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 · 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-Universit¨at 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 H¨older-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. Ess´en, 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.

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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 Calder´on 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

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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

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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 Calder´on 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 

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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 Calder´on 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 Ess´en and Xiao [55] and Janson [76] on the boundary values of these functions on the unit circle, Ess´en, 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) H¨older 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
Let M ≡ max{−1, s}. Then Tps (Rn ) is the collection of all functions in L ploc (Rn ) satisfying f Tps (Rn ) ≡ sup

1/p



x∈Rn

B(x,1)

| f (x)| p dx

+ sup fuM,s (x) < ∞ . x∈Rn

We followed [146, Sect. 1.7.2] (but change the notation because of item (i)); see also [153]. (xix) Local approximation spaces II. Let p ∈ (0, ∞], s ∈ (0, ∞) and M ≡ s. The local approximation space Csp (Rn ) is the collection of all functions f ∈ Lmax{p, 1}(Rn ) such that f Csp (Rn ) ≡ f L p (Rn ) + f pM,s L p (Rn ) < ∞ . We refer to [15, 40, 46, 153] and [146, Sect. 1.7.2]. (xx) Let α ∈ R. The space Qα (Rn ) is defined to be the collection of all f ∈ L2loc (Rn ) such that  f Qα (Rn ) ≡ sup I

 

1 |I|1−

2α n

I I

| f (x) − f (y)|2 dx dy |x − y|n+2α

1/2 < ∞,

where I ranges over all cubes in Rn ; see, for example, [7, 44, 54].

1.3.3 Function Spaces Defined by Fourier Analytic Tools Except the first two all spaces here will be defined by using a decomposition in the Fourier image induced by a smooth dyadic decomposition of unity. Let ψ ∈ Cc∞ (Rn ) be a radial function such that ψ (x) = 1 if |x| ≤ 1 and ψ (x) = 0 if |x| ≥ 3/2. Then by means of

ψ 0 (x) ≡ ψ (x) ,

ψ j (x) ≡ ψ (2− j x) − ψ (2− j+1x) ,

j ∈ N,

x ∈ Rn ,

(1.1)

1.3 A Collection of the Function Spaces Appearing in the Book

11

we obtain a smooth dyadic decomposition of unity, namely, ∞

∑ ψ j (x) = 1

for all x ∈ Rn .

j=0

We put

ϕ0 ≡ Φ ≡ F −1 ψ , ϕ (x) ≡ F −1 [ψ (2ξ )](x) and ϕ j ≡ F −1 ψ j ,

j ∈ Z+ . (1.2)

Then ϕ j (x) = 2 jn ϕ (2 j x), j ∈ Z+ , follows. (xxi) Local Hardy spaces. Let p ∈ (0, ∞). Let ϕ ∈ S (Rn ) such that ϕ (0) = 1. Then h p (Rn ) is the collection of all f ∈ S (Rn ) such that





−1

sup F [ϕ (t ξ ) F f (ξ )]( · )

< ∞. f h p (Rn ) ≡

L p (Rn )

0
(xxii) Bessel-potential spaces (sometimes also called Lebesgue or Liouville spaces). Let s ∈ R and p ∈ (1, ∞). Then H ps (Rn ) is the set of all tempered distributions f ∈ S (Rn ) such that F −1 [(1 + |ξ |2 )s/2 F f (ξ )]( · ) is a regular distribution and





f Hps (Rn ) ≡ F −1 [(1 + |ξ |2)s/2 F f (ξ )]( · ) p n < ∞ . L (R )

(xxiii) Besov spaces (general case). Let p, q ∈ (0, ∞] and s ∈ R. Let {ϕ j } j∈Z+ be as in (1.2). Then Bsp, q(Rn ) is the collection of all f ∈ S (Rn ) such that  f Bsp, q(Rn ) ≡

∞



q ∑ 2 jsq ϕ j ∗ f L p (Rn )

1/q < ∞.

j=0

(xxiv) Triebel-Lizorkin spaces. Let p ∈ (0, ∞), q ∈ (0, ∞] and s ∈ R. Let {ϕ j } j∈Z+ be as in (1.2). Then Fp,s q (Rn ) is the collection of all f ∈ S (Rn ) such that

 1/q



∞



js q



f Fp,s q(Rn ) ≡ ∑ (2 |ϕ j ∗ f |) < ∞.

p n

j=0 L (R )

s (Rn ) is defined to be We refer to [64, 145]. The Triebel-Lizorkin space F∞, q the set of all f ∈ S (Rn ) such that 1/q   ∞ 1 js q f F∞,s q(Rn ) ≡ sup < ∞, (1.3) ∑ [2 |ϕ j ∗ f (x)|] dx |P| P j= P dyadic jP l(P)≤1

where the supremum is taken over all dyadic cubes P with side length l(P) ≤ 1 and jP ≡ −log2 l(P); see [64].

12

1 Introduction

(xxv) Besov-Morrey spaces. Let s ∈ R, q ∈ (0, ∞] and 0 < u ≤ p ≤ ∞. Let s (Rn ) is defined to be the set of all {ϕ j } j∈Z+ be as in (1.2). Then N pqu n f ∈ S (R ) satisfying 

∞

∑2

s (Rn ) ≡ f N pqu

jsq

j=0

sup |B|

q/p−q/u

 B

B

|ϕ j ∗ f (x)| dx

q/u 1/q < ∞,

u

where the supremum is taken over all balls B in Rn ; see [88, 97]. (xxvi) Triebel-Lizorkin-Morrey spaces. Let s ∈ R, q ∈ (0, ∞] and 0 < u ≤ s (Rn ) is defined p ≤ ∞, u = ∞. Let {ϕ j } j∈Z+ be as in (1.2). The class E pqu n to be the collection of all f ∈ S (R ) satisfying

1/p−1/u f E pqu s (Rn ) ≡ sup |B| B

⎧  ⎨ ⎩

B

⎫1/u ⎬

u/q

∞

∑ 2 jsq|ϕ j ∗ f (x)|q

j=0

dx

⎭

< ∞,

where the supremum is taken over all balls B in Rn . We refer, e. g., to [88, 126, 139]. (xxvii) Inhomogeneous Besov-type spaces. Let τ , s ∈ R and p, q ∈ (0, ∞]. Let {ϕ j } j∈Z+ be as in (1.2). 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|



∞

∑

 P

j=( jP ∨0)

(2 |ϕ j ∗ f (x)|) dx js

p

q/p 1/q

< ∞.

(xxviii) Inhomogeneous Triebel-Lizorkin-type spaces. Let τ , s ∈ R, q ∈ (0, ∞] and p ∈ (0, ∞). Let {ϕ j } j∈Z+ be as in (1.2). The inhomogeneous TriebelLizorkin-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 < ∞. j ∑ τ ⎭ P∈Q |P| ⎩ P j=( j ∨0) P

A comment. The definitions of the spaces in (1.3) and (xxv)–(xxviii) are all of the same spirit. The major difference between Besov-Morrey and Triebel-LizorkinMorrey spaces on the one side and the spaces Bs,p,τq (Rn ) and Fp,s, qτ (Rn ) on the other side consists in the starting index for the summation with respect to j. In (xxv) and (xxvi) the summation starts always with 0, whereas in the (xxvii) and (xxviii) the summation starts at jP ∨ 0. Comparing with (1.3) we find that this time there is a difference in the set of admissible cubes. The distribution spaces Fp,s, qτ (Rn ) and Bs,p,τq (Rn ) have some overlap with all 26 different classes we have introduced above; see the next subsection.

1.3 A Collection of the Function Spaces Appearing in the Book

13

(xxix) Homogeneous Besov-type spaces. Let τ , s ∈ R and p, q ∈ (0, ∞]. Let ϕ j (x) ≡ 2 jn ϕ (2 j x), j ∈ Z. 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∈Q |P|



∞

∑

 P

j= jP

(2 |ϕ j ∗ f (x)|) dx js

q/p 1/q

p

with suitable modifications made when p = ∞ or q = ∞. (xxx) Homogeneous Triebel-Lizorkin-type spaces. Let τ , s ∈ R, q ∈ (0, ∞] and p ∈ (0, ∞). Let ϕ j (x) ≡ 2 jn ϕ (2 j x), j ∈ Z. 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 ⎨ js q f F˙p,s, qτ (Rn ) ≡ sup (2 | ϕ ∗ f (x)|) dx j ∑ τ ⎭ P∈Q |P| ⎩ P j= jP with suitable modifications made when q = ∞. (xxxi) Besov-Hausdorff spaces and Triebel-Lizorkin-Hausdorff spaces. The s, τ s, τ n n inhomogeneous classes BH p, q (R ) and FH p, q (R ) will be investigated in Chap. 7. For the homogeneous counterparts, see Sect. 8.4. 1 Let s ∈ R, p ∈ (1, ∞), q ∈ [1, ∞) and τ ∈ [0, (p∨q) ], {ϕ j } j∈Z+ be as in (1.2). s, τ n The Besov-Hausdorff spaces BH p, q (R ) and the Triebel-Lizorkin-Hausdorff s, τ n spaces FH p, q (R ) (q = 1) are defined, respectively, to be the sets of all n f ∈ S (R ) such that  f BHp,s, τq (Rn ) ≡ inf ω

∞

∑2

1/q jsq

ϕ j ∗

j=0

q f [ω (·, 2− j )]−1 L p (Rn )

<∞

and

 1/q

∞



jsq q − j −q



f FHp,s, τq (Rn ) ≡ inf ∑ 2 |ϕ j ∗ f | [ω (·, 2 )]

ω

j=0

< ∞,

L p (Rn )

the infimums here are taken over all nonnegative Borel measurable functions ω on Rn × (0, ∞) with 



Rn

(∞)

[N ω (x)](p∨q) dΛnτ (p∨q) (x) ≤ 1,

and with the restriction that ω (·, 2− j ) is allowed to vanish only where ϕ j ∗ f vanishes, where N ω is the nontangential maximal function of ω

14

1 Introduction (∞)

and Λnτ (p∨q) is the nτ (p ∨q) -dimensional Hausdorff capacity; see Sect. 7.1 below. (xxxii) There is a number of further spaces appearing in the book. But they will be of restricted importance.

1.4 A Table of Coincidences As mentioned above there is some overlap of these different definitions. We are collecting some of these coincidence relations in what follows.

1.4.1 Besov-Morrey Spaces (i) It holds Bs,p,0q (Rn ) = Bsp, q(Rn ) for all s, p, and q; see Lemma 2.1. This implies s, 0 n s n s n B∞, ∞ (R ) = B∞, ∞ (R ) = Z (R ) , s, 0 n B∞, ∞ (R ) s, 0 n B∞, ∞ (R ) Bs,p,0p(Rn )

= = =

s > 0,

Bs∞, ∞ (Rn ) = Cs (Rn ) , Bs∞, ∞ (Rn ) , Bsp, p(Rn ) = Wps (Rn ) ,

s > 0 , s ∈ N , 0 < s < 1, s > 0 , s ∈ N , 1 ≤ p < ∞

(all in the sense of equivalent norms); see, e. g., [145, Sect. 2.2.2] and the references given there. (ii) Let s ∈ R, 0 < u ≤ p ≤ ∞ and q ∈ (0, ∞]. On the one hand we have s Bs,p,0q (Rn ) = N pqp (Rn ) = Bsp,q (Rn )

and

s, 1/u−1/p

Bu, ∞

(Rn ) = N ps∞u (Rn )

(in the sense of equivalent quasi-norms), but on the other hand, it holds s, 1/u−1/p

Bu, q

s (Rn )  N pqu (Rn )

if

0 < u < p < ∞ and 0 < q < ∞ ;

see Corollary 3.3. (iii) Let 0 < p < p0 < ∞, k ∈ N and s> Then

s−k/p, 1p n+k n

B p,q

  k 1 + n max 0, − 1 . p p

(Rn ) = Z s (Rn )

if

p ≤ q ≤ ∞,

1.4 A Table of Coincidences

15

and n 1 n+k s− k+n p + p0 , p n

B p0 ,q

(Rn ) = Z s (Rn )

if

p ≤ q ≤ ∞,

in the sense of equivalent quasi-norms; see Theorem 6.9 below.

1.4.2 Triebel-Lizorkin-Morrey Spaces (i) It holds Fu,s,q0 (Rn ) = Fp,s q(Rn ); see Lemma 2.1. This implies Fp,m,20 (Rn ) = Fp,m2(Rn ) = Wpm (Rn ) ,

m ∈ N, 1 < p < ∞,

Fp,s, 0p(Rn ) = Fp,s p(Rn ) = Wps (Rn ) ,

s > 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 < p < p0 < ∞, k ∈ N and   1 k + n max 0, − 1 . p p

s> 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 · ).

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 2, c Springer-Verlag Berlin Heidelberg 2010 

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 .

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

24

2.1.2 The Calder´on 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 Calder´on 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 Calder´on 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

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

+ +

 

|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)

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

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

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

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 .

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

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)

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

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. 

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

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

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

 (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)<1

see, for example, [64, p. 131]. To show that Tψ is well defined for all t ∈ as,p,τq(Rn ), we need the following conclusion. Lemma 2.7. Let s ∈ R, τ ∈ [0, ∞), p, q ∈ (0, ∞] and Ψ , ψ ∈ S (Rn ) satisfy, respectively, (2.1) and (2.2). Then for all t ∈ as,p,τq (Rn ),

∑

Tψ t =

tQΨQ +

l(Q)=1

∑

tQ ψQ

l(Q)<1

converges in S  (Rn ); moreover, Tψ : as,p,τq (Rn ) → S  (Rn ) is continuous. Proof. To prove Lemma 2.7, we only need to show that there exists an M ∈ Z such that for all t ∈ as,p,τq (Rn ) and φ ∈ S (Rn ),

∑

|tQ || ΨQ , φ | +

l(Q)=1

∑

l(Q)<1

|tQ || ψQ , φ |  tas,p,τq(Rn ) φ SM .

In fact, observe that |tQ | ≤ tas,p,τq(Rn ) |Q|s/n+1/2+τ −1/p for all dyadic cubes Q with l(Q) ≤ 1. Thus,

∑

|tQ || ΨQ , φ | +

l(Q)=1



≤ tas,p,τq(Rn )

∑

|tQ || ψQ , φ |

l(Q)<1

∑



| ΨQ , φ | +

l(Q)=1

∑

s/n+1/2+τ −1/p

|Q|

| ψQ , φ | .

l(Q)<1

Let M > (n/p − n − nτ − s) ∨ 0. We see that

∑

l(Q)=1

| ΨQ , φ | ≤

∑n

k∈Z



Rn

|Ψ (x − k)||φ (x)| dx

 Ψ SM φ SM

∑n (1 + |k|)−n−M

k∈Z

 Ψ SM φ SM .

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

32

On the other hand, by Lemma 2.4, we obtain

∑

|Q|s/n+1/2+τ −1/p| ψQ , φ |

l(Q)<1

 ψ SM+1 φ SM+1

∞

1

∑ ∑n 2− jn(s/n+1+τ −1/p)2− jM (1 + |2− jk|)n+M

j=1 k∈Z

 ψ SM+1 φ SM+1 , 

which completes the proof of Lemma 2.7.

