Morrey and Campanato meet Besov, Lizorkin and Triebel

Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris 2005 Wen Yuan · Wi...

Author: Yuan W. | Sickel W. | Yang D.

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Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris

2005

Wen Yuan · Winfried Sickel · Dachun Yang

Morrey and Campanato Meet Besov, Lizorkin and Triebel

ABC

Wen Yuan School of Mathematical Sciences Beijing Normal University Laboratory of Mathematics and Complex Systems Ministry of Education Beijing 100875 People’s Republic of China [email protected]

Dachun Yang (Corresponding Author) School of Mathematical Sciences Beijing Normal University Laboratory of Mathematics and Complex Systems Ministry of Education Beijing 100875 People’s Republic of China [email protected]

Winfried Sickel Mathematisches Institut Friedrich-Schiller-Universität Jena Ernst-Abbe-Platz 2 Jena 07743 Germany [email protected]

Corresponding author, who is supported by the National Natural Science Foundation (Grant No. 10871025) of China.

ISBN: 978-3-642-14605-3 e-ISBN: 978-3-642-14606-0 DOI: 10.1007/978-3-642-14606-0 Springer Heidelberg Dordrecht London New York Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2010935182 Mathematics Subject Classification (2010): 42B35, 46E35, 42B25, 42C40, 42B15, 47G30, 47H30 c Springer-Verlag Berlin Heidelberg 2010 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, 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. Cover design: WMXDesign GmbH, Heidelberg Printed on acid-free paper springer.com

Preface

This book is based on three developments in the theory of function spaces. As the first we wish to mention Besov and Triebel-Lizorkin spaces. These scales Bsp, q (Rn ) and Fp,s q(Rn ) allow a unified approach to various types of function spaces which have been known before like Hölder-Zygmund spaces, Sobolev spaces, Slobodeckij spaces and Bessel-potential spaces. Over the last 60 years these scales have proved their usefulness, there are hundreds of papers and many books using these scales in various connections. In a certain sense all these spaces are connected with the usual Lebesgue spaces L p (Rn ). The second source we wish to mention is Morrey and Campanato spaces. Since several years there is an increasing interest in function spaces built on Morrey spaces and leading to generalizations of Campanato spaces. This interest originates, at least partly, in some applications in the field of Navier-Stokes equations. The third ingredient is the so-called Q spaces (Qα spaces). These spaces were originally defined as spaces of holomorphic functions on the unit disk, which are geometric in the sense that they transform naturally under conformal mappings. However, about 10 years ago, M. Essén, S. Janson, L. Peng and J. Xiao extended these spaces to the n-dimensional Euclidean space Rn . The aim of the book consists in giving a unified treatment of all these three types of spaces, i.e., we will define and investigate the scales Bs,p,τq(Rn ) and Fp,s, qτ (Rn ) generalizing the three types of spaces mentioned before. Such projects have been undertaken by various mathematicians during the last ten years, which have been investigating Besov-Morrey and Triebel-Lizorkin-Morrey spaces. Let us mention only the names Kozono, Yamazaki, Mazzucato, El Baraka, Sawano, Tang, Xu and two of the authors (W.Y. and D.Y.) in this connection. A more detailed history will be given in the first chapter of the book; see Sect. 1.2.

v

vi

Preface

Let us further mention the approach of Hedberg and Netrusov [70] to general spaces of Besov-Triebel-Lizorkin type. There is some overlap with our treatment. Details will be given in Sect. 4.5. The real persons Besov, Lizorkin and Triebel never met Morrey or Campanato (which we learned from personal communications with Professor Besov and Professor Triebel). However, we hope at least, the meaning of the title is clear. We shall develop a theory of spaces of Besov-Triebel-Lizorkin type built on Morrey spaces. A second aim of the book, just a byproduct of the first, will be a completion of the s (Rn ). By looking into the series of monotheory of the Triebel-Lizorkin spaces F∞,q graphs written by Triebel over the last 30 years, these spaces play an exceptional role, in most of the cases they are even not treated. The only exception is the monos (Rn ) graph [145], where they are introduced essentially as the dual spaces of F1,q (with some restrictions in q). Also after Jawerth and Frazier [64] have found a more appropriate definition, there have been no further contributions developing the theory of these spaces further, e. g., by establishing characterizations by differences or local oscillations (at least we do not know about). In Chaps. 4–6 we shall prove characterizations by differences, local oscillations, and wavelets as well as assertions on the boundedness of pseudo-differential operators, nonlinear composition operators and pointwise multipliers. In this book we only treat unweighted isotropic spaces, with other words, all directions and all points in Rn are of equal value. This means anisotropic and/or weighted spaces are not treated here. Further, we also do not deal with spaces of generalized smoothness or smoothness parameters depending on x (variable exponent spaces). However, some basic properties of corresponding spaces of Besov-TriebelLizorkin type are known in all these situations, we refer to • • • • •

Anisotropic spaces: [3, 13, 14, 148]. Spaces of dominating mixed smoothness: [4, 128, 129, 151]. Weighted spaces: [120, 129]. Spaces of generalized smoothness: [57]. Spaces of variable exponent: [47, 152].

Further investigations could be based also on a generalization of the underlying Morrey spaces, we refer to [29–31]. We believe that our methods could be applied also in these more general situations. But nothing is done at this moment. The book contains eight chapters. Because of the generality of the spaces we use Chap. 1 for helping the reader to get an overview in various directions. First of all we summarize the contents of Chaps. 2–8. Second, we give a list of definitions of the function spaces which occur in the book. Third, we collect the various known coincidences of these spaces. Finally, we add a short history. Chapters 2–6 deal with the definition and basic properties of the spaces Bs,p,τq(Rn ) and Fp,s, qτ (Rn ). Chapter 7 is devoted to the study of Besov-Hausdorff and Triebel-Lizorkin-Hausdorff spaces. Finally, in Chap. 8, parts of the theory of the homogeneous counterparts, B˙ s,p,τq (Rn ) and F˙p,s, qτ (Rn ), of Bs,p,τq (Rn ) and Fp,s, qτ (Rn ) are discussed.

Preface

vii

The book is essentially self-contained. However, sometimes we carry over some results originally obtained for the homogeneous spaces, mainly from [163–165]. The papers [163–165] supplement the book in a certain sense. Most of the results are new in this generality and have been published never before. Beijing and Jena May, 2010

Wen Yuan Winfried Sickel Dachun Yang

Contents

1

Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 1.1 A Short Summary of the Book . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 1.2 A Piece of History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 1.2.1 Besov-Triebel-Lizorkin Spaces . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 1.2.2 Morrey-Campanato Spaces .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 1.2.3 Spaces Built on Morrey-Campanato Spaces. . . .. . . . . . . . . . . . . . . . . 1.2.4 Q Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 1.3 A Collection of the Function Spaces Appearing in the Book .. . . . . . . . . . 1.3.1 Function Spaces Defined by Derivatives and Differences . . . . . . 1.3.2 Function Spaces Defined by Mean Values and Oscillations . . . . 1.3.3 Function Spaces Defined by Fourier Analytic Tools .. . . . . . . . . . . 1.4 A Table of Coincidences .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 1.4.1 Besov-Morrey Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 1.4.2 Triebel-Lizorkin-Morrey Spaces . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 1.4.3 Morrey-Campanato Spaces .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 1.4.4 Homogeneous Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 1.5 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .

2 The Spaces Bs,p,τq (Rn ) and Fp,s, qτ (Rn ) . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.1 The ϕ -Transform for Bs,p,τq(Rn ) and Fp,s, qτ (Rn ). . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.1.1 The Definition and Some Preliminaries . . . . . . . .. . . . . . . . . . . . . . . . . 2.1.2 The Calderón Reproducing Formulae and Some Consequences . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.1.3 Sequence Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.2 Embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.3 The Fatou Property .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .

1 1 3 4 4 5 5 6 6 8 10 14 14 15 16 17 17 21 21 21 24 30 39 48

3 Almost Diagonal Operators and Atomic and Molecular Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 49 3.1 Smooth Atomic and Molecular Decompositions . . . . . . .. . . . . . . . . . . . . . . . . 49 3.2 The Relation of As,p,τq (Rn ) to Besov-Triebel-Lizorkin-Morrey Spaces . . 61

ix

x

Contents

4 Several Equivalent Characterizations . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 65 4.1 Preparations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 65 4.1.1 An Equivalent Definition . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 65 4.1.2 Several Technical Lemmas on Differences . . . . .. . . . . . . . . . . . . . . . . 72 4.1.3 Means of Differences .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 83 4.2 Characterizations by Wavelets. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 84 4.2.1 Wavelets and Besov-Triebel-Lizorkin Spaces . .. . . . . . . . . . . . . . . . . 85 4.2.2 Estimates of Mean-Values of Differences by Wavelet Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 86 4.2.3 The Wavelet Characterization of As,p,qτ (Rn ) . . . . .. . . . . . . . . . . . . . . . . 96 4.2.4 The Wavelet Characterization of Fs∞,q (Rn ) . . . . .. . . . . . . . . . . . . . . . .100 4.3 Characterizations of As,p,τq (Rn ) by Differences . . . . . . . . . .. . . . . . . . . . . . . . . . .102 s, τ n 4.3.1 Characterizations of Fp, q (R ) by Differences ... . . . . . . . . . . . . . . . .102 s, τ 4.3.2 Characterizations of Bp, q (Rn ) by Differences ... . . . . . . . . . . . . . . . .106 4.3.3 The Classes As,p,τq (Rn ) and Their Relations to Q Spaces . . . . . . . .108 4.3.4 The Characterization of Fs∞, q (Rn ) by Differences . . . . . . . . . . . . . .110 4.4 Characterizations via Oscillations . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .111 4.4.1 Preparations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .111 4.4.2 Oscillations and Besov-Type Spaces. . . . . . . . . . . .. . . . . . . . . . . . . . . . .115 4.4.3 Oscillations and Triebel-Lizorkin-Type Spaces . . . . . . . . . . . . . . . . .117 4.4.4 Oscillations and Fs∞, q (Rn ) . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .118 4.5 The Hedberg-Netrusov Approach to Spaces of Besov-Triebel-Lizorkin Type .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .119 4.5.1 Some Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .119 4.5.2 Characterizations by Differences . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .121 4.5.3 Characterizations by Oscillations . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .123 4.6 A Characterization of As,p,τq (Rn ) via a Localization of Asp, q (Rn ) . . . . . . . .125 4.6.1 A Characterization of Bs,p,τq (Rn ) . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .126 4.6.2 A Characterization of Fp,s, qτ (Rn ) . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .132 s (Rn ) . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .134 4.6.3 A Characterization of F∞,q 5 Pseudo-Differential Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .137 μ 5.1 Pseudo-Differential Operators of Class S 1,1 (Rn ) . . . . .. . . . . . . . . . . . . . . . .137 5.2 Composition of Functions in As,p,τq (Rn ) . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .142 6 Key Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .147 6.1 Pointwise Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .147 6.1.1 Smooth Functions are Pointwise Multipliers for As,p,τq (Rn ) . . . . .148 6.1.2 Pointwise Multipliers and Paramultiplication . .. . . . . . . . . . . . . . . . .149 s (Rn )) . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .157 6.1.3 A Characterization of M(F∞,q s 6.1.4 A Characterization of M(Fp,q (Rn )), s < n/p . . .. . . . . . . . . . . . . . . . .159 6.2 Diffeomorphisms .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .160 6.3 Traces . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .162 6.3.1 Traces of Functions in As,p,τq (Rn ) . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .163 s (Rn ) and Some Consequences . . . . .166 6.3.2 Traces of Functions in F∞,q

Contents

xi

6.4 Spaces on Rn+ and Smooth Domains .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .168 6.4.1 Spaces on Rn+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .168 6.4.2 Spaces on Smooth Domains .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .173 7 Inhomogeneous Besov-Hausdorff and Triebel-Lizorkin-Hausdorff Spaces . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .177 7.1 Tent Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .177 7.2 Besov-Hausdorff Spaces and Triebel-Lizorkin-Hausdorff Spaces.. . . . .199 7.3 A ( vmo, h1 )-Type Duality Result . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .237 7.4 Real Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .246 8 Homogeneous Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .251 8.1 The Definition and Some Preliminaries .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .251 8.2 The Wavelet Characterization of B˙ s,p,τq (Rn ) and F˙p,s, qτ (Rn ) .. . . . . . . . . . . . . .255 8.3 The Characterization by Differences of B˙ s,p,τq (Rn ) and F˙p,s, qτ (Rn ). . . . . . . .261 8.4 Homogeneous Besov-Hausdorff and Triebel-Lizorkin-Hausdorff Spaces . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .268 References .. . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .271 Index . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .279

Chapter 1

Introduction

The aim of this chapter is to give the reader a better orientation. For convenience of the reader we summarize the contents of the following chapters first, then we continue with some remarks to the history and finally, we collect the definitions of various function spaces and their coincidence relations.

1.1 A Short Summary of the Book Chapter 2. For all s, τ ∈ R, all p ∈ (0, ∞], and all q ∈ (0, ∞], we introduce the inhomogeneous Besov-type spaces Bs,p,τq(Rn ). Triebel-Lizorkin-type spaces Fp,s, qτ (Rn ) are defined for the same range of parameters except that p has to be less than infins, τ n ity. Also corresponding sequence spaces, bs,p,τq (Rn ) and f p, q (R ) (see Definitions 2.1 s, τ s, τ n and 2.2 below), are introduced. The spaces B p, q (R ) and Fp, q (Rn ) are the inhomogeneous counterparts of B˙ s,p,τq (Rn ) and F˙p,s, qτ (Rn ) introduced earlier in [164,165]. Via the Calderón reproducing formulae we establish the ϕ -transform characterization of these spaces in the sense of Frazier and Jawerth for all admissible values of the parameters s, τ , p, and q (see Theorem 2.1 below). On the one side this generalizes the classical results for Bsp, q (Rn ) and Fp,s q(Rn ) in [64, 65] by taking τ = 0, on the other hand it also implies that Bs,p,τq (Rn ) and Fp,s, qτ (Rn ) are well-defined. This method has to be traced to Frazier and Jawerth ([62,64]; see also [65]), and has been further developed by Bownik [23–25]. We continue by deriving some embedding properties for different metrics by using the ϕ -transform characterization; see Sect. 2.2 below. Finally, the Fatou property of Bs,p,τq (Rn ) and Fp,s, qτ (Rn ) is established. Chapter 3. To begin with, in Definition 3.1, we introduce a class of ε -almost s, τ n diagonal operators on bs,p,τq (Rn ) and f p, q (R ). Any ε -almost diagonal operator is an almost diagonal operator in the sense of Frazier and Jawerth [64]. The main result in the first part of this chapter is given in Theorem 3.1 and concerns the boundedness of s, τ n these operators on bs,p,τq (Rn ) and f p, q (R ), respectively. As an application we establish characterizations by atomic and molecular decompositions (see Theorems 3.2 and 3.3). In case τ = 0, Theorems 3.1, 3.2 and 3.3 reduce to the well-known characterizations of Bsp, q (Rn ) and Fp,s q(Rn ), for which we refer to [25, 64, 65].

D. Yang et al., Morrey and Campanato Meet Besov, Lizorkin and Triebel, Lecture Notes in Mathematics 2005, DOI 10.1007/978-3-642-14606-0 1, c Springer-Verlag Berlin Heidelberg 2010

1

2

1 Introduction

In the second section of this chapter we shall compare the spaces Bs,p,τq(Rn ) and with other approaches to introduce spaces of Besov-Triebel-Lizorkin type s (Rn ) denote the Besov-Morrey spaces; see (xxv) built on Morrey spaces. Let N pqu in Sect. 1.3. Then our main result consists in Fp,s, qτ (Rn )

s, 1/u−1/p

Bu, ∞

0 < u ≤ p ≤ ∞,

(Rn ) = N ps∞u (Rn ) ,

in the sense of equivalent quasi-norms and, if 0 < q < ∞, s, 1/u−1/p

s N pqu (Rn ) ⊂ Bu, q

(Rn ) ,

s, 1/u−1/p

s N pqu (Rn ) = Bu, q

(Rn ) ,

0 < u ≤ p ≤ ∞.

s (Rn ) (p = ∞) denote the Triebel-Lizorkin-Morrey spaces studied in [88, Let E pqu 126, 139]. Then we have s, 1/u−1/p

Fu, q

s (Rn ) = E pqu (Rn ) ,

0 < u ≤ p < ∞,

with equivalent quasi-norms. In particular, if 1 < u ≤ p < ∞ 0, 1/u−1/p

Fu, 2

0 (Rn ) = E p2u (Rn ) = Mup (Rn ) ,

also in the sense of with equivalent norms. Thus, these conclusions combined with Theorem 2.1 also give the ϕ -transform characterization of the spaces N ps∞u (Rn ) and s (Rn ), which seems to be also new. E pqu Chapter 4. Following a well-known but rather long and technical procedure (see, for example, [109] and [145]), we establish some equivalent characterizations of the spaces Bs,p,τq (Rn ) and Fp,s, qτ (Rn ). Step by step we establish the following chain of inequalities. First we shall show that Littlewood-Paley characterizations can be dominated by characterizations by differences. The second step consists in proving that characterizations by differences can be estimated from above either by characterizations by oscillations or in terms of wavelet coefficients. The third step consists in estimating oscillations by wavelet coefficients. Finally, as an application of our atomic characterizations we can close the circle and estimate these expressions in terms of wavelet coefficients by the Littlewood-Paley characterization. Here we obtain generalizations of the well-known corresponding results for Bsp, q (Rn ) and s (Rn ). A few more interFp,s q(Rn ) (p < ∞). They seem to be new for the classes F∞, q s, τ s, τ n n esting localization properties of B p, q (R ) and Fp, q (R ) will given as well. In fact, at least for small s, membership of a continuous function in Fp,s, qτ (Rn ) and Bs,p,τq(Rn ) can be checked by investigating the local behavior of this function in the corresponding space with τ = 0. Chapter 5. Based on the smooth atomic and molecular decompositions, derived in Theorems 3.2 and 3.3, we shall prove here the boundedness of exotic pseudo-differential operators on Bs,p,τq (Rn ) and Fp,s, qτ (Rn ) (see Theorem 5.1) under some restrictions for τ . This has several useful consequences. As applications of Theorem 5.1, we can establish mapping properties of f → ∂ f as well as the so-called lifting property. Furthermore, we study the boundedness of nonlinear composition operators T f : g → f ◦ g on spaces As,p,τq (Rn ) ∩C(Rn ).

1.2 A Piece of History

3

Chapter 6. This chapter is devoted to so-called key theorems; see [146, Chap. 4]. Assertions on pointwise multipliers (see Theorem 6.1), on diffeomorphisms (see Theorem 6.7) and traces (see Theorem 6.8) belong to this group. These theorems are basic for the definitions of Besov-Triebel-Lizorkin-type spaces on domains. We finally introduce Besov-Triebel-Lizorkin-type spaces on Rn+ and on bounded C∞ domains in Rn and discuss a few properties. Chapter 7. The main aim of this chapter consists in defining and investigating a class of spaces which have as duals the classes As,p,τq (Rn ). These spaces are introduced by using the Hausdorff capacity. For this reason we call them Besovs, τ s, τ n n Hausdorff spaces BH p, q (R ) and Triebel-Lizorkin-Hausdorff spaces FH p, q (R ), −s, τ −s, τ respectively. They are the predual spaces of B p , q (Rn ) and Fp , q (Rn ) (see Theorem 7.3 below). If τ = 0, these results reduce to the classical duality assertions for Besov spaces Bsp, q (Rn ) and Triebel-Lizorkin spaces Fp,s q(Rn ). These new s, τ s, τ n n scales BH p, q (R ) and FH p, q (R ) have many properties in common with the classes s, τ s, τ n n B p, q (R ) and Fp, q (R ). In particular, we establish the ϕ -transform characterization, characterizations by smooth atomic and molecular decompositions, boundedness of certain pseudo-differential operators, the lifting property, a pointwise multiplier and a diffeomorphism theorem and finally assertions on traces. However, the most important property is the following: let s ∈ R, p = q ∈ (0, ∞) and τ ∈ [0, 1p ], then τ τ n (0 Bs,p,p (Rn ))∗ = BH p−s, ,p (R ) , τ τ τ where 0 Bs,p,p (Rn ) denotes the closure of Cc∞ (Rn ) ∩ Bs,p,p (Rn ) in Bs,p,p (Rn ) (see Theorem 7.12 below). By taking s = 0, p = 2 and τ = 1/2 we get back the well-known result

( cmo (Rn ))∗ = h1 (Rn ) , where cmo (Rn ) is the local CMO(Rn ) space and h1 (Rn ) is the local Hardy s, τ n space; see Sect. 1.3. For suitable indices, the behavior of the scales BH p, q (R ) s, τ n and FH p, q (R ) under real interpolation is investigated; see Theorem 7.14 below. Chapter 8. In the last chapter we focus on the homogeneous case. The homogeneous spaces, including homogeneous Besov-type spaces B˙ s,p,τq (Rn ), TriebelLizorkin-type spaces F˙p,s, qτ (Rn ) and their preduals, homogeneous Besov-Hausdorff s, τ s, τ n n ˙ p, spaces BH˙ p, q (R ) and Triebel-Lizorkin-Hausdorff spaces F H q (R ), were introduced and investigated in [127,164,165,168]. We gather some corresponding results for these spaces. In particular, we establish their wavelet characterizations (see Theorem 8.2 below).

1.2 A Piece of History Here we will give a very rough overview about the history, mentioning some pioneering work, but without having the aim to reach completeness.

4

1 Introduction

1.2.1 Besov-Triebel-Lizorkin Spaces Nikol’skij [108] introduced in 1951 the Nikol’skij-Besov spaces, nowadays denoted by Bsp,∞ (Rn ). However, he was mentioning that this was based on earlier work of Bernstein [10] (p = ∞) and Zygmund [170] (periodic case, n = 1, 1 < p < ∞). Besov [11, 12] complemented the scale by introducing the third index q in 1959. We also refer to Taibleson [136–138] for the early investigations of Besov spaces. Around s (Rn ), 1970 Lizorkin [91, 92] and Triebel [142] started to investigate the scale Fp,q nowadays named after these two mathematicians. Further, we have to mention the contributions of Peetre [111, 113, 114], who extended around 1973–1975 the range of the admissible parameters p and q to values less than one. Of particular importance for us has been the fundamental paper [64] of Frazier and Jawerth; see also [62,63] and the monograph [65] of Frazier, Jawerth and Weiss in this connection. In these papers, the authors describe the Besov and TriebelLizorkin spaces in terms of a fixed countable family of functions with certain properties, namely, smooth atoms and molecules, which have been a second breakthrough in a certain sense (after the Fourier-analytic one in the seventieth), preparing the nowadays widely used wavelet decompositions. However, these decompositions were prepared by earlier contributions to the Calderón reproducing formula in [32, 38, 150, 155] and the studies in [41, 115]. We refer to the introduction in [64] for more details. The theory is summarized in the monographs [14, 109, 114, 145–149]. A much more detailed history can be found in [146, 148]; see also [153].

1.2.2 Morrey-Campanato Spaces In 1938 Morrey [102] introduced the classes Mup (Rn ) which are generalizations of the ordinary Lebesgue spaces. Next we would like to mention the work of John and Nirenberg, which introduced BMO in 1961 (see [79]). At the beginning of the sixties, in a series of papers, Campanato introduced and studied the spaces L p,λ (Rn ), nowadays named after him; see also Meyers [101]. Peetre [110] gave a survey on the topic (to which we refer also for more detailed comments to the early history) and studied the interpolation properties of these classes. Section 2.4 in the monograph [88] of Kufner, John and Fuˇcik is devoted to the study of Morrey and Campanato spaces and summarizes the state of the art at 1975. Function spaces, defined by oscillations, i. e., local approximation by polynomials, were studied by Brudnij [26, 27], Il’in [13, 14], Christ [40], Bojarski [15], DeVore and Sharpley [46], Wallin [153], Seeger [130], and Triebel [146, Sect. 1.7], to mention only a few. Important for us has been also the general approach of Hedberg and Netrusov [70] to those function spaces.

1.2 A Piece of History

5

1.2.3 Spaces Built on Morrey-Campanato Spaces s (Rn ), 1 < u ≤ p < ∞, 1 < q ≤ ∞, were studied The Besov-Morrey spaces N pqu for the first time by Kozono and Yamazaki [88] in connection with applications to the Navier-Stokes equation. Also in connection with applications to pde the s (Rn ), 1 < u ≤ p < ∞, 1 < q ≤ ∞, were studied by homogeneous version N˙pqu Mazzucato [97]. The next step has been done by Tang and Xu [139]. They ins (Rn ) (the Triebel-Lizorkin counterpart of N s (Rn )) and troduced the scale E pqu pqu made first investigations for the extended range 0 < u ≤ p < ∞, 0 < q ≤ ∞, of parameters for both types of spaces. Later, Sawano and Tanaka [126] presented various decompositions including quarkonial, atomic and molecular characterizations s (Rn ) and A˙s (Rn ), where A ∈ {N , E }. Jia and Wang [78] investigated the of A pqu pqu Hardy-Morrey spaces, which are special cases of Triebel-Lizorkin-Morrey spaces. In [154], Wang obtained the atomic characterization and the trace theorem for Besov-Morrey and Triebel-Lizorkin-Morrey spaces independently of Sawano and Tanaka. Recently, Sawano [125] investigated the Sobolev embedding theorem for Besov-Morrey spaces. Recall that the Besov-Morrey and Triebel-Lizorkin-Morrey spaces cover many classic function spaces, such as Besov spaces, Triebel-Lizorkin spaces, Morrey spaces and Sobolev-Morrey spaces. For the Sobolev-Morrey spaces, we refer to Najafov [103–105]. The Besov-type space Bs,p,τq(Rn ) and its homogeneous version B˙ s,p,τq (Rn ), restricted to the Banach space case, were first introduced by El Baraka in [49–51]. In these papers, El Baraka investigated embeddings as well as Littlewood-Paley characterizations of Campanato spaces. El Baraka showed that the spaces Bs,p,τq(Rn ) cover certain Campanato spaces (see [51]). s (Rn ) (p = ∞) have been studied in [88,126, Triebel-Lizorkin-Morrey spaces E˙pqu 139]. The identity s, τ s (Rn ) = E˙pqu (Rn ) F˙p,q

has been proved in [127]. The Besov-type spaces B˙ s,p,τq (Rn ) and the Triebel-Lizorkin-type spaces F˙p,s, qτ (Rn ) were introduced in [164, 165].

1.2.4 Q Spaces The history of Qα spaces (or simply Q spaces) started in 1995 with a paper by Aulaskari, Xiao and Zhao [7]. Originally they were defined as spaces of holomorphic functions on the unit disk, which are geometric in the sense that they transform naturally under conformal mappings (see [7, 160]). Following earlier contributions of Essén and Xiao [55] and Janson [76] on the boundary values of these functions on the unit circle, Essén, Janson, Peng and Xiao [54] extended these spaces to the n-dimensional Euclidean space Rn . There is a rapidly increasing literature devoted to this subject, we refer to [7, 44, 45, 54, 55, 76, 157–162, 169].

6

1 Introduction

Most recently, in [164, 165], two of the authors (W.Y and D.Y) have introduced the scales of homogeneous Besov-Triebel-Lizorkin-type spaces B˙ s,p,τq (Rn ) and F˙p,s, qτ (Rn ) (p = ∞), which generalize the homogeneous Besov-Triebel-Lizorkin spaces (B˙ sp, q(Rn ), F˙p,s q (Rn )) and Q spaces simultaneously, and hence answered an open question posed by Dafni and Xiao in [44] concerning the relation of these spaces. In fact, it holds α, 1 − α F˙ 2 n (Rn ) = Qα (Rn ) 2,2

if α ∈ (0, 1) (n ≥ 2). Recently, Xiao [161], Li and Zhai [90] applied certain special cases of B˙ s,p,τq(Rn ) and F˙p,s, qτ (Rn ), including the Q spaces, to study the Navier-Stokes equation.

1.3 A Collection of the Function Spaces Appearing in the Book As a service for the reader and also for having convenient references inside the book we give a list of definitions of the spaces of functions (distributions) showing up in this book. Sometimes a few comments will be added. We picked up this idea from [145, Sect. 2.2.2] and [153] and a part of our list is just a copy of the list given in [145]. As a general rule within this book we state that all spaces consist of complexvalued functions. We shall divide our collection into three groups: • Function spaces defined by derivatives and differences. • Function spaces defined by mean values and oscillations (local polynomial

approximations). • Function spaces defined by Fourier analytic tools.

The first item contains the classical approaches to define smoothness. In the second item we recall the definitions of spaces related to Morrey-Campanato spaces. Finally, in the third item we define spaces by Fourier analytic tools, in most of the cases by using a smooth dyadic resolution of unity.

1.3.1 Function Spaces Defined by Derivatives and Differences (i) Lebesgue spaces. Let p ∈ (0, ∞). By L p (Rn ) we denote the space of all measurable functions f such that f L p (Rn ) ≡

1/p | f (x)| dx p

Rn

< ∞.

In case p = ∞ the space L∞ (Rn ) is the collection of all measurable functions f such that f L∞ (Rn ) ≡ ess sup | f (x)| < ∞ . x∈Rn

1.3 A Collection of the Function Spaces Appearing in the Book

7

Of a certain importance for the book are the following modified Lebesguetype spaces. Let τ ∈ [0, ∞) and p ∈ (0, ∞]. Let Lτp (Rn ) be the collection of functions f ∈ L ploc (Rn ) such that f

p Lτ (Rn )

1 ≡ sup |P|τ

P

1/p | f (x)| dx p

,

where the supremum is taken over all dyadic cubes P with side length l(P) ≥ 1. (ii) The space C(Rn ) consists of all uniformly continuous functions f such that f C(Rn ) ≡ sup | f (x)| < ∞ . x∈Rn

(iii) Let m ∈ N. The space Cm (Rn ) consists of all functions f ∈ C(Rn ), having all classical derivatives ∂ α f ∈ C(Rn ) up to order |α | ≤ m and such that f Cm (Rn ) ≡

∑

|α |≤m

∂ α f C(Rn ) < ∞ .

We put C0 (Rn ) ≡ C(Rn ). (iv) Hölder spaces. Let m ∈ Z+ and s ∈ (m, m + 1). Then Cs (Rn ) denotes the collection of all functions f ∈ Cm (Rn ) such that f Cs (Rn ) ≡ f Cm (Rn ) +

∑

sup

|α |=m x=y

|∂ α f (x) − ∂ α f (y)| < ∞. |x − y|s−m

(v) Lipschitz spaces. Let s ∈ (0, 1]. The Lipschitz space Lips(Rn ) consists of all functions f ∈ C(Rn ) such that f Lip s(Rn ) ≡ sup x=y

| f (x) − f (y)| < ∞. |x − y|s

(vi) Zygmund spaces. Let m ∈ N. The Zygmund space Z m (Rn ) consists of all functions f ∈ Cm−1 (Rn ) such that f Z m (Rn ) ≡ f Cm−1 (Rn ) + max sup sup

|α |=m h=0 x∈Rn

|∂ α f (x + 2h) − 2 ∂ α f (x + h) + ∂ α f (x)| < ∞. |h|

In case of s > 0, s ∈ N, we use the convention Z s (Rn ) = Cs (Rn ). (vii) Sobolev spaces. Let p ∈ (1, ∞) and m ∈ N. Then Wpm (Rn ) is the collection of all functions f ∈ L p (Rn ) such that the distributional derivatives ∂ α f are functions belonging to L p (Rn ) for all α , |α | ≤ m. We equip this set with the norm f Wpm (Rn ) ≡

∑

|α |≤m

As usual, we define Wp0 (Rn ) ≡ L p (Rn ).

∂ α f L p (Rn ) .

8

1 Introduction

(viii) Slobodeckij spaces. Let p ∈ [1, ∞) and let s ∈ (0, ∞) be not an integer. Let m ∈ Z+ such that s ∈ (m, m + 1). Then Wps (Rn ) consists of all functions f ∈ Wpm (Rn ) such that f Wps (Rn ) ≡ f Wpm (Rn ) +

∑

Rn ×Rn

|α |=m

|∂ α f (x) − ∂ α f (y)| p dx dy |x − y|n+(m+1−s)p

1/p

< ∞.

