.7, T a d-order Calderón–Zygmund operator with kernel K having the vanishing moments up to order d, and ω ∈ M(R+ ) with

−

n n < m0 (ω) ≤ M0 (ω) < n − + d + δ p p

and −

n < m∞ (ω) ≤ M∞ (ω) < 0. p

Then T can be extended into a bounded linear operator on H K˙ ω (Rn ) and there p,q exists a positive constant C such that, for any f ∈ H K˙ ω (Rn ), p,q

T (f )H K˙ p,q n ≤ Cf H K n . ˙ p,q ω (R ) ω (R ) To show this theorem, we first establish a general result about the boundedness of Calderón–Zygmund operators on Hardy spaces associated with ball quasi-Banach function spaces as follows. Theorem 4.8.17 Let X be a ball quasi-Banach function space, Y a linear space equipped with a quasi-seminorm  · Y , Y0 a linear space equipped with a quasiseminorm  · Y0 , η ∈ (1, ∞), and 0 < θ < s < s0 ≤ 1 such that (i) (ii) (iii) (iv)

for the above θ and s, Assumption 1.2.29 holds true; both  · Y and  · Y0 satisfy Definition 1.2.13(ii); 1B(0,1) ∈ Y0 ; for any f ∈ M (Rn ),   f X1/s ∼ sup fgL1 (Rn ) : gY = 1 and   f X1/s0 ∼ sup fgL1 (Rn ) : gY0 = 1

with the positive equivalence constants independent of f ; (v) M(η) is bounded on both Y and Y0 . Assume that d ∈ Z+ , δ ∈ (0, 1], K is a d-order standard kernel defined as in Definition 1.5.7, and T a d-order Calderón–Zygmund operator with kernel K n n having the vanishing moments up to d. If θ ∈ ( n+d+δ , n+d ], then T can be extended n into a bounded linear operator on HX (R ), namely, there exists a positive constant

280

4 Generalized Herz–Hardy Spaces

C such that, for any f ∈ HX (Rn ), T (f )HX (Rn ) ≤ Cf HX (Rn ) . To show Theorem 4.8.17, the following two embedding theorems about X and HX (Rn ) are the essential tools. Lemma 4.8.18 Let X be a ball quasi-Banach function space, Y ⊂ M (Rn ) a linear space equipped with a quasi-seminorm  · Y , θ ∈ (1, ∞), and s ∈ (0, ∞) satisfy the following four statements: (i)  · Y satisfies Definition 1.2.13(ii); (ii) 1B(0,1) ∈ Y ; (iii) for any f ∈ M (Rn ),   f X1/s ∼ sup fgL1 (Rn ) : gY = 1 , where the positive equivalence constants are independent of f ; (iv) M(θ) is bounded on Y . Assume ε ∈ ( θ1 , 1) and υ := [M(1B(0,1))]ε . Then there exists a positive constant C such that, for any f ∈ X, f Lsυ (Rn ) ≤ Cf X . Proof Let all the symbols be as in the present lemma and f ∈ X. Then, by Lemma 2.3.11 with r := 1, we find that, for any x ∈ Rn , .ε  υ(x) := M 1B(0,1) ∼ (1 + |x|)−εn . This implies that, for any x ∈ B(0, 1), υ(x) ∼ 1 and, for any k ∈ N and x ∈ B(0, 2k ) \ B(0, 2k−1 ), υ(x) ∼ 2−kεn . From these, we further deduce that  Rn

|f (y)|s υ(y) dy 

=

|f (y)| υ(y) dy + s

B(0,1)

 |f (y)|s dy +

∼ B(0,1)

2−kεn

|f (y)| 1B(0,1)(y) dy + s

Rn

k k−1 k∈N B(0,2 )\B(0,2 )

|f (y)|s υ(y) dy



B(0,2k )\B(0,2k−1 )

k∈N

 





 k∈N

2

−kεn

|f (y)|s dy

 Rn

|f (y)|s 1B(0,2k ) (y) dy.

(4.103)

4.8 Boundedness of Calderón–Zygmund Operators

281

Next, applying (1.55), we conclude that, for any k ∈ N and x ∈ B(0, 2k ), M

(θ)

-



.

1B(0,1) (x) ≥

1 |B(0, 2k )|



|B(0, 1)| ∼ |B(0, 2k )|



 B(0,2k )

1 θ

1B(0,1)(y)

θ

1 θ

dy

nk

∼ 2− θ ,

which implies that . nk 1B(0,2k )  2 θ M(θ) 1B(0,1) , where the implicit positive constant is independent of k. Applying this and both the assumptions (i) and (iv) of the present lemma, we find that, for any k ∈ N,  1

B(0,2k )

   . nk   (θ)   2 nkθ    1 M B(0,1)   2 θ 1B(0,1) Y , Y

(4.104)

Y

which is the desired estimate of 1B(0,2k ) . Thus, using (4.103), the assumption (iii) of the present lemma, (4.104), and the assumption ε − θ1 ∈ (0, ∞), we find that  Rn

|f (y)|s υ(y) dy

    |f |s 

X 1/s

#

     1B(0,1) + 2−kεn 1B(0,2k ) Y Y k∈N

#

   1 2−kn(ε− θ )  f sX 1B(0,1)Y 1 + ∼

f sX

  1B(0,1) . Y

$

$

k∈N

(4.105)

On the other hand, by the assumption (ii) of the present lemma, we conclude that   1B(0,1) < ∞. Y Therefore, from (4.105), we infer that  f sLs (Rn ) = |f (y)|s υ(y) dy  f sX , υ

Rn

which completes the proof of Lemma 4.8.18.

 

Lemma 4.8.19 Let X be a ball quasi-Banach function space and N ∈ N. Then the Hardy space HX (Rn ) embeds continuously into S  (Rn ). Namely, there exists a

282

4 Generalized Herz–Hardy Spaces

positive constant C such that, for any f ∈ HX (Rn ) and φ ∈ S(Rn ), |f, φ| ≤ CpN (φ)f HX (Rn ) , where pN is defined as in (4.1). Proof Let all the symbols be as in the present lemma, f ∈ HX (Rn ), and φ ∈ S(Rn ). Then we prove the present lemma by considering the following two cases on φ. Case (1) pN (φ) = 0. In this case, we have φ = 0. This further implies that |f, φ| = 0 = pN (φ)f HX (Rn ) , which completes the proof of Lemma 4.8.19 in this case. Case (2) pN (φ) = 0. In this case, from (4.1), we deduce that  pN

φ(−·) pN (φ)

 =

(4.106) φ(−·) pN (φ)

∈ S(Rn ) and

pN (φ) = 1. pN (φ)

Therefore, pφ(−·) ∈ FN (Rn ) with FN (Rn ) as in (4.2). By this and (4.3), we conclude N (φ) that, for any x ∈ B(0, 1), |f, φ| = |f ∗ φ(−·)(0)|       φ(−·)  (0) ≤ pN (φ)MN (f )(x). = pN (φ) f ∗ pN (φ) This further implies that |f, φ| 1B(0,1) ≤ pN (φ)MN (f ).

(4.107)

In addition, since X is a BQBF space, from both (i) and (iv) of Definition 1.2.13, we deduce that 1B(0,1)X ∈ (0, ∞). This, together with (4.107) and Definitions 1.2.13(ii) and 4.2.2, further implies that −1  |f, φ| ≤ 1B(0,1)X pN (φ) MN (f )X −1  = 1B(0,1)X pN (φ) f HX (Rn ) .

(4.108)

Using this, we find that the present lemma holds true in this case. Combining this and (4.106), we then complete the proof of Lemma 4.8.19.   Furthermore, the following auxiliary lemma about function spaces having absolutely continuous quasi-norms is given in [207, Corollary 3.11(ii)], which is important in the proof of Theorem 4.8.17.

4.8 Boundedness of Calderón–Zygmund Operators

283

Lemma 4.8.20 Let X, r, d, s, {λj }j ∈N , and {aj }j ∈N be as in Lemma4 4.3.6. If X has an absolutely continuous quasi-norm, then f ∈ HX (Rn ) and f := j ∈N λj aj converges in HX (Rn ). To prove Theorem 4.8.17, we also need weighted function spaces. In what p follows, the weighted Hardy space Hυ (Rn ), with p ∈ (0, ∞) and υ ∈ A∞ (Rn ), is p defined as in Definition 4.2.2 with X := Lυ (Rn ) [see also Remark 4.2.3(ii)]. Then, via Lemma 4.8.20 above, we establish the following atomic reconstruction theorem about weighted Hardy spaces, which is one of the key tools used in the proof of Theorem 4.8.17. Lemma 4.8.21 Let 0 < θ < s < s0 ≤ 1, υ ∈ A1 (Rn ), and d ≥ n(1/θ − 1) s be a fixed integer. Assume that {aj }j ∈N is a sequence of (Lυ0 (Rn ), ∞, d)-atoms supported, 4 respectively, in the balls {Bj }j ∈N ⊂ B and {λj }j ∈N ⊂ [0, ∞) such that f := j ∈N λj aj in S  (Rn ) and ⎧ ⎫1    # $s ⎬s  ⎨ λj   1 Bj   ⎭  ⎩ 1Bj Ls0 (Rn ) υ  j ∈N 

< ∞. s

Lυ0 (Rn )

Then f ∈ Hυs0 (Rn ) and f =

4

j ∈N λj aj

holds true in Hυs0 (Rn ).

Proof Let all the symbols be as in the present lemma. Then, applying [233, Remarks 2.4(b), 2.7(b), and 3.4(i)], we find that the following four statements hold true: s

(i) Lυ0 (Rn ) is a BQBF space; (ii) for any {fj }j ∈N ⊂ L1loc (Rn ), ⎧ ⎫1/s    ⎨  s ⎬    M(θ) (fj )   ⎭  ⎩  j ∈N 

s Lυ0 (Rn )

⎛ ⎞1/s        ⎝  |fj |s ⎠      j ∈N

;

s Lυ0 (Rn )

(iii) [Lsυ0 (Rn )]1/s is a BBF space and, for any f ∈ L1loc (Rn ), M(f )([Ls0 (Rn )]1/s )  f ([Ls0 (Rn )]1/s ) ; υ υ (iv) Lsυ0 (Rn ) has an absolutely continuous quasi-norm. Thus, the weighted Lebesgue space Lsυ0 (Rn ) under consideration satisfies all the assumptions of Lemma 4.8.20 with r therein replaced by ∞. This finishes the proof of Lemma 4.8.21.  

284

4 Generalized Herz–Hardy Spaces

We now show Theorem 4.8.17. Proof of Theorem 4.8.17 Let all the symbols be as in the present theorem and f ∈ n n HX (Rn ). Then, by the assumptions θ ∈ ( n+d+δ , n+d ] and δ ∈ (0, 1], we conclude that     1 1 −1 −δ ≥n −1 −1 d>n θ θ and  d≤n

 1 −1 . θ

Thus, d = n(1/θ − 1). From this, the assumption (i) of the present theorem, and Lemma 4.3.8, it follows that there exists {λj }j ∈N ⊂ [0, ∞) and {aj }j ∈N of (X, ∞, d)-atoms supported, respectively, in the balls {Bj }j ∈N ⊂ B such that f =



(4.109)

λj aj

j ∈N

in S  (Rn ) and ⎡ ⎤1  

s s    λj ⎣  1Bj ⎦   f HX (Rn ) .    1Bj X  j ∈N 

(4.110)

X

In addition, by the assumptions (ii) through (v) of the present theorem and Lemma 4.8.18 with Y := Y0 , θ := η, and s := s0 , we find that there exists an ε ∈ (0, 1) such that, for any h ∈ M (Rn ), hLsυ0 (Rn )  hX , where υ := [M(1B(0,1))]ε . Combining this, (4.109), and (4.110), we conclude that f =

 j ∈N

λj aj =

 j ∈N

# λj

1Bj Ls0 (Rn ) υ

1Bj X

$#

1Bj X 1Bj Lsυ0 (Rn )

$ aj

(4.111)

4.8 Boundedness of Calderón–Zygmund Operators

285

in S  (Rn ) and ⎧ ⎫1  ⎡ ⎤s  s 1Bj  s0 n ⎪  ⎪ L (R ) ⎪ ⎨ ⎢ λj 1B υX ⎥ ⎪ ⎬    j ⎢ ⎥  ⎣ 1B  s0 n ⎦ 1Bj ⎪  ⎪  ⎪ j Lυ (R ) ⎩j ∈N ⎪ ⎭   

s

Lυ0 (Rn )

⎡ ⎤1  

s s    λj ⎣   1Bj ⎦   f HX (Rn ) < ∞.   1Bj X  j ∈N 

(4.112)

X

Observe that, for any j ∈ N,

1Bj X 1Bj  s0

s

Lυ (Rn )

aj is an (Lυ0 (Rn ), ∞, d)-atom supported

in Bj . This, together with (4.111),4 (4.112), and Lemma 4.8.21, further implies s0 n that f ∈ Hυs0 (Rn ) and f = υ (R ). Applying this and j ∈N λj aj in H4 Remark 4.8.13(ii) with p := s0 , we find that T (f ) = j ∈N λj T (aj ) holds true in s s Hυ0 (Rn ). Therefore, from Lemma 4.8.19 with X replaced by Lυ0 (Rn ), we further infer that  T (f ) = λj T (aj ) (4.113) j ∈N

in S  (Rn ). Next, we prove that T (f ) ∈ HX (Rn ). To this end, choose an r ∈ (max{2, sη }, ∞) with η1 + η1 = 1. Then, by Lemma 4.3.5 with t = ∞, we conclude that, for any j ∈ N, aj is an (X, r, d)-atom supported in the ball Bj . This, together with Lemma 4.8.14, implies that, for any j ∈ N, T (aj ) is a harmless multiple of an (X, r, d, d + δ + rn )-molecule centered at Bj . Moreover, since n θ > n+d+δ , it follows that d +δ+

  n 1 1 − > n . r θ r

On the other hand, using the assumption (ii) of the present theorem and an argument similar to that used in the proof of (4.98) with (r0 /s) and (X1/s ) therein replaced, respectively, by η and Y , we find that, for any f ∈ L1loc (Rn ),    ((r/s))  (f )  f Y . M Y

286

4 Generalized Herz–Hardy Spaces

From this, both the assumptions (i) and (iv) of the present theorem, Theorem 4.5.11, (4.113), and (4.110), we deduce that T (f ) ∈ HX (Rn ) and

T (f )HX (Rn )

⎡ ⎤1  

s s    λj ⎣   1Bj ⎦   f HX (Rn ) .   1Bj X  j ∈N  X

This finishes the proof that T (f ) ∈ HX (Rn ) and further implies that T is bounded on HX (Rn ). Thus, the proof of Theorem 4.8.17 is completed.   Now, we prove Theorem 4.8.16. Proof of Theorem 4.8.16 Let all the symbols be as in the present theorem. Then, using the assumptions m0 (ω) ∈ (− pn , ∞) and M∞ (ω) ∈ (−∞, 0), and Theorem 1.2.44, we find that, under the assumptions of the present theorem, the global p,q generalized Herz space K˙ ω (Rn ) is a BQBF space. Therefore, to prove the present theorem, it suffices to show that all the assumptions of Theorem 4.8.17 hold true for p,q K˙ ω (Rn ). n First, by the assumptions p, q ∈ ( n+d+δ , ∞) and   n n max{M0 (ω), M∞ (ω)} ∈ − , n − + d + δ , p p we conclude that   n n < min 1, p, q, . n+d +δ max{M0 (ω), M∞ (ω)} + n/p Let  s∈

  n n , min 1, p, q, n+d +δ max{M0 (ω), M∞ (ω)} + n/p

and  θ∈

  n n , min s, . n+d +δ n+d

p,q We now prove that K˙ ω (Rn ) satisfies Theorem 4.8.17(i) for the above θ and s. Indeed, from Lemma 4.3.25, it follows that, for any {fj }j ∈N ⊂ L1loc (Rn ),

⎧ ⎫1/s    ⎨  s ⎬    (θ) M (fj )   ⎭  ⎩  j ∈N  ˙ p,q

Kω (Rn )

⎛ ⎞1/s       ⎝ s⎠   |fj |     j ∈N  ˙ p,q

Kω (Rn )

.

4.8 Boundedness of Calderón–Zygmund Operators

287

This implies that, for the above θ and s, Assumption 1.2.29 holds true for K˙ ω (Rn ) p,q and hence Theorem 4.8.17(i) is satisfied for K˙ ω (Rn ). We next show that there exist two linear spaces Y and Y0 and η,s ∈ (0, ∞) such that the assumptions (ii) through (v) of Theorem 4.8.17 hold true. To this end, let (p/s),(q/s) Y := B˙1/ωs (Rn ), s0 ∈ (0, 1] be such that p,q

  s0 ∈ s, min 1, p, q, (p/s ) ,(q/s0 )

Y0 := B˙1/ωs00

 η < min



n max{M0 (ω), M∞ (ω)} + n/p

,

(Rn ), and η ∈ (1, ∞) be such that

p   n , . n(1 − s/p) − s min{m0 (ω), m∞ (ω)} s

(4.114)

Then, applying Lemma 2.3.2, we find that both 







(p/s) ,(q/s) (p/s ) ,(q/s0 ) (Rn ) and B˙1/ωs00 (Rn ) B˙1/ωs

satisfy Definition 1.2.13(ii). This further implies that Theorem 4.8.17(ii) holds true. Now, we show that the assumption (iii) of Theorem 4.8.17 holds true for the (p/s ) ,(q/s0 ) (Rn ). Indeed, combining Lemma 1.1.6 above Y0 . Namely, 1B(0,1) ∈ B˙1/ωs00 n and the assumptions s < max{M0 (ω),M∞ (ω)}+n/p and M0 (ω) > − pn , we conclude that . m0 ω−s0 = −s0 M0 (ω) > −s0 max{M0 (ω), M∞ (ω)}   n s =− . > −n 1 − p (p/s) Applying this and Theorem 1.2.42 with p, q, and ω therein replaced, respectively, (p/s ) ,(q/s ) by (p/s0 ) , (q/s0 ) , and 1/ωs0 , we find that the Herz space K˙ 1/ωs00 ,0 0 (Rn ) is a BQBF space. This, together with Lemma 2.2.3 with p, q, ω, and ξ therein replaced, respectively, by (p/s0 ) , (q/s0 ) , 1/ωs0 , and 0, and Definition 1.2.13(iv), further implies that   1B(0,1) (p/s ) ,(q/s ) 0 0 ˙ B1/ωs0

(Rn )

   1B(0,1) ˙ (p/s0 ) ,(q/s0 ) K1/ωs0 ,0 



(Rn )

< ∞,

(p/s ) ,(q/s0 ) which completes the proof that 1B(0,1) ∈ B˙1/ωs00 (Rn ). Therefore, Theorem 4.8.17(iii) holds true.

288

4 Generalized Herz–Hardy Spaces 







(p/s) ,(q/s) (p/s ) ,(q/s0 ) Next, we prove that both B˙1/ωs (Rn ) and B˙1/ωs00 (Rn ) satisfy Theorem 4.8.17(iv). Indeed, using Lemma 1.1.6 and the assumption m0 (ω) > − pn , we find that

- . n m0 ωs = sm0 (ω) > − . p/s

(4.115)

On the other hand, from Lemma 1.1.6 again and the assumption M∞ (ω) < 0, we deduce that - . M∞ ωs = sM∞ (ω) < 0. Combining this, (4.115), and Lemma 4.3.27 with p, q, and ω therein replaced, respectively, by p/s, q/s, and ωs , we conclude that, for any f ∈ M (Rn ), f K˙ p/s,q/s (Rn ) ωs

 ∼ sup fgL1 (Rn ) : g ˙(p/s) ,(q/s) B1/ωs

 (Rn )

=1 ,

(4.116)

where the positive equivalence constants are independent of f . Similarly, repeating an argument used in the proof of (4.116) with s replaced by s0 , we find that, for any f ∈ M (Rn ),  f  ˙ p/s0 ,q/s0 Kωs0

(Rn )



∼ sup fgL1 (Rn ) : g ˙(p/s0 ) ,(q/s0 ) B1/ωs0

(Rn )

=1 ,

which, together with (4.116), further implies that Theorem 4.8.16(iv) holds true with (p/s),(q/s) (p/s ) ,(q/s0 ) Y = B˙1/ωs (Rn ) and Y0 = B˙1/ωs00 (Rn ). Finally, we prove that the powered Hardy–Littlewood maximal operator M(η) (p/s),(q/s) (p/s ) ,(q/s0 ) is bounded on both B˙1/ωs (Rn ) and B˙1/ωs00 (Rn ). Indeed, from the n assumption s < max{M0 (ω),M , we deduce that ∞ (ω)}+n/p   s n 1− − s min{m0 (ω), m∞ (ω)} p   n = n − s min{m0 (ω), m∞ (ω)} + p   n > 0, ≥ n − s max{M0 (ω), M∞ (ω)} + p

4.8 Boundedness of Calderón–Zygmund Operators

289

which, combined with Lemma 1.1.6 and (4.114), further implies that . .  max M0 ω−s , M∞ ω−s



= −s min{m0 (ω), m∞ (ω)} < n

 1 1 − . η (p/s)

(4.117)

In addition, applying Lemma 1.1.6 again and the assumptions s
− pn , we conclude that . .  min m0 ω−s , m∞ ω−s

  n s =− . = −s max{M0 (ω), M∞ (ω)} > −n 1 − p (p/s)

From this, (4.117), the assumption η < (p/s) , and Corollary 2.3.5 with p, q, ω, and r therein replaced, respectively, by (p/s) , (q/s) , 1/ωs , and η, it follows that, for any f ∈ L1loc (Rn ),    (η)  M (f ) ˙ (p/s) ,(q/s) B1/ωs

(Rn )

 f  ˙ (p/s) ,(q/s) B1/ωs

(Rn )

(4.118)

. 



(p/s) ,(q/s) This further implies that M(η) is bounded on Y = B˙1/ωs (Rn ). Moreover, repeating an argument used in the proof of (4.118) with s replaced by s0 , and using the assumption s0 > s, we find that, for any f ∈ L1loc (Rn ),

   (η)  M (f ) ˙ (p/s0 ) ,(q/s0 ) B1/ωs0

(Rn )

 f  ˙ (p/s0 ) ,(q/s0 ) B1/ωs0



(Rn )

,



(p/s ) ,(q/s0 ) which implies that M(η) is bounded on Y0 = B˙1/ωs00 (Rn ) and hence Theorem 4.8.17(v) holds true. Therefore, all the assumptions of Theorem 4.8.17 p,q hold true for K˙ ω (Rn ). Therefore, T can be extended into a bounded linear operator p,q p,q on H K˙ ω (Rn ) and, for any f ∈ H K˙ ω (Rn ),

T (f )H K˙ p,q n  f H K n . ˙ p,q ω (R ) ω (R ) This finishes the proof of Theorem 4.8.16.

 

Via Theorem 4.8.16 and Remarks 1.2.2(iv) and 4.0.20(ii), we immediately obtain the following boundedness of Calderón–Zygmund operators on the generalized p,q Hardy–Morrey space H M ω (Rn ); we omit the details.

290

4 Generalized Herz–Hardy Spaces

Corollary 4.8.22 Let d ∈ Z+ , K be a d-order standard kernel defined as in Definition 1.5.7 with some δ ∈ (0, 1], T a d-order Calderón–Zygmund operator with kernel K having the vanishing moments up to order d, and p, q, and ω as in Corollary 4.8.15. Then T can be extended into a bounded linear operator p,q on H M ω (Rn ) and there exists a positive constant C such that, for any f ∈ p,q H M ω (Rn ), T (f )H M p,q n ≤ Cf H M p,q (Rn ) . ω (R ) ω

4.9 Fourier Transform The target of this section is to investigate the Fourier transform of a distribution p,q p,q in the generalized Herz–Hardy space H K˙ ω,0 (Rn ) or H K˙ ω (Rn ). Recall that, in 1974, Coifman [45] characterized all f via entire functions of exponential type for n = 1, where f ∈ H p (R) with p ∈ (0, 1]. Later, a number of authors investigated the characterization of f with distribution f belonging to Hardy spaces in higher dimensions (see, for instance, [16, 48, 121, 122, 221]). In particular, Huang et al. [122] studied the Fourier transform of the distribution belonging to the Hardy space associated with the ball quasi-Banach function space, which plays an important role in this section. We first consider the Fourier transform properties of the Hardy spaces associated with the local generalized Herz spaces. Namely, we have the following theorem. Theorem 4.9.1 Let p, q ∈ (0, 1], ω ∈ M(R+ ) with m0 (ω) ∈ (0, ∞) and m∞ (ω) ∈ p,q n (0, ∞), and p− ∈ (0, max{M0 (ω),M ). Then, for any f ∈ H K˙ ω,0 (Rn ), there ∞ (ω)}+n/p exists a continuous function g on Rn such that f = g in S  (Rn ) and |g(x)|

lim

|x|→0+

|x|

n( p1 −1)

= 0.

−

Moreover, there exists a positive constant C, independent of both f and g, such that, for any x ∈ Rn ,   n( 1 −1) |g(x)| ≤ Cf H K˙ p,q (Rn ) max 1, |x| p− ω,0

and  Rn

|g(x)| min |x|

− pn

−

! , |x|−n dx ≤ Cf H K˙ p,q (Rn ) . ω,0

4.9 Fourier Transform

291

To show this theorem, we first investigate the properties of the Fourier transform of the distribution belonging to the Hardy space HX (Rn ) associated with the general ball quasi-Banach function space X. Indeed, we have the following technical lemma, which is essential obtained by Huang et al. [122, Theorems 2.1, 2.2, and 2.3]. Lemma 4.9.2 Let X be a ball quasi-Banach function space and p− ∈ (0, 1] such that, for any given θ ∈ (0, p− ) and u ∈ (1, ∞), there exists a positive constant C such that, for any {fj }j ∈N ⊂ L1loc (Rn ), ⎧ ⎫1   u ⎨  u ⎬    M(fj )   ⎭  ⎩  j ∈N 

X 1/θ

⎛ ⎞1  u    ⎝ u⎠  ≤C |fj |     j ∈N 

.

(4.119)

X 1/θ

Assume that there exists a p0 ∈ [p− , 1] such that X is p0 -concave and there exists a positive constant C such that, for any B ∈ B,   1 1 1B X ≥ C min |B| p0 , |B| p− .

(4.120)

Then, for any f ∈ HX (Rn ), there exists a continuous function g on Rn such that f = g in S  (Rn ) and |g(x)|

lim

|x|→0+

|x|

n( p1 −1)

= 0.

−

Moreover, there exists a positive constant C, independent of both f and g, such that, for any x ∈ Rn ,   n( p1 −1) n( p1 −1) − 0 |g(x)| ≤ Cf HX (Rn ) max |x| , |x| and  Rn

  1/p0 p n(p −1− p 0 ) − , |x|n(p0 −2) |g(x)|p0 min |x| 0 ≤ Cf HX (Rn ) . dx

Proof Let all the symbols be as in the present lemma, d ≥ n(1/p− − 1) a fixed integer, and f ∈ S  (Rn ). Then, by (4.119) and the known atomic decomposition of HX (Rn ) (see [207, Theorem 3.7] or Lemma 4.3.8), we find that there exists {λj }j ∈N ⊂ [0, ∞) and a sequence {aj }j ∈N of (X, ∞, d)-atoms supported, respectively, in the balls {Bj }j ∈N ⊂ B such that f =

 j ∈N

λj aj

292

4 Generalized Herz–Hardy Spaces

in S  (Rn ) and ⎡ ⎤1  

s s    λj ⎣  1Bj ⎦   f HX (Rn ) .    1Bj X  j ∈N  X

Using this and repeating the proof of [122, Theorems 2.1, 2.2, and 2.3], we conclude that there exists a continuous function g on Rn such that f = g in S  (Rn ), lim

|x|→0+

|g(x)| |x|

n( p1 −1)

= 0,

−

and, for any x ∈ Rn ,   n( 1 −1) n( 1 −1) |g(x)|  f HX (Rn ) max |x| p0 , |x| p− and  Rn

 1/p0  p n(p −1− p 0 ) − , |x|n(p0 −2) dx |g(x)|p0 min |x| 0  f HX (Rn )

with the implicit positive constants independent of both f and g. This then finishes the proof of Lemma 4.9.2.   Remark 4.9.3 We point out that Lemma 4.9.2 has a wide range of applications. Here we give several function spaces to which Lemma 4.9.2 can be applied. (i) Let p ∈ (0, 1] and p− = p0 = p. Then, in this case, using Remark 1.2.31(i) 1

and the fact that, for any ball B ∈ B, 1B Lp (Rn ) = |B| p , we can easily find that the Lebesgue space Lp (Rn ) satisfies all the assumptions of Lemma 4.9.2. Therefore, Lemma 4.9.2 with X := Lp (Rn ) holds true. This result was originally established in [221]. (ii) Let p := (p1 , . . . , pn ) ∈ (0, 1]n , p− := min{p1 , . . . , pn }, and p0 := max{p1 , . . . , pn }. Then, in this case, as was mentioned in [122, Subsection 4.1], the mixed-norm Lebesgue space Lp (Rn ) satisfies all the assumptions of Lemma 4.9.2. Thus, Lemma 4.9.2 with X replaced by Lp (Rn ) holds true. This result was also obtained in [122, Theorem 4.1].

4.9 Fourier Transform

293

(iii) Let p(·) ∈ C log (Rn ) satisfy 0 < p− ≤ p+ ≤ 1, where p− and p+ are defined, respectively, in (1.59) and (1.60). Let p0 = p+ . Then, in this case, as was pointed out in [122, Subsection 4.2], the variable Lebesgue space Lp(·) (Rn ) satisfies all the assumptions of Lemma 4.9.2. This then implies that Lemma 4.9.2 with X := Lp(·) (Rn ) holds true, which coincides with [122, Theorem 4.2]. In order to prove Theorem 4.9.1, we also require the following estimate about the quasi-norm of the characteristic functions of balls on the local generalized Herz p,q space K˙ ω,0 (Rn ), which plays an important role in the proof of Theorem 4.9.1 and is also of independent interest. Lemma 4.9.4 Let p, q, ω, and p− be as in Theorem 4.9.1. Then there exists a positive constant C such that, for any B ∈ B,  1B 

˙ p,q (Rn ) K ω,0

≥ C min |B|, |B|

1 p−

 .

(4.121)

Proof Let all the symbols be as in the present lemma. We first show that, for any B(x0 , 2k0 ) ∈ B with x0 ∈ Rn and k0 ∈ Z, 2

nk0 p

    ω(2k0 )  1B(x0,2k0 )  ˙ p,q

Kω,0 (Rn )

(4.122)

.

To achieve this, we consider the following four cases on x0 . Case (1) x0 = 0. In this case, from the fact that B(0, 2k0 ) \ B(0, 2k0 −1 ) ⊂ B(0, 2k0 ) and the definition of  · K˙ p,q (Rn ) , we deduce that ω,0

2

nk0 p

    ω(2k0 ) ∼ ω(2k0 ) 1B(0,2k0 )\B(0,2k0−1 )       1B(0,2k0 )  ˙ p,q

Kω,0 (Rn )

Lp (Rn )

,

(4.123)

which completes the proof of (4.122) in this case. Case (2) |x0 | ∈ (0, 3 · 2k0 −1 ). In this case, we first claim that  B

3 · 2k0 −2 x0 , 2k0 −2 |x0 |





  ⊂ B(x0 , 2k0 ) ∩ B(0, 2k0 ) \ B(0, 2k0 −1 ) .

Indeed, for any y ∈ B((3 · 2k0 −2 /|x0 |)x0 , 2k0 −2 ), we have      3 · 2k0 −2   3 · 2k0 −2   x0  +  x0  < 2k0 −2 + 3 · 2k0 −2 = 2k0 |y| ≤ y − |x0 | |x0 |

(4.124)

294

4 Generalized Herz–Hardy Spaces

and      3 · 2k0 −2   3 · 2k0 −2     x 0  − y − x0  > 3 · 2k0 −2 − 2k0 −2 = 2k0 −1 . |y| ≥  |x0| |x0 | This implies that  B

3 · 2k0 −2 x0 , 2k0 −2 |x0 |



  ⊂ B(0, 2k0 ) \ B(0, 2k0 −1 ) .

(4.125)

On the other hand, for any y ∈ B((3 · 2k0 −2 /|x0 |)x0 , 2k0 −2 ), we have       3 · 2k0 −2   3 · 2k0 −2 x0  +  x0 − x0  |y − x0 | ≤ y − |x0 | |x0 |     < 2k0 −2 + 3 · 2k0 −2 − |x0 | . This further implies that, for any y ∈ B((3 · 2k0 −2 /|x0 |)x0, 2k0 −2 ) with x0 ∈ [3 · 2k0 −2 , 3 · 2k0 −1 ), |y − x0 | < |x0 | − 2k0 −1 < 2k0 and, for any y ∈ B((3 · 2k0 −2 /|x0 |)x0 , 2k0 −2 ) with |x0 | ∈ (0, 3 · 2k0 −2 ), |y − x0 | < 2k0 − |x0 | < 2k0 . Therefore, B((3 · 2k0 −2 /|x0 |)x0 , 2k0 −2 ) ⊂ B(x0 , 2k0 ). By this and (4.125), we find that    ! 3 · 2k0 −2 x0 , 2k0 −2 ⊂ B(x0 , 2k0 ) ∩ B(0, 2k0 ) \ B(0, 2k0 −1 ) . B |x0 | This finishes the proof of the above claim. Thus, from the definition of the quasinorm  · K˙ p,q (Rn ) , it follows that ω,0

2

nk0 p

   p1   3 · 2k0 −2 k0 −2   x0 , 2 ω(2 ) ∼ ω(2 ) B  |x | k0

k0

0

   1  p  ω(2k0 ) B(x0 , 2k0 ) ∩ B(0, 2k0 ) \ B(0, 2k0 −1 )       1B(x0,2k0 )  ˙ p,q n Kω,0 (R )

and hence (4.122) holds true in this case.

(4.126)

4.9 Fourier Transform

295

Case (3) |x0 | ∈ [3 · 2k0 −1 , 2k0 +1 ). In this case, for any y ∈ B(x0 , 2k0 ), we have |y| ≤ |y − x0 | + |x0 | < 2k0 + 2k0 +1 < 2k0 +2 and |y| ≥ |x0 | − |y − x0 | > 3 · 2k0 −1 − 2k0 = 2k0 −1 . This implies that B(x0 , 2k0 ) ⊂ B(0, 2k0 +2 ) \ B(0, 2k0 −1 ). Applying this and Lemma 1.1.3, we conclude that 2

nk0 q p

q  q  q  p  ω(2k0 ) ∼ ω(2k0 ) B(x0 , 2k0 )  q    q  p  ω(2k0 +2 ) B(x0 , 2k0 ) ∩ B(0, 2k0 +2 ) \ B(0, 2k0 +1 )   q    q p  + ω(2k0 +1 ) B(x0 , 2k0 ) ∩ B(0, 2k0 +1 ) \ B(0, 2k0 )   q    q p  + ω(2k0 ) B(x0 , 2k0 ) ∩ B(0, 2k0 ) \ B(0, 2k0 −1 )   q    1B(x0,2k0 )  ˙ p,q n , (4.127) Kω,0 (R )

which completes the estimation of (4.122) in this case. Case (4) |x0 | ∈ [2k , 2k+1 ) with k ∈ Z ∩ [k0 + 1, ∞). In this case, from Lemma 1.5.2, it follows that, for any 0 < t < τ < ∞, ω(t)  ω(τ )

 min{m0 (ω),m∞ (ω)}−ε t , τ

where ε ∈ (0, min{m0 (ω), m∞ (ω)}) is a fixed positive constant. Using this, Lemma 1.1.3, and the assumptions k ∈ [k0 + 1, ∞) and ε ∈ (0, min{m0 (ω), m∞ (ω)}), we further find that ω(2k0 +1 ) ω(2k0 ) ∼  ω(2k ) ω(2k )



2k0 +1 2k

min{m0 (ω),m∞ (ω)}−ε  1.

(4.128)

On the other hand, applying the assumption |x0 | ∈ [2k , 2k+1 ) with k ∈ Z ∩ [k0 + 1, ∞), we conclude that, for any y ∈ B(x0 , 2k0 ), |y| ≤ |y − x0 | + |x0 | < 2k0 + 2k+1 ≤ 2k−1 + 2k+1 < 2k+2

296

4 Generalized Herz–Hardy Spaces

and |y| ≥ |x0 | − |y − x0 | > 2k − 2k0 ≥ 2k − 2k−1 = 2k−1 . This implies that B(x0 , 2k0 ) ⊂ B(0, 2k+2 ) \ B(0, 2k−1 ).

(4.129)

Thus, from (4.128) and an argument similar to that used in the estimation of (4.127), we deduce that 2

nk0 p

  n    p  ω(2k0 )  ω(2k ) B(x0 , 2k0 )  1B(x0,2k0 )  ˙ p,q

Kω,0 (Rn )

,

which, together with (4.123), (4.126), and (4.127), implies that (4.122) holds true. Next, we show that, for any B(x0 , r) ∈ B with x0 ∈ Rn and r ∈ (0, ∞),   n r p ω(r)  1B(x0,r) K˙ p,q (Rn ) .

(4.130)

ω,0

Indeed, for any r ∈ (0, ∞), there exists a k ∈ Z such that r ∈ [2k , 2k+1 ). Thus, applying Lemma 1.1.3 and (4.122), we find that     n nk r p ω(r) ∼ 2 p ω(2k )  1B(x0,2k ) K˙ p,q (Rn )  1B(x0,r) K˙ p,q (Rn ) . ω,0

ω,0

This finishes the proof of (4.130). Finally, from the definition of p− and Lemma 1.1.12, we infer that, for any r ∈ (0, 1], ω(r)  r M0 (ω)+ε and, for any r ∈ (1, ∞), ω(r)  r m∞ (ω)−ε , where    n n n ε ∈ 0, min − M0 (ω) − , −n + m∞ (ω) + p− p p

(4.131)

4.9 Fourier Transform

297

is a fixed positive constant. Therefore, using the assumption ε ∈ (0, pn− − M0 (ω) − n n p ) and (4.130), we conclude that, for any x0 ∈ R and r ∈ (0, 1], 1 n   n n |B(x0 , r)| p− ∼ r p−  r M0 (ω)+ p +ε  r p ω(r)  1B(x0,r) K˙ p,q (Rn ) .

(4.132)

ω,0

Moreover, applying the fact that ε ∈ (0, −n + m∞ (ω) + pn ), (4.131), and (4.130), we find that, for any x0 ∈ Rn and r ∈ (1, ∞),   n n |B(x0 , r)| ∼ r n  r m∞ (ω)+ p −ε  r p ω(r)  1B(x0,r) K˙ p,q (Rn ) ,

(4.133)

ω,0

which, together with (4.132), further implies that (4.121) holds true and hence completes the proof of Lemma 4.9.4.   Via both Lemmas 4.9.2 and 4.9.4, we now show Theorem 4.9.1. Proof of Theorem 4.9.1 Let all the symbols be as in the present theorem. Then, from the assumption m0 (ω) ∈ (0, ∞) and Theorem 1.2.42, it follows that the local p,q generalized Herz space K˙ ω,0 (Rn ) is a BQBF space. This implies that, in order to complete the proof of the present theorem, we only need to show that the Herz p,q space K˙ ω,0 (Rn ) under consideration satisfies all the assumptions of Lemma 4.9.2. p,q Namely, K˙ ω,0 (Rn ) satisfies (4.119), the p0 -concavity for some p0 ∈ [p− , 1], and (4.120). p,q We first prove that (4.119) holds true for K˙ ω,0 (Rn ). Indeed, by the fact that min{m0 (ω), m∞ (ω)} ∈ (0, ∞) and Remark 1.1.5(iii), we find that max{M0 (ω), M∞ (ω)} ∈ (0, ∞) and hence n ∈ (0, p). max{M0 (ω), M∞ (ω)} + n/p Applying this and Lemma 4.3.10, we conclude that, for any given θ ∈ (0, p− ) and u ∈ (1, ∞), and for any {fj }j ∈N ⊂ L1loc (Rn ), ⎧ ⎫1   u ⎨  u ⎬    M(fj )   ⎩ ⎭   j ∈N 

˙ p,q (Rn )]1/θ [K ω,0

⎛ ⎞1  u    ⎝ u⎠   |fj |     j ∈N 

p,q This further implies that K˙ ω,0 (Rn ) satisfies (4.119).

. ˙ p,q (Rn )]1/θ [K ω,0

(4.134)

298

4 Generalized Herz–Hardy Spaces

In addition, from p, q ∈ (0, 1], it follows that, for any {fj }j ∈N ⊂ K˙ ω,0 (Rn ), p,q

 j ∈N

fj K˙ p,q (Rn ) ω,0

       ≤ |fj |  j ∈N  ˙ p,q

Kω,0

,

(4.135)

(Rn )

p,q p,q which implies that K˙ ω,0 (Rn ) is strictly 1-concave and hence K˙ ω,0 (Rn ) satisfies the p0 -concavity with p0 = 1. Finally, applying Lemma 4.9.4, we conclude that, for any B ∈ B,

  1 1B K˙ p,q (Rn )  min |B|, |B| p− . ω,0

Thus, (4.120) also holds true for K˙ ω,0 (Rn ) with p0 = 1. This, combined with both (4.134) and (4.135), further implies that all the assumptions of Lemma 4.9.2 hold p,q true for K˙ ω,0 (Rn ), and hence finishes the proof of Theorem 4.9.1.   p,q

Next, we investigate the Fourier transform properties of the generalized Herz– p,q Hardy space H K˙ ω (Rn ). Indeed, we have the following conclusion. Theorem 4.9.5 Let p ∈ (0, 1), q ∈ (0, 1], ω ∈ M(R+ ) with m0 (ω) ∈ (− pn , ∞) and   1 < m∞ (ω) ≤ M∞ (ω) < 0, n 1− p p,q n and p− ∈ (0, min{p, max{M0 (ω),M ). Then, for any f ∈ H K˙ ω (Rn ), there ∞ (ω)}+n/p exists a continuous function g on Rn such that f = g in S  (Rn ) and

|g(x)|

lim

|x|→0+

|x|

n( p1 −1)

= 0.

−

Moreover, there exists a positive constant C, independent of both f and g, such that, for any x ∈ Rn , |g(x)| ≤ Cf 

n ˙ p,q HK ω (R )

  n( p1 −1) − max 1, |x|

and  Rn

|g(x)| min |x|

− pn

−

! , |x|−n dx ≤ Cf H K˙ p,q n . ω (R )

4.9 Fourier Transform

299

To show Theorem 4.9.5, we need the following technical estimate for the quasinorm of the characteristic function of balls on the global generalized Herz space p,q K˙ ω (Rn ). Lemma 4.9.6 Let p, q, ω, and p− be as in Theorem 4.9.5. Then there exists a positive constant C such that, for any B ∈ B,  1B K˙ p,q n ≥ C min |B|, |B| ω (R )

1 p−

 (4.136)

.

Proof Let all the symbols be as in the present lemma and B := B(x0 , r) ∈ B with x0 ∈ Rn and r ∈ (0, ∞). Then, there exists a k ∈ Z such that r ∈ [2k , 2k+1 ). This implies that B(x0 , 2k ) \ B(x0 , 2k−1 ) ⊂ B(x0 , r). Therefore, by Lemma 1.1.3 and the definition of the quasi-norm  · K˙ p,q n , we find that ω (R )   n nk r p ω(r) ∼ 2 p ω(2k ) ∼ ω(2k ) 1B(x0,2k )\B(x0,2k−1 ) Lp (Rn )    1B(x0,r) K˙ p,q (Rn ) . ω

(4.137)

Moreover, from both the definition of p− and Lemma 1.1.12, it follows that, for any r ∈ (0, 1], ω(r)  r M0 (ω)+ε

(4.138)

ω(r)  r m∞ (ω)−ε ,

(4.139)

and, for any r ∈ (1, ∞),

where    n n n ε ∈ 0, min − M0 (ω) − , −n + m∞ (ω) + p− p p is a fixed positive constant. We now show (4.136) by considering the following two cases on r. Case (1) r ∈ (0, 1]. In this case, using the assumption ε ∈ (0, pn− − M0 (ω) − pn ), (4.138), and (4.137), we conclude that 1 n   n n |B(x0 , r)| p− ∼ r p−  r M0 (ω)+ p +ε  r p ω(r)  1B(x0,r) K˙ p,q (Rn ) . ω

This finishes the estimation of (4.136) in this case.

(4.140)

300

4 Generalized Herz–Hardy Spaces

Case (2) r ∈ (1, ∞). In this case, applying the fact that ε ∈ (0, −n+m∞ (ω)+ pn ), (4.139), and (4.137), we find that   n n |B(x0 , r)| ∼ r n  r m∞ (ω)+ p −ε  r p ω(r)  1B(x0,r) K˙ p,q (Rn ) , ω

which completes the proof of (4.136) in this case. Combining this and (4.140), we conclude that Lemma 4.9.6 holds true.   Based on the above lemma and the Fourier transform properties of Hardy spaces associated with ball quasi-Banach function spaces established in Lemma 4.9.2, we next prove Theorem 4.9.5. Proof of Theorem 4.9.5 Let all the symbols be as in the present theorem. Then, by the assumptions m0 (ω) ∈ (− pn , ∞) and M∞ (ω) ∈ (−∞, 0), and Theorem 1.2.44, p,q we find that the global generalized Herz space K˙ ω (Rn ) under consideration is a BQBF space. Thus, to finish the proof of the present theorem, it suffices to prove p,q that the Herz space K˙ ω (Rn ) satisfies all the assumptions of Lemma 4.9.2. First, we show that (4.119) holds true. Indeed, from Lemma 4.3.24 with r = θ , it follows that, for any given θ ∈ (0, p− ) and u ∈ (1, ∞), and for any {fj }j ∈N ⊂ L1loc (Rn ), ⎧ ⎫1   u  ⎨  u ⎬    M(fj )   ⎩ ⎭   j ∈N 

n 1/θ ˙ p,q [K ω (R )]

⎛ ⎞1   u    ⎝ u⎠  ≤C |fj |     j ∈N 

,

n 1/θ ˙ p,q [K ω (R )]

p,q which implies that K˙ ω (Rn ) satisfies (4.119). p,q Next, we prove that K˙ ω (Rn ) is strictly 1-concave. Indeed, applying the p,q assumptions p, q ∈ (0, 1], we conclude that, for any {fj }j ∈N ⊂ K˙ ω (Rn ),

 j ∈N

fj K˙ p,q n ω (R )

       ≤ |fj |  j ∈N  ˙ p,q Kω

. (Rn )

p,q This implies that K˙ ω (Rn ) is strictly p0 -concave with p0 := 1. Finally, using Lemma 4.9.6, we find that, for any B ∈ B,

1B K˙ p,q n ω (R )

  1 p  min |B|, |B| − ,

p,q which implies that (4.120) holds true for K˙ ω (Rn ) with p0 := 1. Thus, the Herz p,q space K˙ ω (Rn ) under consideration satisfies (4.119), 1-concavity, and (4.120) with p0 := 1. By this and Lemma 4.9.2, we then obtain Theorem 4.9.5.  