Now we have the following inhomogeneous analogue of [165, Theorem 3.1], which is the so-called ϕ -transform characterization in the sense of Frazier and Jawerth. Theorem 2.1. Let s ∈ R, τ ∈ [0, ∞), p, q ∈ (0, ∞] and Φ , Ψ , ϕ and ψ be as in Lemma 2.3. Then the operators Sϕ : As,p,τq(Rn ) → as,p,τq (Rn ) and Tψ : as,p,τq (Rn ) → As,p,τq (Rn ) are bounded. Furthermore, Tψ ◦ Sϕ is the identity on As,p,τq (Rn ). To prove this theorem, we need some technical lemmas. For a sequence t = {tQ }l(Q)≤1 , r ∈ (0, ∞] and a fixed λ ∈ (0, ∞), set  (tr,∗ λ )Q

≡

|tR |r ∑ −1 λ {R∈Q: l(R)=l(Q)} (1 + l(R) |xR − xQ |)

1/r , Q ∈ Q, l(Q) ≤ 1

and tr,∗ λ ≡ {(tr,∗ λ )Q }l(Q)≤1 . We have the following estimate, which is an immediate consequence of the corresponding result on homogeneous spaces a˙s,p,τq (Rn ) in [165, Lemma 3.4] and the fact that the operator V in (2.15) is an isometric embedding of as,p,τq(Rn ) in a˙s,p,τq(Rn ). For completeness, we give a direct proof. Lemma 2.8. Let s ∈ R, τ ∈ [0, ∞), p, q ∈ (0, ∞] and λ ∈ (n, ∞). Then there exists a constant C ∈ [1, ∞) such that for all t ∈ as,p,τq(Rn ), ∗ tas,p,τq(Rn ) ≤ t p∧q, λ as,p,τq (Rn ) ≤ Ctas,p,τq(Rn ) . ∗ Proof. Let t ∈ as,p,τq (Rn ). Notice that |tQ | ≤ (t p∧q, λ )Q holds for all dyadic cubes Q ∗ s, τ with l(Q) ≤ 1. We then obtain that tas,p,τq(Rn ) ≤ t p∧q, λ a p, q (Rn ) . To see the converses, fix a dyadic cube P. For all l(Q) ≤ 1, let rQ ≡ tQ if Q ⊂ 3P and rQ ≡ 0 otherwise, and let uQ ≡ tQ − rQ . Set r ≡ {rQ }l(Q)≤1 and u ≡ {uQ }l(Q)≤1 . Then for all such Q, we have ∗ (t p∧q, λ )Q

p∧q

= (r∗p∧q, λ )Q + (u∗p∧q, λ )Q . p∧q

p∧q

(2.16)

∗ Applying the fact that for all t = {tQ }l(Q)≤1 , t p∧q, λ bsp,q (Rn ) ∼ tbsp,q(Rn ) ∗ s n s n and t p∧q, λ  f p,q s (Rn ) ∼ t f s (Rn ) , where b p,q (R ) and f p,q (R ) are, respectively, p,q

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

33

the corresponding sequence spaces for the inhomogeneous Besov space Bsp,q(Rn ) s (Rn ) (see [64, Lemma 2.3] for its proof), we and the Triebel-Lizorkin space Fp,q then have ⎧ ⎡ ⎤q/p ⎫1/q ⎪ ⎪ ⎪ ⎪ ⎬ 1 ⎨ ∞ ⎢ ⎥ −s/n−1/2+1/p ∗ p IP ≡ [|Q| (r ) ] ⎣ ⎦ Q ∑ ∑ p∧q, λ ⎪ |P|τ ⎪ ⎪ ⎪ ⎩ j=( jP ∨0) l(Q)=2− j ⎭ Q⊂P

≤

1 r∗ s n |P|τ p∧q, λ b p,q(R )



1 rbsp,q(Rn ) |P|τ

 tbs,p,τq(Rn ) and similarly, ⎧ ⎡ ⎤ p/q ⎫1/p ⎪ ⎪ ⎪ ⎪  ⎨ ⎬ 1 ⎢ ⎥ −s/n−1/2 ∗ q IP ≡ [|Q| (r ) χ (x)] dx  t f p,s, τq(Rn ) . ⎣ ⎦ Q Q ∑ p∧q, λ ⎪ |P|τ ⎪ P Q⊂P ⎪ ⎪ ⎩ ⎭ l(Q)≤1 On the other hand, let Q ⊂ P be a dyadic cube with side length no more then 1. Then l(Q) = 2−i l(P) for some nonnegative integer i ≥ (− jP ) ∨ 0 = −( jP ∧ 0).  is any dyadic cube with l(Q)  = l(Q) = 2−i l(P) and Q  ⊂ P + kl(P)  3P Suppose Q n for some k ∈ Z , where P + kl(P) ≡ {x + kl(P) : x ∈ P}. Then |k| ≥ 2 and  −1 |xQ − x  | ∼ 2i |k|. Thus, 1 + l(Q) Q ⎧ ⎪ ⎪ ⎨

⎤q/p ⎫1/q ⎪ ⎪ ⎬ ∞ 1 ⎢ ⎥ −s/n−1/2+1/p ∗ p JP ≡ [|Q| (u ) ] ⎣ ⎦ Q ∑ ∑ p∧q, λ ⎪ |P|τ ⎪ −i l(P) ⎪ ⎪ ⎩i=−( jP ∧0) l(Q)=2 ⎭ Q⊂P ⎡

⎧ ⎡ ⎪ ⎨ ∞ 1 ⎢  ∑ 2inq/p−iλ q/(p∧q) ⎣ ∑n |k|−λ |P|τ ⎪ ⎩i=−( j ∧0) k∈Z P

|k|≥2

q ⎫1/q ⎤ p∧q ⎪ ⎪ ⎪ ⎬ ⎥ −s/n−1/2+1/p p∧q  × ∑ (|Q| |tQ |) ⎥ ⎦ ⎪ . ⎪ −i l(P)  l(Q)=2 ⎪ ⎭  Q⊂P+kl(P)

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

34

When p ≤ q, by λ > n, we have ⎧ ⎪ ⎪ ⎨

JP

⎞q/p ⎫1/q ⎪ ⎪ ⎬ ∞ inq/p−iλ q/p ⎜ −λ ⎟  tbs,p,τq(Rn ) 2 |k|  tbs,p,τq(Rn ) ; ⎝∑ ⎠ ∑ ⎪ ⎪ ⎪ ⎪ k∈Zn ⎩i=−( jP ∧0) ⎭ |k|≥2 ⎛

when p > q, by H¨older’s inequality and λ > n, we obtain ⎧ ⎡ ⎪ ⎪ ⎢ 1 ⎨ ∞ −λ 2inq/p−iλ ⎢ JP  ∑ ⎣ ∑n |k| τ ⎪ |P| ⎪i=−( j ∧0) k∈Z P ⎩ |k|≥2 ⎧ ⎪ ⎨

⎤⎫1/q ⎪ ⎪ ⎬ ⎥ −s/n−1/2+1/p q⎥  |tQ |) ⎦ ∑ (|Q| ⎪ −i l(P) ⎪  l(Q)=2 ⎭  Q⊂P+kl(P)

⎞⎫1/q ⎪ ⎬ ⎜ ⎟ in−i λ − λ  tbs,p,τq(Rn ) 2 ⎝ ∑ |k| ⎠ ∑ ⎪ ⎪ ⎩i=−( jP ∧0) ⎭ k∈Zn ⎛

∞

|k|≥2

 tbs,p,τq(Rn ) . Therefore, by (2.16), ∗ tmin{p, q}, λ bs,p,τq (Rn )  sup (IP + JP )  tbs,p,τq (Rn ) . P∈Q

To complete the proof, for any i ∈ Z+ , k ∈ Zn+ and dyadic cube P, set  ∈ Q : l(Q)  = 2−i l(P), Q  ⊂ P + kl(P), Q  ∩ (3P) = 0}. A(i, k, P) ≡ {Q /  ∈ A(i, k, P). Simi −1 |xQ − x  | ∼ 2i |k| for any Q ⊂ P and Q Recall that 1 + l(Q) Q larly to the proof of [64, Remark A.3], by (2.11), we obtain that for all x ∈ P and a ∈ (0, p ∧ q],

∑

 −s/n−1/2|t  |) p∧q (|Q| Q

 −1 |xQ − x  |)λ (1 + l(Q)  Q Q∈A(i, k, P) ⎡ ⎛ ⎢ ⎜ ⎜  2−i(λ −n(p∧q)/a)|k|−λ ⎢ ⎣M ⎝

⎞

∑

−i l(P)  l(Q)=2  Q⊂P+kl(P)

⎤ p∧q

⎟ ⎥ ⎥  −s/n |t  |χ   )a ⎟ (|Q| Q Q ⎠ (x + kl(P))⎦

a

,

where herein and in what follows, M denotes the Hardy-Littlewood maximal function on Rn . Let a ≡ 2n(p∧q) n+λ . Then a ∈ (0, p ∧ q). Applying Minkowski’s inequality, Fefferman-Stein’s vector-valued inequality and H¨older’s inequality, we have

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

35

⎧   qp ⎫ 1p ⎨ ⎬ s 1 1 −n−2 ∗ q JP ≡ [|Q| (u ) χ (x)] dx Q Q p∧q, λ ⎭ |P|τ ⎩ P Q⊂P,∑ l(Q)≤1 ⎧ ⎡ ⎛ ⎪ ⎪ ⎪ ⎪ ⎪  ⎜ ∞ 1 ⎨ ⎢ ⎢ ⎜  ⎢ ∑ ⎜ ∑ 2−i(λ −n(p∧q)/a)|k|−λ |P|τ ⎪ P ⎣ i=−( j ∧0) ⎝ k∈Zn ⎪ P ⎪ ⎪ |k|≥2 ⎪ ⎩ ⎫ 1p p q ⎤ ⎤ p∧q ⎞ p∧q q ⎪ ⎪ a ⎪ ⎪ ⎪ ⎥ ⎬ ⎟ ⎥ ⎢ ⎜ ⎟ ⎥ s ⎟ −n a ⎥ ⎢ ⎜ ⎟ ⎥  Q ) ⎠ (x + kl(P))⎦ ⎟ ⎥ dx × ⎣M ⎝ ∑ (|Q| |tQ |χ ⎪ ⎠ ⎦ −i l(P) ⎪  l(Q)=2 ⎪ ⎪  ⎪ Q⊂P+kl(P) ⎭ ⎡

⎛

⎞

 t f p,s, τq(Rn ) . Therefore, by (2.16) again, ∗   s, τ tmin{p, n  sup (IP + JP )  t f s, τ (Rn ) , q}, λ  f p, q (R ) p, q P∈Q



which completes the proof of Lemma 2.8.

 (x) ≡ Φ (−x) and Let Φ and ϕ satisfy, respectively, (2.1) and (2.2). Since Φ  , ϕ in ϕ(x) ≡ ϕ (−x) also satisfy, respectively, (2.1) and (2.2), we may take Φ  n place of Φ and ϕ in Definition 2.1. For any f ∈ S (R ), define the sequence sup( f ) ≡ {supQ ( f )}l(Q)≤1 by setting  supQ ( f ) ≡

|Q|1/2 supy∈Q |ϕ j ∗ f (y)|  ∗ f (y)| supy∈Q |Φ

if

l(Q) = 2− j < 1 ,

if

l(Q) = 1.

For any γ ∈ Z+ , the sequence infγ ( f ) ≡ {infQ, γ ( f )}l(Q)≤1 is defined by setting  infQ, γ ( f ) ≡ |Q|

1/2

max

 = 2 l(Q), Q ⊂Q inf |ϕ j ∗ f (y)| : l(Q) −γ

 y∈Q

if l(Q) = 2− j < 1 and  infQ, γ ( f ) ≡ max

 ∗ f (y)| : l(Q)  = 2 −γ , Q ⊂Q inf |Φ

 y∈Q

if l(Q) = 1. We then have the following lemma.

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

36

Lemma 2.9. Let s ∈ R, τ ∈ [0, ∞), p, q ∈ (0, ∞] and γ ∈ Z+ be sufficiently large. Then there exists a constant C ∈ [1, ∞) such that for all f ∈ As,p,τq (Rn ), C−1 infγ ( f )as,p,τq (Rn ) ≤  f As,p,τq (Rn ) ≤  sup( f )as,p,τq (Rn ) ≤ Cinfγ ( f )as,p,τq (Rn ) . Proof. The inequality  f As,p,τq (Rn ) ≤  sup( f )as,p,τq (Rn ) immediately follows from the definitions of  f As,p,τq(Rn ) and sup( f ).  ∗ f and ϕ j ∗ f , j ∈ N, we obtain Applying [64, Lemma A.4] to the functions Φ − j that for all Q ∈ Q with l(Q) = 2 ≤ 1, (sup( f )∗r, λ )Q ∼ (infγ ( f )∗r, λ )Q , where r = p ∧ q. Thus,  sup( f )∗r, λ as,p,τq(Rn ) ∼ infγ ( f )∗r, λ as,p,τq(Rn ) , which together with Lemma 2.8 yields that  sup( f )as,p,τq (Rn ) ∼ infγ ( f )as,p,τq (Rn ) . To complete the proof of Lemma 2.9, we still need to show that infγ ( f )as,p,τq (Rn )   f As,p,τq(Rn ) . Define a sequence t ≡ {tJ }l(J)≤1 by setting tJ ≡ |J|1/2 inf |ϕi−γ ∗ f (y)| if l(J) = 2−i < 1 y∈J

and −γ ∗ f (y)| if l(J) = 1. tJ ≡ inf |Φ y∈J

Then for all r ∈ (0, ∞), dyadic cubes Q with l(Q) = 2− j ≤ 1 and a fixed λ > n, we have

Q ≤ infQ, γ ( f )χ

∑

⎧ ⎪ ⎨

 Q⊂Q −γ l(Q)  l(Q)=2

⎪ ⎩

∑

J⊂Q l(J)=2−γ l(Q)

tJr

⎫1/r ⎪ ⎬ ⎪ ⎭

χQ 

∑

 Q⊂Q −γ l(Q)  l(Q)=2

where χQ ≡ |Q|−1/2 χQ . We further obtain that infγ ( f )as,p,τq (Rn )  tr,∗ λ as,p,τq (Rn ) .

Q , (tr,∗ λ )Q χ

(2.17)

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

37

s, τ n In fact, for the space f p, q (R ), by (2.17), for each P ∈ Q, we obtain that

⎧ ⎡ ⎤ p/q ⎫1/p ⎪ ⎪  ⎬ ⎨ ∞ 1 jsq q ⎣ ∑ ⎦  2 [inf ( f ) χ (x)] dx Q, γ Q ∑ ⎪ |P|τ ⎪ ⎭ ⎩ P j=( j ∨0) l(Q)=2− j P

⎧ ⎡ ⎪ 1 ⎨ ⎣ ∞  ⎪ P j=(∑ |P|τ ⎩ j ∨0) P

⎫1/p ⎪ ⎬

⎤ p/q

∑

− j−γ  l(Q)=2

⎧ ⎡ ⎪ ∞ 1 ⎨ ⎣ ∼ ⎪ P i=( j ∑ |P|τ ⎩ ∨0)+γ P

Q (x)]q ⎦ 2 jsq [(tr,∗ λ )Q χ

dx

⎫1/p ⎪ ⎬

⎤ p/q

∑

−i  l(Q)=2

Q (x)]q ⎦ 2isq [(tr,∗ λ )Q χ

⎪ ⎭

dx

⎪ ⎭

≡ IP . γn

γn

Notice that P = ∪2m=1 Pm , where {Pm }2m=1 are disjoint dyadic cubes with side length l(Pm ) = 2−γ l(P) = 2−( jP +γ ) . This together with the facts that γ ∈ Z+ and ( jP + γ ) ∨ 0 ≤ ( jP ∨ 0) + γ yields that

IP

⎧ ⎡ ⎪ ∞ 1 ⎨ ⎣ ≤ ∑ |P|τ ⎪ ⎩ P i=( j +γ )∨0 P

⎫1/p ⎪ ⎬

⎤ p/q

∑

−i  l(Q)=2

⎧ ⎡ ⎪ γn  ∞ 1 ⎨2 ⎣ ∑ ≤ ∑ ⎪m=1 Pm i=( j ∨0) |P|τ ⎩ Pm

2isq [(tr,∗ λ )Q χQ (x)]q ⎦

dx

⎪ ⎭ ⎫1/p ⎪ ⎬

⎤ p/q

∑

−i  l(Q)=2

Q (x)]q ⎦ 2isq [(tr,∗ λ )Q χ

dx

⎪ ⎭

 tr,∗ λ  f p,s, τq (Rn ) , which further implies that infγ ( f ) f p,s, τq (Rn )  tr,∗ λ  f p,s, τq (Rn ) . The proof for the space bs,p,τq (Rn ) is similar and we leave the details to the reader. Finally, picking r = (p ∧ q), by Lemmas 2.8 and 2.6, we obtain infγ ( f )as,p,τq (Rn )  tr,∗ λ as,p,τq (Rn )  tas,p,τq(Rn )   f ∗As, τ (Rn )   f As,p,τq(Rn ) , p, q

which completes the proof of Lemma 2.9.