(ix) Besov spaces (classical variant). Let s ∈ (0, ∞) and p, q ∈ [1, ∞]. Let M ∈ N. Then, if s ∈ [M − 1, M), the space Bsp,q (Rn ) is the collection of all functions f ∈ L p (Rn ) satisfying f Bsp,q (Rn ) ≡ f L p (Rn ) +

|h|−sq ΔhM f ( · ) L p (Rn ) q

Rn

dh |h|n

1/q

< ∞.

Besov spaces can be defined in various ways; see in particular item (xx) below. In Chaps. 2–4 we shall prove the equivalence of some of these approaches in a much more general context.

1.3.2 Function Spaces Defined by Mean Values and Oscillations Now we turn to a group of spaces which are related to Morrey-Campanato spaces. (x) Functions of bounded mean oscillations. The space BMO (Rn ) is the set of locally integrable functions f on Rn such that f BMO (Rn ) ≡ sup Q

1 |Q|

Q

| f (x) − fQ | dx < ∞ ,

where the supremum is taken on all cubes Q with sides parallel to the coordinate axes and where 1 fQ ≡ f (x) dx |Q| Q denotes the mean value of the function f on Q. (xi) According to Sarason [122], a function f of BMO (Rn ) which satisfies the limiting condition 1 | f (x) − fQ | dx = 0 lim sup a→0 |Q|≤a |Q| Q is said to be of vanishing mean oscillation. The subspace of BMO (Rn ) consisting of the functions of vanishing mean oscillation is denoted by VMO (Rn ). We note that the space VMO (Rn ) considered by Coifman and Weiss [42] is different from that considered by Sarason, and it coincides with our CMO (Rn ); see the next item.

1.3 A Collection of the Function Spaces Appearing in the Book

9

(xii) We denote by CMO (Rn ) the closure of Cc∞ (Rn ) in BMO (Rn ), and we endow CMO (Rn ) with the norm of BMO (Rn ). (xiii) Functions of local bounded mean oscillations. The space bmo (Rn ) consists of all functions f ∈ BMO (Rn ) which satisfy also the following condition 1 sup | f (x)| dx < ∞ . |Q|≥1 |Q| Q We equip this space with the norm f bmo(Rn ) ≡ f BMO(Rn ) + sup

|Q|=1 Q

| f (x)| dx .

(xiv) Functions of local vanishing mean oscillations. We set vmo (Rn ) ≡ VMO (Rn ) ∩ bmo (Rn ) , and we endow the space vmo (Rn ) with the norm of bmo (Rn ). (xv) We denote by cmo (Rn ) the closure of Cc∞ (Rn ) in bmo (Rn ), and we endow cmo (Rn ) with the norm of bmo (Rn ). (xvi) Morrey spaces. Let 0 < u ≤ p ≤ ∞. The space Mup (Rn ) is defined to be the set of all u-locally Lebesgue-integrable functions f on Rn such that f Mup (Rn ) ≡ sup |B|1/p−1/u

1/u

B

B

| f (x)|u dx

< ∞,

where the supremum is taken over all balls B in Rn ; see [89, Sect. 2.4]. (xvii) Campanato spaces. Let λ ∈ [0, ∞) and p ∈ [1, ∞). The collection of all functions f ∈ L ploc (Rn ) such that 1 f L p,λ (Rn ) ≡ sup λ /n B |B|

1/p

| f (x) − fB | dx p

B

< ∞,

where the supremum is taken over all balls B in Rn . This set becomes a Banach space if functions are considered modulo constants. Furthermore, L p,λ (Rn ) consists of the constant functions only if λ > n + p; see [33–36], [110] and [89, Sect. 2.4]. (xviii) Local approximation spaces I. Let p ∈ [1, ∞) and s ∈ [−n/p, ∞). Let B(x,t) be the ball with center x and radius t. Let M ∈ Z+ . Denote by PM (Rn ) the set of all polynomials of total degree less than or equal to M. For u ∈ (0, ∞] we define the local oscillation of f ∈ Luoc (Rn ) by setting, for all x ∈ Rn and all t ∈ (0, ∞), oscM f (x,t) ≡ inf t −n u

1/u B(x,t)

| f (y) − P(y)|u dy

,

10

1 Introduction

where the infimum is taken over all polynomials P ∈ PM (Rn ) with the usual modification if u = ∞, i. e., oscM ∞ f (x,t) ≡ inf sup | f (y) − P(y)| . y∈B(x,t)

Now we define the associated sharp maximal function fuM,s (x) ≡ sup t −s oscM u f (x,t) . 0 0 , s ∈ N , 1 ≤ p < ∞ ,

Fp,0,20 (Rn ) 0 n F∞, 2 (R )

=

Fp,0 2(Rn ) n

= h (R ) , p

n

0 < p < ∞,

= bmo (R )

(all in the sense of equivalent norms); see, e. g. [145, Sect. 2.2.2] and the references given there. (ii) Let p ∈ (0, ∞) and s ∈ (n max{0, 1p − 1}, ∞). Then 0 (Rn ) = Fp,s ∞ (Rn ) = Csp (Rn ) ; Fp,s, ∞

see [130] and [146, Theorem 1.7.2]. (iii) Let s ∈ R, p ∈ (0, ∞) and q ∈ (0, ∞]. Then s, 1/p

s n Fp, q (Rn ) = F∞, q (R )

with equivalent quasi-norms; see [64] or Proposition 2.4 below. In particular, 0, 1/p

0 n n Fp, 2 (Rn ) = F∞, 2 (R ) = bmo (R ) .

(iv) Let q ∈ (0, ∞] and 0 < u ≤ p ≤ ∞, u = ∞. Then s, 1/u−1/p

Fu, q

s (Rn ) = E pqu (Rn ) .

For s = 0 and 1 < u ≤ p < ∞ this yields 0, 1/u−1/p

Fu, 2

0 (Rn ) = E p2u (Rn ) = Mup (Rn )

and with 1 < u = p < ∞ 0 (Rn ) = M pp (Rn ) = L p (Rn ) , Fp,0,20 (Rn ) = E p2p

all in the sense of equivalent quasi-norms; see Corollary 3.3 below.

(1.4)

16

1 Introduction

(v) Let α ∈ (0, 1) if n ≥ 2 and α ∈ (0, 1/2) if n = 1. Then we have α , 1 − αn

F2,2 2

(Rn ) = Qα (Rn ) ∩ L21 − α (Rn ) , 2

n

in the sense of equivalent norms; see Corollary 4.5 and Remark 4.7. (vi) Let 0 Then

s−k/p, 1p n+k n

Fp,q and

(Rn ) = Z s (Rn )

n 1 n+k s− k+n p + p0 , p n

Fp0 ,q

p ≤ q ≤ ∞,

if

(Rn ) = Z s (Rn )

if

0 < q ≤ ∞,

in the sense of equivalent quasi-norms; see Theorem 6.9 below. (vii) Pointwise multipliers. For a quasi-Banach space X of functions, the space M(X) denotes the associated space of all pointwise multipliers; see Sect. 6.1. Let s ∈ (0, 1). Then s,τ s n M(F1,1 (Rn )) = L∞ (Rn ) ∩ F1,1, unif (R ) ,

τ = 1 − s/n ;

see Corollary 6.2 below.

1.4.3 Morrey-Campanato Spaces (i) Let 0 < u ≤ p ≤ ∞. Then Muu (Rn ) = Lu (Rn )

and

Mu∞ (Rn ) = L∞ (Rn ) .

(ii) Let p ∈ [1, ∞) and λ ∈ (n, n + p). Then L p,n (Rn ) = BMO (Rn ), L p,n (Rn ) = Z

λ −n p

(Rn )

and

L p,n+p(Rn ) = Lip 1(Rn ) ;

see [34, 36] and [89, Theorem 2.4.6.1]. (iii) Let p ∈ [1, ∞). Then −n/p

Tp

(Rn ) = L p (Rn )

and

Tp0 (Rn ) = bmo (Rn ) .

(iv) Let p ∈ [1, ∞) and s ∈ (−n/p, 0). Then −n/s

L p,λ (Rn ) = M p

(Rn ) = Tps (Rn ) ,

see [89, Theorem 2.4.6.1] and [146, Sect. 1.7.2].

s=

λ −n ; p

1.5 Notation

17

(v) Let p ∈ [1, ∞) and s ∈ (0, ∞). Then Tps (Rn ) = Z s (Rn ) ; see [146, Sect. 1.7.2] and the references given there.

1.4.4 Homogeneous Spaces Here we make use of the following interpretation. When comparing a class of functions, which is defined modulo polynomials of a certain order, with the spaces A˙ s,p,τq (Rn ), then we always associate to an element of the first space the equivalence class [ f ] ≡ {g : g = f + p , p is an arbitrary polynomial} . By means of this interpretation the following relations are known. (i) We have 0 n n F˙∞, 2 (R ) = BMO (R ) .

(ii) Let s ∈ R, p ∈ (0, ∞) and q ∈ (0, ∞]. Then s, 1/p s n F˙p, q (Rn ) = F˙∞, q (R )

with equivalent quasi-norms. In particular, 0, 1/p 0 n n F˙p, 2 (Rn ) = F˙∞, 2 (R ) = BMO (R ) .

(iii) Let α ∈ (0, 1) if n ≥ 2 and α ∈ (0, 1/2) if n = 1. Then we have α

α, − F˙2,2 2 n (Rn ) = Qα (Rn ) 1

in the sense of equivalent norms; see [164]. (iv) Let λ ∈ [0, n + 2). Then 0, 2λ /n F˙2, 2 (Rn ) = L 2,λ (Rn ) ;

see [50].

1.5 Notation At the end of this chapter, we make some conventions on notation. Throughout this book, C denotes unspecified positive constants, possibly different at each occurrence; the symbol X Y means that there exists a positive constant

18

1 Introduction

C such that X ≤ CY , and X ∼ Y means C−1Y ≤ X ≤ CY. We also use C(γ , β , · · · ) to denote a positive constant depending on the indicated parameters γ , β , · · · . The real numbers are denoted by R. Many times we shall use the abbreviations a+ ≡ max(0, a), a for the integer part of the real number a, and a∗ ≡ a − a. The symbol χE is used to denote the characteristic function of set E ⊂ Rn . If q ∈ [1, ∞] then by q we mean its conjugate index, i. e., 1/q + 1/q = 1. Further we shall use the abbreviations p ∨ q ≡ max{p, q} and p ∧ q ≡ min{p, q}. When dealing with the classes As,p,τq (Rn ), then four restrictions for the set of parameters s, p, q, τ will occur relatively often. They are connected with the quantities

σ p ≡ max{n(1/p − 1), 0} and σ p, q ≡ max{n(1/ min{p, q} − 1), 0} ,

(1.5)

(restrictions for s) and

τs,p

1 ≡ + p

τs,p,q

1 ≡ + p

1−(σ p +n−s)∗ n s−σ p n 1−(σ p,q +n−s)∗ n s−σ p,q n

if s ≤ σ p , if s > σ p , if if

s ≤ σ p,q , s > σ p,q

(1.6) (1.7)

(restrictions for τ ). Also, set N ≡ {1, 2, · · · } and Z+ ≡ N ∪ {0}. By Cc∞ (Rn ) we denote the set of all infinitely differentiable and compactly supported functions on Rn . The symbol S (Rn ) is used in place of the set of all Schwartz functions ϕ on Rn , i. e., ϕ is infinitely differentiable and ϕ SM ≡

sup

sup |∂ γ ϕ (x)|(1 + |x|)n+M+|γ | < ∞

γ ∈Zn+ , |γ |≤M x∈Rn

for all M ∈ N. The topological dual of S (Rn ), the set of tempered distributions, will be denoted by S (Rn ). For k = (k1 , · · · , kn ) ∈ Zn and j ∈ Z, Q jk denotes the dyadic cube Q jk ≡ {(x1 , · · · , xn ) : ki ≤ 2 j xi < ki + 1 for i = 1, · · · , n} . For the collection of all such cubes we use Q ≡ Q jk : j ∈ Z, k ∈ Zn .

1.5 Notation

19

Furthermore, we denote by xQ the lower left-corner 2− j k of Q = Q jk . When the dyadic cube Q appears as an index, such as ∑Q∈Q and {·}Q∈Q , it is understood that Q runs over all dyadic cubes in Rn . For each cube Q, we denote its side length by l(Q), its center by cQ , and for r > 0, we denote by rQ the cube concentric with Q having the side length rl(Q). Further, the abbreviation jQ ≡ − log2 l(Q) is used. For j ∈ Z, ϕ ∈ S (Rn ) and x ∈ Rn , we set ϕ(x) ≡ ϕ (−x),

ϕ (x) ≡ F ϕ (x) ≡

Rn

ϕ (ξ )e−ix·ξ d ξ ,

ϕ j (x) ≡ 2 jn ϕ (2 j x), and ϕQ (x) ≡ |Q|−1/2 ϕ (2 j x − k) = |Q|1/2 ϕ j (x − xQ )

if

Q = Q jk .

For a dyadic cube Q, we shall work also with the L2 (Rn )-normalized version

χQ (x) ≡ |Q|−1/2 χQ (x) . Let E denote a class of tempered distributions. Then E loc denotes the collection of all f ∈ S (Rn ) such that the product ϕ · f belongs to E for all test functions ϕ ∈ Cc∞ (Rn ). Furthermore, if E is in addition quasi-normed, then E unif is the collection of all f ∈ S (Rn ) such that f E unif ≡ sup ψ ( · − λ ) f ( · ) E < ∞ . λ ∈Rn

Here ψ is a nontrivial function in Cc∞ (Rn ).

Chapter 2

The Spaces Bs,p,τq (Rn ) and Fs,p,τq(Rn )

In this chapter, we introduce the spaces Bs,p,τq (Rn ) and Fp,s, qτ (Rn ), establish their ϕ -transform characterizations, prove some embeddings and the Fatou property.

2.1 The ϕ -Transform for Bs,p,τq (Rn ) and Fs,p,τq (Rn ) Nowadays wavelet decompositions play an important role in the study of function spaces and their applications; see, for example, [99, 100, 149]. The ϕ -transform decomposition of Frazier and Jawerth [62–64] is rather close in spirit to wavelet decompositions. In this section, we establish the ϕ -transform characterizations of the spaces Bs,p,τq (Rn ) and Fp,s, qτ (Rn ), which will play a crucial role in our considerations.

2.1.1 The Definition and Some Preliminaries We start with the definitions of Bs,p,τq(Rn ) and Fp,s, qτ (Rn ). Select a pair of Schwartz functions Φ and ϕ such that ⊂ {ξ ∈ Rn : |ξ | ≤ 2} and |Φ (ξ )| ≥ C > 0 if |ξ | ≤ 5 supp Φ 3

(2.1)

and 1 3 5 supp ϕ ⊂ ξ ∈ Rn : ≤ |ξ | ≤ 2 and |ϕ(ξ )| ≥ C > 0 if ≤ |ξ | ≤ . (2.2) 2 5 3 It is easy to see that

Rn x

γ ϕ (x) dx

= 0 for all multi-indices γ ∈ Zn+ .

Definition 2.1. Let τ , s ∈ R, q ∈ (0, ∞], Φ and ϕ satisfy (2.1) and (2.2), respectively, and put ϕ j ≡ 2 jn ϕ (2 j · ).


21

2 The Spaces Bs,p,τq (Rn ) and Fp,s, qτ (Rn )

22

(i) Let p ∈ (0, ∞]. The inhomogeneous Besov-type space Bs,p,τq (Rn ) is defined to be the set of all f ∈ S (Rn ) such that 1 f Bs,p,τq(Rn ) ≡ sup τ P∈Q |P|

∞

∑

(2 |ϕ j ∗ f (x)|) dx js

P

j=( jP ∨0)

q/p 1/q

p

< ∞, (2.3)

where ϕ0 is replaced by Φ . (ii) Let p ∈ (0, ∞). The inhomogeneous Triebel-Lizorkin-type space Fp,s, qτ (Rn ) is defined to be the set of all f ∈ S (Rn ) such that ⎧ p/q ⎫1/p ⎬ ∞ 1 ⎨ js q f Fp,s, qτ (Rn ) ≡ sup (2 | ϕ ∗ f (x)|) dx < ∞, (2.4) j ∑ τ ⎭ P∈Q |P| ⎩ P j=( j ∨0) P

where ϕ0 is replaced by Φ . Remark 2.1. (i) When p = q ∈ (0, ∞), Bs,p,τq(Rn ) = Fp,s, qτ (Rn ). If we replace dyadic cubes P in Definition 2.1 by arbitrary cubes P, we then obtain equivalent quasi-norms. (ii) The definitions given here are slightly more general than those given in Sect. 1.3. The coincidence will be proved by establishing the independence of the above definitions from Φ and ϕ ; see Corollary 2.1 below. (iii) For τ > 1/p it is necessary to start the summation with respect to j in dependence on the size of the dyadic cube P. If the summation would start always with j = 0, then a Lebesgue point argument shows that only the function f = 0 a. e. belongs to such a space. For simplicity, in what follows, we use As,p,τq(Rn ) to denote either Bs,p,τq(Rn ) or If As,p,τq (Rn ) means Fp,s, qτ (Rn ), then the case p = ∞ is excluded. In the same way we shall use the abbreviation Asp, q (Rn ) in place of Fp,s q(Rn ) and Bsp, q (Rn ), respectively.

Fp,s, qτ (Rn ).

Lemma 2.1. (i) The classes As,p,τq (Rn ) are quasi-Banach spaces, i. e., complete quasi-normed spaces. With d = min{1, p, q} it holds f + gdAs, τ (Rn ) ≤ f dAs, τ (Rn ) + gdAs, τ (Rn ) p, q

p, q

p, q

for all f , g ∈ As,p,τq (Rn ). n s n (ii) If τ = 0, then As,0 p, q (R ) = A p, q (R ).

Proof. To prove (i) the needed arguments are standard, we refer, e. g., to [145, Sect. 2.3.3]. Details are left to the reader. The proof of (ii) is obvious. Sometimes it is of great service if one can restrict supP∈Q in the definition of As,p,τq (Rn ) to a supremum taken with respect to dyadic cubes with side length ≤ 1.

2.1 The ϕ -Transform for Bs,p,τq (Rn ) and Fp,s, qτ (Rn )

23

Lemma 2.2. Let s ∈ R and q ∈ (0, ∞]. (i) Let p ∈ (0, ∞] and τ ∈ [1/p, ∞). A tempered distribution f belongs to Bs,p,τq(Rn ) if, and only if, f #Bs, τ (Rn ) ≡ p, q

sup {P∈Q, |P|≤1}

1 |P|τ

∞

∑

P

j=( jP ∨0)

(2 js |ϕ j ∗ f (x)|) p dx

q/p 1/q < ∞.

Furthermore, the quasi-norms f Bs,p,τq (Rn ) and f #Bs, τ (Rn ) are equivalent. p, q

(ii) Let p ∈ (0, ∞) and τ ∈ [1/p, ∞). A tempered distribution f belongs to Fp,s, qτ (Rn ) if, and only if,

f #F s, τ (Rn ) ≡ p, q

sup {P∈Q, |P|≤1}

⎧ p/q ⎫1/p ⎬ ∞ 1 ⎨ js q (2 | ϕ ∗ f (x)|) dx < ∞. j ⎭ |P|τ ⎩ P j=(∑ j ∨0) P

Furthermore, the quasi-norms f Fp,s, qτ (Rn ) and f #F s, τ (Rn ) are equivalent. p, q

for some r ∈ N. Let {Qm : m = Proof. Let P be a dyadic cube such that |P| = 1, . . . , 2rn } be the collection of all dyadic cubes with volume 1 and such that 2rn

rn

P=

2

Qm .

m=1

Then, with g ∈ L ploc (Rn ), 1 |P|τ

P

1/p |g(x)| dx p

1 = |P|τ 1 ≤ |P|τ ≤

2rn

∑

m=1 Qm

1/p |g(x)| dx p

1 sup 2 τp {Q∈Q, l(Q)≤1} |Q| rn

1 τ {Q∈Q, l(Q)≤1} |Q|

1/p

|g(x)| dx p

Q

1/p

sup

Q

|g(x)| p dx

.

(2.5)

This proves the claim for Fp,s, qτ (Rn ). In case of Bs,p,τq (Rn ) one applies the inequality (2.5) either in combination with (∑ . . .)q/p ≤ ∑ (. . .)q/p if q/p < 1 or in combination with Minkowski’s inequality if q/p ≥ 1. Remark 2.2. Lemma 2.2 does not extend to values τ < 1/p. A proof of this claim will be given at the end of Sect. 4.2.3 under the additional assumption s > σ p .


24

2.1.2 The Calderón Reproducing Formulae and Some Consequences The independence of As,p,τq (Rn ) from the choice of Φ and ϕ will be an immediate corollary of the ϕ -transform characterization of As,p,τq (Rn ). To establish this characterization, we need the Calderón reproducing formulae, which play important roles in the whole book. Let Φ and ϕ satisfy, respectively, (2.1) and (2.2). By [64, pp. 130–131] or [65, Lemma (6.9)], there exist functions Ψ ∈ S (Rn ) satisfying (2.1) and ψ ∈ S (Rn ) satisfying (2.2) such that for all ξ ∈ Rn , ∞

(ξ ) + ∑ ϕ (2− j ξ )ψ (2− j ξ ) = 1. Φ (ξ )Ψ

(2.6)

j=1

Furthermore, we have the following Calderón reproducing formula; see [64, (12.4)]. Lemma 2.3. Let Φ , Ψ ∈ S (Rn ) satisfy (2.1) and ϕ , ψ ∈ S (Rn ) satisfy (2.2) such that (2.6) holds. Then for all f ∈ S (Rn ), ∞

∗ f + ∑ ψj ∗ ϕ j ∗ f f =Ψ ∗Φ j=1

=

∞

∑n Φ ∗ f (k)Ψ (· − k) + ∑ 2− jn ∑n ϕ j ∗ f (2− j k) ψ j (· − 2− j k) j=1

k∈Z

=

∑

l(Q)=1

∞

f , ΦQ ΨQ + ∑

∑

j=1 l(Q)=2− j

k∈Z

f , ϕQ ψQ

(2.7)

in S (Rn ). The following basic estimate will be used throughout the book.

Lemma 2.4. Let M ∈ Z+ , and ψ , ϕ ∈ S (Rn ) with ψ satisfying Rn xγ ψ (x) dx = 0 for all multi-indices γ ∈ Zn+ satisfying |γ | ≤ M. Then there exists a positive constant C ≡ C(M, n) such that for all j ∈ Z+ and x ∈ Rn , ψ j ∗ ϕ (x) ≤ Cψ S ϕ S 2− jM M+1 M+1

1 . (1 + |x|)n+M

(2.8)

Proof. Since ψ has vanishing moments of any order, we see that (−y)α α |ψ j ∗ ϕ (x)| = ϕ (x − y) − ∑ ∂ ϕ (x) ψ j (y) dy Rn α ! 0≤|α |≤M (−y)α |ψ j (y)| dy ϕ (x − y) − ∑ ∂ α ϕ (x) α! |y|≤(1+|x|)/2 0≤|α |≤M


+ +

|y|>(1+|x|)/2

25

|ϕ (x − y)||ψ j (y)| dy

∑

|y|>(1+|x|)/2 0≤|α |≤M

|∂ α ϕ (x)| |y||α | |ψ j (y)| dy

≡ I1 + I2 + I3 . Since ϕ , ψ ∈ S (Rn ), then by the mean value theorem, there exists θ ∈ [0, 1] such that I1

sup |∂ α ϕ (x − θ y)||y|M+1

|y|≤(1+|x|)/2 |α |=M+1

2− jM ψ SM dy (2− j + |y|)n+M

ϕ SM+1 |y|M+1 2− jM ψ SM dy n+2M+1 (2− j + |y|)n+M |y|≤(1+|x|)/2 (1 + |x − θ y|) 2− jM , (1 + |x|)n+M

ϕ SM+1 ψ SM

where for the last inequality we use the fact that 1 + |x − θ y| 1 + |x|. Similarly, I2

ϕ SM 2− jM ψ SM dy n+M (2− j + |y|)n+M |y|>(1+|x|)/2 (1 + |x − y|)

ϕ SM ψ SM

2− jM , (1 + |x|)n+M

and I3

ϕ SM |y||α | 2− j(M+1) ψ SM+1 dy −j n+M+1 |y|>(1+|x|)/2 0≤|α |≤M (1 + |x|)n+M+|α | (2 + |y|)

∑

ϕ SM ψ SM+1

2− jM , (1 + |x|)n+M

which completes the proof of Lemma 2.4.

Remark 2.3. The proof of Lemma 2.4 is similar to that of [164, Lemma 2.2]. To establish the ϕ -transform characterization of As,p,τq (Rn ), we need some technical lemmas first. The following lemma is a slight variant of [65, Lemma (6.10)]. For the convenience of the reader, we give some details. Recall that a function g is called at most polynomially increasing with order m ∈ Z+ , if there exists a positive constant C such that |g(x)| ≤ C(1 + |x|)m for all x ∈ Rn . Lemma 2.5. Let h ∈ S (Rn ) and g ∈ C∞ (Rn ) be at most polynomially increasing with order m ∈ Z+ such that supp h, g ⊂ {ξ ∈ Rn : |ξ | < 2v π } for some v ∈ Z. Then g∗h =

∑n 2−vnh(2−vk)g(· − 2−vk)

k∈Z

holds pointwise as well as in S (Rn ).

(2.9)


26

Proof. First we assume that g ∈ S (Rn ). Then by [65, Lemma (6.10)], (2.9) holds pointwise. We now further show that in this case, (2.9) also holds in S (Rn ). Indeed, for any given α , β ∈ Zn+ , let ϕ α ,β ≡ sup |xα ||∂ β ϕ (x)| x∈Rn

denote the usual Schwartz quasi-norm, where for any α = (α1 , · · · , αn ) ∈ Zn+ and x = (x1 , · · · , xn ) ∈ Rn , xα = xα1 1 · · · xαn n and ∂ α = ( ∂∂x )α1 · · · ( ∂∂xn )αn . Then 1

h(2−v k)g(· − 2−vk)α ,β ≤ |h(2−v k)| sup |x||α | |∂ β g(x − 2−vk)| x∈Rn

|h(2−v k)| sup |x||α | x∈Rn

gS|α |+|β | (1 + |x − 2−vk|)n+|β |+|α |

|h(2−v k)|(1 + |2−vk|)|α | gS|α |+|β | . Since h is a Schwartz function, then |h(2−v k)| (1 + |2−vk|)−n−|α |−1 . Thus, 1

∑n 2−vnh(2−vk)g(· − 2−vk)α ,β ∑n 2−vn (1 + |2−vk|)n+1 gS|α|+|β | < ∞,

k∈Z

k∈Z

which together with the completion of S (Rn ) implies that

∑n 2−vn h(2−vk)g(· − 2−vk) ∈ S (Rn ) ,

k∈Z

and hence (2.9) holds in S (Rn ) if g ∈ S (Rn ). For the general case, we set gδ (x) ≡ η (δ x)g(x) for δ ∈ (0, 1) and x ∈ Rn , where ⊂ {ξ ∈ Rn : |ξ | < 1}. Then gδ ∈ S (Rn ), η ∈ S (Rn ) satisfies η (0) = 1 and supp η and for sufficiently small δ > 0, by the conclusion proved above, we know that gδ ∗ h =

∑n 2−vnh(2−vk)gδ (· − 2−vk)

(2.10)

k∈Z

holds in both pointwise and S (Rn ), which together with Lebesgue’s dominated convergence theorem yields that (2.9) holds pointwise. Next we show that (2.9) also holds in S (Rn ). Notice that for all φ ∈ S (Rn ), | gδ (· − 2−vk), φ |

Rn

Rn

|g(y − 2−vk)||φ (y)| dy (1 + |y − 2−vk|)m dy (1 + |y|)n+m+1

(1 + |2−vk|)m


27

and

∑

2−vn |h(2−v k)|(1 + |2−vk|)m

k∈Zn

2−vn < ∞. ∑ −v n+1 k∈Zn (1 + |2 k|)

This observation together with Lebesgue’s dominated convergence theorem and (2.10) implies that

g ∗ h, φ = lim gδ ∗ h, φ δ →0

= lim

∑

δ →0 k∈Zn

2−vn h(2−v k) gδ (· − 2−vk), φ

∑n 2−vn h(2−vk) g(· − 2−vk), φ .

=

k∈Z

Thus (2.9) holds in S (Rn ), which completes the proof of Lemma 2.5.

Let γ be a fixed integer. Replacing ϕ j by ϕ j−γ (ϕ0 by Φ−γ ) in (2.3) and (2.4), we obtain a new quasi-norm in As,p,τq (Rn ), denoted by f ∗As, τ (Rn ) . p, q

Lemma 2.6. The quasi-norms f ∗As, τ (Rn ) and f As,p,τq(Rn ) are equivalent on p, q

S (Rn ) with equivalent constants depending on γ .

Proof. By similarity, we only consider Bs,p,τq (Rn ) and the case γ > 0. Notice that f ∗Bs, τ (Rn ) p, q

1 ∼ sup τ P∈Q |P|

∞

∑

j=( jP ∨0)−γ

(2 |ϕ j ∗ f (x)|) dx js

P

q/p 1/q

p

,

where ϕ−γ is replaced by Φ−γ . Thus, to show f ∗Bs, τ (Rn ) f Bs,p,τq (Rn ) , it suffices p, q

to prove that for all P ∈ Q with l(P) ≥ 1, 1 IP ≡ |P|τ

0

∑

j=−γ

P

(2 js |ϕ j ∗ f (x)|) p dx

q/p 1/q f Bs,p,τq (Rn )

and that for all P ∈ Q with l(P) < 1, 1 JP ≡ |P|τ

jP −1

∑

j= jP −γ

P

(2 js |ϕ j ∗ f (x)|) p dx

q/p 1/q f Bs,p,τq(Rn ) .

We first estimate IP . By (2.1) and (2.2), there exist η j ∈ S (Rn ), j = −γ , · · · , −1, and ζ1 , ζ2 ∈ S (Rn ) such that

ϕ j = η j ∗ Φ,

j = −γ , · · · , −1, and ϕ = ϕ0 = ζ1 ∗ Φ + ζ2 ∗ ϕ1 .


28

We now consider two cases. When p ∈ [1, ∞], by Minkowski’s inequality, we see that

|ϕ j ∗ f (x)| dx

1/p =

p

P

≤

P

Rn

|η j ∗ Φ ∗ f (x)| dx

1/p

p

|η j (y)|

P

|(Φ ∗ f )(x − y)| p dx

1/p dy,

which further implies that 1 |P|τ

−1

∑

j=−γ

P

1 |P|τ

(2 js |ϕ j ∗ f (x)|) p dx

−1

∑

j=−γ

|η j (y)|

Rn

−1

∑

f Bs,p,τq(Rn )

j=−γ

Rn

q/p 1/q

P

|(Φ ∗ f )(x − y)| dx

|η j (y)| dy

q 1/q

1/p

p

dy

q 1/q

f Bs,p,τq(Rn ) . The estimate for the term j = 0 is similar. When p ∈ (0, 1), for j = −γ , · · · , 0, by Lemma 2.3, we have ∞

∗ f + ∑ ϕ j ∗ ψi ∗ ϕi ∗ f . ϕ j ∗ f = ϕ j ∗Ψ ∗ Φ i=1

∗ f and ϕi ∗ f are C∞ (Rn ) functions with polynomially increasing (see Notice that Φ [134, Chap. 1, Theorem 3.13]). Then applying Lemma 2.5 with v = 0, h = ϕ j ∗ Ψ , ∗ f or v = i, h = ϕ j ∗ ψi and g = ϕi ∗ f , and the monotonicity of the q norms, g=Φ in particular d

∑ |a j | j

≤ ∑ |a j |d ,

0 < d ≤ 1,

{a j } j ⊂ C ,

j

we have |ϕ j ∗ f (x)| p ≤

∑n |ϕ j ∗ Ψ (k)| p |Φ ∗ f (x − k)| p

k∈Z ∞

+∑

∑

i=1 k∈Zn

2−inp |ϕ j ∗ ψi (2−i k)| p |ϕi ∗ f (x − 2−ik)| p .