Chapter 5

Localized Generalized Herz–Hardy Spaces

Recall that, in 1979, Goldberg [89] introduced the local Hardy space hp (Rn ), with p ∈ (0, ∞), as a localization of the classical Hardy spaces H p (Rn ) and showed some properties which are different from the classical Hardy spaces. Indeed, Goldberg proved that the local Hardy space hp (Rn ) contains S(Rn ) as a dense subspace and established the boundedness of pseudo-differential operators of order zero on hp (Rn ). Moreover, due to [89], the localized Hardy space is well defined on manifolds. After that, lots of nice works have been done in the study of localized Hardy spaces and their applications; see, for instance, [27, 51, 181, 273]. Specially, some variants of the local Hardy spaces hp (Rn ) have also been studied; see, for instance, [223] for weighted localized Hardy spaces, [234] for localized Herz–Hardy spaces, [271] for (weighted) localized Orlicz–Hardy spaces, and [272] for localized Musielak–Orlicz–Hardy spaces. Very recently, Sawano et al. [207] introduced the local Hardy space hX (Rn ) associated with the ball quasi-Banach function space X and gave various maximal function characterizations of hX (Rn ). Wang et al. [233] then established atomic, molecular, and various Littlewood–Paley function characterizations of hX (Rn ) as well as showed the boundedness of pseudodifferential operators on these localized Hardy spaces. The main target of this chapter is devoted to introducing the localized generalized Herz–Hardy spaces and then establishing their complete real-variable theory. To achieve this, we begin with showing their various maximal function characterizations via the known maximal function characterizations of the local Hardy space hX (Rn ) associated with the ball quasi-Banach function space X. Then, to overcome the obstacle caused by the deficiency of associate spaces of global generalized Herz spaces, we establish improved atomic and molecular characterizations of hX (Rn ) as well as the boundedness of pseudo-differential operators on hX (Rn ) without recourse to associate spaces (see Theorems 5.3.14, 5.4.11, and 5.6.9 below). Via these improved conclusions, we establish the atomic and molecular characterizations of localized generalized Herz–Hardy spaces and the boundedness of pseudo-differential operators on localized generalized Herz–Hardy spaces. In © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 Y. Li et al., Real-Variable Theory of Hardy Spaces Associated with Generalized Herz Spaces of Rafeiro and Samko, Lecture Notes in Mathematics 2320, https://doi.org/10.1007/978-981-19-6788-7_5

301

302

5 Localized Generalized Herz–Hardy Spaces

addition, to clarify the relation between localized generalized Herz–Hardy spaces and generalized Herz–Hardy spaces, we find the relation between hX (Rn ) and Hardy spaces HX (Rn ) associated with the ball quasi-Banach function space X (see Theorem 5.2.3 below). Applying this and some auxiliary lemmas about generalized Herz spaces, we then obtain the relation between localized generalized Herz–Hardy spaces and generalized Herz–Hardy spaces. As applications, we also establish various Littlewood–Paley function characterizations of hX (Rn ) and hence, together with the construction of the quasi-norm  · K˙ p,q n , we further obtain the ω (R ) Littlewood–Paley function characterizations of localized generalized Herz–Hardy spaces. In addition, we introduce localized generalized Hardy–Morrey spaces. Using the equivalence between generalized Herz spaces and generalized Morrey spaces, we also conclude the corresponding real-variable characterizations and applications of localized generalized Hardy–Morrey spaces in this chapter. For any given N ∈ N and for any f ∈ S  (Rn ), the local grand maximal function mN (f ) is defined by setting, for any x ∈ Rn ,   mN (f )(x) := sup |f ∗ φt (y)| : t ∈ (0, 1), |x − y| < t, φ ∈ FN (Rn ) , (5.1) where FN (Rn ) is as in (4.2). Then we introduce localized generalized Herz–Hardy spaces as follows. Definition 5.0.1 Let p, q ∈ (0, ∞), ω ∈ M(R+ ), and N ∈ N. Then p,q (i) the local generalized Herz–Hardy space hK˙ ω,0 (Rn ), associated with the local p,q generalized Herz space K˙ ω,0 (Rn ), is defined to be the set of all the f ∈ S  (Rn ) such that

f hK˙ p,q (Rn ) := mN (f )K˙ p,q (Rn ) < ∞; ω,0

ω,0

(ii) the local generalized Herz–Hardy space hK˙ ω (Rn ), associated with the global p,q generalized Herz space K˙ ω (Rn ), is defined to be the set of all the f ∈ S  (Rn ) such that p,q

f hK˙ p,q n := mN (f )K n < ∞. ˙ p,q ω (R ) ω (R ) Remark 5.0.2 In Definition 5.0.1, for any given α ∈ R and for any t ∈ (0, ∞), let p,q ω(t) := t α . Then, in this case, the local generalized Herz–Hardy space hK˙ ω,0 (Rn ) α,q goes back to the classical homogeneous local Herz-type Hardy space hK˙ p (Rn ) which was originally introduced in [72, Definition 1.2] (see also [175, Section 2.6]). However, to the best of our knowledge, even in this case, the local generalized Herz– p,q Hardy space hK˙ ω (Rn ) is also new.

5.1 Maximal Function Characterizations

303

Recall that local and global generalized Morrey spaces are given in Remark 1.2.2(vi). We now introduce the following concepts of localized generalized Hardy–Morrey spaces. Definition 5.0.3 Let p, q ∈ (0, ∞), ω ∈ M(R+ ), and N ∈ N. p,q

(i) The local generalized Hardy–Morrey space hM ω,0 (Rn ), associated with the p,q local generalized Morrey space M ω,0 (Rn ), is defined to be the set of all the f ∈ S  (Rn ) such that f hM p,q (Rn ) := mN (f )M p,q (Rn ) < ∞; ω,0

ω,0

p,q

(ii) The local generalized Hardy–Morrey space hM ω (Rn ), associated with the p,q global generalized Morrey space M ω (Rn ), is defined to be the set of all the f ∈ S  (Rn ) such that f hM p,q n := mN (f )M p,q (Rn ) < ∞. ω (R ) ω Remark 5.0.4 (i) We should point out that, to the best of our knowledge, in Definition 5.0.3, even when ω(t) := t α for any t ∈ (0, ∞) and for any given α ∈ R, the local p,q p,q generalized Hardy–Morrey spaces hM ω,0 (Rn ) and hM ω (Rn ) are also new. (ii) In Definition 5.0.3, let p, q ∈ [1, ∞) and ω satisfy max{M0 (ω), M∞ (ω)} ∈ (−∞, 0). Then, in this case, by Remark 1.2.2(vi), we find that the local generalized p,q p,q Hardy–Morrey spaces hM ω,0 (Rn ) and hM ω (Rn ) coincide, respectively, with p,q p,q the local generalized Herz–Hardy spaces hK˙ ω,0 (Rn ) and hK˙ ω (Rn ) in the sense of equivalent norms.

5.1 Maximal Function Characterizations In this section, we establish the maximal function characterizations of localized generalized Herz–Hardy spaces. To begin with, we recall the following definitions of various localized maximal functions (see, for instance, [207, Definition 5.1]). Definition 5.1.1 Let N ∈ N, a, b ∈ (0, ∞), φ ∈ S(Rn ), and f ∈ S  (Rn ). (i) The local radial maximal function m(f, φ) is defined by setting, for any x ∈ Rn , m(f, φ)(x) := sup |f ∗ φt (x)|. t ∈(0,1)

304

5 Localized Generalized Herz–Hardy Spaces

(ii) The local non-tangential maximal function m∗a (f, φ), with aperture a ∈ (0, ∞), is defined by setting, for any x ∈ Rn , m∗a (f, φ)(x) := sup

sup

t ∈(0,1) {y∈Rn : |y−x| n. Assume that X is strictly r-convex and there exists a positive constant C such that, for any g ∈ X and z ∈ Rn ,  1   r   r |g(· − y)| dy  ≤ C(1 + |z|)A gX .   z+[0,1]n  X

Then, for any f ∈ S  (Rn ), m∗∗ b (f, φ)X  m(f, φ)X ,

(5.2)

5.1 Maximal Function Characterizations

307

where the implicit positive constant is independent of f . In particular, when N ∈ N ∩ [b + 1, ∞), if one of the quantities m(f, φ)X , m∗a (f, φ)X , mN (f )X , ∗∗ m∗∗ b (f, φ)X , and mb,N (f )X

is finite, then the others are also finite and mutually equivalent with the positive equivalence constants independent of f . Remark 5.1.6 Let X be a ball quasi-Banach function space satisfying Assumption 1.2.29 for some 0 < θ < s ≤ 1 and Assumption 1.2.33 for the same s. Then we claim that X satisfies all the assumptions of Lemma 5.1.5(ii). Indeed, by Assumption 1.2.33 and [29, Lemma 2.6] with X and p therein replaced, respectively, by X1/s and θs , we find that X1/θ is a ball Banach function space and hence X is strictly θ -convex. On the other hand, from both Assumption 1.2.29 and Lemma 5.1.4 with r := θ , it follows that (5.2) holds true with r := θ and A ∈ ( nθ , ∞). This then implies that all the assumptions of Lemma 5.1.5(ii) hold true for the above X and hence finishes the proof of the above claim. Remark 5.1.7 We should point out that Lemma 5.1.5 has a wide range of applications. Here we present several function spaces to which Lemma 5.1.5 can be applied. (i) Let p ∈ (0, ∞) and b ∈ ( 2n p , ∞). Then, in this case, combining Remarks 1.2.31(i), 1.2.34(i), and 5.1.6, we can easily find that the Lebesgue space Lp (Rn ) satisfies all the assumptions of Lemma 5.1.5. This then implies that Lemma 5.1.5 with X := Lp (Rn ) holds true, which coincides with the known maximal function characterizations of localized Hardy spaces established in [89, Theorem 1]. υ (ii) Let p ∈ (0, ∞), υ ∈ A∞ (Rn ), and b ∈ ( 2nq p , ∞), where qυ is the same as in (1.58). Then, in this case, from Remarks 1.2.31(ii), 1.2.34(ii), and 5.1.6, p we can easily infer that the weighted Lebesgue space Lυ (Rn ) satisfies all the assumptions of Lemma 5.1.5. Therefore, Lemma 5.1.5 with X therein replaced p by Lυ (Rn ) holds true. This result coincides with [18, Theorem 4.2]. (iii) Let p := (p1 , . . . , pn ) ∈ (0, ∞)n and b ∈ ( min{p2n , ∞). Then, in this 1 ,...,pn } case, using Remarks 1.2.31(iii), 1.2.34(iii), and 5.1.6, we can easily conclude that the mixed-norm Lebesgue space Lp (Rn ) satisfies all the assumptions of Lemma 5.1.5. Thus, Lemma 5.1.5 with X := Lp (Rn ) holds true. To the best of our knowledge, this result is totally new. (iv) Let 0 < q ≤ p < ∞ and b ∈ ( 2n q , ∞). Then, in this case, by Remarks 1.2.31(iv), 1.2.34(iv), and 5.1.6, we easily find that the Morrey space p Mq (Rn ) satisfies all the assumptions of Lemma 5.1.5. This further implies p that Lemma 5.1.5 with X replaced by Mq (Rn ) holds true. To the best of our knowledge, this result is totally new.

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5 Localized Generalized Herz–Hardy Spaces

(v) Let p(·) ∈ C log (Rn ) and b ∈ ( p2n− , ∞), where p− is the same as in (1.59). Then, in this case, from Remarks 1.2.31(v), 1.2.34(v), and 5.1.6, we can easily deduce that the variable Lebesgue space Lp(·)(Rn ) satisfies all the assumptions of Lemma 5.1.5. Thus, Lemma 5.1.5 with X := Lp(·) (Rn ) holds true. This result was also pointed out in [186, p. 3747]. Via the above two lemmas, we now show Theorem 5.1.2. Proof of Theorem 5.1.2 Let all the symbols be as in the present theorem. Then, from the assumption m0 (ω) ∈ (− pn , ∞) and Theorem 1.2.42, it follows that the p,q local generalized Herz space K˙ ω,0 (Rn ) under consideration is a BQBF space. This, together with Lemma 5.1.5(i), then finishes the proof of (i). Next, we show (ii). Indeed, by Lemma 5.1.5(ii), we conclude that it suffices to prove that all the assumptions of Lemma 5.1.5(ii) hold true for the Herz space p,q K˙ ω,0 (Rn ) under consideration. Namely, there exist r, A ∈ (0, ∞) with (b−A)r > n p,q p,q satisfying that K˙ ω,0 (Rn ) is strictly r-convex and, for any f ∈ K˙ ω,0 (Rn ) and z ∈ Rn ,  1   r   |f (· − y)|r dy    z+[0,1]n  ˙ p,q

Kω,0

 (1 + |z|)A f K˙ p,q (Rn ) . ω,0

(Rn )

(5.3)

To achieve this, let    n 2n , min p, q, r∈ . b max{M0 (ω), M∞ (ω)} + n/p Then, from Theorem 1.3.3 with s := r, we deduce that the local generalized Herz p,q space K˙ ω,0 (Rn ) is strictly r-convex. On the other hand, let A ∈ ( nr , b − nr ). We then prove that (5.3) holds true for this A. Indeed, by Lemma 1.8.5 and Remark 1.2.30(i), we find that, for any f ∈ L1loc (Rn ),    (r)  M (f ) ˙ p,q

Kω,0 (Rn )

 f K˙ p,q (Rn ) . ω,0

Applying this, Lemma 5.1.4, and the assumption A > nr , we further conclude that, p,q for any f ∈ K˙ ω,0 (Rn ) and z ∈ Rn ,  1   r   r |f (· − y)| dy    z+[0,1]n  ˙ p,q

Kω,0 (Rn )

n

 (1 + |z|) r f K˙ p,q (Rn )  (1 + |z|)A f K˙ p,q (Rn ) . ω,0

ω,0

5.1 Maximal Function Characterizations

309

This finishes the proof of (5.3). Thus, under the assumptions of the present theorem, p,q K˙ ω,0 (Rn ) satisfies all the assumptions of Lemma 5.1.5(ii), which completes the proof of (ii) and hence Theorem 5.1.2.   Remark 5.1.8 We point out that, in Theorem 5.1.2, if ω(t) := t α for any t ∈ (0, ∞) and for any given α ∈ [n(1 − p1 ), ∞), then Theorem 5.1.2 goes back to [175, Theorem 2.6.1]. Via Theorem 5.1.2 and Remark 5.0.4(ii), we immediately obtain the maximal p,q function characterizations of the local generalized Hardy–Morrey space hM ω,0 (Rn ) as follows; we omit the details. Corollary 5.1.9 Let a, b ∈ (0, ∞), p, q ∈ [1, ∞), ω ∈ M(R+ ), N ∈ N, and φ ∈ S(Rn ) satisfy  Rn

φ(x) dx = 0.

(i) Let N ∈ N ∩ [b + 1, ∞) and ω satisfy M∞ (ω) ∈ (−∞, 0) and −

n < m0 (ω) ≤ M0 (ω) < 0. p

Then, for any f ∈ S  (Rn ), m(f, φ)M p,q (Rn )  m∗a (f, φ)M p,q (Rn )  m∗∗ b (f, φ)M p,q (Rn ) , ω,0

ω,0

ω,0

m(f, φ)M p,q (Rn )  mN (f )M p,q (Rn )  mb+1 (f )M p,q (Rn ) ω,0

ω,0



ω,0

m∗∗ n , b (f, φ)M p,q ω,0 (R )

and ∗∗ m∗∗ b (f, φ)M p,q (Rn ) ∼ mb,N (f )M p,q (Rn ) , ω,0

ω,0

where the implicit positive constants are independent of f . (ii) Let ω satisfy −

n < m0 (ω) ≤ M0 (ω) < 0 p

and −

n < m∞ (ω) ≤ M∞ (ω) < 0, p

310

5 Localized Generalized Herz–Hardy Spaces

and b ∈ (2n max{ p1 , q1 }, ∞). Then, for any f ∈ S  (Rn ), m∗∗ b (f, φ)M p,q (Rn )  m(f, φ)M p,q (Rn ) , ω,0

ω,0

where the implicit positive constant is independent of f . In particular, when N ∈ N ∩ [b + 1, ∞), if one of the quantities m(f, φ)M p,q (Rn ) , m∗a (f, φ)M p,q (Rn ) , mN (f )M p,q (Rn ) , ω,0

ω,0

ω,0

∗∗ m∗∗ b (f, φ)M p,q (Rn ) , and mb,N (f )M p,q (Rn ) ω,0

ω,0

is finite, then the others are also finite and mutually equivalent with the positive equivalence constants independent of f . Next, we establish the following maximal function characterizations of the p,q local generalized Herz–Hardy space hK˙ ω (Rn ) with the help of the known maximal function characterizations of the local Hardy space hX (Rn ) presented in Lemma 5.1.5. Theorem 5.1.10 Let p, q, a, b ∈ (0, ∞), ω ∈ M(R+ ), N ∈ N, and φ ∈ S(Rn ) satisfy  Rn

φ(x) dx = 0.

(i) Let N ∈ N ∩ [b + 1, ∞) and ω satisfy m0 (ω) ∈ (− pn , ∞) and M∞ (ω) ∈ (−∞, 0). Then, for any f ∈ S  (Rn ), ∗ ∗∗ m(f, φ)K˙ p,q n  ma (f, φ)K n  mb (f, φ)K n , ˙ p,q ˙ p,q ω (R ) ω (R ) ω (R )

m(f, φ)K˙ p,q n  mN (f )K n  mb+1 (f )K n ˙ p,q ˙ p,q ω (R ) ω (R ) ω (R )  m∗∗ n , ˙ p,q b (f, φ)K ω (R ) and ∗∗ m∗∗ n ∼ mb,N (f )K n , ˙ p,q ˙ p,q b (f, φ)K ω (R ) ω (R )

where the implicit positive constants are independent of f . (ii) Let ω satisfy m0 (ω) ∈ (− pn , ∞) and −

n < m∞ (ω) ≤ M∞ (ω) < 0, p

5.1 Maximal Function Characterizations

311

and     n n n , , max{M0 (ω), M∞ (ω)} + ,∞ . b ∈ 2 max p q p Then, for any f ∈ S  (Rn ), m∗∗ n  m(f, φ)K n , ˙ p,q ˙ p,q b (f, φ)K ω (R ) ω (R ) where the implicit positive constant is independent of f . In particular, when N ∈ N ∩ [b + 1, ∞), if one of the quantities ∗ m(f, φ)K˙ p,q n , ma (f, φ)K n , mN (f )K n , ˙ p,q ˙ p,q ω (R ) ω (R ) ω (R ) ∗∗ m∗∗ n , and mb,N (f )K n ˙ p,q ˙ p,q b (f, φ)K ω (R ) ω (R )

is finite, then the others are also finite and mutually equivalent with the positive equivalence constants independent of f . Proof Let all the symbols be as in the present theorem. Note that ω satisfies m0 (ω) ∈ (− pn , ∞) and M∞ (ω) ∈ (−∞, 0). From this and Theorem 1.2.44, it p,q follows that the global generalized Herz space K˙ ω (Rn ) under consideration is a BQBF space. Thus, applying Lemma 5.1.5(i), we then complete the proof of (i). p,q Now, we prove (ii). Indeed, we only need to show that the Herz space K˙ ω (Rn ) under consideration satisfies all the assumptions of Lemma 5.1.5(ii). For this purpose, let r, A ∈ (0, ∞) be such that  r∈

  2n n , min p, q, b max{M0 (ω), M∞ (ω)} + n/p

and A ∈ ( nr , b − nr ). Then, applying Theorem 1.3.4 with s := r, we conclude that p,q the global generalized Herz space K˙ ω (Rn ) is strictly r-convex. On the other hand, from Lemma 4.1.10 and Remark 1.2.30(i), we deduce that, for any f ∈ L1loc (Rn ),    (r)  M (f ) ˙ p,q

Kω (Rn )

 f K˙ p,q n . ω (R )

By this, Lemma 5.1.4, and the assumption A > p,q f ∈ K˙ ω (Rn ) and z ∈ Rn ,

n r,

we further find that, for any

 1   r   |f (· − y)|r dy    z+[0,1]n  ˙ p,q

Kω (Rn )

n

A  (1 + |z|) r f K˙ p,q n  (1 + |z|) f K n . ˙ p,q ω (R ) ω (R )

312

5 Localized Generalized Herz–Hardy Spaces

Combining this and the fact that K˙ ω (Rn ) is strictly r-convex, we conclude that all p,q the assumptions of Lemma 5.1.4(ii) hold true for K˙ ω (Rn ). This finishes the proof of (ii) and hence Theorem 5.1.10.   p,q

Remark 5.1.11 p,q (i) Let p, q ∈ (0, ∞) and ω ∈ M(R+ ). Then the quasi-norm of hK˙ ω (Rn ) in Definition 5.0.1(ii) depends on N. However, by Theorem 5.1.10, we conclude p,q that the local Hardy space hK˙ ω (Rn ) is independent of the choice of N whenever ω ∈ M(R+ ) satisfies m0 (ω) ∈ (− pn , ∞) and

−

n < m∞ (ω) ≤ M∞ (ω) < 0, p

and N satisfies     n n n , , max{M0 (ω), M∞ (ω)} + N ∈ N ∩ 1 + 2 max ,∞ . p q p (ii) Notice that, if p = q ∈ (0, ∞) and ω(t) := 1 for any t ∈ (0, ∞), then, in this case, m0 (ω) = M0 (ω) = m∞ (ω) = M∞ (ω) = 0 p,q and hK˙ ω (Rn ) coincides with the classical local Hardy space hp (Rn ) in the sense of equivalent quasi-norms. Therefore, Theorem 5.1.10 completely excludes the classical local Hardy space hp (Rn ) and, by Remark 1.2.45, we conclude that the classical local Hardy space hp (Rn ) is the critical case of p,q hK˙ ω (Rn ) considered in Theorem 5.1.10.

Using Theorem 5.1.10 and Remark 5.0.4(ii), we immediately obtain the following maximal function characterizations of the local generalized Hardy–Morrey p,q space hM ω (Rn ); we omit the details. Corollary 5.1.12 Let a, b ∈ (0, ∞), p, q ∈ [1, ∞), ω ∈ M(R+ ), N ∈ N, and φ ∈ S(Rn ) satisfy  φ(x) dx = 0. Rn

(i) Let N ∈ N ∩ [b + 1, ∞) and ω satisfy M∞ (ω) ∈ (−∞, 0) and −

n < m0 (ω) ≤ M0 (ω) < 0. p

5.1 Maximal Function Characterizations

313

Then, for any f ∈ S  (Rn ), ∗ ∗∗ m(f, φ)M p,q n  ma (f, φ)M p,q (Rn )  mb (f, φ)M p,q (Rn ) , ω (R ) ω ω

m(f, φ)M p,q n  mN (f )M p,q (Rn )  mb+1 (f )M p,q (Rn ) ω (R ) ω ω  m∗∗ n , b (f, φ)M p,q ω (R ) and ∗∗ m∗∗ n ∼ mb,N (f )M p,q (Rn ) , b (f, φ)M p,q ω (R ) ω

where the implicit positive constants are independent of f . (ii) Let ω satisfy −

n < m0 (ω) ≤ M0 (ω) < 0 p

and −

n < m∞ (ω) ≤ M∞ (ω) < 0, p

and b ∈ (2n max{ p1 , q1 }, ∞). Then, for any f ∈ S  (Rn ), m∗∗ n  m(f, φ)M p,q (Rn ) , b (f, φ)M p,q ω (R ) ω where the implicit positive constant is independent of f . In particular, when N ∈ N ∩ [b + 1, ∞), if one of the quantities ∗ m(f, φ)M p,q n , ma (f, φ)M p,q (Rn ) , mN (f )M p,q (Rn ) , ω (R ) ω ω ∗∗ m∗∗ n , and mb,N (f )M p,q (Rn ) b (f, φ)M p,q ω (R ) ω

is finite, then the others are also finite and mutually equivalent with the positive equivalence constants independent of f . Remark 5.1.13 Let p, q ∈ (0, ∞) and ω ∈ M(R+ ). Then the quasi-norms of p,q p,q hM ω,0 (Rn ) and hM ω (Rn ) in Definition 5.0.3 depend on N. However, from p,q Corollaries 5.1.9 and 5.1.12, we deduce that the local Hardy spaces hM ω,0 (Rn ) p,q and hM ω (Rn ) are both independent of the choice of N whenever p, q ∈ [1, ∞), ω ∈ M(R+ ) satisfies −

n < m0 (ω) ≤ M0 (ω) < 0 p

314

5 Localized Generalized Herz–Hardy Spaces

and −

n < m∞ (ω) ≤ M∞ (ω) < 0, p

and N satisfies     1 1 , N ∈ N ∩ 1 + 2n max ,∞ . p q

5.2 Relations with Generalized Herz–Hardy Spaces The target of this section is to investigate the relation between localized generalized Herz–Hardy spaces and generalized Herz–Hardy spaces. For this purpose, we first establish a relation between localized Hardy spaces and Hardy spaces associated with ball quasi-Banach function spaces, which extends the results for various concrete function spaces (see Remark 5.2.4 below). Using this general conclusion, we then obtain the relation between localized generalized Herz–Hardy spaces and generalized Herz–Hardy spaces. First, we give the relation between localized Hardy spaces and Hardy spaces associated with ball quasi-Banach function spaces. To this end, we first recall the definition of localized Hardy spaces associated with ball quasi-Banach function spaces as follows, which was introduced in [207, Definition 5.2]. Definition 5.2.1 Let X be a ball quasi-Banach function space and N ∈ N. Then the local Hardy space hX (Rn ) is defined to be the set of all the f ∈ S  (Rn ) such that f hX (Rn ) := mN (f )X < ∞. Remark 5.2.2 (i) Let p ∈ (0, ∞) and X := Lp (Rn ). Then, in this case, hX (Rn ) is just the local Hardy space hp (Rn ) which was originally introduced in [89]. p (ii) Let p ∈ (0, ∞), υ ∈ A∞ (Rn ), and X := Lυ (Rn ). Then, in this case, hX (Rn ) p is just the weighted local Hardy space hυ (Rn ) which was originally studied in [18]. (iii) Let p := (p1 , . . . , pn ) ∈ (0, ∞)n and X := Lp (Rn ). Then the mixed-norm local Hardy space hp (Rn ) is defined as in Definition 5.2.1 with X therein replaced by Lp (Rn ). To the best of our knowledge, hp (Rn ) is totally new. p (iv) Let 0 < q ≤ p < ∞ and X := Mq (Rn ). Then, in this case, HX (Rn ) is just p the local Hardy–Morrey space hMq (Rn ) which was originally introduced in [205, Section 4].

5.2 Relations with Generalized Herz–Hardy Spaces

315

(v) Let p(·) : Rn → (0, ∞) and X := Lp(·) (Rn ). Then, in this case, hX (Rn ) is just the variable local Hardy space hp(·) (Rn ) which was originally introduced in [186, Section 9]. Then we have the following conclusion. Theorem 5.2.3 Let X be a ball quasi-Banach function space and let r ∈ (0, ∞) be such that X is strictly r-convex and M is bounded on X1/r . Assume ϕ ∈ S(Rn ) satisfying that 1B(0,1) ≤  ϕ ≤ 1B(0,2). Then there exist two positive constants C1 and C2 such that, for any f ∈ S  (Rn ), C1 f hX (Rn ) ≤ f ∗ ϕX + f − f ∗ ϕHX (Rn ) ≤ C2 f hX (Rn ) . Remark 5.2.4 We should point out that Theorem 5.2.3 has a wide range of applications. Here we give several function spaces to which Theorem 5.2.3 can be applied. (i) Let p ∈ (0, ∞). Then, in this case, applying Remark 1.2.31(i) and the fact that, for any r ∈ (0, p], Lp (Rn ) is strictly r-convex, we can easily conclude that the Lebesgue space Lp (Rn ) satisfies all the assumptions of Theorem 5.2.3. Thus, Theorem 5.2.3 with X := Lp (Rn ) holds true. This result goes back to [89, Lemma 4]. (ii) Let p ∈ (0, ∞) and υ ∈ A∞ (Rn ). Then, in this case, from Remarks 1.2.31(i) p and the fact that Lυ (Rn ) is strictly r-convex for any r ∈ (0, p], we can easily p deduce that the weighted Lebesgue space Lυ (Rn ) satisfies all the assumptions of Theorem 5.2.3. Therefore, Theorem 5.2.3 with X therein replaced by p Lυ (Rn ) holds true. If further assume that p := 1, then, in this case, the aforementioned result goes back to [18, Proposition 4.1]. (iii) Let p := (p1 , . . . , pn ) ∈ (0, ∞)n . Then, in this case, using Remark 1.2.31(iii) and the fact that, for any r ∈ (0, min{p1 , . . . , pn }], Lp (Rn ) is strictly r-convex, we can easily find that the mixed-norm Lebesgue space Lp (Rn ) satisfies all the assumptions of Theorem 5.2.3. This then implies that Theorem 5.2.3 with X := Lp (Rn ) holds true. To the best of our knowledge, this result is totally new. (iv) Let 0 < q ≤ p < ∞. Then, in this case, by Remark 1.2.31(iv) and the fact that, p for any r ∈ (0, q], Mq (Rn ) is strictly r-convex, we easily conclude that the p Morrey space Mq (Rn ) satisfies all the assumptions of Theorem 5.2.3. Thus, p Theorem 5.2.3 with X replaced by Mq (Rn ) holds true. This result was also obtained in [205, Proposition 4.8]. (v) Let p(·) ∈ C log (Rn ). Then, in this case, from Remarks 1.2.31(v), 1.2.34(v), and 5.1.6, we can easily infer that the variable Lebesgue space Lp(·) (Rn ) satisfies all the assumptions of Theorem 5.2.3. This further implies that

316

5 Localized Generalized Herz–Hardy Spaces

Theorem 5.2.3 with X := Lp(·) (Rn ) holds true. This result coincides with [186, Lemma 9.1]. To prove this theorem, we need the following Plancherel–Pólya–Nikol’skij inequality which is a consequence of [229, p. 16, Theorem] (see also [186, Lemma 2.6]). Lemma 5.2.5 Let ϕ ∈ S(Rn ) be such that  ϕ has compact support, and let r ∈ (0, ∞). Then there exists a positive constant C, independent of ϕ, such that, for any f ∈ S  (Rn ) and x, y ∈ Rn , n

|f ∗ ϕ(y)| ≤ C (1 + |x − y|) r M(r) (f ∗ ϕ) (x). Via Lemma 5.2.5, we next show Theorem 5.2.3. Proof of Theorem 5.2.3 Let all the symbols be as in the present theorem and f ∈ S  (Rn ). We first prove that f hX (Rn )  f ∗ ϕX + f − f ∗ ϕHX (Rn ) .

(5.4)

Indeed, from Definitions 4.2.2 and 5.2.1, (4.3), and (5.1), we deduce that f − f ∗ ϕhX (Rn ) ≤ f − f ∗ ϕHX (Rn ) .

(5.5)

On the other hand, by the assumption that M is bounded on X1/r and Remark 1.2.30(i), we conclude that M(r) is bounded on X. This, together with Lemma 5.1.4, further implies that, for any g ∈ X and z ∈ Rn ,  1   r n   r |g(· − y)| dy   (1 + |z|) r gX .   z+[0,1]n  X

Applying this, the assumption that X is strictly r-convex, and Lemma 5.1.5(ii), we find that f ∗ ϕhX (Rn ) ∼ m(f ∗ ϕ, ϕ)X , which, combined with (5.5), implies that f hX (Rn )  f ∗ ϕhX (Rn ) + f − f ∗ ϕhX (Rn )  m(f ∗ ϕ, ϕ)X + f − f ∗ ϕHX (Rn ) .

(5.6)

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317

Now, we estimate m(f ∗ϕ, ϕ). Indeed, using Lemma 5.2.5 with y replaced by x −y, we conclude that, for any t ∈ (0, 1) and x ∈ Rn , |(f ∗ ϕ) ∗ ϕt (x)|      f ∗ ϕ(x − y)ϕt (y) dy  =  

Rn

n

Rn

(1 + |y|) r M(r) (f ∗ ϕ) (x) |ϕt (y)| dy

  y  n  y  1 (r)   r   1 + M ∗ ϕ) (x) (f   ϕ  dy tn t t Rn  n ∼ M(r) (f ∗ ϕ) (x) (1 + |y|) r |ϕ(y)| dy



  M(r) (f ∗ ϕ) (x)

Rn

Rn

1 dy ∼ M(r) (f ∗ ϕ) (x). (1 + |y|)n+1

Therefore, from (5.1), it follows that m (f ∗ ϕ, ϕ)  M(r) (f ∗ ϕ) . Then, combining this, Definition 1.2.13(ii), the assumption that M is bounded on X1/r , and Remark 1.2.30(i), we conclude that     m (f ∗ ϕ, ϕ)X  M(r) (f ∗ ϕ)  f ∗ ϕX . X

By this and (5.6), we further obtain f hX (Rn )  f ∗ ϕX + f − f ∗ ϕHX (Rn ) , which completes the proof of (5.4). Conversely, we show f ∗ ϕX + f − f ∗ ϕHX (Rn )  f hX (Rn ) .

(5.7)

To achieve this, we first estimate the term f − f ∗ ϕHX (Rn ) . Indeed, applying Definition 4.1.1(i), we find that M (f − f ∗ ϕ, ϕ) = sup |(f − f ∗ ϕ) ∗ ϕt | t ∈(0,∞)

= sup |f ∗ ϕt − f ∗ (ϕ ∗ ϕt )| . t ∈(0,∞)

(5.8)

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5 Localized Generalized Herz–Hardy Spaces

We next claim that, for any t ∈ [2, ∞), f ∗ ϕt − f ∗ (ϕ ∗ ϕt ) = 0. Indeed, notice that, for any t ∈ (0, ∞), f ∗ ϕt − f ∗ (ϕ ∗ ϕt ) ∈ S  (Rn ). Thus, for any t ∈ (0, ∞), we have F (f ∗ ϕt − f ∗ (ϕ ∗ ϕt )) = f ϕ (t·) (1 −  ϕ) .

(5.9)

Using the assumption  ϕ 1B(0,1) = 1, we conclude that, for any t ∈ (0, ∞) and x ∈ B(0, 1), ϕ (tx)[1 −   ϕ (x)] = 0. On the other hand, for any t ∈ [2, ∞) and x ∈ [B(0, 1)] , we have |tx| ≥ t ≥ 2. This, together with the assumption that  ϕ 1[B(0,2)] = 0, further implies that, for any t ∈ [2, ∞) and x ∈ [B(0, 1)] ,

ϕ (tx)[1 −   ϕ (x)] = 0. Therefore, for any t ∈ [2, ∞), we have ϕ (t·) (1 −   ϕ ) = 0. By this and (5.9), we find that, for any t ∈ [2, ∞), F (f ∗ ϕt − f ∗ (ϕ ∗ ϕt )) = 0 in S  (Rn ), and hence f ∗ ϕt − f ∗ (ϕ ∗ ϕt ) = 0, which completes the proof of the above claim. Now, we define ψ ∈ S(Rn ) by setting, for any x ∈ Rn , ψ(x) :=

1 x . ϕ 2n 2

Then, from (5.8) and the above claim, we deduce that M (f − f ∗ ϕ, ϕ) = sup |f ∗ ϕt − f ∗ (ϕ ∗ ϕt )| t ∈(0,2)

≤ sup |f ∗ ψt | + sup |f ∗ (ϕ ∗ ϕt )| t ∈(0,1)

t ∈(0,2)

= m(f, ψ) + sup |f ∗ (ϕ ∗ ϕt )| . t ∈(0,2)

(5.10)

5.2 Relations with Generalized Herz–Hardy Spaces

319

Combining this and both Lemmas 4.1.4(ii) and 5.1.5(ii), we find that f − f ∗ ϕHX (Rn ) ∼ M(f − f ∗ ϕ, ϕ)X        m(f, ψ)X +  sup |f ∗ (ϕ ∗ ϕt )|  t ∈(0,2) X       ∼ f hX (Rn ) +  sup |f ∗ (ϕ ∗ ϕt )| .  t ∈(0,2)

(5.11)

X

Next, we prove that        sup |f ∗ (ϕ ∗ ϕt )|  f hX (Rn ) .  t ∈(0,2)

(5.12)

X

Indeed, for any t ∈ (0, ∞), we have ϕ ∗ ϕt = F ( ϕ (−·) ϕ (−t·)) . Thus, for any α, β ∈ Zn+ , t ∈ (0, 2), and x ∈ Rn , we obtain  β α  x ∂ (ϕ ∗ ϕt ) (x)  - ..  ∼ F ∂ β (·)α  ϕ (−·) ϕ (−t·) (x)   β- α .  ∂ (·)  ϕ (−·) ϕ (−t·) (x) dx  Rn



  γ ∈Zn + γ ≤β

Rn

  α x  ϕ (−x)∂ γ ( ϕ (−t·)) (x)

 .  ϕ (−tx)∂ β−γ (·)α  ϕ (−·) (x) dx +        .   t |γ | x α  ϕ (−x) + ∂ β−γ (·)α  ϕ (−·) (x) dx γ ∈Zn + γ ≤β

 

Rn

Rn

1 dx ∼ 1. (1 + |x|)n+1

(5.13)

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5 Localized Generalized Herz–Hardy Spaces

Fix an N ∈ N∩( nr +1, ∞). Then, applying (5.13), we conclude that, for any α ∈ Zn+ with |α| ≤ N, any t ∈ (0, 2), and x ∈ Rn ,   ·    1 ∗ ϕ (x) (1 + |x|)N+n ∂ α (ϕ ) t 2n 2   ·     β α 1  (x)  (ϕ ∗ ϕt ) x ∂ 2n 2 β∈Zn+ , |β|≤N





β∈Zn+ , |β|≤N

 x    β α   1. x ∂ (ϕ ∗ ϕt ) 2

This, together with (4.1), further implies that, for any t ∈ (0, 2),  pN

·  1 ∗ ϕ  1, (ϕ ) t 2n 2

where the implicit positive constant is independent of t. From this, (4.2), and (5.1), it follows that sup |f ∗ (ϕ ∗ ϕt )|  mN (f ).

t ∈(0,2)

Combining this and Definition 1.2.13(ii), we further conclude that        sup |f ∗ (ϕ ∗ ϕt )|  mN (f )X ∼ f hX (Rn ) . t ∈(0,2)  X

This finishes the estimation of (5.12). From this and (5.11), we further infer that f − f ∗ ϕHX (Rn )  f hX (Rn ) . In addition, applying (5.10) and Definition 5.1.1(i), we find that     |f ∗ ϕ| = f ∗ ψ 1  ≤ m (f, ψ) . 2

From this, Definition 1.2.13(ii), and Lemma 5.1.5(ii), it follows that f ∗ ϕX ≤ m (f, ψ)X ∼ f hX (Rn ) . This, together with (5.14), further implies that f ∗ ϕX + f − f ∗ ϕHX (Rn )  f hX (Rn ) ,

(5.14)

5.2 Relations with Generalized Herz–Hardy Spaces

321

which completes the proof of (5.7). Combining both (5.4) and (5.7), we obtain f hX (Rn ) ∼ f ∗ ϕX + f − f ∗ ϕHX (Rn ) .  

This finishes the proof of Theorem 5.2.3. (Rn )

(Rn ),

Based on the above relation between hX and HX we now establish the p,q following relation between the local generalized Herz–Hardy space hK˙ ω,0 (Rn ) and p,q the generalized Herz–Hardy space H K˙ ω,0 (Rn ). Theorem 5.2.6 Let p, q ∈ (0, ∞), ω ∈ M(R+ ) with m0 (ω) ∈ (− pn , ∞) and m∞ (ω) ∈ (− pn , ∞), and ϕ ∈ S(Rn ) with 1B(0,1) ≤  ϕ ≤ 1B(0,2). Then there exist two positive constants C1 and C2 such that, for any f ∈ S  (Rn ), C1 f hK˙ p,q (Rn ) ≤ f ∗ ϕK˙ p,q (Rn ) + f − f ∗ ϕH K˙ p,q (Rn ) ω,0

ω,0

ω,0

≤ C2 f hK˙ p,q (Rn ) . ω,0

Proof Let all the symbols be as in the present theorem. Then, combining the assumption m0 (ω) ∈ (− pn , ∞) and Theorem 1.2.42, we find that, under the p,q assumptions of the present theorem, the local generalized Herz space K˙ ω,0 (Rn ) is a BQBF space. This implies that, to complete the proof of the present theorem, we p,q only need to show that the Herz space K˙ ω,0 (Rn ) under consideration satisfies all the assumptions of Theorem 5.2.3. Indeed, let    n r ∈ 0, min p, q, . max{M0 (ω), M∞ (ω)} + n/p Then, applying Theorem 1.3.3 with s := r, we conclude that K˙ ω,0 (Rn ) is strictly r-convex. On the other hand, from Lemma 1.8.5, it follows that, for any f ∈ L1loc (Rn ), p,q

M(f )[K˙ p,q (Rn )]1/r  f [K˙ p,q (Rn )]1/r . ω,0

ω,0

Therefore, all the assumptions of Theorem 5.2.3 hold true for K˙ ω,0 (Rn ) and hence the proof of Theorem 5.2.6 is completed.   p,q

Theorem 5.2.6, together with Remark 5.0.4(ii), further implies that the following p,q conclusion holds true for the local generalized Hardy–Morrey space hM ω,0 (Rn ); we omit the details.

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5 Localized Generalized Herz–Hardy Spaces

Corollary 5.2.7 Let p, q ∈ [1, ∞), ω ∈ M(R+ ) with −

n < m0 (ω) ≤ M0 (ω) < 0 p

and −

n < m∞ (ω) ≤ M∞ (ω) < 0, p

and ϕ ∈ S(Rn ) with 1B(0,1) ≤ ϕ  ≤ 1B(0,2). Then there exist two positive constants C1 and C2 such that, for any f ∈ S  (Rn ), C1 f hM p,q (Rn ) ≤ f ∗ ϕM p,q (Rn ) + f − f ∗ ϕH M p,q (Rn ) ω,0

ω,0

ω,0

≤ C2 f hM p,q (Rn ) . ω,0

Next, we establish the relation between the local generalized Herz–Hardy space p,q p,q hK˙ ω (Rn ) and the generalized Herz–Hardy space H K˙ ω (Rn ) as follows. Theorem 5.2.8 Let p, q ∈ (0, ∞), ω ∈ M(R+ ) with m0 (ω) ∈ (− pn , ∞) and −

n < m∞ (ω) ≤ M∞ (ω) < 0, p

and ϕ ∈ S(Rn ) with 1B(0,1) ≤  ϕ ≤ 1B(0,2). Then there exist two positive constants C1 and C2 such that, for any f ∈ S  (Rn ), C1 f hK˙ p,q n ≤ f ∗ ϕK n + f − f ∗ ϕH K n ˙ p,q ˙ p,q ω (R ) ω (R ) ω (R ) ≤ C2 f hK˙ p,q n . ω (R ) Proof Let all the symbols be as in the present theorem. Then, since ω ∈ M(R+ ) satisfies that m0 (ω) ∈ (− pn , ∞) and M∞ (ω) ∈ (−∞, 0), from Theorem 1.2.44, p,q it follows that the global generalized Herz space K˙ ω (Rn ) under consideration is a BQBF space. Thus, to finish the proof of the present theorem, it suffices to prove that p,q all the assumptions of Theorem 5.2.3 hold true for K˙ ω (Rn ). Namely, there exists p,q n an r ∈ (0, ∞) such that K˙ ω (R ) is strictly r-convex and the Hardy–Littlewood p,q maximal operator M is bounded on [K˙ ω (Rn )]1/r .