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

38

Using Lemmas 2.7, 2.8 and 2.9 to replace Lemmas 3.3, 3.4 and 3.5 in [165] and repeating the proof of Theorem 3.1 in [165] then complete the proof of Theorem 2.1; see also the proof of Theorem 2.2 in [64, pp. 50–51]. For the reader’s convenience, we give some details. Proof of Theorem 2.1. By similarity, we only consider the spaces Bs,p,τq(Rn ). Let f ∈ Bs,p,τq (Rn ). Since for all dyadic cubes Q = Q jk with l(Q) ≤ 1, |(Sϕ f )Q | = | f , ϕQ | = |Q|1/2 |ϕ j ∗ f (2− j k)| ≤ supQ ( f ),  . Then by Lemma 2.9, where and in what follows, when j = 0, ϕ0 is replaced by Φ we obtain Sϕ f bs,p,τq (Rn ) ≤ supQ ( f )bs,p,τq (Rn )   f Bs,p,τq(Rn ) . Next we prove that the operator Tψ is bounded from bs,p,τq(Rn ) to Bs,p,τq (Rn ). Let t = {tQ }l(Q)≤1 ∈ bs,p,τq(Rn ). Then by Lemma 2.7, Tψ t =

∑

tQΨQ +

l(Q)=1

∑

tQ ψQ

l(Q)<1

converges in S  (Rn ). Set f ≡ Tψ t =

∑

tQΨQ +

l(Q)=1

∑

tQ ψQ .

l(Q)<1

By (2.1) and (2.2), we obtain that

ϕ j ∗ f =

j+1

∑

∑

i= j−1 l(Q)=2−i

tQ ϕ j ∗ ψQ

for all j ∈ Z+ , where we set ψQ ≡ ΨQ if l(Q) = 1 and ψQ ≡ 0 if l(Q) = 2. Notice that ϕ j ∗ ψQ ∈ S (Rn ) for all j ∈ Z+ and dyadic cubes Q with l(Q) = 2−i . Then for r ∈ (0, ∞) and a fixed number λ > n, we have ' (−λ / min{1, r} |ϕ j ∗ ψQ (x)|  |Q|−1/2 1 + 2i|x − xQ | . Therefore, if x ∈ Q∗ ⊂ Q ⊂ Q∗∗ , where Q∗ , Q and Q∗∗ are respectively dyadic cubes with l(Q∗ ) = 2−i−1 , l(Q) = 2−i and l(Q∗∗ ) = 2−i+1 , using (2.11) when r ∈ (0, 1], or H¨older’s inequality when r ∈ (1, ∞) together with the fact that when r ∈ (1, ∞) and λ > n, ∑ (1 + 2i|x − xQ|)−λ  1, l(Q)=2−i

2.2 Embeddings

39

we obtain that ⎧ ⎨

⎫1/r ⎬ |tQ | −r/2 ∗ |ϕ j ∗ f (x)|  ∑ |Q| χ (x) . Q ⎩ ∑ ⎭ (1 + 2i|x − xQ |)λ i= j−1 l(Q)=2−i j+1

r

Therefore, for x ∈ Q∗ , we have ) * |ϕ j ∗ f (x)|  |Q|−1/2 (tr,∗ λ )Q∗ + (tr,∗ λ )Q + (tr,∗ λ )Q∗∗ χQ∗ (x). Taking r = min{p, q}, then by Lemma 2.8, we obtain Tψ tBs,p,τq(Rn ) =  f Bs,p,τq(Rn )  tr,∗ λ bs,p,τq(Rn )  tbs,p,τq(Rn ) , which completes the proof of Theorem 2.1.



As an immediate conclusion of Theorem 2.1, we obtain the next important property of our spaces As,p,τq(Rn ). Corollary 2.1. The definition of the spaces As,p,τq(Rn ) is independent of the choices of Φ and ϕ .

2.2 Embeddings From Definition 2.1, it is easy to deduce the following basic properties of the spaces As,p,τq (Rn ); see also Proposition 2.3.2/2 in [145]. In what follows, the symbol ⊂ stands for continuous embedding. To begin with we collect so-called elementary embeddings. Proposition 2.1. Let τ , s ∈ R, p, q ∈ (0, ∞] and ε ∈ (0, ∞). (i) The scale As,p,τq(Rn ) is monotone with respect to q, namely, if q1 ≤ q2 , then As,p,τq1 (Rn ) ⊂ As,p,τq2 (Rn ); (ii) The scale As,p,τq (Rn ) is monotone with respect to s, namely, for all q1 , q2 ∈ (0, ∞], ε, τ n s, τ n As+ p, q1 (R ) ⊂ A p, q2 (R );

(iii) For all p ∈ (0, ∞), Bs,p,τmin{p, q}(Rn ) ⊂ Fp,s, qτ (Rn ) ⊂ Bs,p,τmax{p, q}(Rn ); (iv) If τ ∈ (−∞, 0), then As,p,τq (Rn ) = {0}.

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

40

Proof. The properties (i), (ii) are simple corollaries of both the monotonicity of the q -norm on q and H¨older’s inequality. The property (iii) follows from the (generalized) Minkowski’s inequality. Property (iv) is obvious. We leave the details to the reader.  Proposition 2.2. Let p, q ∈ (0, ∞]. (i) Let s ∈ R and τ ∈ [1/p, ∞). Then the scale As,p,τq (Rn ) is monotone with respect to τ , i. e., if 1/p ≤ τ0 ≤ τ1 < ∞, then As,p,τq1 (Rn ) ⊂ As,p,τq0 (Rn ) follows. (ii) Let s ∈ (σ p , ∞), q1 , q2 ∈ (0, ∞] and assume 0 ≤ τ0 < τ1 < 1/p. Then the spaces As,p,τq11 (Rn ) and As,p,τq02 (Rn ) are incomparable, i. e., As,p,τq11 (Rn ) \ As,p,τq02 (Rn ) = 0/

and

As,p,τq01 (Rn ) \ As,p,τq12 (Rn ) = 0/ .

Proof. Part (i) is an immediate consequence of Lemma 2.2. Part (ii) will be proved at the end of Sect. 4.2.3.  Remark 2.5. There is, of course, no monotonicity with respect to p (as in case of the spaces L p (Rn )). However, if we would restrict us to subspaces defined by means of the condition supp f ⊂ Q (where Q is a fixed cube in Rn ), then one can prove also monotonicity properties with respect to p (and with respect to τ ∈ [0, ∞)). Proposition 2.3. Let s ∈ R, τ ∈ [0, ∞) and p, q ∈ (0, ∞]. Then we have S (Rn ) ⊂ As,p,τq (Rn ) ⊂ S  (Rn ) . Proof. With τ = 0 the claim has been proved by Triebel in [145, Sect. 2.3.2]. Thus, it remains to deal with τ > 0. Step 1. We shall prove S (Rn ) ⊂ As,p,τq(Rn ). Let f ∈ S (Rn ) and Φ , ϕ be as in Definition 2.1. To prove this embedding, we need to show that there exists an M ∈ N such that  f As,p,τq(Rn )   f SM for all f ∈ S (Rn ). Let P be an arbitrary dyadic cube. If jP > 0, applying Lemma 2.4 with M > max{0, n(1/p − 1), s + nτ }, we obtain 1 |P|τ





∞

∑

j= jP

(2 |ϕ j ∗ f (x)|) dx js

P

1  ϕ SM+1  f SM+1 τ |P|

q/p 1/q

p



∞

∑



j= jP

2 jsp− jM p dx P (1 + |x|)(n+M)p

 ϕ SM+1  f SM+1 2 jP (s+nτ −M)  ϕ SM+1  f SM+1 . If jP ≤ 0, then |P|−τ ≤ 1. Notice that for all x ∈ Rn , |Φ ∗ f (x)|  Φ SM  f SM (1 + |x|)−(n+M) .

q/p 1/q

2.2 Embeddings

41

Applying Lemma 2.4, again with M > max{0, s, n(1/p − 1)}, we conclude 1 |P|τ 



∞

∑

j=0

 P

 P

(2 |ϕ j ∗ f (x)|) dx js

q/p 1/q

p

|Φ ∗ f (x)| dx



1/p

∞

∑

+

p

j=1

 P

(2 |ϕ j ∗ f (x)|) dx js

q/p 1/q

p

 Φ SM  f SM + ϕ SM+1  f SM+1 . Thus,  f Bs,p,τq (Rn )   f SM+1 , namely, S (Rn ) ⊂ Bs,p,τq (Rn ). Now we use Proposition 2.1(iii) and find S (Rn ) ⊂ Bs,p,τp∧q(Rn ) ⊂ Fp,s, qτ (Rn ), which completes the proof of S (Rn ) ⊂ As,p,τq (Rn ). Step 2. Proof of As,p,τq (Rn ) ⊂ S  (Rn ). The proof of this assertion will be postponed and given below of Corollary 2.2.  Proposition 2.4. Let s ∈ R, τ ∈ [0, ∞) and p, q ∈ (0, ∞]. (i) If 0 < p1 ≤ p2 ≤ ∞, then s, τ + p1 − p1 2 1

A p2 , q

(Rn ) ⊂ As,p1τ, q (Rn );

(ii) If τ ∈ [0, 1/p], then As

p 1−τ p , q

(Rn ) ⊂ As,p,τq (Rn );

(iii) Let p < ∞. If τ = 1/p, then s, 1/p

s n Fp, q (Rn ) = F∞, q (R ).

Proof. Parts (i) and (ii) are consequences of H¨older’s inequality. Concerning (iii) we apply first Lemma 2.2. From this and the inhomogeneous version of [64, Corollary 5.7] (see the comment on page 133 in [64]) it is evident that the idens, 1/p s n  tity Fp, q (Rn ) = F∞, q (R ) holds. Remark 2.6. (i) The three assertions in Proposition 2.4, in particular the last one, make clear that p and τ interact. (ii) The homogeneous counterpart of Proposition 2.4(iii) has been proved in [165, Proposition 3.1(viii)].

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

42

Sobolev-type Embeddings The main result of this section is the following Sobolev-type embedding, which is an immediate consequence of the corresponding result on homogeneous spaces in [165, Proposition 3.3] and the fact that the operator V in (2.15) is an isometric embedding of as,p,τq(Rn ) in a˙s,p,τq (Rn ). For completeness, we give a direct proof. Proposition 2.5. Let τ ∈ [0, ∞), r, q ∈ (0, ∞] and −∞ < s1 < s0 < ∞. (i) If 0 < p0 < p1 ≤ ∞ such that s0 − n/p0 = s1 − n/p1, then bsp00,,τq (Rn ) ⊂ bsp11,,τq (Rn ). (ii) If 0 < p0 < p1 < ∞ such that s0 − n/p0 = s1 − n/p1, then f ps00 ,, τr (Rn ) ⊂ f ps11 ,, τq (Rn ). Proof. By Theorem 2.1, it suffices to prove the corresponding conclusions on sequence spaces as,p,τq (Rn ). The embedding bsp00,,τq (Rn ) ⊂ bsp11,,τq (Rn ) is immediately deduced from (2.13) and (2.11). s ,τ To prove f p00 , r (Rn ) ⊂ f ps11,, τq (Rn ), by Proposition 2.1(i), we only need to show s0 , τ n that f p0 , ∞ (R ) ⊂ f ps11 ,, τq (Rn ). Let t ∈ f ps00,, τ∞ (Rn ). Without loss of generality, we may assume that t f s0 , τ (Rn ) = 1. p0 , ∞

For any λ ∈ (0, ∞) and P ∈ Q, pick N ∈ Z such that )

1 − 2−qn/p1

*1/q

λ 21+n/p1

*1/q λ ) . < |P|τ 2nN/p1 ≤ 1 − 2−qn/p1 2

Step 1. Let N ≥ ( jP ∨ 0). Since |Q|−s0 /n−1/2 |tQ | ≤ |Q|τ −1/p0 t f s0 , τ (Rn ) = 2− jn(τ −1/p0) p0 , ∞

for all Q ∈ Q with l(Q) = 2− j ≤ 1, this together with s0 − n/p0 = s1 − n/p1 yields that ⎧ ⎪ ⎨

⎫1/q ⎬ ,q ⎪ + − jq(s0 −s1 ) −s0 /n  2 sup |t | χ (x) |Q| Q Q ∑ ⎪ ⎪ ⎩ j=( jP ∨0) ⎭ l(Q)=2− j N

Q⊂P

) *−1/q ≤ 2−( jP ∨0)nτ 2nN/p1 1 − 2−qn/p1 ) *−1/q ≤ |P|τ 2nN/p1 1 − 2−qn/p1 ≤ λ /2,

2.2 Embeddings

43

and ⎫1/q ⎬ ,q ⎪ + − jq(s0 −s1 ) −s0 /n  2 sup |t | χ (x) |Q| Q Q ∑ ⎪ ⎪ ⎭ ⎩ j=N+1 l(Q)=2− j ⎧ ⎪ ⎨

∞

Q⊂P

) *−p1 (s0 −s1 )/qn ) *−1/q 1 − 2−q(s0−s1 ) ≤ 2 p1 (s0 −s1 )/n 1 − 2−qn/p1 + , Q (x) . ×|P|τ p1 (s0 −s1 )/n λ −p1 (s0 −s1 )/n sup |Q|−s0 /n |tQ |χ Q⊂P l(Q)≤1

Notice that for all dyadic cubes P, ⎧ ⎫1/q ⎪ ⎨ ⎬ + ,q ⎪ Q (x) |Q|−s1 /n |tQ |χ ∑ ⎪ ⎪ ⎩ Q⊂P ⎭ l(Q)≤1

=

⎧ ⎪ ⎨

∞

∑ ⎪ ⎩ j=( jP ∨0)

2− jq(s0 −s1 ) sup

l(Q)=2− j Q⊂P

+

⎫1/q ⎬ ,q ⎪ Q (x) |Q|−s0 /n |tQ |χ . ⎪ ⎭

Then by s0 − n/p0 = s1 − n/p1, we have ⎧ ⎫ ⎧ ⎫1/q  ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎨ ⎬ ⎬ + , q   x∈P: ∑ |Q|−s1 /n|tQ |χQ (x) ⎪ > λ ⎪ ⎪ ⎪ ⎩ Q⊂P ⎪ ⎪ ⎭ ⎭ ⎩ l(Q)≤1 ⎧ ⎨ + ,  Q (x) > 2−1−p1 (s0 −s1 )/n   x ∈ P : sup |Q|−s1 /n |tQ |χ Q⊂P ⎩ l(Q)≤1