(2.11)


29

Thus, using Lemma 2.4 and [164, Lemma 2.2] with M > (n(1/p − 1)∨0), we obtain 1 |P|τ

P

(2 js |ϕ j ∗ f (x)|) p dx

1/p

∑ |ϕ j ∗ Ψ (k)|

f Bs,p,τq (Rn ) f Bs,p,τq (Rn )

k∈Zn

p

∞

1/p

+∑2

−inp

∑ |ϕ j ∗ ψi(2

−i

k)|

p

k∈Zn

i=1

∞ 2− jM p 2−iM p ∑n (1 + |k|)(n+M)p + ∑ 2−inp ∑n (2− j + |2−ik|)(n+M)p i=1 k∈Z k∈Z

f Bs,p,τq (Rn ) . To prove JP f Bs,p,τq(Rn ) , denote by P(i) the dyadic cube containing P with

l(P(i)) = 2i l(P). We now consider two cases. If jP ≥ γ + 1, by jP(γ ) = jP − γ and P ⊂ P(γ ), we have

JP

⎧ ⎨

jP −1

1 |P(γ )|τ ⎩ j=∑ j

P(γ )

P(γ )

(2 js |ϕ j ∗ f (x)|) p dx

⎫ q/p ⎬1/q ⎭

f Bs,p,τq (Rn ) .

If 1 ≤ jP ≤ γ , by a similar argument to the estimate for IP , we see that 1 |P|τ

jP −1

∑

j= jP −γ

(2 |ϕ j ∗ f (x)|) dx js

P

q/p 1/q f Bs,p,τq(Rn ) ,

p

which together with the previous estimates yields f ∗Bs, τ (Rn ) f Bs,p,τq (Rn ) . p, q

To prove the converse estimate that f Bs,p,τq(Rn ) f ∗Bs, τ (Rn ) , it suffices to show p, q

that for all P ∈ Q with l(P) ≥ 1, 1 |P|τ

|Φ ∗ f (x)| dx

1/p

p

P

f ∗Bs, τ (Rn ) . p, q

(2.12)

Indeed, similarly to the estimates for IP , if p ∈ [1, ∞], using the fact that there exist ρ j ∈ S (Rn ), j = −γ , · · · , 1, such that

Φ ∗ f = ρ−γ ∗ Φ−γ ∗ f +

1

∑

j=−γ +1

ρj ∗ ϕj ∗ f

(see, for example, [64, p. 130]), and Minkowski’s inequality, we have (2.12); if p ∈ (0, 1), Lemmas 2.3, 2.4 and 2.5, and (2.11) also yield (2.12), which completes the proof of Lemma 2.6.


30

2.1.3 Sequence Spaces Now we introduce the corresponding inhomogeneous sequence spaces of Bs,p,τq(Rn ) and Fp,s, qτ (Rn ), which are indexed by the set of dyadic cubes Q with l(Q) ≤ 1. Definition 2.2. Let τ , s ∈ R and q ∈ (0, ∞]. (i) Let p ∈ (0, ∞]. The inhomogeneous sequence space bs,p,τq (Rn ) is defined to be the set of all sequences t ≡ {tQ }l(Q)≤1 ⊂ C such that tbs,p,τq(Rn ) < ∞, where ⎧ ⎪ ⎪ ⎨

⎤ q ⎫ 1q p⎪ ⎪ ⎬ ∞ 1 ⎢ ⎥ j(s+n/2−n/p)q p s, τ tb p, q(Rn ) ≡ sup 2 |t | . (2.13) ⎣ ⎦ Q ∑ ∑ τ ⎪ P∈Q |P| ⎪ ⎪ ⎪ l(Q)=2− j ⎩ j=( jP ∨0) ⎭ ⎡

Q⊂P

s, τ n (ii) Let p ∈ (0, ∞). The inhomogeneous sequence space f p, q (R ) is defined to be the set of all sequences t ≡ {tQ }l(Q)≤1 ⊂ C such that t f p,s, τq(Rn ) < ∞, where

t f p,s, τq(Rn )

⎧ ⎡ ⎤ p ⎫ 1p q ⎪ ⎪ ⎬ ⎨ ∞ 1 j(s+n/2)q q ⎣ ∑ ⎦ ≡ sup 2 |t | χ (x) dx . (2.14) Q Q ∑ τ ⎪ P∈Q |P| ⎪ ⎭ ⎩ P j=( jP ∨0) l(Q)=2− j

s, τ s, τ n n Similarly, we use as,p,τq (Rn ) to denote either bs,p,τq (Rn ) or f p, q (R ). If a p, q (R ) s, τ n means f p, q (R ), then the case p = ∞ is excluded. Under the additional restriction p ≥ q also the sequence spaces F˙p,s, qτ (Rn ) allow a total discretization. This fact is an immediate consequence of [37, Proposition 2.2].

Remark 2.4. Let τ ∈ [0, ∞), s ∈ R, p ∈ (0, ∞) and q ∈ (0, ∞]. If p ≥ q, then there s, τ n exists a positive constant C, depending only on p and q, such that for all t ∈ f p, q (R ), C−1 t f p,s, τq(Rn )

⎧ ⎪ 1 ⎨ ∞ ≤ sup ∑ ∑ (|Q|−s/n−1/2+1/q|tQ |)q τ P∈Q |P| ⎪ ⎩ j=( j ∨0) l(Q)=2− j P

Q⊂P

⎤ p/q−1 ⎫1/p ⎪ ⎬ 1 −s/n−1/2+1/q q⎦ ⎣ × (|R| |tR |) ∑ ⎪ |Q| R∈Q ⎭ ⎡

R⊂Q

≤ Ct f p,s, τq(Rn ) . The homogeneous counterpart of as,p,τq (Rn ), denoted by a˙s,p,τq (Rn ), was already introduced in [165]. The relation between as,p,τq(Rn ) and a˙s,p,τq(Rn ) is trivial. In fact, define V : as,p,τq (Rn ) → a˙s,p,τq (Rn ) by setting


(V t)Q ≡

31

if l(Q) ≤ 1 , otherwise.

tQ 0

(2.15)

Then V is an isometric embedding of as,p,τq (Rn ) in a˙s,p,τq (Rn ). Define W : a˙s,p,τq (Rn ) → as,p,τq(Rn ) by setting (W t)Q = tQ if l(Q) ≤ 1. Then W is continuous and W ◦ V is the identity on as,p,τq (Rn ). Next we establish the relation between As,p,τq (Rn ) and as,p,τq (Rn ). Let Φ , Ψ , ϕ and ψ be as in Lemma 2.3. Recall that the ϕ -transform Sϕ is defined by setting (Sϕ f )Q ≡ f , ΦQ if l(Q) = 1 and (Sϕ f )Q ≡ f , ϕQ if l(Q) < 1, the inverse ϕ -transform Tψ is defined by

∑

Tψ t ≡

∑

tQΨQ +

l(Q)=1

tQ ψQ ;

l(Q) max{2n + s, 2n + s + nd(τ − 1)}.

Thus, (i) is proved. Step 2. Let p ∈ (0, 1). For t > 1 we have at (x) | f (x)| + t nτ −n f L1τ (Rn ) .

(4.17)

4.1 Preparations

81

Substep 2.1. Small cubes. Let |P| < 1. By using (4.17) the estimate of TN,P is split into two parts. According to the first one we have

∞

2(n+s−N)md ∑ |P|τ d m=( j ∨0) ≤

t −sq

#

1

P

f dL p (Rn ) τ

2m−( jP ∨0)+2

q

P

∞

2(n+s−N)md ∑ |P|τ d m= jP

p

| f (x)| p dx

2m− jP +2

−sq

t 1

dt t

dt t

dq

d

q

f dL p (Rn ) τ

if N > s + n + nτ and s > 0. Now we turn to the estimate of the second part. Let ε > 0. We obtain ∞

2(n+s−N)md ∑ |P|τ d m=( j ∨0)

2m−( jP ∨0)+2

t

−sq

1

P

P

∞

2(n+s−N)md d/p f dL1 (Rn ) ∑ |P| τ |P|τ d m= jP f dL1 (Rn ) τ

∞

|P|d/p |P|τ d

#

∑

[t

nτ −n

2m− jP +2 1

q f L1τ (Rn ) ] dx p

dt t (−s+nτ −n)q t

p

dt t

dq

dq

2(n+s−N)md 2d(m− jP )[(nτ −s−n)++ε ]

m= jP

f dL1 (Rn ) 2−d jP ( p −τ )n 2−d jP [(nτ −s−n)++ε ] 2 jP [n+s−N+(nτ −s−n)++ε ]d τ d f L1 (Rn ) τ 1

& ' n n N > max n + s + (nτ − s − n)+ + ε , n + s − + . p τ

if

Substep 2.2. Large cubes. Let |P| = 2rn for some r ∈ N. If N > s + n > n, then the first part (related to | f (x)|, see (4.17)) can be estimated from above by f dL p (Rn ) . τ For the estimate of the second part we argue as in Substep 2.1 and find dq dt t −sq [t nτ −n f L1τ (Rn ) ] p dx t 1 P d m+2 q 2 |P|d/p ∞ (n+s−N)md d (−s+nτ −n)q dt f L1 (Rn ) 2 t ∑ τ t |P|τ d m=r 1

∞

2(n+s−N)md ∑ |P|τ d m=r

f dL1 (Rn ) 2rnd( p −τ ) 1

τ

f dL1 (Rn ) , τ

#

2m+2

∞

q

∑ 2(n+s−N)md 2md[(nτ −s−n)++ε ]

m=r

p

82

if

4 Several Equivalent Characterizations

& ' n N > max n + s + (nτ − s − n)+ + ε , n + s + (nτ − s − n)+ + ε + − nτ . p

It remains to estimate ∑rm=0 . Here we need the assumption τ ≥ 1/p. Then, arguing as before, we find

2(n+s−N)md ∑ |P|τ d m=0 r

f dL1 (Rn ) τ f dL1 (Rn ) τ

2m+2

t −sq

#

1 r

∑2

P

(n+s−N)md

[t nτ −n f L1τ (Rn ) ] p dx

2m+2

t

dt t

(−s+nτ −n)q

1

m=0 r

qp

dt t

dq

dq

∑ 2(n+s−N)md 2md[(nτ −s−n)++ε ]

m=0

f dL1 (Rn ) ,

(4.18)

τ

if N > n + s + (nτ − s − n)+ + ε . The estimate of TN,P can be done in the same way. This finishes the proof of (ii). Step 3. Let p ∈ (0, 1) and assume τ ∈ [0, ∞). We only need to modify the estimate (4.18). To all dyadic cubes P of sidelength l(P) ≥ 1 we associate to f an extension EP f such that EP f denotes an extension of the restriction of f to 2P and EP f Bs0

n p,∞ (R )

≤ 2 EP f Bs0

p,∞ (2P)

.

s

0 For the definition of B p,∞ (2P) we refer to Sect. 6.4. Then

2(n+s−N)md ∑ |P|τ d m=0 r

2m+2

t 1

2(n+s−N)md = ∑ |P|τ d m=0 r

q

#

2m+2

[at (x)] dx p

P

p

dt t

dq

t −sq

1

# × t −n P

−sq

t/2 1. Then it is obvious that |h| 1. Then '1/p & | f (x) − f (y)| p 1 dx dy I1 + I2 , sp+n |P|τ P P |x − y| where '1/p | f (x + h) − f (x)| p dx dh , |h|sp+n |h| 0, it is not difficult to Remark 4.12. Let q ∈ (0, ∞). Since F∞,q s (Rn ) if, and only if f ∈ L∞ (Rn ) and prove that under the above restrictions, f ∈ F∞,q

f ♣,osl < ∞. F s (Rn ) ∞,q

4.5 The Hedberg-Netrusov Approach to Spaces of Besov-Triebel-Lizorkin Type In [70] Hedberg and Netrusov developed an axiomatic approach to function spaces of Besov-Triebel-Lizorkin type. This general theory covers our spaces Fp,s, qτ (Rn ) and Bs,p,τq (Rn ), at least under some restrictions. We shall consider two different realizations of the Hedberg-Netrusov approach.

4.5.1 Some Preliminaries Recall, Morrey spaces Mup (Rn ), 0 < u ≤ p ≤ ∞ have been defined in item (xvi) in Sect. 1.3. Let q ∈ (0, ∞] and s ∈ R. Then we define two types of quasi-Banach spaces of locally Lebesgue-integrable functions. By sq (Mup (Rn )) we denote the collection of all sequences { f j }∞j=0 of those functions such that { f j }∞j=0 sq (Mup (Rn ))

≡

∞

∑2

j=0

1/q jsq

f j qM p (Rn ) u

< ∞.

120


Furthermore, by Mu (Rn )(sq ) we denote the collection of all sequences { f j }∞j=0 of locally Lebesgue-integrable functions such that p

$ 1/q $ $ ∞ $ $ $ jsq q $ { f j }∞j=0 Mup (Rn )(sq ) ≡ $ 2 | f | j $ ∑ $ $ j=0 $

< ∞.

p Mu (Rn )

Of course, both sq (Mup (Rn )) and Mup (Rn )(sq ), are quasi-Banach spaces. With d = min{1, u, q} it holds { f j }∞j=0 + {g j }∞j=0ds (M p (Rn )) ≤ { f j }∞j=0 ds (M p (Rn )) + {g j }∞j=0 ds (M p (Rn )) , u

q

u

q

u

q

and { f j }∞j=0 + {g j }∞j=0 dM p (Rn )(s ) ≤ { f j }∞j=0 dM p (Rn )(s ) + {g j }∞j=0 dM p (Rn )(s ) u

u

q

u

q

q

for all sequences { f j }∞j=0 and {g j }∞j=0 of locally Lebesgue-integrable functions. The maximal operator # Mr f (x) ≡ sup a−n

1/r

B(0,a)

a>0

| f (x + y)|r dy

is bounded on these two types of spaces if r is chosen in an appropriate way. Indeed, with w ≡ p/r we find # 1/r 1

B

r/u

r

Mr f Mup (Rn ) = sup |B| w − u

B

[M(| f |r )(x)]u/r dx

.1/r = M(| f |r )M w (Rn ) u/r .1/r | f |r M w (Rn ) u/r

f

p Mu (Rn )

holds if 0 < r < u ≤ p ≤ ∞; see Chiarenza and Frasca [39]. Immediately one obtains {Mr f j }∞j=0 sq (Mup (Rn )) { f j }∞j=0 sq (Mup (Rn ))

(4.37)

under the same restrictions. The vector-valued situation has been treated by Tang and Xu [139]. We have {Mr f j }∞j=0 Mup (Rn )(sq ) ≤ { f j }∞j=0 Mup (Rn )(sq ) ,

(4.38)

if 0 < r < min{u, q} ≤ u ≤ p ≤ ∞. Finally, we have to estimate the norms of the shift operators. Let

4.5 The Hedberg-Netrusov Approach to Spaces of Besov-Triebel-Lizorkin Type

S+ ({ f j }∞j=0 ) ≡ { f j+1 }∞j=0

and

121

S− ({ f j }∞j=0 ) ≡ { f j−1 }∞j=0 ,

f−1 = 0. Obviously, (S+ )N |L (E) 2−Ns and (S− )N |L (E) 2Ns , where E stands either for sq (Mup (Rn )) or for Mup (Rn )(sq ), and with constants behind independent of N. These observations can be summarized by saying that both types of spaces belong to the class S(s, s, r) with r as above; see [70, p. 6].

4.5.2 Characterizations by Differences Now we employ [70, Proposition 1.1.12, Theorem 1.1.14] with E = Mup (Rn )(sq ). s (Rn ) have been defined in item Recall that the Triebel-Lizorkin-Morrey spaces E pqu (xxvi) in Sect. 1.3. Corollary 4.10. Let v ∈ (0, ∞], u ∈ (0, ∞), p ∈ [u, ∞], and s ∈ R such that &

1 1 1 − 1, − r ∈ (0, min{u, q}) and s > n max r r v

' .

Then the following assertions are equivalent for functions in Lrloc (Rn ): s (Rn ); (i) f ∈ E pqu v (ii) f ∈ L loc (Rn ) and

f E s

n pqu (R )

$ $ ≡ $ f Lv (B(x,1)) $M p (Rn ) u $ $ $ ∞ q/v 1/q $ $ $ j(s+ nv )q $ +$ |ΔhM f (x)|v dh $ $ ∑2 B(x,2− j ) $ $ j=1

< ∞. p

Mu (Rn )

The quasi-norms f E pqu s (Rn ) and f s E

n pqu (R )

are equivalent.

For us it will be convenient to reformulate this a bit. We shall use the abbreviation # bv,t (x) ≡

1 tn

B(x,t)

|ΔhM f (x)|v dh

1/v .

122


Then, using the monotonicity of bv,t with respect to t we find $ $ $ ∞ q/v 1/q $ $ $ $ ∑ 2 j(s+ nv )q $ |ΔhM f (x)|v dh $ $ B(x,2− j ) $ j=1 $ p n Mu (R ) $# $ 1/q $ $ 1 dt $ $ q ∼$ t −sq bv,t (x) . $ $ 0 $ p n t Mu (R )

p

Replacing Mu (Rn ) by its original definition we get the following supplement to Corollary 4.10. Corollary 4.11. Let u, p, q, v, r, s be as in Corollary 4.10. Then the following assertions are equivalent for functions in Lrloc (Rn ): s (Rn ); (i) f ∈ E pqu (ii) f ∈ Lvloc (Rn ) and

f E s (Rn ) pqu

≡ sup P∈Q

1 1

1

|P| u − p

+ sup P∈Q

P

1

B(x,1)

1 1

|P| u − p

P

The quasi-norms f E pqu s (Rn ) and f s E

n pqu (R )

1/u

u/v | f (y)| dy v

1 0

t −sq bqv,t (x)

dx dt t

1/u

u/q dx

< ∞.

are equivalent.

s, 1/u−1/p

s (Rn ) (see Corollary Remark 4.13. In view of the identity Fu, q (Rn ) = E pqu 3.3(i)), the above corollary supplements the results of Sect. 4.3.1. s The next corollary deals with Besov-Morrey spaces N pqu (Rn ); see item (xxv) in Sect. 1.3. Again we employ Proposition 1.1.12 and Theorem 1.1.14 in [70], this time with E = sq (Mup (Rn )).

Corollary 4.12. Let s ∈ R, v ∈ (0, ∞] and 0 < r n max − 1, − . r r v Then the following assertions are equivalent for functions in Lrloc (Rn ): s (Rn ); (i) f ∈ N pqu v (ii) f ∈ L loc (Rn ) and

f N s

n pqu (R )

$ $ ≡ $ f Lv (B(x,1)) $M p (Rn ) u ⎛ $# 1/v$ $ $q ∞ n )q $ $ j(s+ M v v $ +⎝∑ 2 |Δh f (x)| dh $ − j) $ $ B(x,2 j=1

p Mu (Rn )

⎞1/q ⎠

0.

(4.40)

For each dyadic cube Q = 2− j ([0, 1)n + k), we set

ΨQ ≡ |Q|−1/2Ψ (2 j (· − xQ )) and fQ ≡

CΨ−1 |Q|−1/2

Rn

f (y)ΨQ (y) dy,

where xQ denotes the center of Q. Then there exists a positive constant C such that for all f ∈ Cs+nτ −n/p(Rn ), + f ( f − fQ )ΨQ B ≤ C |Q|τ −1/2 f ♥ s+n τ −n/p s, τ n n C (R ) . B (R ) p, q

(4.41)

Here f ♥ is defined as in Sect. 4.1.3 with M = 1. s, τ B (Rn ) p, q

Proof. Observe that by (4.40), Rn

( f (y) − fQ )ΨQ (y) dy =

Rn

f (y)ΨQ (y) dy − fQ

Rn

ΨQ (y) dy = 0.

(4.42)

Let PQ ≡ {x ∈ Rn : |2 j xi − ki | ≤ 1, i = 1, · · · , n}. Then supp ΨQ ⊂ PQ and |PQ | ∼ |Q|. As a consequence of (4.40) and (4.42), we obtain % % % % | f (x) − fQ | = %% f (x) − fQ − (CΨ )−1 |Q|−1/2 ( f (y) − fQ )ΨQ (y) dy%% ≤ (CΨ )−1 |Q|−1/2

Rn

Rn

| f (x) − f (y)||ΨQ (y)| dy

(4.43)

for arbitrary x ∈ Rn . Since 0 < s + nτ − n/p < 1, the definition of Cs+nτ −n/p(Rn ) (see item (iv) in Sect. 1.3) implies that

4.6 A Characterization of As,p,τq (Rn ) via a Localization of Asp, q (Rn )

127

| f (x) − f (y)| ≤ f Cs+nτ −n/p (Rn ) |x − y|s+nτ −n/p . Thus, if x ∈ 3PQ , by the support condition of Φ , we have | f (x) − fQ | |Q|(s+nτ −n/p)/n f Cs+nτ −n/p (Rn ) Ψ L∞ (Rn ) .

(4.44)

Notice that for all x ∈ Rn ,

Δh1 [( f − fQ )ΨQ ](x) = ( f (x + h) − fQ ) ΨQ (x + h) − ( f (x) − fQ) ΨQ (x) (4.45) = ΨQ (x) ( f (x + h) − f (x)) + ( f (x + h) − fQ ) (ΨQ (x + h) − ΨQ(x)) . (4.46) The formula (4.45) will be used in case t ≤ 2(l(Q) ∧ 1), whereas the decomposition (4.46) will be applied if 2l(Q) < t < 2 (l(Q) < 1). Oriented on these decompositions we introduced the following notation

I1 ≡ I2 ≡ I3 ≡

2(l(Q)∧1)

t −sq

Rn

0

&

2(l(Q)∧1)

· · ·t −n

0

&

2

2l(Q)

··· t

−n

··· t

−n

t −n

1q dt |ΨQ (x)| p | f (x + h) − f (x)| p dh dx , t t/2≤|h| 0, we have ( f − fQ )ΨQ L p (Rn ) |Q|−1/2 |Q|1/p |Q|(s+nτ −n/p)/n f Cs+nτ −n/p (Rn ) |Q|τ −1/2 f Cs+nτ −n/p (Rn ) .

4.6 A Characterization of As,p,τq (Rn ) via a Localization of Asp, q (Rn )

131

When |Q| > 1, using (4.43), we obtain | f (x) − fQ | |Q|−1 Ψ L∞ (Rn )

PQ

| f (x) − f (y)| dy.

Because of p ≥ 1 we can apply Hölder’s inequality and find ( f − fQ )ΨQ L p (Rn ) |Q|−3/2 1

3

|Q| p − 2

PQ

τ −1/2

|Q|

p

#

PQ

PQ

| f (x) − f (y)| dy 1/p

| f (x)| p dx

1

1/p dx

3

+ |Q| p − 2

1/p

# PQ

| f (y)| p dy

f Lτp (Rn ) .

This proves one direction of the claim in (i). Step 2. We suppose f ∈ Lτp (Rn ) and f < ∞. Applying Lemma 4.14 in s, τ B (Rn ) p, q

combination with Theorem 4.7(i) we conclude that f ∈ Bs,p,τq(Rn ) and < ∞, f Bs,p,τq(Rn ) f Lτp (Rn ) + f s, τ B (Rn ) p, q

which completes the proof of (i). Step 3. Proof of (ii). This time we work with Theorem 4.7(ii) and argue as above. Only one further comment is needed. Because f ∈ C(Rn ) we have N p ( f ) ≤ f C(Rn ) . The most interesting example is given by the choice Ψ = X , where X denotes the characteristic function of the cube [−1, 1]n. This function does not satisfy the condition (4.52). However, there are some simple modifications to justify such a choice. We shall work with a modified system of dyadic cubes in this context. Let Pj,k ≡ {x ∈ Rn : 2− j (ki − 1) ≤ xi < 2− j (ki + 1) , i = 1, . . . , n} ,

j ∈ Z, k ∈ Zn ,

and denote by Q ∗ the collection of all such cubes. It is well known that X ∈ Bsp, q (Rn ) if and only if either s < 1/p and q is arbitrary or s = 1/p and q = ∞; see [119, Lemma 2.3.1/3]. Corollary 4.18. Let p ∈ [1, ∞), q ∈ (0, ∞], and s and τ be as in (4.53) (i) Let s ∈ (0, 1/p). Then f ∈ Bs,p,τq (Rn ) if, and only if f ∈ Lτp (Rn ) and $# $ $ $ 1 1/2−τ $ $ ≡ sup |P| f (y) dy X < ∞ , (4.56) f f − s, τ P$ $ B p, q (Rn ) |P| P P∈Q∗ Bsp, q (Rn )

132


where XP is defined as in Lemma 4.13. Furthermore, f Lτp (Rn ) + f s, τ B (Rn ) p, q

and f Bs,p,τq(Rn ) are equivalent. (ii) Let p ∈ (1, ∞). Then the assertions in (i) remain true if s = 1/p and q = ∞. √ Proof. Let K be a natural number such that 2K ≥ 3 n. Then, let Ψ be the characteristic function of the cube centered at the origin and having side-length 2K. Under the given restrictions on s we can apply Theorem 4.16 with respect to this function Ψ . Next, observe sup |Q|1/2−τ ( f − fQ )ΨQ Bsp, q (Rn )

Q∈Q

$# $ $ $ 1 $ f − ∼ sup |P|1/2−τ $ f (y) dy X , P$ $ |P| P P∈Q∗ Bsp, q (Rn )

which proves the claim.

4.6.2 A Characterization of Fs,p,τq (Rn ) We proceed as in the previous subsection. Lemma 4.15. Let p ∈ (1, ∞), q ∈ (0, ∞), τ ∈ [0, ∞) and s be as in (4.39). Let

Ψ ∈ Fp,s q(Rn ) ∩ L∞ (Rn ) be a function such that (4.40) is satisfied. Then there exists a positive constant C such that for all f ∈ Cs+nτ −n/p (Rn ), ♣ τ −1/2 f ( f − fQ )ΨQ ♣ ≤ C |Q| + f s+n τ −n/p s, τ n F C (R ) . F (Rn )

(4.57)

p, q

♣ If q ∈ [1, ∞), (4.57) also holds with ( f − fQ )ΨQ ♣ F and f F s, τ (Rn ) replaced, rep, q

♥ ♣ ♥ spectively, by ( f − fQ )ΨQ ♥ F and f F s, τ (Rn ) . Here f F s, τ (Rn ) and f F s, τ (Rn ) are

defined as in Sect. 4.1.3 with M = 1.

p, q

p, q

p, q

Proof. By similarity, we only give the proof for ( f − fQ )ΨQ ♣ F . We comment on the modifications needed in comparison with the proof of Lemma 4.13. Again we employ (4.45) and (4.46). This leads to the following definitions of I1 through I4 : $ $ $& 2(l(Q)∧1) q ' 1q $ # $ $ dt $ I1 ≡ $ t −sq t −n |ΨQ (x)|| f (x + h) − f (x)| dh $ $ 0 t t/2≤|h| max {J − n, J − n + n(τ − 1/p)}, where J is as in Definition 3.2. Let M > max{s, s + n(τ − 1/p)} and extM be as in (6.29). We prove that extM is a linear and bounded extension operator from As,p,τq(Rn+ ) into As,p,τq (Rn ). Notice that the assumption s > max {J − n, J − n + n(τ − 1/p)} together with Proposition 2.7 implies that As,p,τq(Rn+ ) ⊂ Lτp (Rn+ ). Thus, extM f makes sense for f ∈ As,p,τq (Rn+ ). As in the proof of [146, Theorem 4.5.2], we denote by oscM u f the oscillations n in the sense of (4.33). By f the oscillations based on R based on Rn and OscM + u [146, p. 224, (7)–(9)], we have the following estimates: Let x = (x , xn ) ∈ Rn and t ∈ (0, 2]. If xn > t, then M oscM 1 (extM f )(x,t) = Osc1 f (x,t);

if xn < −t, then oscM 1 (extM f )(x,t)

M

∑ OscM1 f ((x , −λk xn ),Ct)

k=0

for some positive constant C; if |xn | ≤ t, then M

oscM 1 (extM f )(x,t) Osc1 f ((x , |xn |),Ct)

172

6 Key Theorems

for some positive constant C. By Theorems 4.10 and 4.13, the definition of extM f and Remark 4.11, we obtain that f Bs,p,τq (Rn ) ≤ extM f Bs,p,τq (Rn ) +

1 f Lτp (Rn ) + sup τ + P∈Q |P| n

)

C(l(P)∧1)

t −sq

0

P⊂R+

×

P

p OscM−1 f (x, Mt) dx 1

q/p

dt t

*1/q

f Bs,p,τq(Rn ) +

and f Fp,s, qτ (Rn ) ≤ extM f Fp,s, qτ (Rn ) +

1 f Lτp (Rn ) + sup τ + |P| P∈Q n

) P

C(l(P)∧1)

t −sq

0

P⊂R+

q dt × OscM−1 f (x, Mt) 1 t

*1/p

p/q dx

f Fp,s, qτ (Rn ) . +

Then ext ≡ extM is the desired extension operator in the case s > max {J − n, J − n + n(τ − 1/p)}. For the case when s ≤ max {J − n, J − n + n(τ − 1/p)}, choose σ ∈ R such that s + σ > max {J − n, J − n + n(τ − 1/p)}. Let M > max{s + σ , s + σ + n(τ − 1/p)}. From the proved conclusion and Proposition 6.5, we deduce that ext ≡ Jσ ◦ extM ◦ J−σ is the desired extension operator, which completes the proof of Theorem 6.11.

6.4 Spaces on Rn+ and Smooth Domains

173

6.4.2 Spaces on Smooth Domains We now deal with the spaces on a bounded C∞ domain Ω in Rn . Let D(Ω ) be the set of all C∞ (Ω ) functions supported in Ω and denoted by D (Ω ) its topological dual. Observe that φ ∈ D(Ω ) can be extended to S (Rn ) by setting φ ≡ 0 outside Ω . Then the restriction operator Re : S (Rn ) → D (Ω ) can be defined naturally as an adjoint operator. The Besov-type spaces and Triebel-Lizorkin-type spaces on Ω are defined as follows. Definition 6.2. Let s ∈ R, τ ∈ [0, ∞) and p, q ∈ (0, ∞]. The space As,p,τq (Ω ) is defined to be the restriction of As,p,τq (Rn ) on Ω , quasi-normed by f As,p,τq(Ω ) ≡ inf gAs,p,τq(Rn ) , where the infimum is taken over all g ∈ As,p,τq(Rn ) with g|Ω = f in the sense of D (Ω ). The space As,p,τq (Ω ) is also a quasi-Banach space. Let Cm (Ω ) be the set of all functions f on Ω such that f = g|Ω for some functions g ∈ Cm (Rn ) with ∂ α gL∞ (Rn ) ∼ ∂ α f L∞ (Ω ) for all |α | ≤ m. Similarly to Theorem 6.10, we also obtain the pointwise multiplication theorem for As,p,τq (Ω ). Theorem 6.12. Let m ∈ N, s, τ , p, q be as in Theorem 6.1. If m is sufficiently large, then there exists a positive constant C(m) such that for all g ∈ BCm (Ω ) and all f ∈ As,p,τq (Ω ), g f As,p,τq(Ω ) ≤ C(m)

∑

|α |≤m

∂ α gL∞ (Ω ) f As,p,τq(Ω ) .