5.3 Atomic Characterizations

323

To this end, let   r ∈ 0, min p, q,

n max{M0 (ω), M∞ (ω)} + n/p

 .

p,q Then, by Theorem 1.3.4 with s therein replaced by r, we find that K˙ ω (Rn ) is strictly r-convex. On the other hand, applying Lemma 4.1.10, we conclude that, for any f ∈ L1loc (Rn ),

M(f )[K˙ p,q n 1/r  f [K n 1/r . ˙ p,q ω (R )] ω (R )] This further implies that all the assumptions of Theorem 5.2.3 hold true for p,q K˙ ω (Rn ) and then finishes the proof of Theorem 5.2.8.   Via Theorem 5.2.8 and Remark 5.0.4(ii), we immediately obtain the followp,q ing relation between the local Hardy space hM ω (Rn ) and the Hardy space p,q p,q n H M ω (R ) associated with the global generalized Morrey space M ω (Rn ); we omit the details. Corollary 5.2.9 Let p, q, ω, and ϕ be as in Corollary 5.2.7. Then there exist two positive constants C1 and C2 such that, for any f ∈ S  (Rn ), C1 f hM p,q n ≤ f ∗ ϕM p,q (Rn ) + f − f ∗ ϕH M p,q (Rn ) ω (R ) ω ω ≤ C2 f hM p,q n . ω (R )

5.3 Atomic Characterizations The main target of this section is to establish the atomic characterization of the local p,q p,q generalized Herz–Hardy spaces hK˙ ω,0 (Rn ) and hK˙ ω (Rn ). Indeed, by the known atomic characterization of localized Hardy spaces associated with ball quasi-Banach p,q function spaces, we obtain the atomic characterization of hK˙ ω,0 (Rn ). On the other p,q hand, in order to show the atomic characterization of hK˙ ω (Rn ), recall that the associate spaces of global generalized Herz spaces are still unknown. To overcome this difficulty, we first prove an improved atomic characterization of the local Hardy space hX (Rn ), with X being a ball quasi-Banach function space, without recourse to the associate space X . From this improved conclusion of hX (Rn ), we deduce the p,q desired atomic characterization of hK˙ ω (Rn ). To begin with, we establish the atomic characterization of the local generalized p,q Herz–Hardy space hK˙ ω,0 (Rn ). For this purpose, we first introduce the definitions p,q,r,d,s of local atoms and the local atomic Hardy space hK˙ ω,0 (Rn ) associated with the p,q local generalized Herz space K˙ (Rn ) as follows. ω,0

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5 Localized Generalized Herz–Hardy Spaces

Definition 5.3.1 Let p, q ∈ (0, ∞), ω ∈ M(R+ ) with m0 (ω) ∈ (− pn , ∞), r ∈ [1, ∞], and d ∈ Z+ . Then a measurable function a is called a localp,q (K˙ ω,0 (Rn ), r, d)-atom if (i) there exists a ball B(x0 , r0 ) ∈ B, with x0 ∈ Rn and r0 ∈ (0, ∞), such that supp (a) := {x ∈ Rn : a(x) = 0} ⊂ B(x0 , r0 ); (ii) aLr (Rn ) ≤

|B(x0 ,r0 )|1/r 1B(x0 ,r0 ) K˙ p,q (Rn ) ; ω,0

(iii) when r0 ∈ (0, 1), then, for any α ∈ Zn+ satisfying |α| ≤ d,  Rn

a(x)x α dx = 0.

Definition 5.3.2 Let p, q ∈ (0, ∞), ω ∈ M(R+ ) with m0 (ω) ∈ (− pn , ∞) and m∞ (ω) ∈ (− pn , ∞),   s ∈ 0, min 1, p, q,

n max{M0 (ω), M∞ (ω)} + n/p

 ,

d ≥ n(1/s − 1) be a fixed integer, and   r ∈ max 1, p,

  n ,∞ . min{m0 (ω), m∞ (ω)} + n/p

p,q,r,d,s Then the local generalized atomic Herz–Hardy space hK˙ ω,0 (Rn ), associated p,q n ˙ with the local generalized Herz space Kω,0 (R ), is defined to be the set of all the p,q f ∈ S  (Rn ) such that there exists a sequence {aj }j ∈N of local-(K˙ ω,0 (Rn ), r, d)atoms supported, respectively, in the balls {Bj }j ∈N ⊂ B and a sequence {λj }j ∈N ⊂ [0, ∞) satisfying that

f =



λj aj

j ∈N

in S  (Rn ) and ⎧ ⎫1    # $s ⎬s   ⎨ λj   1 Bj   ⎭  ⎩ 1Bj K˙ p,q (Rn )  j ∈N  ˙ p,q ω,0

Kω,0 (Rn )

< ∞.

5.3 Atomic Characterizations

325

p,q,r,d,s Moreover, for any f ∈ hK˙ ω,0 (Rn ),

f hK˙ p,q,r,d,s (Rn ) ω,0

⎧ ⎫1  ⎧  ⎪ # $s ⎪ ⎨⎨ ⎬s  λj   := inf  1 Bj  p,q  ⎩  ⎪ ⎭ 1  Bj K ˙ (Rn ) ⎪  ˙ p,q ⎩ j ∈N ω,0

Kω,0 (Rn )

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

,

where the infimum is taken over all the decompositions of f as above. Then we have the following atomic characterization of the local generalized p,q Herz–Hardy space hK˙ ω,0 (Rn ). Theorem 5.3.3 Let p, q, ω, d, s, and r be as in Definition 5.3.2. Then hK˙ ω,0 (Rn ) = hK˙ ω,0 p,q

p,q,r,d,s

(Rn )

with equivalent quasi-norms. To show the above atomic characterization, we require the known atomic characterization of the local Hardy space hX (Rn ) associated with ball quasi-Banach function space X. First, we recall the following definition of local-(X, r, d)-atoms, which is just [233, Definition 4.6]. Definition 5.3.4 Let X be a ball quasi-Banach function space, r ∈ [1, ∞], and d ∈ Z+ . Then a measurable function a is called a local-(X, r, d)-atom if (i) there exists a ball B(x0 , r0 ) ∈ B, with x0 ∈ Rn and r0 ∈ (0, ∞), such that supp (a) := {x ∈ Rn : a(x) = 0} ⊂ B(x0 , r0 ); 1/r 0 ,r0 )| (ii) aLr (Rn ) ≤ |B(x 1B(x ,r ) X ; 0 0

(iii) when r0 ∈ (0, 1), then, for any α ∈ Zn+ satisfying |α| ≤ d,  Rn

a(x)x α dx = 0.

Remark 5.3.5 Let X be a ball quasi-Banach function space, r, t ∈ [1, ∞], and d ∈ Z+ . (i) Assume that a is a local-(X, t, d)-atom supported in a ball B ∈ B and r ≤ t. Then it is obvious that a is also a local-(X, r, d)-atom supported in B. (ii) Obviously, for any (X, r, d)-atom a supported in the ball B ∈ B, a is also a local-(X, r, d)-atom supported in B. Via local-(X, r, d)-atoms, we now present the following definition of local atomic Hardy spaces associated with ball quasi-Banach function spaces (see, for instance, [233]).

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5 Localized Generalized Herz–Hardy Spaces

Definition 5.3.6 Let X be a ball quasi-Banach function space, r ∈ (1, ∞], d ∈ Z+ , and s ∈ (0, 1]. Then the local atomic Hardy space hX,r,d,s (Rn ), associated with X, is defined to be the set of all the f ∈ S  (Rn ) such that there exists a sequence {aj }j ∈N of local-(X, r, d)-atoms supported, respectively, in the balls {Bj }j ∈N ⊂ B and a sequence {λj }j ∈N ⊂ [0, ∞) satisfying that f =



λj aj

j ∈N

in S  (Rn ) and ⎡ ⎤1  

s s    λj ⎣  1Bj ⎦  < ∞.    1Bj X  j ∈N  X

Moreover, for any f ∈ hX,r,d,s (Rn ),

f hX,r,d,s (Rn )

⎧⎡ ⎤1  

s s ⎪ ⎨   λj ⎣  := inf  1 Bj ⎦    ⎪ 1  B X ⎩ j ∈N j 

⎫ ⎪ ⎬ ⎪ ⎭

,

X

where the infimum is taken over all the decompositions of f as above. Then we state the following atomic characterization of the local Hardy space hX (Rn ), which was obtained in [233, Theorem 4.8] and plays an essential role in the proof of Theorem 5.3.3. Lemma 5.3.7 Let X be a ball quasi-Banach function space satisfying both Assumption 1.2.29 with 0 < θ < s ≤ 1 and Assumption 1.2.33 with the same s and r ∈ (1, ∞], and let d ≥ n(1/θ − 1) be a fixed integer. Then hX (Rn ) = hX,r,d,s (Rn ) with equivalent quasi-norms. Remark 5.3.8 We point out that Lemma 5.3.7 has a wide range of applications. Here we present several function spaces to which Lemma 5.3.7 can be applied. (i) Let p := (p1 , . . . , pn ) ∈ (0, ∞)n , 5  d≥ n

1 −1 min{1, p1 , . . . , pn }

6

5.3 Atomic Characterizations

327

be a fixed integer, r ∈ (max{1, p1 , . . . , pn }, ∞], and s ∈ (0, min{1, p1 , . . . , pn }). Then, in this case, combining both Remarks 1.2.31(iii) and 1.2.34(iii), we can easily find that the mixed-norm Lebesgue space Lp (Rn ) satisfies all the assumptions of Lemma 5.3.7. Thus, Lemma 5.3.7 with X := Lp (Rn ) holds true. To the best of our knowledge, this result is totally new. (ii) Let 0 < q ≤ p < ∞, 5  d≥ n

1 −1 min{1, q}

6

be a fixed integer, r ∈ (max{1, p}, ∞], and s ∈ (0, min{1, q}). Then, in this case, from both Remarks 1.2.31(iv) and 1.2.34(iv), we easily infer that the p Morrey space Mq (Rn ) satisfies all the assumptions of Lemma 5.3.7. This then p implies that Lemma 5.3.7 with X := Mq (Rn ) holds true. To the best of our knowledge, this result is totally new. (iii) Let p(·) ∈ C log (Rn ) satisfy 0 < p− ≤ p+ < ∞, where p− and p+ are defined, respectively, in (1.59) and (1.60). Let 5  6 1 d≥ n −1 p− be a fixed nonnegative integer, r ∈ (max{1, p+ }, ∞], and s := min{1, p− }. Then, in this case, applying both Remarks 1.2.31(v) and 1.2.34(v), we can easily conclude that the variable Lebesgue space Lp(·) (Rn ) satisfies all the assumptions of Lemma 5.3.7. Therefore, Lemma 5.3.7 with X := Lp(·) (Rn ) holds true. This result coincides with [222, Corollary 4.1]. Via Lemma 5.3.7, we now show Theorem 5.3.3. Proof of Theorem 5.3.3 Let p, q, ω, d, s, and r be as in the present theorem. Then, from the assumption m0 (ω) ∈ (− pn , ∞) and Theorem 1.2.42, we deduce that the p,q local generalized Herz space K˙ ω,0 (Rn ) under consideration is a BQBF space. This implies that, to complete the proof of the present theorem, we only need to prove p,q that the Herz space K˙ ω,0 (Rn ) under consideration satisfies all the assumptions of Lemma 5.3.7. First, let θ ∈ (0, s) satisfy 6   5  6 5  1 1 1 −1 ≤n −1 < n −1 + 1. n s θ s

(5.15)

328

5 Localized Generalized Herz–Hardy Spaces

p,q We now show that K˙ ω,0 (Rn ) satisfies Assumption 1.2.29 with the above θ and s. Indeed, by Lemma 4.3.11, we conclude that, for any {fj }j ∈N ⊂ L1loc (Rn ),

⎧ ⎫1/s    ⎨  s ⎬    M(θ) (fj )   ⎩ ⎭   j ∈N  ˙ p,q

Kω,0 (Rn )

⎛ ⎞1/s       ⎝   |fj |s ⎠     j ∈N  ˙ p,q

,

(5.16)

Kω,0 (Rn )

p,q which implies that Assumption 1.2.29 holds true for K˙ ω,0 (Rn ) with the above θ and s. p,q Next, we prove that K˙ ω,0 (Rn ) satisfies Assumption 1.2.33 with the above s and p,q r. Indeed, from Lemma 1.8.6, we deduce that [K˙ ω,0 (Rn )]1/s is a BBF space and, for any f ∈ L1loc (Rn ),

   ((r/s))  (f ) M

˙ p,q (Rn )]1/s ) ([K ω,0

 f ([K˙ p,q (Rn )]1/s ) . ω,0

(5.17)

This implies that, for the above s and r, Assumption 1.2.33 holds true with X := p,q K˙ ω,0 (Rn ). Finally, using (5.15), we find that d ≥ n(1/θ − 1). This, together with p,q (5.16), the fact that [K˙ ω,0 (Rn )]1/s is a BBF space, and (5.17), further implies that, p,q under the assumptions of the present theorem, K˙ ω,0 (Rn ) satisfies all the assumptions of Lemma 5.3.7. Therefore, we have p,q p,q,r,d,s hK˙ ω,0 (Rn ) = hK˙ ω,0 (Rn )

with equivalent quasi-norms, which completes the proof of Theorem 5.3.3.

 

As an application, we now establish the atomic characterization of the local p,q generalized Hardy–Morrey space hM ω,0 (Rn ). To achieve this, we first introduce the definition of the local atoms associated with the local generalized Morrey space p,q M ω,0 (Rn ) as follows. Definition 5.3.9 Let p, q ∈ [1, ∞), ω ∈ M(R+ ) with M∞ (ω) ∈ (−∞, 0) and −

n < m0 (ω) ≤ M0 (ω) < 0, p

r ∈ [1, ∞], and d ∈ Z+ . Then a measurable function a on Rn is called a localp,q (M ω,0 (Rn ), r, d)-atom if (i) there exists a ball B(x0 , r0 ) ∈ B, with x0 ∈ Rn and r0 ∈ (0, ∞), such that supp (a) := {x ∈ Rn : a(x) = 0} ⊂ B(x0 , r0 ); (ii) aLr (Rn ) ≤

|B(x0 ,r0 )|1/r 1B(x0 ,r0 ) M p,q (Rn ) ; ω,0

5.3 Atomic Characterizations

329

(iii) when r0 ∈ (0, 1), then, for any α ∈ Zn+ satisfying that |α| ≤ d,  Rn

a(x)x α dx = 0. p,q

Now, we give the following atomic characterization of hM ω,0 (Rn ), which is a corollary of both Theorem 5.3.3 and Remark 1.2.2(vi); we omit the details. Corollary 5.3.10 Let p, q ∈ [1, ∞), ω ∈ M(R+ ) with −

n < m0 (ω) ≤ M0 (ω) < 0 p

and −

n < m∞ (ω) ≤ M∞ (ω) < 0, p

s ∈ (0, 1), d ≥ n(1/s − 1) be a fixed integer, and  r∈

 n ,∞ . min{m0 (ω), m∞ (ω)} + n/p p,q,r,d,s

Then the local generalized atomic Hardy–Morrey space hM ω,0 (Rn ), associated p,q n with the local generalized Morrey space M ω,0 (R ), is defined to be the set of all the p,q f ∈ S  (Rn ) such that there exists a sequence {aj }j ∈N of local-(M ω,0 (Rn ), r, d)atoms supported, respectively, in the balls {Bj }j ∈N ⊂ B and a sequence {λj }j ∈N ⊂ [0, ∞) satisfying that f =



λj aj

j ∈N

in S  (Rn ) and ⎧ ⎫1    # $s ⎨ ⎬s  λj   1 Bj   ⎭  ⎩ 1Bj M p,q (Rn )  j ∈N  ω,0

< ∞.

p,q M ω,0 (Rn )

p,q,r,d,s

Moreover, for any f ∈ hM ω,0

f hM p,q,r,d,s (Rn ) ω,0

(Rn ),

⎧ ⎫1  ⎧  ⎪ # $s ⎪ ⎨⎨ ⎬s  λj   := inf  1 Bj  ⎩ ⎪ ⎭  1Bj M p,q (Rn ) ⎪ ⎩ j ∈N  ω,0

⎫ ⎪ ⎪ ⎬ p,q

M ω,0 (Rn )

⎪ ⎪ ⎭

,

330

5 Localized Generalized Herz–Hardy Spaces

where the infimum is taken over all the decompositions of f as above. Then p,q

p,q,r,d,s

hM ω,0 (Rn ) = hM ω,0

(Rn )

with equivalent quasi-norms. The remainder of this section is devoted to establishing the atomic characterip,q zation of the local generalized Herz–Hardy space hK˙ ω (Rn ). To this end, we first p,q introduce the following local-(K˙ ω (Rn ), r, d)-atoms and the local atomic Hardy p,q space associated with K˙ ω (Rn ). Definition 5.3.11 Let p, q ∈ (0, ∞), ω ∈ M(R+ ) with m0 (ω) ∈ (− pn , ∞) and M∞ (ω) ∈ (−∞, 0), r ∈ [1, ∞], and d ∈ Z+ . Then a measurable function a is p,q called a local-(K˙ ω (Rn ), r, d)-atom if (i) there exists a ball B(x0 , r0 ) ∈ B, with x0 ∈ Rn and r0 ∈ (0, ∞), such that supp (a) := {x ∈ Rn : a(x) = 0} ⊂ B(x0 , r0 ); (ii) aLr (Rn ) ≤

|B(x0 ,r0 )|1/r 1B(x0 ,r0 ) K˙ p,q (Rn ) ; ω

(iii) when r0 ∈ (0, 1), then, for any α ∈ Zn+ satisfying |α| ≤ d,  Rn

a(x)x α dx = 0.

Definition 5.3.12 Let p, q ∈ (0, ∞), ω ∈ M(R+ ) with m0 (ω) ∈ (− pn , ∞) and −

n < m∞ (ω) ≤ M∞ (ω) < 0, p

  s ∈ 0, min 1, p, q,

n max{M0 (ω), M∞ (ω)} + n/p

 ,

d ≥ n(1/s − 1) be a fixed integer, and   r ∈ max 1, p,

  n ,∞ . min{m0 (ω), m∞ (ω)} + n/p

p,q,r,d,s Then the local generalized atomic Herz–Hardy space hK˙ ω (Rn ), associated p,q n ˙ with the global generalized Herz space Kω (R ), is defined to be the set of all the p,q f ∈ S  (Rn ) such that there exists a sequence {aj }j ∈N of local-(K˙ ω (Rn ), r, d)atoms supported, respectively, in the balls {Bj }j ∈N ⊂ B and a sequence {λj }j ∈N ⊂ [0, ∞) satisfying that

f =

 j ∈N

λj aj

5.3 Atomic Characterizations

331

in S  (Rn ) and ⎧ ⎫1    # $s  ⎨ ⎬s  λj   1 Bj   ⎩  p,q ⎭ 1Bj K˙ ω (Rn )  j ∈N  ˙ p,q

< ∞.

Kω (Rn )

p,q,r,d,s Moreover, for any f ∈ hK˙ ω (Rn ),

f hK˙ p,q,r,d,s (Rn ) ω

⎧⎧ ⎫1    # $s ⎪ ⎨⎨ ⎬s  λj   := inf  1 Bj  ⎪   p,q ⎭ ⎩ 1  Bj K ˙ ω (Rn ) ⎩ j ∈N  ˙ p,q

Kω (Rn )

⎫ ⎪ ⎬ ⎪ ⎭

,

where the infimum is taken over all the decompositions of f as above. Then we have the following atomic characterization of the local generalized p,q Herz–Hardy space hK˙ ω (Rn ). Theorem 5.3.13 Let p, q, ω, d, s, and r be as in Definition 5.3.12. Then hK˙ ωp,q (Rn ) = hK˙ ωp,q,r,d,s (Rn ) with equivalent quasi-norms. To prove this atomic characterization, we first establish an atomic characterization of the local Hardy space hX (Rn ) via borrowing some ideas from [233, Theorem 4.8] and get rid of the usage of associate spaces. Namely, we have the following conclusion. Theorem 5.3.14 Let X be a ball quasi-Banach function space satisfy: (i) Assumption 1.2.29 holds true with 0 < θ < s ≤ 1; (ii) for the above s, X1/s is a ball Banach function space and there exists a linear space Y ⊂ M (Rn ) equipped with a seminorm  · Y such that, for any f ∈ M (Rn ),   f X1/s ∼ sup fgL1 (Rn ) : gY = 1 ,

(5.18)

where the positive equivalence constants are independent of f ; (iii) for the above s and Y , there exists an r ∈ (1, ∞] and a positive constant C such that, for any f ∈ L1loc (Rn ),    ((r/s))  (f ) ≤ Cf Y . M Y

332

5 Localized Generalized Herz–Hardy Spaces

Then hX (Rn ) = hX,r,d,s (Rn ) with equivalent quasi-norms. Remark 5.3.15 We should point out that Theorem 5.3.14 is an improved version of the known atomic characterization of hX (Rn ) obtained in [233, Theorem 4.8]. Indeed, if Y ≡ (X1/s ) in Theorem 5.3.14, then this theorem goes back to [233, Theorem 4.8]. To show Theorem 5.3.14, we require the following atomic decomposition of the local Hardy space hX (Rn ), which was obtained in [233, pp. 37–39]. Lemma 5.3.16 Let X be a ball quasi-Banach function space satisfying Assumption 1.2.29 with 0 < θ < s ≤ 1, d ≥ n(1/θ − 1) be a fixed integer, and f ∈ hX (Rn ). Then there exists {λj }j ∈N ⊂ [0, ∞) and {aj }j ∈N of local-(X, ∞, d)atoms supported, respectively, in the balls {Bj }j ∈N ⊂ B satisfying that f =



λj aj

j ∈N

in S  (Rn ) and ⎡ ⎤1  

s s    λj ⎣  1Bj ⎦   f hX (Rn ) ,    1Bj X  j ∈N  X

where the implicit positive constant is independent of f . Now, we turn to prove Theorem 5.3.14. Proof of Theorem 5.3.14 Let X, r, d, and s be as in the present theorem. We first prove that hX (Rn ) ⊂ hX,r,d,s (Rn ). For this purpose, let f ∈ hX (Rn ). Then, applying the assumption (i) of the present theorem and Lemma 5.3.16, we find that there exists {λj }j ∈N ⊂ [0, ∞) and {aj }j ∈N of local-(X, ∞, d)-atoms supported, respectively, in the balls {Bj }j ∈N ⊂ B satisfying that f =



λj aj

(5.19)

j ∈N

in S  (Rn ) and ⎡ ⎤1  

s s    λj ⎣  1Bj ⎦   f hX (Rn ) .    1Bj X  j ∈N  X

(5.20)

5.3 Atomic Characterizations

333

In addition, for any j ∈ N, from Remark 5.3.5(i) with a := aj and t := ∞, it follows that aj is a local-(X, r, d)-atom supported in the ball Bj . This, combined with (5.19), (5.20), and Definition 5.3.6, further implies that f ∈ hX,r,d,s (Rn ) and

f hX,r,d,s (Rn )

⎡ ⎤1  

s s    λj ⎣  ≤ 1Bj ⎦   f hX (Rn ) ,   1Bj X  j ∈N 

(5.21)

X

which completes the proof that hX (Rn ) ⊂ hX,r,d,s (Rn ). Conversely, we next show that hX,r,d,s (Rn ) ⊂ hX (Rn ). Indeed, let f ∈ X,r,d,s h (Rn ), {λl,j }l∈{1,2}, j ∈N ⊂ [0, ∞), and {al,j }l∈{1,2}, j ∈N be a sequence of local-(X, r, d)-atoms supported, respectively, in the balls {Bl,j }l∈{1,2}, j ∈N ⊂ B satisfying that, for any j ∈ N, r(B1,j ) ∈ (0, 1) and r(B2,j ) ∈ [1, ∞), f =

2  

(5.22)

λl,j al,j

l=1 j ∈N

in S  (Rn ), and ⎡ ⎤1   2

s s    λl,j ⎣  1Bl,j ⎦  < ∞.    1Bl,j X  l=1 j ∈N 

(5.23)

X

/ In what follows, let φ ∈ S(Rn ) be such that supp (φ) ⊂ B(0, 1) and Rn φ(x) dx = 0. Then, using (5.22) and repeating an argument similar to that used in the estimation of (4.24) with {λj }j ∈N and {aj }j ∈N therein replaced, respectively, by {λl,j }l∈{1,2}, j ∈N, and {al,j }l∈{1,2}, j ∈N, we conclude that, for any t ∈ (0, ∞), |f ∗ φt | ≤

2  

  λl,j al,j ∗ φt  .

l=1 j ∈N

This, together with Definition 5.1.1(i), further implies that m(f, φ) ≤

2   l=1 j ∈N

λl,j m(al,j , φ).

334

5 Localized Generalized Herz–Hardy Spaces

Using this and Definition 1.2.13(ii), we find that                m(f, φ)X   λ1,j m(a1,j , φ) +  λ2,j m(a2,j , φ)  j ∈N  j ∈N  X

X

=: IV1 + IV2 .

(5.24)

We first estimate IV1 . Indeed, for any given j ∈ N, by the assumption r(B1,j ) ∈ (0, 1) and both Definitions 5.3.4 and 4.3.1, we conclude that aj is an (X, r, d)-atom supported in Bj . Thus, using both Definitions 5.1.1(i) and 4.1.1(i) and repeating the arguments similar to those used in the estimations of (4.26), (4.28), and (4.30) with {λj }j ∈N , {aj }j ∈N , and {Bj }j ∈N therein replaced, respectively, by {λ1,j }j ∈N , {a1,j }j ∈N , and {B1,j }j ∈N , we find that ⎡   ⎤1  

s s        λ 1,j   ⎣ IV1   λ1,j M(a1,j , φ) 1B1,j ⎦  .      1  B1,j X j ∈N   j ∈N  X

(5.25)

X

This is the desired estimate of IV1 . On the other hand, we deal with IV2 . To this end, we first claim that, for any given j ∈ N and for any x ∈ (2B2,j ) , m(a2,j , φ)(x) = 0. Indeed, fix a j ∈ N. Then, applying the assumption r(B2,j ) ∈ [1, ∞), we find that, for any t ∈ (0, 1), x ∈ (2B2,j ) , and y ∈ B2,j , |x − y| ≥ r(B2,j ) ≥ 1 > t, which implies that y ∈ [B(x, t)] . This, together with the assumption supp (φ) ⊂ B(0, 1), further implies that, for any t ∈ (0, 1) and x ∈ (2B2,j ) ,  a2,j ∗ φt (x) =  =

Rn

φt (x − y)a2,j (y) dy

B2,j ∩B(x,t )

φt (x − y)a2,j (y) dy = 0.

Therefore, by Definition 5.1.1(i), we further conclude that, for any given j ∈ N and for any x ∈ (2B2,j ) , m(a2,j , φ)(x) = 0, which completes the proof of the above claim. Combining this claim and both Definitions 5.1.1(i) and 4.1.1(i), we obtain                IV2 ∼  λ m(a , φ)1  λ M(a , φ)1 2,j 2,j 2B2,j  2,j 2,j 2B2,j  .     j ∈N j ∈N X

X

5.3 Atomic Characterizations

335

From this and an argument similar to that used in the estimation of II1 in the proof of Theorem 4.3.18 with {λj }j ∈N , {aj }j ∈N , and {Bj }j ∈N therein replaced, respectively, by {λ2,j }j ∈N , {a2,j }j ∈N , and {B2,j }j ∈N , it follows that ⎡ ⎤1  

s s    λ2,j  ⎣ ⎦ IV2   1B2,j  ,   1B2,j X   j ∈N

(5.26)

X

which is the desired estimate of IV2 . Thus, by both the assumptions (i) and (ii) of the present theorem, Remark 5.1.6, Lemma 5.1.5(ii), (5.24), (5.25), (5.26), and (5.23), we find that ⎡ ⎤1   2

s s    λl,j ⎣  ⎦ f hX (Rn ) ∼ m(f, φ)X   1Bl,j  < ∞. (5.27)   1Bl,j X  l=1 j ∈N  X

This further implies that f ∈ hX (Rn ) and hence hX (Rn ) ⊂ hX,r,d,s (Rn ). Moreover, from (5.27), the choice of {λl,j }l∈{1,2},j ∈N, and Definition 5.3.6, we deduce that f hX (Rn )  f hX,r,d,s (Rn ) , which, combined with (5.21), implies that f hX (Rn ) ∼ f hX,r,d,s (Rn ) . Thus, we have hX (Rn ) = hX,r,d,s (Rn ) with equivalent quasi-norms, which completes the proof of Theorem 5.3.14.

 

Via the above atomic characterization of local Hardy spaces associated with ball quasi-Banach function spaces, we now prove Theorem 5.3.13. Proof of Theorem 5.3.13 Let p, q, ω, r, s, and d be as in the present theorem. Then, combining the assumptions m0 (ω) ∈ (− pn , ∞) and M∞ (ω) ∈ (−∞, 0), p,q and Theorem 1.2.44, we conclude that the global generalized Herz space K˙ ω (Rn ) under consideration is a BQBF space. From this and Theorem 5.3.14, it follows that, to finish the proof of the present theorem, we only need to show that the assumptions p,q (i) through (iii) of Theorem 5.3.14 hold true for K˙ ω (Rn ). p,q First, we show that Theorem 5.3.14(i) holds true for K˙ ω (Rn ). To this end, let θ ∈ (0, s) be such that 6   5  6 5  1 1 1 −1 ≤n −1 < n −1 + 1. n s θ s

(5.28)

336

5 Localized Generalized Herz–Hardy Spaces

Then, applying Lemma 4.3.25, we find that, for any {fj }j ∈N ⊂ L1loc (Rn ), ⎧ ⎫1/s    ⎨  s ⎬    (θ) M (fj )   ⎩ ⎭   j ∈N  ˙ p,q

Kω (Rn )

⎛ ⎞1/s       ⎝ s⎠   |fj |     j ∈N  ˙ p,q

.

Kω (Rn )

This implies that, for the above θ and s, K˙ ω (Rn ) satisfies Assumption 1.2.29 and hence Theorem 5.3.14(i) holds true. p,q Next, we prove that K˙ ω (Rn ) satisfies Theorem 5.3.14(ii). Indeed, from the assumptions m0 (ω) ∈ (− pn , ∞) and M∞ (ω) ∈ (−∞, 0), and Lemma 1.1.6, it follows that p,q

- . n m0 ωs = sm0 (ω) > − p/s and - . M∞ ωs = sM∞ (ω) < 0. Applying these, the assumptions p/s, q/s ∈ (1, ∞), and Theorem 1.2.48 with p, p/s,q/s q, and ω replaced, respectively, by p/s, q/s, and ωs , we find that K˙ ωs (Rn ) is a BBF space. Moreover, by Lemma 1.3.2, we conclude that 

K˙ ωp,q (Rn )

1/s

p/s,q/s = K˙ ωs (Rn ).

Thus, [K˙ ω (Rn )]1/s is a BBF space. On the other hand, from (4.42), it follows that, for any f ∈ M (Rn ), p,q

 f [K˙ p,q ∼ sup fgL1 (Rn ) : g ˙(p/s) ,(q/s) n 1/s ω (R )] B1/ωs

(Rn )

 =1 .

Therefore, [K˙ ω (Rn )]1/s is a BBF space and (5.18) holds true with p,q





(p/s) ,(q/s) Y := B˙1/ωs (Rn ). p,q These further imply that the Herz space K˙ ω (Rn ) under consideration satisfies Theorem 5.3.14(ii). p,q Finally, we show that Theorem 5.3.14(iii) holds true for K˙ ω (Rn ). Indeed, using Lemma 4.3.23, we conclude that, for any f ∈ L1loc (Rn ),

   ((r/s))  (f ) ˙ (p/s) ,(q/s) M B1/ωs

(Rn )

 f  ˙ (p/s) ,(q/s) B1/ωs

(Rn )

,

5.3 Atomic Characterizations

337

which implies that Theorem 5.3.14(iii) holds true for K˙ ω (Rn ) with p,q





(p/s) ,(q/s) Y := B˙1/ωs (Rn ).

Moreover, by (5.28), we find that d ≥ n(1/θ − 1). This, together with the fact that the assumptions (i) through (iii) of Theorem 5.3.14 hold true for the Herz space p,q K˙ ω (Rn ) under consideration, further implies that hK˙ ωp,q (Rn ) = hK˙ ωp,q,r,d,s (Rn ) with equivalent quasi-norms, which completes the proof of Theorem 5.3.13.

 

As an application of Theorem 5.3.13, we now establish the atomic characterizap,q tion of the local generalized Hardy–Morrey space hM ω (Rn ). To this end, we first p,q n introduce the definition of local-(M ω (R ), r, d)-atoms as follows. Definition 5.3.17 Let p, q, ω, r, and d be as in Definition 5.3.9. Then a measurable p,q function a on Rn is called a local-(M ω (Rn ), r, d)-atom if (i) there exists a ball B(x0 , r0 ) ∈ B, with x0 ∈ Rn and r0 ∈ (0, ∞), such that supp (a) := {x ∈ Rn : a(x) = 0} ⊂ B(x0 , r0 ); (ii) aLr (Rn ) ≤

|B(x0 ,r0 )|1/r 1B(x0 ,r0 ) M p,q (Rn ) ; ω

(iii) when r0 ∈ (0, 1), then, for any α ∈ Zn+ satisfying |α| ≤ d,  Rn

a(x)x α dx = 0. p,q

Then we have the following atomic characterization of hM ω (Rn ), which can be deduced from Theorem 5.3.13 and Remark 1.2.2(vi) immediately; we omit the details. Corollary 5.3.18 Let p, q, ω, r, d, and s be as in Corollary 5.3.10. Then the p,q,r,d,s local generalized atomic Hardy–Morrey space hM ω (Rn ), associated with the p,q n global generalized Morrey space M ω (R ), is defined to be the set of all the f ∈ p,q S  (Rn ) such that there exists a sequence {aj }j ∈N of local-(M ω (Rn ), r, d)-atoms supported, respectively, 4 in the balls {Bj }j ∈N ⊂ B and a sequence {λj }j ∈N ⊂ [0, ∞) satisfying that f = j ∈N λj aj in S  (Rn ) and ⎧ ⎫1    # $s ⎨ ⎬s  λj   1 Bj   ⎭ ⎩  1Bj M p,q n ω (R )  j ∈N 

< ∞. p,q

M ω (Rn )

338

5 Localized Generalized Herz–Hardy Spaces p,q,r,d,s

Moreover, for any f ∈ hM ω

f hM p,q,r,d,s (Rn ) ω

(Rn ),

⎧⎧ ⎫1    # $s ⎪ ⎨⎨ ⎬s  λj   := inf  1 Bj  p,q ⎪ ⎭  ⎩  1  n Bj M ω (R ) ⎩ j ∈N 

⎫ ⎪ ⎬ p,q M ω (Rn )

⎪ ⎭

,

where the infimum is taken over all the decompositions of f as above. Then n p,q,r,d,s hM p,q (Rn ) ω (R ) = hM ω

with equivalent quasi-norms.

5.4 Molecular Characterizations In this section, we investigate the molecular characterization of localized generalized Herz–Hardy spaces via viewing generalized Herz spaces as special cases of ball quasi-Banach function spaces. Precisely, we establish the molecular characterp,q ization of the local generalized Herz–Hardy space hK˙ ω,0 (Rn ) via using the known molecular characterization of localized Hardy spaces associated with ball quasiBanach function spaces obtained in [233, Theorem 5.2] (see also Lemma 5.4.5 below). On the other hand, to prove the molecular characterization of the local p,q generalized Herz–Hardy space hK˙ ω (Rn ), recall that the associate spaces of the global generalized Herz spaces are still unknown. To overcome this obstacle, we first establish an improved molecular characterization of localized Hardy spaces associated with ball quasi-Banach function spaces (see Theorem 5.4.11 below) without recourse to associate spaces. Combining this molecular characterization p,q and the fact that the global generalized Herz space K˙ ω (Rn ) is a special ball quasiBanach function space, we then obtain the desired molecular characterization of p,q hK˙ ω (Rn ). p,q We first investigate the molecular characterization of hK˙ ω,0 (Rn ). To begin with, we introduce the local molecules associated with the local generalized Herz space p,q K˙ ω,0 (Rn ) as follows. Definition 5.4.1 Let p, q ∈ (0, ∞), ω ∈ M(R+ ) with m0 (ω) ∈ (− pn , ∞), r ∈ [1, ∞], d ∈ Z+ , and τ ∈ (0, ∞). Then a measurable function m on Rn is called a p,q local-(K˙ ω,0 (Rn ), r, d, τ )-molecule centered at a ball B(x0 , r0 ) ∈ B, with x0 ∈ Rn and r0 ∈ (0, ∞), if (i) for any i ∈ Z+ ,  m1S (B(x i

0 ,r0 ))

 

Lr (Rn )

≤ 2−τ i

|B(x0 , r0 )|1/r ; 1B(x0 ,r0 ) K˙ p,q (Rn ) ω,0

5.4 Molecular Characterizations

339

(ii) when r0 ∈ (0, 1), then, for any α ∈ Zn+ with |α| ≤ d,  Rn

m(x)x α dx = 0.

Then we establish the following molecular characterization of the local generalp,q ized Herz–Hardy space hK˙ ω,0 (Rn ). Theorem 5.4.2 Let p, q ∈ (0, ∞), ω ∈ M(R+ ) with m0 (ω) ∈ (− pn , ∞) and m∞ (ω) ∈ (− pn , ∞),   s ∈ 0, min 1, p, q,

n max{M0 (ω), M∞ (ω)} + n/p

 ,

d ≥ n(1/s − 1) be a fixed integer,   r ∈ max 1, p,

  n ,∞ , min{m0 (ω), m∞ (ω)} + n/p

and τ ∈ (0, ∞) with τ > n(1/s − 1/r). Then f ∈ hK˙ ω,0 (Rn ) if and only if ˙ p,q (Rn ), r, d, τ )f ∈ S  (Rn ) and there exists a sequence {mj }j ∈N of local-(K ω,0 molecules centered, respectively, at the balls {Bj }j ∈N ⊂ B and a sequence {λj }j ∈N ⊂ [0, ∞) such that p,q

f =



λj mj

j ∈N

in S  (Rn ) and ⎧ ⎫1    # $s  ⎨ ⎬s  λj   1 Bj   ⎩ ⎭  1Bj K˙ p,q (Rn )  j ∈N  ˙ p,q ω,0

< ∞.

Kω,0 (Rn )

Moreover, there exist two positive constants C1 and C2 such that, for any f ∈ p,q hK˙ ω,0 (Rn ),

C1 f hK˙ p,q (Rn ) ω,0

⎧ ⎫1  ⎧ ⎪  # $s ⎪ ⎨⎨ ⎬s  λi   ≤ inf  1 Bj  ⎪ p,q ⎭  ⎩  1Bj K˙ (Rn ) ⎪ ⎩ j ∈N  ˙ p,q ω,0

Kω,0

≤ C2 f hK˙ p,q (Rn ) , ω,0

where the infimum is taken over all the decompositions of f as above.

⎫ ⎪ ⎪ ⎬ (Rn )

⎪ ⎪ ⎭

340

5 Localized Generalized Herz–Hardy Spaces

To obtain this molecular characterization, we first recall the following definition of local-(X, r, d, τ )-molecules with X being a ball quasi-Banach function space, which is just [233, Definition 5.1]. Definition 5.4.3 Let X be a ball quasi-Banach function space, r ∈ [1, ∞], d ∈ Z+ , and τ ∈ (0, ∞). Then a measurable function m on Rn is called a local-(X, r, d, τ )molecule centered at a ball B(x0 , r0 ) ∈ B, with x0 ∈ Rn and r0 ∈ (0, ∞), if (i) for any i ∈ Z+ ,  m1S (B(x i

0 ,r0 ))

 

Lr (Rn )

≤ 2−τ i

|B(x0 , r0 )|1/r ; 1B(x0 ,r0 ) X

(ii) when r0 ∈ (0, 1), then, for any α ∈ Zn+ with |α| ≤ d,  Rn

m(x)x α dx = 0.

Remark 5.4.4 Let X be a ball quasi-Banach function space, r ∈ [1, ∞], d ∈ Z+ , and τ ∈ (0, ∞). Then it is obvious that (i) for any local-(X, r, d)-atom a supported in B ∈ B, a is a local-(X, r, d, τ )molecule centered at B; (ii) for any (X, r, d, τ )-molecule m centered at a ball B ∈ B, m is a local(X, r, d, τ )-molecule centered at B. The following molecular characterization of hX (Rn ) was established in [233, Theorem 5.2], which is an essential tool in the proof of Theorem 5.4.2. Lemma 5.4.5 Let X be a ball quasi-Banach function space satisfying both Assumption 1.2.29 with 0 < θ < s ≤ 1 and Assumption 1.2.33 with the same s and r ∈ (1, ∞], τ ∈ (n(1/θ − 1/r), ∞), and let d ≥ n(1/θ − 1) be a fixed integer. Then f ∈ hX (Rn ) if and only if f ∈ S  (Rn ) and there exists a sequence {mj }j ∈N of local-(X, r, d, τ )-molecules centered, respectively, 4at the balls {Bj }j ∈N ⊂ B and a sequence {λj }j ∈N ⊂ [0, ∞) such that both f = j ∈N λj mj in S  (Rn ) and ⎡ ⎤1  

s s    λj ⎣  ⎦ 1Bj  < ∞.    1Bj X  j ∈N  X

Moreover, for any f ∈ hX (Rn ),

f hX (Rn )

⎧⎡ ⎤1  

s s ⎪ ⎨   λi ⎣  ∼ inf  1 Bj ⎦  ⎪   1  Bj X ⎩ j ∈N 

⎫ ⎪ ⎬ ⎪ ⎭ X

5.4 Molecular Characterizations

341

with the positive equivalence constants independent of f , where the infimum is taken over all the decompositions of f as above. Remark 5.4.6 We point out that Lemma 5.4.5 has a wide range of applications. Here we give several function spaces to which Lemma 5.4.5 can be applied. (i) Let p := (p1 , . . . , pn ) ∈ (0, ∞)n , 5  d≥ n

1 −1 min{1, p1 , . . . , pn }

6

be a fixed integer, r ∈ (max{1, p1 , . . . , pn }, ∞], s ∈ (0, min{1, p1 , . . . , pn }), and     1 1 − τ∈ n ,∞ . min{1, p1 , . . . , pn } r Then, in this case, by both Remarks 1.2.31(iii) and 1.2.34(iii), we can easily find that the mixed-norm Lebesgue space Lp (Rn ) satisfies all the assumptions of Lemma 5.4.5. Thus, Lemma 5.4.5 with X := Lp (Rn ) holds true. To the best of our knowledge, this result is totally new. (ii) Let 0 < q ≤ p < ∞, 5  d≥ n

1 −1 min{1, q}

6

be a fixed integer, r ∈ (max{1, p}, ∞], s ∈ (0, min{1, q}), and   τ∈ n

  1 1 − ,∞ . min{1, q} r

Then, in this case, from both Remarks 1.2.31(iv) and 1.2.34(iv), we can p easily deduce that the Morrey space Mq (Rn ) satisfies all the assumptions of p Lemma 5.4.5. This then implies that Lemma 5.4.5 with X := Mq (Rn ) holds true. To the best of our knowledge, this result is totally new. (iii) Let p(·) ∈ C log (Rn ) satisfy 0 < p− ≤ p+ < ∞, where p− and p+ are defined, respectively, in (1.59) and (1.60). Let 5  d≥ n

1 −1 min{1, p− }

6

be a fixed integer, r ∈ (max{1, p+ }, ∞], s := min{1, p− }, and   τ∈ n

  1 1 − ,∞ . min{1, p− } r

342

5 Localized Generalized Herz–Hardy Spaces

Then, in this case, using both Remarks 1.2.31(v) and 1.2.34(v), we can easily conclude that the variable Lebesgue space Lp(·) (Rn ) satisfies all the assumptions of Lemma 5.4.5. Therefore, Lemma 5.4.5 with X := Lp(·) (Rn ) holds true. To the best of our knowledge, this result is totally new. Based on the above lemma, we next show Theorem 5.4.2. Proof of Theorem 5.4.2 Let p, q, ω, r, d, and s be as in the present theorem. Since ω satisfies that m0 (ω) ∈ (− pn , ∞), from Theorem 1.2.42, it follows that p,q the local generalized Herz space K˙ ω,0 (Rn ) under consideration is a BQBF space. Thus, to complete the present theorem, it suffices to prove that all the assumptions of p,q Lemma 5.4.5 hold true for the Herz space K˙ ω,0 (Rn ) under consideration. Namely, p,q K˙ ω,0 (Rn ) satisfies both Assumption 1.2.29 with the above s and some θ ∈ (0, s), and Assumption 1.2.33 with the above s and r. Indeed, let θ ∈ (0, s) satisfy that 6   5  6 5  1 1 1 −1 ≤n −1 < n −1 +1 n s θ s

(5.29)

and 

 1 1 − τ >n . θ r

(5.30)

Then, by Lemma 4.3.25, we find that, for any {fj }j ∈N ⊂ L1loc (Rn ), ⎧ ⎫1/s    ⎨  s ⎬    (θ) M (fj )   ⎩ ⎭   j ∈N  ˙ p,q

Kω,0 (Rn )

⎛ ⎞1/s       ⎝ s⎠   |fj |     j ∈N  ˙ p,q

,

Kω,0 (Rn )

which implies that, for the above θ and s, Assumption 1.2.29 holds true p,q for K˙ ω,0 (Rn ). On the other hand, applying Lemma 1.8.6, we conclude that p,q [K˙ ω,0 (Rn )]1/s is a BBF space and, for any f ∈ L1loc (Rn ),    ((r/s))  (f ) M

˙ (Rn )]1/s ) ([K ω,0 p,q

 f ([K˙ p,q (Rn )]1/s ) . ω,0

p,q This implies that K˙ ω,0 (Rn ) satisfies Assumption 1.2.33 with the above s and r. Finally, from (5.29), we deduce that d ≥ n(1/θ − 1). Combining this, (5.30), p,q and the fact that K˙ ω,0 (Rn ) satisfies both Assumptions 1.2.29 and 1.2.33, we find p,q that the Herz space K˙ ω,0 (Rn ) under consideration satisfies all the assumptions of Lemma 5.4.5 and then complete the proof of Theorem 5.4.2.  