⎫ ⎬ ) * p1 (s0 −s1 )/qn ) *1/q −qn/p1 −q(s0−s1 ) −τ (p1 /p0 −1) p1 /p0  × 1−2 |P| λ 1−2 , ⎭

and hence t ps11 , τ

f p1 , q (Rn )



∞ 1  sup λ p0 −1 τ p 0 0 P∈Q |P| ⎧ ⎫ ⎨ ⎬ + ,  −s1 /n   ×  x ∈ P : sup |Q| |tQ |χQ (x) > λ  d λ ⎭ Q⊂P ⎩ l(Q)≤1

∼ t ∼ 1.

p0 s ,τ f p01 , ∞ (Rn )

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

44

Step 2. Let N < ( jP ∨ 0). Notice that ⎧ ⎪ ⎨

∞

∑ sup ⎪ ⎩ j=( jP ∨0) l(Q)=2− j ≤

⎧ ⎪ ⎨

+

Q⊂P

∞

⎫1/q ⎬ ,q ⎪ Q (x) |Q|−s1 /n |tQ |χ ⎪ ⎭

∑ sup ⎪ ⎩ j=N+1 l(Q)=2− j

+

Q⊂P

⎫1/q ⎬ ,q ⎪ Q (x) |Q|−s1 /n |tQ |χ . ⎪ ⎭

By the same argument as above, we also obtain t f s1 , τ (Rn )  1, which completes p1 , q the proof of Proposition 2.5.  As a corollary of Theorem 2.1 and Proposition 2.5, we have the following Sobolev-type embedding conclusions for Bs,p,τq (Rn ) and Fp,s, qτ (Rn ), which generalize the classical results on Besov spaces Bsp, q (Rn ) and Triebel-Lizorkin spaces Fp,s q(Rn ) (see [145, p. 129]). In fact, if τ = 0, Corollary 2.2 is just [145, Theorem 2.7.1]. Corollary 2.2. Let τ ∈ [0, ∞), r, q ∈ (0, ∞] and −∞ < s1 < s0 < ∞. (i) If 0 < p0 < p1 ≤ ∞ such that s0 − n/p0 = s1 − n/p1, then Bsp00,,τq (Rn ) ⊂ Bsp11,,τq (Rn ). (ii) If 0 < p0 < p1 < ∞ such that s0 − n/p0 = s1 − n/p1, then Fps00,, rτ (Rn ) ⊂ Fps11,, qτ (Rn ). Corollary 2.2 can be used to complete the proof of Proposition 2.3. Proof of Proposition 2.3 (continued). To show As,p,τq (Rn ) ⊂ S  (Rn ), we need to prove that there exists an M ∈ N such that for all f ∈ As,p,τq (Rn ) and φ ∈ S (Rn ), | f , φ |   f As,p,τq(Rn ) φ SM . Let Φ , Ψ , ϕ and ψ be as in Lemma 2.3. Then by the Calder´on reproducing formulae in Lemmas 2.3 and 2.4, we obtain | f , φ | ≤



∞

Rn

|Ψ ∗ f (x)||Φ ∗ φ (x)| dx + ∑

 φ SM+1 ∼ φ SM+1

∞

∑ 2− jM

j=0 ∞

∑ 2− jM

j=0



n j=1 R

 Rn

|ψ j ∗ f (x)|(1 + |x|)−(n+M) dx

∑n

k∈Z

|ψ j ∗ f (x)||ϕ j ∗ φ (x)| dx

 Q0k

|ψ j ∗ f (x)|(1 + |x|)−(n+M) dx,

where we used ψ0 to replace Ψ , and M ∈ N will be determined later.

2.2 Embeddings

45 n

Notice that there exists 2n disjoint dyadic cubes {Ql }2l=1 with l(Ql ) = 1 such that n n / {Ql }2l=1 and x ∈ Q0k , then |x| ≥ 1. the ball B(0, 1) ⊂ (∪2l=1 Ql ). Obviously, if Q0k ∈ Moreover, if setting

χm (k) ≡ χ{k∈Zn : 2m ≤|cQ

0k

|<2m+1 } (k),

where cQ0k denotes the center of Q0k , we then have ∑k∈Zn χm (k)  2mn . If p ∈ [1, ∞], let M > max{−s, 0}. Then applying H¨older’s inequality yields that | f , φ |  φ SM+1  φ SM+1



∞

j=0

+



∞ |ψ j ∗ f (x)| ∑2 ∑ Ql (1 + |x|)n+M dx + ∑ ∑n χm (k) j=0 m=0 k∈Z l=1  n 1/p  ∞ 2 − jM − js jsp p 2 |ψ j ∗ f (x)| dx ∑2 ∑2

∞

∑ ∑

2n

− jM

Ql

l=1

χm (k)2− js−m(n+M)

m=0 k∈Zn ∞

 φ SM+1  f As,p,τq (Rn ) ∑ 2

 

− jM

Q0k

2

− js



 Q0k

···

1/p

··· +

j=0

∞

∑ ∑



χm (k)2

− js−m(n+M)

m=0 k∈Zn

 φ SM+1  f As,p,τq (Rn ) . s−n/p+n, τ

The case p ∈ (0, 1) can be deduced from the embedding As,p,τq (Rn ) ⊂ A1, q in Corollary 2.2, which completes the proof of Proposition 2.3.

(Rn ) 

Embeddings into H¨older-Zygmund and Lebesgue-type Spaces Later on, see Chap. 4 below, we shall need also embeddings into H¨older and Lebesgue-type spaces. Recall, that Z s (Rn ) denotes the H¨older-Zygmund spaces; see item (vi) in Sect. 1.3. If s > 0, then it is well known that s (Rn ) Z s (Rn ) = Bs∞,∞ (Rn ) = F∞,∞

in the sense of equivalent norms; see [145, p. 51]. Proposition 2.6. Let s ∈ R, τ ∈ [0, ∞) and p, q ∈ (0, ∞]. Then s+nτ −n/p As,p,τq(Rn ) ⊂ F∞, (Rn ); ∞

in particular, if s + nτ − n/p > 0.

As,p,τq (Rn ) ⊂ Z s+nτ −n/p(Rn )

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

46

Proof. Recall that

 f F∞,s ∞(Rn ) ≡ sup 2 js ϕ j ∗ f L∞ (Rn ) j≥0

with ϕ0 ≡ Φ , where Φ and ϕ are as in Definition 2.1. Let φ ∈ S (Rn ) such that φ ≡ 1 on {ξ ∈ Rn : |ξ | ≤ 2}. Then by (2.1) and (2.2), we obtain that

ϕj ∗ f ≡ φj ∗ ϕj ∗ f. Thus, for all j ∈ Z+ and x ∈ Rn , by [62, p. 782, (2.11)], we have |ϕ j ∗ f (x)| ≤

∑n



Q jk

k∈Z

≤ 

|φ j (x − y)||ϕ j ∗ f (y)| dy

sup |ϕ j ∗ f (z)| ∑n z∈Q

k∈Z

jk

∑

jn/p

2

k∈Zn

×



|φ j (x − y)| dy

Q jk



∑ (1 + |l|)

−n−1

Q jk

|ϕ j ∗ f (z)| dz p

Q j,k+l

l∈Zn



1/p



|φ j (x − y)| dy

 2 j(n/p−s−nτ ) f As,p,τq (Rn )



 2 j(n/p−s−nτ ) f As,p,τq (Rn ) ,

Rn

|φ j (x − y)| dy



which completes the proof of Proposition 2.6. For τ ∈ [0, ∞) and p ∈ (0, ∞], set  f Lτp (Rn ) ≡

1 τ |P| {P∈Q, |P|≥1}



sup

P

1/p | f (x)| p dx

,

(2.18)

where the supremum is taken over all dyadic cubes P with side length l(P) ≥ 1. Denote by Lτp (Rn ) the set of all functions f satisfying  f Lτp (Rn ) < ∞. Obviously, when τ = 0, then Lτp (Rn ) = L p (Rn ). The scale is monotone in τ . If τ0 < τ1 , then Lτp0 (Rn ) ⊂ Lτp1 (Rn ) follows. We also claim that Lτp (Rn ) ⊂ S  (Rn ) when p ∈ [1, ∞]. In fact, by H¨older’s inequality, we see that for all φ ∈ S (Rn ), | f , φ | ≤ ≤



Rn

∑

| f (x)||φ (x)| dx 

k∈Zn

Q0k

| f (x)| p dx

  f Lτp (Rn ) φ SM

Q0k

1



|φ (x)| p dx

∑n (1 + |k|)n+M

k∈Z

  f Lτp (Rn ) φ SM , which proves the above claim.

1/p 

1/p

2.2 Embeddings

47

Proposition 2.7. Let τ ∈ [0, ∞). (i) Let p ∈ [1, ∞]. Then p τ 0, τ n n n B0, p, 1 (R ) ⊂ Fp, 1 (R ) ⊂ Lτ (R ) .

(ii) Let p ∈ (0, 1). Then . σ ,τ σ ,τ B p,pp (Rn ) = Fp, pp (Rn ) ⊂ L1τ (Rn ) ∩ Lτp (Rn ) . Proof. Step 1. Proof of (i). Let {ϕ j } j∈Z+ be the system for the definition of the spaces As,p,τq (Rn ); see Sect. 1.3.3. Then, for each dyadic cube P, f=

∞

∑ ϕj ∗ f

j=0

in the sense of L p (P). Thus, if |P| ≥ 1, then 1 1  f L p (P) ≤ |P|τ |P|τ

  P

p

∞

∑ |ϕ j ∗ f (x)|

1/p ≤  f F 0, τ (Rn )

dx

(2.19)

p, 1

j=0

follows. The first embedding in (i) is an obvious consequence of Proposition 2.1(iii). Step 2. Proof of (ii). We obtain σ ,τ

Fp, pp (Rn ) ⊂ F1,0,1τ (Rn ) ⊂ L1τ (Rn ) , where we used Proposition 2.1, Corollary 2.2 and Step 1. Thus, we get f=

∞

∑ ϕj ∗ f

j=0

in L1 (P). H¨older’s inequality yields that this identity takes place in L p (P) as well. Similarly as in (2.19) we conclude 1 1  f L p (P) ≤ τ |P| |P|τ 1 ≤ |P|τ

  P



p

∞

∑ |ϕ j ∗ f (x)|

j=0 ∞

∑ |ϕ j ∗ f (x)|

1/p dx

1/p p

dx

P j=0

≤  f F 0, τ (Rn ) p, p

provided that |P| ≥ 1. This finishes the proof of Proposition 2.7.



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

48

Remark 2.7. For τ = 0 much more is known about these various types of embeddings. In particular, many times if, and only if assertions are known. We refer to [133] and the references given there.

2.3 The Fatou Property Next we show that the spaces As,p,τq(Rn ) satisfy the following Fatou property, which is often applied in connection with nonlinear problems. We are going to use the Fatou property in Sect. 6.1. Proposition 2.8. Let s ∈ R, τ ∈ [0, ∞), p, q ∈ (0, ∞] and { fm }m∈Z+ ⊂ As,p,τq (Rn ) be a sequence such that fm  f (weak convergence in S  (Rn )) as m → ∞ and sup  fm As,p,τq(Rn ) < ∞.

m∈Z+

Then f ∈ As,p,τq(Rn ) and  f As,p,τq(Rn ) ≤ sup  fm As,p,τq(Rn ) . m∈Z+

Proof. We follow [60]. By similarity we only consider the space Fp,s, qτ (Rn ). From assumption, it follows that for all j ∈ Z+ and x ∈ Rn ,

ϕ j ∗ fm (x) = fm (x − ·)(ϕ j ) → f (x − ·)(ϕ j ) = ϕ j ∗ f (x) as m → ∞, where when j = 0, ϕ0 is replaced by Φ . Notice that for each Q ∈ Q and N ≥ ( jQ ∨ 0), Fatou’s lemma yields  Q

⎡ ⎣

⎤p q

N

∑

j=( jQ ∨0)

2

jsq

q⎦

|ϕ j ∗ f (x)|

dx ≤ lim inf m→∞

 Q

⎡ ⎣

⎤p q

N

∑

2

jsq

q⎦

|ϕ j ∗ fm (x)|

dx.

j=( jQ ∨0)

This combined with Levi’s lemma and Definition 2.1 yields the desired conclusion, which completes the proof of Proposition 2.8.  Remark 2.8. (i) There are many spaces which do not have the Fatou property. The most obvious examples are L1 (Rn ) and C(Rn ); see [59]. (ii) The Fatou property of the spaces Fp,s q (Rn ) (p < ∞) and Bsp, q (Rn ) has been proved by Franke [59]; see also Franke and Runst [60]. Bourdaud and Meyer [19] gave an independent proof restricted to Besov spaces.

Chapter 3

Almost Diagonal Operators and Atomic and Molecular Decompositions

In the first part of this chapter we shall show that under certain restrictions on the parameters our spaces As,p,τq (Rn ) allow characterizations by smooth molecules and smooth atoms. This opens the door to the discretization of our distribution spaces As,p,τq (Rn ). Afterwards, based on these discretizations, we are able to coms (Rn ) (see item (xxv) pare the classes As,p,τq(Rn ) with the Besov-Morrey spaces N pqu s in Sect. 1.3) and the Triebel-Lizorkin-Morrey spaces E pqu(Rn ) (see item (xxv) in Sect. 1.3).

3.1 Smooth Atomic and Molecular Decompositions As an application of Theorem 2.1, we study boundedness of operators on As,p,τq(Rn ) by first considering their boundedness on the corresponding space as,p,τq (Rn ) of sequences. We first show that almost diagonal operators are bounded on as,p,τq (Rn ) for appropriate indices, which generalize the basic results of Frazier and Jawerth with respect to asp, q (Rn ); see [64, Theorem 3.3] and [65, Theorem (6.20)]. We shall say that an operator A is associated with the matrix {aQP }l(Q), l(P)≤1 , if for all sequences t ≡ {tQ }l(Q)≤1 ⊂ C,  At ≡ {(At)Q }l(Q)≤1 ≡



∑

l(P)≤1

.

aQPtP l(Q)≤1

Furthermore, we shall use the abbreviation  n if as,p,τq (Rn ) = bs,p,τq (Rn ) , J ≡ min{1,n p} s, τ s, τ n n min{1, p, q} if a p, q (R ) = f p, q (R ) .