Theorem 6.12 is an immediate corollary of Definition 6.2 and Theorem 6.1 To obtain the extension property for As,p,τq (Ω ), we need some preparations. Since Ω is bounded, there exists a finite collection {Bm }km=1 of open balls and a C∞ domain Ω0 such that Ω0 ⊂ Ω and Ω ⊂ Ω0 ∪ (∪km=1 Bm ) . Furthermore, there exist k-diffeomorphisms ψ1 , · · · , ψk on Rn satisfying that, for all m ∈ {1, · · · , k}, ψm (Bm ∩ Ω ) = ψm (Bm ) ∩ Rn+ (6.30)

174

6 Key Theorems

and

ψm (Bm ∩ ∂ Ω ) = ψm (Bm ) ∩ ∂ Rn+ ; see [146, Sect. 5.1.3] or [124, Sect. 5]. Let φ0 , · · · , φk ∈ Cc∞ (Rn ) be the C∞ (Rn ) resolution of unity satisfying that supp φ0 ⊂ Ω0 , supp φm ⊂ Bm for m ∈ {1, · · · , k} and ∑km=0 φm ≡ 1 in a neighborhood of Ω . Now we establish the extension theorem for As,p,τq (Ω ). The proof of Theorem 6.13 is similar to that for [124, Theorem 5.4]. For the sake of convenience of the reader, we give the details. Theorem 6.13. Let p ∈ [1, ∞), q ∈ (0, ∞], s ∈ R and τ ∈ [0, ∞). There exists a linear and bounded operator Ext from As,p,τq(Ω ) into As,p,τq (Rn ) such that Re ◦ Ext is the identity in As,p,τq (Ω ). Proof. Let ext be the extension operator obtained in Theorem 6.11. Let k ∈ N be sufficiently large. For each m ∈ {1, · · · , k}, let φ m be a bump function such that φ m ≡ 1 in a neighborhood of supp φm and has support in Bm . For f ∈ As,p,τq(Ω ), we choose a representation g ∈ As,p,τq(Rn ) of f such that gAs,p,τq(Rn ) f As,p,τq(Ω ) . Define Ext f ≡ φ0 · g +

k

∑ φm ·

m=1

, + ext (φm · g) ◦ ψm−1|Rn+ ◦ ψm .

From this and the support conditions of φm , we deduce that for all ϕ ∈ S (Rn ), Ext f , ϕ = φ0 · g, ϕ +

k

∑

m=1

= φ0 · g, ϕ +

k

∑

m=1

-+

, . ext (φm · g) ◦ ψm−1|Rn+ ◦ ψm , φm · ϕ -

. ext (φm · g) ◦ ψm−1|Rn+ , |J(ψm−1 )| · [φm · ϕ ] ◦ ψm−1 ,

which together with (6.30) implies that Ext f is independent of the choice of g. It was also proved in [124] that for all test functions h ∈ D(Ω ), Ext f |Ω , h = f , h. In fact, let Eh be the extended function of h by setting Eh ≡ h on Ω and Eh ≡ 0 outside Ω . Then Eh ∈ S (Rn ) and we have Ext f |Ω , h = Ext f , Eh = φ0 · g, Eh

. k + ∑ ext (φm · g) ◦ ψm−1|Rn+ , |J(ψm−1 )| · [φm · Eh] ◦ ψm−1 , m=1

6.4 Spaces on Rn+ and Smooth Domains

175

which together with the fact that [φm · Eh] ◦ ψm−1 is supported in Rn+ yields that Ext f |Ω , h = φ0 · g, Eh +

k

∑

/

(φm · g) ◦ ψm−1, |J(ψm−1 )| · [φm · Eh] ◦ ψm−1

0

m=1

= φ0 · g, Eh +

k

∑

/

φm · g, φm · Eh

0

m=1

= g, Eh = f , h. Thus, Ext f |Ω = f in D (Ω ). By the definition of Ext f , the pointwise multiplication property in Theorem 6.1, the diffeomorphism property in Theorem 6.7 and the extension conclusion in Theorem 6.11, we obtain that Ext f As,p,τq (Rn ) gAs,p,τq(Rn ) f As,p,τq(Ω ) , which further yields that the operator Ext is the desired one, and then completes the proof of Theorem 6.13.

Remark 6.12. Let Ω be a bounded Lipschitz domain. Then Rychkov [121] has s (Ω ) and proved the existence of a universal extension operator for all spaces Fp,q Bsp,q (Ω ), simultaneously.

Chapter 7

Inhomogeneous Besov-Hausdorff and Triebel-Lizorkin-Hausdorff Spaces

Similarly to [164, Sects. 4, 5] and [165, Sects. 5, 6], in this section, we introduce s, τ n the inhomogeneous Besov-Hausdorff space BH p, q (R ) and the Triebel-Lizorkins, τ n Hausdorff space FH p, q (R ), whose dual spaces are, respectively, certain Besov-type space and Triebel-Lizorkin-type space when p ∈ (1, ∞), q ∈ [1, ∞), s ∈ R and 1 τ ∈ [0, (p∨q) ]. Recall that (p ∨ q) denotes the conjugate index of p ∨ q, namely, 1 p∨q

s, τ s, τ 1 n n + (p∨q) = 1. The spaces BH p, q (R ) and FH p, q (R ) have some properties simi-

lar to those of As,p,τq (Rn ), which include the ϕ -transform characterization, embedding properties, smooth atomic and molecular decompositions.

7.1 Tent Spaces We begin with recalling the notion of Hausdorff capacities; see [1, 2, 163]. Definition 7.1. Let d ∈ (0, ∞) and E ⊂ Rn . The d-dimensional Hausdorff capacity of E is defined by (∞) Λd (E)

≡ inf

∑ j

rdj

:E⊂

B(x j , r j ) ,

(7.1)

j

where the infimum is taken over all covers of E by countable families of open balls with radius r j . (∞)

The notion of Λd

(∞)

in Definition 7.1 when d = 0 also makes sense, and Λ0

is

(∞) monotone, countably subadditive; however, Λ0 does not vanish on the empty set, (∞) (∞) it has the property that for all sets E ⊂ Rn , Λ0 (E) ≥ 1 and Λ0 (E) = 1 if E is

bounded.


177

178

7 Inhomogeneous Besov-Hausdorff and Triebel-Lizorkin-Hausdorff Spaces

(∞) , was introduced in [163], which is A dyadic version of Hausdorff capacity, Λ d defined by

∑ l(I j )

(∞) Λd (E) ≡ inf

:E⊂

d

j

◦

,

Ij

j

where now the infimum ranges only over covers of E by dyadic cubes {I j } j and A◦ denotes the interior of the set A. (∞) (∞) are equivalent, i. e., there exist positive, finite constants Recall that Λd and Λ d C1 and C2 , only depending on the dimension n, such that (∞)

(∞)

(∞)

C1Λd (E) ≤ Λd (E) ≤ C2Λd (E) for all E ⊂ Rn .

(7.2)

We also recall the notions of the Choquet integral with respect to the Hausdorff (∞) (∞) capacities Λd and Λd ; see [1, 2]. For any function f : Rn → [0, ∞], define

(∞)

Rn

f (x) d Λd (x) ≡

∞ 0

(∞)

Λd ({x ∈ Rn : f (x) > λ }) d λ .

This functional is not sublinear, so sometimes we need to use an equivalent integral (∞) , which is sublinear, and satisfies Fatou’s lemma that for all with respect to Λ d (∞) -measurable functions { fm }∞ , nonnegative Λ m=1

d

(∞)

lim inf fm dΛ d

Rn m→∞

≤ lim inf m→∞

(∞)

Rn

. fm dΛ d

n For any measurable function f on Rn+1 + and all x ∈ R , we define the nontangential maximal function N f (x) by

N f (x) ≡ sup | f (y, t)|. |y−x| k}. (∞)

Then by (7.3) and the monotone property of Λ0 , we have

(∞)

[N ω (x)](p∨q) dΛ0 (x) ∞

(∞) = {x ∈ Rn : [N ω (x)](p∨q) > λ } d λ Λ0

1≥

= ≥ =

Rn

0 ∞

∑

k+1

k=0 k ∞ k+1

∑

(∞)

{x ∈ Rn : [N ω (x)](p∨q) > λ } d λ

(∞)

{x ∈ Rn : [N ω (x)](p∨q) > k + 1} d λ

Λ0 Λ0

k=0 k ∞ (∞) Λ0 (Ek+1 ). k=0

∑

(∞)

Notice that Λ0 (E) ≥ 1 for any set E. The argument above yields that (∞)

Λ0 (Ek ) = 0 for all k ∈ N, which implies that for all x ∈ Rn and k ∈ Z, ω k (x) ≤ 1. Thus, s,τ f BTp,q (Rn+1 ) ≡ Z+

q 2−ksq f k p n L (R ) k=0 ∞

∑

1/q

180


1/q ∞ −ksq k q s,τ ≡ ∑2 |f | f FTp,q (Rn+1 Z+ ) k=0

and

.

L p (Rn )

s,τ (ii) It is easy to check that · ATp,q (Rn+1 ) is a quasi-norm, namely, there exists a Z+

s,τ nonnegative constant ρ ∈ [0, 1] such that for all f1 , f2 ∈ ATp,q (Rn+1 Z+ ),

ρ s,τ s,τ s,τ f1 + f2 ATp,q (Rn+1 ) ≤ 2 ( f 1 ATp,q (Rn+1 ) + f 2 ATp,q (Rn+1 ) ). Z+

Z+

(7.4)

Z+

In fact, let ω1 , ω2 be nonnegative Borel measurable functions on Rn+1 + satisfying (7.3) such that

1/q

∞

∑ 2 jsq f j [ω j ]−1 qL p (Rn )

s,τ ≤ 2 fi BTp,q (Rn+1 ) Z+

j=0

for i ∈ {1, 2}. Notice that ω ≡ 2

1 − (p∨q)

s,τ f1 + f2 BTp,q (Rn+1 ) Z+

max{ω1 , ω2 } still satisfies (7.3). Then 1/q

∞

∑2

jsq

j=0

j j q ( f1 + f2 )[ω j ]−1 L p (Rn )

s,τ s,τ f1 BTp,q (Rn+1 ) + f 2 BTp,q (Rn+1 ) . Z+

Z+

s,τ The proof of FTp,q (Rn+1 Z+ ) is similar. (iii) If ω satisfies (7.3), then

(∞)

[ω (x, t)](p∨q) Λnτ (p∨q) (B(x, t)) =

Rn

(∞)

[ω (x, t)](p∨q) χB(x,t) (y) dΛnτ (p∨q) (y)

Rn

(∞)

(N ω (y))(p∨q) dΛnτ (p∨q) (y)

1, (∞)

which together with Λnτ (p∨q) (B(x, t)) = t nτ (p∨q) further implies that ω (x, t) t −nτ . (∞) (iv) Let 0 < a ≤ b ≤ 1/τ . We claim that Rn [N ω (x)]a dΛnτ a (x) < ∞ induces

(∞)

Rn

[N ω (x)]b dΛnτ b (x) < ∞.

To this end, without loss of generality, we may assume Rn

(∞)

[N ω (x)]a dΛnτ a (x) ≤ 1.

For all l ∈ Z, set El ≡ {x ∈ Rn : N ω (x) > 2l }. Then 1≥

(∞)

Rn

[N ω (x)]a dΛnτ a (x) ∼

∑ 2laΛnτ a (El ). (∞)

l∈Z

7.1 Tent Spaces

181

For each l ∈ Z, there exists a countable ball cover {B(x jl , r jl )} j of El such that

Λnτ a (El ) ∼ ∑ rnjlτ a . (∞)

j

Thus, ∑l∈Z 2la ∑ j rnjlτ a 1. For all j and l, 2l rnjlτ 1. Then 2lb rilnτ b 2la rilnτ a since a ≤ b. Therefore,

∑ 2lbΛnτ b (El )

(∞)

(∞)

[N ω (x)]b dΛnτ b (x) ∼

Rn

l∈Z

∑ 2lb ∑ rnjlτ b

j

l∈Z

∑ 2 ∑ rnjlτ a

la

j

l∈Z

1. This proves our claim. s,τ Similarly to [164,165], we have the following atoms for the spaces ATp,q (Rn+1 Z+ ).

Definition 7.3. Let s ∈ R, p ∈ (1, ∞), q ∈ [1, ∞) and τ ∈ (0, on

Rn+1 Z+

is called an

s,τ ATp,q (Rn+1 Z+ )-atom

1 ]. (p∨q)

A function a

associated a ball B, if a is supported in

T (B) ≡ {(x,t) ∈ Rn+1 Z+ : B(x,t) ⊂ B} and satisfies that Rn

p/q

∞

∑2

jsq

−j

|a (x)| χT (B) (x, 2 ) j

q

dx ≤ |B|−τ p

j=0

s,τ s,τ n+1 if ATp,q (Rn+1 Z+ ) = FTp,q (RZ+ ), or ∞

∑ 2 jsq

j=0

Rn

|a j (x)| p χT (B) (x, 2− j ) dx

q/p

≤ |B|−τ q

s,τ s,τ n+1 if ATp,q (Rn+1 Z+ ) = BTp,q (RZ+ ).

Lemma 7.1. Let s ∈ R, p ∈ (1, ∞), q ∈ [1, ∞) and τ ∈ (0, a positive constant C such that all

s,τ ATp,q (Rn+1 Z+ )-atoms a

s,τ a ATp,q (Rn+1 ) ≤ C. Z+

1 ]. (p∨q)

Then there exists

s,τ belong to ATp,q (Rn+1 Z+ ) with

182


s,τ Proof. By similarity, we only consider the space FTp,q (Rn+1 Z+ ). Suppose a is an s,τ n+1 FTp,q (RZ+ )-atom associated with a ball B ≡ B(xB , rB ). Let ε be a positive real number such that nτ + ε > nτ (p ∨ q) . We set

−nτ (p∨q) κ rB min

ω (x, t) ≡

nτ +ε 1/(p∨q)

rB

|x − xB|2 + t 2

1,

,

where the positive constant κ will be determined later. Notice that for all x ∈ Rn , the √ distance between the cone Γ (x) and (xB , 0) is |x − xB|/ 2. Thus the nontangential maximal function of ω is bounded by

−nτ (p∨q)

N ω (x) ≤ κ rB

nτ +ε 1/(p∨q) √ 2rB min 1, . |x − xB |

Therefore, by nτ + ε > nτ (p ∨ q) ,

κ −1

(∞)

[N ω (x)](p∨q) dΛnτ (p∨q) (x) nτ +ε √ 2rB −nτ (p∨q) (∞) ≤ rB min 1, d Λnτ (p∨q) (x) |x − xB | Rn n τ +ε √ ∞ 2rB (∞) −nτ (p∨q) n = Λnτ (p∨q) min 1, x ∈ R : rB >λ dλ |x − xB | 0 ≤ ≤

Rn

r−nτ (p∨q) B 0

√

(∞) nτ (p∨q) −1/(nτ +ε ) Λnτ (p∨q) B xB , 2rB (λ rB ) dλ

r−nτ (p∨q) √ B

nτ (p∨q) −1/(nτ +ε ) nτ (p∨q)

2rB (λ rB

0

)

dλ

= C, where the constant C is independent of rB . Choose κ = C−1 to make ω satisfy (7.3). Notice that if (x, 2−k ) ∈ T (B), then [ω k (x)]−1 ∼ rBnτ . Then we have

Rn

p/q

∞

∑2

ksq

k=0

∼ rBnτ p

−q

|a (x)| [ω (x)] k

q

∑2

ksq

k

dx p/q

Rn

−k

|a (x)| χT (B) (x, 2 ) k

q

dx

k∈Z

1, s,τ (Rn+1 which yields a ∈ FTp,q Z+ ) and completes the proof of Lemma 7.1.

7.1 Tent Spaces

183

s,τ To obtain the atomic decomposition characterization of ATp,q (Rn+1 Z+ ), we need the following lemma.

Lemma 7.2. Let s ∈ R, p ∈ (1, ∞) and τ ∈ (0, s,τ ∑ j g j FTp,p (Rn+1 ) < ∞, then

1 p ].

s,τ If {g j } j ⊂ FTp,p (Rn+1 Z+ ) and

Z+

s,τ (Rn+1 g ≡ ∑ g j ∈ FTp,p Z+ ) j

and there exists a positive constant C, independent of {g j } j , such that s,τ s,τ g FTp,p (Rn+1 ) ≤ C ∑ g j FTp,p (Rn+1 ) . Z+

Z+

j

s,τ Proof. Without lost of generality, we may assume that λ j = g j FTp,p (Rn+1 ) > 0 for Z+

s,τ all j. Let f j ≡ λ j−1 g j . Then f j FTp,p (Rn+1 ) = 1 and g = ∑ j λ j f j . For any ε > 0, take Z+

ω j ≥ 0 such that

Rn

and

(∞)

[N ω j (x)] p dΛnτ p (x) ≤ 1

1/p ∞ ∑ 2ksp | f jk | p [ω kj ]−p k=0

≤ 1 + ε.

L p (Rn )

Since p > 1, then p p/p |g| p = ∑ λ j f j ≤ ∑ λ j | f j | p [ω j ]−p . ∑ λ j [ω j ] p j j j Notice that ∑ j λ j < ∞. Define

ω

1/p −1/p = C1 C2

∑λj

−1/p

j

1/p

∑ λ j [ω j ]

p

,

j

where C1 and C2 are as in (7.2). Notice that the vanishing of ω implies the vanishing of all ω j , which only happen whenever all the g j vanish, namely, when g is zero. Then by (7.2), the subadditivity of the nontangential maximal function, and the (∞) , we obtain sublinear property of the integral with respect to Λ d

Rn

[N ω (x)]

p

(∞) dΛnτ p (x)

(∞) (x) [N ω (x)] p dΛ nτ p −1 (∞) ≤ C2−1 ∑ λ j λ ∑ j [N ω j (x)] p dΛnτ p (x)

≤ C1−1

Rn

j

≤ 1.

j

Rn

184


Furthermore, we have

∞

∑ 2ksp|gk (x)| p [ω k (x)]−p dx Rn k=0

∑λj

p/p

∑λj

j

p/p

j

∑λj

∑ 2ksp ∑ λ j | f jk (x)| p [ω kj (x)]−p

Rn k=0

∑λj

j

p

∞

dx

j

∞

∑ 2ksp| f jk (x)| p [ω kj (x)]−p dx Rn k=0

(1 + ε ) p .

j

s,τ Therefore, g FTp,p (Rn+1 ) ∑ j λ j , which completes the proof of Lemma 7.2. Z+

As an important tool of this section, we need the following Lemma 4.1 in [44]. Lemma 7.3. Let d ∈ (0, n] and {I j } be a sequence of dyadic cubes in Rn such that ∑ j |I j |d/n < ∞. Then there exists a sequence {Jk } of dyadic cubes with mutually disjoint interiors, ∪k Jk = ∪ j I j and

∑ |Jk |d/n ≤ ∑ |I j |d/n. j

k

Moreover, if a set O ⊂ (∪ j I j ) , then the tent T (O) ⊂

T ((Jk )∗ ) ,

k

√ where (Jk )∗ is the cube with the same center as Jk but 5 n times the side length. s,τ (Rn+1 We then have the following atomic decomposition of ATp,q Z+ ). The proof is similar to that of [164, Theorem 4.1(i)]; see also [44, Theorem 5.4].

Proposition 7.1. Let s ∈ R, p ∈ (1, ∞), q ∈ [1, ∞) and τ ∈ (0, s,τ ATp,q (Rn+1 Z+ ),

1 ]. (p∨q)

s,τ ATp,q (Rn+1 Z+ )-atoms

then there exists a sequence {am }m of sequence {λm }m ⊂ C such that f = ∑m λm am pointwise and . ∑ |λm | ≤ C f ATp,qs,τ (Rn+1 Z+ ) m

s,τ In particular, if p = q ∈ (1, ∞), then f = ∑m λm am also in ATp,p (Rn+1 Z+ ).

If f ∈

and an 1 -

7.1 Tent Spaces

185

s,τ Conversely, if p = q ∈ (1, ∞) and there exist a sequence {am }m of ATp,p (Rn+1 Z+ )1 atoms and an -sequence {λm }m ⊂ C such that f = ∑m λm am pointwise, then f = s,τ ∑m λm am also in ATp,p (Rn+1 Z+ ) and s,τ f ATp,p (Rn+1 ) ≤ C ∑ |λm |, Z+

m

where C is a positive constant independent of f . s,τ Proof. By similarity, we only consider the space FTp,q (Rn+1 Z+ ). s,τ n+1 Let f ∈ FTp,q (RZ+ ). Let ω be a nonnegative Borel measurable function satisfying (7.3) and

Rn

p/q

∞

∑2

ksq

−q

| f (x)| [ω (x)] k

q

k

dx ≤ 2 f p

s,τ

FTp,q (Rn+1 Z )

.

+

k=0

For each l ∈ Z, let

El ≡ {x ∈ Rn : N ω (x) > 2l }.

(∞) (El ) < ∞, which together with From (7.2) and (7.3), it follows that Λ nτ (p∨q) Lemma 7.3 and its proof in [44, p. 386–387] yields that there exists a sequence {I j, l } j of dyadic cubes with disjoint interiors such that

∑[l(I j, l )]nτ (p∨q) j

and T (El ) ⊂

(∞)

≤ 2Λ (E ) nτ (p∨q) l

S∗ (I j, l ),

j

where

S∗ (I j, l ) ≡ (y, t) ∈ Rn+1 + : y ∈ I j, l , 0 < t < 2diam(I j, l ) .

The advantage is that {S∗(I j, l )} j have disjoint interiors for different values of j. Define c T j, l ≡ S∗ (I j, l )

S∗ (Ii, m )

,

m>l i

where for any set E ⊂ Rn , E c ≡ Rn \ E. Then T j, l have disjoint interiors for different values of j or l. Notice that

T (El ) = {(x,t) ∈ Rn+1 + : ω (x,t) > 0}

l

and for each j and l, S∗ (I j, l ) is contained in an (n + 1)-dimensional cube of side length 2diam(I j, l ). By an argument similar to that in [44, p. 396], we know that ∪l ∪ j T j, l contains {(x,t) ∈ Rn+1 + : ω (x,t) > 0} \ T∞ ,

186


where T∞ is a set of zero nτ (p ∨ q) -Hausdorff capacity and hence also zero (n + 1)dimensional Lebesgue measure. This observation further implies that f = ∑ f χTj, l a. e. on Rn+1 Z+ (or more precisely, quasi-everywhere with respect to nτ (p ∨ q) Hausdorff capacity). √ Recall that I ∗j, l = 5 nI j, l . Let

a j, l

⎧ p/q ⎫−1/p ⎬ ⎨ ∞ ksq k q −k ≡ f χTj, l [l(I ∗j, l )]nτ p 2 | f (x)| χ (x, 2 ) dx , T ∑ j, l ⎭ ⎩ Rn k=0

and

λ j, l ≡

⎧ ⎨ ⎩

[l(I ∗j, l )]nτ p

∑ 2ksq | f k (x)|q χTj, l (x, 2−k )

Rn

⎫1/p ⎬

p/q

∞

dx

k=0

⎭

.

We see that f = ∑ j, l λ j, l a j, l pointwise. Since√S∗ (I j, l ) ⊂ T (B j, l ), where B j, l is the ball with the same center as I j, l and radius 5 nl(I j, l )/2, then supp a j, l ⊂ T (B j, l ). s,τ It is easy to see that each a j, l is an FTp,q (Rn+1 Z+ )-atom. 1 Next we verify that {λ j, l } j, l is -summable. Notice that ω ≤ 2l+1 on T j, l ⊂ (T (El+1 ))c . When p ≥ q, by Hölder’s inequality and (2.11),

∑ |λ j, l | ≤ ∑ 2(l+1)[l(I ∗j, l )]nτ j, l

j, l

⎧ ⎨

×

⎩

Rn

∑ 2ksq| f k (x)|q [wk (x)]−q χTj, l (x, 2−k )

∑2

q

dx

k=0

≤

⎫1 ⎬p

p

∞

(l+1)p

nτ p

l(I ∗j, l )

⎭

1 p

j, l

×

⎧ ⎨ ⎩

∑ ∑ 2ksq | f k (x)|q [ω k (x)]−q χTj, l (x, 2−k )

Rn j, l

k=0

∑2

s,τ f FTp,q (Rn+1 ) Z+

s,τ f FTp,q (Rn+1 ) Z+ Z+

l

s,τ f FTp,q (Rn+1 ) .

⎫1 ⎬p

p

∞

Rn

l p

1

(∞) Λnτ p (El )

p

1 p (∞) [N ω (x)] p dΛnτ p (x)

q

dx

⎭

7.1 Tent Spaces

187

When p < q, by Hölder’s inequality and Minkowski’s inequality,

∑ |λ j, l | ≤ ∑ 2 j, l

(l+1)q

nτ q

[l(I ∗j, l )]

1 q

j, l

⎧ ⎪ ⎨

⎡

∑ ⎪ ⎩ j, l

×

⎣

p

∞

∑ 2ksq| f k (x)|q [ω k (x)]−q χTj, l (x, 2−k )

Rn

k=0

≤

∑2

(l+1)q

⎧ ⎨

q

[l(I ∗j, l )]nτ q

⎤ qp ⎫ 1q ⎪ ⎬ ⎦ dx ⎪ ⎭

1 q

j, l

×

⎩

Rn

∑ ∑ 2ksq | f k (x)|q [ω k (x)]−q χTj, l (x, 2−k ) j, l k=0

∑2

Z+

s,τ f FTp,q (Rn+1 )

l

Z+

f

s,τ FTp,q (Rn+1 Z+ )

Rn

lq

q

dx

⎭

1

f FTp,q s,τ (Rn+1 )

⎫1 ⎬p

p

∞

q

(∞) Λnτ q (El )

1 q (∞) [N ω (x)]q dΛnτ q (x)

.

s,τ In particular, if p = q ∈ (1, ∞), by Lemma 7.2, f = ∑ j λ j a j also in FTp,q (Rn+1 Z+ ). On the other hand, assume that p = q ∈ (1, ∞) and there exist a sequence {a j } j s,τ 1 of FTp,q (Rn+1 Z+ )-atoms and an -sequence {λ j } j ⊂ C such that f = ∑ j λ j a j pointwise. By Lemma 7.2 again, we obtain that the summation f = ∑ j λ j a j converges in s,τ FTp,q (Rn+1 Z+ ), which completes the proof of Proposition 7.1. s,τ (Rn+1 For all f ∈ ATp,p Z+ ), set

s,τ ||| f |||ATp,p (Rn+1 ) ≡ inf Z+

∑ |λm | : m

f = ∑ λm am ,

(7.5)

m

where the infimum is taken over all possible atomic decomposition of f . s,τ Proposition 7.1 implies that the norm ||| · |||ATp,p (Rn+1 ) is equivalent to the quasi-norm Z+

s,τ n+1 s,τ · ATp,p (Rn+1 ) , which together with Lemma 7.2 further yields that ATp,p (RZ+ ) Z+

s,τ becomes a Banach space under the norm ||| · |||ATp,p (Rn+1 ) . Z+

s,τ As the dual spaces of ATp,q (Rn+1 Z+ ), we now introduce the following two classes of tent spaces.

Definition 7.4. Let s ∈ R, p ∈ (1, ∞), q ∈ (1, ∞] and τ ∈ (0, ∞). The tent space s,τ n+1 k (Rn+1 AWp,q Z+ ) is defined to be the set of all functions f on RZ+ such that { f }k∈Z+

188


s,τ n+1 s,τ are Lebesgue measurable and f AWp,q (Rn+1 ) < ∞, where when AWp,q (RZ+ ) = Z+

s,τ BWp,q (Rn+1 Z+ ),

1 s,τ f BWp,q ≡ sup τ (Rn+1 Z+ ) |B| B

∞

∑2

k

Rn

k=0

−k

| f (x)| χT (B) (x, 2 ) dx

ksq

p

q/p 1/q ,

s,τ s,τ n+1 (Rn+1 and when AWp,q Z+ ) = FWp,q (RZ+ ),

⎧ p/q ⎫1/p ⎬ ∞ 1 ⎨ ksq k q −k s,τ f FWp,q 2 | f (x)| χ (x, 2 ) dx , n+1 ≡ sup ∑ T (B) (RZ ) τ ⎭ + B |B| ⎩ Rn k=0 and the supremum runs over all balls B in Rn . We need the following technical lemma. Lemma 7.4. Let s ∈ R, p ∈ (1, ∞), q ∈ (1, ∞], τ ∈ (0, ∞) and a ∈ (0, ∞). Then there s,τ exists a positive constant C such that for all f ∈ AWp,q (Rn+1 Z+ ) and nonnegative Borel n+1 measurable functions ω on R+ , when p ≤ q, ∞

∑ 2ksq

k=0

Rn

| f k (x)| p [ω k (x)]ap dx

q/p ≤ C f

(∞) q [N ω (x)]aq dΛnτ q (x) s,τ BWp,q (Rn+1 Z+ ) Rn

and Rn

∞

∑2

p/q ksq

| f (x)| [ω (x)] k

q

k

dx ≤ C f p

aq

s,τ

FWp,q (Rn+1 Z+ ) Rn

k=0

(∞)

[N ω (x)]ap dΛnτ p (x);

when p > q, ∞

∑ 2ksq

k=0

Rn

| f k (x)| p [ω k (x)]ap dx

q ≤ C f s,τ n+1 BWp,q (RZ )

q/p

[N ω (x)]

ap

Rn

+

q/p

(∞) dΛnτ p (x)

and

Rn

∞

∑2

p/q ksq

| f (x)| [ω (x)] k

q

k=0

≤ C f p s,τ n+1 FWp,q (RZ ) +

k

aq

dx

[N ω (x)]

aq

Rn

p/q

(∞) dΛnτ q (x)

.

7.1 Tent Spaces

189

s,τ Proof. By similarity, we only consider FWp,q (Rn+1 Z+ ). For all l ∈ Z, set

Ol ≡ {x ∈ Rn : N ω (x) > 2l }. Without loss of generality, we may assume that the integrals on the right-hand side (∞) of the desired inequalities are finite. Hence Λnτ (p∧q) (Ol ) < ∞. Let {I lj } j be some dyadic cube covering of Ol with

∑ |I lj |τ (p∧q) Λnτ (p∧q)(Ol ). (∞)

j

Then Lemma 7.3 tells us that there exists a sequence {Jil }i of dyadic cubes with mutually disjoint interiors such that

∑ |Jil |τ (p∧q) ≤ ∑ |I lj |τ (p∧q) i

j

and T (Ol ) ⊂

T ((Jil )∗ )

.

i

Notice that if ω k (y) > 2l , then N ω (x) > 2l for all x ∈ B(y, 2−k ), and hence (y, 2−k ) ∈ T (Ol ). We have l k l+1 } ⊂ T (Ol ). Al ≡ {(y, 2−k ) ∈ Rn+1 + : 2 < ω (y) ≤ 2

(7.6)

When p ≤ q, by (2.11), (7.6) and Definition 7.4, we have

Rn

=

p/q

∞

∑ 2ksq | f k (x)|q [ω k (x)]aq

Rn

∑2

∑ ∑2

lap Rn

−k

| f (x)| [ω (x)] χAl (x, 2 ) k

q

k

aq

∑2 ∑

∑2

Rn

i

ksq

−k

| f (x)| χAl (x, 2 ) k

q

∞

∑2

k=0

−k

| f (x)| χT ((Jl )∗ ) (x, 2 ) k

q

i

+

i

∑2 ∑

l∈Z

j

dx p/q

ksq

∑ 2lap/q ∑[l((Jil )∗ )]nτ p f FpW˙ p,s, qτ (Rn+1 Z ) lap

dx

p/q

∞

k=0

lap

l∈Z

ksq

l∈Z k=0

l∈Z

p/q

∞

l∈Z

dx

k=0

[l(I lj )]nτ p f p s,τ n+1 FW (R ) p,q

Z+

dx

190


p s,τ ∑ 2lapΛnτ p (Ol ) f FW n+1 p,q (RZ ) (∞)

l∈Z

+

(∞) f p s,τ n+1 [N ω (x)]ap dΛnτ p (x). FWp,q (RZ ) Rn +

When p > q, by Minkowski’s inequality, (7.6) and Definition 7.4, we obtain ⎧ p/q ⎫1/p ⎬ ⎨ ∞ ksq k q k aq 2 | f (x)| [ ω (x)] dx ∑ ⎭ ⎩ Rn k=0

⎧ ⎪ ⎨

⎡

∑ ⎪ ⎩l∈Z ⎧ ⎪ ⎨

∑ ⎪ ⎩l∈Z

2laq ⎣

p/q

∞

∑ 2ksq| f k (x)|q χAl (x, 2−k )

Rn

k=0

⎡

2laq ∑ ⎣ i

p/q

∞

∑ 2ksq | f k (x)|q χT ((Jil )∗ ) (x, 2−k )

Rn

k=0

⎤q/p ⎫1/q ⎪ ⎬ ⎦ dx ⎪ ⎭

1/q

∑2 ∑

⎤q/p ⎫1/q ⎪ ⎬ dx⎦ ⎪ ⎭

laq

i

l∈Z

[l((Jil )∗ )]nτ q f q s,τ n+1 ) FW (R p,q

s,τ f FWp,q (Rn+1 ) Z+

f FWp,q s,τ (Rn+1 ) Z+

Z+

1/q

∑ 2laqΛnτ q (Ol ) (∞)

l∈Z

Rn

1/q

[N ω (x)]

aq

(∞) dΛnτ q (x)

,


In the following theorem, we establish the dual relation between the tent spaces s,τ s,τ n+1 (Rn+1 ATp,q Z+ ) and AWp,q (RZ+ ). Theorem 7.1. Let s ∈ R, p ∈ (1, ∞), q ∈ [1, ∞) and τ ∈ (0, space of

s,τ (Rn+1 ATp,q Z+ )

is

τ n+1 AWp−s, ,q (RZ+ )

f , g =

1 (p∨q) ].