5.4 Molecular Characterizations

343

As an application of Theorem 5.4.2, we now establish the molecular characterip,q zation of the local generalized Hardy–Morrey space hM ω,0 (Rn ) via introducing the p,q n definition of local-(M ω,0 (R ), r, d, τ )-molecules as follows. Definition 5.4.7 Let p, q ∈ [1, ∞), ω ∈ M(R+ ) with M∞ (ω) ∈ (−∞, 0) and −

n < m0 (ω) ≤ M0 (ω) < 0, p

r ∈ [1, ∞], d ∈ Z+ , and τ ∈ (0, ∞). Then a measurable function m on Rn is p,q called a local-(M ω,0 (Rn ), r, d, τ )-molecule centered at a ball B(x0 , r0 ) ∈ B, with x0 ∈ Rn and r0 ∈ (0, ∞), if (i) for any i ∈ Z+ ,   m1S (B(x ,r ))  r n ≤ 2−τ i i 0 0 L (R )

|B(x0 , r0 )|1/r ; 1B(x0 ,r0 ) M p,q (Rn ) ω,0

(ii) when r0 ∈ (0, 1), then, for any α ∈ Zn+ with |α| ≤ d,  Rn

m(x)x α dx = 0.

Then, combining Remark 5.0.4(ii) and Theorem 5.4.2, we immediately conclude p,q the following molecular characterization of hM ω,0 (Rn ); we omit the details. Corollary 5.4.8 Let p, q ∈ [1, ∞), ω ∈ M(R+ ) with −

n < m0 (ω) ≤ M0 (ω) < 0 p

and −

n < m∞ (ω) ≤ M∞ (ω) < 0, p

s ∈ (0, 1), d ≥ n(1/s − 1) be a fixed integer,  r∈

 n ,∞ , min{m0 (ω), m∞ (ω)} + n/p

344

5 Localized Generalized Herz–Hardy Spaces p,q

and τ ∈ (0, ∞) with τ > n(1/s − 1/r). Then f ∈ hM ω,0 (Rn ) if and only if p,q f ∈ S  (Rn ) and there exists a sequence {mj }j ∈N of local-(M ω,0 (Rn ), r, d, τ )molecules centered, respectively, at cubes {Bj }j ∈N ⊂ B and a sequence {λj }j ∈N ⊂ [0, ∞) such that both f =



λj mj

j ∈N

in S  (Rn ) and ⎧ ⎫1    # $s ⎨ ⎬s  λj   1 Bj   p,q ⎭ ⎩  1Bj M (Rn )  j ∈N  ω,0

< ∞.

p,q M ω,0 (Rn )

Moreover, there exist two positive constants C1 and C2 such that, for any f ∈ p,q hM ω,0 (Rn ),

C1 f hM p,q (Rn ) ω,0

⎧ ⎧ ⎫1  ⎪  # $s ⎪ ⎨ ⎨ ⎬s  λi   ≤ inf  1 Bj  p,q  ⎩ ⎪  ⎭ 1  n B ⎪ j M ⎩ j ∈N  ω,0 (R )

⎫ ⎪ ⎪ ⎬ p,q

M ω,0 (Rn )

⎪ ⎪ ⎭

≤ C2 f hM p,q (Rn ) , ω,0

where the infimum is taken over all the decompositions of f as above. Next, we are devoted to establishing the molecular characterization of the local p,q generalized Herz–Hardy space hK˙ ω (Rn ). For this purpose, we first introduce the p,q definition of local-(K˙ ω (Rn ), r, d, τ )-molecules as follows. Definition 5.4.9 Let p, q ∈ (0, ∞), ω ∈ M(R+ ) with m0 (ω) ∈ (− pn , ∞) and −

n < m∞ (ω) ≤ M∞ (ω) < 0, p

r ∈ [1, ∞], d ∈ Z+ , and τ ∈ (0, ∞). Then a measurable function m on Rn is p,q called a local-(K˙ ω (Rn ), r, d, τ )-molecule centered at a ball B(x0 , r0 ) ∈ B, with n x0 ∈ R and r0 ∈ (0, ∞), if (i) for any i ∈ Z+ ,  m1S (B(x i

 

0 ,r0 )) Lr (Rn )

≤ 2−τ i

|B(x0 , r0 )|1/r ; 1B(x0 ,r0 ) K˙ p,q n ω (R )

5.4 Molecular Characterizations

345

(ii) when r0 ∈ (0, 1), then, for any α ∈ Zn+ with |α| ≤ d,  Rn

m(x)x α dx = 0.

p,q Via local-(K˙ ω (Rn ), r, d, τ )-molecules, we have the following molecular charp,q acterization of the local generalized Herz–Hardy space hK˙ ω (Rn ).

Theorem 5.4.10 Let p, q ∈ (0, ∞), ω ∈ M(R+ ) with m0 (ω) ∈ (− pn , ∞) and −

n < m∞ (ω) ≤ M∞ (ω) < 0, p

  s ∈ 0, min 1, p, q,

n max{M0 (ω), M∞ (ω)} + n/p

 ,

d ≥ n(1/s − 1) be a fixed integer,   r ∈ max 1, p,

  n ,∞ , min{m0 (ω), m∞ (ω)} + n/p

and τ ∈ (0, ∞) with τ > n(1/s − 1/r). Then f ∈ hK˙ ω (Rn ) if and only if ˙ ωp,q (Rn ), r, d, τ )f ∈ S  (Rn ) and there exists a sequence {mj }j ∈N of local-(K molecules centered, respectively, at the balls {Bj }j ∈N ⊂ B and a sequence {λj }j ∈N ⊂ [0, ∞) such that both p,q

f =



λj mj

j ∈N

in S  (Rn ) and ⎧ ⎫1    # $s ⎬s   ⎨ λj   1 Bj   ⎭ ⎩  1Bj K˙ p,q n ω (R )  j ∈N  ˙ p,q

< ∞.

Kω (Rn )

Moreover, there exist two positive constants C1 and C2 such that, for any f ∈ p,q hK˙ ω (Rn ),

C1 f hK˙ p,q n ω (R )

⎧⎧ ⎫1    # $s ⎪ ⎨⎨ ⎬s  λi   ≤ inf  1 Bj  p,q ⎪ ⎭  ⎩  1  n Bj K ˙ ω (R ) ⎩ j ∈N  ˙ p,q

Kω (Rn )

≤ C2 f hK˙ p,q n , ω (R ) where the infimum is taken over all the decompositions of f as above.

⎫ ⎪ ⎬ ⎪ ⎭

346

5 Localized Generalized Herz–Hardy Spaces

To establish the above molecular characterization, we first show the following improved molecular characterization of localized Hardy spaces associated with ball quasi-Banach function spaces. Theorem 5.4.11 Let X be a ball quasi-Banach function space satisfy: (i) Assumption 1.2.29 holds true with 0 < θ < s ≤ 1; (ii) for the above s, X1/s is a ball Banach function space and there exists a linear space Y ⊂ M (Rn ) equipped with a seminorm  · Y such that, for any f ∈ M (Rn ),   f X1/s ∼ sup fgL1 (Rn ) : gY = 1 , where the positive equivalence constants are independent of f ; (iii) for the above s and Y , there exists an r ∈ (1, ∞] and a positive constant C such that, for any f ∈ L1loc (Rn ),     ((r/s)) (f ) ≤ Cf Y . M Y

Let τ ∈ (0, ∞) satisfy τ > n(1/θ − 1/r), and d ≥ n(1/θ − 1) be a fixed integer. Then f ∈ hX (Rn ) if and only if f ∈ S  (Rn ) and there exists a sequence {mj }j ∈N of local-(X, r, d, τ )-molecules centered, respectively, at the balls {Bj }j ∈N ⊂ B and a sequence {λj }j ∈N ⊂ [0, ∞) such that both f =



(5.31)

λj mj

j ∈N

in S  (Rn ) and ⎡ ⎤1  

s s    λj ⎣  1Bj ⎦  < ∞.    1Bj X  j ∈N 

(5.32)

X

Moreover, there exist two positive constants C1 and C2 such that, for any f ∈ hX (Rn ),

C1 f hX (Rn )

⎧⎡ ⎤1  

s s ⎪  ⎨  λi ⎣  ≤ inf  1 Bj ⎦    ⎪ 1  Bj X ⎩ j ∈N 

⎫ ⎪ ⎬ ⎪ ⎭

X

≤ C2 f hX (Rn ) , where the infimum is taken over all the decompositions of f as above.

(5.33)

5.4 Molecular Characterizations

347

Proof Let all the symbols be as in the present theorem. We first prove the necessity. To achieve this, let f ∈ hX (Rn ). Then, from the assumptions (i) through (iii) of the present theorem, and Theorem 5.3.14, we deduce that f ∈ hX,r,d,s (Rn ). Thus, by Definition 5.3.6, we conclude that there exists a sequence {λj }j ∈N ⊂ [0, ∞) and a sequence {aj }j ∈N of local-(X, r, d)-atoms supported, respectively, in the balls {Bj }j ∈N such that both f =



(5.34)

λj aj

j ∈N

in S  (Rn ) and ⎡ ⎤1  

s s    λj ⎣  1Bj ⎦  < ∞.    1Bj X  j ∈N 

(5.35)

X

Using Remark 5.4.4(i), we find that, for any j ∈ N, aj is a local-(X, r, r, τ )molecule centered at Bj . This, combined with both (5.34) and (5.35), implies that the necessity holds true. Furthermore, by the choice of {λj }j ∈N , Definition 5.3.6, and Theorem 5.3.14 again, we conclude that ⎧⎡ ⎤1  

s s ⎪  ⎨  λi ⎣  inf  1 Bj ⎦    ⎪ ⎩ j ∈N 1Bj X 

⎫ ⎪ ⎬ ⎪ ⎭

≤ f hX,r,d,s (Rn ) ∼ f hX (Rn ) ,

(5.36)

X

where the infimum is taken over all the sequences {λj }j ∈N ⊂ [0, ∞) and {mj }j ∈N of local-(X, r, d, τ )-molecules centered, respectively, at the balls {Bj }j ∈N ⊂ B such that both (5.31) and (5.32) hold true. Conversely, we show the sufficiency. Let f ∈ S  (Rn ), {λl,j }l∈{1,2}, j ∈N ⊂ [0, ∞), and {ml,j }l∈{1,2}, j ∈N be a sequence of local-(X, r, d, τ )-molecules centered, respectively, at the balls {Bl,j }l∈{1,2}, j ∈N ∈ B such that, for any j ∈ N, r(B1,j ) ∈ (0, 1) and r(B2,j ) ∈ [1, ∞), f =

2  

(5.37)

λl,j ml,j

l=1 j ∈N

in S (Rn ), and ⎡ ⎤1   2

s s    λl,j ⎣  1Bl,j ⎦  < ∞.    1Bl,j X  l=1 j ∈N  X

(5.38)

348

5 Localized Generalized Herz–Hardy Spaces

In what follows, fix a φ ∈ S(Rn ) satisfying that supp (φ) ⊂ B(0, 1) and  Rn

φ(x) dx = 0.

Then, applying (5.37) and an argument similar to that used in the estimation of (4.24) with {λj }j ∈N and {aj }j ∈N therein replaced, respectively, by {λl,j }l∈{1,2}, j ∈N and {ml,j }l∈{1,2}, j ∈N, we find that, for any t ∈ (0, ∞), |f ∗ φt | ≤

2  

  λl,j ml,j ∗ φt  .

l=1 j ∈N

Combining this and Definition 5.1.1(i), we further obtain m(f, φ) ≤

2  

λl,j m(ml,j , φ).

l=1 j ∈N

This, together with Definition 1.2.13(ii), implies that                m(f, φ)X   λ1,j m(m1,j , φ) +  λ2,j m(m2,j , φ)  j ∈N  j ∈N  X

=: V1 + V2 .

X

(5.39)

We next estimate V1 and V2 , respectively. First, we deal with V1 . Indeed, for any j ∈ N, by Remark 5.4.4(ii) with m := m1,j , we conclude that m1,j is an (X, r, d, τ )-molecule centered at B1,j . Therefore, from Definitions 5.1.1(i), 4.1.1(i), and 1.2.13(ii), and some arguments similar to those used in the estimations of (4.62), (4.63), (4.66), and (4.67), it follows that ⎡   ⎤1/s   

s       λ1,j  ⎣    ⎦ V1   λj M(m1,j , φ)   1B1,j (5.40)  ,   1B1,j X j ∈N    j ∈N X X

which is the desired estimate of V1 . Conversely, we next estimate V2 . To this end, for any j ∈ N and k ∈ Z+ , let μj,k := 2−k(τ + r ) n

12k B2,j X 1B2,j X

5.4 Molecular Characterizations

349

and 1B2,j X

n

aj,k := 2k(τ + r )

12k B2,j X

m2,j 1Sk (B2,j ) .

Then, for any j ∈ N, we have 

m2,j =

m2,j 1Sk (B2,j ) =

k∈Z+



μj,k aj,k

(5.41)

k∈Z+

almost everywhere in Rn . In addition, by the Tonelli theorem, the Hölder inequality, and Definition 5.4.3(i) with m therein replaced by m2,j with j ∈ N, we conclude that, for any t ∈ (0, ∞) and x ∈ Rn ,   n k∈Z+ R

=

  μj,k aj,k (y) |φt (x − y)| dy



  m2,j (y) |φt (x − y)| dy

k∈Z+ Sk (B2,j )

≤

   m2,j 1S (B )  r n φt  r  n k 2,j L (R ) L (R )

k∈Z+

≤

 |B2,j |1/r φt Lr  (Rn ) 2−kτ ∼ 1. 1B2,j X k∈Z+

From this, (5.41), and the Fubini theorem, we deduce that, for any j ∈ N, t ∈ (0, ∞), and x ∈ Rn ,   m2,j ∗ φt (x) = μj,k aj,k (y)φt (x − y) dy Rn k∈Z +

=





μj,k

k∈Z+

=



Rn

aj,k φj,k (x − y) dy

μj,k aj,k ∗ φt (x).

k∈Z+

This, together with Definition 5.1.1, further implies that, for any j ∈ N, m(m2,j , φ) ≤

 k∈Z+

μj,k m(aj,k , φ).

350

5 Localized Generalized Herz–Hardy Spaces

By this and Definition 1.2.13(ii), we conclude that         V2   λ2,j μj,k m(aj,k , φ)  . j ∈N k∈Z+ 

(5.42)

X

We now claim that, for any j ∈ N and k ∈ Z+ , aj,k is a local-(X, r, d)-atom supported in 2k B2,j . Indeed, applying Definition 5.4.3(i), we find that, for any j ∈ N and k ∈ Z+ , 1/r   |2k B2,j |1/r aj,k  r n ≤ 2 nkr |B2,j | = . L (R ) 12k B2,j X 12k B2,j

(5.43)

On the other hand, for any j ∈ N and k ∈ Z+ , observe that supp (aj,k ) ⊂ 2k B2,j and r(2k B2,j ) = 2k r(B2,j ) ≥ 2k ≥ 1, which, combined with (5.43) and Definition 5.3.4, imply that aj,k is a local(X, r, d)-atom supported in 2k B2,j . This finishes the proof of the above claim. Combining this claim, the fact that, for any j ∈ N and k ∈ Z+ , r(2k B2,j ) ≥ 1, the claim obtained in the proof of Theorem 5.3.14, (5.42), and Definitions 5.1.1(i) and 4.1.1(i), we conclude that         V2   λ2,j μj,k m(aj,k , φ)12k+1 B2,j    j ∈N k∈Z+ X          λ μ M(a , φ)1 k+1 2,j j,k j,k 2 B2,j  .  j ∈N k∈Z+  X

Using this and an argument similar to that used in the estimation of II1 in the proof of Theorem 4.3.18 with {λj }j ∈N and {aj }j ∈N therein replaced, respectively, by {λ2,j μj,k }j ∈N, k∈Z+ and {aj,k }j ∈N, k∈Z+ , we find that ⎡ ⎤1  

s s     λ2,j μj,k ⎣  V2   12k B2,j ⎦    12k B2,j X  j ∈N k∈Z+  X ⎡ ⎤1  

s s     λ2,j ⎣  −ks(τ + nr ) ⎦ ∼ 2 12k B2,j  .   1B2,j X k∈Z+  j ∈N  X

5.4 Molecular Characterizations

351

This, together with an argument similar to that used in the estimation of (4.66) with {λj }j ∈N and {Bj }j ∈N therein replaced, respectively, by {λ2,j }j ∈N and {B2,j }j ∈N , further implies that ⎡ ⎤1/s   

s    λ2,j ⎣  V2   1B1,j ⎦  ,   1B2,j X  j ∈N 

(5.44)

X

which is the desired estimate of V2 . Then, from both the assumptions (i) and (ii) of the present theorem, Remark 5.1.6, Lemma 5.1.5(ii), (5.39), (5.40), (5.44), and (5.38), it follows that

f hX (Rn )

⎡ ⎤1   2

s s    λl,j ⎣  ⎦ ∼ m(f, φ)X   1Bl,j  < ∞.   1Bl,j X  l=1 j ∈N 

(5.45)

X

This further implies that f ∈ hX (Rn ), and hence finishes the proof of the sufficiency. Moreover, by (5.45) and the choice of {λl,j }l∈{1,2}, j ∈N, we conclude that ⎧⎡ ⎫ ⎤1  

s s ⎪ ⎪ ⎨   ⎬ λi   f hX (Rn )  inf ⎣ , 1 Bj ⎦   ⎪ ⎪ 1Bj X ⎩  j ∈N  ⎭ X

where the infimum is taken over all the sequences {λj }j ∈N ⊂ [0, ∞) and {mj }j ∈N of local-(X, r, d, τ )-molecules centered, respectively, at the balls {Bj }j ∈N ⊂ B such that both (5.31) and (5.32) hold true. From this and (5.36), we deduce that (5.33) holds true, which completes the proof of Theorem 5.4.11.   Remark 5.4.12 We should point out that Theorem 5.4.11 is an improved version of the known molecular characterization of hX (Rn ) established by Wang et al. in [233, Theorem 5.2]. Indeed, if Y ≡ (X1/s ) in Theorem 5.4.11, then this theorem goes back to [233, Theorem 5.2]. With the help of the above improved molecular characterization of hX (Rn ), we now prove Theorem 5.4.10. Proof of Theorem 5.4.10 Let p, q, ω, r, d, and s be as in the present theorem. Then, using the assumptions m0 (ω) ∈ (− pn , ∞) and M∞ (ω) ∈ (−∞, 0), p,q and Theorem 1.2.44, we find that the global generalized Herz space K˙ ω (Rn ) under consideration is a BQBF space. Therefore, to complete the proof of the p,q present theorem, we only need to show that K˙ ω (Rn ) satisfies (i) through (iii) of

352

5 Localized Generalized Herz–Hardy Spaces

Theorem 5.4.11. Indeed, let θ ∈ (0, s) satisfy that 6   5  6 5  1 1 1 −1 ≤n −1 < n −1 +1 n s θ s

(5.46)

and  τ >n

 1 1 − . θ r

(5.47)

For the above θ , s, and r, by the proof of Theorem 5.3.13, we conclude that the following three statements hold true: (i) for any {fj }j ∈N ⊂ L1loc (Rn ), ⎧ ⎫1/s    ⎨  s ⎬    (θ) M (fj )   ⎭  ⎩  j ∈N  ˙ p,q

Kω (Rn )

⎛ ⎞1/s       ⎝ s⎠   |fj |     j ∈N  ˙ p,q

;

Kω (Rn )

(ii) [K˙ ω (Rn )]1/s is a BBF space and, for any f ∈ M (Rn ), p,q

f [K˙ p,q n 1/s ω (R )]

 ∼ sup fgL1 (Rn ) : g ˙ (p/s) ,(q/s) B1/ωs

 (Rn )

=1

with the positive equivalence constants independent of f ; (iii) for any f ∈ L1loc (Rn ),     ((r/s)) (f ) ˙ (p/s) ,(q/s) M B1/ωs

(Rn )

 f  ˙ (p/s) ,(q/s) B1/ωs

(Rn )

.

These further imply that the Herz space K˙ ω (Rn ) under consideration satisfies the assumptions (i) through (iii) of Theorem 5.4.11. In addition, from (5.46) and the assumption d ≥ n(1/s − 1), it follows that d ≥ n(1/s − 1) = n(1/θ − 1). This, together with (5.47) and the fact that (i) through (iii) of Theorem 5.4.11 hold true, then finishes the proof of Theorem 5.4.10.   p,q

As an application of Theorem 5.4.10, we next establish the molecular characterp,q ization of the local generalized Hardy–Morrey space hM ω (Rn ). To begin with, we introduce the following concept of local molecules associated with the global p,q generalized Morrey space M ω (Rn ).

5.4 Molecular Characterizations

353

Definition 5.4.13 Let p, q, ω, r, d, and τ be as in Definition 5.4.7. Then a p,q measurable function m on Rn is called a local-(M ω (Rn ), r, d, τ )-molecule n centered at a ball B(x0 , r0 ) ∈ B, with x0 ∈ R and r0 ∈ (0, ∞), if (i) for any i ∈ Z+ ,   m1S (B(x ,r ))  r n ≤ 2−τ i i 0 0 L (R )

|B(x0 , r0 )|1/r ; 1B(x0 ,r0 ) M p,q n ω (R )

(ii) when r0 ∈ (0, 1), then, for any α ∈ Zn+ with |α| ≤ d,  Rn

m(x)x α dx = 0.

Then, by Remark 5.0.4(ii) and Theorem 5.4.10, we conclude the molecular p,q characterization of the local generalized Hardy–Morrey space hM ω (Rn ) as follows; we omit the details. Corollary 5.4.14 Let p, q, ω, r, d, and s be as in Corollary 5.4.8. Then f ∈ p,q hM ω (Rn ) if and only if f ∈ S  (Rn ) and there exists a sequence {mj }j ∈N of p,q local-(M ω (Rn ), r, d, τ )-molecules centered, respectively, at cubes {Bj }j ∈N ⊂ B and a sequence {λj }j ∈N ⊂ [0, ∞) such that both f =



λj mj

j ∈N

in S  (Rn ) and ⎧ ⎫1    # $s ⎬s  ⎨ λj   1 Bj   p,q ⎭ ⎩  1Bj M ω (Rn )  j ∈N 

< ∞. p,q

M ω (Rn )

Moreover, there exist two positive constants C1 and C2 such that, for any f ∈ p,q hM ω (Rn ),

C1 f hM p,q n ω (R )

⎧⎧ ⎫1    # $s ⎪ ⎨⎨ ⎬s  λi   ≤ inf  1 Bj  p,q ⎪ ⎭   ⎩ 1  Bj M ω (Rn ) ⎩ j ∈N 

⎫ ⎪ ⎬ p,q

M ω (Rn )

≤ C2 f hM p,q n , ω (R ) where the infimum is taken over all the decompositions of f as above.

⎪ ⎭

354

5 Localized Generalized Herz–Hardy Spaces

5.5 Littlewood–Paley Function Characterizations The main target of this section is to prove the Littlewood–Paley function characterip,q p,q zations of the local generalized Herz–Hardy spaces hK˙ ω,0 (Rn ) and hK˙ ω (Rn ). To achieve this, we first establish various Littlewood–Paley function characterizations of the local Hardy space hX (Rn ) with X being a ball quasi-Banach function space. To begin with, we present the following concepts of the local Littlewood–Paley functions. Definition 5.5.1 Let ϕ0 ∈ S(Rn ) satisfy 1B(0,1) ≤  ϕ0 ≤ 1B(0,2), and ϕ ∈ S(Rn ) satisfy 1B(0,4)\B(0,2) ≤  ϕ ≤ 1B(0,8)\B(0,1). Then, for any f ∈ S  (Rn ), the local Lusin area function S loc (f ), the local Littlewood–Paley g-function g loc (f ), and the local Littlewood–Paley gλ∗ -function (gλ∗ ) loc (f ) with λ ∈ (0, ∞) are defined, respectively, by setting, for any x ∈ Rn , 

1

S loc (f )(x) := |f ∗ ϕ0 (x)| + 0

dy dt |f ∗ ϕt (y)| n+1 t B(x,t ) 2



1

g loc (f )(x) := |f ∗ ϕ0 (x)| + 0

dt |f ∗ ϕt (x)| t 2

 12 ,

 12 ,

and -

gλ∗

. loc

(f )(x) #



1

:= |f ∗ ϕ0 (x)| + 0

Rn

t t + |x − y|

λn

dy dt |f ∗ ϕt (y)|2 n+1 t

$1 2

.

We now establish the following Lusin area function characterization of the local Hardy space hX (Rn ). Theorem 5.5.2 Let X be a ball quasi-Banach function space satisfying both Assumptions 1.2.29 and 1.2.33 with the same s. Then f ∈ hX (Rn ) if and only if f ∈ S  (Rn ) and S loc (f ) ∈ X. Moreover, there exist two positive constants C1 and C2 such that, for any f ∈ hX (Rn ), C1 f hX (Rn ) ≤ S loc (f )X ≤ C2 f hX (Rn ) .

5.5 Littlewood–Paley Function Characterizations

355

In order to show this theorem, we require some auxiliary lemmas and concepts. First, we need the following Calderón reproducing formula which was given in [233, (5.3)] (see also [91, Proposition 1.1.6]). Lemma 5.5.3 Let ϕ0 and ϕ be as in Theorem 5.5.2. Then there exist a, b, c ∈ (0, ∞) and ψ0 , ψ ∈ S(Rn ) such that - . 0 ⊂ B(0, a), supp ψ - .  ⊂ B(0, c) \ B(0, b), supp ψ and, for any f ∈ S  (Rn ), 

1

f = f ∗ ϕ0 ∗ ψ0 +

f ∗ ϕt ∗ ψt

0

dt t

in S  (Rn ), namely,  f = f ∗ ϕ0 ∗ ψ0 + lim

ε→0+ ε

1

f ∗ ϕt ∗ ψt

dt t

in S  (Rn ). We also need the following auxiliary estimate about convolutions, which is just [233, (5.9)]. Lemma 5.5.4 Let f ∈ S  (Rn ) and ϕ ∈ S(Rn ). Then there exists a positive integer m, depending only on f , and a positive constant C, independent of f , such that, for any t ∈ (0, 1] and x ∈ Rn , |f ∗ ϕt (x)| ≤ Ct −n−m (1 + |x|)m . Let S be the Lusin area operator as in Definition 4.6.1. Then the following Lp boundedness of S plays an important role in the proof of Theorem 5.5.2, which was obtained in [85, Theorem 7.8]. Lemma 5.5.5 Let p ∈ (1, ∞) and S be as in Definition 4.6.1. Then there exists a positive constant C such that, for any f ∈ Lp (Rn ), S(f )Lp (Rn ) ≤ C f Lp (Rn ) . Moreover, the following conclusion is useful in the proof of Theorem 5.5.2, which characterizes the Hardy space HX (Rn ) via the Lusin area function S and was showed in [207, Theorem 3.21].

356

5 Localized Generalized Herz–Hardy Spaces

Lemma 5.5.6 Let X be a ball quasi-Banach function space satisfying both Assumptions 1.2.29 and 1.2.33 with the same s ∈ (0, 1], ϕ be as in Definition 5.5.1, and S be as in Definition 4.6.1 with the above ϕ. Then f ∈ HX (Rn ) if and only if f ∈ S  (Rn ), f vanishes weakly at infinity, and S(f ) ∈ X. Moreover, for any f ∈ HX (Rn ), f HX (Rn ) ∼ S(f )X with the positive equivalence constants independent of f . In what follows, for any f ∈ S  (Rn ) and x ∈ Rn , let  S loc (f )(x) :=



1

0

dy dt |f ∗ ϕt (y)| n+1 t B(x,t ) 2

 12 (5.48)

.

S loc Then, in order to prove Theorem 5.5.2, we also need a technical estimate about  as follows, which was obtained in [233, p. 53]. Lemma 5.5.7 Let X be a ball quasi-Banach function space, d ∈ Z+ , and θ ∈ (0, ∞). Then there exists a positive constant C such that, for any local-(X, ∞, d)atom a supported in the ball B ∈ B with r(B) ∈ [1, ∞),  S loc (a)1(4B) ≤ C

1 M(θ) (1B ) . 1B X

Furthermore, to show Theorem 5.5.2, we require some conclusions about X-tent spaces. We first recall some basic concepts. Let α ∈ (0, ∞) and f : Rn+1 →C + be a measurable function. Then the Lusin area function Aα (f ), with aperture α, is defined by setting, for any x ∈ Rn ,  A

(α)

(f )(x) :=

dy dt |f ∗ ϕt (y)| n+1 t Γα (x) 2

1 2

,

where, for any x ∈ Rn and α ∈ (0, ∞), Γα (x) is defined as in (4.5). For any given ball quasi-Banach function space X, the X-tent space TXα (Rn+1 + ), with aperture α, n+1 is defined to be the set of all the measurable functions f : R+ → C such that f T α (Rn+1 ) := Aα (f )X < ∞. X

+

In addition, let α ∈ (0, ∞). Then, for any ball B(x, r) ∈ B with x ∈ Rn and r ∈ (0, ∞), let ! Tα (B) := (y, t) ∈ Rn+1 + : 0 < t < r/α, |y − x| < r − αt .

5.5 Littlewood–Paley Function Characterizations

357

When α := 1, we denote Tα (B) simply by T (B). Then the following definition of atoms is just [207, Definition 3.17]. Definition 5.5.8 Let X be a ball quasi-Banach function space, p ∈ (1, ∞), and α ∈ (0, ∞). A measurable function a : Rn+1 → C is called a (TXα , p)-atom if + there exists a ball B ∈ B such that (i) supp (a) := {(y, t) ∈ Rn+1 + : a(y, t) = 0} ⊂ Tα (B); (ii) Aα (a)Lp (Rn ) ≤

|B|1/p 1B X .

Moreover, if a is a (TXα , p)-atom for any p ∈ (1, ∞), then a is called a (TXα , ∞)atom. Repeating an argument similar to that used in the proof of [115, Lemma 4.8] with Lϕ (Rn ) therein replaced by X, we obtain the following auxiliary conclusion about (TX1 , ∞)-atoms, which plays a key role in the proof of the Lusin area function characterization of hX (Rn ); we omit the details. Lemma 5.5.9 Let d ∈ Z+ , ψ ∈ S(Rn ) satisfy that, for any γ ∈ Zn+ with |γ | ≤ d, / γ Rn ψ(x)x dx = 0, and X be a ball quasi-Banach function space. Assume that 1 a is a (TX , ∞)-atom supported in T (B) with B ∈ B. Then, for any τ ∈ (0, ∞), /∞ dt 0 a(·, t) ∗ ψt t is a harmless constant multiple of an (X, ∞, d, τ )-molecule centered at the ball B. Moreover, the following atomic characterization of X-tent spaces obtained in [207, Theorem 3.19] is also an essential tool in the proof of Theorem 5.5.2. Lemma 5.5.10 Let f : Rn+1 → C be a measurable function and X a ball + quasi-Banach function space. Assume that X satisfies both Assumptions 1.2.29 and 1.2.33 with the same s ∈ (0, 1]. Then f ∈ TX1 (Rn+1 + ) if and only if there exists a sequence {λj }j ∈N ⊂ [0, ∞) and a sequence {aj }j ∈N of (TX1 , ∞)-atoms supported, respectively, in {T (Bj )}j ∈N with {Bj }j ∈N ⊂ B such that, for almost every (x, t) ∈ Rn+1 + , f (x, t) =



λj aj (x, t)

j ∈N

and |f (x, t)| =

 j ∈N

λj |aj (x, t)|.

358

5 Localized Generalized Herz–Hardy Spaces

Moreover,

f T 1 (Rn+1 ) X

+

⎡ ⎤1/s   

s    λj ⎣  ∼ 1 Bj ⎦  ,   1Bj X  j ∈N  X

where the positive equivalence constants are independent of f . Via these preparations, we now show Theorem 5.5.2. Proof of Theorem 5.5.2 Let all the symbols be as in the present theorem. We first prove the sufficiency. To this end, let f ∈ S  (Rn ) satisfy S loc (f ) ∈ X. Then, applying Lemma 5.5.3, we find that there exist a, b, c ∈ (0, ∞) and ψ0 , ψ ∈ S(Rn ) 0 ) ⊂ B(0, a), supp (ψ)  ⊂ B(0, c) \ B(0, b), and such that supp (ψ 

1

f = f ∗ ϕ0 ∗ ψ0 +

f ∗ ϕt ∗ ψt

0

dt t

(5.49)

in S  (Rn ). From both Definitions 5.5.1 and 1.2.13(ii), it follows that  1      2 dy dt 2    f ∗ ϕt 1{τ ∈(0,1)} 1 n+1 =  f ∗ ϕt (y)1{τ : τ ∈(0,1)}(t)  TX (R+ )   Γ (·) t n+1 X    1   12   dy dt   |f ∗ ϕt (y)|2 n+1 =   t   0 B(·,t ) X

≤ S loc (f )X < ∞. This, combined with Lemma 5.5.10, further implies that there exists a sequence {λ1,j }j ∈N ⊂ [0, ∞) and a sequence {a1,j }j ∈N of (TX1 , ∞)-atoms supported, respectively, in {T (B1,j )}j ∈N with {B1,j }j ∈N ⊂ B such that, for almost every (x, t) ∈ Rn+1 + f ∗ ϕt (x)1{τ : τ ∈(0,1)}(t) =



λ1,j a1,j (x, t),

(5.50)

j ∈N

|f ∗ ϕt (x)| 1{τ : τ ∈(0,1)}(t) =

 j ∈N

λ1,j a1,j |(x, t)| ,

(5.51)

5.5 Littlewood–Paley Function Characterizations

359

and ⎡ ⎤1/s   ∞ 

s      λ1,j ⎣  ⎦ 1B1,j   ∼ f ∗ ϕt 1{τ ∈(0,1)}T 1 (Rn+1 ) + X   1B1,j X  j =1  X

 S loc (f )X .

(5.52)

Next, we show that 

1

f ∗ ϕt ∗ ψt

0

 dt λ1,j = t



j ∈N

1

aj (·, t) ∗ ψt

0

dt t

(5.53)

in S  (Rn ). Indeed, by Lemma 5.5.4, we find that there exists an m ∈ N, depending only on f , such that, for any t ∈ (0, 1] and x ∈ Rn , |f ∗ ϕt (x)|  t −n−m (1 + |x|)m .

(5.54)

This further implies that, for any t ∈ (0, 1] and x ∈ Rn , |f ∗ ϕt | ∗ |ψt | (x)  |f ∗ ϕt (y)| |ψt (x − y)| dy = Rn

   1  x − y  t (1 + |y|) n ψ  dy t t Rn  ∼ t −n−m (1 + |x − ty|)m |ψ(y)| dy −n−m



m

Rn

t

−n−m



(1 + |x|)

 t −n−m (1 + |x|)m







m Rn

Rn

|ψ(y)| dy +

(1 + |y|) |ψ(y)| dy m

Rn

1 dy ∼ t −n−m (1 + |x|)m . (1 + |y|)n+1

From this and (5.51), we deduce that 

1

 Rn j ∈N

0

1



= 0

  dx dt λ1,j a1,j  ∗ |ψt | (x) |η(x)| t 

Rn j ∈N

  dx dt λ1,j a1,j (x, t) |ψt (−·)| ∗ |η| (x) t

360

5 Localized Generalized Herz–Hardy Spaces



1

= 

0

= 

0

 

1 1

Rn

Rn

Rn

0 1

∼

|f ∗ ϕt (x)| |ψt (−·)| ∗ |η| (x) |f ∗ ϕt | ∗ |ψt | (x) |η(x)|

dx dt t

t −n−m (1 + |x|)m (1 + |x|)−n−m−1 

1 t n+m+1

0

dx dt t

dt

Rn

dx dt t

1 dx < ∞. (1 + |x|)n+1

Combining this, (5.49), (5.50), and the Fubini theorem, we further conclude that  dt η(x) dx f ∗ ϕt ∗ ψt (x) t 0 Rn  1 dx dt = f ∗ ϕt ∗ ψt (x)η(x) n t 0 R  1 dx dt f ∗ ϕt (x)ψt (−·) ∗ η(x) = n t 0 R  1  dx dt λ1,j a1,j (x, t)ψt (−·) ∗ η(x) = n t 0 R





1

 =

Rn

⎡ ⎣

j ∈N

 j ∈N

⎤



1

λ1,j

a1,j ∗ ψt (x)

0

dt ⎦ η(x) dx, t

which implies that (5.53) holds true.  ⊂ B(0, c) \ B(0, b). This implies that, for any In addition, observe that supp (ψ) γ ∈ Zn+ ,  Rn

ψ(x)x γ dx = (−2πi)−|γ |

 Rn

ψ(x) (2πix)γ dx

. = (−2πi)−|γ | F ψ [2πi (·)]γ (0)  = (−2πi)−|γ | ∂ γ ψ(0) = 0.

In what follows, let d ≥ n(1/θ − 1) be a fixed integer and τ ∈ (n( θ1 − 1r ), ∞) a/ fixed positive constant. Then, from Lemma 5.5.9, we deduce that, for any j ∈ N, ∞ dt 0 a1,j (·, t) ∗ ψt t is a harmless constant multiple of an (X, ∞, d, τ )-molecule centered at B1,j . On the other hand, by (5.51), we find that, for any j ∈ N and

5.5 Littlewood–Paley Function Characterizations

361

t ∈ [1, ∞) and for almost every x ∈ Rn , a1,j (x, t) = 0. This further implies that, for any j ∈ N,  0

1

dt = a1,j (·, t) ∗ ψt (y) t



∞

a1,j (·, t) ∗ ψt (y)

0

dt . t

/1 Therefore, for any j ∈ N, 0 a1,j (·, t) ∗ ψt dtt is a harmless constant multiple of an (X, ∞, d, τ )-molecule centered at B1,j . Combining this, (5.53), Lemma 5.4.5, and (5.52), we further conclude that    

1

f ∗ ϕt ∗ ψt (·)

0

 dt   t hX (Rn )

⎡ ⎤1/s   ∞ 

s    λ1,j ⎣   1B1,j ⎦   S loc (f )X .   1B1,j X  j =1 

(5.55)

X

Next, we prove that f ∗ ϕ0 ∗ ψ0 hX (Rn )  S loc (f )X .

(5.56)

To achieve this, for any α ∈ Zn , let Qα :=√2α + [0, 2)n , e := (1, . . . , 1) ∈ Rn denote the unit of Rn , and Bα := B(e + α, n). Take arrangements of {Qα }α∈Zn and {Bα }α∈Zn , which are denoted, respectively, by {Q2,j }j ∈N and {B2,j }j ∈N . Then it holds true that, for any j ∈ N, Q2,j ⊂ B2,j , {Qj }j ∈N are pairwise disjoint cubes, "

Q2,j = Rn , and

j ∈N



1B2,j ≤ 3n .

j ∈N

For any j ∈ N, if f ∗ ϕ0 ∗ ψ0 1Q2,j L∞ (Rn ) = 0, define λ2,j := 0 and a2,j := 0; if f ∗ ϕ0 ∗ ψ0 1Q2,j L∞ (Rn ) = 0, define     λ2,j := f ∗ ϕ0 ∗ ψ0 1Q2,j L∞ (Rn ) 1B2,j X and a2,j :=

f ∗ ϕ0 ∗ ψ0 1Q2,j f ∗ ϕ0 ∗ ψ0 1Q2,j L∞ (Rn ) 1B2,j X

.

Then, for any x ∈ Rn , f ∗ ϕ0 ∗ ψ0 (x) =

 j ∈N

λ2,j a2,j (x)

(5.57)

362

5 Localized Generalized Herz–Hardy Spaces

and |f ∗ ϕ0 ∗ ψ0 (x)| =



  λ2,j a2,j (x) .

(5.58)

j ∈N

4 We now prove that f ∗ ϕ0 ∗ ψ0 = j ∈N λ2,j a2,j in S  (Rn ). Indeed, applying (5.54) with ϕ and t therein replaced, respectively, by ϕ0 ∗ ψ0 and 1, we find that, for any x ∈ Rn , |f ∗ ϕ0 ∗ ψ0 (x)|  (1 + |x|)m , which, together with (5.58), further implies that, for any given η ∈ S(Rn ), 

 Rn j ∈N

  λ2,j a2,j (x) |η(x)| dx



=  

Rn

Rn

|f ∗ ϕ0 ∗ ψ0 (x)| |η(x)| dx  (1 + |x|)m |η(x)| dx ∼

Rn

1 dx < ∞. (1 + |x|)n+1

From this, (5.57), and the Fubini theorem, we deduce that, for any given η ∈ S(Rn ),  Rn

f ∗ ϕ0 ∗ ψ0 (x)η(x) dx

=



 Rn j ∈N

λ2,j a2,j (x)η(x) dx =



 λ2,j

j ∈N

Rn

a2,j (x)η(x) dx.

4 This further implies that f ∗ ϕ0 ∗ ψ0 = j ∈N λ2,j a2,j holds true in S  (Rn ). In addition, notice that, for any j ∈ N, supp (a2,j ) ⊂ B2,j and   a2,j 

L∞ (Rn )

≤

1 . 1B2,j X

These further imply that, for any j ∈ N, 1/r   a2,j  r n ≤ |B2,j | . L (R ) 1B2,j X

√ Moreover, for any j ∈ N, we have r(B2,j ) = n ≥ 1. Therefore, applying Definition 5.3.4, we find that, for any j ∈ N, a2,j is a local-(X, 4 r, d)-atom supported in the ball B2,j . Combining this, the fact that f ∗ϕ0 ∗ψ0 = j ∈N λ2,j a2,j

5.5 Littlewood–Paley Function Characterizations

363

holds true in S  (Rn ), Lemma 5.3.7, and Definition 5.3.6, we conclude that f ∗ ϕ0 ∗ ψ0 hX (Rn ) ∼ f ∗ ϕ0 ∗ ψ0 hX,r,d,s (Rn ) ⎡ ⎤1  

s s    λ2,j ⎣   1B2,j ⎦    1B2,j X  j ∈N  X ⎡ ⎤1   s     s ⎣ f ∗ ϕ0 ∗ ψ0 1Q  ∞ n 1B ⎦  ∼  . 2,j L (R ) 2,j    j ∈N 

(5.59)

X

Next, we estimate f ∗ ϕ0 ∗ ψ0 1Q2,j L∞ (Rn ) 1B2,j for any j ∈ N. Indeed, for any given j ∈√N and x ∈ B2,j , and for any y ∈ Q2,j , we have |y − x| < 2r(B2,j ) = 2 n. This, combined with Lemma 5.2.5 with ϕ, r, and y therein replaced, respectively, by ϕ0 , θ , and z, implies that, for any j ∈ N and x ∈ B2,j ,   f ∗ ϕ0 ∗ ψ0 1Q  ∞ n 2,j L (R ) ≤ ≤

sup

√ y∈Rn , |y−x|λ} X < ∞.

Remark 6.1.4 Let X be a ball quasi-Banach function space and E ⊂ Rn a measurable set. Then it is easy to show that 1E W X = 1E X . The following conclusion obtained in [278, Lemma 2.13] shows that the weak ball quasi-Banach function space is also a ball quasi-Banach function space. This plays a key role in the proof of Theorem 6.1.1. Lemma 6.1.5 Let X be a ball quasi-Banach function space. Then the weak ball quasi-Banach function space W X is also a ball quasi-Banach function space.

6.1 Maximal Function Characterizations

405

p,q To show the maximal function characterizations of W H K˙ ω,0 (Rn ), we also require a lemma about the boundedness of the Hardy–Littlewood maximal operator on weak local generalized Herz spaces as follows.

Lemma 6.1.6 Let p, q ∈ (0, ∞) and ω ∈ M(R+ ) satisfy m0 (ω) ∈ (− pn , ∞) and m∞ (ω) ∈ (− pn , ∞). Then, for any given   r ∈ 0, min p,

n max{M0 (ω), M∞ (ω)} + n/p

 ,

there exists a positive constant C such that, for any f ∈ L1loc (Rn ), M(f )[W K˙ p,q (Rn )]1/r ≤ Cf [W K˙ p,q (Rn )]1/r . ω,0

ω,0

To prove Lemma 6.1.6, we first state the following auxiliary conclusion in ball quasi-Banach function spaces, which was obtained in [219, Theorem 4.4]. Lemma 6.1.7 Let X be a ball quasi-Banach function space and there exists a p− ∈ (0, ∞) such that, for any given r ∈ (0, p− ) and u ∈ (1, ∞), there exists a positive constant C such that, for any {fj }j ∈N ⊂ L1loc (Rn ), ⎧ ⎫1    ⎨  - . ⎬ u  u   M fj   ⎩ ⎭   j ∈N 

X 1/r

⎛ ⎞1       u u ⎝ fj  ⎠  ≤C     j ∈N 

.

X 1/r

Then, for any given r ∈ (0, p− ) and u ∈ (1, ∞), there exists a positive constant C such that, for any {fj }j ∈N ⊂ L1loc (Rn ), ⎧ ⎫1    ⎨  - . ⎬ u   u  M fj   ⎩ ⎭    j ∈N

(W X)1/r

⎛ ⎞1       u ⎝ u fj  ⎠  ≤C     j ∈N 

.

(W X)1/r

Via this conclusion, we now prove Lemma 6.1.6. Proof of Lemma 6.1.6 Let all the symbols be as in the present lemma. Then, by the assumption m0 (ω) ∈ (− pn , ∞) and Theorem 1.2.42, we conclude that the local p,q generalized Herz space K˙ ω,0 (Rn ) under consideration is a BQBF space. Moreover, let   n p− := min p, . max{M0 (ω), M∞ (ω)} + n/p

406

6 Weak Generalized Herz–Hardy Spaces

Then, for any given r ∈ (0, p− ) and u ∈ (1, ∞), applying Lemma 4.3.10, we find that, for any {fj }j ∈N ⊂ L1loc (Rn ), ⎧ ⎫1    ⎨  - . ⎬ u    u M fj   ⎩ ⎭    j ∈N

˙ p,q (Rn )]1/r [K ω,0

⎛ ⎞1       u ⎝ u⎠    fj      j ∈N 

.

˙ p,q (Rn )]1/r [K ω,0

This, combined with the fact that K˙ ω,0 (Rn ) is a BQBF space and Lemma 6.1.7 with p,q X := K˙ ω,0 (Rn ), f1 := f , and fj := 0 for any j ∈ N ∩ [2, ∞), further implies that, for any given r ∈ (0, p− ) and for any f ∈ L1loc (Rn ), p,q

M(f )[W K˙ p,q (Rn )]1/r  f [W K˙ p,q (Rn )]1/r , ω,0

ω,0

 

which then completes the proof of Lemma 6.1.6.