(3.1)

Definition 3.1. Let s ∈ R, p, q ∈ (0, ∞] and ε ∈ (0, ∞). The operator A, associated with the matrix {aQP }l(Q), l(P)≤1, is called ε -almost diagonal on as,p,τq(Rn ) if the matrix {aQP }l(Q), l(P)≤1 satisfies

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 3, c Springer-Verlag Berlin Heidelberg 2010 

49

50

3 Almost Diagonal Operators and Atomic and Molecular Decompositions

|aQP |/ωQP (ε ) < ∞,

sup

(3.2)

l(Q), l(P)≤1

where 

  −J−ε l(Q) s |xQ − xP | ωQP (ε ) ≡ 1+ l(P) max(l(P), l(Q))   (n+ε )/2   l(Q) l(P) (n+ε )/2+J−n × min , . l(P) l(Q) We remark that each ε -almost diagonal operator is also an almost diagonal operator in the sense of Frazier and Jawerth; see [64, Sects. 3 and 12]. The following conclusion is an immediate corollary of the corresponding result on homogeneous spaces in [165, Theorem 4.1] and the fact that the operator V in (2.15) is an isometric embedding of as,p,τq(Rn ) in a˙s,p,τq (Rn ). For completeness, we give a direct proof. Theorem 3.1. Let ε ∈ (0, ∞), s ∈ R, p, q ∈ (0, ∞] and τ ∈ [0, 1/p + ε /(2n)). Then all ε -almost diagonal operators on as,p,τq (Rn ) are bounded on as,p,τq (Rn ). Proof. By similarity, we only give the proof for bs,p,τq (Rn ). Let t = {tQ }l(Q)≤1 ∈ bs,p,τq (Rn ) and A be a ε -almost diagonal operator associated with the matrix {aQR }l(Q), l(R)≤1 and ε ∈ (0, ∞). Without loss of generality, we may assume s = 0. Indeed, if the conclusion holds for s = 0, let  tR ≡ [l(R)]−stR and B be the operator associated with the matrix {bQR }l(Q), l(R)≤1, where bQR ≡ (l(R)/l(Q))s aQR for all dyadic cubes Q and R with l(Q), l(R) ≤ 1. Then we have t b0, τ (Rn )   t b0, τ (Rn ) ∼ tbs,p,τq(Rn ) , Atbs,p,τq(Rn ) = B p, q

p, q

which yields the desired conclusion. We now consider the case when min{p, q} > 1. For all Q ∈ Q with l(Q) ≤ 1, we write A ≡ A0 + A1 with (A0t)Q ≡

∑

aQRtR and (A1t)Q ≡

{R: 1≥l(R)≥l(Q)}

∑

aQRtR .

{R: l(R)
By Definition 3.1, we see that for all such Q, |(A0t)Q | 

∑

{R: 1≥l(R)≥l(Q)}



l(Q) l(R)

 n+ε 2

|tR | n+ε , (1 + [l(R)]−1|xQ − xR|)

3.1 Smooth Atomic and Molecular Decompositions

51

and therefore A0tb0, τ (Rn )

⎧ ⎡ ⎛  n+ε   ⎨ ∞ l(Q) 2 1 ⎣ ⎝  sup ∑ ∑ ∑ τ P l(Q)=2− j {R: l(Q)≤l(R)≤[l(P)∧1]} l(R) P∈Q |P| ⎩ j=( j ∨0) p, q

P

⎤ qp ⎫ 1q ⎪ ⎬ Q (x) |tR |χ ⎠ dx⎦ × n+ε ⎪ (1 + [l(R)]−1|xQ − xR |) ⎭ ⎞p

⎧ ⎡ ⎛ ⎞ p ⎤ qp ⎫ 1q ⎪ ⎪  ⎨ ⎬ ∞ 1 ⎣ ⎝ ⎠ ⎦ + sup · · · dx ∑ ∑ ∑ τ ⎪ P∈Q |P| ⎪ ⎩ j=( j ∨0) P l(Q)=2− j {R: l(R)>[l(P)∧1]} ⎭ P

≡ I1 + I2 . For all i ∈ Z+ and m ∈ N, set U0, i ≡ {R ∈ Q : l(R) = 2−i and |xQ − xR | < l(R)} and Um, i ≡ {R ∈ Q : l(R) = 2−i and 2m−1 l(R) ≤ |xQ − xR| < 2m l(R)}. Then we have #Um, i  2mn , where #Um, i denotes the cardinality of Um, i . Notice that ε , |tR | ≤ |R|1/2−1/p+τ tb0, τ (Rn ) for all R ∈ Q with l(R) ≤ 1. Thus, by 0 ≤ τ < 1p + 2n p, q

⎧ ⎡ ⎛   n+ε  jP ∨0 ∞ l(Q) 2 1 ⎨ ∞ ⎣ ⎝ I2  tb0, τ (Rn ) sup ∑ ∑ ∑ ∑ ∑ τ p, q P l(Q)=2− j i=0 m=0 R∈Um, i l(R) P∈Q |P| ⎩ j=( j ∨0) P

⎤q/p ⎫1/q ⎪ ⎬ |R|1/2−1/p+τ χQ (x) ⎠ dx⎦ × n+ε ⎪ (1 + [l(R)]−1|xQ − xR |) ⎭ ⎞p

 tb0, τ (Rn ) . p, q

For I1 , let r and u be the same as in the proof of Lemma 2.8. We see that ⎧ ⎡ ⎛ ⎪ ⎪  j 1 ⎨ ∞ ⎣ ⎝ I1  sup 2(i− j)(n+ε )/2 ∑ ∑ ∑ τ P P∈Q |P| ⎪ − j ⎪ l(Q)=2 i=( jP ∨0) ⎩ j=( jP ∨0) ⎤q/p ⎫1/q ⎪ ⎬ Q (x) |rR |χ ⎠ ⎦ × ∑ dx ⎪ (1 + [l(R)]−1|xQ − xR|)n+ε ⎭ l(R)=2−i ⎞p

52

3 Almost Diagonal Operators and Atomic and Molecular Decompositions

⎧ ⎡ ⎛ ⎪  j 1 ⎨ ∞ ⎣ ⎝ + sup 2(i− j)(n+ε )/2 ∑ ∑ ∑ τ P∈Q |P| ⎪ ⎩ j=( j ∨0) P l(Q)=2− j i=( j ∨0) P

P

⎤q/p ⎫1/q ⎪ ⎬ Q (x) |uR |χ ⎠ ⎦ × ∑ dx ⎪ (1 + [l(R)]−1 |xQ − xR|)n+ε ⎭ l(R)=2−i ⎞p

≡ J1 + J2 . Applying [64, Remark A.3] with a = 1, for all x ∈ Q, we have ⎞ ⎛ |rR | ∑ (1 + [l(R)]−1|xQ − xR|)n+ε  M ⎝ ∑ −i |rR |χR⎠ (x). l(R)=2−i l(R)=2 Hence H¨older’s inequality and the L p (Rn )-boundedness for p ∈ (1, ∞] of the HardyLittlewood maximal operator yield ⎧ ⎡ ⎛ ⎪  ⎨ j ∞ 1 ⎣ ⎝ ∑ 2(i− j)ε /2 J1  sup ∑ τ P∈Q |P| ⎪ ⎩ j=( j ∨0) P i=( j ∨0) P

P

⎤q/p ⎫1/q ⎪ ⎬ R ⎠ (x)⎠ dx⎦ × M ⎝ ∑ |rR |χ ⎪ ⎭ l(R)=2−i ⎛

⎞

⎞p

⎧ ⎡ ⎛ ⎞p ⎤q/p ⎫1/q ⎪ ⎪  ⎨ ⎬ ∞ 1 ⎣ ⎝ ⎠ ⎦   sup |t | χ (x) dx ∑ ∑ R R τ ⎪ P∈Q |P| ⎪ ⎩i=( j ∨0) 3P l(R)=2−i ⎭ P

 tb0, τ (Rn ) , p, q

where the last inequality follows from Minkowski’s inequality if q > p or (2.11) if q ≤ p. To estimate J2 , we notice that if R ∩ (3P) = 0, / then R ⊂ P + kl(P) and (P + kl(P)) ∩ (3P) = 0/ for some k ∈ Zn with |k| ≥ 2 and 1 + [l(R)]−1 |xQ − xR | ∼ |k|l(P)/l(R).

3.1 Smooth Atomic and Molecular Decompositions

53

Therefore, by H¨older’s inequality, ⎧ ⎡ ⎛ ⎪ ⎪  ⎨ j ∞ 1 ⎜ − jqε /2 ⎢ J2  sup 2 2−i(n+ε )/2 ⎣ ⎝ ε ∑ ∑ ∑ τ +1+ n ⎪ P P∈Q |P| − j ⎪ l(Q)=2 i=( jP ∨0) ⎩ j=( jP ∨0) ⎤q/p ⎫1/q ⎪ ⎪ ⎬ ⎟ ⎥ |tR |χQ (x)⎠ dx⎦ ⎪ ⎪ ⎭ ⎞p

×

∑ |k|−n−ε ∑ −i

k∈Zn |k|≥2

l(R)=2 R⊂P+kl(P)

⎧ ⎡ ⎤q ⎫1/q ⎪ ⎪ ⎨ ⎬ j ∞ 1 − jqε /2 ⎢ −iε /2 −n−ε ⎥  tb0, τ (Rn ) sup 2 2 |k| ⎣ ⎦ ε ∑ ∑ ∑ p, q ⎪ P∈Q |P| n ⎪ ⎩ j=( j ∨0) ⎭ k∈Zn i=( j ∨0) P

P

|k|≥2

 tb0, τ (Rn ) . p, q

Thus, A0tb0, τ (Rn )  tb0, τ (Rn ) . p, q

p, q

Some similar computations to I1 also yield that A1tb0, τ (Rn )  tb0, τ (Rn ) . p, q

p, q

The case that min{p, q} ≤ 1 is a simple consequence of the case that min{p, q} > 1. Indeed, choosing a δ ∈ (0, p ∧ q), then p/δ > 1 and q/δ > 1.  be an operator associated with the matrix Let A   { aQP }l(Q), l(P)≤1 ≡ |aQP |δ (l(Q)/l(P))n/2−δ n/2 . l(Q), l(P)≤1

 is a  Then A ε -almost diagonal operator with s = 0 and  ε = δ ε. Define  t ≡ {[l(Q)]n/2−δ n/2|tQ |δ }l(Q)≤1 . Then

1/δ  t  0, τδ

b p/δ , q/δ (Rn )

= tb0, τ (Rn ) . p, q

Since δ < 1, by (2.11), we see that 1/δ  t  0, τδ Atb0, τ (Rn )  A

b p/δ , q/δ (Rn )

p, q

.

Applying the conclusions for min{p, q} > 1 yields 1/δ  Atb0, τ (Rn )  A t  ˙ 0, τδ p, q

b p/δ , q/δ (Rn )

  t

which completes the proof of Theorem 3.1.

1/δ

0, τδ

b p/δ , q/δ (Rn )

∼ tb0, τ (Rn ) , p, q



54

3 Almost Diagonal Operators and Atomic and Molecular Decompositions

As an application of Theorem 3.1, we establish characterizations of As,p,τq (Rn ) by inhomogeneous smooth atomic and molecular decompositions. For the forerunners with respect to the inhomogeneous spaces Asp, q (Rn ) we refer to [64] and [25]. For a real number s we denote by s the largest integer which is less than or equal to s. Let J be defined as in (3.1) if as,p,τq (Rn ) is replaced by As,p,τq (Rn ). Recall, xQ denotes the center of the cube Q. Definition 3.2. Let s ∈ R, τ ∈ [0, ∞), p, q ∈ (0, ∞], N = max{J − n − s, −1} and s∗ = s − s. Let Q be a dyadic cube with l(Q) ≤ 1. (i) A function mQ is called an inhomogeneous smooth synthesis molecule for As,p,τq (Rn ) supported near Q if there exist a real number δ ∈ (max{s∗ , (s + nτ )∗ }, 1] and a real number M ∈ (J, ∞) 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,

(3.3) (3.4)

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

if l(Q) < 1, (3.5)

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

if |γ | ≤ s + nτ , (3.6)

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 (3.7) |z|≤|x−y|

if |γ | = s + nτ . A collection {mQ }l(Q)≤1 is called a family of inhomogeneous smooth synthesis molecules for As,p,τq(Rn ), if each mQ is an inhomogeneous smooth synthesis molecule for As,p,τq(Rn ) supported near Q. (ii) A function bQ is called an inhomogeneous smooth analysis molecule for As,p,τq (Rn ) supported near Q if there exist a ρ ∈ ((J − s)∗ , 1] and an M ∈ (J, ∞) such that  Rn

xγ bQ (x) dx = 0

if |γ | ≤ s + nτ  and l(Q) < 1,

|bQ (x)| ≤ (1 + |x − xQ|)−M if l(Q) = 1,

(3.8) (3.9)

|bQ (x)| ≤ |Q|−1/2 (1 + [l(Q)]−1 |x − xQ |)− max{M, M+n+s+nτ −J} if l(Q) < 1, (3.10) |∂ γ bQ (x)| ≤ |Q|−1/2−|γ |/n (1 + [l(Q)]−1|x − xQ |)−M

if |γ | ≤ N,

(3.11)

3.1 Smooth Atomic and Molecular Decompositions

55

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 (3.12) |z|≤|x−y|

if |γ | = N. A collection {bQ }l(Q)≤1 is called a family of inhomogeneous smooth synthesis molecules for As,p,τq(Rn ), if each bQ is an inhomogeneous smooth analysis molecule for As,p,τq(Rn ) supported near Q. We add a remark concerning the interpretation of certain restrictions. If s + nτ < 0, then (3.6) and (3.7) are void. Analogously, if J − n − s < 0, then N = −1 and (3.11) and (3.12) are void. Also we wish to point out that only when l(Q) < 1, the inhomogeneous smooth synthesis and analysis molecules for As,p,τq (Rn ) supported near the dyadic cube Q have to fulfil the vanishing moment condition, which is the only difference between the homogeneous and the inhomogeneous case. To establish the inhomogeneous smooth atomic and molecular decomposition characterizations, we need some elementary lemmas. Lemma 3.1. Let s, p, q, J, M, N and ρ be as in Definition 3.2. Assume that     M−J ρ − (J − s)∗ 1 τ ∈ 0, + ∧ p 2n n if N ≥ 0,

   M−J s+n−J 1 τ ∈ 0, + ∧ p 2n n 

if N < 0, and δ ∈ (max{(s+ nτ )∗ , s∗ }, 1]. Then there exist a positive real number ε1 and a positive constant C such that ε1 > 2n(τ − 1p ) and for all families {mQ }l(Q)≤1 of inhomogeneous smooth synthesis molecules for As,p,τq(Rn ) and families {bQ }l(Q)≤1 of inhomogeneous smooth analysis molecules for As,p,τq (Rn ), |mP , bQ | ≤ CωQP (ε1 ). Namely, the operator associated with the matrix {aQP }l(Q), l(P)≤1 defined by aQP ≡ mP , bQ  is ε1 -almost diagonal on as,p,τq (Rn ). The proof of Lemma 3.1 is a slight modification of the homogeneous analogue in [165, Lemma 4.2]; see also [64, Corollary B.3]. We give some details. Proof. The proof for the case l(Q), l(P) < 1 is the same as those for [165, Lemma 4.2] and [64, Corollary B.3]. In fact, if l(Q) = 2−v ≤ 2−μ = l(P), applying [64, Lemma B.1] when s+nτ ≥ 0, or [64, Lemma B.2] when s+nτ < 0, with R = M, θ = δ , k = v, j = μ , L = s + nτ , S = M + n + s + nτ − J, x1 = xQ , g = mP (xP − ·)

56

3 Almost Diagonal Operators and Atomic and Molecular Decompositions

and h = bQ , we obtain the desired estimate. If l(Q) = 2−v ≥ 2−μ = l(P), applying [64, Lemma B.1] when N ≥ 0, or [64, Lemma B.2] when N < 0, with R = M, θ = ρ , k = μ , j = v, L = N, S = M − s, x1 = xP , g = bQ (xQ − ·) and h = mP , we also obtain the desired estimate. Notice that in the above argument (see also the proofs of [165, Lemma 4.2] and [64, Corollary B.3]), the vanishing moment conditions for mQ are only used when we take h = mP and apply [64, Lemma B.1], i. e., when l(P) < l(Q). This never happens in the inhomogeneous case if |P| = 1. A similar observation holds for bQ if |Q| = 1. Thus, we obtain the cases l(Q) < l(P) = 1 and l(P) < l(Q) = 1. The remaining estimate for |mP , bQ | when |P| = |Q| = 1 is an immediate consequence of [64, Lemma B.2]. This finishes the proof of Lemma 3.1.