Then the dual

under the following pairing ∞

∑ f k (x)gk (x) dx. Rn

(7.7)

k=0

s,τ (Rn+1 Proof. By similarity, we only consider FTp,q Z+ ).

τ n+1 We first show that each function g ∈ FWp−s, , q (RZ+ ) induces a bounded linear

s,τ functional on FTp,q (Rn+1 Z+ ) via the pairing in (7.7). Indeed, let ω be a nonnegative Borel measurable function on Rn+1 + satisfying (7.3). Then by Lemma 7.4, we have

⎧ ⎨ ⎩

Rn

∞

∑2

k=0

−ksq

q

q

|gk (x)| [ω k (x)]

⎫1/p ⎬

p /q dx

⎭

g FW −s, τ (Rn+1 ) . p , q

Z+

7.1 Tent Spaces

191

s,τ Therefore, for all f ∈ FTp,q (Rn+1 older’s inequality, we have Z+ ), by H¨

∞ k k f (x)g (x) dx ∑ Rn k=0 ≤

≤

∞

⎩

p/q

∞

−ksq

1/q

dx

q

q

−ksq

dx

∑ 2ksq | f k (x)|q [ω k (x)]−q

dx

dx

⎭

⎫1/p ⎬

k=0

|g (x)| [ω (x)]

⎫1/p ⎬

p /q

p/q

q

k

⎭

|gk (x)| [ω k (x)]

∞

q

k

⎫1/p ⎬

k=0

Rn

∞

∑2

Rn

⎧ ⎨

k=0

k=0

⎧ ⎨

⎩

k

∞

∑2

−q

| f (x)| [ω (x)] q

∑ 2ksq | f k (x)|q [ω k (x)]−q

Rn

⎩

1/q

k

k=0

⎧ ⎨

×

∑2

Rn

ksq

⎭

g FW −s, τ (Rn+1 ) . p , q

Z+

Taking the infimum over all admissible ω gives the desired conclusion. Next we prove the converse. Let L be a bounded linear functional on s,τ n FTp,q (Rn+1 Z+ ). Fix a ball B ≡ B(xB , rB ) in R . For ε ∈ (0, rB ), define T ε (B) ≡ T (B) ∩ {(x,t) : ε ≤ t ≤ 1}. If f is supported in T ε (B) with f ∈ L p (q (T ε (B))), namely,

Rn

p/q

∞

∑ |f

k

−k

(x)| χT ε (B) (x, 2 ) q

dx < ∞,

k=0

then fixing ω as in the proof of Lemma 7.1, we have (ω (x,t))−1 ∼ rBnτ for all (x,t) ∈ T (B), and f p s,τ n+1 FTp,q (RZ ) +

rBnτ p rBnτ p

Rn

%

p/q

∞

∑ 2ksq | f k (x)|q χT ε (B) (x, 2−k )

k=0

1

ε (s∨0)p

+1

& Rn

∞

dx p/q

∑ | f k (x)|q χT ε (B)(x, 2−k )

dx.

k=0

Hence L induces a bounded linear functional on L p (q (T ε (B))), and acts via the inner-product with a unique function gB ∈ L p (q (T ε (B))) (see [145, p. 177]). For −j all j ∈ N, taking B j = B(0, j) and ε j = 2− j , we get a unique gB j ∈ L p (q (T 2 (B j )))

192

7 Inhomogeneous Besov-Hausdorff and Triebel-Lizorkin-Hausdorff Spaces −j

for each j. Moreover, by the uniqueness, gB j+1 = gB j on T 2 (B j ); letting j → ∞, we get a unique function g on Rn × Z+ that is locally in L p (q (Rn × Z+ )), and such that L( f ) =

∞

∑ f k (x)gk (x) dx,

(7.8)

Rn k=0

s,τ ε whenever f ∈ FTp,q (Rn+1 Z+ ) with support in some finite tent T (B). We claim that s,τ s,τ n+1 the subspace of such f is dense in FTp,q (Rn+1 Z+ ). In fact, for any f ∈ FTp,q (RZ+ ), set f j ≡ f χT 2− j (B ) , then f j → f pointwise as j → ∞. Notice that | f − f j | ≤ 2| f |. By j

s,τ (Rn+1 Lebesgue’s dominated convergence theorem we obtain that f j → f in FTp,q Z+ ) k −k −k+1 as j → ∞. Define g(x, t) ≡ g (x) when t ∈ [2 , 2 ) for all k ∈ Z+ . Thus, if we τ n+1 can show that g ∈ FWp−s, , q (RZ+ ), then by taking limits we will get the representation of L via the pairing (7.7). τ n+1 n To verify g ∈ FWp−s, , q (RZ+ ), fix a ball B ⊂ R . For every ε > 0, set

fε (x,t) ≡ t sq |g(x,t)|q −1 χT ε (B) (x,t) sgn g(x,t) p −1 ∞

q

∑ 2−ksq |gk (x)|q χT ε (B) (x, 2−k )

×

,

k=0

where sgn g(x,t) ≡ 1 when g(x,t) > 0, sgn g(x,t) ≡ −1 when g(x,t) < 0 and sgn g(x,t) ≡ 0 when g(x,t) = 0. Then fε is supported in T ε (B). Recall that if we choose ω as in the proof of Lemma 7.1, then for all (x,t) ∈ T (B), [ω (x,t)]−1 ∼ |B|τ . Therefore, s,τ |L( fε )| ≤ L fε FTp,q (Rn+1 Z+ ) ⎧ p/q ⎫1/p ⎨ ⎬ ∞ ksq k q k −q L 2 | f (x)| χ [ ω (x)] dx ∑ ε T ε (B)(x, 2−k ) ⎩ Rn k=0 ⎭

∼ L |B|τ

⎧ ⎨ ⎩

Rn

∞

∑2

−ksq

k=0

⎫1/p ⎬

p /q

q

|gk (x)| χT ε (B)(x, 2−k )

dx

⎭

,

which together with the fact that L( fε ) =

∞

∑

Rn k=0

fεk (x)gk (x) dx

=

Rn

∞

∑2

k=0

−ksq

p /q

q

|g (x)| χT ε (B)(x, 2−k ) k

yields

|B|−τ

⎧ ⎨ ⎩

Rn

∞

∑2

k=0

−ksq

q

|gk (x)| χT ε (B)(x, 2−k )

⎫1/p ⎬

p /q dx

⎭

L .

dx

7.1 Tent Spaces

193

Notice that the above inequality is true for all ε > 0 with a constant independent of ε . We get the same inequality for the integral over T (B), which is independent of the choice of B. Then taking infimum over all balls B in Rn , we see that g ∈ τ n+1 FWp−s, , q (RZ+ ), which completes the proof of Theorem 7.1. By Remark 7.1(i), Theorem 7.1 is also correct for τ = 0. s,τ (Rn+1 Recall that Proposition 7.1 implies that ATp,p Z+ ) is a Banach space. s,τ (Rn+1 When p = q, we also determine the predual space of ATp,p Z+ ). Denote by s,τ s,τ n+1 n+1 0 AWp,p (RZ+ ) the closure of all functions in AWp,p (RZ+ ) with compact support. We then have the following conclusion. Theorem 7.2. Let s ∈ R, p ∈ (1, ∞) and τ ∈ (0, 1p ]. Then the dual space of the tent s,τ −s,τ n+1 (Rn+1 space 0 AWp,p Z+ ) is ATp ,p (RZ+ ) under the pairing (7.7).

To prove this theorem, we need some technical lemmas. Lemma 7.5. Let s ∈ R, p ∈ (1, ∞) and τ ∈ (0, constant C such that for all f s,τ C−1 f ATp,p (Rn+1 ) ≤ Z+

g

s,τ ∈ ATp,p (Rn+1 Z+ ),

−s,τ AW (Rn+1 ) Z+ p ,p

1 p ].

Then there exists a positive

∞ sup f k (x)gk (x) dx ∑ n R k=0 ≤1, g has compact support

s,τ ≤ C f ATp,p (Rn+1 ) . Z+

s,τ s,τ Proof. Recall that the norm ||| · |||ATp,p (Rn+1 ) is equivalent to · ATp,p (Rn+1 ) . By Z+

Z+

τ n+1 Theorem 7.1 and the Hahn-Banach theorem, there exists an h ∈ AWp−s, ,p (RZ+ ) with h AW −s,τ (Rn+1 ) ≤ 1 such that p ,p

Z+

∞ k k f ATp,p s,τ s,τ f (x)h (x) dx n+1 ∼ ||| f ||| n+1 ∼ . ∑ (RZ ) ATp,p (RZ ) n + + R k=0

For j ∈ N, let g(x, 2−k ) ≡ h(x, 2−k )χ{|x|≤ j, 1/ j≤2−k ≤1} (x, 2−k ). Then g AW −s,τ (Rn+1 ) ≤ h AW −s,τ (Rn+1 ) ≤ 1 p ,p

Z+

p ,p

Z+

and g has compact support. Furthermore, Lebesgue’s dominated convergence theorem implies that if j is large enough, then ∞ k k s,τ s,τ f (x)g (x) dx f ATp,p , n+1 ∼ ||| f ||| n+1 ∼ ∑ (RZ ) ATp,p (RZ ) Rn k=0 + + which completes the proof of Lemma 7.5.

194


The proof of the following lemma is a modification of [42, Lemma 4.2]. Lemma 7.6. Let p ∈ (1, ∞), τ ∈ (0, sequence in

0,τ ATp,p (Rn+1 Z+ ).

1 p ]

and { fm }m∈N be a uniformly bounded

0,τ Then there exist a function f ∈ ATp,p (Rn+1 Z+ ) and a sub-

τ n+1 sequence { fmi }i∈N of { fm }m∈N such that for all g ∈ AWp0, ,p (RZ+ ) with compact support, fmi , g → f , g

as i → ∞, where f , g is defined as in (7.7), and f AT 0,τ (Rn+1 ) ≤ C sup fm AT 0,τ (Rn+1 ) Z+

p,p

p,p

m∈N

Z+

with C being a positive constant independent of f . Proof. Without loss of generality, we may assume that fm AT 0,τ (Rn+1 ) ≤ 1 for all p,p

Z+

m ∈ N. By Proposition 7.1 and its proof, each fm has an atomic decomposition representation fm =

∑ ∑

λm, j,Q am, j,Q

j∈Z Q∈I (m) j

0,τ in ATp,p (Rn+1 Z+ ), where I j

(m)

⊂ Q(Rn ), λm ≡ {λm, j,Q }

∑ ∑

(m)

j∈Z, Q∈I j

⊂ C satisfies that

|λm, j,Q | 1

j∈Z Q∈I (m) j

and each am, j,Q is an AT˙p0,τ (Rn+1 supported in T (BQ ), where and in what Z+ )-atom √ n follows, for all Q ∈ Q(R ), BQ ≡ B(cQ , 5 nl(Q)/2). λm ≡ { λm, j,Q } j∈Z, Q∈Q(Rn ) ⊂ C by setting, For all m ∈ N, define a sequence (m) for all j ∈ Z, λm, j,Q ≡ λm, j,Q when Q ∈ I and λm, j,Q ≡ 0 otherwise, and a set j

{ am, j,Q } j∈Z, Q∈Q(Rn ) of functions on Rn+1 by setting, for all j ∈ Z, am, j,Q ≡ am, j,Q Z (m)

when Q ∈ I j

and am, j,Q ≡ 0 otherwise. We see that for each m ∈ N, λm 1 =

∑ ∑

| λm, j,Q | =

j∈Z Q∈Q(Rn )

∑ ∑

|λm, j,Q | 1

j∈Z Q∈I (m) j

0,τ (Rn+1 and each am, j,Q is still an ATp,p Z+ )-atom supported in T (BQ ). Moreover,

fm =

∑ ∑n

j∈Z Q∈Q(R ) 0,τ in ATp,p (Rn+1 Z+ ).

λm, j,Q am, j,Q

7.1 Tent Spaces

195

Since

∑ ∑n

| λm, j,Q | 1

j∈Z Q∈Q(R )

holds for all m ∈ N, a diagonalization argument yields that there exist a sequence

λ ≡ {λ j,Q } j∈Z, Q∈Q(Rn ) ∈ 1 and a subsequence { λmi }i∈N of { λm }m∈N such that λmi , j,Q → λ j,Q as i → ∞ for all j ∈ Z and Q ∈ Q(Rn ), and λ 1 1. On the other hand, recall that supp am, j,Q ⊂ T (BQ ) for all m ∈ N and j ∈ Z. From Definition 7.3, it follows that { am, j,Q L p ( p (T (BQ ))) }m∈N is a uniformly bounded sep p quence in L ( (T (BQ ))), where L p ( p (T (BQ ))) consists of all functions on T (BQ ) equipped with the norm that F L p ( p (T (BQ ))) ≡

1/p

∞

∑ |F(x, 2

−j

Rn i=0

−j

)| χT (BQ ) (x, 2 ) dx p

.

Then by the Alaoglu theorem, there exist a unique function a j,Q ∈ L p ( p (T (BQ ))) ami , j,Q }i∈N again, such that for all and a subsequence of { ami , j,Q }i∈N , denoted by { functions g ∈ L p ( p (T (BQ ))), ami , j,Q , g → a j,Q , g 0,τ as i → ∞ and each a j,Q is also a constant multiple of an ATp,p (Rn+1 Z+ )-atom supported in T (2BQ ) with the constant independent of j and Q. Applying a diagonalization argument again, we conclude that there exists a subsequence, denoted by { ami , j,Q }i∈N p p again, such that for all g ∈ L ( (T (BQ ))),

ami , j,Q , g → a j,Q , g as i → ∞ for all j ∈ Z and Q ∈ Q(Rn ). Let f≡

∑ ∑

λ j,Q a j,Q .

j∈Z Q∈Q(Rn ) 0,τ (Rn+1 By Proposition 7.1, we see that f ∈ ATp,p Z+ ) and

f AT 0,τ (Rn+1 ) 1. p,p

Z+

τ n+1 Let g ∈ AWp0, ,p (RZ+ ) such that

supp g ⊂ B(0, 2M ) × {2−M , 2−M+1 , · · · , 2M }

196


for some M ∈ N. Without loss of generality, we may assume that g AW 0,τ We need to show that fmi , g → f , g as i → ∞. It is easy to see that g L p ( p (T (B(0,2M ))) g AW 0,τ

p ,p

(Rn+1 Z ) +

p ,p

(Rn+1 Z ) +

= 1.

∼ 1.

Thus, ami , j,Q , g → a j,Q , g as i → ∞ for all j ∈ Z and Q ∈ Q(Rn ). 0,τ Recall that a AT 0,τ (Rn+1 ) ≤ C for all ATp,p (Rn+1 Z+ )-atoms a, where C is a positive Z+

p,p

constant independent of a. By

∑ ∑n

j∈Z Q∈Q(R )

| λmi , j,Q | 1,

we see that for any ε > 0, there exists an L ∈ N such that

∑

{ j∈Z:

| j|>L} {Q∈Q(Rn ):

∑

| jQ |>L or

| λmi , j,Q | < ε /C Q[−2L ,2L )n }

and hence

∑

∑

j∈Z Q∈Q(Rn ) | j|>L | j |>L orQ[−2L ,2L )n Q

≤

∑

∑

| λmi , j,Q || ami , j,Q , g|


≤ C < ε.

∑

| λmi , j,Q | ami , j,Q AT 0,τ (Rn+1 ) g AW 0,τ

∑


p,p

Z+

p,p

(Rn+1 Z ) +

| λmi , j,Q |

Similarly, by ∑ j∈Z ∑Q∈Q(Rn ) |λ j,Q | 1, there exists an L ∈ N such that

∑

∑

|λ j,Q ||a j,Q , g| < ε ,


which yields lim fmi , g = f , g

i→∞

and completes the proof of Lemma 7.6.

7.1 Tent Spaces

197

We are now ready to prove Theorem 7.2. s,τ Proof of Theorem 7.2. By Theorem 7.1 and the definition of 0 AWp,p (Rn+1 Z+ ), we have that −s,τ s,τ n+1 s,τ n+1 n+1 ∗ 0 AWp,p (RZ+ ) ⊂ AWp,p (RZ+ ) = (ATp ,p (RZ+ )) ,

which implies that τ −s,τ n+1 n+1 ∗∗ s,τ ∗ ATp−s, ⊂ (0 AWp,p (Rn+1 ,p (RZ+ ) ⊂ (ATp ,p (RZ+ ) Z+ )) .

To show τ n+1 s,τ n+1 ∗ ATp−s, ,p (RZ+ ) ⊂ (0 AWp,p (RZ+ )) ,

we first claim that if this is true when s = 0, then it is also true for all s ∈ R. To see this, for all u ∈ R, define an operator Au by setting, for all functions f on Rn+1 Z+ , x ∈ Rn and j ∈ Z+ , (Au f )(x, 2− j ) ≡ 2 ju f (x, 2− j ).

s,τ s+u,τ n+1 Obviously, Au is an isometric isomorphism from AWp,p (Rn+1 Z+ ) to AWp,p (RZ+ ) s,τ s+u,τ s,τ n+1 n+1 ∗ and from ATp,p (Rn+1 Z+ ) to ATp,p (RZ+ ). If L ∈ (0 AWp,p (RZ+ )) , then 0,τ ∗ L ◦ As ∈ (0 AWp,p (Rn+1 Z+ ))

and hence, by the above assumption, there exists a function g ∈ ATp0,,pτ (Rn+1 Z+ ) such that L ◦ As (F) =

∞

∑ f j (x)g j (x) dx Rn j=0

0,τ s,τ n+1 for all F ∈ 0 AWp,p (Rn+1 Z+ ). Notice that As ◦ A−s is the identity on 0 AWp,p (RZ+ )

s,τ 0,τ n+1 and A−s is an isometric isomorphism from 0 AWp,p (Rn+1 Z+ ) onto 0 AWp,p (RZ+ ). Therefore,

L( f ) = L ◦ As ◦ A−s( f ) =

∞

∑ (A−s f ) j (x)g j (x) dx =

Rn j=0

∞

∑ f j (x)(A−s g) j (x) dx

Rn j=0

s,τ 0,τ −s,τ n+1 n+1 for all f ∈ 0 AWp,p (Rn+1 Z+ ). Since g ∈ ATp ,p (RZ+ ), we have A−s g ∈ ATp ,p (RZ+ ) and A−s g AT −s,τ (Rn+1 ) = g AT 0,τ (Rn+1 ) . p ,p

Z+

p ,p

Z+

Thus, the above claim is true. Next we prove that 0,τ 0,τ ∗ n+1 (Rn+1 (0 AWp,p Z+ )) ⊂ ATp ,p (RZ+ ).

198


0,τ ∗ To this end, we choose L ∈ (0 AWp,p (Rn+1 Z+ )) . It suffices to show that there exists a 0,τ n+1 g ∈ ATp0, ,pτ (Rn+1 Z+ ) such that for all f ∈ AWp,p (RZ+ ) with compact support, L has a

0,τ form as in (7.7). In fact, for f ∈ AWp,p (Rn+1 Z+ ) with compact support, if h, f = 0

τ n+1 holds for all h ∈ ATp0, ,p (RZ+ ), then Theorem 7.1 implies that f must be the zero

0,τ 0,τ n+1 element of AWp,p (Rn+1 Z+ ). Thus, ATp ,p (RZ+ ) is a total set of linear functionals on

0,τ n+1 0 AWp,p (RZ+ ).

To complete the proof of Theorem 7.2, we need the following functional analysis result (see [48, p. 439, Exercise 41]): Let X be a locally convex linear topological space and Y be a linear subspace of X ∗ . Then Y is X -dense in X ∗ if and only if Y is a total set of functionals on X . From this functional result and the fact that τ 0,τ n+1 n+1 ATp0, ,p (RZ ) is a total set of linear functionals on 0 AWp,p (RZ ), we deduce that + + τ 0,τ n+1 n+1 ∗ ATp0, ,p (RZ+ ) is weak ∗-dense in (0 AWp,p (RZ+ )) . Then there exists a sequence

{g(m) }m∈N in ATp0,,pτ (Rn+1 Z+ ) such that

g(m) , f → L( f ) 0,τ as m → ∞ for all f in 0 AWp,p (Rn+1 Z+ ). Applying the Banach-Steinhaus theorem, we (m) conclude that the sequence { g AT 0,τ (Rn+1 ) }m∈N is uniformly bounded. Then by p ,p

Z+

τ n+1 Lemmas 7.6 and 7.5, we obtain a subsequence {g(mi ) }i∈N and g ∈ ATp0, ,p (RZ+ ) such that

L( f ) = lim g(mi ) , f = g, f i→∞

0,τ (Rn+1 for all f ∈ AWp,p Z+ ) with compact support and

g AT 0,τ

p ,p

(Rn+1 Z ) +

sup f

≤1 0,τ AW (Rn+1 ) Z+ p ,p f has compact support

sup f

≤1 0,τ AW (Rn+1 ) Z+ p ,p f has compact support

L ( which completes the proof of Theorem 7.2.

|g, f | |L( f )|

0,τ n+1 ∗ 0 AWp ,p (RZ+ ))

,

Remark 7.2. It is still unclear whether or not Theorem 7.2 is true for the spaces s,τ −s,τ n+1 n+1 0 ATp,q (RZ+ ) and AWp ,q (RZ+ ) when p = q. The difficulty lies in the fact that

s,τ the space ATp,q (Rn+1 Z+ ) when p = q is only known to be a quasi-Banach space so far. Thus, Lemma 7.5 in the case that p = q seems not available, due to the Hahn-Banach theorem is not valid for these spaces.

7.2 Besov-Hausdorff Spaces and Triebel-Lizorkin-Hausdorff Spaces

199

7.2 Besov-Hausdorff Spaces and Triebel-Lizorkin-Hausdorff Spaces In this section, we determine the predual spaces of As,p,τq (Rn ). Let Φ and ϕ satisfy, respectively, (2.1) and (2.2). We define an operator ρϕ by setting

ρϕ ( f )(x, 2− j ) ≡ ϕ j ∗ f (x) for all f ∈ S (Rn ), x ∈ Rn and j ∈ Z+ , where when j = 0, ϕ0 is replaced by Φ . n Conversely. for all functions F on Rn+1 Z+ and x ∈ R , we define a map πϕ by

πϕ (F)(x) ≡

∞

∑

n k=0 R

F(y, 2−k )ϕk (x − y) dy =

∞

∑

n k=0 R

F k (y)ϕk (x − y) dy,

(7.9)

which makes sense due to the following technical lemma. Lemma 7.7. Let s ∈ R, p ∈ (1, ∞), q ∈ (1, ∞] and τ ∈ [0, ∞), Φ and ϕ satisfy, respectively, (2.1) and (2.2), and for all ξ ∈ Rn , ∞

' (ξ )|2 + ∑ |ϕ'(2− j ξ )|2 = 1. |Φ j=1

s,τ s, τ n (Rn+1 Then πϕ is a bounded and surjective linear operator from AWp,q Z+ ) to A p, q (R ). s,τ (Rn+1 Proof. By similarity, we only give the proof for the space FWp,q Z+ ). Let F ∈ s,τ n+1 FWp,q (RZ+ ). Notice that there exists a constant γ > 1 such that for all cubes P, P × (0, l(P)] ⊂ T (γ P). Therefore, we have

⎧ p/q ⎫1/p ⎬ ∞ 1 ⎨ jsq j q s,τ F FWp,q 2 |F (x)| dx . n+1 ∼ sup ∑ (RZ ) τ ⎭ + P∈Q |P| ⎩ P j= jP ∨0

(7.10)

We claim that (7.9) holds in S (Rn ). For all m ∈ Z+ and k ∈ Zn , setting R−1 ≡ 0, / Rm ≡ [−2m+1 , 2m+1 )n and

χRm \Rm−1 (k) ≡ χ{k∈Zn : Q0k ⊂Rm \Rm−1 } (k), we then have ∑k∈Zn χRm \Rm−1 (k) 2mn . Then for all φ ∈ S (Rn ) and i ∈ Z+ , by Lemma 2.4 and (7.10), we obtain Rn

|φ ∗ ϕi (x − y)||F i (y)| dy

∞

∑ ∑n

m=0 k∈Z

χRm \Rm−1 (k)

x+Q0k

2−iM |F i (y)| dy (1 + |x − y|)n+M

200


∞

∑ ∑n χRm \Rm−1 (k)2−iM 2−m(n+M)2−is

m=0 k∈Z

× 2

%

&1/p

2 |F (y)| dy isp

x+Q0k −iM −is

i

p

s,τ F FWp,q (Rn+1 ) ,

2

Z+

where M can be any positive number. If we choose M > max{0, −s}, then, as l → ∞,

∑

n |i|>l R

|φ ∗ ϕi (x − y)||F i (y)| dy → 0,

which implies that (7.9) holds in S (Rn ). Now we verify that s,τ πϕ (F) Fp,s, qτ (Rn ) F FWp,q (Rn+1 ) . Z+

By the above claim and Lemma 2.4, we see that πϕ (F) Fp,s, qτ (Rn ) ⎧ p/q ⎫1/p ⎬ ∞ 1 ⎨ jsq = sup 2 | ϕ ∗ π (F)(x)| dx j ϕ ∑ τ ⎭ P∈Q |P| ⎩ P j= j ∨0 P

⎧ q p/q ⎫1/p ⎬ ∞ ∞ 1 ⎨ js i ≤ sup 2 | ϕ ∗ φ (x − y)||F (y)| dy dx j i ∑ ∑ τ ⎭ P dyadic |P| ⎩ P j= jP ∨0 i=0 Rn ∞ 1 sup ∑ τ P j= j ∨0 P∈Q |P| P q p/q ⎫1/p ⎬ ∞ 2 js 2−|i− j|M 2−(i∧ j)M |F i (y)| × ∑ dy dx . n ⎭ (2−(i∧ j) + |x − y|)n+M i=0 R

Similarly to the proof of Lemma 4.1, applying (7.10), we have s,τ πϕ (F) Fp,s, qτ (Rn ) F FWp,q (Rn+1 ) ; Z+

s,τ see also [164, p. 2797]. Thus, πϕ is a bounded linear operator from FWp,q (Rn+1 Z+ ) to s, τ n Fp, q (R ). By the Calderón reproducing formula in Lemma 2.3, the composite operator πϕ ρϕ is the identity on Fp,s, qτ (Rn ), which implies the surjectivity of πϕ , and hence completes the proof of Lemma 7.7.


201

From Lemma 7.7, we also deduce that s,τ f Fp,s, τq (Rn ) ∼ ρϕ ( f ) FWp,q (Rn+1 ) . Z+

We now introduce the following spaces. Definition 7.5. Let s ∈ R, p ∈ (1, ∞), q ∈ [1, ∞) and τ ∈ [0, respectively, (2.1) and (2.2).

1 ], Φ (p∨q)

and ϕ satisfy,

s, τ n n (i) The Besov-Hausdorff space BH p, q (R ) is defined to be the set of all f ∈ S (R ) such that s,τ f BHp,s, τq (Rn ) ≡ ρϕ ( f ) BTp,q (Rn+1 ) < ∞. Z+

(ii) The Triebel-Lizorkin-Hausdorff space set of all f ∈ S (Rn ) such that

s, τ n FH p, q (R )

(q = 1) is defined to be the

s,τ f FHp,s, τq (Rn ) ≡ ρϕ ( f ) FTp,q (Rn+1 ) < ∞. Z+

s, τ s, τ s, τ n n n For simplicity, we use AH p, q (R ) to denote either BH p, q (R ) or FH p, q (R ).

Remark 7.3. (i) From (7.4), we deduce that · AHp,s, τq(Rn ) is a quasi-norm.

s, τ n s n (ii) By Remark 7.1(i), when τ = 0, AH p, q (R ) = A p,q (R ).

s, τ n To show that the space AH p, q (R ) is independent of the choices of Φ and ϕ , we need a technical lemma. For all β ∈ [1, ∞) and x ∈ Rn , define the β -nontangential maximal function Nβ f of a measurable function f on Rn+1 + by

Nβ f (x) ≡ sup | f (y, t)|. |y−x| 1. Then by Lemma 2.4, Set ψk ≡ 0 if k < 0. Notice that ψk ∗ ψ for all x ∈ Rn , we have ∞ k+1 j ∗ ϕ j ∗ f ≤ j (x − y)||ϕ j ∗ f (y)| dy |ψk ∗ ψ ∑ ψk ∗ ψ ∑ n j=0 j=(k−1)∨0 R

k+1

∑

n j=(k−1)∨0 R

k+1

∑

2−|k− j|M 2−(k∧ j)M |ϕ j ∗ f (y)| dy (2−(k∧ j) + |x − y|)n+M

2−|k− j|M 2(k∧ j)n

j=(k−1)∨0 ∞

+∑2

−lM−ln+(k∧ j)n

l=1

where M ∈ N will be determined later.

|x−y| λ }) ,

224


where Nβ ω (x) ≡

sup ω (y,t).

|y−x| λ } =

B(y,t)

t∈(0,∞) y∈Rn

ω (y,t)>λ

and that {x ∈ Rn : Nβ ω (x) > λ } =

B(y, β t).

t∈(0,∞) y∈Rn

ω (y,t)>λ

By the Linderöf covering lemma, there exists a countable subset {Bl }∞ l=0 of {B(y,t) : t ∈ (0, ∞), y ∈ Rn satisfy ω (y,t) > λ }

such that {x ∈ R : Nβ ω (x) > λ } = n

∞

(β Bl )

l=0

and

∞

{x ∈ R : N ω (x) > λ } ⊃ n

Bl .

l=0

By Lemma 7.15, it suffices to prove that (∞) Λd ({x ∈

R : Nβ ω (x) n

> λ }, {β Bl }∞ l=0 )

β

d

(∞) Λd

∞

Bl , {Bl }∞ l=0

.

l=0

Let {B∗k }∞ k=0 be a ball covering of ∪l∈N Bl such that ∞

∑ rBd ∗k ≤ 2Λd

(∞)

∞ (∪∞ l=0 Bl , {Bl }l=0 )

k=0

and that B∗k engulfs Bl whenever they intersect, where rB∗k denotes the radius of B∗k . Therefore, β B∗k engulfs β Bl whenever they intersect and {x ∈ R : Nβ ω (x) > λ } ⊂ n

∞ k=0

(β B∗k )

.