Via the above lemmas and the known maximal function characterizations of Hardy spaces associated with ball quasi-Banach function spaces proved in [207, Theorem 3.1] (see also Lemma 4.1.4 above), we next show Theorem 6.1.1. Proof of Theorem 6.1.1 Let all the symbols be as in the present theorem. Then, from the assumption m0 (ω) ∈ (− pn , ∞) and Theorem 1.2.42, it follows that the p,q local generalized Herz space K˙ ω,0 (Rn ) under consideration is a BQBF space. By p,q this and Lemma 6.1.5 with X := K˙ ω,0 (Rn ), we conclude that the weak space p,q W K˙ ω,0 (Rn ) is also a BQBF space. This, combined with Lemma 4.1.4(i), finishes the proof of (i). Next, we show (ii). Indeed, let  r∈

  n n , min p, . b max{M0 (ω), M∞ (ω)} + n/p

Then, using Lemma 6.1.6, we find that, for any f ∈ L1loc (Rn ), M(f )[W K˙ p,q (Rn )]1/r  f [W K˙ p,q (Rn )]1/r . ω,0

ω,0

From this, the fact that b ∈ ( nr , ∞), and Lemma 4.1.4(ii), we deduce that (ii) holds true and hence complete the proof of Theorem 6.1.1.   Via both Theorem 6.1.1 and Remark 6.0.25, we immediately obtain the following maximal function characterizations of the weak generalized Hardy–Morrey spaces p,q W H M ω,0 (Rn ); we omit the details. Corollary 6.1.8 Let / a, b ∈ (0, ∞), p, q ∈ [1, ∞), ω ∈ M(R+ ), N ∈ N, and φ ∈ S(Rn ) satisfy Rn φ(x) dx = 0.

6.1 Maximal Function Characterizations

407

(i) Let N ∈ N ∩ [b + 1, ∞) and ω satisfy M∞ (ω) ∈ (−∞, 0) and −

n < m0 (ω) ≤ M0 (ω) < 0. p

Then, for any f ∈ S  (Rn ), M(f, φ)W M p,q (Rn )  Ma∗ (f, φ)W M p,q (Rn )  Mb∗∗ (f, φ)W M p,q (Rn ) , ω,0

ω,0

ω,0

M(f, φ)W M p,q (Rn )  MN (f )W M p,q (Rn )  Mb+1 (f )W M p,q (Rn ) ω,0

ω,0

ω,0

Mb∗∗ (f, φ)W M p,q (Rn ) , ω,0

 and

Mb∗∗ (f, φ)W M p,q (Rn ) ∼ M∗∗ b,N (f )W M p,q (Rn ) , ω,0

ω,0

where the implicit positive constants are independent of f . (ii) Let ω ∈ M(R+ ) satisfy −

n < m0 (ω) ≤ M0 (ω) < 0 p

and −

n < m0 (ω) ≤ M0 (ω) < 0. p

Assume b ∈ ( pn , ∞). Then, for any f ∈ S  (Rn ), Mb∗∗ (f, φ)W M p,q (Rn )  M(f, φ)W M p,q (Rn ) , ω,0

ω,0

where the implicit positive constant is independent of f . In particular, when N ∈ N ∩ [b + 1, ∞), if one of the quantities M(f, φ)W M p,q (Rn ) , Ma∗ (f, φ)W M p,q (Rn ) , MN (f )W M p,q (Rn ) , ω,0

ω,0

ω,0

Mb∗∗ (f, φ)W M p,q (Rn ) , and M∗∗ b,N (f )W M p,q (Rn ) ω,0

ω,0

is finite, then the others are also finite and mutually equivalent with the positive equivalence constants independent of f . We now establish the maximal function characterizations of the weak generalized p,q Herz–Hardy space W H K˙ ω (Rn ) as follows.

408

6 Weak Generalized Herz–Hardy Spaces

n Theorem / 6.1.9 Let p, q, a, b ∈ (0, ∞), ω ∈ M(R+ ), N ∈ N, and φ ∈ S(R ) satisfy Rn φ(x) dx = 0.

(i) Let N ∈ N ∩ [b + 1, ∞) and ω satisfy m0 (ω) ∈ (− pn , ∞) and M∞ (ω) ∈ (−∞, 0). Then, for any f ∈ S  (Rn ), ∗ ∗∗ M(f, φ)W K˙ p,q n  Ma (f, φ)W K n  Mb (f, φ)W K n , ˙ p,q ˙ p,q ω (R ) ω (R ) ω (R )

M(f, φ)W K˙ p,q n  MN (f )W K n  Mb+1 (f )W K n ˙ p,q ˙ p,q ω (R ) ω (R ) ω (R )  Mb∗∗ (f, φ)W K˙ p,q n , ω (R ) and ∗∗ Mb∗∗ (f, φ)W K˙ p,q n ∼ Mb,N (f )W K n , ˙ p,q ω (R ) ω (R )

where the implicit positive constants are independent of f . (ii) Let ω ∈ M(R+ ) satisfy m0 (ω) ∈ (− pn , ∞) and −

n < m∞ (ω) ≤ M∞ (ω) < 0. p

Assume b ∈ (max{ pn , max{M0 (ω), M∞ (ω)} + S  (Rn ),

n p }, ∞).

Then, for any f ∈

Mb∗∗ (f, φ)W K˙ p,q n  M(f, φ)W K n , ˙ p,q ω (R ) ω (R ) where the implicit positive constant is independent of f . In particular, when N ∈ N ∩ [b + 1, ∞), if one of the quantities ∗ M(f, φ)W K˙ p,q n , Ma (f, φ)W K n , MN (f )W K n , ˙ p,q ˙ p,q ω (R ) ω (R ) ω (R ) ∗∗ Mb∗∗ (f, φ)W K˙ p,q n , and Mb,N (f )W K n ˙ p,q ω (R ) ω (R )

is finite, then the others are also finite and mutually equivalent with the positive equivalence constants independent of f . Remark 6.1.10 p,q (i) Let p, q ∈ (0, ∞) and ω ∈ M(R+ ). Then the quasi-norm of W H K˙ ω (Rn ) in Definition 6.0.20(ii) depends on N. However, from Theorem 6.1.9, we infer p,q that the weak Hardy space W H K˙ ω (Rn ) is independent of the choice of N whenever ω ∈ M(R+ ) satisfies m0 (ω) ∈ (− pn , ∞) and

−

n < m∞ (ω) ≤ M∞ (ω) < 0, p

6.1 Maximal Function Characterizations

409

and N satisfies     n n N ∈ N ∩ 1 + max , max{M0 (ω), M∞ (ω)} + ,∞ . p p (ii) Notice that, if p = q ∈ (0, ∞) and ω(t) := 1 for any t ∈ (0, ∞), then, in this case, m0 (ω) = M0 (ω) = m∞ (ω) = M∞ (ω) = 0 p,q and W H K˙ ω (Rn ) coincides with the classical weak Hardy space W H p (Rn ) in the sense of equivalent quasi-norms. Therefore, Theorem 6.1.9 completely excludes the classical weak Hardy space W H p (Rn ) and, based on Remark 1.2.45, we find that the classical weak Hardy space W H p (Rn ) is p,q the critical case of W H K˙ ω (Rn ) considered in Theorem 6.1.9.

To prove this maximal function characterizations, we first show the boundedness of the Hardy–Littlewood maximal operator on weak global generalized Herz spaces as follows. Lemma 6.1.11 Let p, q ∈ (0, ∞) and ω ∈ M(R+ ) satisfy m0 (ω) ∈ (− pn , ∞) and m∞ (ω) ∈ (− pn , ∞). Then, for any given   r ∈ 0, min p,

n max{M0 (ω), M∞ (ω)} + n/p

 ,

there exists a positive constant C such that, for any f ∈ L1loc (Rn ), M(f )[W K˙ p,q n 1/r ≤ Cf [W K n 1/r . ˙ p,q ω (R )] ω (R )] Proof Let all the symbols be as in the present lemma. Then, from the assumptions m0 (ω) ∈ (− pn , ∞) and M∞ (ω) ∈ (−∞, 0), and Theorem 1.2.44, it follows that the p,q global generalized Herz space K˙ ω (Rn ) under consideration is a BQBF space. Let  n . p− := min p, max{M0 (ω), M∞ (ω)} + n/p 

Then, for any given r ∈ (0, p− ) and u ∈ (1, ∞), applying Lemma 4.3.24, we find that, for any {fj }j ∈N ⊂ L1loc (Rn ), ⎧ ⎫1    ⎨  - . ⎬ u  u   M fj   ⎭  ⎩   j ∈N

˙ ω (Rn )]1/r [K p,q

⎛ ⎞1       u u ⎝ fj  ⎠       j ∈N 

.

˙ ω (Rn )]1/r [K p,q

410

6 Weak Generalized Herz–Hardy Spaces

This, combined with the fact that K˙ ω (Rn ) is a BQBF space and Lemma 6.1.7 with p,q X := K˙ ω (Rn ), f1 := f , and fj := 0 for any j ∈ N ∩ [2, ∞), further implies that, for any given r ∈ (0, p− ) and for any f ∈ L1loc (Rn ), p,q

M(f )[W K˙ p,q n 1/r  f [W K n 1/r , ˙ p,q ω (R )] ω (R )]  

which then completes the proof of Lemma 6.1.11. Using Lemma 6.1.11, we next show Theorem 6.1.9.

Proof of Theorem 6.1.9 Let all the symbols be as in the present theorem. Then, applying the assumptions m0 (ω) ∈ (− pn , ∞) and M∞ (ω) ∈ (−∞, 0), and Thep,q orem 1.2.44, we conclude that the global generalized Herz space K˙ ω (Rn ) under consideration is a BQBF space. This, together with Lemma 6.1.5, further implies p,q that the weak Herz space W K˙ ω (Rn ) is a BQBF space. Thus, by Lemma 4.1.4(i), we then complete the proof of (i). Next, we prove (ii). Indeed, let  r∈

  n n , min p, . b max{M0 (ω), M∞ (ω)} + n/p

(6.1)

Then, from Lemma 6.1.11, it follows that, for any f ∈ L1loc (Rn ), M(f )[W K˙ p,q n 1/r  f [W K n 1/r . ˙ p,q ω (R )] ω (R )]

(6.2)

In addition, by (6.1), we find that b ∈ ( nr , ∞). Combining this, (6.2), and Lemma 4.1.4(ii), we then complete the proof of (ii) and hence Theorem 6.1.9.   As an application of Theorem 6.1.9, we next present the following result about the maximal function characterizations of the weak generalized Hardy–Morrey p,q space W H M ω (Rn ), which is just a immediately corollary of both Theorem 6.1.9 and Remark 6.0.25; we omit the details. Corollary 6.1.12 /Let a, b ∈ (0, ∞), p, q ∈ [1, ∞), ω ∈ M(R+ ), N ∈ N, and φ ∈ S(Rn ) satisfy Rn φ(x) dx = 0. (i) Let N ∈ N ∩ [b + 1, ∞) and ω satisfy M∞ (ω) ∈ (−∞, 0) and −

n < m0 (ω) ≤ M0 (ω) < 0. p

Then, for any f ∈ S  (Rn ), ∗ ∗∗ M(f, φ)W M p,q n  Ma (f, φ)W M p,q (Rn )  Mb (f, φ)W M p,q (Rn ) , ω (R ) ω ω

M(f, φ)W M p,q n  MN (f )W M p,q (Rn )  Mb+1 (f )W M p,q (Rn ) ω (R ) ω ω  Mb∗∗ (f, φ)W M p,q n , ω (R )

6.1 Maximal Function Characterizations

411

and ∗∗ Mb∗∗ (f, φ)W M p,q n ∼ Mb,N (f )W M p,q (Rn ) , ω (R ) ω

where the implicit positive constants are independent of f . (ii) Let ω ∈ M(R+ ) satisfy −

n < m0 (ω) ≤ M0 (ω) < 0 p

and −

n < m0 (ω) ≤ M0 (ω) < 0. p

Assume b ∈ ( pn , ∞). Then, for any f ∈ S  (Rn ), Mb∗∗ (f, φ)W M p,q n  M(f, φ)W M p,q (Rn ) , ω (R ) ω where the implicit positive constant is independent of f . In particular, when N ∈ N ∩ [b + 1, ∞), if one of the quantities ∗ M(f, φ)W M p,q n , Ma (f, φ)W M p,q (Rn ) , MN (f )W M p,q (Rn ) , ω (R ) ω ω ∗∗ Mb∗∗ (f, φ)W M p,q n , and Mb,N (f )W M p,q (Rn ) ω (R ) ω

is finite, then the others are also finite and mutually equivalent with the positive equivalence constants independent of f . Remark 6.1.13 Let p, q ∈ (0, ∞) and ω ∈ M(R+ ). Then the quasi-norms of p,q p,q W H M ω,0 (Rn ) and W H M ω (Rn ) in Definition 6.0.24 depend on N. However, p,q from Corollaries 6.1.8 and 6.1.12, it follows that the spaces W H M ω,0 (Rn ) and p,q W H M ω (Rn ) are both independent of the choice of N whenever p, q ∈ [1, ∞), ω ∈ M(R+ ) satisfies −

n < m0 (ω) ≤ M0 (ω) < 0 p

and −

n < m∞ (ω) ≤ M∞ (ω) < 0, p

and N satisfies   n N ∈ N ∩ 1 + ,∞ . p

412

6 Weak Generalized Herz–Hardy Spaces

6.2 Relations with Weak Generalized Herz Spaces In this section, via the relation between ball quasi-Banach function spaces and associated Hardy spaces obtained in [207, Theorem 3.4] (see also Lemma 4.2.6 above), we investigate the relations between weak generalized Herz spaces and p,q associated Hardy spaces. We first establish the relation between W K˙ ω,0 (Rn ) p,q and the associated Hardy space W H K˙ ω,0 (Rn ) as follows. Indeed, the following p,q theorem shows that, under some reasonable and sharp assumptions, W K˙ ω,0 (Rn ) = p,q W H K˙ ω,0 (Rn ) with equivalent quasi-norms. Theorem 6.2.1 Let p ∈ (1, ∞), q ∈ (0, ∞), and ω ∈ M(R+ ) satisfy −

n n < m0 (ω) ≤ M0 (ω) <  p p

and − where

1 p

+

1 p

n n < m∞ (ω) ≤ M∞ (ω) <  , p p

= 1. Then

p,q (i) W K˙ ω,0 (Rn ) → S  (Rn ). p,q p,q (ii) If f ∈ W K˙ ω,0 (Rn ), then f ∈ W H K˙ ω,0 (Rn ) and there exists a positive constant C, independent of f , such that

f W H K˙ p,q (Rn ) ≤ C f W K˙ p,q (Rn ) . ω,0

ω,0

p,q (iii) If f ∈ W H K˙ ω,0 (Rn ), then there exists a locally integrable function g p,q belonging to W K˙ ω,0 (Rn ) such that g represents f , which means that f = g in S  (Rn ),

f W H K˙ p,q (Rn ) = gW H K˙ p,q (Rn ) , ω,0

ω,0

and there exists a positive constant C, independent of f , such that gW K˙ p,q (Rn ) ≤ C f W H K˙ p,q (Rn ) . ω,0

ω,0

Proof Let all the symbols be as in the present theorem. Then, from the assumption m0 (ω) ∈ (− pn , ∞) and Theorem 1.2.42, it follows that the local generalp,q ized Herz space K˙ ω,0 (Rn ) under consideration is a BQBF space. This, together with Lemma 6.1.5, further implies that the weak local generalized Herz space p,q W K˙ ω,0 (Rn ) is a BQBF space. Therefore, to finish the proof of the present theorem,

6.2 Relations with Weak Generalized Herz Spaces

413

we only need to show that W K˙ ω,0 (Rn ) satisfies all the assumptions of Lemma 4.2.6. Namely, there exists an r ∈ (1, ∞) such that the Hardy–Littlewood maximal p,q operator M is bounded on [W K˙ ω,0 (Rn )]1/r . Indeed, applying the assumptions p,q

  n min{m0 (ω), m∞ (ω)} ∈ − , ∞ p and 

n max{M0 (ω), M∞ (ω)} ∈ −∞,  p

 ,

and Remark 1.1.5(iii), we conclude that n ∈ (1, ∞). max{M0 (ω), M∞ (ω)} + n/p Combining this and the assumption p ∈ (1, ∞), we further find that  min p,

n max{M0 (ω), M∞ (ω)} + n/p

Thus, we can choose an   r ∈ 1, min p,

 ∈ (1, ∞).

n max{M0 (ω), M∞ (ω)} + n/p

 .

For this r, from Lemma 6.1.6, we infer that, for any f ∈ L1loc (Rn ), M(f )[W K˙ p,q (Rn )]1/r  f [W K˙ p,q (Rn )]1/r . ω,0

ω,0

This further implies that there exists an r ∈ (1, ∞) such that M is bounded p,q on [W K˙ ω,0 (Rn )]1/r . Thus, all the assumptions of Lemma 4.2.6 hold true for p,q   W K˙ ω,0 (Rn ), and hence the proof of Theorem 6.2.1 is then completed. Using Theorem 6.2.1 and Remark 6.0.25, we immediately obtain the following p,q p,q conclusion, which shows that W M ω,0 (Rn ) = W H M ω,0 (Rn ) with equivalent quasinorms under some reasonable and sharp assumptions; we omit the details. Corollary 6.2.2 Let p ∈ (1, ∞), q ∈ [1, ∞), and ω ∈ M(R+ ) satisfy −

n < m0 (ω) ≤ M0 (ω) < 0 p

414

6 Weak Generalized Herz–Hardy Spaces

and −

n < m∞ (ω) ≤ M∞ (ω) < 0. p

Then (i) W M ω,0 (Rn ) → S  (Rn ). p,q p,q (ii) If f ∈ W M ω,0 (Rn ), then f ∈ W H M ω,0 (Rn ) and there exists a positive constant C, independent of f , such that p,q

f W H M p,q (Rn ) ≤ C f W M p,q (Rn ) . ω,0

ω,0

p,q

(iii) If f ∈ W H M ω,0 (Rn ), then there exists a locally integrable function g p,q belonging to W M ω,0 (Rn ) such that g represents f , which means that f = g in S  (Rn ), f W H M p,q (Rn ) = gW H M p,q (Rn ) , ω,0

ω,0

and there exists a positive constant C, independent of f , such that gW M p,q (Rn ) ≤ C f W H M p,q (Rn ) . ω,0

ω,0

Similarly, we next prove that, under some reasonable and sharp assumptions, the p,q weak generalized Herz–Hardy space W H K˙ ω (Rn ) coincides with the weak global p,q n ˙ generalized Herz space W Kω (R ) with equivalent quasi-norms as follows. Theorem 6.2.3 Let p ∈ (1, ∞), q ∈ (0, ∞), and ω ∈ M(R+ ) with −

n n < m0 (ω) ≤ M0 (ω) <  p p

and −

n < m∞ (ω) ≤ M∞ (ω) < 0. p

Then p,q (i) W K˙ ω (Rn ) → S  (Rn ). p,q p,q (ii) If f ∈ W K˙ ω (Rn ), then f ∈ W H K˙ ω (Rn ) and there exists a positive constant C, independent of f , such that

f W H K˙ p,q n ≤ C f W K n . ˙ p,q ω (R ) ω (R )

6.2 Relations with Weak Generalized Herz Spaces

415

(iii) If f ∈ W H K˙ ω (Rn ), then there exists a locally integrable function g p,q belonging to W K˙ ω (Rn ) such that g represents f , which means that f = g  n in S (R ), p,q

f W H K˙ p,q n = gW H K n , ˙ p,q ω (R ) ω (R ) and there exists a positive constant C, independent of f , such that gW K˙ p,q n ≤ C f W H K n . ˙ p,q ω (R ) ω (R ) Proof Let all the symbols be as in the present theorem. Notice that ω satisfies both m0 (ω) ∈ (− pn , ∞) and M∞ (ω) ∈ (−∞, 0). From these and Theorem 1.2.44, p,q it follows that the global generalized Herz space K˙ ω (Rn ) under consideration is a BQBF space. By this and Lemma 6.1.5, we conclude that the weak global p,q generalized Herz space W K˙ ω (Rn ) is also a BQBF space. On the other hand, let   r ∈ 1, min p,

n max{M0 (ω), M∞ (ω)} + n/p

 .

Then, applying Lemma 6.1.11, we find that, for any f ∈ L1loc (Rn ), M(f )[W K˙ p,q n 1/r  f [W K n 1/r . ˙ p,q ω (R )] ω (R )] This, together with the facts that r ∈ (1, ∞) and W K˙ ω (Rn ) is a BQBF space, and p,q Lemma 4.2.6 with X := W K˙ ω (Rn ), finishes the proof of Theorem 6.2.1.   p,q

Via both Theorem 6.2.3 and Remark 6.0.25, we immediately obtain the following p,q relation between the weak generalized Hardy–Morrey space W H M ω (Rn ) and the p,q weak global generalized Morrey space W M ω (Rn ); we omit the details. Corollary 6.2.4 Let p, q, and ω be as in Corollary 6.2.2. Then (i) W M ω (Rn ) → S  (Rn ). p,q p,q (ii) If f ∈ W M ω (Rn ), then f ∈ W H M ω (Rn ) and there exists a positive constant C, independent of f , such that p,q

f W H M p,q n ≤ C f W M p,q (Rn ) . ω (R ) ω p,q

(iii) If f ∈ W H M ω (Rn ), then there exists a locally integrable function g p,q belonging to W M ω (Rn ) such that g represents f , which means that f = g  n in S (R ), f W H M p,q n = gW H M p,q (Rn ) , ω (R ) ω

416

6 Weak Generalized Herz–Hardy Spaces

and there exists a positive constant C, independent of f , such that gW M p,q n ≤ C f W H M p,q (Rn ) . ω (R ) ω

6.3 Atomic Characterizations The main target of this section is to characterize weak generalized Herz–Hardy spaces via atoms. For this purpose, we first establish the atomic characterization p,q of the weak generalized Herz–Hardy space W H K˙ ω,0 (Rn ) via the known atomic characterization of weak Hardy spaces associated with ball quasi-Banach function spaces directly. However, due to the deficiency of associate spaces of global generalized Herz spaces, the atomic characterization of Hardy spaces associated with ball quasi-Banach function spaces mentioned above is not applicable to establish the atomic characterization of the weak generalized Herz–Hardy space p,q W H K˙ ω (Rn ) [see Remark 1.2.19(vi) for the details]. To overcome this obstacle, we first establish an improved atomic characterization of weak Hardy spaces associated with ball quasi-Banach function spaces (see both Propositions 6.3.11 and 6.3.15 below) without recourse to associate spaces. Then, using this improved p,q conclusion, we obtain the desired atomic characterization of W H K˙ ω (Rn ). p,q Recall that the concept of (K˙ ω,0 (Rn ), r, d)-atoms is given in Definition 4.3.1. We now introduce the weak generalized atomic Herz–Hardy spaces associated with the local generalized Herz spaces as follows. Definition 6.3.1 Let p, q ∈ (0, ∞), ω ∈ M(R+ ) with m0 (ω) ∈ (− pn , ∞) and m∞ (ω) ∈ (− pn , ∞), p− ∈ 0,

n } min{p, q, max{M0 (ω),M ∞ (ω)}+n/p

max{1, p, q}

,

d ≥ n(1/p− − 1) be a fixed integer, and   r ∈ max 1, p,

  n ,∞ . min{m0 (ω), m∞ (ω)} + n/p

Then the weak generalized atomic Herz–Hardy space W H K˙ ω,0 (Rn ), associated p,q with the weak local generalized Herz space W K˙ ω,0 (Rn ), is defined to be the set of all the f ∈ S  (Rn ) such that there exists a sequence {ai,j }i∈Z, j ∈N of p,q (K˙ ω,0 (Rn ), r, d)-atoms supported, respectively, in the balls {Bi,j }i∈Z, j ∈N ⊂ B and p,q,r,d

6.3 Atomic Characterizations

417

 independent of f , satisfying that, for three positive constants c ∈ (0, 1], A, and A, any i ∈ Z, 

1cBi,j ≤ A,

j ∈N

f =

 i∈Z j ∈N

   i 1Bi,j  ˙ p,q n ai,j A2 K (R ) ω,0

in S  (Rn ), and ⎧    ⎪ ⎨     i 1Bi,j  sup 2   ⎪ i∈Z ⎩ j ∈N  ˙ p,q

Kω,0

Moreover, for any f ∈ W H K˙ ω,0

p,q,r,d

f W H K˙ p,q,r,d (Rn ) ω,0

⎫ ⎪ ⎬ (Rn )

⎪ ⎭

< ∞.

(Rn ),

⎧   ⎪ ⎨    i  := inf sup 2  1Bi,j   ⎪ i∈Z  ˙ p,q ⎩j ∈N

Kω,0

⎫ ⎪ ⎬ (Rn )

⎪ ⎭

,

where the infimum is taken over all the decompositions of f as above. Then we have the following atomic characterization of the weak generalized p,q Herz–Hardy space W H K˙ ω,0 (Rn ). Theorem 6.3.2 Let p, q, ω, r, and d be as in Definition 6.3.1. Then p,q p,q,r,d W H K˙ ω,0 (Rn ) = W H K˙ ω,0 (Rn )

with equivalent quasi-norms. To show this theorem, we first recall the following definition of weak Hardy spaces associated with ball quasi-Banach function spaces, which is just [278, Definition 2.21]. Definition 6.3.3 Let X be a ball quasi-Banach function space and N ∈ N. Then the weak Hardy space W HX (Rn ) is defined to be the set of all the f ∈ S  (Rn ) such that f W HX (Rn ) := MN (f )W X < ∞.

418

6 Weak Generalized Herz–Hardy Spaces

Remark 6.3.4 (i) Let p ∈ (0, ∞) and X := Lp (Rn ). Then, in this case, the space W HX (Rn ) is just the classical weak Hardy space W H p (Rn ) which was originally introduced in [153, p. 114]. p (ii) Let p ∈ (0, ∞), υ ∈ A∞ (Rn ), and X := Lυ (Rn ). Then, in this case, the p n space W HX (R ) is just the weighted weak Hardy space W Hυ (Rn ) which was introduced in [194, Definition 3]. (iii) Let p := (p1 , . . . , pn ) ∈ (0, ∞)n and X := Lp (Rn ). Then, in this case, the space W HX (Rn ) is just the weak mixed-norm Hardy space W H p (Rn ) which was introduced in [278, Definition 7.27]. p (iv) Let 0 < q ≤ p < ∞ and X := Mq (Rn ). Then, in this case, the space p n W HX (R ) is just the weak Hardy–Morrey space W H Mq (Rn ) which was originally introduced in [109, Definition 4.1]. (v) Let p(·) : Rn → (0, ∞) and X := Lp(·) (Rn ). Then, in this case, the space W HX (Rn ) is just the variable weak Hardy space W H p(·) (Rn ) which was originally introduced in [267, Definition 2.13]. The following conclusion shows that, under some assumptions of the ball quasiBanach function space X, distributions in the weak Hardy space W HX (Rn ) can be decomposed into linear combinations of atoms, which was proved in [278, Theorem 4.2] and plays an essential role in the proof of Theorem 6.3.2 above. Lemma 6.3.5 Let X be a ball quasi-Banach function space and let r ∈ (0, 1), r0 , s, p− ∈ (0, ∞), and θ ∈ (1, ∞) be such that the following four statements hold true: (i) there exists a positive constant C such that, for any {fj }j ∈N ⊂ L1loc (Rn ), ⎧ ⎫r  ⎨     - .1/r ⎬    M fj ⎩ ⎭   j ∈N 

X 1/r

⎛ ⎞r       1/r     ⎠  ≤ C ⎝ fj    j ∈N

;

X 1/r

1

(ii) X is θ -concave and M bounded on X θp− ; (iii) X1/s is a ball Banach function space and M bounded on (X1/s ) ; (iv) M is bounded on (W X)1/r0 . Assume that d ≥ n(1/p− − 1) is a fixed nonnegative integer and f ∈ W HX (Rn ). Then there exists {ai,j }i∈Z, j ∈N of (X, ∞, d)-atoms supported, respectively, in the  balls {Bi,j }i∈Z, j ∈N ⊂ B and three positive constants c ∈ (0, 1], A, and A, independent of f , such that, for any i ∈ Z, 

1cBi,j ≤ A,

j ∈N

f =

 i∈Z j ∈N

   i 1Bi,j  ai,j A2 X

6.3 Atomic Characterizations

419

in S  (Rn ), and ⎧ ⎨

  ⎫   ⎬   i sup 2  1Bi,j   ⎭  f W HX (Rn ) , ⎩ i∈Z j ∈N  X

where the implicit positive constant is independent of f . The following atomic reconstruction theorem of the weak Hardy space W HX (Rn ) is just [278, Theorem 4.7], which also plays a vital role in the proof of Theorem 6.3.2. Lemma 6.3.6 Let X be a ball quasi-Banach function space and let p− ∈ (0, 1) be such that the following three statements hold true: (i) for any given θ ∈ (0, p− ) and u ∈ (1, ∞), there exists a positive constant C such that, for any {fj }j ∈N ⊂ L1loc (Rn ), ⎧ ⎫1   u  ⎨  u ⎬    M(fj )   ⎩ ⎭    j ∈N

X 1/θ

⎧ ⎫1     ⎨ ⎬u    u ≤C |fj |  ⎩ ⎭   j ∈N 

;

X 1/θ

(ii) for any s ∈ (0, p− ), X1/s is a ball Banach function space; (iii) there exists an s0 ∈ (0, p− ), an r0 ∈ (s0 , ∞), and a C ∈ (0, ∞) such that, for any f ∈ L1loc (Rn ),    ((r0 /s0 ) )  (f ) M

(X 1/s0 )

≤ C f (X1/s0 ) .

Let d ≥ n(1/p− − 1) be a fixed integer, c ∈ (0, 1], r ∈ (max{1, r0 }, ∞],  ∈ (0, ∞). Assume that {ai,j }i∈Z, j ∈N is a sequence of (X, r, d)-atoms and A, A supported, respectively, in the balls {Bi,j }i∈Z, j ∈N ⊂ B such that, for any i ∈ Z, 

1cBi,j ≤ A,

j ∈N

f :=

4

4 i∈Z

 i 1Bi,j X ai,j converges in S  (Rn ), and

j ∈N A2

⎧ ⎨

  ⎫   ⎬    1 sup 2i  < ∞. B i,j   i∈Z ⎩ j ∈N  ⎭ X

420

6 Weak Generalized Herz–Hardy Spaces

Then f ∈ W HX (Rn ) and f W HX (Rn )

       i  sup 2  1Bi,j   , i∈Z j ∈N  X

where the implicit positive constant is independent of f . Remark 6.3.7 We point out that both Lemmas 6.3.5 and 6.3.6 have a wide range of applications. Here we present two function spaces to which both Lemmas 6.3.5 and 6.3.6 can be applied (see also [278, Section 7]). (i) Let p := (p1 , . . . , pn ) ∈ (0, ∞)n , 5  6 max{1, p1 , . . . , pn } d≥ n −1 min{p1 , . . . , pn } be a fixed integer, and r ∈ (max{1, p1 , . . . , pn }, ∞]. Then, in this case, as was mentioned in [278, Subsection 7.2], the mixed-norm Lebesgue space Lp (Rn ) satisfies all the assumptions of both Lemmas 6.3.5 and 6.3.6. Thus, both Lemmas 6.3.5 and 6.3.6 with X := Lp (Rn ) hold true. These results coincide with [278, Theorem 7.30]. (ii) Let 0 < q ≤ p < ∞, 5  d≥ n

1 −1 min{1, q}

6

be a fixed integer, and r ∈ (max{1, p}, ∞]. Then, in this case, as was p pointed out in [278, Subsection 7.1], the Morrey space Mq (Rn ) satisfies all the assumptions of both Lemmas 6.3.5 and 6.3.6. This then implies that both p Lemmas 6.3.5 and 6.3.6 with X := Mq (Rn ) hold true. These results coincide with [278, Theorem 7.11]. Based on the above two lemmas, we now show the atomic characterization of the p,q weak generalized Herz–Hardy space W H K˙ ω,0 (Rn ). Proof of Theorem 6.3.2 Let p, q, ω, r, p− , and d be as in the present theorem. We p,q p,q,r,d p,q first prove W H K˙ ω,0 (Rn ) ⊂ W H K˙ ω,0 (Rn ). To this end, let f ∈ W H K˙ ω,0 (Rn ). Now, we claim that, for the above r and p− , all the assumptions of Lemma 6.3.5 p,q hold true with X := K˙ ω,0 (Rn ). Assume that this claim holds true for the moment. p,q Then, from Lemma 6.3.5 with X therein replaced by K˙ ω,0 (Rn ), it follows that p,q there exists {ai,j }i∈Z, j ∈N of (K˙ ω,0 (Rn ), ∞, d)-atoms supported, respectively, in

6.3 Atomic Characterizations

421

 the balls {Bi,j }i∈Z, j ∈N ⊂ B and three positive constants c ∈ (0, 1], A, and A, independent of f , such that, for any i ∈ Z, 

1cBi,j ≤ A,

j ∈N

f =

 i∈Z j ∈N

in S  (Rn ), and ⎧    ⎪ ⎨     i 1Bi,j  sup 2   ⎪ i∈Z ⎩ j ∈N  ˙ p,q

Kω,0

   i 1Bi,j  ˙ p,q n ai,j A2 K (R )

⎫ ⎪ ⎬ (Rn )

(6.3)

ω,0

⎪ ⎭

 f W H K˙ p,q (Rn ) < ∞.

(6.4)

ω,0

In addition, by Lemma 4.3.5 with t := ∞, we conclude that, for any i ∈ Z and p,q j ∈ N, ai,j is a (K˙ ω,0 (Rn ), r, d)-atom supported in the ball Bi,j . This, combined p,q,r,d with (6.3), (6.4), and Definition 6.3.1, further implies that f ∈ W H K˙ (Rn ) ω,0

and then finishes the proof that W H K˙ ω,0 (Rn ) ⊂ W H K˙ ω,0 (Rn ). p,q p,q,r,d Thus, to complete the proof that W H K˙ ω,0 (Rn ) ⊂ W H K˙ ω,0 (Rn ), it remains to show the above claim. Indeed, applying the assumption m0 ∈ (− pn , ∞) and p,q Theorem 1.2.42, we find that the local generalized Herz space K˙ ω,0 (Rn ) under consideration is a BQBF space. Using Lemma 4.3.10 with u := 1r , we conclude that, for any {fj }j ∈N ⊂ L1loc (Rn ), p,q

⎧ ⎫r  ⎨    - .1/r ⎬    M f j ⎩ ⎭    j ∈N

˙ p,q (Rn )]1/r [K ω,0

p,q,r,d

⎛ ⎞r       1/r  fj  ⎠  ⎝      j ∈N

,

˙ p,q (Rn )]1/r [K ω,0

p,q which implies that the assumption (i) of Lemma 6.3.5 holds true for K˙ ω,0 (Rn ). We next prove that there exists a θ , an s, and an r0 such that (ii), (iii), and (iv) of Lemma 6.3.5 hold true. Indeed, let    1 n . θ ∈ max{1, p, q}, min p, q, p− max{M0 (ω), M∞ (ω)} + n/p

Then, from the assumptions p/θ, q/θ ∈ (0, 1), the reverse Minkowski inequality, and Lemma 1.3.1, we deduce that, for any {fj }j ∈N ⊂ M (Rn ),  j ∈N

fj [K˙ p,q (Rn )]1/θ ω,0

       ≤ |fj |  j ∈N 

,

˙ (Rn )]1/θ [K ω,0 p,q

(6.5)

422

6 Weak Generalized Herz–Hardy Spaces

which implies that K˙ ω,0 (Rn ) is θ -concave. On the other hand, using Lemma 1.8.5 with r := θp− , we conclude that, for any f ∈ L1loc (Rn ), p,q

M(f )

1

˙ p,q (Rn )] θp− [K ω,0

 f 

1

˙ p,q (Rn )] θp− [K ω,0

.

From this and (6.5), we further infer that, for the above θ and p− , Lemma 6.3.5(ii) holds true. Let    n s ∈ 0, min 1, p, q, . max{M0 (ω), M∞ (ω)} + n/p Then, by Lemma 1.8.6 with r := ∞, we find that [K˙ ω,0 (Rn )]1/s is a BBF space and, for any f ∈ L1loc (Rn ), p,q

M(f )([K˙ p,q (Rn )]1/s )  f ([K˙ p,q (Rn )]1/s ) , ω,0

ω,0

which imply that Lemma 6.3.5(iii) holds true with the above s. Finally, let   r0 ∈ 0, min p,

n max{M0 (ω), M∞ (ω)} + n/p

 .

Then, from Lemma 6.1.6 with r := r0 , we deduce that, for any f ∈ L1loc (Rn ), M(f )[W K˙ p,q (Rn )]1/r0  f [W K˙ p,q (Rn )]1/r0 . ω,0

ω,0

This implies that Lemma 6.3.5(iv) holds true with the above r0 . Therefore, the Herz p,q space K˙ ω,0 (Rn ) under consideration satisfies all the assumptions of Lemma 6.3.5 with r and p− in the present theorem. This then finishes the proof of the above claim and further implies that p,q p,q,r,d W H K˙ ω,0 (Rn ) ⊂ W H K˙ ω,0 (Rn ).

Moreover, combining (6.3), (6.4), and Definition 6.3.1 again, we conclude that, for p,q any f ∈ W H K˙ ω,0 (Rn ), f W H K˙ p,q,r,d (Rn )  f W H K˙ p,q (Rn ) . ω,0

ω,0

(6.6)

p,q,r,d p,q Conversely, we now prove that W H K˙ ω,0 (Rn ) ⊂ W H K˙ ω,0 (Rn ). Indeed, p,q p,q,r,d from the proof that W H K˙ ω,0 (Rn ) ⊂ W H K˙ ω,0 (Rn ), it follows that the local p,q generalized Herz space K˙ (Rn ) under consideration is a BQBF space. Thus, in ω,0

p,q,r,d p,q order to finish the proof that W H K˙ ω,0 (Rn ) ⊂ W H K˙ ω,0 (Rn ), we only need to

6.3 Atomic Characterizations

423

show that the assumptions (i) through (iii) of Lemma 6.3.6 hold true for K˙ ω,0 (Rn ). Indeed, for any given θ ∈ (0, p− ) and u ∈ (1, ∞), applying Lemma 4.3.10 with r := θ , we find that, for any {fj }j ∈N ⊂ L1loc (Rn ), p,q

⎧ ⎫1   u ⎨  u ⎬    M(fj )   ⎩ ⎭    j ∈N

˙ p,q (Rn )]1/θ [K ω,0

⎛ ⎞1  u    ⎝ u⎠   |fj |     j ∈N 

.

˙ p,q (Rn )]1/θ [K ω,0

p,q This implies that Lemma 6.3.6(i) holds true for K˙ ω,0 (Rn ). In addition, from p,q Lemma 1.8.6, we deduce that, for any s ∈ (0, p− ), [K˙ ω,0 (Rn )]1/s is a BBF space. This implies that Lemma 6.3.6(ii) holds true. Finally, we show that there exists an s0 and an r0 such that Lemma 6.3.6(iii) holds true with these s0 and r0 . To do this, let s0 ∈ (0, p− ) and

  r0 ∈ max 1, p,

  n ,r . max{M0 (ω), M∞ (ω)} + n/p

Then, using Lemma 1.8.6 again with s and r therein replaced, respectively, by s0 and r0 , we conclude that, for any f ∈ L1loc (Rn ),    ((r0/s0 ) )  (f ) M

˙ (Rn )]1/s0 ) ([K ω,0 p,q

 f ([K˙ p,q (Rn )]1/s0 ) , ω,0

which implies that, for the above s0 and r0 , Lemma 6.3.6(iii) holds true for p,q K˙ ω,0 (Rn ). Therefore, all the assumptions of Lemma 6.3.6 hold true for the Herz p,q p,q,r,d space K˙ ω,0 (Rn ) under consideration. This then implies that W H K˙ ω,0 (Rn ) ⊂ p,q W H K˙ (Rn ). Moreover, using Definition 6.3.1 and Lemma 6.3.6 again, we further ω,0

p,q,r,d find that, for any f ∈ W H K˙ ω,0 (Rn ),

f W H K˙ p,q (Rn )  f W H K˙ p,q,r,d (Rn ) , ω,0

ω,0

which, together with (6.6), further implies that p,q p,q,r,d W H K˙ ω,0 (Rn ) = W H K˙ ω,0 (Rn )

with equivalent quasi-norms. This finishes the proof of Theorem 6.3.2.

 

As an application, we now establish the atomic characterization of the weak p,q generalized Hardy–Morrey space W H M ω,0 (Rn ). Recall that the definition of the p,q (M ω,0 (Rn ), r, d)-atoms is as in Definition 4.3.12. Then, applying Theorem 6.3.2 and Remark 6.0.25, we obtain the following atomic characterization of the weak p,q generalized Hardy–Morrey space W H M ω,0 (Rn ) immediately; we omit the details.

424

6 Weak Generalized Herz–Hardy Spaces

Corollary 6.3.8 Let p, q ∈ [1, ∞), p− ∈ (0, min{p, q}/ max{p, q}), ω ∈ M(R+ ) with −

n < m0 (ω) ≤ M0 (ω) < 0 p

and −

n < m∞ (ω) ≤ M∞ (ω) < 0, p

d ≥ n(1/p− − 1) be a fixed integer, and  r∈

 n ,∞ . min{m0 (ω), m∞ (ω)} + n/p p,q,r,d

Then the weak generalized atomic Hardy–Morrey space W H M ω,0 (Rn ), associp,q ated with the weak local generalized Morrey space W M ω,0 (Rn ), is defined to be the set of all the f ∈ S  (Rn ) such that there exists a sequence {ai,j }i∈Z, j ∈N of p,q (M ω,0 (Rn ), r, d)-atoms supported, respectively, in the balls {Bi,j }i∈Z, j ∈N ⊂ B  independent of f , satisfying that, and three positive constants c ∈ (0, 1], A, and A, for any i ∈ Z, 

1cBi,j ≤ A,

j ∈N

f =

 i∈Z j ∈N

   i 1Bi,j  p,q n ai,j A2 M (R ) ω,0

in S  (Rn ), and ⎧   ⎪  ⎨     i sup 2  1Bi,j   i∈Z ⎪ ⎩ j ∈N  p,q,r,d

Moreover, for any f ∈ W H M ω,0

f W H M p,q,r,d (Rn ) ω,0

⎫ ⎪ ⎬ p,q

M ω,0 (Rn )

⎪ ⎭

< ∞.

(Rn ),

⎧   ⎪  ⎨      := inf sup 2i  1 Bi,j   i∈Z ⎪  ⎩ j ∈N

⎫ ⎪ ⎬ p,q M ω,0 (Rn )

⎪ ⎭

,

6.3 Atomic Characterizations

425

where the infimum is taken over all the decompositions of f as above. Then p,q

p,q,r,d

W H M ω,0 (Rn ) = W H M ω,0

(Rn )

with equivalent quasi-norms. The remainder of this section is devoted to establishing the atomic characterizap,q tion of the weak generalized Herz–Hardy space W H K˙ ω (Rn ). To this end, we first introduce the following definition of weak generalized atomic Herz–Hardy spaces p,q via (K˙ ω (Rn ), r, d)-atoms introduced in Definition 4.3.14. Definition 6.3.9 Let p, q ∈ (0, ∞), ω ∈ M(R+ ) with m0 (ω) ∈ (− pn , ∞) and − p− ∈ 0,

n < m∞ (ω) ≤ M∞ (ω) < 0, p n min{p, q, max{M0 (ω),M } ∞ (ω)}+n/p

max{1, p, q}

,

d ≥ n(1/p− − 1) be a fixed integer, and   r ∈ max 1, p,

  n ,∞ . min{m0 (ω), m∞ (ω)} + n/p

p,q,r,d (Rn ), associated Then the weak generalized atomic Herz–Hardy space W H K˙ ω p,q n ˙ with the weak global generalized Herz space W Kω (R ), is defined to be the set of all the f ∈ S  (Rn ) such that there exists a sequence {ai,j }i∈Z, j ∈N of p,q (K˙ ω (Rn ), r, d)-atoms supported, respectively, in the balls {Bi,j }i∈Z, j ∈N ⊂ B and  independent of f , satisfying that, for three positive constants c ∈ (0, 1], A, and A, any i ∈ Z,



1cBi,j ≤ A,

j ∈N

f =

 i∈Z j ∈N

   i 1Bi,j  ˙ p,q n ai,j A2 K (R ) ω

in S  (Rn ), and ⎧ ⎨

       i sup 2  1Bi,j   ⎩ i∈Z j ∈N  ˙ p,q Kω

⎫ ⎬ (Rn )

⎭

< ∞,

426

6 Weak Generalized Herz–Hardy Spaces

p,q,r,d Moreover, for any f ∈ W H K˙ ω (Rn ),

f W H K˙ p,q,r,d (Rn ) ω

⎧    ⎨     i := inf sup 2  1Bi,j   i∈Z ⎩ j ∈N  ˙ p,q

Kω (Rn )

⎫ ⎬ ⎭

,

where the infimum is taken over all the decompositions of f as above. Then we have the following atomic characterization of the weak generalized p,q Herz–Hardy space W H K˙ ω (Rn ). Theorem 6.3.10 Let p, q, ω, r, and d be as in Definition 6.3.9. Then W H K˙ ωp,q (Rn ) = W H K˙ ωp,q,r,d (Rn ) with equivalent quasi-norms. Due to the deficiency of associate spaces of global generalized Herz spaces, we can not prove this theorem via the atomic characterization of weak Hardy spaces associated with ball quasi-Banach function spaces obtained in [278, Theorems 4.2 and 4.7] (see also both Lemmas 6.3.5 and 6.3.6) directly. To overcome this difficulty, we establish an improved atomic characterization of W HX (Rn ) associated with the ball quasi-Banach function space X without recourse to the associate space X . Indeed, we have the following atomic decomposition theorem. Proposition 6.3.11 Let X be a ball quasi-Banach function space and Y ⊂ M (Rn ) a linear space equipped with a quasi-seminorm  · Y , and let r ∈ (0, 1), r0 , s, p− ∈ (0, ∞), and θ, θ0 ∈ (1, ∞) be such that the following seven statements hold true: (i) there exists a positive constant C such that, for any {fj }j ∈N ⊂ L1loc (Rn ), ⎧ ⎫r  ⎨    - .1/r ⎬    M f j ⎩ ⎭   j ∈N 

X 1/r

⎛ ⎞r       1/r  fj  ⎠  ⎝ ≤C     j ∈N

X 1/r

1

(ii) (iii) (iv) (v) (vi)

X is θ -concave and M bounded on X θp− ; M is bounded on (W X)1/r0 ;  · Y satisfies Definition 1.2.13(ii); 1B(0,1) ∈ Y ; for any f ∈ M (Rn ),   f X1/s ∼ sup fgL1 (Rn ) : gY = 1 ,

where the positive equivalence constants are independent of f ; (vii) M(θ0 ) is bounded on Y .