As an immediate consequence, we have the following analogues of the corresponding results on the homogeneous case in [165, Corollary 4.1]. Corollary 3.1. Let s, p, q, τ and ε1 be as in Lemma 3.1 and Φ , ϕ as in Definition 2.1. Suppose that {mQ }l(Q)≤1 and {bQ }l(Q)≤1 are families of inhomogeneous smooth synthesis and analysis molecules for As,p,τq (Rn ), respectively. Then the operators associated with the matrices {aQP }l(Q), l(P)≤1 = {mQ , ϕP }l(Q), l(P)≤1 and

{bQP }l(Q), l(P)≤1 = {ϕP , bQ }l(Q), l(P)≤1 ,

with ϕP replaced by ΦP when l(P) = 1, are both ε1 -almost diagonal on as,p,τq (Rn ). To prove that  f , bQ  is well defined for all inhomogeneous smooth analysis molecules for As,p,τq(Rn ), we need the following lemma. Its proof can be given as the proof of [25, Lemma 5.4]. Lemma 3.2. Let φ be an inhomogeneous smooth analysis (or synthesis) molecule n for As,p,τq(Rn ) supported near Q ∈ Q. Then there exist a sequence {φk }∞ k=1 ⊂ S (R ) and a positive constant C such that for every k ∈ N, Cφk is also an inhomogeneous smooth analysis (or synthesis) molecule for As,p,τq (Rn ) supported near Q and φk (x) → φ (x) uniformly on Rn as k → ∞. Now we have the following smooth molecular characterization of As,p,τq (Rn ). For the corresponding results of the homogeneous case, see [165, Theorem 4.2]. Theorem 3.2. Let s ∈ R, p, q ∈ (0, ∞] and τ be as in Lemma 3.1. (i) Let {mQ }l(Q)≤1 be a family of inhomogeneous smooth synthesis molecules. Then there exists a positive constant C such that for all t = {tQ }l(Q)≤1 ∈ as,p,τq (Rn ), ! ! ! ! ! ! ! ∑ tQ mQ ! !l(Q)≤1 !

s, τ A p, q (Rn )

≤ Ctas,p,τq(Rn ) .

3.1 Smooth Atomic and Molecular Decompositions

57

(ii) Let {bQ }l(Q)≤1 be a family of inhomogeneous smooth analysis molecules. Then there exists a positive constant C such that for all f ∈ As,p,τq (Rn ), ! ! !{ f , bQ }l(Q)≤1 ! s, τ n ≤ C f  s, τ n . A p, q (R ) a (R ) p, q

Proof. (i) Let Φ , Ψ , ϕ and ψ be as in Lemma 2.3. It is easy to see that mP ∈ S  (Rn ). Then by the Calder´on reproducing formula in Lemma 2.3, we have mP =

∑

mP , ϕQ ψQ ,

l(Q)≤1

where and in what follows, we use ΦQ and ΨQ to replace, respectively, ϕQ and ψQ when l(Q) = 1. Let φ ∈ S (Rn ), A and B be operators associated with, respectively, the matrices {aQP }l(Q), l(P)≤1 ≡ {mP , ϕQ }l(Q), l(P)≤1 and {bQP }l(Q), l(P)≤1 ≡ {ψQ , φP }l(Q), l(P)≤1 . Then Corollary 3.1 implies that A and B are ε1 -almost diagonal. Then, by Theorem 3.1, A and B are bounded on as,p,τq(Rn ). Notice that the operators |A| and |B| associated with, respectively, the matrices {|aQP |}l(Q), l(P)≤1 ≡ {|mP , ϕQ |}l(Q), l(P)≤1 and

{|bQP |}l(Q), l(P)≤1 ≡ {|ψQ , φP |}l(Q), l(P)≤1 ,

are also ε1 -almost diagonal on as,p,τq (Rn ). We then obtain

∑ ∑

|aQP ||tP ||ψQ , φ | =

l(Q)≤1 l(P)≤1

∑ ∑

|aQP ||tP ||bQP0 |

l(Q)≤1 l(P)≤1

= |((|B||A|)(|t|))P0 | ≤ (|B||A|)(|t|)as,p,τq(Rn )  tas,p,τq(Rn ) , where P0 = [0, 1)n , and hence Tψ At =

∑ ∑

aQPtP ψQ =

l(Q)≤1 l(P)≤1

∑

l(P)≤1

tP

∑

l(Q)≤1

aQP ψQ =

∑

tP mP

l(P)≤1

holds in S  (Rn ). Then, applying Theorem 2.1, Corollary 3.1 and Theorem 3.1, we have ! ! ! ! ! ! ≤ Tψ AtAs,p,τq(Rn )  Atas,p,τq(Rn )  tas,p,τq(Rn ) . ! ∑ tP mP ! ! s, τ !l(P)≤1 A p, q (Rn )

58

3 Almost Diagonal Operators and Atomic and Molecular Decompositions

(ii) By Lemma 2.3 and Theorem 2.1, we see that  f , bQ  =

∑

tP ψP , bQ ,

l(P)≤1

where t ≡ {tP }l(P)≤1 ≡ { f , ϕP }l(P)≤1 satisfies tas,p,τq(Rn )   f As,p,τq(Rn ) . Let aQP ≡ ψP , bQ  and A be the operator associated with the matrix {aQP }l(Q), l(P)≤1 . We see that for all Q ∈ Q with l(Q) ≤ 1,  f , bQ  =

∑

aQPtP = (At)Q .

l(P)≤1

Then Corollary 3.1 and Theorem 3.1 yield that { f , bQ }Q as,p,τq (Rn ) = Atas,p,τq(Rn )  tas,p,τq(Rn ) , which completes the proof of Theorem 3.2.



Now we turn to the notion of a smooth atom for As,p,τq(Rn ); see [165, Definition 4.3] for the homogeneous case. Definition 3.3. A function aQ is called an inhomogeneous smooth atom for As,p,τq (Rn ) supported near a dyadic cube Q with l(Q) ≤ 1 if supp aQ ⊂ 3Q,

 Rn

xγ aQ (x) dx = 0

(3.13)

if |γ | ≤ max{J − n − s, −1} and l(Q) < 1, (3.14)

and ∂ γ aQ L∞ (Rn ) ≤ |Q|−1/2−|γ |/n

if |γ | ≤ max{s + nτ + 1, 0}.

(3.15)

A collection {aQ }l(Q)≤1 is called a family of inhomogeneous smooth atoms for As,p,τq (Rn ), if each aQ is an inhomogeneous smooth atom for As,p,τq(Rn ) supported near Q. Remark 3.1. We point out that in Definition 3.3, " the moment condition (3.14) of  and smooth atoms can be strengthened into that Rn xγ aQ (x) dx = 0 if |γ | ≤ N l(Q) < 1, the regularity condition (3.15) can be strengthened into that ∂ γ aQ L∞ (Rn ) ≤ |Q|−1/2−|γ |/n  where K  and N  are arbitrary fixed integer satisfying for all γ ∈ Zn+ with |γ | ≤ K,  ≥ max{s + nτ + 1, 0} and N  ≥ max{J − n − s, −1}. K

3.1 Smooth Atomic and Molecular Decompositions

59

We also remark that, unlike the homogeneous case, an inhomogeneous smooth atom for As,p,τq (Rn ) supported near a dyadic cube Q has vanishing moments only when l(Q) < 1. It is also clear that every inhomogeneous smooth atom for As,p,τq (Rn ) is a multiple of an inhomogeneous smooth synthesis molecule for As,p,τq (Rn ). We then have the following smooth atomic characterization of As,p,τq (Rn ) . Recall, the quantities σ p , σ p,q , τs,p and τs,p,q have been defined in (1.5)–(1.7). Theorem 3.3. Let s ∈ R and p, q ∈ (0, ∞] (p < ∞ if As,p,τq(Rn ) = Fp,s, qτ (Rn )). Let τ satisfy the inequality # 0≤τ <

τs,p if As,p,τq (Rn ) = Bs,p,τq (Rn ) , τs,p,q if As,p,τq (Rn ) = Fp,s, qτ (Rn ) .

(3.16)

Then for each f ∈ As,p,τq (Rn ), there exist a family {aQ }l(Q)≤1 of inhomogeneous smooth atoms for As,p,τq (Rn ), a sequence t ≡ {tQ }Q≤1 ⊂ C of coefficients such that f = ∑l(Q)≤1 tQ aQ in S  (Rn ), t ∈ as,p,τq (Rn ) and tas,p,τq(Rn ) ≤ C f As,p,τq(Rn ) , where C is a positive constant independent of t and f . Conversely, there exists a positive constant C such that for all families {aQ }l(Q)≤1 of inhomogeneous smooth atoms for As,p,τq (Rn ) and t ≡ {tQ }Q≤1 ∈ as,p,τq(Rn ), ! ! ! ! ! ! ! ∑ tQ aQ ! !l(Q)≤1 !

As,p,τq (Rn )

≤ Ctas,p,τq(Rn ) .

The proof of Theorem 3.3 is a simple modification of [165, Theorem 4.3]; see also [64, Theorem 4.1]. For completeness, we give the details. Proof. The second claim follows immediately from Theorem 3.2. We only need to prove the first one. Let Φ , ϕ , Ψ and ψ be as in (2.6). For f ∈ As,p,τq (Rn ), by Lemma 2.3, we write f = ∑l(Q)≤1 tQ ψQ in S  (Rn ), where t ≡ {tQ }l(Q)≤1 ≡ { f , ϕQ }l(Q)≤1 satisfies tas,p,τq(Rn )   f As,p,τq(Rn ) . Choose θ ∈ S (Rn ) such that supp θ ⊂ {x ∈ Rn : "  By (2.1) |x| ≤ 1}, |θ$(ξ )| ≥ C > 0 if 1/2 ≤ |ξ | ≤ 2 and Rn xγ θ (x) dx = 0 if |γ | ≤ N. n and (2.2), there exists a η ∈ S (R ) such that ψ = θ ∗ η . Set gk ≡

 Q0k

θ (· − y)η (y) dy

60

3 Almost Diagonal Operators and Atomic and Molecular Decompositions

for all k ∈ Zn . Then supp gk ⊂ 3Q0k ,

"

Rn x

γg

k (x) dx

 = 0 if |γ | ≤ N,

|∂ γ gk (x)| ≤ C(M, γ )(1 + |k|)−M for any M ∈ N and ψ = ∑k∈Zn gk . Thus, for all j ∈ Z+ and l ∈ Zn ,

ψQ jl ≡ |Q jl |−1/2

∑n gk (2 j x − l).

k∈Z

∗ For Q = Q jl with l(Q) ≤ 1, set rQ ≡ C(tmin{p,q}, λ ) and

aQ ≡ |Q|−1/2

∑n tQ jl gk−l (2 j x − l)/rQ,

l∈Z

where C is a positive constant and will be determined later. From the above representation of f , we deduce that f = ∑Q rQ aQ in S  (Rn ). Let λ > n, M and C be large enough. It is easy to check that each aQ is a smooth atom for As,p,τq (Rn ). Moreover, by Lemma 2.8, we see that ras,p,τq(Rn )  tas,p,τq(Rn )   f As,p,τq(Rn ) , which completes the proof of Theorem 3.3.



Remark 3.2. (i) Observe that in case s ≤ σ p the interval [0, 1/p] is always admissible. Furthermore, if s tends to infinity, also τ can become arbitrarily large. (ii) If τ = 0, Theorem 3.3 reduces to the known results on Bsp, q (Rn ) and Fp,s q(Rn ); see [23, 25, 64, 146]. Atomic decompositions have the essential advantage that they are localized, whereas the Fourier-analytic characterizations, used in the definition of the classes As,p,τq (Rn ), are not. We formulate a first simple consequence. By Asp, q (P) we denote the corresponding space on the cube P, defined by restrictions. We refer to [145,148] and Definition 6.2 below. Corollary 3.2. Let s, p, q and τ be as in Theorem 3.3. Then there exists a positive constant C such that for all f ∈ As,p,τq (Rn ), sup {P∈Q, |P|≥1}

1  f Asp, q(P) ≤ C f As,p,τq(Rn ) . |P|τ

Proof. For given f let ∑l(Q)≤1 tQ aQ be an optimal atomic decomposition of f in the sense of Theorem 3.3, i. e., f = ∑ tQ aQ l(Q)≤1

3.2 The Relation of As,p,τq (Rn ) to Besov-Triebel-Lizorkin-Morrey Spaces

61

and  f As,p,τq(Rn ) ∼ tas,p,τq(Rn ) and {aQ }l(Q)≤1 is a family of inhomogeneous smooth atoms for As,p,τq (Rn ). Now we fix a dyadic cube P such that |P| ≥ 1. Then

∑

f = fP ≡

tQ aQ

l(Q)≤1 3Q∩3P=0/

coincide on P. Define a sequence t P = {tQp }l(Q)≤1 by # tQP

≡

tQ 0

if 3Q ∩ 3P = 0/ , otherwise.

By applying the definition of Asp, q (P) and Theorem 3.3 with τ = 0 (observe that {aQ }l(Q)≤1 is a family of inhomogeneous smooth atoms for As,p,0q (Rn ) as well) we obtain  f Asp, q(P) =  f P Asp, q(P) ≤  f P Asp, q(Rn ) =  f P As, 0 (Rn )  t P as, 0 (Rn ) . p, q

p, q

Since ( jP ∨ 0) = 0 we find that 1 t P as, 0 (Rn ) ≤ tas,p,τq(Rn ) . p, q |4P|τ



This proves the claim.

Remark 3.3. Under the given restrictions in Corollary 3.2 we observe the following: any f ∈ As,p,τq(Rn ) belongs locally to Asp,q (Rn ) and the quasi-norm  f Asp,q (P) may only grow in dependence of |P| and τ .

τ n 3.2 The Relation of As, p, q (R ) to Besov-Triebel-Lizorkin-Morrey Spaces

We need another set of sequence spaces. Let p, q, s, τ be as in Definition 2.2. Define the spaces as,p,τq (Rn ) to be set of all {tQ }l(Q)≤1 ⊂ C satisfying tas,p,τq(Rn ) < ∞, where bs,p,τq(Rn ), when as,p,τq(Rn ) =  ⎧ ⎪ ⎨

⎤q/p ⎫1/q ⎪ ⎬ 1 ⎣ ⎝ j(s+n/2)q ⎠ ⎦ tbs, τ (Rn ) ≡ ∑ 2 sup |t | χ (x) dx , Q Q ∑ τq p, q ⎪ ⎪ P l(Q)=2− j P∈Q |P| ⎩ j=0 ⎭ ∞

⎡



⎛

⎞p

62

3 Almost Diagonal Operators and Atomic and Molecular Decompositions

s, τ n and when as,p,τq (Rn ) = fp, q (R ),

⎧ ⎡ ⎤ p/q ⎫1/p ⎪ ⎪  ⎬ ⎨ ∞ 1 j(s+n/2)q q ⎣∑ ∑ 2 ⎦ dx t fp,s, τq(Rn ) ≡ sup |t | χ (x) . Q Q τ ⎪ ⎪ P j=0 l(Q)=2− j P∈Q |P| ⎩ ⎭ There is an essential difference in the definitions of  · bs, τ (Rn ) and  · bs,p,τq(Rn ) . p, q The latter one does not contain an integration, it is totally discrete. The situation for the f -classes is different. Proposition 3.1. Let s ∈ R. (i) Let p ∈ (0, ∞). Let either τ ∈ [0, 1/p) and q ∈ (0, ∞] or τ = 1/p and q = ∞. s, τ n Then t fp,s, τq(Rn ) and t f p,s, τq(Rn ) are equivalent quasi-norms in f p, q (R ). (ii) If p ∈ (0, ∞] and τ ∈ [0, 1/p], then tbs, τ (Rn ) and tbs,p,τ∞(Rn ) are equivalent p, ∞

quasi-norms in bs,p,τ∞(Rn ); however, if q ∈ (0, ∞), then tbs, τ (Rn ) and tbs,p,τq(Rn ) p, q

are not equivalent quasi-norms in bs,p,τq(Rn ).