We then have 2β

d

(∞) Λd

∞

Bl , {Bl }∞ l=0

≥

∞

∑ (β rB∗k )d

l=0

l=0

225

(∞) 0

1 {x ∈ Rn : Nβ ω (x) > λ }, {β Bl }∞ l=0 ,

≥ Λd


Proof of Theorem 7.5. Without loss of generality, we may assume s = 0, since this s, τ n case implies the general case. In fact, let t ≡ {tQ }l(Q)≤1 ∈ aH p, q (R ) and A be a ε -almost diagonal operator associated with the matrix {aQP }l(Q),l(P)≤1 and ε ∈ (0, ∞). If the conclusion holds for s = 0, let tP ≡ l(P)−stP and B be the operator associated with the matrix {bQP }l(Q),l(P)≤1 , where bQP ≡ (l(P)/l(Q))s aQP for all l(Q), l(P) ≤ 1. Then we have t aH 0,τ (Rn ) t aH 0,τ (Rn ) ∼ t aHp,s, τq (Rn ) , At aHp,s, τq(Rn ) = B p,q

p,q

which deduces the desired conclusions. s, τ n By similarity, we only consider f H p, q (R ). By the Aoki theorem (see [8]), there 0,τ κ exists a κ ∈ (0, 1] such that · 0,τ n becomes a norm in f H p,q (Rn ). Let t ∈ f Hp,q (R )

0,τ f H˙ p,q (Rn ). For Q ∈ Q, we write A ≡ A0 + A1 with

∑

aQPtP

∑

aQPtP .

(A0t)Q ≡

{P∈Q: l(Q)≤l(P)}

and (A1t)Q ≡

{P∈Q: l(P) nτ , by Fefferman-Stein’s vector valued inequality, we obtain A0t κ

⎧ ⎧ ⎡ ⎪ ⎨ j ⎨ ∞ q n+ε −2 ⎣ ∑ inf |Q| χ 2(i− j) 2 Q ∑ ∑ ∑ ∑ ω ⎩ ⎪ ⎩ j=0 Q∈Q i=−∞ P∈Am,i (Q) m=0 j ∞

0,τ

f Hp,q (Rn )

⎫κ ⎤q ⎫ 1q ⎪ ⎬ ⎬ |tP ⎦ × ⎪ (1 + 2i|xQ − xP |)n+ε ⎭ p n⎭ L (R ) ⎡ |[ω (·, 2− j )]−1

⎧ ⎨ ∞ j n+ε − 2q ⎣ ∑ |Q| χ Q ∑ ∑ 2(i− j) 2 ⎩ ∑ ∑ i=−∞ P∈Am,i (Q) m=0 j=0 Q∈Q j ⎤q ⎫ 1q κ ⎬ − j −1 |tP |[ωm (·, 2 )] ⎦ × (1 + 2i|xQ − xP |)n+ε ⎭ p n ∞

L (R )


227

∞ j ∞ m(nτ −ε )κ ∑2 ∑ ∑ χQ ∑ 2(i− j)ε /2 j=0 Q∈Q i=−∞ m=0 j

⎞⎤q ⎫ 1q κ ⎬ P [ω (·, 2−i )]−1 ⎠⎦ × M ⎝ ∑ |tP |χ ⎭ P∈Am,i (Q) p ⎛

L (Rn )

t κ 0,τ n . f Hp,q (R ) The proof for A1t is similar. Indeed, we have |(A1t)Q |

∑

%

{P∈Q: l(P)≤l(Q)}

Thus, A1t f H 0,τ (Rn ) p,q

l(P) l(Q)

& n+ε 2

|tP | . (1 + l(Q)−1|xQ − xP|)n+ε

⎧ ⎡ ⎨ ∞ ∞ q −2 ⎣ ∑ ∑ 2−l n+2 ε inf |Q| χ Q ∑ ∑ ω ⎩ j=0 Q∈Q j l=0 P∈Q j+l ⎤q ⎫ 1q ⎬ − j −1 |tP |[ω (·, 2 )] ⎦ × n+ε j ⎭ (1 + 2 |xQ − xP |)

.

L p (Rn )

Let 0, j,l (Q) ≡ {P ∈ Q j+l : 2 j |xP − xQ | ≤ A and

√ n/2}

√ √ m, j,l (Q) ≡ {P ∈ Q j+l : 2m−1 n/2 < 2 j |xP − xQ | ≤ 2m n/2} A

for all j ∈ Z and m, l ∈ Z+ . Set m (x, s) ≡ 2−(m+l)nτ sup{ω (y, s) : y ∈ Rn , |y − x| < ω

√ m+l+1 n2 s}

m for all m ∈ Z+ and (x, s) ∈ Rn+1 + . Similarly, we have that a constant multiple of ω satisfies (7.3) and m (x, 2− j )]−1 ω (y, 2− j−l ) 2(m+l)nτ [ω m, j,l (Q). Choosing ε > 2nτ , for m, l ∈ Z+ , x ∈ Q with Q ∈ Q j , y ∈ P with P ∈ A similarly to the estimate of A0t f H 0,τ (Rn ) , we also have p,q

A1t f H 0,τ (Rn ) t f H 0,τ (Rn ) , p,q

which completes the proof of Theorem 7.5.

p,q

228


From Theorem 7.5, we deduce the smooth atomic and molecular decomposition s, τ n characterizations of AH p, q (R ). We begin with the smooth synthesis molecules, the s, τ n smooth analysis molecules and the smooth atoms for AH p, q (R ) as follows. Definition 7.7. Let s ∈ R, τ ∈ [0, ∞), p ∈ (1, ∞), q ∈ [1, ∞), s∗ = s − s and N ≡ max(−s + 2nτ , −1). Let Q be a dyadic cube with l(Q) ≤ 1. (i) A function mQ is said to be an inhomogeneous smooth synthesis molecule for s, τ n ∗ ∗ AH p, q (R ) supported near Q if there exist a δ ∈ (max{s , (s + nτ ) }, 1] and an M ∈ (n + 2nτ , ∞) such that Rn

xγ mQ (x) dx = 0

if |γ | ≤ N and l(Q) < 1,

|mQ (x)| ≤ (1 + |x − xQ|)−M

if l(Q) = 1,

|mQ (x)| ≤ |Q|−1/2 (1 + [l(Q)]−1 |x − xQ |)− max(M, M−s) |∂ γ mQ (x)| ≤ |Q|−1/2−|γ |/n (1 + [l(Q)]−1|x − xQ |)−M

if l(Q) < 1, if |γ | ≤ s + 3nτ ,

and |∂ γ mQ (x) − ∂ γ mQ (y)| ≤ |Q|−1/2−|γ |/n−δ /n|x − y|δ × sup (1 + [l(Q)]−1 |x − z − xQ|)−M |z|≤|x−y|

if |γ | = s + 3nτ . A collection {mQ }l(Q)≤1 is called a family of inhomogeneous smooth synthes, τ n sis molecules for AH p, q (R ), if each mQ is an inhomogeneous smooth synthesis s, τ n molecule for AH p, q(R ) supported near Q. (ii) A function bQ is said to be an inhomogeneous smooth analysis molecule for s, τ n ∗ AH p, q (R ) supported near Q if there exist a ρ ∈ ((−s) , 1] and an M ∈ (n + 2nτ , ∞) such that Rn

xγ bQ (x) dx = 0

if |γ | ≤ s + 3nτ and l(Q) < 1,

|bQ (x)| ≤ (1 + |x − xQ|)−M if l(Q) = 1, |bQ (x)| ≤ |Q|−1/2 (1 + [l(Q)]−1|x − xQ |)− max(M, M+n+s+nτ −J) if l(Q) < 1, |∂ γ bQ (x)| ≤ |Q|−1/2−|γ |/n(1 + [l(Q)]−1 |x − xQ |)−M

if |γ | ≤ N,

and |∂ γ bQ (x) − ∂ γ bQ (y)| ≤ |Q|−1/2−|γ |/n−ρ /n|x − y|ρ × sup (1 + [l(Q)]−1|x − z − xQ|)−M if |γ | = N. |z|≤|x−y|


229

A collection {bQ }l(Q)≤1 is called a family of inhomogeneous smooth synthes, τ n sis molecules for AH p, q (R ), if each bQ is an inhomogeneous smooth analysis s, τ n molecule for AH p, q(R ) supported near Q. Definition 7.8. A function aQ is called an inhomogeneous smooth atom for s, τ n AH p, q (R ) supported near a dyadic cube Q with l(Q) ≤ 1 if supp aQ ⊂ 3Q, Rn

xγ aQ (x) dx = 0

if |γ | ≤ max{−s + 2nτ , −1} and l(Q) < 1,

and ∂ γ aQ L∞ (Rn ) ≤ |Q|−1/2−|γ |/n

if |γ | ≤ max{s + 3nτ + 1, 0}.

A collection {aQ }l(Q)≤1 is called a family of inhomogeneous smooth atoms for s, τ s, τ n n AH p, q (R ), if each aQ is an inhomogeneous smooth atom for AH p, q (R ) supported near Q. s, τ n We remark that the smooth molecules and atoms for AH p, q (R ) are also the s, τ n smooth molecules and atoms for A p, q (R ) in Definitions 3.2 and 3.3. Similarly to the proofs of Theorems 3.2 and 3.3, we have the following decoms, τ n position characterizations of AH p, q (R ).

Theorem 7.6. Let p ∈ (1, ∞), q ∈ [1, ∞), s ∈ R and τ ∈ [0,

1 (p∨q) ].

s, τ n (i) If {mQ }l(Q)≤1 is a family of smooth synthesis molecules for AH p, q (R ), then s, τ there exists a positive constant C such that for all t ≡ {tQ }l(Q)≤1 ∈ aH p, q (Rn ),

∑ tQ mQ l(Q)≤1

s, τ AHp, q (Rn )

≤ C t aHp,s, τq(Rn ) .

s, τ n (ii) If {bQ }l(Q)≤1 is a family of smooth analysis molecules for AH p, q (R ), then there s, τ n exists a positive constant C such that for all f ∈ AH p, q (R ),

{ f , bQ }l(Q)≤1 aHp,s, τq (Rn ) ≤ C f AHp,s, τq (Rn ) . s, τ n Theorem 7.7. Let s, p, q, τ be as in Theorem 7.6. Then for each f ∈ AH p, q (R ), s, τ n there exist a family {aQ }l(Q)≤1 of smooth atoms for AH p, q (R ), a coefficient sequence t ≡ {tQ }l(Q)≤1 , and a positive constant C depending only on p, q, s, τ such that f = ∑ tQ aQ l(Q)≤1

230


in S (Rn ) and t aHp,s, τq(Rn ) ≤ C f AHp,s, τq (Rn ) . Conversely, there exists a positive constant C depending only on p, q, s, τ such s, τ n that for all families {aQ }l(Q)≤1 of smooth atoms for AH p, q (R ) and s, τ n t ≡ {tQ }l(Q)≤1 ∈ aH p, q (R ),

∑ tQ aQ l(Q)≤1

s, τ AHp, q (Rn )

≤ C t aHp,s, τq(Rn ) .

Based on these smooth atomic and molecular decomposition characterizations, similarly to the arguments in Chaps. 5 and 6, we obtain that the mapping properties of pseudo-differential operators in Theorem 5.1, lifting properties in Proposition 5.1, pointwise multiplier properties in Theorem 6.1 and diffeomorphism properties s, τ n in Theorem 6.7 have counterparts for the spaces AH p, q (R ). 1 Theorem 7.8. Let s, μ ∈ R, p ∈ (1, ∞), q ∈ [1, ∞), τ ∈ [0, (p∨q) ] and N be as in

μ

Definition 7.7. Assume that a ∈ S1,1 (Rn ) and a(x, D) be the corresponding pseudodifferential operator. If s > 2nτ , then a(x, D) is a bounded linear operator from s+ μ ,τ s, τ n AH p,q (Rn ) to AH p, q (R ). If s ≤ 2nτ and (5.1) holds, then a(x, D) is a bounded s+ μ ,τ s, τ n linear operator from AH p,q (Rn ) to AH p, q (R ). 1 Proposition 7.5. Let s, σ ∈ R, p ∈ (1, ∞), q ∈ [1, ∞) and τ ∈ [0, (p∨q) ]. Then the

s, τ s−σ ,τ n (Rn ). lifting operator Iσ maps AH p, q (R ) isomorphically onto AH p,q

1 Theorem 7.9. Let s ∈ R, p ∈ (1, ∞), q ∈ [1, ∞) and τ ∈ [0, (p∨q) ]. If m ∈ N is sufficiently large, then there exists a positive constant C(m) such that for all s, τ n g ∈ BCm (Rn ) and f ∈ AH p, q (R ),

g f AHp,s, τq(Rn ) ≤ C(m)

∑

|α |≤m

α

∂ g L∞ (Rn ) f AHp,s, τq (Rn ) .

1 Theorem 7.10. Let m ∈ N, s ∈ R, p ∈ (1, ∞), q ∈ [1, ∞), τ ∈ [0, (p∨q) ] and ψ be an m-diffeomorphism. If m ∈ N is sufficiently large, then Dψ is an isomorphic mapping s, τ n of AH p, q (R ) onto itself. s, τ n Also, we establish the trace property for the space AH p, q (R ).

Theorem 7.11. Let n ≥ 2, p ∈ (1, ∞), q ∈ [1, ∞), s ∈ ( 1p + 2nτ , ∞) and τ ∈ n−1 [0, n(p∨q) ]. Then there exists a surjective and continuous operator n τ s− 1 , n−1

p s, τ n Tr : f ∈ AH p, q (R ) → Tr( f ) ∈ AH p,q

(Rn−1 )

s, τ n such that Tr( f )(x ) = f (x , 0) for all x ∈ Rn−1 and smooth atoms f for AH p, q (R ).


231

To prove this theorem, we need a technical lemma; see [168, Lemma 4.1]. Lemma 7.17. Let d ∈ (0, n] and Ω be an open set in Rn . Define (∞) Λd,∗ (Ω )

≡ inf

∞

∑

rdj

j=1 (∞)

∞

dist(x j , ∂ Ω ) :Ω⊂ B(xr , r j ), r j > 10000 j=1

.

(∞)

Then Λd (Ω ) and Λd,∗ (Ω ) are equivalent for all Ω . (∞)

(∞)

Proof. The inequality Λd (Ω ) ≤ Λd,∗ (Ω ) is trivial from the definitions. To prove the converse, we choose a ball covering {B(x j , r j )}∞j=1 of Ω such that ∞

∑ rdj ≤ 2Λd

(∞)

(Ω ).

j=1

Let {B(X j , R j )}∞j=1 be a Whitney covering of Ω satisfying

Ω = ∪∞j=1 B(X j , R j ), R j /1000 ≤ dist(X j , ∂ Ω ) ≤ R j /100 and ∑ j∈N χR j ≤ Cn ; see, for example, [68, Proposition 7.3.4]. Set

J1 ≡ j ∈ N : (B(X j , R j ) ∩ B(xk , rk )) = 0/ and R j ≤ 4rk for some k ∈ N and J2 ≡ (N \ J1). Notice that if k ∈ N satisfies (B(X j , R j ) ∩ B(xk , rk )) = 0/ for some j ∈ J2 , then B(xk , rk ) ⊂ B(X j , 2R j ), since rk < R j /4. With this in mind, we define

K2 ≡ k ∈ N : (B(xk , rk ) ∩ B(X j , R j )) = 0/ for some j ∈ J2 , and K1 ≡ (N \ K2 ). It is easy to see that ∞

B(xk , rk ) ⊂

k=1

k∈K1

B(xk , rk )

B(X j , 2R j ) .

j∈J2

Furthermore, for each k ∈ N, the cardinality of the set

j ∈ J2 : (B(xk , rk ) ∩ B(X j , R j )) = 0/

(7.19)

232


is bounded by a constant depending only on the dimension. Thus, we have ∞

∑ rkd = ∑ rkd + ∑ rkd

k=1

k∈K1

∼

k∈K2

⎛

∑ rkd + ∑ ⎝ j∈J2

k∈K1

∼

⎞

j∈J2

rkd ⎠

∑

|B(xk , rk )| ⎠ .

k∈K2 , (B(xk ,rk )∩B(X j ,R j )) =0/

⎛

∑ rkd + ∑ ⎝

k∈K1

∑

⎞ d n

k∈K2 , (B(xk ,rk )∩B(X j ,R j )) =0/

Notice that

B(X j , R j ) ⊂ Ω ⊂

∞

B(xk , rk ) .

k=1

Then for each j ∈ J2 , we have B(X j , R j ) ⊂

⎧ ⎨ ⎩

k∈K2 , (B(xk ,rk )∩B(X j ,R j )) =0/

⎫ ⎬ B(xk , rk ) ⎭

d

Since d ∈ (0, n], by the monotonicity of n , we see that ⎛

⎞

∑

⎝

|B(xk , rk )| ⎠ d n

k∈K2 , (B(xk ,rk )∩B(X j ,R j )) =0/

⎞d

⎛

n

∑

≥⎝

|B(xk , rk )|⎠

k∈K2 , (B(xk ,rk )∩B(X j ,R j )) =0/ d

≥ |B(X j , R j )| n . As a consequence,

∞

∑ rkd ∑ rkd + ∑ Rdj ,

k=0

k∈K1

j∈J2

which combined with (7.19) yields that (∞)

Λd,∗ (Ω ) ≤

∞

∑ rkd + ∑ (2R j )d ∑ rkd + ∑ Rdj ∑ rkd Λd

k∈K1

j∈J2

This finishes the proof of Lemma 7.17.

k∈K1

j∈J2

(∞)

(Ω ).

k=0


233

s, τ n Proof of Theorem 7.11. For similarity, we concentrate on BH p, q (R ). By s, τ n Theorem 7.7, any f ∈ BH p, q (R ) admits a decomposition

∑

f=

tQ aQ

l(Q)≤1 s, τ n in S (Rn ), where each aQ is a smooth atom for BH p, q (R ) and t ≡ {tQ }l(Q)≤1 ⊂ C satisfies t bHp,s, τq(Rn ) f BHp,s, τq (Rn ) .

Since s > 1/p + 2nτ , there is no need to postulate any moment condition on aQ . Define 1 tQ 2 Tr( f ) ≡ ∑ tQ aQ (∗ , 0) = ∑ 1 [l(Q)] aQ (∗ , 0). 2 l(Q)≤1 l(Q)≤1 [l(Q)] By the support condition of atoms, the above summation can be re-written as 2

Tr( f ) ≡ ∑

∑

tQ ×[(i−1)l(Q ),il(Q )) [l(Q )]

i=0 Q ∈Q(Rn−1 ) l(Q )≤1

1 2

1

[l(Q )] 2 aQ ×[(i−1)l(Q ),il(Q )) (∗ , 0). (7.20)

We need to show that (7.20) converges in S (Rn−1 ) and Tr( f )

n τ s− 1p , n−1

BHp,q

(Rn−1 )

f BHp,s, τq (Rn ) .

By Theorem 7.7 again, we only need to prove that for each Q ∈ Q(Rn−1 ) with l(Q ) ≤ 1, 1 [l(Q )] 2 aQ ×[(i−1)l(Q ),il(Q )) (∗ , 0) n s− 1 , n−1 τ

is a smooth atom for BH p,q p

(Rn−1 ) supported near Q and for all i ∈ {0, 1, 2},

. / [l(Q )]− 12 tQ ×[(i−1)l(Q ),il(Q )) Q ∈Q(Rn−1 )

n τ s− 1p , n−1

bHp,q

(Rn−1 )

< ∞.

(7.21)

Indeed, it was already proved in Sect. 6.3 that 1

[l(Q )] 2 aQ ×[(i−1)l(Q ),il(Q )) (∗ , 0) s− 1 , n τ

is a smooth atom for BH p,q p n−1 (Rn−1 ). By similarity, we only prove (7.21) when i = 1. Let ω be a nonnegative function on Rn+1 + satisfying (7.3) and ⎧ ⎪ ⎨

∞

⎡

⎣ ∑ ⎪∑ ⎩ j=0

Q∈Q j (Rn )

s

1

|Q|−( n + 2 )p |tQ | p

⎤ q ⎫ 1q p⎪ ⎬ [ω (x, 2− j )]−p dx⎦ t bHp,s, τq (Rn ) . ⎪ Q ⎭

234


For all λ ∈ (0, ∞), set

/ . Eλ ≡ x ∈ Rn : [N ω (x)](p∨q) > λ .

Then there exists a ball cover {Bm }m such that nτ (p∨q)

Λnτ (p∨q) (Eλ ) ∼ ∑ rBm (∞)

.

m

(∞) be the {(n − 1) nτ (p ∨ q) }-Hausdorff capacity in Rn−1 and define ω Let Λ n−1 nτ (p∨q)

on Rn+ by setting, for all x ∈ Rn−1 and t ∈ (0, ∞),

(x ,t) ≡ C sup ω ((x , xn ),t), ω |xn | λ , then [N ω (x , 0)](p∨q) > λ , and hence (x , 0) ∈ Bm Therefore, if [N ω for some m, which further implies that . / n−1 (p∨q) ∗ (x )] : [N ω >λ ⊂ Bm , Eλ ≡ x ∈ R m

where B∗m is the projection of Bm from Rn to Rn−1 . Thus, we obtain

Rn−1

(∞)

(x )](p∨q) dΛ [N ω (x ) = nτ (p∨q)

∞ 0

(∞) Λnτ (p∨q) (Eλ ) d λ

0

Λnτ (p∨q) (Eλ ) d λ

∞

(∞)

1. Furthermore, . / [l(Q )]− 12 tQ ×[0,l(Q )) Q ∈Q(Rn−1 ),l(Q )≤1

⎧ ⎪ ⎨

n τ s− 1p , n−1

bHp,q

∞

⎡

∑ ⎪ ⎩ j=0

⎣

⎧ ⎪ ⎨

⎡

∞

Q ∈Q j (Rn−1 )

⎣ ⎪∑ ⎩ j=0

∑ ∑

Q ∈Q j (Rn−1 )

np

[l(Q )]−sp− 2 +1 |tQ ×[0,l(Q )) | p

np

[l(Q )]−sp− 2 |tQ ×[0,l(Q )) | p

(Rn−1 )

⎤ q ⎫ 1q p⎪ ⎬ − j −p ⎦ (x , 2 )] dx [ω ⎪ Q ⎭

⎤ q ⎫ 1q p⎪ ⎬ − j −p [ω (x, 2 )] dx⎦ ⎪ Q ⎭

t bHp,s, τq(Rn ) , n τ s− 1 , n−1

s, τ p n which implies that Tr is bounded from BH p, q (R ) to BH p,q

(Rn−1 ).


235 s− 1 , n τ

Let us show that Tr is surjective. To this end, for any f ∈ BH p,q p n−1 (Rn−1 ), by Theorem 7.7, there exist a sequence {aQ }Q ∈Q(Rn−1 ),l(Q )≤1 of smooth atoms for n τ s− 1 , n−1

BH p,q p

(Rn−1 ) and coefficients t ≡ {tQ }Q ∈Q(Rn−1 ),l(Q )≤1 such that

∑

f=

tQ aQ

Q ∈Q(Rn−1 ),l(Q )≤1

in S (Rn−1 ) and t

n τ s− 1p , n−1

bHp,q

(Rn−1 )

f

n τ s− 1p , n−1

BHp,q

(Rn−1 )

.

Let ϕ ∈ Cc∞ (R) with supp ϕ ⊂ (− 12 , 12 ) and ϕ (0) = 1. For all Q ∈ (Rn−1 ) and x ∈ R, set ϕQ (x) ≡ ϕ (2− log2 l(Q ) x). Under this notation, we define F≡

∑

tQ aQ ⊗ ϕQ .

Q ∈Q(Rn−1 ),l(Q )≤1 1

It is easy to check that for all Q ∈ Q(Rn−1 ) with l(Q ) ≤ 1, [l(Q )]− 2 aQ ⊗ ϕQ s, τ n is a smooth atom for BH p, q (R ) supported near Q × [0, l(Q )). Thus, to show F ∈ s, τ n BH p, q (R ), by Theorem 7.7, it suffices to prove that 1 {[l(Q )] 2 tQ }Q ∈Q(Rn−1 ),l(Q )≤1

s, τ

bHp, q (Rn )

satisfy Let ω

Rn−1

f

n τ s− 1p , n−1

BHp,q

(Rn−1 )

.

(∞)

(x )](p∨q) dΛ [N ω (x ) ≤ 1 nτ (p∨q)

such that ⎧ ⎪ ⎨

∞

∑ ⎪ ⎩ j=0

⎡ ⎣

|Q |−(

s−1/p 1 n−1 + 2 )p

|tQ | p

Q ∈Q j (Rn−1 )

is equivalent to t ists a ball

∑

s− 1p , n τ

. By Lemma 7.17, for each λ ∈ (0, ∞), there ex-

bHp,q n−1 (Rn−1 ) covering {B∗m }m ≡ {B(xB∗m , rB∗m )}m

∑(rB∗m )nτ (p∨q) m

⎤ q ⎫ 1q p⎪ ⎬ − j −p ⎦ [ω (x , 2 )] dx ⎪ Q ⎭

of Eλ such that (∞)

∼Λ nτ (p∨q) (Eλ )

and rB∗m > dist(xB∗m , ∂ Eλ )/10000

236


for all m. For all x = (x , xn ) ∈ Rn and t ∈ (0, ∞), define (x ,t)χ[0,t) (xn ). ω (x,t) ≡ ω 1

Notice that if N ω (x , xn ) > λ (p∨q) , then 1

(y ,t) = ω (y , yn ,t) > λ (p∨q) ω 1

(y ) > λ (p∨q) and thus, for some |(y , yn ) − (x , xn )| < t and yn ∈ [0,t). Then N ω ∗ y ∈ Bm for some m. Since, for all z ∈ B(y ,t), 1

(z ) ≥ ω (y ,t) > λ (p∨q) , Nω

we see that

B(y ,t) ⊂ Eλ ⊂

B∗m

,

m

and hence t ≤ 10000rB∗m . Since xn ∈ [0,t), we have (x , xn ) ∈ (20000B∗m) × [0, 20000rB∗m ) and Eλ ⊂

(20000B∗m) × [0, 20000rB∗m ),

m

which further implies that (∞) (∞) Λnτ (p∨q) (Eλ ) ∑(rB∗m )nτ (p∨q) Λnτ (p∨q) (Eλ )

m

and

Rn

(∞)

[N ω (x , xn )](p∨q) dΛnτ (p∨q) (x) =

∞ 0 ∞ 0

(∞)

Λnτ (p∨q) (Eλ ) d λ (∞) Λnτ (p∨q) (Eλ ) d λ

Rn−1

(∞)

(x )](p∨q) dΛ [N ω (x ) nτ (p∨q)

1. Therefore, we have 1 {[l(Q )] 2 tQ }Q ∈Q(Rn−1 ),l(Q )≤1

⎧ ⎪ ⎨

s, τ n bHp, q (R )

∞

∑ ⎪ ⎩ j=0

⎡ ⎣

∑

Q ∈Q j (Rn−1 )

s

1

p

[l(Q )]−( n + 2 )pn+ 2 |tQ | p

⎤ qp ⎫ 1q ⎪ ⎬ − j −p ⎦ [ω (x, 2 )] dx ⎪ Q ×[0,l(Q )) ⎭

7.3 A ( vmo , h1 )-Type Duality Result

⎧ ⎪ ⎨

∞

⎡

∑ ⎪ ⎩ j=0

t

⎣

∑

Q ∈Q

j

|Q |−(

s−1/p 1 n−1 + 2 )p

|tQ | p

(Rn−1 )

n τ s− 1p , n−1

bHp,q

f

237

⎤ q ⎫ 1q p⎪ ⎬ − j −p ⎦ (x , 2 )] dx [ω ⎪ Q ⎭

(Rn−1 )

n τ s− 1p , n−1

BHp,q

(Rn−1 )

,

s, τ n which implies that F ∈ BH p, q (R ) and

F BHp,s, τq (Rn ) f

n τ s− 1p , n−1

BHp,q

(Rn−1 )

.

Furthermore, the definition of F implies Tr(F) = f , which completes the proof of Theorem 7.11.

7.3 A ( vmo, h1 )-Type Duality Result s,τ When p = q ∈ (1, ∞), applying the atomic decomposition of ATp,p (Rn ) in Proposis,τ n tion 7.1, we also find a predual space of AH p,p (R ). In what follows, we denote by s,τ s,τ n ∞ n n 0 A p,p (R ) the closure of Cc (R ) in A p,p (R ). Recall that

τ (Rn ); Cc∞ (Rn ) ⊂ S (Rn ) ⊂ As,p,p

see Proposition 2.3. Theorem 7.12. Let s ∈ R, p ∈ (1, ∞) and τ ∈ [0, 1p ]. Then the dual space of

s,τ n 0 A p,p (R )

ear map

τ −s,τ n n is AH p−s, ,p (R ) in the following sense: if f ∈ AH p ,p (R ), then the lin-

ν →

Rn

f (x)ν (x) dx,

(7.22)

τ defined initially for all ν ∈ Cc∞ (Rn ), has a bounded extension to 0 As,p,p (Rn ) with the operator norm no more than a constant multiple of f AH −s,τ (Rn ) . p ,p

τ τ n Conversely, if L ∈ (0 As,p,p (Rn ))∗ , then there exists an f ∈ AH p−s, ,p (R ) with f AH −s,τ (Rn ) no more than a constant multiple of L such that L has the form p ,p

(7.22) for all ν ∈ Cc∞ (Rn ).

We remark that Theorem 7.12 generalizes the classical result that ( cmo (Rn ))∗ = h1 (Rn )

238


by taking s = 0, p = 2 and τ = 1/2, where cmo (Rn ) is the local CMO(Rn ) space and h1 (Rn ) is the local Hardy space; see, for example, [43]. The homogeneous counterpart of Theorem 7.12 was already established in [166]. To prove Theorem 7.12, we need several functional analysis results and some technical conclusions. We first obtain the corresponding result for sequence spaces. τ τ (Rn ) be the subspace of as,p,p (Rn ) consisting of all sequences with finite Let 0 as,p,p non-vanishing elements. Proposition 7.6. Let s ∈ R, p ∈ (1, ∞) and τ ∈ [0, 1p ]. Then τ τ n (0 as,p,p (Rn ))∗ = aH p−s, ,p (R ) τ n in the following sense: for each t = {tQ }l(Q)≤1 ∈ aH p−s, ,p (R ), the map

λ ≡ {λQ }l(Q)≤1 → λ ,t ≡

∑

λQtQ

(7.23)

l(Q)≤1

τ (Rn ) with the operator norm no more induces a continuous linear functional on 0 as,p,p τ (Rn ))∗ is of than a constant multiple of t aH −s,τ (Rn ) . Conversely, every L ∈ (0 as,p,p p ,p

τ n the form (7.23) for a certain t ∈ aH p−s, ,p (R ) and t aH −s,τ (Rn ) is no more than a p ,p

constant multiple of L .

Proof. Since Proposition 7.6 when τ = 0 is just the classic result on TriebelLizorkin spaces, we only need consider the case that τ > 0. By Proposition 7.4 τ and the definition of 0 as,p,p (Rn ), we have that s,τ n s,τ n 0 a p (R ) ⊂ a p (R )

τ n ∗ = (aH p−s, ,p (R )) ,

which implies that τ −s,τ n n ∗∗ τ ⊂ (0 as,p,p (Rn ))∗ . aH p−s, ,p (R ) ⊂ (aH p ,p (R ))

To show τ τ n (Rn ))∗ ⊂ aH p−s, (0 as,p,p ,p (R ),

we first claim that if this is true when s = 0, then it is also true for all s ∈ R. In fact, for all u ∈ R, define an operator Tu by setting, for all sequences t ≡ {tQ }l(Q)≤1 ⊂ C u and dyadic cubes Q satisfying l(Q) ≤ 1, (Tut)Q ≡ |Q|− n tQ . Then Tu is an isometric τ τ s,τ s+u,τ n n n (Rn ) to as+u, isomorphism from as,p,p p,p (R ) and from aH p,p (R ) to aH p,p (R ). If s,τ 0, τ L ∈ (0 a p,p(R)∗ , then L ◦ Ts ∈ (0 a p,p(Rn ))∗ and hence there exists a sequence τ n λ ≡ {λQ }l(Q)≤1 ∈ aH p0, ,p (R )


239

such that

∑

L ◦ Ts (t) =

tQ λQ

l(Q)≤1

τ s,τ n n for all t ∈ 0 a0, p (R ). Since Ts ◦ T−s is the identity on 0 a p,p (R ) and T−s is an isomets,τ 0,τ n n ric isomorphism from 0 a p,p(R ) onto 0 a p,p (R ), then

L(t) = L ◦ Ts ◦ T−s (t) =

∑

(T−st)Q λQ =

l(Q)≤1

∑

tQ (T−s λ )Q

l(Q)≤1

τ τ −s,τ n n for all t ∈ 0 as,p,p (Rn ). Since λ ∈ aH p0, ,p (R ), we see that T−s λ ∈ aH p ,p (R ) and

T−s λ aH −s,τ (Rn ) = λ aH 0,τ p ,p

p ,p

(Rn )

.