;

6.3 Atomic Characterizations

427

Assume that d ≥ n(1/p− − 1) is a fixed nonnegative integer and f ∈ W HX (Rn ). Then there exists {ai,j }i∈Z, j ∈N of (X, ∞, d)-atoms supported, respectively, in the  balls {Bi,j }i∈Z, j ∈N ⊂ B and three positive constants c ∈ (0, 1], A, and A, independent of f , such that, for any i ∈ Z, 

1cBi,j ≤ A,

j ∈N

f =

 i∈Z j ∈N

   i 1Bi,j  ai,j A2 X

in S  (Rn ), and ⎧ ⎨

  ⎫   ⎬   i 1Bi,j  sup 2   ⎭  f W HX (Rn ) , i∈Z ⎩ j ∈N  X

where the implicit positive constant is independent of f . Remark 6.3.12 We should point out that Proposition 6.3.11 is an improved version of the known atomic decomposition of W HX (Rn ) obtained in [278, Theorem 4.2]. Indeed, if Y ≡ (X1/s ) in Proposition 6.3.11, then this proposition goes back to [278, Theorem 4.2]. To show this proposition, we first prove the following technical lemma by borrowing some ideas from the proof of [278, Lemma 4.3], which shows a crucial fact that distributions in weak Hardy spaces vanish weakly at infinity. Lemma 6.3.13 Let X be a ball quasi-Banach function space, Y ⊂ M (Rn ) a linear space equipped with a quasi-seminorm  · Y , θ ∈ (1, ∞), and s ∈ (0, ∞) satisfy the following four statements: (i)  · Y satisfies Definition 1.2.13(ii); (ii) 1B(0,1) ∈ Y ; (iii) for any f ∈ M (Rn ),   f X1/s ∼ sup fgL1 (Rn ) : gY = 1 , where the positive equivalence constants are independent of f ; (iv) M(θ) is bounded on Y . Assume f ∈ W HX (Rn ). Then f vanishes weakly at infinity. In order to prove Lemma 6.3.13, we require the following auxiliary estimate about Ap -weights (see, for instance, [70, Corollary 7.6(3)]).

428

6 Weak Generalized Herz–Hardy Spaces

Lemma 6.3.14 Let p ∈ [1, ∞) and υ ∈ Ap (Rn ). Then there exist two positive constants δ and C such that, for any ball B1 ∈ B and any ball B2 ⊂ B1 , υ(B2 ) ≤C υ(B1 )



|B2 | |B1 |

δ .

We next show Lemma 6.3.13. Proof of Lemma 6.3.13 Let all the symbols be as in the present lemma. Then, from the assumptions (i) through (iv) of the present proposition and Lemma 4.8.18, we deduce that there exists an ε ∈ (0, 1) such that, for any g ∈ M (Rn ), gLsυ (Rn )  gX ,

(6.7)

where υ := [M(1B(0,1))]ε . Let ϕ ∈ S(Rn ) and N ∈ N. Then, by (4.3), we conclude that, for any t ∈ (0, ∞) and x, y ∈ Rn with |y − x| < t, |f ∗ ϕt (x)|  MN (f )(y), which further implies that |f ∗ ϕt (x)| 1B(x,t )  MN (f )1B(x,t ). Applying this, Definitions 2.3.7(ii) and 6.1.3, (6.7), Remark 6.1.4 with X := Lsυ (Rn ), and Definition 6.3.3, we find that, for any t ∈ (0, ∞) and x ∈ Rn , |f ∗ ϕt (x)|  

MN (f )1B(x,t )W Lsυ (Rn ) 1B(x,t )W Lsυ (Rn ) MN (f )W X f W HX (Rn ) ∼ . 1B(x,t )Lsυ (Rn ) 1B(x,t )Lsυ (Rn )

Therefore, for any given φ ∈ S(Rn ) and for any t ∈ (0, ∞), we have    

Rn

  f ∗ ϕt (x)φ(x) dx  

  ∼

Rn

1 |φ(x)| dx 1B(x,t )Lsυ (Rn )

Rn

1B(x,1)Lsυ (Rn ) 1 |φ(x)| dx. 1B(x,1)Lsυ (Rn ) 1B(x,t )Lsυ (Rn )

(6.8)

6.3 Atomic Characterizations

429

Moreover, using Lemma 2.3.11 with r := 1, we find that, for any x ∈ Rn ,   1B(x,1) s n = L (R )



υ



. ε M 1B(0,1) (y) dy

1 s

B(x,1)



(1 + |y|)−nε dy

∼

1 s

B(x,1)

 (1 + |x|)− s |B(x, 1)| s ∼ (1 + |x|)− s . 1

nε

nε

(6.9)

On the other hand, from Lemma 2.3.9 with f := 1B(0,1) and δ := ε, it follows that υ ∈ A1 (Rn ). Then, for any t ∈ [1, ∞) and x ∈ Rn , by Lemma 6.3.14 with p := 1, B1 := B(x, t), and B2 := B(x, 1), we find that there exists a δ ∈ (0, ∞), independent of t and x, such that 1 δ   1B(x,1)Lsυ (Rn ) nδ υ(B(x, 1)) s |B(x, 1)| s ∼  ∼ t− s . s n 1B(x,t )Lυ (R ) υ(B(x, t)) |B(x, t)| Combining this, (6.8), and (6.9), we further conclude that, for any given φ ∈ S(Rn ),   f ∗ ϕt (x)φ(x) dx  Rn  nδ nε nδ  t− s (1 + |x|) s |φ(x)| dx  t − s → 0

   

Rn

as t → ∞. This implies that f ∗ϕt → 0 in S  (Rn ) as t → ∞, and hence f vanishes weakly at infinity, which then completes the proof of Lemma 6.3.13.   Applying Lemma 6.3.13, we now prove the atomic decomposition of W HX (Rn ) as follows. Proof of Proposition 6.3.11 Let all the symbols be as in the present proposition and f ∈ W HX (Rn ). Then, by the assumptions (iv) through (vii) of the present proposition and Lemma 6.3.13, we find that f vanishes weakly at infinity. Using this, (i) through (iii) of the present proposition, and repeating the proof of [278, Theorem 4.2], we conclude that there exists a sequence {ai,j }i∈Z, j ∈N of (X, ∞, d)atoms supported, respectively, in the balls {Bi,j }i∈Z, j ∈N ⊂ B and three positive  independent of f , such that, for any i ∈ Z, constants c ∈ (0, 1], A, and A, 

1cBi,j ≤ A,

j ∈N

f =

 i∈Z j ∈N

   i 1Bi,j  ai,j A2 X

430

6 Weak Generalized Herz–Hardy Spaces

in S  (Rn ), and ⎧   ⎫  ⎬ ⎨     i sup 2  1Bi,j   ⎭  f W HX (Rn ) . ⎩ i∈Z j ∈N  X

 

This finishes the proof of Proposition 6.3.11.

On the other hand, we establish the following atomic reconstruction theorem of W HX (Rn ) without recourse to associate spaces. Proposition 6.3.15 Let X be a ball quasi-Banach function space and let p− ∈ (0, 1) be such that the following three statements hold true: (i) for any given θ ∈ (0, p− ) and u ∈ (1, ∞), there exists a positive constant C such that, for any {fj }j ∈N ⊂ L1loc (Rn ), ⎧ ⎫1   u  ⎨  u ⎬    M(fj )   ⎩ ⎭   j ∈N 

X 1/θ

⎧ ⎫1     ⎨ ⎬u    u ≤C |fj |  ⎩ ⎭   j ∈N 

;

X 1/θ

(ii) for any s ∈ (0, p− ), X1/s is a ball Banach function space; (iii) there exists an s0 ∈ (0, p− ), an r0 ∈ (s0 , ∞), a C ∈ (0, ∞), and a linear space Y0 ⊂ M (Rn ) equipped with a quasi-seminorm  · Y0 such that, for any f ∈ M (Rn ),   f X1/s0 ∼ sup fgL1 (Rn ) : gY0 = 1 with the positive equivalence constants independent of f and, for any f ∈ L1loc (Rn ),    ((r0/s0 ) )  (f ) M

Y0

≤ C f Y0 .

Let d ≥ n(1/p− − 1) be a fixed integer, c ∈ (0, 1], r ∈ (max{1, r0 }, ∞],  ∈ (0, ∞). Assume that {ai,j }i∈Z, j ∈N is a sequence of (X, r, d)-atoms and A, A supported, respectively, in the balls {Bi,j }i∈Z, j ∈N ⊂ B such that, for any i ∈ Z, 

1cBi,j ≤ A,

j ∈N

f :=

 i∈Z j ∈N

   i 1Bi,j  ai,j A2 X

6.3 Atomic Characterizations

431

converges in S  (Rn ), and ⎧ ⎨

  ⎫   ⎬    i sup 2  1Bi,j   ⎭ < ∞. ⎩ i∈Z j ∈N  X

Then f ∈ W HX (Rn ) and f W HX (Rn )

      i  sup 2  1Bi,j   , i∈Z j ∈N  X

where the implicit positive constant is independent of f . Remark 6.3.16 We should point out that Proposition 6.3.15 is an improved version of the known atomic reconstruction of W HX (Rn ) established in [278, Theorem 4.8]. Indeed, if Y0 ≡ (X1/s0 ) in Proposition 6.3.15, then this proposition goes back to [278, Theorem 4.8]. To prove Proposition 6.3.15, we first establish the following technical lemma via borrowing some ideas from [278, Lemma 4.8]. Lemma 6.3.17 Let s ∈ (0, ∞), r ∈ (s, ∞], X be a ball quasi-Banach function space, and Y ⊂ M (Rn ) a linear space equipped with a quasi-seminorm  · Y . Assume that the following two statements hold true: (i) for any f ∈ M (Rn ),   f X1/s ∼ sup fgL1 (Rn ) : gY = 1 , where the positive equivalence constants are independent of f ; (ii) there exists a positive constant C such that, for any f ∈ L1loc (Rn ),    ((r/s))  (f ) ≤ C f Y . M Y

Let {λj }j ∈N ⊂ [0, ∞), {Bj }j ∈N ⊂ B, and {aj }j ∈N ⊂ M (Rn ) satisfy that, for any j ∈ N, supp (aj ) ⊂ Bj and aj Lr (Rn ) ≤ |Bj |1/r . Then there exists a positive constant C, independent of {λj }j ∈N , {Bj }j ∈N , and {aj }j ∈N , such that ⎛ ⎛ ⎞1  ⎞1    s s      s  s    ⎝     λj aj ⎠  ≤ C ⎝ λj 1Bj ⎠        j ∈N    j ∈N X

X

432

6 Weak Generalized Herz–Hardy Spaces

Proof Let all the symbols be as in the present lemma and g ∈ M (Rn ) with gY = 1. Then, by the Tonelli theorem, the assumption that, for any j ∈ N, supp (aj ) ⊂ Bj , the Hölder inequality, and the assumption that, for any j ∈ N, aj Lr (Rn ) ≤ |Bj |1/r , we find that      s   λj aj (x) g(x) dx     n   R j ∈N     aj (x)s |g(x)| dx λsj ≤ j ∈N

≤

 Rn

=

 j ∈N

≤



j ∈N

Bj

 s    λsj aj  Lr/s (Rn ) g1Bj L(r/s) (Rn )  s   λsj aj Lr (Rn ) g1Bj L(r/s) (Rn )  s   λsj Bj  r g1Bj L(r/s) (Rn ) .

(6.10)

Observe that, from (1.54) and (1.55) with θ := (r/s) and f := g, it follows that, for any j ∈ N and x ∈ Bj , # M

((r/s))

(g)(x) ≥

1 |Bj |

$1/(r/s)

 |g(y)|

(r/s)

dy

Bj

 −1/(r/s)   g1B  (r/s) n . ∼ Bj  j L (R )

(6.11)

On the other hand, by the assumption (ii), we find that, for any f, g ∈ M (Rn ), fgL1 (Rn )  f X1/s gY . This, combined with (6.10), (6.11), the Tonelli theorem, the assumption (ii) of the present lemma, and the assumption that gY = 1, further implies that        s   λj aj (x) g(x) dx    n   R j ∈N    −1/(r/s)   Bj  g1B  (r/s) n dx  λsj j L (R ) Bj

j ∈N



 j ∈N

 λsj



M((r/s) ) (g)(x) dx Bj

6.3 Atomic Characterizations

 ∼

433

  λj 1B (x)s M((r/s)) (g)(x) dx j

Rn j ∈N

      s  λj 1B    j   j ∈N 

X 1/s

⎛ s ⎞1   s          ((r/s))  λj 1B s ⎠  (g)  ⎝  . M j   Y  j ∈N  X

Applying this, Definition 1.2.28(i) with p := 1s , the arbitrariness of g, and the assumption (i) of the present lemma, we conclude that ⎛ ⎛ 1  ⎞1  ⎞1    s s s               ⎝   s s s λj aj  ⎠  =  λj aj   λj 1B  ⎠   ⎝   . j          j ∈N  j ∈N   j ∈N X 1/s X

X

 

This then finishes the proof of Lemma 6.3.17. Via Lemma 6.3.17, we now prove the atomic reconstruction theorem.

 be as in the present proposition Proof of Proposition 6.3.15 Let X, r, d, c, A, and A and {ai,j }i∈Z, j ∈N a sequence of (X, r, d)-atoms supported, respectively, in the balls {Bi,j }i∈Z, j ∈N ⊂ B such that, for any i ∈ Z, 

1cBi,j ≤ A,

j ∈N

 i∈Z j ∈N

   i 1Bi,j  ai,j A2 X

converges in S  (Rn ), and ⎧ ⎨

  ⎫   ⎬    i sup 2  1Bi,j   ⎭ < ∞. i∈Z ⎩ j ∈N  X

Then, repeating the proof of [278, Theorem 4.7] via replacing [278, Lemma 4.8] therein by Lemma 6.3.17 here, we obtain f ∈ W HX (Rn ) and f W HX (Rn )

       i  sup 2  1Bi,j   , i∈Z j ∈N  X

which then completes the proof of Proposition 6.3.15.

 

434

6 Weak Generalized Herz–Hardy Spaces

Via the improved atomic characterization of W HX (Rn ) established in both Propositions 6.3.11 and 6.3.15 above, we next show the atomic characterization p,q of the weak generalized Herz–Hardy space W H K˙ ω (Rn ). Proof of Theorem 6.3.10 Let p, q, ω, r, p− , and d be as in the present theorem. p,q p,q,r,d We first prove that W H K˙ ω (Rn ) ⊂ W H K˙ ω (Rn ). To this end, we show p,q n ˙ that, the global generalized Herz space Kω (R ) under consideration satisfies all the assumptions of Proposition 6.3.11. Indeed, from the assumptions m0 (ω) ∈ p,q (− pn , ∞) and M∞ (ω) ∈ (−∞, 0), and Theorem 1.2.44, it follows that K˙ ω (Rn ) is a BQBF space. Applying Lemma 4.3.24 with u := 1r , we conclude that, for any {fj }j ∈N ⊂ L1loc (Rn ), ⎧ ⎫r  ⎨     - .1/r ⎬    M f j ⎩ ⎭    j ∈N

n 1/r ˙ p,q [K ω (R )]

⎛ ⎞r       1/r  fj  ⎠  ⎝     j ∈N 

,

n 1/r ˙ p,q [K ω (R )]

which implies that Proposition 6.3.11(i) holds true. Now, we choose a    n 1 . min p, q, θ ∈ max{1, p, q}, p− max{M0 (ω), M∞ (ω)} + n/p Then, by the assumptions p/θ, q/θ ∈ (0, 1), the reverse Minkowski inequality, and Lemma 1.3.2, we find that, for any {fj }j ∈N ⊂ M (Rn ),  j ∈N

fj [K˙ p,q n 1/θ ω (R )]

       ≤ |f | j   j ∈N 

.

n 1/θ ˙ p,q [K ω (R )]

p,q This implies that K˙ ω (Rn ) is θ -concave. Moreover, from Lemma 4.1.10 with r := θp− , we deduce that, for any f ∈ L1loc (Rn ),

M(f )

1

n θp ˙ p,q [K ω (R )] −

 f 

1

n θp ˙ p,q [K ω (R )] −

,

p,q which, together with the fact that K˙ ω (Rn ) is θ -concave, further implies that Proposition 6.3.11(ii) holds true. In addition, let

  r0 ∈ 0, min p,

n max{M0 (ω), M∞ (ω)} + n/p

 .

Then, using Lemma 6.1.11 with r := r0 , we conclude that, for any f ∈ L1loc (Rn ), M(f )[W K˙ p,q n 1/r  f [W K n 1/r , ˙ p,q ω (R )] 0 ω (R )] 0

6.3 Atomic Characterizations

435

which implies that Proposition 6.3.11(iii) holds true with the above r0 . Finally, let   s ∈ 0, min 1, p, q,

n max{M0 (ω), M∞ (ω)} + n/p



and θ0 ∈ (1, ∞) satisfy p   n , . θ0 < min n(1 − s/p) − s min{m0 (ω), m∞ (ω)} s 

Then, repeating an argument similar to that used in the proof of Theorem 4.8.16 with η therein replaced by θ0 , we find that (i) for any f ∈ M (Rn ),  ∼ sup fgL1 (Rn ) : g ˙ (p/s) ,(q/s) f [K˙ p,q n 1/s ω (R )] B1/ωs

 (Rn )

=1

with positive equivalence constants independent of f ; (ii)  ·  ˙ (p/s) ,(q/s) n satisfies Definition 1.2.13(ii); B1/ωs

(R ) 



(p/s) ,(q/s) (iii) 1B(0,1) ∈ B˙1/ωs (Rn ); (p/s),(q/s) (iv) M(θ0 ) is bounded on B˙1/ωs (Rn ).

These imply that the assumptions (iv) through (vii) of Proposition 6.3.11 hold true with the above s and θ0 . Thus, all the assumptions of Proposition 6.3.11 hold true p,q for the Herz space K˙ ω (Rn ) under consideration. p,q p,q Let f ∈ W H K˙ ω (Rn ). Then, from Proposition 6.3.11 with X := K˙ ω (Rn ), p,q we infer that there exists {ai,j }i∈Z, j ∈N of (K˙ ω (Rn ), ∞, d)-atoms supported, respectively, in the balls {Bi,j }i∈Z, j ∈N ⊂ B and three positive constants c ∈ (0, 1],  independent of f , such that, for any i ∈ Z, A, and A, 

1cBi,j ≤ A,

j ∈N

f =

 i∈Z j ∈N

   i 1Bi,j  ˙ p,q n ai,j A2 K (R ) ω

(6.12)

in S  (Rn ), and ⎧ ⎨

       sup 2i  1 Bi,j   i∈Z ⎩ j ∈N  ˙ p,q

Kω (Rn )

⎫ ⎬ ⎭

 f W H K˙ p,q n . ω (R )

(6.13)

436

6 Weak Generalized Herz–Hardy Spaces

p,q In addition, using Lemma 4.3.5 with X := K˙ ω (Rn ) and t := ∞, we find that, p,q n for any i ∈ Z and j ∈ N, ai,j is a (K˙ ω (R ), r, d)-atom supported in the ball Bi,j . Combining this, (6.12), (6.13), and Definition 6.3.9, we conclude that f ∈ p,q,r,d (Rn ) and W H K˙ ω

f W H K˙ p,q,r,d (Rn )  f W H K˙ p,q n . ω (R )

(6.14)

ω

This then finishes the proof that W H K˙ ω (Rn ) ⊂ W H K˙ ω (Rn ). p,q,r,d p,q Conversely, we next show that W H K˙ ω (Rn ) ⊂ W H K˙ ω (Rn ). For this p,q purpose, we first prove that the Herz space K˙ ω (Rn ) under consideration satisfies the assumptions (i), (ii), and (iii) of Proposition 6.3.15 with p− as in the present theorem. Indeed, for any given θ ∈ (0, p− ) and u ∈ (1, ∞), from Lemma 4.3.24 with r := θ , it follows that, for any {fj }j ∈N ⊂ M (Rn ), p,q

⎧ ⎫1    ⎨  - . ⎬ u  u   M fj   ⎩ ⎭   j ∈N 

n 1/θ ˙ p,q [K ω (R )]

p,q,r,d

⎛ ⎞1      u u ⎝ fj  ⎠       j ∈N 

,

n 1/θ ˙ p,q [K ω (R )]

which implies that Proposition 6.3.15(i) holds true. Next, we show that, for any p,q s ∈ (0, p− ), [K˙ ω (Rn )]1/s is a BBF. Indeed, fix an s ∈ (0, p− ). Then, from the assumptions m0 (ω) ∈ (− pn , ∞) and M∞ (ω) ∈ (−∞, 0), and Lemma 1.1.6, it follows that - . n m0 ωs = sm0 (ω) > − p/s and - . M∞ ωs = sM∞ (ω) < 0. Applying these, the assumptions p/s, q/s ∈ (1, ∞), and Theorem 1.2.48 with p, p/s,q/s q, and ω replaced, respectively, by p/s, q/s, and ωs , we find that K˙ ωs (Rn ) is p,q n a BBF space. From this and Lemma 1.3.2, we further infer that [K˙ ω (R )]1/s is a BBF. This then implies that Proposition 6.3.15(ii) holds true. Finally, let s0 ∈ (0, p− ) and   r0 ∈ max 1, p,

  n ,∞ . max{M0 (ω), M∞ (ω)} + n/p

6.3 Atomic Characterizations

437

Then, repeating an argument similar to that used in the proof of Theorem 4.3.16 with s and r therein replaced, respectively, by s0 and r0 , we conclude that, for any f ∈ M (Rn ), 



f [K˙ p,q n 1/s ∼ sup fgL1 (Rn ) : g ˙ (p/s0 ) ,(q/s0 ) ω (R )] 0 B1/ωs0

(Rn )

=1

and, for any f ∈ L1loc (Rn ),    ((r0 /s0 ) )  (f ) ˙ (p/s0 ) ,(q/s0 ) M B1/ωs0

(Rn )

 f  ˙ (p/s0 ) ,(q/s0 ) B1/ωs0

(Rn )

.

p,q These imply that the global generalized Herz space K˙ ω (Rn ) under consideration (p/s ) ,(q/s0 ) (Rn ). satisfies Proposition 6.3.15(iii) with the above s0 , r0 , and Y0 := B˙1/ωs00 Therefore, the assumptions (i) through (iii) of Proposition 6.3.15 hold true for p,q K˙ ω (Rn ). p,q p,q,r,d (Rn ), we find that In addition, by the proof that W H K˙ ω (Rn ) ⊂ W H K˙ ω p,q p,q n ˙ ˙ Kω (R ) is a BQBF space. Combining this, the fact that Kω (Rn ) satisfies (i), p,q (ii), and (iii) of Proposition 6.3.15, Proposition 6.3.15 with X := K˙ ω (Rn ), and p,q,r,d p,q Definition 6.3.9, we conclude that W H K˙ ω (Rn ) ⊂ W H K˙ ω (Rn ) and, for any p,q,r,d n f ∈ W H K˙ ω (R ),

f W H K˙ p,q n  f  ˙ p,q,r,d (Rn ) . WHK ω (R ) ω

This, combined with (6.14), further implies that W H K˙ ωp,q (Rn ) = W H K˙ ωp,q,r,d (Rn ) with equivalent quasi-norms, which then completes the proof of Theorem 6.3.10.   Finally, combining Theorem 6.3.10 and Remark 6.0.25, we have the following p,q atomic characterization of W H M ω (Rn ); we omit the details. Corollary 6.3.18 Let p, q, ω, r, and d be as in Corollary 6.3.8. Then the weak p,q,r,d generalized atomic Hardy–Morrey space W H M ω (Rn ), associated with the p,q n weak local generalized Morrey space W M ω (R ), is defined to be the set of all the p,q f ∈ S  (Rn ) such that there exists a sequence {ai,j }i∈Z, j ∈N of (M ω (Rn ), r, d)atoms supported, respectively, in the balls {Bi,j }i∈Z, j ∈N ⊂ B and three positive

438

6 Weak Generalized Herz–Hardy Spaces

 independent of f , satisfying that, for any i ∈ Z, constants c ∈ (0, 1], A, and A, 

1cBi,j ≤ A,

j ∈N

f =

 i∈Z j ∈N

   i 1Bi,j  p,q n ai,j A2 M (R ) ω

in S  (Rn ), and ⎧ ⎨

      i sup 2  1Bi,j   i∈Z ⎩ j ∈N 

p,q M ω (Rn )

p,q,r,d

Moreover, for any f ∈ W H M ω f W H M p,q,r,d (Rn ) ω

⎫ ⎬ ⎭

< ∞,

(Rn ),

⎧    ⎨     i := inf sup 2  1Bi,j   i∈Z ⎩ j ∈N 

⎫ ⎬ p,q M ω (Rn )

⎭

,

where the infimum is taken over all the decompositions of f as above. Then n p,q,r,d (Rn ) W H M p,q ω (R ) = W H M ω

with equivalent quasi-norms.

6.4 Molecular Characterizations In this section, we establish the molecular characterization of weak generalized Herz–Hardy spaces. Indeed, applying the molecular characterization of weak Hardy spaces W HX (Rn ) associated with ball quasi-Banach function spaces X obtained in [278], we immediately obtain the molecular characterization of the weak p,q generalized Herz–Hardy space W H K˙ ω,0 (Rn ). However, due to the deficiency of p,q the associate space of the global generalized Herz space K˙ ω (Rn ), we can not prove the molecular characterization of the weak generalized Herz–Hardy space p,q W H K˙ ω (Rn ) via the known molecular characterization of W HX (Rn ) directly [see Remark 1.2.19(vi) for the details]. To overcome this difficulty, we first prove an improved molecular reconstruction theorem of weak Hardy spaces associated with ball quasi-Banach function spaces (see Proposition 6.4.5 below) without recourse to associate spaces. Then, combining this improved conclusion and the atomic p,q characterization of W H K˙ ω (Rn ) established in the last section, we obtain the p,q desired molecular characterization of W H K˙ ω (Rn ).

6.4 Molecular Characterizations

439

p,q Recall that the definition of (K˙ ω,0 (Rn ), r, d, τ )-molecules is given in Definition 4.5.1. We now characterize the weak generalized Herz–Hardy space p,q W H K˙ ω,0 (Rn ) via these molecules as follows.

Theorem 6.4.1 Let p, q ∈ (0, ∞), ω ∈ M(R+ ) satisfy m0 (ω) ∈ (− pn , ∞) and m∞ (ω) ∈ (− pn , ∞), p− ∈ 0,

n min{p, q, max{M0 (ω),M } ∞ (ω)}+n/p



max{1, p, q}

,

d ≥ n(1/p− − 1) be a fixed integer,   r ∈ max 1, p,

  n ,∞ , min{m0 (ω), m∞ (ω)} + n/p

and τ ∈ (n( p1− − 1r ), ∞). Then f belongs to the weak generalized Herz–Hardy p,q space W H K˙ ω,0 (Rn ) if and only if f ∈ S  (Rn ) and there exists a sequence p,q {mi,j }i∈Z, j ∈N of (K˙ ω,0 (Rn ), r, d, τ )-molecules centered, respectively, at the balls  independent {Bi,j }i∈Z, j ∈N ⊂ B and three positive constants c ∈ (0, 1], A, and A, of f , such that, for any i ∈ Z, 

1cBi,j ≤ A,

j ∈N

f =

 i∈Z j ∈N

   i 1Bi,j  ˙ p,q n mi,j A2 K (R ) ω,0

in S  (Rn ), and ⎧   ⎪  ⎨     i sup 2  1Bi,j   ⎪ i∈Z ⎩ j ∈N  ˙ p,q

Kω,0

⎫ ⎪ ⎬ (Rn )

⎪ ⎭

< ∞.

Moreover, there exist two positive constants C1 and C2 such that, for any f ∈ p,q W H K˙ ω,0 (Rn ),

C1 f W H K˙ p,q (Rn ) ω,0

⎧   ⎪   ⎨   i ≤ inf sup 2  1Bi,j   ⎪ ⎩ i∈Z j ∈N  ˙ p,q

Kω,0 (Rn )

⎫ ⎪ ⎬ ⎪ ⎭

≤ C2 f W H K˙ p,q (Rn ) , ω,0

where the infimum is taken over all the decompositions of f as above.

440

6 Weak Generalized Herz–Hardy Spaces

To show this molecular characterization, we first recall the following molecular characterization of weak Hardy spaces W HX (Rn ) associated with ball quasiBanach function spaces X, which was obtained by Zhang et al. in [278, Theorem 5.3]. Lemma 6.4.2 Let X be a ball quasi-Banach function space and let p− ∈ (0, 1) and p+ ∈ [p− , ∞) be such that the following three statements hold true: (i) for any given θ ∈ (0, p− ) and u ∈ (1, ∞), there exists a positive constant C such that, for any {fj }j ∈N ⊂ L1loc (Rn ), ⎧ ⎫1   u  ⎨  u ⎬    M(fj )   ⎭  ⎩  j ∈N 

X 1/θ

⎧ ⎫1    ⎬u   ⎨   u ≤C |fj |  ⎭  ⎩  j ∈N 

;

X 1/θ

(ii) for any s ∈ (0, p− ), X1/s is a ball Banach function space; (iii) for any given s ∈ (0, p− ) and r ∈ (p+ , ∞), there exists a positive constant C such that, for any f ∈ L1loc (Rn ),    ((r/s))  (f ) M

(X 1/s )

≤ C f (X1/s ) .

Let d ≥ n(1/p− − 1) be a fixed integer, r ∈ (max{1, p+ }, ∞], τ ∈ (n( p1− − n  r ), ∞), c ∈ (0, 1], and A, A ∈ (0, ∞). Assume that {mi,j }i∈Z, j ∈N is a sequence of (X, r, d, τ )-molecules centered, respectively, at the balls {Bi,j }i∈Z, j ∈N ⊂ B such that, for any i ∈ Z, 

1cBi,j ≤ A,

j ∈N

f :=

 i∈Z j ∈N

   i 1Bi,j  mi,j A2 X

converges in S  (Rn ), and ⎧ ⎨

  ⎫   ⎬   i 1Bi,j  sup 2   ⎭ < ∞. i∈Z ⎩ j ∈N  X

6.4 Molecular Characterizations

441

Then f ∈ W HX (Rn ) and f W HX (Rn )

⎧ ⎨

  ⎫   ⎬    i  sup 2  1Bi,j   ⎭, ⎩ i∈Z j ∈N  X

where the implicit positive constant is independent of f . With the help of the above lemma, we next show Theorem 6.4.1. Proof of Theorem 6.4.1 Let all the symbols be as in the present theorem. We p,q first prove the necessity. To achieve this, let f ∈ W H K˙ ω,0 (Rn ). Then, applying p,q,r,d Theorem 6.3.2, we find that f ∈ W H K˙ ω,0 (Rn ). From this and Definition 6.3.1, p,q it follows that there exists a sequence {ai,j }i∈Z, j ∈N of (K˙ ω,0 (Rn ), r, d)-atoms supported, respectively, in the balls {Bi,j }i∈Z, j ∈N ⊂ B and three positive constants  independent of f , such that, for any i ∈ Z, c ∈ (0, 1], A, and A, 

1cBi,j ≤ A,

j ∈N

f =

 i∈Z j ∈N

   i 1Bi,j  ˙ p,q n ai,j A2 K (R )

(6.15)

ω,0

in S  (Rn ), and ⎧ ⎪ ⎨

      i 1Bi,j  sup 2   i∈Z ⎪  ˙ p,q ⎩ j ∈N

Kω,0 (Rn )

⎫ ⎪ ⎬ ⎪ ⎭

< ∞.

(6.16)

p,q On the other hand, by Remark 4.5.4 with X := K˙ ω,0 (Rn ), we conclude that, p,q for any i ∈ Z and j ∈ N, ai,j is a (K˙ ω,0 (Rn ), r, d, τ )-molecule centered at Bi,j . Combining this, (6.15), and (6.16), we then complete the proof the necessity. Moreover, using both Definition 6.3.1 and Theorem 6.3.2 again, we find that

⎧   ⎪   ⎨   i inf sup 2  1Bi,j   ⎪  ˙ p,q ⎩ i∈Z j ∈N

Kω,0

⎫ ⎪ ⎬ (Rn )

⎪ ⎭

 f W H K˙ p,q,r,d, (Rn ) ω,0

∼ f W H K˙ p,q (Rn ) , ω,0

(6.17)

where the infimum is taken over all the decompositions of f as in the present theorem.

442

6 Weak Generalized Herz–Hardy Spaces

Conversely, we next prove the sufficiency. For this purpose, assume that f ∈ p,q S  (Rn ), {mi,j }i∈Z, j ∈N is a sequence of (K˙ ω,0 (Rn ), r, d, τ )-molecules centered,  ∈ (0, ∞) respectively, at the balls {Bi,j }i∈Z, j ∈N ⊂ B, c ∈ (0, 1], and A, A satisfying that, for any i ∈ Z, 

1cBi,j ≤ A,

j ∈N

f =

 i∈Z j ∈N

   i 1Bi,j  ˙ p,q n mi,j A2 K (R ) ω,0

in S  (Rn ), and ⎧    ⎪ ⎨     i sup 2  1Bi,j   ⎪ i∈Z ⎩ j ∈N  ˙ p,q

Kω,0

⎫ ⎪ ⎬ (Rn )

⎪ ⎭

< ∞.

Let  p+ := max 1, p,

 n . min{m0 (ω), m∞ (ω)} + n/p

Then, repeating an argument similar to that used in the proof of Theorem 6.3.2 with s0 and r0 therein replaced, respectively, by s and r, we conclude that the local p,q generalized Herz space K˙ ω,0 (Rn ) under consideration satisfies the following four statements: p,q (i) K˙ ω,0 (Rn ) is a BQBF space; (ii) for any given θ ∈ (0, p− ) and u ∈ (1, ∞), and for any {fj }j ∈N ⊂ L1loc (Rn ),

⎧ ⎫1   u ⎨  u ⎬    M(fj )   ⎩ ⎭   j ∈N 

˙ p,q (Rn )]1/θ [K ω,0

⎧ ⎫1    ⎨ ⎬u    u  |fj |  ⎩ ⎭   j ∈N 

;

˙ p,q (Rn )]1/θ [K ω,0

p,q (iii) for any s ∈ (0, p− ), [K˙ ω,0 (Rn )]1/s is a BBF space; (iv) for any given s ∈ (0, p− ) and r ∈ (p+ , ∞), and for any f ∈ L1loc (Rn ),

   ((r/s))  (f ) M

˙ (Rn )]1/s ) ([K ω,0 p,q

 f ([K˙ p,q (Rn )]1/s ) . ω,0

p,q These, together with Lemma 6.4.2 with X := K˙ ω,0 (Rn ), further imply that f ∈ p,q W H K˙ ω,0 (Rn ). This then finishes the proof of the sufficiency. Moreover, applying

6.4 Molecular Characterizations

443

p,q Lemma 6.4.2 again with X := K˙ ω,0 (Rn ) and the choice of {mi,j }i∈Z, j ∈N, we find that ⎧  ⎫   ⎪ ⎪ ⎨  ⎬     f W H K˙ p,q (Rn )  inf sup 2i  , 1 Bi,j   ω,0 i∈Z ⎪  ˙ p,q n ⎪ ⎩ j ∈N ⎭

Kω,0 (R )

where the infimum is taken over all the decompositions of f as in the present p,q theorem. From this and (6.17), we further infer that, for any f ∈ W H K˙ ω,0 (Rn ),

f W H K˙ p,q (Rn ) ω,0

⎧   ⎪  ⎨      ∼ inf sup 2i  1 Bi,j   i∈Z ⎪ ⎩ j ∈N  ˙ p,q

Kω,0 (Rn )

⎫ ⎪ ⎬ ⎪ ⎭

,

where the infimum is taken over all the decompositions of f as in the present theorem. This finishes the proof of Theorem 6.4.1.   From Theorem 6.4.1 and Remark 6.0.25 above, we deduce the following molecp,q ular characterization of the weak generalized Hardy–Morrey space W H M ω,0 (Rn ); we omit the details. Corollary 6.4.3 Let p, q ∈ [1, ∞), p− ∈ (0, min{p, q}/ max{p, q}), ω ∈ M(R+ ) with −

n < m0 (ω) ≤ M0 (ω) < 0 p

and −

n < m∞ (ω) ≤ M∞ (ω) < 0, p

d ≥ n(1/p− − 1) be a fixed integer,  r∈

 n ,∞ , min{m0 (ω), m∞ (ω)} + n/p

and τ ∈ (n( p1− − 1r ), ∞). Then f belongs to the weak generalized Hardy–Morrey p,q space W H M ω,0 (Rn ) if and only if f ∈ S  (Rn ) and there exists a sequence p,q {mi,j }i∈Z, j ∈N of (M ω,0 (Rn ), r, d, τ )-molecules centered, respectively, at the balls  independent {Bi,j }i∈Z, j ∈N ⊂ B and three positive constants c ∈ (0, 1], A, and A,

444

6 Weak Generalized Herz–Hardy Spaces

of f , such that, for any i ∈ Z, 

1cBi,j ≤ A,

j ∈N

f =

 i∈Z j ∈N

   i 1Bi,j  p,q n mi,j A2 M (R ) ω,0

in S  (Rn ), and ⎧    ⎪ ⎨      1 sup 2i  Bi,j   i∈Z ⎪ ⎩ j ∈N 

⎫ ⎪ ⎬ p,q M ω,0 (Rn )

⎪ ⎭

< ∞.

Moreover, there exist two positive constants C1 and C2 such that, for any f ∈ p,q W H M ω,0 (Rn ),

C1 f W H M p,q (Rn ) ω,0

⎧    ⎪ ⎨      ≤ inf sup 2i  1 Bi,j   i∈Z ⎪  ⎩ j ∈N

⎫ ⎪ ⎬ p,q

M ω,0 (Rn )

⎪ ⎭

≤ C2 f W H M p,q (Rn ) , ω,0

where the infimum is taken over all the decompositions of f as above. Now, we turn to establish the molecular characterization of the weak generalized p,q p,q Herz–Hardy space W H K˙ ω (Rn ). Recall that the concept of (K˙ ω (Rn ), r, d, τ )molecules is introduced in Definition 4.5.9 above. Then we show the following conclusion. Theorem 6.4.4 Let p, q ∈ (0, ∞), ω ∈ M(R+ ) satisfy m0 (ω) ∈ (− pn , ∞) and −

n < m∞ (ω) ≤ M∞ (ω) < 0, p

r ∈ (max{1, p, min{m0 (ω),mn∞ (ω)}+n/p }, ∞], p− ∈ 0,

n } min{p, q, max{M0 (ω),M ∞ (ω)}+n/p

max{1, p, q}

,

6.4 Molecular Characterizations

445

d ≥ n(1/p− − 1) be a fixed integer, and τ ∈ (n( p1− − 1r ), ∞). Then f p,q belongs to the weak generalized Herz–Hardy space W H K˙ ω (Rn ) if and only if p,q  n f ∈ S (R ) and there exists a sequence {mi,j }i∈Z, j ∈N of (K˙ ω (Rn ), r, d, τ )molecules centered, respectively, at the balls {Bi,j }i∈Z, j ∈N ⊂ B and three positive  independent of f , such that, for any i ∈ Z, constants c ∈ (0, 1], A, and A, 

1cBi,j ≤ A,

j ∈N

f =

 i∈Z j ∈N

   i 1Bi,j  ˙ p,q n mi,j A2 K (R ) ω

in S  (Rn ), and ⎧    ⎨     i sup 2  1Bi,j   ⎩ i∈Z j ∈N  ˙ p,q Kω

⎫ ⎬ (Rn )

⎭

< ∞.

Moreover, there exist two positive constants C1 and C2 such that, for any f ∈ p,q W H K˙ ω (Rn ), ⎧   ⎫  ⎨  ⎬    i  C1 f W H K˙ p,q ≤ inf sup 1 2 n B i,j   ω (R ) i∈Z ⎩ j ∈N  ˙ p,q n ⎭ Kω (R )

≤ C2 f W H K˙ p,q n , ω (R ) where the infimum is taken over all the decompositions of f as above. In order to show this theorem, we first establish the following molecular reconstruction theorem of the weak Hardy space W HX (Rn ), which improves [278, Theorem 5.3] via removing the assumption about associate spaces. Proposition 6.4.5 Let X be a ball quasi-Banach function space and let p− ∈ (0, 1) and p+ ∈ [p− , ∞) be such that the following three statements hold true: (i) for any given θ ∈ (0, p− ) and u ∈ (1, ∞), there exists a positive constant C such that, for any {fj }j ∈N ⊂ L1loc (Rn ), ⎧ ⎫1   u  ⎨  u ⎬    M(fj )   ⎭  ⎩   j ∈N

X 1/θ

⎧ ⎫1    ⎬u   ⎨   ≤C |fj |u  ⎭  ⎩  j ∈N 

X 1/θ

;

446

6 Weak Generalized Herz–Hardy Spaces

(ii) for any s ∈ (0, p− ), X1/s is a ball Banach function space and there exists a linear space Ys ⊂ M (Rn ) equipped with a quasi-seminorm  · Ys such that, for any f ∈ M (Rn ),   f X1/s ∼ sup fgL1 (Rn ) : gYs = 1 with the positive equivalence constants independent of f ; (iii) for any given s ∈ (0, p− ) and r ∈ (p+ , ∞), there exists a positive constant C such that, for any f ∈ L1loc (Rn ),    ((r/s))  (f ) M

Ys

≤ C f Ys .

Let d ≥ n(1/p− − 1) be a fixed integer, r ∈ (max{1, p+ }, ∞], τ ∈ (n( p1− − n  r ), ∞), c ∈ (0, 1], and A, A ∈ (0, ∞). Assume that {mi,j }i∈Z, j ∈N is a sequence of (X, r, d, τ )-molecules centered, respectively, at the balls {Bi,j }i∈Z, j ∈N ⊂ B such that, for any i ∈ Z, 

1cBi,j ≤ A,

j ∈N

f :=

 i∈Z j ∈N

   i 1Bi,j  mi,j A2 X

converges in S  (Rn ), and ⎧ ⎨

  ⎫   ⎬    i 1Bi,j  sup 2   ⎭ < ∞. ⎩ i∈Z j ∈N  X

Then f ∈ W HX (Rn ) and f W HX (Rn )

        sup 2i  1 Bi,j  ,  i∈Z j ∈N  X

where the implicit positive constant is independent of f .  be as in the present proposition and Proof Let X, r, d, τ , c, A, and A {mi,j }i∈Z, j ∈N a sequence of (X, r, d, τ )-molecules centered, respectively, at the balls {Bi,j }i∈Z, j ∈N ⊂ B such that, for any i ∈ Z,  j ∈N

1cBi,j ≤ A,

6.4 Molecular Characterizations

447

the summation  i∈Z j ∈N

   i 1Bi,j  mi,j A2 X

converges in S  (Rn ), and ⎧ ⎨

  ⎫   ⎬   i sup 2  1Bi,j   ⎭ < ∞. ⎩ i∈Z j ∈N  X

Then, repeating the proof of [278, Theorem 5.3] via replacing [278, Lemma 4.8] therein by Lemma 6.3.17 here, we conclude that f :=

 i∈Z j ∈N

   i 1Bi,j  mi,j ∈ W HX (Rn ) A2 X

and f W HX (Rn )

⎧ ⎨

  ⎫   ⎬   i  sup 2  1Bi,j   ⎭, i∈Z ⎩ j ∈N  X

which then completes the proof of Proposition 6.4.5.

 

Remark 6.4.6 We should point out that Proposition 6.4.5 is an improved version of the known molecular reconstruction theorem obtained by Zhang et al. in [278, Theorem 5.3]. Indeed, if Ys ≡ (X1/s ) in Proposition 6.4.5, then this proposition is just [278, Theoreem 5.3]. Using this proposition and the atomic characterization of the weak generalized p,q Herz–Hardy space W H K˙ ω (Rn ) obtained in the last section, we now prove the p,q molecular characterization of W H K˙ ω (Rn ). Proof of Theorem 6.4.4 Let all the symbols be as in the present theorem. We first p,q show the necessity. Indeed, let f ∈ W H K˙ ω (Rn ). Then, applying Theorem 6.3.10, p,q,r,d we find that f ∈ W H K˙ ω (Rn ). This, together with Definition 6.3.9, further p,q implies that there exists a sequence {ai,j }i∈Z, j ∈N of (K˙ ω (Rn ), r, d)-atoms supported, respectively, in the balls {Bi,j }i∈Z, j ∈N ⊂ B and three positive constants  independent of f , such that, for any i ∈ Z, c ∈ (0, 1], A, and A, 

1cBi,j ≤ A,

j ∈N

f =

 i∈Z j ∈N

   i 1Bi,j  ˙ p,q n ai,j A2 K (R ) ω

(6.18)

448

6 Weak Generalized Herz–Hardy Spaces

in S  (Rn ), and ⎧ ⎨

      i sup 2  1Bi,j   ⎩ i∈Z j ∈N  ˙ p,q Kω

⎫ ⎬ (Rn )

⎭

< ∞.