Proof. Step 1. Without restrictions concerning τ we have t f p,s, τq(Rn ) ≤ t fp,s, τq(Rn )

and

tbs,p,τq(Rn ) ≤ tbs, τ (Rn ) . p, q

Step 2. To prove t fp,s, τq(Rn ) and t f p,s, τq(Rn ) are equivalent, it suffices to show that for all dyadic cubes P with jP ≥ 1, ⎧ ⎡ ⎤ p/q ⎫1/p ⎪ ⎪  ⎬ ⎨ jP −1 1 ⎣ ∑ 2 j(s+n/2)q ∑ |tQ |q χQ (x)⎦ dx IP ≡  t f p,s, τq(Rn ) . ⎪ |P|τ ⎪ ⎭ ⎩ P j=0 l(Q)=2− j Indeed, for each j = 0, · · · , jP − 1, there exists a unique Q j ∈ Q such that l(Q j ) = 2− j and P ⊂ Q j . Then, by geometric properties of dyadic cubes, τ < 1/p and |tQ j | ≤ t f p,s, τq(Rn ) |Q j |s/n+1/2+τ −1/p, we have IP

1 = |P|τ



∑ 2 j(s+n/2)q|P|q/p|tQ j |q

j=0

1 ≤ t f p,s, τq(Rn ) τ |P|  t f p,s, τq(Rn ) , which yields the claim.

1/q

jP −1



jP −1

∑2

j=0

1/q − jn(τ −1/p)q

|P|

q/p

3.2 The Relation of As,p,τq (Rn ) to Besov-Triebel-Lizorkin-Morrey Spaces

63

Step 3. Next we consider the Besov-type spaces. Using the same type of arguments as in Step 2 it is easy to see that the converse inequality tbs, τ

n p, ∞(R )

 tbs,p,τ∞(Rn )

holds as long as τ ≤ 1/p. However, if q ∈ (0, ∞), the converse inequality does not hold. In fact, for all j ∈ Z+ , set R j ≡ [2− j , 2− j+1 )n . Define a sequence {tQ }l(Q)≤1 by setting tR j ≡ 2− j(s+n/2+nτ −n/p) for all j ∈ Z+ and tQ ≡ 0 otherwise. Then, similarly to the arguments in [127, Theorem 1.1], it is easy to check that tbs,p,τq(Rn ) < ∞ but tbs, τ (Rn ) = ∞, which p, q completes the proof of Proposition 3.1.

From the proof of Proposition 3.1 we also deduce a further equivalent quasi-norm of  f bs,p,τq(Rn ) . Lemma 3.3. Let s ∈ R, p, q ∈ (0, ∞] and τ ∈ [0, 1/p). Then ⎧ ⎡ ⎛ ⎞ p ⎤q/p ⎫1/q ⎪ ⎪  ⎨ ⎬ ∞ 1 j(s+n/2)q ⎣ ⎝ ⎠ ⎦ tbs,p,τq(Rn ) ∼ sup 2 |t | χ (x) dx . ∑ ∑ Q Q τ ⎪ P l(Q)=2− j P∈Q |P| ⎪ ⎩ j=0 ⎭ In Sect. 1.3 we recalled the definition of Morrey spaces Mup (Rn ) (see item s (Rn ) (see item (xxv)) and Triebel-Lizorkin(xvi)), Besov-Morrey spaces N pqu s n s (Rn ) denote either the BesovMorrey spaces E pqu(R ) (see item (xxvi)). Let A pqu s n s Morrey space N pqu (R ) or the Triebel-Lizorkin-Morrey space E pqu (Rn ) (p = ∞). Sawano and Tanaka [126] presented various decompositions including quarkonial, s (Rn ). Now it is easy to compare the atomic and molecular characterizations of A pqu s, τ s (Rn ). We only need to compare the associated sequence spaces A p, q(Rn ) and A pqu 0, 1/u−1/p

spaces. Notice that au, q (Rn ) is just the sequence space a pqu(Rn ) defined in 0, 1/u−1/p [126, Definition 4.2]. There it has been shown that au, q (Rn ) is the sequence space of all admissible coefficient sequences of atomic decompositions of funcs (Rn ); see [126, Theorem 4.9]. Arguing as in [127, Theorem 1.1], tions f ∈ A pqu taking into account also the wavelet characterizations of Besov-Morrey and TriebelLizorkin-Morrey spaces (see [123] and [126, Theorem 3.9]), Proposition 3.1 and Theorem 3.3 yield the following. Corollary 3.3. Let s ∈ R. (i) If q ∈ (0, ∞] and 0 < u ≤ p ≤ ∞, u = ∞, then s, 1/u−1/p

Fu, q

s (Rn ) = E pqu (Rn )

64

3 Almost Diagonal Operators and Atomic and Molecular Decompositions

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

Fu, 2

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

holds with equivalent norms. (ii) If 0 < u = p ≤ ∞ and q ∈ (0, ∞], then s, 1/u−1/p

Bu, q

s (Rn ) = N pqu (Rn ) = Bsp,q (Rn )

with equivalent quasi-norms. If 0 < u ≤ p ≤ ∞, then s, 1/u−1/p

Bu, ∞

(Rn ) = N ps∞u(Rn )

with equivalent quasi-norms. If 0 < u < p < ∞ and q ∈ (0, ∞), then s, 1/u−1/p

s N pqu (Rn ) ⊂ Bu, q

and the embedding is proper.

(Rn )

Chapter 4

Several Equivalent Characterizations

Using the characterizations of As,p,τq (Rn ) by atoms and molecules obtained in Chap. 3, in this chapter, for certain p, q, s, τ , we establish characterizations of As,p,τq (Rn ) by wavelets, differences and oscillations (local approximation by polynomials). In addition we consider a localization property of As,p,τq(Rn ) by using Asp, q (Rn ).

4.1 Preparations Here we collect some assertions which will be of some use for us when establishing characterizations of As,p,τq (Rn ) by wavelets, differences and oscillations.

4.1.1 An Equivalent Definition In this section, we give an equivalent definition of As,p,τq(Rn ) with the conditions on Φ and ϕ different from those in Definition 2.1. The following Definition 4.1 is more convenient than Definition 2.1 in applications. Definition 4.1. Let L ∈ (0, 1/2] and Θ (Rn , L) be the set of all sequences {φ j }∞j=0 ⊂ S (Rn ) such that for all j ∈ N, supp φ0 ⊂ {ξ ∈ Rn : |ξ | ≤ 2} and supp φj ⊂ {ξ ∈ Rn : 2 j L ≤ |ξ | ≤ 2 j+1 }, (4.1) and that for every M ∈ N, there exists a positive constant C(n, M) such that for all j ∈ Z+ and ξ ∈ Rn , |φ j (ξ )| ≤ C(n, M)

2 jn (1 + 2 j |ξ |)n+M

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 4, c Springer-Verlag Berlin Heidelberg 2010 

(4.2)

65

66

4 Several Equivalent Characterizations

and

∞

∑ φj (ξ ) = 1

for all ξ ∈ Rn .

(4.3)

j=0

It was proved in [145, p. 45] that Θ (Rn , L) is not empty. Replacing {Φ , ϕ j : j ∈ N} in Definition 2.1 by {φ j }∞j=0 ∈ Θ (Rn , L), we then obtain two classes of spaces, denoted respectively by Bs,p,τq (Θ (Rn , L)) and Fp,s, qτ (Θ (Rn , L)). Similarly, we use As,p,τq (Θ (Rn , L)) to denote either Bs,p,τq(Θ (Rn , L)) or Fp,s, qτ (Θ (Rn , L)), and use  · As,p,τq(Θ (Rn ,L)) to denote the quasi-norm of As,p,τq(Θ (Rn , L)). Lemma 4.1. The space As,p,τq(Θ (Rn , L)) is independent of the choices of L ∈ (0, 1/2] and {φ j }∞j=0 ∈ Θ (Rn , L). Proof. Let L1 , L2 ∈ (0, 1/2], {φ j }∞j=0 ∈ Θ (Rn , L1 ) and {ψ j }∞j=0 ∈ Θ (Rn , L2 ). By symmetry, we only need to show that for all f ∈ As,p,τq (Θ (Rn , L2 )),  f As,p,τq(Θ (Rn ,L1 ))   f As,p,τq(Θ (Rn ,L2 )) .

(4.4)

Set ψ−r ≡ 0 for all r ∈ N. From (4.1), it follows that for all j ∈ Z+ , log2 L2  2

∑L

φj =

φ j ∗ ψ j+r ,

(4.5)

r= log2 21

where log2 L21 and log2 L22  respectively denote the maximal integer no more than

log2 L21 and the minimal integer no less than log2 L22 . Thus,  f Bs,p,τq(Θ (Rn ,L1 )) log2 L2  2



∑L

r= log2 21



1 sup τ |P| P∈Q

∞

∑

j=( jP ∨0)

2 jsq

 P

|φ j ∗ ψ j+r ∗ f (x)| p dx

q/p1/q .

When r = log2 L21 , · · · , 0, then j + r ≤ j. Applying [65, Lemma (6.10)], for all x ∈ Rn , we have

φ j ∗ ψ j+r ∗ f (x) =

∑n 2− jnφ j (x − 2− j k) ψ j+r ∗ f (2− j k),

k∈Z

which together with the fact φ ∈ S (Rn ) yields that

IP

1 ≡ |P|τ



∞

∑

j=( jP ∨0)

 2 jsq P

 q  1q |φ j ∗ ψ j+r ∗ f (x)| p dx

p

(4.6)

4.1 Preparations

67

⎧ ⎡ ⎤ q ⎫ 1q p⎪ ⎪  p ⎪ ⎪  ⎬ |ψ j+r ∗ f (2− j k)| 1 ⎨ ∞ ⎥ jsq ⎢  2 dx , ⎣ ⎦ ∑ ∑ ∑ j −j n+M ⎪ |P|τ ⎪ Q jl k∈Zn (1 + 2 |x − 2 k|) ⎪ ⎪ l∈Zn ⎩ j=( jP ∨0) ⎭ Q ⊂P jl

where M ∈ N will be determined later. Let δ ∈ (0, min{1, p, q}). Since 1 + 2 j |x − 2− j k| ∼ 1 + 2 j |2− j l − 2− j k| for all l, k ∈ Zn , j ∈ Z+ and x ∈ Q jl , then for each j, applying [64, Lemma A. 4] to the function ψ j+r ∗ f (2− j y) and (2.11), we find that there exists a natural number γ such that for all x ∈ Q jl , |ψ j+r ∗ f (2− j k)|

∑n (1 + 2 j |x − 2− j k|)n+M

k∈Z



1/δ 1  ∑ sup |ψ j+r ∗ f (z)|δ j −j −j (n+M)δ z∈Q jk k∈Zn (1 + 2 |2 l − 2 k|)   ⎫1/δ ⎧ ⎪ ⎪ ⎪  ⊂ Q jk , l(Q)  = 2−γ l(Q jk ) ⎪ ⎪ ⎪ max inf |ψ j+r ∗ f (z)|δ : Q ⎪ ⎪ ⎬ ⎨  z∈Q  ∑ ⎪ ⎪ (1 + |l − k|)(n+M)δ ⎪ ⎪ k∈Zn ⎪ ⎪ ⎪ ⎪ ⎭ ⎩  

2 jn ∑n (1 + |l − k|)(n+M)δ k∈Z

1/δ

 Q jk

|ψ j+r ∗ f (z)|δ dz

.

(4.7)

Noticing that 3P is a union of dyadic cubes, by (4.7), we have

IP

⎧ ⎡ ⎪ ⎨ ∞ 1 ⎢  ∑ 2 jsq ⎣ |P|τ ⎪ ⎩ j=( j ∨0) P

×

2 jn (1 + |l − k|)(n+M)δ

∑

l∈Zn Q jl ⊂P



 Q jl

∑

k∈Zn

χ{k∈Zn : Q jk ⊂3P} (k)

⎤ q ⎫ 1q p⎪ p ⎪   ⎬ δ ⎥ δ |ψ j+r ∗ f (z)| dz dx⎦ ⎪ Q jk ⎪ ⎭

⎧ ⎡ ⎤ q ⎫ 1q p p⎪ ⎪   ⎪ ⎪  ⎬ δ 1 ⎨ ∞ ⎥ jsq ⎢ n + τ 2 χ (k) · · · dx ⎣ ⎦ ∑ Q ∑n {k∈Z : Q jk ∩3P=0}/ ⎪ j=(∑ ⎪ |P| ⎪ jl ⎪ l∈Zn k∈Z ⎩ jP ∨0) ⎭ Q ⊂P jl

≡ I1 + I2 .