Thus, the above claim is true. Next we prove that 0,τ τ n ∗ n (0 a0, p,p (R )) ⊂ aH p ,p (R ). τ 0,τ n n Notice that 0 a0, p,p (R ) consists of all sequences in a p,p (R ) with finite non-vanishing 0,τ elements. We know that every L ∈ (0 a p,p(Rn ))∗ is of the form

λ →

∑

λQtQ

l(Q)≤1 0,τ n for a certain t ≡ {tQ }l(Q)≤1 ⊂ C. In fact, for any m ∈ N, let m 0 a p,p (R ) denote the set 0,τ of all sequences λ ≡ {λQ }l(Q)≤1 ∈ a p,p (Rn ), where λQ = 0 if Q ∩ [−2m , 2m )n = 0/ 0,τ n ∗ or l(Q) < 2−m . Then L ∈ (m 0 a p,p (R )) . It is easy to see that each linear functional 0, τ n ∗ in (m 0 a p,p (R )) has the form (7.23). Thus, there exists tm ≡ {(tm )Q }l(Q)≤1 , where 0,τ n (tm )Q = 0 if Q ∩ (−2m , 2m ]n = 0/ or l(Q) < 2−m , such that L(λ ) for all λ ∈ m 0 a p,p (R ) has the form (7.23) with t replaced by tm . By this construction, we are easy to see that (tm+1 )Q = (tm )Q if Q ⊂ [−2m , 2m )n and 2−m ≤ l(Q) ≤ 1. Thus, if let tQ ≡ (tm )Q when Q ⊂ [−2m , 2m )n and 2−m ≤ l(Q) ≤ 1, then t ≡ {tQ }l(Q)≤1 is the desired sequence. To complete the proof of Proposition 7.6, we need to show that

t aH 0,τ

p ,p

(Rn )

L (

0,τ n ∗ 0 a p,p (R ))

.

To this end, for all m ∈ N, define χm by setting χm (Q) ≡ 1 if Q ⊂ [−2m , 2m )n and τ n 2−m ≤ l(Q) ≤ 1, χm (Q) ≡ 0 otherwise. Then for all λ ≡ {λQ }l(Q)≤1 ∈ a0, p,p (R ) with λ a0,τ (Rn ) ≤ 1, we have p,p

τ n λm ≡ {λQ χm (Q)}l(Q)≤1 ∈ 0 a0, p,p (R )

240


and λm a0,τ (Rn ) ≤ 1. Thus, using Fatou’s lemma yields p,p

∑

∑

|λQ ||tQ | ≤ lim

m→∞

l(Q)≤1

= lim

m→∞

|λQ |tQ χm (Q)tQ |tQ | l(Q)≤1

∑

≤ lim L ( m→∞

≤ L (

|λQ |χm (Q)|tQ |

l(Q)≤1

0,τ n ∗ 0 a p,p (R ))

0,τ n ∗ 0 a p,p (R ))

λm a0,τ (Rn ) p,p

.

(7.24)

Notice that for all m ∈ N, τ n tm ≡ {tQ χm (Q)}l(Q)≤1 ∈ aH p0,,p (R ).

For each m, we define function F (m) by setting, for all x ∈ Rn and j ∈ Z+ , F (m) (x, 2− j ) ≡

∑

Q (x). |tQ |χm (Q)χ

Q∈Q j

Then (m) AT 0,τ F (m) ∈ ATp0,,pτ (Rn+1 Z+ ) and F

p ,p

(Rn+1 Z ) +

∼ tm aH 0,τ

p ,p

(Rn )

.

Applying Theorem 7.2, we see that F

(m)

AT 0,τ

p ,p

(Rn+1 Z ) +

∞ (m) −j −j sup F (x, 2 )G(x, 2 ) dx ∑ Rn j=0 ∞ sup ∑ ∑ |tQ |χm (Q)|Q|−1/2 G(x, 2− j ) dx , j=0 Q∈Q Q j

0,τ where the supremum is taken over all functions G ∈ AWp,p (Rn+1 Z+ ) with compact support satisfying G AW 0,τ (Rn+1 ) ≤ 1. If we set p,p

Z+

−1/2

λQ ≡ |Q|

Q

G(x, 2− j ) dx

and λ ≡ {λQ }l(Q)≤1 , then using Hölder’s inequality, we obtain λ a0,τ (Rn ) G AW 0,τ (Rn+1 ) 1, p,p

p,p

Z+


241

and hence tm aH 0,τ

p ,p

(Rn )

∼ F (m) AT 0,τ (Rn+1 ) Z+ p ,p sup

∑

|λQ ||tQ | : λ ∈

τ n a0, p,p (R ),

l(Q)≤1

λ a0,τ (Rn ) ≤ 1 , p,p

which together with (7.24) yields tm aH 0,τ

p ,p

(Rn )

∼ F (m) AT 0,τ

p ,p

(Rn+1 Z ) +

L (

0,τ n ∗ 0 a p,p (R ))

.

τ n+1 n To show t ∈ aH p0, ,p (R ), let F be the function on RZ+ defined by setting, for all n x ∈ R and j ∈ Z+ , Q (x). F(x, 2− j ) ≡ ∑ |tQ |χ Q∈Q j

Notice that t aH 0,τ

p ,p

(Rn )

∼ F AT 0,τ

p ,p

It suffices to prove that F ∈ ATp0,,pτ (Rn+1 Z+ ). Recall that F (m) AT 0,τ (Rn+1 ) L ( p ,p

Z+

(Rn+1 Z ) +

.

0,τ n ∗ 0 a p,p (R ))

.

0,τ (Rn+1 By Lemma 7.6, there exist a subsequence {F (mi ) }i∈N and F ∈ ATp,p Z+ ) such

τ n+1 that for all G ∈ AWp0, ,p (RZ+ ) with compact support,

G F (mi ) , G → F, as i → ∞ and its quasi-norm 0,τ n+1 L F AT (R ) ( p,p

Z+

0,τ n ∗ 0 a p,p (R ))

,

which together with the uniqueness of the weak limit and the fact that F (m) → F 0,τ (Rn+1 pointwise as m → ∞ yields that F = F in ATp,p Z+ ) and F AT 0,τ (Rn+1 ) L ( p,p

Z+

This finishes the proof of Proposition 7.6.

0,τ n ∗ 0 a p,p (R ))

.

s,τ s,τ n n n Let 0 A p,p (R ) denote the closure of S (R ) in A p,p (R ). As an immediate cons, τ n sequence of Proposition 7.6 and the ϕ -transform characterizations of AH p, q (R ) s, τ n and A p, q (R ), we have the following theorem, which generalizes the classical results on Besov spaces and Triebel-Lizorkin spaces when p = q; see, for example, [145, p. 180].

242


Theorem 7.13. Let s ∈ R, p ∈ (1, ∞) and τ ∈ [0, 1p ]. Then the dual space of s,τ A (Rn ) is AH −s,τ (Rn ) in the following sense: If f ∈ AH −s,τ (Rn ), then the linear p ,p

0 p,p

p ,p

s,τ n map defined as in (7.22) for all ν ∈ S (Rn ), has a bounded extension to 0 A p,p (R ) with operator norm no more than a constant multiple of f AH −s,τ (Rn ) . p ,p

s,τ −s,τ n n ∗ Conversely, if L ∈ (0 A p,p (R )) , then there exists an f ∈ AH p ,p (R ) with f AH −s,τ (Rn ) no more than a constant multiple of L such that L has the form p ,p

(7.22) for all ν ∈ S (Rn ). Proof. Since the case that τ = 0 is known (see [145, p. 180]), we only need consider s,τ the case that τ > 0. By Theorem 7.3 and the definition of A (Rn ), we have that 0 p,p

s,τ n 0 A p,p (R )

τ τ n ∗ ⊂ As,p,p (Rn ) = (AH p−s, ,p (R )) ,

which implies that τ −s,τ s,τ n n ∗∗ n ∗ AH p−s, ⊂ (0 A ,p (R ) ⊂ (AH p ,p (R )) p,p (R )) .

To show

s,τ −s,τ n n ∗ (0 A p,p (R )) ⊂ AH p ,p (R ),

let Φ and ϕ satisfy, respectively, (2.1) and (2.2) such that (2.6) holds with Ψ and ψ s,τ replaced, respectively, by Φ and ϕ . If L ∈ ( A (Rn ))∗ , then applying Theorem 7.4, 0 p,p

we see that

τ ≡ L ◦ Tϕ ∈ (0 as,p,p (Rn ))∗ . L

τ n By Proposition 7.6, there exists a λ ≡ {λQ }l(Q)≤1 ∈ aH p−s, ,p (R ) such that

L(t) =

∑

tQ λQ

l(Q)≤1

τ for all t ≡ {tQ }l(Q)≤1 ∈ 0 as,p,p (Rn ) and

L (0 as,p,pτ (Rn ))∗ L λ aH −s,τ (Rn ) p ,p

Notice that L ◦ Sϕ = L ◦ Tϕ ◦ Sϕ = L. Thus, for all f ∈ S (Rn ), if letting g ≡ Tϕ (λ ) ≡

∑

l(Q)≤1

λQ ϕQ ,

. s,τ n ∗ (0 A p,p (R ))


then

243

∑

◦ Sϕ ( f ) = L( f ) = L

(Sϕ f )Q λQ = f , g.

l(Q)≤1

Furthermore, by Theorem 7.4 again, we have g AH −s,τ (Rn ) λ aH −s,τ (Rn ) L p ,p

p ,p

. s,τ n ∗ (0 A p,p (R ))

This finishes the proof of Theorem 7.13.

Now we are ready to prove Theorem 7.12. Proof of Theorem 7.12. By Proposition 2.3, we see that τ (Rn ), Cc∞ (Rn ) ⊂ S (Rn ) ⊂ As,p,p

and hence s,τ n 0 A p,p (R )

s,τ n ⊂ 0 A p,p (R ).

Therefore, to obtain Theorem 7.12, by Theorem 7.13, it suffices to prove that s,τ n 0 A p,p (R )

τ ⊂ 0 As,p,p (Rn ).

s,τ s,τ n n Let f ∈ 0 A p,p (R ) and ε > 0. By the definition of 0 A p,p (R ), there exists a funcn tion g ∈ S (R ) such that

f − g As,p,pτ (Rn ) < ε /2. Thus, to complete the proof, it suffices to find a function h ∈ Cc∞ (Rn ) such that g − h As,p,pτ (Rn ) < ε /2. By the proof of Proposition 2.3, we know that for all ϕ ∈ S (Rn ), ϕ As,p,pτ (Rn ) ≤ C ϕ SM when M > max{0, s + nτ , n(1/p − 1)} + 1. On the other hand, since g ∈ S (Rn ), for each fixed M ∈ N, there exists a function h ∈ Cc∞ (Rn ) such that g − h SM < ε /(2C). Thus, we have g − h As,p,pτ (Rn ) ≤ C g − h SM < ε /2, which completes the proof of Theorem 7.12.

244


s,τ s,τ n n The proof of Theorem 7.12 implies that 0 A p,p (R ) = 0 A p,p (R ). We have the following interesting remark.

Remark 7.7. (i) We first claim that when τ > 0, the dual property in Theorem 7.12 is not possible τ n to be correct for 0 Bs,p,τq(Rn ) and BH p−s, ,q (R ) with p ∈ (1, ∞), q ∈ [1, ∞) and q > p, which is quite different from the case that τ = 0. Recall that when τ = 0, p ∈ (1, ∞) and q ∈ [1, ∞), s, τ n 0 B p, q (R )

τ n = Bs,p,τq (Rn ) and (Bs,p,τq (Rn ))∗ = B−s, p ,q (R );

see [145, p. 244]. To show the claim, by Remark 7.5 (see also [168, Propositions 2.2(i) and 2.3(i)]), we know that if 1 < p0 < p1 < ∞, −∞ < s1 < s0 < ∞, q ∈ [1, ∞) and

1 1 τ ∈ 0, min , (p0 ∨ q) (pq ∨ q)

such that s0 − n/p0 = s1 − n/p1, then BH ps00,,qτ (Rn ) ⊂ BH ps11,,qτ (Rn ) ⇐⇒ τ (p0 ∨ q) = τ (p1 ∨ q) . When τ > 0, the sufficient and necessary condition that τ (p0 ∨ q) = τ (p1 ∨ q) is equivalent to that q ≥ p1 . If we assume that Theorem 7.12 is correct for s, τ −s,τ n n 0 B p, q (R ) and BH p ,q (R ) with τ > 0 and certain 1 < p < q < ∞, then by this assumption together with an argument by duality and the embedding s−n/p+n/q,τ

Bs,p,τq (Rn ) ⊂ Bq,q

(Rn )

in Corollary 2.2, we see that −s+n/p−n/q,τ

BHq ,q

(Rn ) ⊂ BH ps,τ,q (Rn ),

which is not true since q 0 and p = q, only when 1 ≤ q 0, the dual property in Theorem 7.12 is not τ n possible to be correct for all 0 Fp,s, qτ (Rn ) and FH p−s, ,q (R ) with p, q ∈ (1, ∞) and q > p. In fact, by Remark 7.5, we know that the embedding FH ps00,,rτ (Rn ) ⊂ FH ps11,,qτ (Rn ) is true only when

τ (p0 ∨ r) ≤ τ (p1 ∨ q) + τ (1/p0 − 1/p1)(p0 ∨ r) (p1 ∨ q) .


245

If we assume that τ n (0 Fp,s, qτ (Rn ))∗ = FH p−s, ,q (R )

for all s ∈ R, τ > 0 and 1 q together with an argument by duality, we have −s+n/p−n/q,τ

FHq ,r

τ n (Rn ) ⊂ FH p−s, ,q (R ),

which is not possible by the above conclusion. Thus, the claim is also true. It is also unclear that when p = q, for which range of p, q ∈ (1, ∞), the conclusion of Theorem 7.12 is true. By Theorem 7.12 and Corollary 2.2, a dual argument yields that following conclusion, which improves Proposition 7.3(ii) in the case that p = q. Proposition 7.7. Let s0 , s1 ∈ R, p0 , p1 ∈ (1, ∞) and τ ∈ [0, p1 ] such that p0 < p1 and s0 − n/p0 = s1 − n/p1. Then FH ps00,,pτ 0 (Rn ) ⊂ FH ps11,,pτ 1 (Rn ). Also, from Proposition 7.6, Theorem 3.1 and a dual argument, we deduce the following result. Proposition 7.8. Let ε ∈ (0, ∞), s ∈ R, p ∈ (1, ∞) and τ ∈ [0, 1/p]. Then all the s,τ ε -almost diagonal operators are bounded on aH p,p (Rn ). We remark that Proposition 7.8 improves Theorem 7.5 in the case that p = q, since in Theorem 7.5, we need an additional condition that ε > 2nτ . From Proposition 7.8 and the arguments in Sect. 7.2, we deduce that when p = q, s, τ n the smooth atomic and molecular decomposition characterizations of AH p, q (R ) in Theorems 7.6 and 7.7 can be improved. Precisely, in the case that p = q, via replacing the conditions N ≡ max{s + 2nτ , −1}, M ∈ (n + 2nτ , ∞) and |γ | ≤ s + 3nτ in Definition 7.7, respectively, by N ≡ max{s, −1}, M ∈ (n, ∞) and |γ | ≤ s+ nτ , we obtain a class of “weaker” molecules. Proposition 7.8 and the arguments in Sect. 7.2 then yield that Theorem 7.6 is still true for these new molecules in the case that p = q. Also, via replacing the conditions |γ | ≤ max{−s + 2nτ , −1} and |γ | ≤ max{s + 3nτ + 1, −1} in Definition 7.8, respectively, by |γ | ≤ max{−s, −1} and |γ | ≤ max{s + nτ + 1, −1},

246


we obtain a class of “weaker” atoms. Proposition 7.8 and the arguments in Sect. 7.2 then yield that in the case that p = q, Theorem 7.7 is still true for these new atoms in the case that p = q. Via these improvements, in the case that p = q, the conditions s > 2nτ or s ≤ 2nτ in Theorem 7.8 can be replaced, respectively, by s > 0 or s ≤ 0, and the condition s ∈ (1/p + 2nτ , ∞) in Theorem 7.11 can be replaced by s ∈ (1/p, ∞). We omit the details.

7.4 Real Interpolation In this section we are concerned with the interpolation properties of the spaces s, τ n AH p, q (R ). Nowadays interpolation theory is a well established tool in various branches of mathematics, but in particular in the theory of partial differential equations. s, τ n To establish the real interpolation properties of AH p, q (R ), we need some preparations (see, for example, [145, pp. 62–63]). Let H be a linear complex Hausdorff space and A0 and A1 be complex quasi-Banach spaces such that A0 , A1 ⊂ H . Let A0 +A1 be the set of all elements a ∈ H such that a can be represented as a = a0 +a1 with a0 ∈ A0 and a1 ∈ A1 . As usual, for t ∈ (0, ∞) and a ∈ A0 + A1 , Peetre’s celebrated K-functional is defined by K(t, a; A0 , A1 ) ≡ inf( a0 A0 + t a1 A1 ), where the infimum is taken over all representations of a of the above form. Let θ ∈ (0, 1) and q ∈ (0, ∞]. The interpolation space (A0 , A1 )θ ,q is defined to be the set of all a ∈ A0 + A1 such that a (A0,A1 )θ ,q < ∞, where a (A0,A1 )θ ,q ≡

%

∞ 0

[t

−θ

dt K(t, a; A0 , A1 )] t

&1/q

q

with suitable modifications when q = ∞. s, τ n Lemma 7.9 is the basis for the real interpolation theory of AH p, q (R ), which n shows that S (R ) can be identified as the Hausdorff space H mentioned above. The following result partially generalizes [145, Theorem 2.4.2]. Theorem 7.14. Let θ ∈ (0, 1), q0 , q1 , q ∈ [1, ∞), p ∈ (1, ∞), τ ∈ [0, 1/p ] and s0 , s1 ∈ (0, ∞) satisfy s0 = s1 , s = (1 − θ )s0 + θ s1 and

τ (p ∨ q) = τ (p ∨ q0 ) = τ (p ∨ q1 ) . Then s0 ,τ s1 ,τ s,τ (Rn ), AH p,q (Rn ))θ ,q = BH p,q (Rn ). (AH p,q 0 1

7.4 Real Interpolation

247

Proof. When τ = 0, Theorem 7.14 is just the classic result obtained in [145, Theorem 2.4.2]. We only consider the case when τ > 0, under which the restriction that

τ (p ∨ q) = τ (p ∨ q0 ) = τ (p ∨ q1 ) implies that p ≥ max{q0, q1 , q}. Let q2 ∈ [q0 ∨ q1 , p]. We first show that s0 ,τ s1 ,τ s,τ (BH p,q (Rn ), BH p,q (Rn ))θ ,q ⊂ BH p,q (Rn ). 2 2

(7.25)

Without loss of generality, we may assume that s0 > s1 . Notice that (0, ∞) =

∞

[2(k−1)(s0 −s1 ) , 2k(s0 −s1 ) ).

k=−∞

Then ∞ 0

s0 ,τ s1 ,τ t −θ q [K(t, f ; BH p,q (Rn ), BH p,q (Rn ))]q 2 2

∞

dt t

s0 ,τ s1 ,τ (Rn ), BH p,q (Rn ))]q . ∑ 2−θ qk(s0−s1) [K(2k(s0 −s1 ) , f ; BHp,q 2 2

k=0

s0 ,τ s1 ,τ n n Write f ≡ f0 + f1 with f0 ∈ BH p,q 2 (R ) and f 1 ∈ BH p,q2 (R ) such that

f0 BH s0 ,τ (Rn )) + 2k(s0−s1 ) f1 BH s1 ,τ (Rn )) p,q2

≤ 2K(2

p,q2

k(s0 −s1 )

,

s0 ,τ s1 ,τ f ; BH p,q (Rn ), BH p,q (Rn )). 2 2

(7.26)

There exist ω0 , ω1 satisfying (7.3) with q replaced by q2 such that for i = 0, 1, q ksi qi −k −1 i 2 ϕ ∗ f [ ω (·, 2 )] p i i k ∑ ∞

L (Rn )

k=0

f i qi

s ,τ

i (Rn )) BHp,q 2

.

(7.27)

Set ω ≡ (ω0 + ω1 )/2. Then (7.27) remains true if we replace ωi by ω , which together with (7.26) further yields that 2ks0 ϕk ∗ f [ω (·, 2−k )]−1 p n L (R ) ≤ 2ks0 ϕk ∗ f0 [ω (·, 2−k )]−1

L p (Rn )

≤ f0 BH s0 ,τ (Rn ) + 2 p,q2

K(2

k(s0 −s1 )

,

k(s0 −s1 )

+ 2k(s0 −s1 ) 2ks1 ϕk ∗ f1 [ω (·, 2−k )]−1

f1 BH s1 ,τ (Rn )

s0 ,τ f ; BH p,q (Rn ), 2

p,q2

s1 ,τ BH p,q (Rn )). 2

L p (Rn )

248


Notice that ω also satisfies (7.3). We then have f qBH s, τ (Rn ) p, q

∞

q

∑ 2ksq ϕk ∗ f [ω (·, 2−k )]−1 L p (Rn )

k=0 ∞

q k(s−s0 )q k(s0 −s1 ) s0 ,τ n s1 ,τ n K(2 2 , f ; BH (R ), BH (R )) ∑ p,q2 p,q2

k=0 ∞ 0

s0 ,τ s1 ,τ t −θ q [K(t, f ; BH p,q (Rn ), BH p,q (Rn ))]q 2 2

dt . t

This implies that (7.25) holds. Let r ∈ [1, min{q0 , q1 }]. Next we prove that s,τ s0 ,τ s1 ,τ (Rn ) ⊂ (BH p,r (Rn ), BH p,r (Rn ))θ ,q . BH p,q

(7.28)

Since s > s1 , applying Hölder’s inequality concludes that s, τ n s1 ,τ n BH p, q (R ) ⊂ BH p,r (R ),

which further implies that s0 ,τ s1 ,τ (Rn ), BH p,r (Rn )) t f BH s1 ,τ (Rn ) t f BHp,s, τq(Rn ) . K(t, f ; BH p,r p,r

Thus,

1 0

s0 ,τ s1 ,τ t −θ q [K(t, f ; BH p,r (Rn ), BH p,r (Rn ))]q

dt q f BH s, τ (Rn ) . p, q t

It remains to estimate I≡

∞ 1

s0 ,τ s1 ,τ t −θ q [K(t, f ; BH p,r (Rn ), BH p,r (Rn ))]q

dt . t

Similarly, we have I

∞

s0 ,τ s1 ,τ (Rn ), BH p,r (Rn ))]q . ∑ 2−θ qk(s0 −s1) [K(2k(s0 −s1 ) , f ; BHp,r

k=0

Let Φ and ϕ satisfy, respectively, (2.1) and (2.2). Assume further that (2.6) holds with Ψ and ψ replaced, respectively, by Φ and ϕ . Then by the Calderón reproducing formula in Lemma 2.3, we can write f ≡ f0 + f1 with f0 ≡

k

∑ ϕj ∗ f

j=0

and f1 ≡

∞

∑

j=k+1

ϕj ∗ f,

7.4 Real Interpolation

249

where when j = 0, ϕ0 is replaced by Φ . Then I

∞

∞

∑ 2kq(s−s0) f0 BHp,rs0 ,τ (Rn ) + ∑ 2kqs−kqs0 2kq(s0 −s1) f1 BHp,rs1 ,τ (Rn ) ≡ I1 + I2. q

k=0

q

k=0

Notice that ϕm ∗ ϕ j ≡ 0 if |m − j| > 1. For I1 , we have I1 =

∞

∞

∑ 2kq(s−s0) inf ∑ 2mrs0 ω

k=0

∞

≤ inf ∑ 2 ω

k=0 ∞

≤ inf ∑ 2 ω

∞

mrs0

|ϕm ∗ f0 | [ω (x, 2 p

Rn

m=0

−m −p

)]

r/p q/r r/p q/r

dx

j+1

k

∑

kq(s−s0 )

|ϕm ∗ f0 | p [ω (x, 2−m )]−p dx

Rn

m=0

∑2

kq(s−s0 )

∑

2mrs0

j=0 m=( j−1)∨0

k=0

×

|ϕm ∗ ϕ j ∗ ϕ j ∗ f | [ω (x, 2 p

Rn

−m −p

)]

r/p q/r .

dx

Let t0 ∈ (s, s0 ). By Hölder’s inequality with r/q + r/σ = 1, we further have q/σ ∞

ω

×

∑

∑

|ϕm ∗ ϕ j ∗ ϕ j ∗ f | [ω (x, 2 p

Rn ∞

×

∞

2mqt0

|ϕm ∗ ϕ j ∗ ϕ j ∗ f | [ω (x, 2

inf ∑ 2 ω

∑

p

Rn

j+1

jq(s−t0 )

j=0

2mqt0

dx

j=0 m=( j−1)∨0

k=0

)]

q/p

∑

j=0 m=( j−1)∨0

j+1

k

inf ∑ 2kq(s−t0 ) ∑ ω

−m −p

j+1

k

∑

2mσ (s0 −t0 )

j=0 m=( j−1)∨0

k=0

j+1

k

I1 inf ∑ 2kq(s−s0)

∑

2

mqt0

−m −p

)]

Rn

m=( j−1)∨0

q/p dx

|ϕm ∗ ϕ j ∗ ϕ j ∗ f | [ω (x, 2 p

−m −p

)]

q/p .

dx

Similarly to the proof of [164, Propostion 5.1], we obtain that the last line of the above inequalities can be dominated by f BHp,s, τq (Rn ) , and hence I1 f BHp,s, τq (Rn ) . The proof of the estimate that I2 f BHp,s, τq (Rn ) is similar. In fact, for I2 , we also have ∞

I2 ≤ inf ∑ 2kq(s−s1 ) ω

k=0

∞

j+1

∑

∑

2mrs1

j=k+1 m=( j−1)∨0

×

Rn

|ϕm ∗ ϕ j ∗ ϕ j ∗ f | [ω (x, 2 p

−m −p

)]

r/p q/r dx

.

250


Let t1 ∈ (s1 , s). By Hölder’s inequality with r/q + r/σ = 1 again, we obtain that

∞

I2 inf ∑ 2 ω

×

k=0

Rn ∞

j=0

∞

q/σ

j+1

∑

∑

2

mσ (s1 −t1 )

j=k+1 m=( j−1)∨0

|ϕm ∗ ϕ j ∗ ϕ j ∗ f | p [ω (x, 2−m )]−p dx

inf ∑ 2 ω

kq(s−s1 )

jq(s−t1 )

j+1

∑

m=( j−1)∨0

2

mqt1

q/p

j+1

k

∑

∑

j=0 m=( j−1)∨0

|ϕm ∗ ϕ j ∗ ϕ j ∗ f | [ω (x, 2 p

Rn

2mqt1

−m −p

)]

q/p dx

.

Similarly to the estimate of I1 , we have I2 f BHp,s, τq (Rn ) , which further yields (7.28). By (7.25), (7.28), [145, Remark 2.4.1/4] and the trivial embedding that for i = 0, 1, si ,τ si ,τ si ,τ BH p,r (Rn ) ⊂ BH p,q (Rn ) ⊂ BH p,q (Rn ), i 2

we see that s, τ n s0 ,τ n s1 ,τ n BH p, q (R ) ⊂ (BH p,r (R ), BH p,r (R ))θ ,q

s0 ,τ s1 ,τ ⊂ (BH p,q (Rn ), BH p,q (Rn ))θ ,q 0 1 s0 ,τ s1 ,τ ⊂ (BH p,q (Rn ), BH p,q (Rn ))θ ,q 2 2 s, τ n ⊂ BH p, q (R ).

This proves Theorem 7.14 for Besov-Hausdorff spaces. The interpolation conclusion for Triebel-Lizorkin-Hausdorff spaces follows form that for Besov-Hausdorff spaces and the trivial embedding that s,τ s,τ s, τ n n (Rn ) ⊂ FH p, BH p,min{p,q} q (R ) ⊂ BH p,max{p,q} (R ),

which completes the proof of Theorem 7.14.

Recall that when τ = 0, the Besov-Hausdorff and Triebel-Lizorkin-Hausdorff spaces are just, respectively, Besov spaces and Triebel-Lizorkin spaces. Then Theorem 7.14 partially generalizes the classical real interpolation conclusions in [145, Theorem 2.4.2].

Chapter 8

Homogeneous Spaces

In this chapter we deal with the homogeneous counterpart of As,p,τq (Rn ). The homogeneous Besov-type spaces B˙ s,p,τq (Rn ) and Triebel-Lizorkin-type spaces F˙p,s, qτ (Rn ) were introduced and investigated in [127, 164–167].

8.1 The Definition and Some Preliminaries To recall definitions of B˙ s,p,τq (Rn ) and F˙p,s, qτ (Rn ) in [164,165], we need some notation. Following Triebel’s [145], we set S∞ (Rn ) ≡ ϕ ∈ S (Rn ) :

Rn

ϕ (x)xγ dx = 0 for all multi-indices γ ∈ Zn+

and use S∞ (Rn ) to denote the topological dual of S∞ (Rn ), namely, the set of all continuous linear functionals on S∞ (Rn ) endowed with weak ∗-topology. Let ϕ ∈ S (Rn ) such that supp ϕ ⊂ {ξ ∈ Rn : 1/2 ≤ |ξ | ≤ 2}, |ϕ(ξ )| ≥ C > 0 if 3/5 ≤ |ξ | ≤ 5/3. (8.1) Then ϕ ∈ S∞ (Rn ). Moreover, it is well known that there exists a function ψ ∈ S (Rn ) satisfying (8.1) such that

∑ ϕ(2 j ξ )ψ (2 j ξ ) = 1

j∈Z

for all ξ ∈ Rn \ {0}; see [65, Lemma (6.9)]. Let P(Rn ) denote the set of all polynomials on Rn . We endow S (Rn )/P(Rn ) with the quotient topology (namely, O is open in S (Rn )/P(Rn ) if and only if π −1 (O) is open in S (Rn )), where π is the quotient map form S (Rn ) to S (Rn )/P(Rn ). The following assertion is well known. For completeness, we give its proof.


251

252

8 Homogeneous Spaces

Proposition 8.1. S (Rn )/P(Rn ) is identified with S∞ (Rn ) as topological spaces, namely, there exists a homeomorphism T mapping S (Rn )/P(Rn ) onto S∞ (Rn ). Proof. Define the map T : S (Rn )/P(Rn ) → S∞ (Rn ) by setting T ([ f ]) ≡ f |S∞ (Rn )

for all [ f ] = f + P(Rn ) ∈ S (Rn )/P(Rn ).