(6.19)

p,q In addition, from Remark 4.5.4 with X := K˙ ω (Rn ), it follows that, for any i ∈ Z p,q and j ∈ N, ai,j is a (K˙ ω (Rn ), r, d, τ )-molecule centered at Bi,j . This, combined with (6.18) and (6.19), finishes the proof of the necessity. Moreover, using both Definition 6.3.9 and Theorem 6.3.10 again, we conclude that

  ⎧   ⎨    inf sup 2i  1 Bi,j   ⎩ i∈Z   j ∈N

⎫ ⎬ n ˙ p,q K ω (R )

⎭

 f W H K˙ p,q,r,d, (Rn ) ω

∼ f W H K˙ p,q n , ω (R )

(6.20)

where the infimum is taken over all the decompositions of f as in the present theorem. Conversely, we prove the sufficiency. To this end, let f ∈ S  (Rn ), {mi,j }i∈Z, j ∈N p,q be a sequence of (K˙ ω (Rn ), r, d, τ )-molecules centered, respectively, at the balls  ∈ (0, ∞) such that, for any i ∈ Z, {Bi,j }i∈Z, j ∈N ⊂ B, c ∈ (0, 1], and A, A 

1cBi,j ≤ A,

j ∈N

f =

 i∈Z j ∈N

   i 1Bi,j  ˙ p,q n mi,j A2 K (R ) ω

in S  (Rn ), and ⎧    ⎨      1 sup 2i  Bi,j   i∈Z ⎩ j ∈N  ˙ p,q

Kω (Rn )

⎫ ⎬ ⎭

< ∞.

We next claim that the global generalized Herz space K˙ ω (Rn ) under consideration satisfies all the assumptions of Proposition 6.4.5. Assume that this claim holds p,q true for the moment. Then, applying Proposition 6.4.5 with X := K˙ ω (Rn ), we p,q n conclude that f ∈ W H K˙ ω (R ), which completes the proof of the sufficiency. Thus, to finish the proof of the sufficiency, we only need to show the above claim. For this purpose, let p,q

 p+ := max 1, p,

 n . min{m0 (ω), m∞ (ω)} + n/p

6.4 Molecular Characterizations

449

Then, repeating an argument similar to that used in the proof of Theorem 6.3.10 with s0 and r0 therein replaced, respectively, by s and r, we find that the global p,q generalized Herz space K˙ ω (Rn ) under consideration satisfies the following four statements: p,q (i) K˙ ω (Rn ) is a BQBF space; (i) for any given θ ∈ (0, p− ) and u ∈ (1, ∞), and for any {fj }j ∈N ⊂ L1loc (Rn ),

⎧ ⎫1   u ⎨  u ⎬    M(fj )   ⎩ ⎭    j ∈N

n 1/θ ˙ p,q [K ω (R )]

⎧ ⎫1    ⎨ ⎬u    u  |fj |  ⎩ ⎭   j ∈N 

;

n 1/θ ˙ p,q [K ω (R )]

p,q (ii) for any s ∈ (0, p− ), [K˙ ω (Rn )]1/s is a BBF space and, for any f ∈ M (Rn ),

 f [K˙ p,q fgL1 (Rn ) : g ˙ (p/s) ,(q/s) ∼ sup n 1/s ω (R )] B1/ωs

 (Rn )

with the positive equivalence constants independent of f ; (iii) for any given s ∈ (0, p− ) and r ∈ (p+ , ∞), and for any f ∈ L1loc (Rn ),    ((r/s))  (f ) ˙ (p/s) ,(q/s) M B1/ωs

(Rn )

 f  ˙ (p/s) ,(q/s) B1/ωs

(Rn )

.

p,q These further imply that the global generalized Herz space K˙ ω (Rn ) under consideration satisfies all the assumptions of Proposition 6.4.4 with the above p− , (p/s),(q/s) (Rn ) for any s ∈ (0, p− ). This then finishes the above p+ , and Ys := B˙1/ωs claim and further implies that the sufficiency holds true. p,q In addition, from Proposition 6.4.5 again with X := K˙ ω (Rn ) and the choice of {mi,j }i∈Z, j ∈N , we deduce that

f W H K˙ p,q n ω (R )

⎧    ⎨     i  inf sup 2  1Bi,j   ⎩ i∈Z j ∈N  ˙ p,q Kω

⎫ ⎬ (Rn )

⎭

,

where the infimum is taken over all the decompositions of f as in the present p,q theorem. This, together with (6.20), further implies that, for any f ∈ W H K˙ ω (Rn ), f W H K˙ p,q n ω (R )

⎧    ⎨     i ∼ inf sup 2  1Bi,j   i∈Z ⎩ j ∈N  ˙ p,q Kω

⎫ ⎬ (Rn )

⎭

,

where the infimum is taken over all the decompositions of f as in the present   theorem. This then finishes the proof of Theorem 6.4.4.

450

6 Weak Generalized Herz–Hardy Spaces

As an application, we now establish the following molecular characterization of weak Hardy spaces associated with global generalized Morrey spaces, which is just a simple corollary of Theorem 6.4.4 and Remark 6.0.25; we omit the details. Corollary 6.4.7 Let p, q, ω, r, d, and τ be as in Corollary 6.4.3. Then f p,q belongs to the weak generalized Hardy–Morrey space W H M ω (Rn ) if and only p,q  n if f ∈ S (R ) and there exists a sequence {mi,j }i∈Z, j ∈N of (M ω (Rn ), r, d, τ )molecules centered, respectively, at the balls {Bi,j }i∈Z, j ∈N ⊂ B and three positive  independent of f , such that, for any i ∈ Z, constants c ∈ (0, 1], A, and A, 

1cBi,j ≤ A,

j ∈N

f =

 i∈Z j ∈N

   i 1Bi,j  p,q n mi,j A2 M (R ) ω

in S  (Rn ), and ⎧    ⎨     i 1Bi,j  sup 2   i∈Z ⎩ j ∈N 

⎫ ⎬ p,q M ω (Rn )

⎭

< ∞.

Moreover, there exist two positive constants C1 and C2 such that, for any f ∈ p,q W H M ω (Rn ),   ⎫ ⎧   ⎬ ⎨   i  ≤ inf 2 1 C1 f W H M p,q sup n B i,j  ω (R ) ⎩ i∈Z  j ∈N  p,q n ⎭ M ω (R )

≤ C2 f W H M p,q n , ω (R ) where the infimum is taken over all the decompositions of f as above.

6.5 Littlewood–Paley Function Characterizations The main target of this section is to characterize weak generalized Herz–Hardy spaces via various Littlewood–Paley functions. Precisely, using the Lusin area function S, the g-function g, and the gλ∗ -function gλ∗ presented in Definitions 4.6.1 and 4.6.2, we establish several equivalent characterizations of the weak generalized p,q p,q Herz–Hardy spaces W H K˙ ω,0 (Rn ) and W H K˙ ω (Rn ). To begin with, we state the following Littlewood–Paley function characterizations of the Hardy space p,q W H K˙ ω,0 (Rn ).

6.5 Littlewood–Paley Function Characterizations

451

Theorem 6.5.1 Let p, q ∈ (0, ∞), ω ∈ M(R+ ) with m0 (ω) ∈ (− pn , ∞) and m∞ (ω) ∈ (− pn , ∞),  p− := min p, q,  p+ := max p,

 n , max{M0 (ω), M∞ (ω)} + n/p  n , min{m0 (ω), m∞ (ω)} + n/p

and   λ ∈ max

  2 2 2 , 1− + ,∞ . min{1, p− } max{1, p+ } min{1, p− }

Assume that, for any f ∈ S  (Rn ), S(f ) and gλ∗ (f ) are as in Definition 4.6.1, and g(f ) is as in Definition 4.6.2. Then the following four statements are mutually equivalent: (i) (ii) (iii) (iv)

f f f f

p,q ∈ W H K˙ ω,0 (Rn ); p,q  n ∈ S (R ), f vanishes weakly at infinity, and S(f ) ∈ W K˙ ω,0 (Rn ); p,q ∈ S  (Rn ), f vanishes weakly at infinity, and g(f ) ∈ W K˙ ω,0 (Rn ); p,q ∈ S  (Rn ), f vanishes weakly at infinity, and gλ∗ (f ) ∈ W K˙ ω,0 (Rn ).

p,q Moreover, for any f ∈ W H K˙ ω,0 (Rn ),

f W H K˙ p,q (Rn ) ∼ S(f )W K˙ p,q (Rn ) ∼ g(f )W K˙ p,q (Rn ) ω,0 ω,0 ω,0  ∗    ∼ gλ (f ) W K˙ p,q (Rn ) , ω,0

where the positive equivalence constants are independent of f . In order to prove this theorem, we need the following Littlewood–Paley function characterizations of weak Hardy spaces associated with ball quasi-Banach function spaces, which are just [242, Theorems 3.12, 3.16, and 3.21]. Lemma 6.5.2 Let X be a ball quasi-Banach function space and let p− ∈ (0, ∞), p+ ∈ [p− , ∞), and θ0 ∈ (1, ∞) be such that the following four statements hold true: (i) for any given θ ∈ (0, p− ) and u ∈ (1, ∞), there exists a positive constant C such that, for any {fj }j ∈N ⊂ L1loc (Rn ), ⎧ ⎫1   u ⎨  u ⎬    M(fj )   ⎭  ⎩   j ∈N

X 1/θ

⎧ ⎫1    ⎬u  ⎨   u ≤C |fj |  ⎭  ⎩  j ∈N 

X 1/θ

452

6 Weak Generalized Herz–Hardy Spaces

and ⎧ ⎫1   u ⎨  u ⎬    M(fj )   ⎩ ⎭    j ∈N

(W X)1/θ

⎧ ⎫1     ⎨ ⎬u    u ≤C |fj |  ⎩ ⎭   j ∈N 

;

(W X)1/θ

(ii) X is θ0 -concave; (iii) for any s ∈ (0, min{1, p− }), X1/s is a ball Banach function space; (iv) for any given s ∈ (0, min{1, p− }) and r ∈ (p+ , ∞), there exists a positive constant C such that, for any f ∈ L1loc (Rn ),    ((r/s))  (f ) M

(X 1/s )

≤ C f (X1/s ) .

Let   λ ∈ max

  2 2 2 , 1− + ,∞ . min{1, p− } max{1, p+ } min{1, p− }

Then the following four statements are mutually equivalent: (i) (ii) (iii) (iv)

f f f f

∈ W HX (Rn ); ∈ S  (Rn ), f vanishes weakly at infinity, and S(f ) ∈ W X; ∈ S  (Rn ), f vanishes weakly at infinity, and g(f ) ∈ W X; ∈ S  (Rn ), f vanishes weakly at infinity, and gλ∗ (f ) ∈ W X.

Moreover, for any f ∈ W HX (Rn ),   f W HX (Rn ) ∼ S(f )W X ∼ g(f )W X ∼ gλ∗ (f )W X with the positive equivalence constants independent of f . Remark 6.5.3 We point out that Lemma 6.5.2 has a wide range of applications. Here we give several function spaces to which Lemma 6.5.2 can be applied (see also [242, Section 5]). (i) Let p := (p1 , . . . , pn ) ∈ (0, ∞)n and λ ∈ (0, ∞) satisfy  λ > max

 2 2 2 . , 1− + min{1, p1 , . . . , pn } max{1, p1 , . . . , pn } min{1, p1 , . . . , pn }

Then, in this case, as was pointed out in [242, Subsection 5.2], the mixed-norm Lebesgue space Lp (Rn ) satisfies all the assumptions of Lemma 6.5.2. This implies that Lemma 6.5.2 with X := Lp (Rn ) holds true. This result coincides with [242, Theorems 5.15 and 5.16].

6.5 Littlewood–Paley Function Characterizations

453

(ii) Let 0 < q ≤ p < ∞ and   λ ∈ max

  2 2 2 , 1− + ,∞ . min{1, q} max{1, p} min{1, q}

Then, in this case, as was mentioned in [242, Subsection 5.1], the Morrey space p Mq (Rn ) satisfies all the assumptions of Lemma 6.5.2. Thus, Lemma 6.5.2 with p X := Mq (Rn ) holds true. This result coincides with [242, Theorems 5.6 and 5.7]. (iii) Let p(·) ∈ C log (Rn ) satisfy 0 < p− ≤ p+ < ∞, and   λ ∈ max

  2 2 2 , 1− + ,∞ , min{1, p− } max{1, p+ } min{1, p− }

where p− and p+ are defined, respectively, in (1.59) and (1.60). In this case, as was pointed out in [242, Subsection 5.3], the variable Lebesgue space Lp(·) (Rn ) satisfies all the assumptions of Lemma 6.5.2. Therefore, Lemma 6.5.2 with X := Lp(·) (Rn ) holds true. This result coincides with [242, Theorems 5.18 and 5.19]. Moreover, to prove Theorem 6.5.1, we still require the following auxiliary lemma about the Fefferman–Stein vector-valued inequality on weak local generalized Herz spaces. Lemma 6.5.4 Let p, q ∈ (0, ∞) and ω ∈ M(R+ ) satisfy m0 (ω) ∈ (− pn , ∞) and m∞ (ω) ∈ (− pn , ∞). Then, for any given   r ∈ 0, min p,

n max{M0 (ω), M∞ (ω)} + n/p



and u ∈ (1, ∞), there exists a positive constant C such that, for any {fj }j ∈N ⊂ L1loc (Rn ), ⎧ ⎫1   u ⎨  u ⎬    M(fj )   ⎭  ⎩   j ∈N

˙ p,q (Rn )]1/r [W K ω,0

⎧ ⎫1    ⎬u  ⎨   u ≤C |fj |  ⎭  ⎩  j ∈N 

. ˙ p,q (Rn )]1/r [W K ω,0

Proof Let all the symbols be as in the present lemma. Then, by the assumption m0 (ω) ∈ (− pn , ∞) and Theorem 1.2.42, we conclude that the local generalized p,q Herz space K˙ (Rn ) under consideration is a BQBF space. Moreover, let ω,0

 n . p− := min p, max{M0 (ω), M∞ (ω)} + n/p 

454

6 Weak Generalized Herz–Hardy Spaces

Then, for any given r ∈ (0, p− ) and u ∈ (1, ∞), applying Lemma 4.3.10, we find that, for any {fj }j ∈N ⊂ L1loc (Rn ), ⎧ ⎫1    ⎨  - . ⎬ u    u M fj   ⎩ ⎭    j ∈N

˙ p,q (Rn )]1/r [K ω,0

⎛ ⎞1       u ⎝ u⎠    fj      j ∈N 

.

˙ p,q (Rn )]1/r [K ω,0

This, combined with the fact that K˙ ω,0 (Rn ) is a BQBF space and Lemma 6.1.7 with p,q X := K˙ ω,0 (Rn ), further implies that, for any given r ∈ (0, p− ) and u ∈ (1, ∞), and for any {fj }j ∈N ⊂ L1loc (Rn ), p,q

⎧ ⎫1    ⎨  - . ⎬ u    u M fj   ⎩ ⎭    j ∈N

˙ (Rn )]1/r [W K ω,0 p,q

⎛ ⎞1      u ⎝ u fj  ⎠       j ∈N 

,

˙ (Rn )]1/r [W K ω,0 p,q

 

which then completes the proof of Lemma 6.5.4. Via the above two lemmas, we next prove Theorem 6.5.1.

Proof of Theorem 6.5.1 Let all the symbols be as in the present theorem. Then, by the assumption m0 (ω) ∈ (− pn , ∞) and Theorem 1.2.42, we conclude that the local p,q generalized Herz space K˙ ω,0 (Rn ) under consideration is a BQBF space. Therefore, from Lemma 6.5.2, it follows that, to finish the proof of the present theorem, we only p,q need to show that K˙ ω,0 (Rn ) satisfies all the assumptions of Lemma 6.5.2. Indeed, for any given θ ∈ (0, p− ) and u ∈ (1, ∞), using Lemma 6.5.4 with r := θ , we find that, for any {fj }j ∈N ⊂ L1loc (Rn ), ⎧ ⎫1   u  ⎨  u ⎬    M(fj )   ⎩ ⎭   j ∈N 

˙ p,q (Rn )]1/θ [W K ω,0

⎧ ⎫1    ⎨ ⎬u    u  |fj |  ⎩ ⎭   j ∈N 

.

(6.21)

˙ p,q (Rn )]1/θ [W K ω,0

In addition, let θ0 ∈ (max{1, p, q}, ∞). Then, repeating an argument similar to that used in the proof of Theorem 6.3.2 with s0 and r0 therein replaced, respectively, by s and r, we conclude that the following four statements hold true: (i) for any given θ ∈ (0, p− ) and u ∈ (1, ∞), and for any {fj }j ∈N ⊂ L1loc (Rn ), ⎧ ⎫1   u ⎨  u ⎬    M(fj )   ⎭  ⎩  j ∈N 

˙ p,q (Rn )]1/θ [K ω,0

⎧ ⎫1    ⎬u  ⎨   u  |fj |  ⎭  ⎩   j ∈N

˙ p,q (Rn )]1/θ [K ω,0

;

6.5 Littlewood–Paley Function Characterizations

455

p,q (ii) K˙ ω,0 (Rn ) is θ0 -concave; p,q (iii) for any s ∈ (0, min{1, p− }), [K˙ ω,0 (Rn )]1/s is a BBF space; (iv) for any given s ∈ (0, min{1, p− }) and r ∈ (p+ , ∞), and for any f ∈ L1loc (Rn ),

   ((r/s))  (f ) M

˙ p,q (Rn )]1/s ) ([K ω,0

 f ([K˙ p,q (Rn )]1/s ) . ω,0

These, combined with (6.21), further imply that the Herz space K˙ ω,0 (Rn ) under consideration satisfies all the assumptions of Lemma 6.5.2 with p− and p+ as in the present theorem. This then implies that (i), (ii), (iii), and (iv) of the present theorem p,q are mutually equivalent and, for any f ∈ W H K˙ ω,0 (Rn ), p,q

f W H K˙ p,q (Rn ) ∼ S(f )W K˙ p,q (Rn ) ∼ g(f )W K˙ p,q (Rn ) ω,0 ω,0 ω,0  ∗  ∼ g (f ) ˙ p,q n λ

W Kω,0 (R )

with the positive equivalence constants independent of f , which completes the proof of Theorem 6.5.1.   Combining Theorem 6.5.1 and Remark 6.0.25, we immediately obtain the following Littlewood–Paley function characterizations of the weak generalized p,q Hardy–Morrey space W H M ω,0 (Rn ); we omit the details. Corollary 6.5.5 Let p, q ∈ [1, ∞), ω ∈ M(R+ ) with −

n < m0 (ω) ≤ M0 (ω) < 0 p

and −

n < m∞ (ω) ≤ M∞ (ω) < 0, p

and     2(min{m0 (ω), m∞ (ω)} + n/p) λ ∈ max 2, 3 − ,∞ . n Then the following four statements are mutually equivalent: (i) (ii) (iii) (iv)

f f f f

p,q

∈ W M ω,0 (Rn ); p,q ∈ S  (Rn ), f vanishes weakly at infinity, and S(f ) ∈ W M ω,0 (Rn ); p,q ∈ S  (Rn ), f vanishes weakly at infinity, and g(f ) ∈ W M ω,0 (Rn ); p,q ∈ S  (Rn ), f vanishes weakly at infinity, and gλ∗ (f ) ∈ W M ω,0 (Rn ).

456

6 Weak Generalized Herz–Hardy Spaces p,q

Moreover, for any f ∈ W M ω,0 (Rn ), f W M p,q (Rn ) ∼ S(f )W M p,q (Rn ) ∼ g(f )W M p,q (Rn ) ω,0 ω,0 ω,0  ∗  p,q ∼ g (f ) n , W M ω,0 (R )

λ

where the positive equivalence constants are independent of f . Now, we turn to investigate the Littlewood–Paley function characterizations p,q of the weak generalized Herz–Hardy space W H K˙ ω (Rn ). Indeed, we have the following conclusion. Theorem 6.5.6 Let p, q ∈ (0, ∞), ω ∈ M(R+ ) with m0 (ω) ∈ (− pn , ∞) and −

n < m∞ (ω) ≤ M∞ (ω) < 0, p

 p− := min p, q,  p+ := max p,

 n , max{M0 (ω), M∞ (ω)} + n/p  n , min{m0 (ω), m∞ (ω)} + n/p

and   λ ∈ max

  2 2 2 , 1− + ,∞ . min{1, p− } max{1, p+ } min{1, p− }

Assume that, for any f ∈ S  (Rn ), S(f ) and gλ∗ (f ) are as in Definition 4.6.1, and g(f ) is as in Definition 4.6.2. Then the following four statements are mutually equivalent: (i) (ii) (iii) (iv)

f f f f

p,q ∈ W H K˙ ω (Rn ); p,q  n ∈ S (R ), f vanishes weakly at infinity, and S(f ) ∈ W K˙ ω (Rn ); p,q  n ˙ ∈ S (R ), f vanishes weakly at infinity, and g(f ) ∈ W Kω (Rn ); p,q ∈ S  (Rn ), f vanishes weakly at infinity, and gλ∗ (f ) ∈ W K˙ ω (Rn ).

p,q Moreover, for any f ∈ W H K˙ ω (Rn ),

f W H K˙ p,q n ∼ S(f )W K n ∼ g(f )W K n ˙ p,q ˙ p,q ω (R ) ω (R ) ω (R )  ∗  ∼ gλ (f )W K˙ p,q (Rn ) , ω

where the positive equivalence constants are independent of f . To prove these Littlewood–Paley function characterizations, we first establish the following representation formula of the quasi-norm  · W K˙ p,q n . ω (R )

6.5 Littlewood–Paley Function Characterizations

457

Lemma 6.5.7 Let p, q ∈ (0, ∞) and ω ∈ M(R+ ). Then, for any f ∈ M (Rn ),   τξ (f ) ˙ p,q n , f W K˙ p,q n = sup W K (R ) ω (R ) ξ ∈Rn

ω,0

where, for any ξ ∈ Rn , the operator τξ is defined as in (4.69). Proof Let all the symbols be as in the present lemma. Then, using Definition 6.0.18(ii), Remark 1.2.2(ii), and Definition 6.0.18(i), we find that, for any f ∈ M (Rn ), f W K˙ p,q n = ω (R ) =

sup λ∈(0,∞)

    λ 1{y∈Rn : |f (y)|>λ}K˙ p,q (Rn ) ω



sup λ∈(0,∞)

=

   λ sup 1{y∈Rn: |f (y)|>λ} (· − ξ )K˙ p,q (Rn ) ξ ∈Rn



sup

λ∈(0,∞), ξ ∈Rn



= sup

ξ ∈Rn



ω,0

  λ 1{y∈Rn : |τξ (f )(y)|>λ} K˙ p,q (Rn )



ω,0

    sup λ 1{y∈Rn : |τξ (f )(y)|>λ}K˙ p,q (Rn ) ω,0

λ∈(0,∞)

  = sup τξ (f )K˙ p,q (Rn ) . ξ ∈Rn

ω,0

 

This finishes the proof of Lemma 6.5.7.

Based on this formula and the Littlewood–Paley function characterizations of p,q W H K˙ ω,0 (Rn ) obtained in Theorem 6.5.1 above, we next prove Theorem 6.5.6. Proof of Theorem 6.5.6 Let all the / symbols be as in the present theorem, f ∈ S  (Rn ), and φ ∈ S(Rn ) satisfy Rn φ(x) dx = 0. We first show that (i) implies p,q (ii). Indeed, assume f ∈ W H K˙ ω (Rn ). Let M denote the radial maximal function defined as in Definition 4.1.1(i). Then, for any ξ ∈ Rn , using Theorem 6.1.1(ii) with f therein replaced by τξ (f ), Lemma 4.6.9(ii), Lemma 6.5.7 with f therein replaced by M(f, φ), and Theorem 6.1.9(ii), we conclude that   τξ (f ) ˙ p,q (Rn ) WHK ω,0    . ∼ M τξ (f ) , φ W K˙ p,q (Rn ) ∼ τξ (M (f, φ))W K˙ p,q (Rn ) ω,0

ω,0

 M (f, φ)W K˙ p,q n ∼ f W H K n < ∞, ˙ p,q ω (R ) ω (R )

(6.22)

which further implies that τξ (f ) ∈ W H K˙ ω,0 (Rn ). Applying this with ξ := 0, we p,q further find that f ∈ W H K˙ ω,0 (Rn ). This, together with Theorem 6.5.1, implies that f vanishes weakly at infinity. On the other hand, from Lemma 4.6.9(iii) with p,q

458

6 Weak Generalized Herz–Hardy Spaces

A := S, the fact that τξ (f ) ∈ W H K˙ ω,0 (Rn ) for any ξ ∈ Rn , Theorem 6.5.1, and (6.22), it follows that p,q

   . τξ (S (f )) ˙ p,q n ∼ S τξ (f )  ˙ p,q n W Kω,0 (R ) W Kω,0 (R )   ∼ τξ (f )W H K˙ p,q (Rn )  f W H K˙ p,q n . ω (R ) ω,0

Using this and Lemma 6.5.7 again with f replaced by S(f ), we find that S (f )W K˙ p,q n  f W H K n < ∞, ˙ p,q ω (R ) ω (R )

(6.23)

which completes the proof that (i) implies (ii). Conversely, we next prove that (ii) implies (i). Indeed, assume that f vanishes p,q weakly at infinity and S(f ) ∈ W K˙ ω (Rn ). Then, by Lemma 4.6.9(iv), we conclude that, for any ξ ∈ Rn , τξ (f ) vanishes weakly at infinity. In addition, from Lemma 4.6.9(iii) with A := S and Lemma 6.5.7 with f therein replaced by S(f ), we deduce that, for any ξ ∈ Rn ,    . S τξ (f )  ˙ p,q n = τξ (S (f )) ˙ p,q n W K (R ) W K (R ) ω,0

ω,0

≤ S (f )W K˙ p,q n < ∞, ω (R )

(6.24)

which, combined with the fact that τξ (f ) vanishes weakly at infinity and Theop,q rem 6.5.1, further implies that τξ (f ) ∈ W H K˙ ω,0 (Rn ). Moreover, for any ξ ∈ Rn , using Lemma 4.6.9(ii), Theorem 6.1.1(ii) with f replaced by τξ (f ), Theorem 6.5.1 again, and (6.24), we conclude that   τξ (M (f, φ))

˙ (Rn ) WK ω,0 p,q

 . = M τξ (f ) , φ W K˙ p,q (Rn ) ω,0    .   ∼ τξ (f ) ˙ p,q n ∼ S τξ (f ) W H Kω,0 (R )

˙ (Rn ) WK ω,0 p,q

 S (f )W K˙ p,q n , ω (R ) which, together with Theorem 6.1.9(ii) and Lemma 6.5.7 with f replaced by M(f, φ), further implies that f W H K˙ p,q n ∼ M (f, φ)W K n ˙ p,q ω (R ) ω (R ) !   ∼ sup τξ (M (f, φ))W K˙ p,q (Rn ) ξ ∈Rn

 S (f )W K˙ p,q n < ∞, ω (R )

ω,0

(6.25)

6.6 Boundedness of Calderón–Zygmund Operators

459

which completes the proof that (ii) implies (i). Therefore, (i) is equivalent to (ii). Moreover, combining (6.23) and (6.25), we find that f W K˙ p,q n ∼ S (f )W H K n . ˙ p,q ω (R ) ω (R ) Now, repeating an argument similar to that used in the estimations of both (6.23) and (6.25) above with S therein replaced by g or gλ∗ , we conclude that (i) is equivalent to (iii) and (i) is also equivalent to (iv). Moreover, for any A ∈ {g, gλ∗ } p,q and f ∈ W H K˙ ω (Rn ), it holds true that f W H K˙ p,q n ∼ A (f )W K n . ˙ p,q ω (R ) ω (R ) This then finishes the proof of Theorem 6.5.6.

 

Using the above theorem and Remark 6.0.25, we immediately obtain the following Littlewood–Paley function characterizations of the weak generalized p,q Hardy–Morrey space W H M ω (Rn ); we omit the details. Corollary 6.5.8 Let p, q, ω, and λ be as in Corollary 6.5.5. Then the following four statements are mutually equivalent: (i) (ii) (iii) (iv)

f f f f

p,q

∈ W M ω (Rn ); p,q ∈ S  (Rn ), f vanishes weakly at infinity, and S(f ) ∈ W M ω (Rn ); p,q ∈ S  (Rn ), f vanishes weakly at infinity, and g(f ) ∈ W M ω (Rn ); p,q ∈ S  (Rn ), f vanishes weakly at infinity, and gλ∗ (f ) ∈ W M ω (Rn ). p,q

Moreover, for any f ∈ W M ω (Rn ), f W M p,q n ∼ S(f )W M p,q (Rn ) ∼ g(f )W M p,q (Rn ) ω (R ) ω ω  ∗    ∼ gλ (f ) W M p,q (Rn ) , ω

where the positive equivalence constants are independent of f .

6.6 Boundedness of Calderón–Zygmund Operators Let δ ∈ (0, 1), d ∈ Z+ , K be a standard kernel as in Definition 1.5.7 with δ ∈ (0, 1), and T a d-order Calderón–Zygmund operator as in Definition 1.5.8 with kernel K. The main target of this section is to investigate the boundedness of operator T from generalized Herz–Hardy spaces to weak generalized Herz–Hardy spaces. First, we establish two boundedness criteria of T from Hardy spaces HX (Rn ) to weak Hardy spaces W HX (Rn ), which improve the boundedness of convolutional type Calderón– Zygmund operators from HX (Rn ) to W HX (Rn ) obtained in [278, Theorem 6.3]. As applications, we obtain the boundedness of T from generalized Herz–Hardy spaces n to weak generalized Herz–Hardy spaces even in the critical case p = n+d+δ .

460

6 Weak Generalized Herz–Hardy Spaces

Next, we show the following boundedness of Calderón–Zygmund operators about Hardy-type spaces associated with ball quasi-Banach function spaces. Theorem 6.6.1 Let d ∈ Z+ , K be a d-order standard kernel with δ ∈ (0, 1), and T a d-order Calderón–Zygmund operator with kernel K having the vanishing moments up to d. Let X be a ball quasi-Banach function space, Y a linear space equipped with a quasi-seminorm ·Y , and Y0 a linear space equipped with a quasiseminorm  · Y0 , and let r1 ∈ (0, ∞), η ∈ (1, ∞), and 0 < θ < s < s0 ≤ 1 be such that (i) (ii) (iii) (iv)

for the above θ and s, Assumption 1.2.29 holds true; both  · Y and  · Y0 satisfy Definition 1.2.13(ii); 1B(0,1) ∈ Y0 ; for any f ∈ M (Rn ),   f X1/s ∼ sup fgL1 (Rn ) : gY = 1 and   f X1/s0 ∼ sup fgL1 (Rn ) : gY0 = 1

with the positive equivalence constants independent of f ; (v) M(η) is bounded on both Y and Y0 ; (vi) M is bounded on (W X)1/r1 ; (vii) there exists a positive constant C such that, for any {fj }j ∈N ⊂ L1loc (Rn ), ⎧ ⎫ n    ⎨  - . n+d+δ ⎬ n+d+δ    n M fj   ⎩  ⎭  j∈N 

n+d+δ (W X) n

⎛ ⎞ n        n+d+δ n+d+δ  ⎝  n fj  ⎠ ≤C     j∈N 

. n+d+δ X n

n If θ ∈ (0, n+d ], then T is well defined on HX (Rn ) and there exists a positive constant C such that, for any f ∈ HX (Rn ),

T (f )W HX (Rn ) ≤ Cf HX (Rn ) . Remark 6.6.2 We should point out that Theorem 6.6.1 is an improved version of the known boundedness of convolutional type Calderón–Zygmund operators from HX (Rn ) to W HX (Rn ) established by Zhang et al. in [278, Theorem 6.3]. Indeed, in Theorem 6.6.1, if d = 0, K(x, y) ≡ K1 (x − y) for some K1 ∈ L1loc (Rn \ {0}), Y ≡ (X1/s ) , and Y0 ≡ (X1/s0 ) , then T coincides with the convolution δ-type Calderón–Zygmund operator as in [278, Theorem 6.3], and this proposition goes back to [278, Theorem 6.3]. To establish this boundedness criterion, we first show the following boundedness of Calderón–Zygmund operators under stronger assumptions than Theorem 6.6.1.

6.6 Boundedness of Calderón–Zygmund Operators

461

Proposition 6.6.3 Let d ∈ Z+ , K be a d-order standard kernel with δ ∈ (0, 1), and T a d-order Calderón–Zygmund operator with kernel K having the vanishing moments up to order d. Let X be a ball quasi-Banach function space satisfying Assumption 1.2.29 with some 0 < θ < s ≤ 1 and Assumption 1.2.33 with the same s and some r0 ∈ (1, ∞). Assume that there exists an r1 ∈ (0, ∞) such that M is bounded on (W X)1/r1 , and there exists a positive constant C such that, for any {fj }j ∈N ⊂ L1loc (Rn ), ⎧ ⎫ n    ⎨  - . n+d+δ ⎬ n+d+δ    n M fj   ⎩  ⎭  j ∈N  n+d+δ (W X) n ⎛ ⎞ n        n+d+δ n+d+δ  ⎝  fj  n ⎠ ≤C .     j ∈N  n+d+δ X

(6.26)

n

n If θ ∈ (0, n+d ] and X has an absolutely continuous quasi-norm, then T has a unique extension on HX (Rn ) and there exists a positive constant C such that, for any f ∈ HX (Rn ),

T (f )W HX (Rn ) ≤ Cf HX (Rn ) . Remark 6.6.4 We point out that Proposition 6.6.3 has a wide range of applications. Here we present two function spaces to which Proposition 6.6.3 can be applied. In what follows, let d and δ be the same as in Proposition 6.6.3. n (i) Let p := (p1 , . . . , pn ) ∈ [ n+d+δ , ∞)n . Then, in this case, as was pointed out in both Remark 1.2.27 and [278, Subsection 7.2], the mixed-norm Lebesgue space Lp (Rn ) satisfies all the assumptions of Proposition 6.6.3. Thus, Proposition 6.6.3 with X := Lp (Rn ) holds true. If further assume that K is a convolutional type operator defined as in Proposition 4.8.2, and d := 0, then, in this case, the aforementioned result coincides with [278, Theorem 7.36]. n (ii) Let p(·) ∈ C log (Rn ) satisfy n+d+δ ≤ p− ≤ p+ < ∞, where p− and p+ are defined, respectively, in (1.59) and (1.60). Then, in this case, combining Remarks 1.2.27, 1.2.31(v), and 1.2.34(v) and [267, Proposition 7.8], we can easily find that the variable Lebesgue space Lp(·) (Rn ) satisfies all the assumptions of Proposition 6.6.3. This then implies that Proposition 6.6.3 with X := Lp(·) (Rn ) holds true. If further assume that p+ ∈ (0, 1], K is a convolutional type operator defined as in Proposition 4.8.2, and d := 0, then, in this case, the aforementioned result is just [267, Theorem 7.4].

To show this proposition, we need the following technical estimate about radial maximal functions.

462

6 Weak Generalized Herz–Hardy Spaces

Lemma 6.6.5 Let d ∈ Z+ , K be a d-order standard kernel with δ ∈ (0, 1), and T a d-order Calderón–Zygmund with kernel K having the vanishing moments up to order d. Assume that X is a ball quasi-Banach function space, φ ∈ S(Rn ), and r ∈ [2, ∞). Then there exists a positive constant C such that, for any (X, r, d)-atom a supported in the ball B ∈ B, M (T (a) , φ) 1(4B) ≤ C

n+d+δ 1 [M (1B )] n , 1B X

where the radial maximal function M is defined as in Definition 4.1.1(i). Proof Let all the symbols be as in the present lemma and a an (X, r, d)-atom supported in the ball B := B(x0 , r0 ) with x0 ∈ Rn and r0 ∈ (0, ∞). Then, combining Definition 4.3.4(iii), the fact that T has the vanishing moments up to order d, and Definition 4.8.1, we conclude that, for any γ ∈ Zn+ with |γ | ≤ d,  Rn

T (a)(x)x γ dx = 0.

(6.27)

Fix a t ∈ (0, ∞) and an x ∈ [B(x0 , 4r0 )] . Then, from (6.27), it follows that |T (a) ∗ φt (x)|     φt (x − y)T (a)(y) dy  =  Rn

  ⎡ ⎤         γ 1  ⎢ ⎥ ∂ (φ( x−· ))(x ) x − y 0 t γ⎥  ⎢ = n (y − x0 ) ⎦ T (a)(y) dy  φ − ⎣ t γ!  t Rn  γ ∈Zn +   |γ |≤d ⎧     ⎪ ⎪      γ   ∂ (φ( x−· ))(x ) 1 ⎨ x − y 0 t γ  ≤ n − (y − x0 )  φ  t ⎪ t γ! ⎪  γ ∈Zn ⎩ |y−x0 |λ/3} B(ξ,2 )\B(ξ,2 )  1  +

1 q

L (Rn )



ω(2k )

q

k∈Z

q     × 1 1B(ξ,2k )\B(ξ,2k−1)  4 1  1 n r r {y∈R : { j∈N [M(fj,k,3 )(y)] } >λ/3}

1 q

L (Rn )

=: IIλ,1 + IIλ,2 + IIλ,3 .

(6.50)

We next estimate IIλ,1 , IIλ,2 , and IIλ,3 , respectively. To achieve this, let ε ∈ (0, {min{m0 (ω), m∞ (ω)} + n, − max{M0 (ω), M∞ (ω)}}) be a fixed positive constant. Then, applying Lemma 1.5.2, we conclude that, for any 0 < t < τ < ∞,  min{m0 (ω),m∞ (ω)}−ε t τ

(6.51)

 max{M0 (ω),M∞ (ω)}+ε t . τ

(6.52)

ω(t)  ω(τ ) and, for any 0 < τ < t < ∞, ω(t)  ω(τ )

For the simplicity of the presentation, let m := min{m0 (ω), m∞ (ω)} − ε and M := max{M0 (ω), M∞ (ω)} + ε. Thus, we have m ∈ (−n, ∞) and M ∈ (−∞, 0). Now, we deal with IIλ,1 . Indeed, using the fact that the Hardy–Littlewood maximal operator M satisfies the size condition (1.91) (see, for instance, [197, Remark 4.4]), together with the Minkowski integral inequality, we find that, for

6.6 Boundedness of Calderón–Zygmund Operators

477

any k ∈ Z and x ∈ Rn satisfying 2k−1 ≤ |x − ξ | < 2k , ⎧ ⎧ ⎫1 ⎫1 r r ⎬ r ⎨  |f ⎨  r ⎬ (y)| j,k,1  dy M(fj,k,1 )(x) n ⎩ ⎩ ⎭ ⎭ Rn |x − y| j ∈N

j ∈N

⎧ ⎨  |f

 

Rn

⎩

j ∈N

 ∼ ∼

j,k,1 (y)| |x − y|n

[ B(ξ,2k−2 )

⎫1 r ⎬ r ⎭

4

j ∈N |fj (y)| |x − y|n

k−2  

[

r ] 1r

4

i i−1 i=−∞ B(ξ,2 )\B(ξ,2 )

dy

dy

j ∈N |fj (y)| |x − y|n

r ] 1r

dy. (6.53)

On the other hand, for any k ∈ Z, i ∈ Z ∩ (−∞, k − 2], and x, y ∈ Rn satisfying 2k−1 ≤ |x − ξ | < 2k and 2i−1 ≤ |y − ξ | < 2i , we have |x − y| ≥ |x − ξ | − |y − ξ | > 2k−1 − 2i > 2k−2 , which, together with (6.53), implies that, for any k ∈ Z and x ∈ Rn satisfying 2k−1 ≤ |x − ξ | < 2k , ⎧ ⎨  ⎩

j ∈N

⎫1 r r ⎬ M(fj,k,1 )(x) ⎭

⎛  ⎞1   r     −nk ⎝ r⎠  2 |fj | 1B(ξ,2i )\B(ξ,2i−1)     i=−∞  j ∈N  k−2 

.

(6.54)

L1 (Rn )

In addition, by Lemma 1.1.3 and (6.52), we conclude that, for any k ∈ Z and i ∈ Z ∩ (−∞, k − 2], ω(2k−2 ) ω(2k ) ∼  ω(2i ) ω(2i )



2k−2 2i

M ∼ 2(k−i)M .

478

6 Weak Generalized Herz–Hardy Spaces

From this and (6.54), it follows that, for any k ∈ Z, ⎧  ⎫1   r ⎨   ⎬    r M(fj,k,1 ) ω(2k )  1B(ξ,2k )\B(ξ,2k−1)  ⎩  ⎭  j ∈N 

L1 (Rn )

 ω(2k )

k−2 

  2−nk 1B(ξ,2k )\B(ξ,2k−1) L1 (Rn )

i=−∞

⎛  ⎞1   r    ⎝  × |fj |r ⎠ 1B(ξ,2i )\B(ξ,2i−1 )     j ∈N 

L1 (Rn )

⎛  ⎞1   r k−2   k   ω(2 )   i r⎠ ⎝ ω(2 ∼ ) |f | 1 i )\B(ξ,2i−1 )   j B(ξ,2   ω(2i ) i=−∞  j ∈N 

L1 (Rn )

⎛  ⎞1   r k−2      (k−i)M i ⎝ r⎠  2 ω(2 )  |fj | 1B(ξ,2i )\B(ξ,2i−1)    i=−∞  j ∈N 

(6.55)

.

L1 (Rn )

Applying this, the fact that, for any λ ∈ (0, ∞),

1

4 1 {y∈Rn: { j∈N [M(fj,k,1 )(y)]r } r >λ/3}

⎫1 ⎧ ⎬r ⎨    3 r M(fj,k,1 ) ≤ , ⎭ λ⎩

(6.56)

j ∈N

Lemma 1.2.11, and the assumption M ∈ (−∞, 0), we further conclude that, for any q ∈ (0, 1] and λ ∈ (0, ∞),

IIλ,1

⎧ q ⎛ ⎫1   r   ⎬ ⎨    q   r  −1 ⎜ k ω(2 )  M(fj,k,1 ) λ ⎝ 1B(ξ,2k )\B(ξ,2k−1)  ⎩  ⎭ k∈Z  j ∈N  1  λ−1

 # k−2   k∈Z

L (Rn )

2(k−i)M ω(2i )

i=−∞

⎛  ⎞1   r    ⎝  r⎠ × |fj | 1B(ξ,2i )\B(ξ,2i−1)     j ∈N 

L1 (Rn )

⎤q ⎫ q1 ⎪ ⎬ ⎥ ⎦ ⎪ ⎭

⎞ q1 ⎟ ⎠

6.6 Boundedness of Calderón–Zygmund Operators

−1

λ

 k−2  

479

 q 2(k−i)Mq ω(2i )

k∈Z i=−∞

⎛ q ⎞1   r    ⎝  r⎠ × |fj | 1B(ξ,2i )\B(ξ,2i−1)     j ∈N  1 ∼ λ−1

×

⎧ ⎪ ⎨  ⎪ ⎩ i∈Z

∞ 

L (Rn )

⎫ q1 ⎪ ⎬ ⎪ ⎭

⎛ q ⎞1   r  q   ⎝  k r⎠ ω(2 )  |fj | 1B(ξ,2i )\B(ξ,2i−1)     j ∈N  1

L (Rn )

1 q

2(k−i)Mq

k=i+2

 ⎛ ⎞1   r     −1 ⎝ r⎠ ∼λ  |fj | (· + ξ )    ˙ 1,q  j ∈N

(6.57)

.

Kω,0 (Rn )

Moreover, using (6.56), (6.55), the Hölder inequality, and the assumption M ∈ (−∞, 0), we find that, for any q ∈ (1, ∞) and λ ∈ (0, ∞),

IIλ,1

⎧ q ⎛ ⎫1   r  ⎬  q ⎨      ⎜ r ω(2k )  M(fj,k,1 )  λ−1 ⎝ 1B(ξ,2k )\B(ξ,2k−1)  ⎩  ⎭  j ∈N  1 k∈Z  λ−1

2(k−i)M ω(2i )

i=−∞

 ⎛ ⎞1   r     ⎝ |fj |r ⎠ 1B(ξ,2i )\B(ξ,2i−1)  ×     j ∈N −1

λ

×

L1 (Rn )

⎧ # k−2 ⎨  ⎩

k∈Z

k−2  i=−∞

2

⎟ ⎠

L (Rn )

 # k−2   k∈Z

⎞ q1

2

(k−i)Mq  2

$

⎤q ⎫ q1 ⎪ ⎬ ⎥ ⎦ ⎪ ⎭

q q

i=−∞

(k−i)Mq 2



⎛ q ⎞1   r  q   ⎝  i r⎠ |fj | 1B(ξ,2i )\B(ξ,2i−1)  ω(2 )     j ∈N  1

L (Rn )

⎫ q1 ⎪ ⎬ ⎪ ⎭

480

6 Weak Generalized Herz–Hardy Spaces

∼ λ−1

×

⎧ ⎪ ⎨  ⎪ ⎩ i∈Z

∞ 

⎛ q ⎞1   r   q ⎝  i r⎠ ω(2 )  |fj | 1B(ξ,2i )\B(ξ,2i−1)     j ∈N  1

L (Rn )

1 2

(k−i)Mq 2

q

k=i+2

 ⎛ ⎞1   r     −1 ⎝ r⎠ ∼λ  |fj | (· + ξ )    ˙ 1,q  j ∈N

(6.58)

,

Kω,0 (Rn )

which, together with (6.57), then completes the estimate of IIλ,1 . We next deal with IIλ,2 . Indeed, from Lemma 6.6.10, it follows that, for any k ∈ Z and λ ∈ (0, ∞),     1 1B(ξ,2k )\B(ξ,2k−1)  4 1   n r r {y∈R : {

j∈N [M(fj,k,2 )(y)]

} >λ/3}

L1 (Rn )

      ≤ 1 4 1 {y∈Rn: { j∈N [M(fj,k,2 )(y)]r } r >λ/3}  ⎛ ⎞1   r       λ−1 ⎝ |fj,k,2 |r ⎠     j ∈N 

L1 (Rn )

L1 (Rn )

 ⎡⎛ ⎞1   r     −1 ⎢⎝ r⎠  λ ⎣ |fj | 1B(ξ,2k+1)\B(ξ,2k )      j ∈N

L1 (Rn )

⎛  ⎞1   r    ⎝  r⎠ + |fj | 1B(ξ,2k )\B(ξ,2k−1)     j ∈N 

L1 (Rn )

⎛  ⎞1   r    ⎝  r⎠ + |fj | 1B(ξ,2k−1)\B(ξ,2k−2)     j ∈N 

⎤ ⎥ ⎦.

L1 (Rn )

In addition, by Lemma 1.1.3, we conclude that, for any k ∈ Z, ω(2k+1 ) ∼ ω(2k ) ∼ ω(2k−1 ).

(6.59)

6.6 Boundedness of Calderón–Zygmund Operators

481

Therefore, using (6.59), we find that, for any λ ∈ (0, ∞),

IIλ,2

⎛ ⎧ ⎛ q ⎞1   r ⎨   ⎜⎪ q   ⎝  −1 ⎜ k+1 r⎠ ω(2 )  λ ⎜ |fj | 1B(ξ,2k )\B(ξ,2k−1)  ⎪   ⎝⎩  j ∈N  1 k∈Z

⎫ q1 ⎪ ⎬

L (Rn )

+

+

⎧ ⎪ ⎨  ⎪ ⎩k∈Z

⎛ q ⎞1   r  q   ⎝  k r⎠ |fj | 1B(ξ,2k )\B(ξ,2k−1)  ω(2 )     j ∈N  1

⎧ ⎪ ⎨  ⎪ ⎩k∈Z

⎫ q1 ⎪ ⎬

L (Rn )

⎪ ⎭

⎛ q ⎞1   r  q   ⎝  k−1 r⎠ ω(2 )  |fj | 1B(ξ,2k )\B(ξ,2k−1)     j ∈N  1

L (Rn )

 ⎛ ⎞1   r     −1 ⎝ r⎠ ∼λ  |fj | (· + ξ )    ˙ 1,q  j ∈N

⎪ ⎭

⎞ ⎫ q1 ⎪ ⎬ ⎟ ⎟ ⎟ ⎪ ⎭ ⎠

(6.60)

,

Kω,0 (Rn )

which completes the estimate of IIλ,2 . Finally, we estimate IIλ,3 . Indeed, applying the fact that the Hardy–Littlewood maximal operator M satisfies the size condition (1.91) (see, for instance, [197, Remark 4.4]), together with the Minkowski integral inequality, we conclude that, for any k ∈ Z and x ∈ Rn satisfying 2k−1 ≤ |x − ξ | < 2k , ⎧ ⎨  ⎩

j ∈N

⎫1 ⎧ ⎫1 r r ⎬ r ⎨  |f r ⎬ j,k,3 (y)| M(fj,k,3 )(x)  dy n ⎩ ⎭ ⎭ Rn |x − y| j ∈N

⎧ ⎨  |f

 

Rn

⎩

j ∈N

 ∼ ∼

j,k,3 (y)| |x − y|n

[ [B(ξ,2k+1 )]

⎫1 r ⎬ r ⎭

4

j ∈N |fj (y)| |x − y|n

∞   i i−1 i=k+2 B(ξ,2 )\B(ξ,2 )

[

4

dy r ] 1r

dy

j ∈N |fj (y)| |x − y|n

r ] 1r

dy. (6.61)

482

6 Weak Generalized Herz–Hardy Spaces

On the other hand, for any k ∈ Z, i ∈ Z ∩ [k + 2, ∞), and x, y ∈ Rn satisfying 2k−1 ≤ |x − ξ | < 2k and 2i−1 ≤ |y − ξ | < 2i , we have |x − y| ≥ |y − ξ | − |x − ξ | > 2i−1 − 2k ≥ 2i−2 , which, together with (6.61), implies that, for any k ∈ Z and x ∈ Rn satisfying 2k−1 ≤ |x − ξ | < 2k , ⎧ ⎨  ⎩

j ∈N

⎫1 r r ⎬ M(fj,k,3 )(x) ⎭

⎛  ⎞1   r     −ni ⎝ r⎠  2  |fj | 1B(ξ,2i )\B(ξ,2i−1)    i=k+2  j ∈N  ∞ 

(6.62)

.

L1 (Rn )

In addition, from Lemma 1.1.3 and (6.51), we deduce that, for any k ∈ Z and i ∈ Z ∩ [k + 2, ∞), ω(2k+2 ) ω(2k ) ∼  i ω(2 ) ω(2i )



2k+2 2i

m ∼ 2(k−i)m .

Applying this and (6.62), we conclude that, for any k ∈ Z, ⎧  ⎫1   ⎨   ⎬r   r k  M(fj,k,3 ) ω(2 )  1B(ξ,2k )\B(ξ,2k−1)  ⎩  ⎭  j ∈N 

L1 (Rn )

 ω(2k )

∞ 

  2−ni 1B(ξ,2k )\B(ξ,2k−1) L1 (Rn )

i=k+2

 ⎛ ⎞1   r     ⎝ r⎠ × |fj | 1B(ξ,2i )\B(ξ,2i−1)      j ∈N

L1 (Rn )

⎛  ⎞1   r ∞   k   ω(2 )   i r⎠ ⎝ ω(2 ∼ 2(k−i)n ) |f | 1 i )\B(ξ,2i−1 )   j B(ξ,2 i   ω(2 ) i=k+2  j ∈N  ⎛  ⎞1   r ∞        2(k−i)(m+n) ω(2i ) ⎝ |fj |r ⎠ 1B(ξ,2i )\B(ξ,2i−1)    i=k+2  j ∈N 

L1 (Rn )

. L1 (Rn )

6.6 Boundedness of Calderón–Zygmund Operators

483

From this, the fact that, for any λ ∈ (0, ∞),

1

4 1 {y∈Rn : { j∈N [M(fj,k,3 )(y)]r } r >λ/3}

⎫1 ⎧ r r ⎬ 3 ⎨  M(fj,k,3 ) ≤ ⎭ λ⎩ j ∈N

and an argument similar to that used in the estimations of both (6.57) and (6.58), it follows that, for any λ ∈ (0, ∞),

IIλ,3

 ⎛ ⎞1   r     −1 ⎝ r⎠ λ  |fj | (· + ξ )    ˙ 1,q  j ∈N

(6.63)

.

Kω,0 (Rn )

This finishes the estimate of IIλ,3 . Combining (6.50), (6.57), (6.58), (6.60), and (6.63), we further conclude that, for any λ ∈ (0, ∞), ⎧  ⎫1   r ⎨    ⎬ r   M(fj ) λ (· + ξ ) ⎩  ⎭  j ∈N  ˙ 1,q

Kω,0 (Rn )

⎛  ⎞1   r    ⎝  r⎠  |fj | (· + ξ )    j ∈N  ˙ 1,q

.

Kω,0 (Rn )

(6.64) Thus, from the arbitrariness of ξ , it follows that ⎧ ⎫1     ⎨  ⎬r     r M(fj )   ⎩ ⎭    j ∈N

n ˙ 1,q WK ω (R )

⎛ ⎞1   r    ⎝ r⎠   |fj |     j ∈N  ˙ 1,q

.

Kω (Rn )

In particular, letting ξ := 0 in (6.64), we have ⎧ ⎫1    ⎬r  ⎨   r   M(fj )   ⎭  ⎩  j ∈N 

˙ (Rn ) WK ω,0 1,q

⎛ ⎞1   r    ⎝ r⎠   |fj |     j ∈N  ˙ 1,q

.

Kω,0 (Rn )

Therefore, both (6.48) and (6.49) hold true in this case, and hence we complete the proof of Proposition 6.6.11.   In addition, to show Theorem 6.6.9, we also need the following lemma about the convexification of weak local generalized Herz spaces, which is just a simple corollary of both [278, Remark 2.14] and Lemma 1.3.1; we omit the details.

484

6 Weak Generalized Herz–Hardy Spaces

Lemma 6.6.12 Let p, q, s ∈ (0, ∞) and ω ∈ M(R+ ). Then 

p,q W K˙ ω,0 (Rn )

1/s

p/s,q/s = W K˙ ωs ,0 (Rn )

with the same quasi-norms. Then we show Theorem 6.6.9. Proof of Theorem 6.6.9 Let all the symbols be as in the present theorem. Then, combining the assumption m0 (ω) ∈ (− pn , ∞) and Theorems 1.2.42 and 1.4.1, p,q we find that the local generalized Herz space K˙ ω,0 (Rn ) under consideration is a BQBF space having an absolutely continuous quasi-norm. Thus, to complete the proof of the present theorem, we only need to show that all the assumptions of p,q Proposition 6.6.3 hold true for K˙ ω,0 (Rn ). Indeed, let    n s ∈ 0, min 1, p, q, max{M0 (ω), M∞ (ω)} + n/p and   θ ∈ 0, min s,

n n+d

 (6.65)

.

Then, applying Lemma 4.3.25, we conclude that, for any {fj }j ∈N ⊂ L1loc (Rn ), ⎧ ⎫1   s ⎨   s⎬  .   M(θ) fj   ⎩ ⎭   ˙ p,q  j ∈N

Kω,0 (Rn )

⎛ ⎞1       s s ⎝ fj  ⎠       j ∈N  ˙ p,q

.

Kω,0 (Rn )

This implies that Assumption 1.2.29 holds true with the above θ and s. Next, we prove that Assumption 1.2.33 holds true with the above s and some r0 ∈ (1, ∞). Indeed, let   r0 ∈ max 1, p,

  n ,∞ . min{m0 (ω), m∞ (ω)} + n/p

Then, from Lemma 1.8.6 with r := r0 , we deduce that [K˙ ω,0 (Rn )]1/s is a BBF space and, for any f ∈ L1loc (Rn ), p,q

   ((r0 /s) )  (f ) M

˙ (Rn )]1/s ) ([K ω,0 p,q

 f ([K˙ p,q (Rn )]1/s ) ,

which implies that Assumption 1.2.33 holds true.

ω,0

6.6 Boundedness of Calderón–Zygmund Operators

485

Let   r1 ∈ 0, min p,

n max{M0 (ω), M∞ (ω)} + n/p

 .

Then, using Lemma 6.1.6 with r := r1 , we find that, for any f ∈ L1loc (Rn ), M(f )[W K˙ p,q (Rn )]1/r1  f [W K˙ p,q (Rn )]1/r1 , ω,0

ω,0

which implies that, for the above r1 , the Hardy–Littlewood maximal operator M is p,q bounded on [W K˙ ω,0 (Rn )]1/r1 . p,q Finally, we prove that (6.26) holds true with X := K˙ ω,0 (Rn ). Indeed, from the assumptions   n min{m0 (ω), m∞ (ω)} ∈ − , ∞ p and   n max{M0 (ω), M∞ (ω)} ∈ −∞, n − + d + δ , p and Lemma 1.1.6, it follows that



! n n min m0 ω n+d+δ , m∞ ω n+d+δ =

n n min{m0 (ω), m∞ (ω)} > − n+d +δ (n + d + δ)p/n

(6.66)

and



! n n max M0 ω n+d+δ , M∞ ω n+d+δ n max{M0 (ω), M∞ (ω)} n+d +δ n n = α}X , where the implicit positive constants are independent of both f and α. This, together with the proof of [242, Theorem 4.5], further implies that, for any t ∈ (0, ∞), . t −θ K t, f ; HX (Rn ), L∞ (Rn )  M (f )X1/(1−θ )  M (f )W X1/(1−θ ) ∼ f W H

X1/(1−θ )

(Rn ) .

Using this and Definition 6.7.1, we find that f θ,∞  f W H

X1/(1−θ )

(Rn )

< ∞.

This further implies that f ∈ (HX (Rn ), L∞ (Rn ))θ,∞ and then finishes the proof of (6.73). Combining this and (6.72), we conclude that . HX (Rn ), L∞ (Rn ) θ,∞ = W HX1/(1−θ ) (Rn ) and hence complete the proof of Proposition 6.7.7.

 

Via the improved real interpolation result obtained above, we now prove Theorem 6.7.6. Theorem 6.7.6 Let all the symbols be as in the present theorem. Then, by the assumptions m0 (ω) ∈ (− pn , ∞) and M∞ (ω) ∈ (−∞, 0), and Theorem 1.2.44, p,q we find that the global generalized Herz space K˙ ω (Rn ) under consideration is a BBF space. Therefore, in order to finish the proof of the present theorem, we only need to show that the assumptions (i) through (viii) of Proposition 6.7.7 hold true p,q for K˙ ω (Rn ). Indeed, let   n p− := 1, p, q, . max{M0 (ω), M∞ (ω)} + n/p

6.7 Real Interpolations

499

Then, for any given θ ∈ (0, p− ) and u ∈ (1, ∞), from Lemma 4.3.10 with r := θ , it follows that, for any {fj }j ∈N ⊂ L1loc (Rn ), ⎧ ⎫1    ⎨  - . ⎬ u    u M fj   ⎩ ⎭    j ∈N

n 1/θ ˙ p,q [K ω (R )]

⎛ ⎞1       u ⎝ u⎠    fj      j ∈N 

.

n 1/θ ˙ p,q [K ω (R )]

This implies that Proposition 6.7.7(i) holds true with the above p− . Next, we prove that Proposition 6.7.7(ii) holds true with some θ0 ∈ (1, ∞). Indeed, let θ0 ∈ (max{1, p, q}, ∞). Combining this, the reverse Minkowski inequality, and Lemma 1.3.2 with s := θ0 , we conclude that, for any {fj }j ∈N ⊂ M (Rn ),          fj  ˙ p,q n 1/θ ≤  fj  ,   [Kω (R )] 0  ˙ p,q n 1/θ j ∈N j ∈N [Kω (R )]

0

p,q which further implies that the Herz space K˙ ω (Rn ) under consideration is θ0 concave and hence Proposition 6.7.7(ii) holds true. Let    n r ∈ 0, min p, . max{M0 (ω), M∞ (ω)} + n/p

Then, from Lemma 6.1.11, it follows that, for any f ∈ L1loc (Rn ), M (f )[W K˙ p,q n 1/r  f [W K n 1/r . ˙ p,q ω (R )] ω (R )] This implies that the Hardy–Littlewood maximal operator M is bounded on the p,q weak Herz space [W K˙ ω (Rn )]1/r and hence further implies that, for the above r, Proposition 6.7.7(iii) holds true. p,q Next, we prove that, for any s ∈ (0, p− ), [K˙ ω (Rn )]1/s is a BBF space. Indeed, for any s ∈ (0, p− ), by the assumptions m0 (ω) ∈ (− pn , ∞) and M∞ (ω) ∈ (−∞, 0), and Lemma 1.1.6, we find that - . n m0 ωs = sm0 (ω) > − p/s and - . M∞ ωs = sM∞ (ω) < 0, which, combined with the assumptions p/s, q/s ∈ (1, ∞), Theorem 1.2.48 with p, q, and ω therein replaced, respectively, by p/s, q/s, and ωs , and Lemma 1.3.2,

500

6 Weak Generalized Herz–Hardy Spaces

p,q further imply that [K˙ ω (Rn )]1/s is a BBF space. This finishes the proof that p,q n K˙ ω (R ) satisfies Proposition 6.7.7(iv). Finally, we show that there exists a linear space Y ⊂ M (Rn ), an s0 ∈ (0, p− ), and an r0 ∈ (s0 , ∞) such that (v) through (viii) of Proposition 6.7.7 hold true. To this end, let s0 ∈ (0, p− ) and

  r0 ∈ max 1, p,

  n ,∞ . min{m0 (ω), m∞ (ω)} + n/p

Then, repeating an argument similar to that used in the proof of Theorem 4.8.16 with η therein replaced by (r0 /s0 ) , we conclude that the following four statements hold true: (i)  ·  ˙ (p/s0 ) ,(q/s0 ) B1/ωs0

(Rn )

satisfies Definition 1.2.13(ii);

(p/s ) ,(q/s )

0 (ii) 1B(0,1) ∈ B˙1/ωs00 (Rn ); (iii) for any f ∈ M (Rn ),





f [K˙ p,q n 1/s ∼ sup fgL1 (Rn ) : g ˙ (p/s0 ) ,(q/s0 ) ω (R )] 0 B1/ωs0

(Rn )

=1

with the positive equivalence constants independent of f ; (iv) for any f ∈ L1loc (Rn ),    ((r0/s0 ) )  (f ) ˙ (p/s0 ) ,(q/s0 ) M B1/ωs0

(Rn )

 f  ˙ (p/s0 ) ,(q/s0 ) B1/ωs0

(Rn )

.

These imply that (v) through (viii) of Proposition 6.7.7 hold true with the above s0 (p/s ) ,(q/s0 ) and r0 , and Y := B˙1/ωs00 (Rn ). Therefore, the assumptions (i) through (viii) p,q of Proposition 6.7.7 hold true for the global generalized Herz space K˙ ω (Rn ) under consideration. This then implies that, for any θ ∈ (0, 1), . p/(1−θ),q/(1−θ) n H K˙ ωp,q (Rn ), L∞ (Rn ) θ,∞ = W H K˙ ω1−θ (R ), which completes the proof of Theorem 6.7.6.

 

As an application, by Theorem 6.7.6, Remark 4.0.20(ii), and Remark 6.0.25, we conclude the following real interpolation theorem which shows that the real p,q interpolation space between H M ω (Rn ) and L∞ (Rn ) is just the weak generalized p/(1−θ),q/(1−θ) n (R ); we omit the details. Hardy–Morrey space W H M ω1−θ Corollary 6.7.11 Let p, q, and ω be as in Corollary 6.7.5. Then, for any θ ∈ (0, 1), . p/(1−θ),q/(1−θ) n n ∞ n H M p,q (R ). ω (R ), L (R ) θ,∞ = W H M ω1−θ

Chapter 7

Inhomogeneous Generalized Herz Spaces and Inhomogeneous Block Spaces

The targets of this chapter are threefold. The first one is to introduce the inhomogeneous counterparts of generalized Herz spaces as in Definition 1.2.1 and then find both dual spaces and associate spaces of these inhomogeneous local generalized Herz spaces. The second one is to introduce inhomogeneous block spaces and prove the duality between inhomogeneous global generalized Herz spaces and these inhomogeneous block spaces. Moreover, we also establish the boundedness of some important operators on these block spaces. The last one is, as applications, to investigate the boundedness and the compactness characterizations of commutators on inhomogeneous generalized Herz spaces.

7.1 Inhomogeneous Generalized Herz Spaces In this section, we first introduce the inhomogeneous counterparts of local and global generalized Herz spaces and investigate their basic properties. Indeed, we obtain the relation between these Herz spaces and the corresponding homogeneous generalized Herz spaces. Then, under some reasonable and sharp assumptions, we show that inhomogeneous generalized Herz spaces are ball (quasi-)Banach function spaces. Furthermore, we show the convexity and the absolutely continuity of quasinorms of these Herz spaces as well as establish a boundedness criterion of sublinear operators and the Fefferman–Stein vector-valued inequalities on these Herz spaces. Finally, we find both the dual and the associate space of the inhomogeneous local p,q generalized Herz space Kω,0 (Rn ), and establish the extrapolation theorems of both the inhomogeneous local and the inhomogeneous global generalized Herz spaces. We begin this section with the following definitions of inhomogeneous generalized Herz spaces.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 Y. Li et al., Real-Variable Theory of Hardy Spaces Associated with Generalized Herz Spaces of Rafeiro and Samko, Lecture Notes in Mathematics 2320, https://doi.org/10.1007/978-981-19-6788-7_7

501

502

7 Inhomogeneous Generalized Herz Spaces and Inhomogeneous Block Spaces

Definition 7.1.1 Let p, q ∈ (0, ∞] and ω ∈ M(R+ ). p,q

(i) The inhomogeneous local generalized Herz space Kω,0 (Rn ) is defined to be the p set of all the f ∈ L loc (Rn ) such that  1 q q    q q ω(2k ) f 1B(0,2k )\B(0,2k−1 ) Lp (Rn ) f Kp,q (Rn ) := f 1B(0,1) Lp (Rn ) + ω,0

k∈N

is finite. p,q (ii) The inhomogeneous global generalized Herz space Kω (Rn ) is defined to be p n the set of all the f ∈ L loc (R ) such that   f 1B(ξ,1)q p n f Kp,q n : = sup L (R ) ω (R ) ξ ∈Rn

+



q  q ω(2 ) f 1B(ξ,2k )\B(ξ,2k−1) Lp (Rn )

1 q

k

k∈N

is finite. Remark 7.1.2 (i) Obviously, using Definition 7.1.1, we conclude that, for any f ∈ M (Rn ), f Kp,q n = sup f (· + ξ )Kp,q (Rn ) . ω (R ) ξ ∈Rn

ω,0

p,q

(ii) Observe that the inhomogeneous local generalized Herz space Kω,0 (Rn ) is always nontrivial. Indeed, by Definition 7.1.1(i), we easily know that 1B(0,1) ∈ p,q Kω,0 (Rn ). However, as is showed in Theorem 7.1.5 later, when ω ∈ M(R+ ) satisfying that m∞ (ω) ∈ (0, ∞), then   Kωp,q (Rn ) = f ∈ M (Rn ) : f = 0 almost everywhere in Rn , p,q

and hence the inhomogeneous global generalized Herz space Kω (Rn ) is trivial in this case. (iii) Let p = q ∈ (0, ∞] and ω(t) := 1 for any t ∈ (0, ∞). Then, in this case, p,q

Kω,0 (Rn ) = Kωp,q (Rn ) = Lp (Rn ) with the same quasi-norms. (iv) In Definition 7.1.1(i), for any given α ∈ R and for any t ∈ (0, ∞), let ω(t) := t α . Then, in this case, the inhomogeneous local generalized p,q Herz space Kω,0 (Rn ) coincides with the classical inhomogeneous Herz space α,q Kp (Rn ), which was originally introduced in [163, Definition 1.1(b)] (see also

7.1 Inhomogeneous Generalized Herz Spaces

503

[175, Chapter 1]), with the same quasi-norms. In particular, when p ∈ (1, ∞), q = 1, and α = n(1 − p1 ), the inhomogeneous local generalized Herz space p,q Kω,0 (Rn ) is just the Beurling algebra Ap (Rn ) which was originally introduced in [13] (see also [86, Definition 1.1]). (v) We should point out that, even when ω(t) := t α for any t ∈ (0, ∞) and for p,q any given α ∈ R, the inhomogeneous global generalized Herz space Kω (Rn ) is also new. The following conclusion gives the relation between the local generalized Herz space and its inhomogeneous counterpart. Theorem 7.1.3 Let p, q ∈ (0, ∞] and ω ∈ M(R+ ) satisfy m0 (ω) ∈ (0, ∞) and m∞ (ω) ∈ (0, ∞). Then Kω,0 (Rn ) = K˙ ω,0 (Rn ) ∩ Lp (Rn ). p,q

p,q

Moreover, there exist two positive constants C1 and C2 such that, for any f ∈ p,q Kω,0 (Rn ), C1 f Kp,q (Rn ) ≤ f K˙ p,q (Rn ) + f Lp (Rn ) ≤ C2 f Kp,q (Rn ) . ω,0

ω,0

ω,0

Proof Let all the symbols be as in the present theorem. We first show that p,q p,q Kω,0 (Rn ) ∩ Lp (Rn ) ⊂ K˙ ω,0 (Rn ).

(7.1)

p,q

To this end, let f ∈ Kω,0 (Rn ) ∩ Lp (Rn ). Then, applying both Definitions 1.2.1 and 7.1.1, we find that f Kp,q (Rn ) ω,0

1  q q     q k  f 1B(0,1)Lp (Rn ) + ω(2 ) f 1B(0,2k )\B(0,2k−1) Lp (Rn ) k∈N

 f Lp (Rn ) + f K˙ p,q (Rn ) < ∞. ω,0

(7.2)

This finishes the proof of (7.1). p,q p,q Conversely, we prove Kω,0 (Rn ) ⊂ K˙ ω,0 (Rn ) ∩ Lp (Rn ). Indeed, from Lemma 1.1.12, it follows that, for any k ∈ N, ω(2k )  2k[m∞ (ω)−ε]

504

7 Inhomogeneous Generalized Herz Spaces and Inhomogeneous Block Spaces

and, for any k ∈ Z \ N, ω(2k )  2k[m0 (ω)−ε] , where ε ∈ (0, min{m0 (ω), m∞ (ω)}) is a fixed positive constant. p,q We now claim that, for any f ∈ Kω,0 (Rn ),     f 1[B(0,1)] 

Lp (Rn )

 f Kp,q (Rn ) .

(7.3)

ω,0

Indeed, by the assumption that, for any k ∈ N, ω(2k )  2k[m∞ (ω)−ε] , we conclude p,q that, for any f ∈ Kω,0 (Rn ),  p   f 1[B(0,1)]  p =

 k∈N





L (Rn )

ω(2k )

−p 

ω(2k )

p  p f 1  B(0,2k )\B(0,2k−1) Lp (Rn )

 p  p 2kp[−m∞ (ω)+ε] ω(2k ) f 1B(0,2k )\B(0,2k−1) Lp (Rn ) ,

(7.4)

k∈N

which, together with Lemma 1.2.11 and −m∞ (ω) + ε ∈ (−∞, 0), further implies that, for any q ∈ (0, p],  p   f 1[B(0,1)]  p

L (Rn )



 

2

kq[−m∞ (ω)+ε]



q  q ω(2 ) f 1B(0,2k )\B(0,2k−1) Lp (Rn )

p q

k

k∈N



 

q  q ω(2k ) f 1B(0,2k )\B(0,2k−1) Lp (Rn )

p q

k∈N p

 f Kp,q (Rn ) .

(7.5)

ω,0

This finishes the proof of (7.3) when q ∈ (0, p]. On the other hand, applying (7.4), the Hölder inequality, and the assumption −m∞ (ω) + ε ∈ (−∞, 0), we find that, p,q for any q ∈ (p, ∞) and f ∈ Kω,0 (Rn ),  p   f 1[B(0,1)]  p

L (Rn )



  k∈N

2

kp[−m∞ (ω)+ε](q/p)

[(q/p) ]−1

7.1 Inhomogeneous Generalized Herz Spaces

 ×



505

q  q ω(2 ) f 1B(0,2k )\B(0,2k−1) Lp (Rn )

p q

k

k∈N

∼

 

q  q ω(2k ) f 1B(0,2k )\B(0,2k−1) Lp (Rn )

p q

p

 f Kp,q (Rn ) . ω,0

k∈N

Combining this and (7.5), we further conclude that (7.3) holds true for any q ∈ (0, ∞) and hence complete the prove of the above claim. Thus, from p,q Definition 7.1.1(i), we deduce that, for any f ∈ Kω,0 (Rn ),  p  p   p f Lp (Rn ) = f 1B(0,1)Lp (Rn ) + f 1[B(0,1)]  p

L (Rn )

p

 f Kp,q (Rn ) < ∞, ω,0

(7.6) p,q

which further implies that Kω,0 (Rn ) ⊂ Lp (Rn ). p,q p,q p,q We next show that Kω,0 (Rn ) ⊂ K˙ ω,0 (Rn ). To achieve this, let f ∈ Kω,0 (Rn ). Then, by the assumption that, for any k ∈ Z \ N, ω(2k )  2k[m0 (ω)−ε] , Lemma 1.2.11, and the assumption m0 (ω) − ε ∈ (0, ∞), we conclude that, for any q ∈ [p, ∞),   k∈Z\N



ω(2k )



q  q f 1  B(0,2k )\B(0,2k−1) Lp (Rn )

 q 2kq[m0 (ω)−ε] f 1B(0,2k )\B(0,2k−1) Lp (Rn )

k∈Z\N



⎧ ⎨  ⎩

 p 2kp[m0 (ω)−ε] f 1B(0,2k )\B(0,2k−1) Lp (Rn )

k∈Z\N

⎡

  p f 1  ⎣ B(0,2k )\B(0,2k−1) p L

⎫q ⎬p ⎭

⎤q

p

  ⎦ ∼ f 1B(0,1)q p n . L (R ) (Rn )

(7.7)

k∈Z\N

On the other hand, using the assumption that, for any k ∈ Z \ N, ω(2k )  2k[m0 (ω)−ε] , the Hölder inequality, and the assumption m0 (ω) − ε ∈ (0, ∞) again, we find that, for any q ∈ (0, p),   k∈Z\N



ω(2k )



k∈Z\N

q  f 1

q  B(0,2k )\B(0,2k−1) Lp (Rn )

 q 2kq[m0 (ω)−ε] f 1B(0,2k )\B(0,2k−1) Lp (Rn )

506

7 Inhomogeneous Generalized Herz Spaces and Inhomogeneous Block Spaces



⎧ ⎨  ⎩

2kq[m0

(ω)−ε](p/q)

k∈Z\N

⎫(p/q)−1 ⎡ ⎤q p ⎬   p   ⎣ f 1B(0,2k )\B(0,2k−1) Lp (Rn ) ⎦ ⎭ k∈Z\N

⎤q p    p q   ⎦ ⎣ f 1B(0,2k )\B(0,2k−1) Lp (Rn ) ∼ ∼ f 1B(0,1)Lp (Rn ) , ⎡

k∈Z\N

which, combined with (7.7), further implies that f K˙ p,q (Rn ) ω,0

=

⎧ ⎨   ⎩

k∈Z\N

⎫1 ⎬q q    q k   f 1B(0,2k )\B(0,2k−1) Lp (Rn ) + ω(2 ) ··· ⎭ k∈N

  q  f 1B(0,1) p L

+ (Rn )



q  q ω(2 ) f 1B(0,2k )\B(0,2k−1) Lp (Rn )

1 q

k

k∈N

∼ f Kp,q (Rn ) < ∞.

(7.8)

ω,0

This implies that Kω,0 (Rn ) ⊂ K˙ ω,0 (Rn ). Therefore, we complete the proof that p,q

p,q

Kω,0 (Rn ) = K˙ ω,0 (Rn ) ∩ Lp (Rn ). p,q

p,q

p,q

Moreover, from (7.2), (7.6), and (7.8), it follows that, for any f ∈ Kω,0 (Rn ), f Kp,q (Rn ) ∼ f K˙ p,q (Rn ) + f Lp (Rn ) ω,0

ω,0

with positive equivalence constants independent of f . This finishes the proof of Theorem 7.1.3.   Remark 7.1.4 We should point out that, in Theorem 7.1.3, when ω(t) := t α for any t ∈ (0, ∞) and for any given α ∈ R, Theorem 7.1.3 goes back to [175, Proposition 1.1.2]. For the triviality of inhomogeneous global generalized Herz spaces, we have the following conclusion which shows that, under some assumptions, the Herz space p,q Kω (Rn ) is trivial. Theorem 7.1.5 Let p, q ∈ (0, ∞] and ω ∈ M(R+ ) satisfy m∞ (ω) ∈ (0, ∞). Then   Kωp,q (Rn ) = f ∈ M (Rn ) : f (x) = 0 for almost every x ∈ Rn .

7.1 Inhomogeneous Generalized Herz Spaces

507

Proof Let all the symbols be as in the present theorem and f be a measurable function on Rn such that    x ∈ Rn : |f (x)| > 0  = 0. Then, repeating an argument similar to that used in the proof of Theorem 1.2.7 p,q p,q with K˙ ω (Rn ) and Definition 1.2.1 therein replaced, respectively, by Kω (Rn ) and Definition 7.1.1, we obtain f Kp,q n = ∞. ω (R )  

This then finishes the proof of Theorem 7.1.5.

Remark 7.1.6 Let p, q ∈ (0, ∞] and ω ∈ M(R+ ) be such that m∞ (ω) ∈ (0, ∞). Then, combining Theorems 1.2.7 and 7.1.5, we know that, in this case, K˙ ωp,q (Rn ) = Kωp,q (Rn ). p,q

Now, we show that the inhomogeneous local generalized Herz space Kω,0 (Rn ) is a ball quasi-Banach function space. Theorem 7.1.7 Let p, q ∈ (0, ∞] and ω ∈ M(R+ ). Then the inhomogeneous local p,q generalized Herz space Kω,0 (Rn ) is a ball quasi-Banach function space. Proof Let p, q ∈ (0, ∞] and ω ∈ M(R+ ). Obviously, the inhomogeneous local p,q generalized Herz space Kω,0 (Rn ) is a quasi-normed linear space satisfying both (i) and (ii) of Definition 1.2.13. Moreover, Definition 1.2.13(iii) is a simple corollary of the monotone convergence theorem. Therefore, using Proposition 1.2.36, we further p,q p,q find that Kω,0 (Rn ) is complete and hence Kω,0 (Rn ) is a quasi-Banach space. Thus, to finish the proof of the present theorem, it remains to show that the inhop,q mogeneous local generalized Herz space Kω,0 (Rn ) satisfies Definition 1.2.13(iv). To this end, let B(x0 , r) ∈ B with x0 ∈ Rn and r ∈ (0, ∞). Then, for any k ∈ N ∩ (ln(r + |x0 |)/ ln 2 + 1, ∞), we have 2k−1 > r + |x0 |. This implies that, for any x ∈ B(x0 , r), |x| ≤ |x − x0 | + |x0 | < r + |x0 | < 2k−1 . Therefore, x ∈ B(0, 2k−1 ) and hence B(x0 , r) ⊂ B(0, 2k−1 ). By this, we further conclude that   B(x0 , r) ∩ B(0, 2k ) \ B(0, 2k−1 ) = ∅.

508

7 Inhomogeneous Generalized Herz Spaces and Inhomogeneous Block Spaces

Combining this and Definition 7.1.1(i), we find that   1B(x ,r) q p,q n 0 K (R ) ω,0

 q = 1B(x0,r) 1B(0,1)Lp (Rn )  q   q + ω(2k ) 1B(x0,r) 1B(0,2k )\B(0,2k−1) Lp (Rn ) k∈N∩[1,

ln(r+|x0 |) ] ln 2

< ∞, p,q

which implies that 1B(x0 ,r) ∈ Kω,0 (Rn ) and hence finishes the proof of Theorem 7.1.7.   However, the following example shows that inhomogeneous global generalized Herz spaces may not be ball quasi-Banach function spaces. Example 7.1.8 Let p, q ∈ (0, ∞], α1 ∈ R, α2 ∈ [0, ∞), and  ω(t) :=

t α1 (1 − ln t) when t ∈ (0, 1], t α2 (1 + ln t) when t ∈ (1, ∞).

p,q

Then 1B(0,1) ∈ / Kω (Rn ) and hence, in this case, the inhomogeneous global p,q generalized Herz space Kω (Rn ) is not a ball quasi-Banach function space. Proof Let all the symbols be as in the present example and, for any k ∈ N, ξk ∈ Rn satisfy |ξk | = 2k + 1. Then, by the proof of Example 1.2.41, we find that, for any k ∈ N, B(0, 1) ⊂ B(ξk , 2k+1 ) \ B(ξk , 2k ). From this, Remark 7.1.2(i), and Definition 7.1.1(i), we deduce that       1B(0,1) p,q n ≥ 1B(0,1)(· + ξk ) p,q n = ω(2k+1 ) 1B(0,1) p n K (R ) K (R ) L (R ) ω

ω,0

∼2

(k+1)α2

[1 + (k + 1) ln 2] → ∞ p,q

as k → ∞. This further implies that 1B(0,1) ∈ / Kω (Rn ) and hence the p,q inhomogeneous global generalized Herz space Kω (Rn ) under consideration is not a BQBF space. This then finishes the proof of Example 7.1.8.   Under a reasonable and sharp assumption, we next prove that the inhomogeneous p,q global generalized Herz space Kω (Rn ) is also a ball quasi-Banach function space. Namely, the following conclusion holds true.

7.1 Inhomogeneous Generalized Herz Spaces

509

Theorem 7.1.9 Let p, q ∈ (0, ∞] and ω ∈ M(R+ ) satisfy M∞ (ω) ∈ (−∞, 0). p,q Then the inhomogeneous global generalized Herz space Kω (Rn ) is a ball quasiBanach function space. Proof Let p, q ∈ (0, ∞] and ω ∈ M(R+ ) with M∞ (ω) ∈ (−∞, 0). Obviously, p,q the inhomogeneous global generalized Herz space Kω (Rn ) is a quasi-normed linear space satisfying both (i) and (ii) of Definition 1.2.13. We now show that p,q Definition 1.2.13(iii) holds true for Kω (Rn ). Indeed, for any given {fm }m∈N ⊂ n n M (R ) and f ∈ M (R ) satisfying 0 ≤ fm ↑ f almost everywhere as m → ∞, and for any given α ∈ (0, f Kp,q n ), from Remark 7.1.2(i), we deduce that there ω (R ) exists a ξ ∈ Rn such that f (· + ξ )Kp,q (Rn ) > α. ω,0

Using this and the monotone convergence theorem, we conclude that there exists an N ∈ N such that, for any m ∈ N ∩ (N, ∞), fm (· + ξ )Kp,q (Rn ) > α. ω,0

By this and Remark 7.1.2(i) again, we find that, for any m ∈ N ∩ (N, ∞), α < sup fm (· + ξ )Kp,q (Rn ) = fm Kp,q n . ω (R ) ω,0

ξ ∈Rn

Letting α → f Kp,q n and m → ∞, we have ω (R ) f Kp,q n ≤ lim fm Kp,q (Rn ) . ω (R ) ω m→∞

On the other hand, it is easy to show that lim fm Kp,q n ≤ f Kp,q (Rn ) . ω (R ) ω

m→∞

Therefore, we conclude that fm Kp,q n ↑ f Kp,q (Rn ) ω (R ) ω p,q

as m → ∞ and hence Definition 1.2.13(iii) holds true for Kω (Rn ). From this and Proposition 1.2.36, it follows that the inhomogeneous global generalized Herz space p,q Kω (Rn ) is quasi-Banach space. p,q Next, we prove that Definition 1.2.13(iv) holds true for Kω (Rn ). Indeed, by Lemma 1.1.12, we find that, for any k ∈ N, ω(2k )  2k[M∞ (ω)+ε] ,

510

7 Inhomogeneous Generalized Herz Spaces and Inhomogeneous Block Spaces

where ε ∈ (0, −M∞ (ω)) is a fixed positive constant. Combining this, Definition 7.1.1(i), and the assumption M∞ (ω) + ε ∈ (−∞, 0), we further conclude that, for any B(x0 , r) ∈ B with x0 ∈ Rn and r ∈ (0, ∞), and ξ ∈ Rn ,   1B(x ,r) (· + ξ )q p,q n 0 K (R ) ω,0

 q = 1B(x0,r) 1B(ξ,1)Lp (Rn ) q   q + ω(2k ) 1B(x0,r) 1B(ξ,2k )\B(ξ,2k−1) Lp (Rn ) k∈N

   q  kq[M (ω)+ε] ∞ < ∞.  1B(x0,r) Lp (Rn ) 1 + 2 k∈N

This, together with Remark 7.1.2(i), further implies that, for any x0 ∈ Rn and r ∈ (0, ∞),   1B(x ,r)  p,q n  1. 0 K (R ) ω

p,q

Therefore, 1B(x0 ,r) ∈ Kω (Rn ) for any x0 ∈ Rn and r ∈ (0, ∞). This implies p,q that Definition 1.2.13(iv) holds true for Kω (Rn ), and hence finishes the proof of Theorem 7.1.9.   Remark 7.1.10 By both Examples 1.1.9 and 7.1.8, we find that the assumption M∞ (ω) ∈ (−∞, 0) in Theorem 7.1.9 is sharp. Indeed, if M∞ (ω) ∈ [0, ∞), then both Examples 1.1.9 and 7.1.8 show that there exists an ω ∈ M(R+ ) such that p,q Kω (Rn ) is not a ball quasi-Banach function space. Moreover, the following theorem indicates that the inhomogeneous local generp,q alized Herz space Kω,0 (Rn ) is a ball Banach function space when p, q ∈ [1, ∞]. Theorem 7.1.11 Let p, q ∈ [1, ∞] and ω ∈ M(R+ ). Then the inhomogeneous p,q local generalized Herz space Kω,0 (Rn ) is a ball Banach function space. Proof Let p, q ∈ [1, ∞] and ω ∈ M(R+ ). Then, using Theorem 7.1.7, we find p,q that the inhomogeneous local generalized Herz space Kω,0 (Rn ) is a BQBF space. p,q Moreover, Kω,0 (Rn ) obviously satisfies the triangle inequality due to p, q ∈ [1, ∞]. Thus, to complete the proof of the present theorem, it remains to show that (1.39) p,q holds true with X := Kω,0 (Rn ). To achieve this, we first claim that, for any given p,q k0 ∈ Z, (1.39) holds true with X := Kω,0 (Rn ) and B := B(0, 2k0 ). We show this claim by considering the following two cases on k0 .

7.1 Inhomogeneous Generalized Herz Spaces

511

Case 1) k0 ∈ Z\N. In this case, we have B(0, 2k0 ) ⊂ B(0, 1). By this, the Hölder p,q inequality, and Definition 7.1.1(i), we conclude that, for any f ∈ Kω,0 (Rn ),  B(0,2k0 )

    |f (y)| dy  f 1B(0,2k0 ) 

Lp (Rn )

   f 1B(0,1)Lp (Rn )

 f Kp,q n , ω (R )

(7.9)

which implies that the above claim holds true in this case. Case 2) k0 ∈ N. In this case, from the Hölder inequality, the fact that, for any given α ∈ (0, ∞) and m ∈ N, and for any {aj }j ∈N ⊂ C, ⎞α ⎛ m m !  α−1 ⎠ ⎝ |aj | ≤ max 1, m |aj |α , j =1

j =1 p,q

and Definition 7.1.1(i), we deduce that, for any f ∈ Kω,0 (Rn ),  B(0,2k0 )

|f (y)| dy

     f 1B(0,2k0 ) 

Lp (Rn )

⎡

⎤1 p k0   p  p     ⎣ ⎦ f 1B(0,1) Lp (Rn ) + f 1B(0,2k )\B(0,2k−1) Lp (Rn ) ∼ k=1

⎫1 ⎧ k0  ⎬q ⎨ q   q  q ω(2k ) f 1B(0,2k )\B(0,2k−1) Lp (Rn )  f 1B(0,1)Lp (Rn ) + ⎭ ⎩ k=1

 f Kp,q (Rn ) . ω,0

This, combined with (7.9), then finishes the proof of the above claim. Applying the above claim, we then show that (1.39) holds true for any B ∈ B. Notice that, for any B ∈ B, there exists a k ∈ Z such that B ⊂ B(0, 2k ). By this and p,q the above claim, we further conclude that, for any f ∈ Kω,0 (Rn ), 

 |f (y)| dy ≤ B

B(0,2k )

|f (y)| dy  f Kp,q (Rn ) , ω,0

p,q

which implies that (1.39) holds true with X := Kω,0 (Rn ), and hence finishes the proof of Theorem 7.1.9.  

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7 Inhomogeneous Generalized Herz Spaces and Inhomogeneous Block Spaces

Similarly, we now show that the inhomogeneous global generalized Herz space p,q Kω (Rn ) is also a ball Banach function space when p, q ∈ [1, ∞]. Theorem 7.1.12 Let p, q ∈ [1, ∞] and ω ∈ M(R+ ) satisfy M∞ (ω) ∈ (−∞, 0). p,q Then the inhomogeneous global generalized Herz space Kω (Rn ) is a ball Banach function space. Proof Let all the symbols be as in the present theorem. Then, by the assumption M∞ (ω) ∈ (−∞, 0) and Theorem 7.1.9, we find that the inhomogeneous global p,q p,q generalized Herz space Kω (Rn ) is a BQBF space. Moreover, notice that Kω (Rn ) satisfies the triangle inequality when p, q ∈ [1, ∞]. Thus, to finish the proof of the present theorem, we only need to show that, for any given B ∈ B, (1.39) holds p,q true with X := Kω (Rn ). Indeed, from Definition 7.1.1(ii), we deduce that, for any p,q f ∈ Kω (Rn ), f Kp,q (Rn ) ≤ f Kp,q n ω (R ) ω,0

p,q

and hence f ∈ Kω,0 (Rn ). Using this, Theorem 7.1.11, and Definition 7.1.1(ii) p,q again, we further conclude that, for any f ∈ Kω (Rn ),  B

|f (y)| dy  f Kp,q (Rn )  f Kp,q n . ω (R ) ω,0

p,q

This implies that (1.39) holds true with X := Kω (Rn ), and then finishes the proof of Theorem 7.1.12.  

7.1.1 Convexities The main target of this subsection is to show the convexity of the inhomogeneous generalized Herz spaces. For this purpose, we first investigate the relations between these Herz spaces and their convexifications as follows, which are useful in the study of the inhomogeneous generalized Herz–Hardy spaces in next chapters. Lemma 7.1.13 Let p, q, s ∈ (0, ∞) and ω ∈ M(R+ ). Then 

p,q

Kω,0 (Rn )

with the same quasi-norms.

1/s

p/s,q/s

= Kωs ,0

(Rn )

7.1 Inhomogeneous Generalized Herz Spaces

513

Proof Let all the symbols be as in the present lemma. Then we find that, for any f ∈ M (Rn ), f [Kp,q (Rn )]1/s ω,0  1 s   = |f | s  p,q

Kω,0 (Rn )

 =

=

|y|