Notice that for any f ∈ S (Rn ) and P1 , P2 ∈ P(Rn ), ( f + P1)|S∞ (Rn ) = ( f + P2 )|S∞ (Rn ) . Then T is well defined. To show that T is a homeomorphism, we need to prove that T is injective, surjective, continuous and its inverse T −1 is also continuous. Step 1. T is injective. It suffices to show that if T ([ f ]) = 0 in S∞ (Rn ), then [ f ] = [0] in S (Rn )/P(Rn ), equivalently, if f ∈ S (Rn ) satisfying f |S∞ (Rn ) = 0, then f ∈ P(Rn ). In fact, since f |S∞ (Rn ) = 0, we know that for all ϕ ∈ S∞ (Rn ),

f , ϕ = 0, and hence f, ϕ = 0. We then claim that supp f ⊂ {0}. To see this, by [67, p. 12, Definition 1.4.1], supp f = Rn \ {x ∈ Rn : f = 0 on a neighborhood of x}. If supp f {0}, we can find x0 ∈ Rn \ {0} such that for any ε > 0, there exists a ϕ ∈ S (Rn ) satisfying supp ϕ ⊂ B(x0 , ε ) and f, ϕ = 0. Since x0 = 0, if ε is sufficiently small, then ϕ ≡ 0 in a neighborhood of 0. Thus, for all α , ∂ α ϕ(0) = 0, namely, ϕ ∈ S∞ (Rn ). Then f, ϕ = 0 contradicts f |S∞ (Rn ) = 0. This finishes the proof of the above claim. By [67, p. 36, Theorem 3.2.1], there exists N ∈ Z+ such that f =

∑

|α |≤N

Cα ∂ α δ ,

α = C ∂αδ where Cα ∈ C and δ is dirac function. This observation together with x α n yields that f ∈ P(R ) and then, T is injective. Step 2. T is surjective. Notice that S (Rn ) is a locally convex space. Then by [116, p. 61, Theorem 3.6], for each f ∈ S∞ (Rn ), there exists a f ∈ S (Rn ) such that f|S∞ (Rn ) = f . Thus, T ([ f]) = f . Step 3. T is continuous. It suffices to show that for all open sets V ⊂ S∞ (Rn ), T −1 (V ) is open in S (Rn )/P(Rn ). Let

i : S (Rn ) → S∞ (Rn )

8.1 The Definition and Some Preliminaries

253

be the map defined as i( f ) ≡ f |S∞ (Rn ) for all f ∈ S (Rn ). Then i is a continuous, surjective and closed map. Since i is continuous, for an open set V ⊂ S∞ (Rn ), i−1 (V ) is open in S (Rn ). Then

π −1 (π ◦ i−1 (V )) = i−1 (V ) + P(Rn ) = ∪P∈P(Rn ) (i−1 (V ) + P) is open in S (Rn ). By the definition of quotient topology, π ◦ i−1 (V ) is open in S (Rn )/P(Rn ). Thus, T −1 (V ) = π ◦ i−1 (V ) is open in S (Rn )/P(Rn ) and T is continuous. Step 4. T −1 is continuous from S∞ (Rn ) to S (Rn )/P(Rn ). It suffices to show that for all open sets V ⊂ S (Rn )/P(Rn ), T (V ) is open in S∞ (Rn ). We first claim that for all sets O ⊂ S∞ (Rn ), if i−1 (O) is open in S (Rn ), then O is open in S∞ (Rn ). Observe that i−1 (Oc ) = [i−1 (O)]c . In fact, for all x ∈ i−1 (Oc ), i(x) ∈ Oc . If x ∈ / [i−1 (O)]c , then x ∈ i−1 (O). Thus, i(x) ∈ O, which contradicts i(x) ∈ c −1 / i−1 (O), O . Thus, i (Oc ) ⊂ [i−1 (O)]c . On the other hand, if x ∈ [i−1 (O)]c , then x ∈ −1 c −1 c −1 c hence i(x) ∈ / O. Thus, x ∈ i (O ) and [i (O)] ⊂ i (O ). This observation implies that i−1 (O) is open in S (Rn ) if and only if i−1 (Oc ) is closed in S (Rn ). Then the above claim follows from the fact that i is a closed map. Since T −1 ◦ i = π is continuous, for all open sets V ⊂ S (Rn )/P(Rn ), i−1 (T (V )) = i−1 ◦ T (V ) = (T −1 ◦ i)−1 (V ) = π −1 (V ) is open in S (Rn ), which together with the above claim implies that T (V ) is open in S∞ (Rn ). Combining Steps 1 through 4, we obtain that T is a homeomorphism, which completes the proof of Proposition 8.1. Following Triebel’s [145], we use the distribution space S∞ (Rn ) in the following Definition 8.1. Definition 8.1. Let s ∈ R, τ ∈ [0, ∞), q ∈ (0, ∞] and ϕ ∈ S (Rn ) satisfy (8.1). (i) Let p ∈ (0, ∞]. The Besov-type space B˙ s,p,τq(Rn ) is defined to be the set of all f ∈ S∞ (Rn ) such that f B˙ s,p,τq(Rn ) < ∞, where 1 f B˙ s,p,τq(Rn ) ≡ sup τ |P| P∈Q

∞

∑

j= jP

(2 |ϕ j ∗ f (x)|) dx js

P

q/p 1/q

p

with suitable modifications made when p = ∞ or q = ∞. (ii) Let p ∈ (0, ∞). The Triebel-Lizorkin-type space F˙p,s, qτ (Rn ) is defined to be the set of all f ∈ S∞ (Rn ) such that f F˙p,s, τq (Rn ) < ∞, where ⎧ p/q ⎫1/p ⎨ ⎬ ∞ 1 f F˙p,s, τq (Rn ) ≡ sup (2 js |ϕ j ∗ f (x)|)q dx ∑ τ ⎭ P∈Q |P| ⎩ P j= jP with suitable modification made when q = ∞.

254


Remark 8.1. These spaces are called homogeneous because of the following fact: There exists a positive constant C such that for all λ ∈ (0, ∞) and f ∈ B˙ s,p,τq (Rn ) or f ∈ F˙p,s, qτ (Rn ), f (λ ·)B˙ s,p,τq (Rn ) ≤ Cλ s−n/p+nτ f B˙ s,p,τq(Rn ) and f (λ ·)F˙p,s, τq (Rn ) ≤ Cλ s−n/p+nτ f F˙p,s, τq (Rn ) . Let A˙ s,p,τq (Rn ) denote either B˙ s,p,τq (Rn ) or F˙p,s, qτ (Rn ). It was proved in [165, Corollary 3.1] that the spaces A˙ s,p,τq (Rn ) are independent of the choices of ϕ . Furthermore, S∞ (Rn ) ⊂ A˙ s,p,τq (Rn ) ⊂ S∞ (Rn ); see [165, Propositions 3.1(ix) and 3.4]. These spaces unify and generalize the classical homogeneous Besov spaces, Triebel-Lizorkin spaces, Q spaces and Morrey spaces; see [127, 164, 165]. An important tool to study A˙ s,p,τq (Rn ) is the following Calderón reproducing formula; see [62, Lemma 2.1] and [164, Lemma 2.1]. Lemma 8.1. Let ϕ , ψ ∈ S (Rn ) satisfying (8.1) such that

∑ ϕ(2 j ξ )ψ (2 j ξ ) = 1

j∈Z

for all ξ ∈ Rn \ {0}. Then for any f ∈ S∞ (Rn ), f=

∑ ψ j ∗ ϕ j ∗ f = ∑ 2− jn ∑n ϕ j ∗ f (2− j k) ψ j (·−2− j k) = ∑ ∑ − j f , ϕQ ψQ

j∈Z

j∈Z

k∈Z

j∈Z l(Q)=2

in S∞ (Rn ). Moreover, for any f ∈ S∞ (Rn ), the above equalities also hold in S∞ (Rn ). The corresponding sequence spaces were introduced in [165, Definition 3.1]. Definition 8.2. Let s ∈ R, τ ∈ [0, ∞) and q ∈ (0, ∞]. (i) Let p ∈ (0, ∞]. The sequence space b˙ s,p,τq (Rn ) is defined to be the set of all sequences t ≡ {tQ }Q∈Q ⊂ C such that tb˙ s,p,τq(Rn ) < ∞, where ⎧ ⎞ p ⎤q/p ⎫1/q ⎡ ⎛ ⎪ ⎪ ⎨ ⎬ ∞ 1 jsq ⎠ ⎦ ⎣ ⎝ 2 |t | χ (x) dx . tb˙ s,p,τq(Rn ) ≡ sup Q Q ∑ ∑ τ ⎪ ⎪ P l(Q)=2− j P∈Q |P| ⎩ ⎭ j= j P

s, τ n (ii) Let p ∈ (0, ∞). The sequence space f˙p, q (R ) is defined to be the set of all sequences t ≡ {tQ }Q∈Q ⊂ C such that t f˙p,s, τq(Rn ) < ∞, where

⎧ p/q ⎫1/p ⎨ ⎬ q 1 −s/n |t | χ (x) dx . t f˙p,s, τq(Rn ) ≡ sup |Q| Q Q ∑ τ ⎭ P∈Q |P| ⎩ P Q⊂P

8.2 The Wavelet Characterization of B˙ s,p,τq (Rn ) and F˙p,s, qτ (Rn )

255

Similarly to Remark 2.4, from [37, Proposition 2.2] again, we deduce the diss, τ n cretization of f˙p, q (R ) in the following Remark 8.2. Remark 8.2. Let τ ∈ [0, ∞), s ∈ R, p ∈ (0, ∞) and q ∈ (0, ∞]. If p ≥ q, then there s, τ n exists a positive constant C, depending only on p and q, such that for all t ∈ f˙p, q (R ), ⎧ ⎪ 1 ⎨ ∞ −1 s, τ C t f˙p, q(Rn ) ≤ sup ∑ ∑ (|Q|−s/n−1/2+1/q|tQ |)q τ P∈Q |P| ⎪ ⎩ j= jP l(Q)=2− j Q⊂P

⎤ p/q−1⎫1/p ⎪ ⎬ 1 −s/n−1/2+1/q q⎦ ⎣ × (|R| |t |) R ∑ ⎪ |Q| R∈Q ⎭ ⎡

R⊂Q

≤ Ct f˙p,s, τq(Rn ) . The spaces A˙ s,p,τq (Rn ) also have the following ϕ -transform characterization; see [164, Theorem 3.1]. Theorem 8.1. Let s ∈ R, τ ∈ [0, ∞), p, q ∈ (0, ∞], ϕ and ψ be as in Lemma 8.1. Then Sϕ : A˙ s,p,τq (Rn ) → a˙s,p,τq (Rn ) and Tψ : a˙s,p,τq (Rn ) → A˙ s,p,τq (Rn ) are bounded; moreover, Tψ ◦ Sϕ is the identity on A˙ s,p,τq(Rn ). We remark that B˙ s,p,τq (Rn ) and F˙p,s, qτ (Rn ) have some similar properties to Bs,p,τq(Rn ) and Fp,s, qτ (Rn ) such as Sobolev-type embedding properties, smooth atomic and molecular decomposition characterizations, boundedness of pseudo-differential operators with homogeneous symbols and trace theorems. These properties have been studied in [127, 164, 165]. Also, the maximal function and local mean characterizations of B˙ s,p,τq (Rn ) and F˙p,s, qτ (Rn ) were already obtained in [167]. However, similarly to homogeneous Besov spaces B˙ sp,q (Rn ) and Triebel-Lizorkin s (Rn ) (see [145, p. 238]), some of the most striking features of the spaces spaces F˙p,q s, τ B p, q (Rn ) and Fp,s, qτ (Rn ) have no counterparts, such as the pointwise multipliers theorem and the diffeomorphism property. Thus, we cannot expect to find counterparts of Theorems 6.1 and 6.7.

8.2 The Wavelet Characterization of B˙ s,p,τq (Rn ) and F˙p,s, qτ (Rn ) s, τ n Let a˙s,p,τq(Rn ) denote either b˙ s,p,τq (Rn ) or f˙p, q (R ). We now focus on the wavelet chars, τ n acterization of A˙ p, q (R ). Differently from those in Chap. 4, what we deal with below are so called “wavelets with two humps”. These wavelet basis have no compact

256


support of their own, and their Fourier transforms have compact support; see, for example, [99, Sect. 6.11] or [65, Sect. 7]. We begin with the one-dimensional case. Let ϕ ∈ S (R) be as in [65, Theorem, (7.11)]. That is, ϕ is a real valued function such that

8 2 2 8 supp ϕ ⊂ − π , − π π, π 3 3 3 3

(8.2)

and the collection # ! " ϕ jk : j, k ∈ Z ≡ 2 j/2 ϕ (2 j · −k) : j, k ∈ Z

(8.3)

is an orthonormal basis of L2 (R). We call ϕ the mother function of the wavelet basis {ϕ jk : j, k ∈ Z}. Define Sϕ f ≡ { f , ϕQ }Q∈Q ≡ { f (ϕQ )}Q∈Q for f ∈ S∞ (R), and let

Tϕ t ≡

∑ tQ ϕQ

Q∈Q

τ when t = {tQ }Q∈Q . To obtain the wavelet characterization of A˙ s,p,q (R), we need to τ (R) is an prove that the coefficient sequence of a wavelet expansion of an f ∈ A˙ s,p,q s,τ element of a˙ p,q (R).

Theorem 8.2. Let s ∈ R, p, q ∈ (0, ∞] and τ be as in Lemma 3.1. The operator Sϕ τ τ τ τ is bounded from A˙ s,p,q (R) to a˙s,p,q (R) and Tϕ is bounded from a˙s,p,q (R) to A˙ s,p,q (R). s,τ ˙ Furthermore, Tϕ ◦ Sϕ and Sϕ ◦ Tϕ are, respectively, the identities on A p,q (R) and τ a˙s,p,q (R). The proof of Theorem 8.2 is similar to that for [65, Theorem (7.20)]. For the reader’s convenience, we give the details. Proof of Theorem 8.2. By (8.3), {ϕQ }Q∈Q is an orthonormal basis of L2 (R). Thus, f=

∑ f , ϕQ ϕQ

Q∈Q

holds in L2 (R). It was further proved in [65, p. 71] that the identity f=

∑ f , ϕQ ϕQ

Q∈Q

also holds in S∞ (R) and S∞ (R), which implies that Tϕ ◦ Sϕ is identity transformation. Recall that ϕ ∈ S (R) and satisfies (8.2). It is easy to check that each ϕQ is a τ (R) in [165, Definition 4.2] (up to a constant smooth synthesis molecule for A˙ s,p,q


257

factor that is independent of Q). Then applying the homogeneous smooth molecular τ (R) obtained in [165, Theorem 4.2], we know decomposition characterization of A˙ s,p,q s,τ that for all t ∈ a˙ p,q (R), $ $ $ $ $ $ $ ∑ tQ ϕQ $ $Q∈Q $

s,τ A˙ p,q (R)

ta˙s,p,qτ (R) ,

τ τ (R) to A˙ s,p,q (R). By (8.2) again, which further implies that Tϕ is bounded from a˙s,p,q s,τ for all t ∈ a˙ p,q (R) and P ∈ Q,

&

%

∑ tQ ϕQ ,tP

= tP ,

Q∈Q

τ (R). which shows that Sϕ ◦ Tϕ is the identity on a˙s,p,q To obtain the boundedness of Sϕ , let ψ ∈ S (R) such that

⊂ [−2, −1/2] ∪ [1/2, 2] supp ψ and for all ξ ∈ R \ {0},

∑ |ψ (2k ξ )|2 = 1.

k∈Z

τ Let f ∈ A˙ s,p,q (R). Then by the Calderón reproducing formula in [165, Lemma 2.1],

f=

∑ f , ψQ ψQ

Q∈Q

τ holds in S∞ (R). Moreover, by the ϕ -transform characterization of A˙ s,p,q (R) in [165, Theorem 3.1], we further have

{ f , ψQ }Q∈Q a˙s,p,qτ (R) f A˙ s,p,qτ (R) . By [165, Lemma 2.1] again, we obtain that

ϕQ =

∑ ϕQ , ψP ψP

P∈Q

in S∞ (Rn ). Thus,

f , ϕQ =

∑ ϕQ , ψP f , ψP ≡ ∑ aQP f , ψP ,

P∈Q

P∈Q

where aQP ≡ ϕQ , ψP . Since ϕQ is a constant multiple of a homogeneous smooth synthesis molecule in [165, Definition 4.2], using [165, Corollary 4.1], we know that

258


τ the matrix operator A ≡ {aQP }Q,P∈Q is ε1 -almost diagonal on a˙s,p,q (R) as in [165, Definition 4.1]. Then [165, Theorem 4.1] tells us

{ f , ϕQ }Q∈Q a˙s,p,qτ (R) { f , ψQ }Q∈Q a˙s,p,qτ (R) f A˙ s,p,qτ (R) , τ τ which yields that Sϕ is bounded from A˙ s,p,q (R) to a˙s,p,q (R), and then, completes the proof of Theorem 8.2.

For the n-dimensional case, the well-known tensor product ansatz yields a wavelet basis "

# 2 jn/2ϕ i (2 j x − k) : j ∈ Z, k ∈ Zn , i ∈ {1, · · · , 2n − 1} .

The 2n − 1 functions ϕ i belong to the Schwartz class S (Rn ) and the Fourier transforms ϕi of ϕ i vanish in a neighborhood of 0 and have compact support; moreover, {2 jn/2ϕ i (2 j x − k) : j ∈ Z, k ∈ Zn , i ∈ {1, · · · , 2n − 1}} yields an orthonormal basis of L2 (Rn ); see, [99, p. 168] or [65, p. 73]. We remark that Theorem 8.2 still holds in this case. Remark 8.3. Theorem 8.2 generalizes the corresponding results on homogeneous Besov and Triebel-Lizorkin spaces established in [65, Sect. 7] by taking τ = 0. Next we establish the difference characterization and the wavelet characterization of A˙ s,p,τq(Rn ) in the sense of Chap. 4, namely, wavelets with compact supports. We need some preparations. Recall that L ploc (Rn ) consists of all p-locally integrable functions and p ≡ max{p, 1}. Proposition 8.2. Let p, q ∈ (0, ∞], τ ∈ [0, ∞) and s ∈ (0, ∞). Then A˙ s,p,τq(Rn ) ⊂ L ploc (Rn ) in the sense of S∞ (Rn ). Proof. Notice that Proposition 2.1(i) and (iii) are also correct for A˙ s,p,τq (Rn ). It sufτ τ fices to consider B˙ s,p,∞ (Rn ). Let f ∈ B˙ s,p,∞ (Rn ). We need to prove that there exists a n function g such that f = g in S∞ (R ) and P

|g(x)| p dx < ∞

for all P ≡ [−2m , 2m )n and m ∈ N. Let L ∈ N be sufficiently large and jP −1

(∂ γ ψ j )(−2− j k) (−x)γ I(x) ≡ ∑ 2− jn ∑ ϕ j ∗ f (2− j k) ψ j (x − 2− j k) − ∑ γ! j=−∞ k∈Zn |γ |≤L


and II(x) ≡

259

∞

∑ ψ j ∗ ϕ j ∗ f (x).

j= jP

By Lemma 8.1, for any φ ∈ S∞

(Rn ),

we have

f , φ = I, φ + II, φ , and hence f ≡ I + II'in S∞ (Rn ). We first estimate P |II(x)| p dx. If p ∈ (1, ∞], by Minkowski’s inequality and s > 0, we see that ( P

)1/p |II(x)| dx

≤

p

(

∞

∑

P

j= jP

|ψ j ∗ ϕ j ∗ f (x)| p dx

)1/p

τ 2− jP s |P|τ f B˙ s,p,∞ (Rn ) .

If p ∈ (0, 1), since II ≡

∞

∑ 2− jn ∑n ϕj ∗ f (2− j k)ψ j (· − 2− j k)

j= jP

k∈Z

in S∞ (Rn ), by (2.11), for all x ∈ P, we have

∞

1 |II(x)| ∑ ∑ |ϕ j ∗ f (2 k)| j x − k|)(n+δ )p (1 + |2 j= jP k∈Zn −j

p

1/p ,

where δ ∈ (0, ∞) will be determined later. Decomposing

∑n ≡

k∈Z

and noticing that

∞

∑n

k∈Z |x−2− j k|≤l(P)

+∑

i=1

∑

k∈Zn 2i−1 l(P) max{nτ − n, n/p − n − s}. Thus, P

'

P |I(x)|

Next we estimate such that |I(x)| ≤

jP −1

∑

τ |II(x)| dx 2− jP s |P|τ −1/p+1 f B˙ s,p,∞ (Rn ) .

By the mean value theorem, there exists θ ∈ [0, 1]

∑n |ϕj ∗ f (2− j k)|

2− jn

j=−∞

p dx.

k∈Z

jP −1

∑ ∑

sup |x|L+1 |(∂ γ ψ j )(θ x − 2− j k)|

|γ |=L+1

|ϕ j ∗ f (2− j k)|2 j(L+1) |x|L+1 (1 + |2 j θ x − k|)−(n+δ ).

j=−∞ k∈Zn

Noticing that τ |ϕ j ∗ f (2− j k)| ≤ 2− js− jnτ + jn/p f B˙ s,p,∞ (Rn ) ,

we then have |I(x)|

jP −1

∑ ∑ 2− js− jnτ + jn/p2 j(L+1)|x|L+1 (1 + |2 j θ x − k|)−(n+δ ) f B˙s,p,∞τ (Rn )

j=−∞ k∈Zn

τ 2− jP s+ jP (L+1) |x|L+1 |P|τ −1/p f B˙ s,p,∞ (Rn ) ,

where we choose δ > 0 and L > s + nτ − n/p − 1. Thus, (

)1/p |I(x)| dx p

P

τ 2− jP s |P|τ |P|1/p−1/p f B˙ s,p,∞ (Rn ) ,

which completes the proof of Proposition 8.2.

By Proposition 8.2, in what follows, when f ∈ A˙ s,p,τq (Rn ), we also use f to denote its representative in L ploc (Rn ). Theorem 8.3. Let s ∈ (0, ∞), p ∈ (0, ∞), q ∈ (0, ∞], τ ∈ [0, ∞) and N1 , M ∈ N such that M ≤ N1 , s < {M ∧ (M + n(1/p − τ ))} and N1 ≥ s + nτ . Then for all f ∈ A˙ s,p,τq (Rn ), ≡ C−1 f A˙ s,p,τq(Rn ) ≤ f A˙ s, τ (Rn ) p, q

2n −1 $

∑

i=1

$ ! $ $ $ f , ψi, j,k j∈Z, k∈Zn $

where ψi, j,k are wavelets in Sect. 4.2. We give the proof of Theorem 8.3 in the next section.

s, τ

a˙ p, q (Rn )

≤ C f A˙ s,p,τq(Rn ) ,

8.3 The Characterization by Differences of B˙ s,p,τq (Rn ) and F˙p,s, qτ (Rn )

261

8.3 The Characterization by Differences of B˙ s,p,τq (Rn ) and F˙p,s, qτ (Rn ) Let at be as in (4.12). For all f ∈ L1loc (Rn ), set f ♣ s, τ B˙ p, q (Rn )

1 ≡ sup τ P∈Q |P|

2l(P)

t

−sq

(

)q/p P

0

[at (x)] dx p

dt t

1/q

and f ♣ s, τ F˙p, q (Rn )

1 ≡ sup τ |P| P∈Q

( P

2l(P)) 0

dt t −sq [at (x)]q t

1/p

) p/q dx

.

We have the following difference characterizations for B˙ s,p,τq (Rn ) and F˙p,s, qτ (Rn ). Theorem 8.4. Let s, p, q, τ , N1 and M be as in Theorem 8.3. Then f A˙ s,p,τq(Rn ) is equivalent to f ♣s, τ n for all f ∈ A˙ s,p,τq (Rn ). A˙ p, q (R )

(Rn , L) be the collection To prove Theorems 8.3 and 8.4, let L ∈ (0, 1/2] and Θ of all Schwartz functions ϕ satisfying supp ϕ ⊂ {ξ ∈ Rn : L ≤ |ξ | ≤ 2} and

∑ ϕ(2− j ξ ) = 1

j∈Z

for all ξ ∈ Rn \ {0}. Similarly to the proof of Lemma 4.1, we obtain the following conclusion. Lemma 8.2. The space A˙ s,p,τq (Rn ) is independent of the choices of L ∈ (0, 1/2] and (Rn , L). ϕ ∈Θ We need to construct a representation of ϕ j ∗ f by an integral mean of differences of f . Let ψ ∈ S∞ (Rn ) such that ⊂ {ξ ∈ Rn : 1/2 ≤ |ξ | ≤ 2} and supp ψ

∑ ψ (2− j ξ ) = 1

j∈Z

for all ξ ∈ Rn \ {0}. Define ϕ by setting, for all ξ ∈ Rn , M

ϕ(ξ ) ≡ (−1)M+1 ∑ (−1)i i=0

(

M i

) ((M − i)ξ ). ψ

(8.4)

262


(Rn , 1/(2M)). Furthermore, for all locally integrable It is easy to check that ϕ ∈ Θ functions f , M ϕ j ∗ f (x) = (−1)M+1 Δ−2 (8.5) − j y f (x) ψ (y) dy. Rn

Via these constructions, we have the following result. Lemma 8.3. Let p, q ∈ (0, ∞], τ ∈ [0, ∞) and s ∈ (0, ∞). There exists a positive constant C such that for all f ∈ A˙ s,p,τq (Rn ), f A˙ s,p,τq(Rn ) ≤ C f ♣ . s, τ A˙ (Rn ) p, q

Proof. By Proposition 8.2, we know that each f ∈ A˙ s,p,τq (Rn ) is a locally integrable function in the sense of S∞ (Rn ). Let ϕ be as in (8.4). Then (8.5) holds for all f ∈ A˙ s,p,τq (Rn ), which together with Lemma 8.2 yields that 1 f B˙ s,p,τq(Rn ) sup τ P∈Q |P|

∞

(

j= jP

P

∑ 2 jsq

Rn

M |Δ−2 − j y f (x)||ψ (y)| dy

qp 1q

)p dx

and ⎧ ⎫1 )q qp ⎬ p ( ∞ 1 ⎨ M f F˙p,s, qτ (Rn ) sup dx . − j y f (x)||ψ (y)| dy ∑ 2 jsq Rn |Δ−2 τ ⎭ P∈Q |P| ⎩ P j= jP Then a modification of the proof of Lemma 4.3 gives us the desired inequalities. Next we show that f ♣ f . s, τ A˙ s, τ (Rn ) A˙ (Rn ) p, q

p, q

Lemma 8.4. Let s ∈ (0, ∞), p ∈ (0, ∞), q ∈ (0, ∞], τ ∈ [0, ∞) and M ∈ N such that M ≤ N1 and s < {M ∧ (M + n(1/p − τ ))}. Then there exists a positive constant C such that for all f ∈ A˙ s,p,τq (Rn ), f ♣ ≤ C f . s, τ A˙ s, τ (Rn ) A˙ (Rn ) p, q

p, q

Proof. By similarity, we only consider the spaces F˙p,s, qτ (Rn ). Let f ∈ F˙p,s, qτ (Rn ). By Proposition 8.2 and [156, Theorem 8.4], similarly to the argument in Sect. 4.3, we see that f=

2n −1

∑ ∑ ∑ ai, j,k ψi, j,k

i=1 j∈Z k∈Zn

8.3 The Characterization by Differences of B˙ s,p,τq (Rn ) and F˙p,s, qτ (Rn )

263

holds in L loc (Rn ) when p ∈ (0, ∞), where ai, j,k ≡ f , ψi, j,k . Therefore, for all P ∈ Q, p

1 |P|τ

(

2l(P))

t P

−sq

0

dt [at (x)] t

1/p

) p/q

q

dx

⎫ ⎧ ,

q - p/q ⎬1/p ∞ 1 ⎨ 2msq 2mn |ΔhM f (x)| dx dx ⎭ |P|τ ⎩ P m=∑ 2−m−2 ≤|h| jR and

2n 2 22 −1 2 2 2 2 ∑ ∑ ∑ ai, j,m ψi, j,m (y)2 dy < ε 2−k(M+n) 2− jR (s+nτ −n/p−M). −k 2 2 B(x,M2 ) i=1 | j|>N m∈Zn

Thus, by the support condition of ψi, j,m and s < M, we see that ψi, j,m (y) = 0 if y∈ / MR and

IR

⎧ , ∞ 1 ⎨ ksq kn 2 2 inf |R|τ ⎩ R k=∑ P∈PM−1 (Rn ) B(x,M2−k ) j −1 R

2n 2 -q p ⎫ 1p q ⎬ 22 −1 2 2 2 × 2 ∑ ∑ ∑ ai, j,m ψi, j,m (y) − P(y)2 dy dx + ε, ⎭ 2 i=1 | j|≤N m∈IMR, j 2

268


where IMR, j is the collection of all m ∈ Zn such that | supp ψi, j,m ∩ (MR)| > 0 for some i ∈ {1, · · · , 2n−1 }. Similarly to the proof of Lemma 4.10, by (4.20) and employing the Taylor remainders of order M of φi, j,m , we have

IR

⎧ , 2n −1 ∞ 1 ⎨ ksq kn 2 2 ∑ ∑ ∑ |R|τ ⎩ R k= j −1 B(x,M2−k ) i=1 | j|≤N R

∑

×

|ai, j,m |2

2 jM |x − y|M χ2− j ([−N3 ,N3 ]n +m) (x) jn/2 (1+|2 j x − m|)n+δ

m∈IMR, j

-q qp dy

⎫ 1p ⎬ dx +ε ⎭

⎧ ,n ⎪ ∞ 2 −1 jR 1 ⎨ ksq 2 ∑ ∑ ∑ τ ⎪ R k= j −1 |R| ⎩ i=1 j=−N R

∑

×

m∈IMR, j

-q qp |ai, j,m |2 jn/2 2 jM 2−kM χ2− j ([−N3 ,N3 ]n +m) (x)

⎫ 1p ⎪ ⎬ dx ⎪ ⎭

⎧ ,n ∞ 2 −1 N 1 ⎨ ksq + τ 2 ∑ ∑ ∑ |R| ⎩ R k= j −1 i=1 j= jR R

×

∑

m∈IMR, j

2 jM 2−kM χ2− j ([−N3 ,N3 ]n +m) (x) |ai, j,m |2 jn/2 (1 + |2 j x − m|)n+δ

-q qp

⎫ 1p ⎬ dx + ε, ⎭

where δ ∈ (0, ∞) can be sufficient large. Similarly to the estimates of I and II in the proof of Lemma 8.4, we obtain that IR g s, τ n + ε . By Theorem 8.3 and the F˙p, q (R ) arbitrariness of ε , we further have IR gF˙p,s, τq (Rn ) ∼ f F˙p,s, qτ (Rn ) , which implies that

gA˙ s, τ (Rn ) f A˙ s,p,τq (Rn ) p, q

and then completes the proof of Theorem 8.5.

8.4 Homogeneous Besov-Hausdorff and Triebel-Lizorkin-Hausdorff Spaces Similarly to Chap. 7, we also determine the predual spaces of A˙ s,p,τq (Rn ), which are called homogeneous Besov-Hausdorff space or Triebel-Lizorkin-Hausdorff space.

8.4 Homogeneous Besov-Hausdorff and Triebel-Lizorkin-Hausdorff Spaces

269

Definition 8.3. Let ϕ be as in Definition 8.1, s ∈ R, p ∈ (1, ∞), q ∈ [1, ∞) and τ ∈ 1 n ˙ s,τ n [0, (p∨q) ]. Then the space AH p,q (R ) is defined to be the set of all f ∈ S∞ (R ) such s,τ s,τ that f ˙ s,τ n , where when AH˙ p,q (Rn ) = BH˙ p,q (Rn ), AHp,q (R )

s,τ f BH˙ p,q (Rn ) ≡ inf

ω

∑2

j∈Z

$ $q ϕ j ∗ f [ω (·, 2− j )]−1 $L p (Rn )

jsq $

1 q

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Morrey and Campanato meet Besov, Lizorkin and Triebel

Meet and Greet

When Science and Christianity Meet

Where Frankfurt and Strawson Meet

Where Texts and Children Meet

When Jews and Christians meet

New Thoughts on Besov Spaces

New Thoughts on Besov Spaces

Pointing: Where Language, Culture, and Cognition Meet

Pointing. Where Language, Culture, and Cognition Meet

Merry Meet

Jesus Wept: When Faith and Depression Meet

Dunc and Amos Meet the Slasher

Pointing: Where Language, Culture, and Cognition Meet

Topsy and Tim Meet the Police

Swap Meet

Merry Meet

A Besov Estimate for Multilinear Singular Integrals

Meet Miss Universe

Meet Strawberry Shortcake

Meet the Trojans

Where Old Ghosts Meet

Meet the Sun Halfway

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The Twain Shall Meet

When Two Paths Meet

Meet Me at Midnight

Meet the Sun Halfway

Meet Blueberry Muffin

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