68

4 Several Equivalent Characterizations

Since 3P ⊂ ∪{i∈Zn : |i|≤√n} (P + il(P)), where P + il(P) = {x + il(P) : x ∈ P}, if we choose M > max{(1/δ −1)n, n(p/(δ q)−1)}, by p/δ > 1, H¨older’s inequality p and the L δ (Rn )-boundedness of the Hardy-Littlewood operator M, we have ⎧ ⎡  ⎪  ⎨ ∞ 1 ⎢ 2 jsq ⎣ ∑ I1  ∑ ∑ χ{k∈Zn: Q jk ⊂(P+il(P))}(k) τ⎪ ∑ Q jl k∈Zn i∈Z√n |P| ⎩ j=( jP ∨0) l∈Zn |i|≤ n Q jl ⊂P ⎤q/p ⎫1/q ⎪ ⎪   p/ δ ⎬ 2 jn ⎥ δ | ψ ∗ f (z)| χ (z) dz dx × ⎦ j+r P+il(P) ⎪ (1 + |l − k|)(n+M)δ Q jk ⎪ ⎭ ⎧ ⎡  ⎪  χ{k∈Zn : Q jk ⊂(P+il(P))}(k) 1 ⎨ ∞ ⎢ jsq  ∑ 2 ⎣ ∑ ∑ ∑ τ⎪ (1 + |l − k|)(n+M)δ Q jl k∈Zn i∈Z√n |P| ⎩ j=( jP ∨0) l∈Zn |i|≤ n

Q jl ⊂P

|i|≤ n

Q jl ⊂P

⎤q/p ⎫1/q ⎪ ⎪ ⎬  p/δ ⎥ δ −j ×M(|ψ j+r ∗ f | χP+il(P) )(x + (k − l)2 ) dx⎦ ⎪ ⎪ ⎭ ⎧ ⎡ ⎪  χ{k∈Zn : Q jk ⊂(P+il(P))}(k) 1 ⎨ ∞ jsq ⎢  ∑ 2 ⎣ ∑ ∑ ∑ τ ⎪ (1 + |l − k|)(n+M)δ Q jl k∈Zn i∈Z√n |P| ⎩ j=( jP ∨0) l∈Zn ⎤q/p⎫1/q ⎪ ⎪ ⎬   p/δ ⎥ δ −j × M(|ψ j+r ∗ f | χP+il(P))(x + (k − l)2 ) dx⎦ ⎪ ⎪ ⎭   ∞ 1  ∑ 2 jsq ∑ τ P+il(P) i∈Z√n |P| j=( j ∨0) P

|i|≤ n

  p/δ × M(|ψ j+r ∗ f |δ χP+il(P))(x) dx



∑

i∈Z√n |i|≤ n

1 |P|τ



∞

∑

j=( jP ∨0)

2

jsq



q/p ⎫1/q ⎬ ⎭

|ψ j+r ∗ f (x)| dx

q/p 1/q

p

P+il(P)

  f Bs,p,τq (Θ (Rn ,L2 )) . / then Q jk ⊂ P + il(P) for a unique i ∈ Zn For I2 , we remark that if Q jk ∩ 3P = 0, j with |i| ≥ 2; moreover, 1 + |l − k| ∼ 2 |i|l(P) for any l ∈ Zn satisfying Q jl ⊂ P. Similarly to the estimate of I1 , by H¨older’s inequality, we obtain

4.1 Preparations

I2

69

⎧ ⎡ ⎪ ⎨ ∞ 1 ⎢  ∑ 2 jsq ⎣ |P|τ ⎪ ⎩ j=( j ∨0) P

×



∑

Q jl

l∈Zn Q jl ⊂P

∑

{i∈Zn : |i|≥2} k∈Zn

χ{k∈Zn : Q jk ⊂(P+il(P))}(k)2 jn  p/δ

 2( jP − j)(n+M)δ

|i|(n+M)δ

∑



Q jk

⎧ ⎡ ⎪ ⎨ ∞ 1 ⎢  2 jsq ⎣ ⎪ j=(∑ |P|τ ⎩ j ∨0) P

|ψ j+r ∗ f (z)|δ χP+il(P)(z) dz

∑





Q jl

l∈Zn Q jl ⊂P

⎤q/p⎫1/q ⎪ ⎪ ⎬ ⎥ dx⎦ ⎪ ⎪ ⎭

∑

∑

{i∈Zn : |i|≥2} k∈Zn

χ{k∈Zn : Q jk ⊂(P+il(P))}(k)  p/δ

×

2( jP − j)(n+M)δ M(|ψ j+r ∗ f |δ χP+il(P) )(x + (k − l)2− j ) |i|(n+M)δ

⎧ ⎡ ⎪ ⎨ ∞ 1 ⎢  ∑ 2 jsq ⎣ |P|τ ⎪ ⎩ j=( j ∨0) P

×

2( jP − j)(n+M)δ |i|(n+M)δ

1  |P|τ ×



∞

∑ 

P+il(P)

∑

∑

χ{k∈Zn : Q jk ⊂(P+il(P))}(k)

Q jl {i∈Zn : |i|≥2} k∈Zn

l∈Zn Q jl ⊂P

⎤q/p⎫1/q ⎪ ⎪ ⎬   p/δ ⎥ δ −j M(|ψ j+r ∗ f | χP+il(P))(x + (k − l)2 ) dx⎦ ⎪ ⎪ ⎭ 

∑ n

2 jsq

j=( jP ∨0)



∑



⎤q/p ⎫1/q ⎪ ⎪ ⎬ ⎥ dx⎦ ⎪ ⎪ ⎭

|i|−(n+M)δ

{i∈Z : |i|≥2}

 p/δ M(|ψ j+r ∗ f |δ χP+il(P) )(x) dx

q/p ⎫1/q ⎬ ⎭

,

p

which by (2.11) or H¨older’s inequality, p/δ > 1 and the L δ (Rn )-boundedness of the Hardy-Littlewood operator M, we have 1 I2  |P|τ ×  f



∑ n

|i|−(n+M)δ min{q/p,1}

{i∈Z : |i|≥2}

 P+il(P)

2 jsq

j=( jP ∨0)

  p/δ q/p δ M(|ψ j+r ∗ f | χP+il(P))(x) dx

Bs,p,τq (Θ (Rn ,L2 ))

.

Thus, IP   f Bs,p,τq(Θ (Rn ,L2 )) for all r = log2 L21 , · · · , 0.

∞

∑

1/q

70

4 Several Equivalent Characterizations

When r = 1, · · · , log2 L22 , the proof is similar. In fact, in this case, we use [65, Lemma (6.10)] again to obtain that for all x ∈ Rn ,

φ j ∗ ψ j+r ∗ f (x) =

∑n 2−( j+r)nφ j (x − 2− j−r k) ψ j+r ∗ f (2− j−r k).

(4.8)

k∈Z

With (4.6) replaced by (4.8), following the previous arguments, we also obtain IP   f Bs,p,τq(Θ (Rn ,L2 )) , which further yields  f Bs,p,τq(Θ (Rn ,L1 ))   f Bs,p,τq(Θ (Rn ,L2 )) . For the Triebel-Lizorkin-type spaces, applying (4.5) again, yields  f Fp,s, qτ (Θ (Rn ,L1 )) log2 L2 



∑

2

L r= log2 21

⎧   qp ⎫ 1p  ⎬ ∞ 1 ⎨ jsq q sup 2 | φ ∗ ψ ∗ f (x)| dx . j j+r ∑ τ ⎭ P∈Q |P| ⎩ P j=( j ∨0) P

Similarly, when r = log2 L21 , · · · , 0, by (4.6) and (4.7), we see that ⎧   p/q ⎫1/p  ⎬ ∞ 1 ⎨ jsq q JP ≡ 2 | φ ∗ ψ ∗ f (x)| dx j j+r ⎭ |P|τ ⎩ P j=(∑ jP ∨0) ⎧ ⎡  ⎪ ∞ χ{k∈Zn : Q jk ⊂3P} (k) 1 ⎨ ⎢ jsq  χQ jl (x) ∑ ⎣ ∑ 2 ∑ τ (n+M)δ n ⎪ |P| ⎩ R j=( j ∨0) l∈Zn k∈Zn (1 + |l − k|) P Q jl ⊂P

 × max

 q/δ  ⊂ Q jk , l(Q)  = 2−γ l(Q jk ) inf |ψ j+r ∗ f (z)|δ : Q

 z∈Q

⎧ ⎡ ⎛ ⎪  ⎨ ∞ 1 ⎢ ⎜ + τ 2 jsq ⎝ ∑ χ{k∈Zn : Q jk ∩3P=0} ⎣ / (k) ∑ n |P| ⎪ R ⎩ k∈Zn j=( j ∨0) P

×

2 jn (1 + |l − k|)(n+M)δ

≡ J1 + J2 .

∑

l∈Zn Q jl ⊂P

⎥ ⎦

χQ jl (x)

⎫ ⎞q/δ ⎤ p/q ⎪1/p ⎪ ⎪  ⎬ ⎟ ⎥ δ |ψ j+r ∗ f (z)| dz⎠ ⎥ dx ⎦ ⎪ Q jk ⎪ ⎪ ⎭

⎫1/p ⎪ ⎪ ⎬

⎤ p/q dx

⎪ ⎪ ⎭

4.1 Preparations

Define

71

 tQ jk ≡ max

  ⊂ Q jk , l(Q)  = 2 l(Q jk ) inf |ψ j+r ∗ f (z)| : Q δ

−γ

 z∈Q

if Q jk ⊂ 3P and tQ jk ≡ 0 otherwise. Then by [64, Lemma A. 2], if M > n(1/δ − 1), we obtain that for each x ∈ Q jl ,   χ{k∈Zn : Q jk ⊂3P} (k) t  M ∑ tQ jk χQ jk (x). ∑ (n+M)δ Q jk k∈Zn (1 + |l − k|) k∈Zn Thus, choosing

M > max{(1/δ − 1)n, n(q/(δ p) − 1)}

and applying the Fefferman-Stein vector-valued maximal inequality, we have ⎫ ⎧ ⎡    q/δ ⎤ p/q ⎪1/p ⎪  ⎬ ⎨ ∞ 1 ⎣ ∑ 2 jsq M ∑ tQ χQ ⎦ dx (x) J1  jk jk ⎪ |P|τ ⎪ ⎭ ⎩ Rn j=( jP ∨0) k∈Zn   f Fp,s, τq (Θ (Rn ,L2 )) . Similarly to the estimate of I2 , by (2.11) or H¨older’s inequality, p/δ > 1, q/δ > 1 and the Fefferman-Stein vector-valued maximal inequality, we see that ⎧ ⎡ ⎛ ⎪  ⎨ ∞ 1 ⎢ ⎜ J2  ⎣ ∑ 2 jsq ⎝ ∑ ∑ χ{k∈Zn : Q jk ⊂(P+il(P))}(k) n |P|τ ⎪ R ⎩ j=( jP ∨0) {i∈Zn : |i|≥2} k∈Zn ⎫1/p ⎤ q/δ p/q ⎪ ⎪  ⎬ 2( jP − j)(n+M)δ jn ⎥ δ × ∑ χQ jl (x) 2 | ψ ∗ f (z)| dz dx ⎦ j+r ⎪ |i|(n+M)δ Q jk ⎪ l∈Zn ⎭ Q jl ⊂P ⎧ ⎡ ⎛ ⎪ ∞ 1 ⎨ ⎢ ⎜  2 jsq ⎝ |i|−(n+M)δ ⎣ ∑ ∑ |P|τ ⎪ n ⎩ Rn j=( jP ∨0) {i∈Z : |i|≥2} ⎫ ⎞q/δ ⎤ p/q ⎪1/p ⎪ ⎪ ⎬ ⎟ ⎥ δ ⎥ ×M(|ψ j+r ∗ f | χP+il(P) )(x + il(P))⎠ ⎦ dx ⎪ ⎪ ⎪ ⎭   f Fp,s, qτ (Θ (Rn ,L2 )) . A similar argument also holds for r = 1, · · · , log2 L22  if we replace (4.6) by (4.8). Thus, we obtain  f Fp,s, τq (Θ (Rn ,L1 ))   f Fp,s, τq (Θ (Rn ,L2 )) , which completes the proof of Lemma 4.1.

 

72

4 Several Equivalent Characterizations

From Lemma 4.1, we immediately deduce that the spaces As,p,τq(Θ (Rn , L)) and coincide.

As,p,τq (Rn )

Corollary 4.1. Let L ∈ (0, 1/2] and p, q, s, τ be as in Definition 2.1. Then we have the coincidence of the spaces As,p,τq (Rn ) and As,p,τq (Θ (Rn , L)) in the sense of equivalent quasi-norms. Proof. It suffices to construct a pair of Schwartz functions Φ and ϕ such that {Φ , ϕ j : j ∈ N} ∈ Θ (Rn , 1/2) and satisfy, respectively, (2.1) and (2.2).  ≥ 0, supp ψ  ⊂ {ξ ∈ Rn : 1/2 ≤ |ξ | ≤ 2} and Indeed, let ψ ∈ S (Rn ) such that ψ  (ξ ) ≥ C > 0 if 3/5 ≤ |ξ | ≤ 5/3, where C is a positive constant independent of ξ . ψ Define ϕ and Φ by setting   (ξ ) ϕ(ξ ) ≡ ψ

−1

∑ ψ (2 ξ ) −i

∞

 (ξ ) ≡ 1 − ∑ ϕ j (ξ ). and Φ j=1

i∈Z

It is easy to check that ϕ and Φ are desired functions, which completes the proof of Corollary 4.1.  

4.1.2 Several Technical Lemmas on Differences To determine the relation between the Littlewood-Paley characterization and characterization of As,p,τq (Rn ) by differences, we establish some technical lemmas first.  ≡ 1 on {ξ ∈ Rn : |ξ | ≤ 2L} and Let L ∈ (0, 1/2] and Φ ∈ S (Rn ) such that Φ  ⊂ {ξ ∈ Rn : |ξ | ≤ 2}. supp Φ Define ϕ j by setting

 (2− j ξ ) − Φ  (2− j+1ξ ) ϕ j (ξ ) = Φ

for all j ∈ N and ξ ∈ Rn . Then {Φ , ϕ j : j ∈ N} ∈ Θ (Rn , L). We then let 1  f  ≡ sup s, τ τ B p, q (Rn ) |P| P∈Q



∞

∑

2 jsq

j=( jP ∨0)

 P

| f (x) − Φ j ∗ f (x)| p dx

q/p 1/q

and ⎧   p/q ⎫1/p ⎨ ⎬ ∞ 1 jsq q  f  ≡ sup 2 | f (x) − Φ ∗ f (x)| dx . s, τ j ∑ τ Fp, q (Rn ) ⎭ P∈Q |P| ⎩ P j=( j ∨0) P

Notice that ϕ j = Φ j − Φ j−1 for all j ≥ 1. From this, it is easy to deduce the following lemma. We omit the details.

4.1 Preparations

73

Lemma 4.2. Let s ∈ R, τ ∈ [0, ∞) and p, q ∈ (0, ∞]. Then there exists a positive constant C such that  1/p  # 1  p sup |Φ ∗ f (x)| dx .  f As,p,τq (Rn ) ≤ C  f As, τ (Rn ) + τ p, q P {P∈Q: l(P)≥1} |P| Next, using a trick of Nikol’skij [109, Sect. 5.2.1], we obtain a representation of f − Φ j ∗ f as an integral mean of differences of f . Recall that for all x, h ∈ Rn ,

ΔhM f (x) ≡

M

∑ (−1) j

#

j=0

M j

 f (x + (M − j)h).

(4.9)

Let ψ ∈ S (Rn ) such that  (x) = 1 if |x| ≤ 1 and ψ  (x) = 0 if |x| ≥ 2.  (x) ≥ 0, ψ ψ For a fixed M ∈ N, we define Φ by setting, for all ξ ∈ Rn ,  (ξ ) ≡ (−1)M+1 Φ

M−1 #

∑

j=0

M j

  ((M − j)ξ ). (−1) j ψ

(4.10)

Then the function Φ satisfies  ⊂ {ξ ∈ Rn : |ξ | ≤ 2}. supp Φ

 (ξ ) = 1 if |ξ | ≤ 1/M and Φ Define ϕ j by setting

 (2− j ξ ) − Φ  (2− j+1ξ ) ϕ j (ξ ) ≡ Φ for all j ∈ N and ξ ∈ Rn . Then {Φ , ϕ j : j ∈ N} ∈ Θ (Rn , 1/(2M)) and for all f ∈ Lτp (Rn ) with p ∈ [1, ∞] and j ∈ Z+ , f (x) − Φ j ∗ f (x) = (−1)M

 Rn

  M Δ−2 f (x) ψ (y) dy. − jy

(4.11)

Denote by L1loc (Rn ) the set of all locally integrable functions on Rn . For f ∈ P ∈ Q and N ∈ N, we define

L1loc (Rn ),

at (x) ≡ t −n

 t/2≤|h|
|ΔhM f (x)| dh

(4.12)

74

4 Several Equivalent Characterizations

and ⎛

TN,P

∞

2(n+s−N)md ≡⎝ ∑ |P|τ d m=( j ∨0)



⎜ TN,P ≡ ⎝

∞

m=( jP

P

⎧  ⎨ 

2(n+s−N)md ⎩ |P|τ d ∨0)

∑

t −sq

#

1

P

⎛

2m−( jP ∨0)+2

P

[at (x)] p dx

 qp

dt t

 dq ⎞ d1 ⎠ ;

(4.13) 1 ⎞ d ⎫ d p p m−( j ∨0)+2 q ⎬ P 2 dt ⎟ t −sq [at (x)]q dx ⎠ . ⎭ t 1

Here s, p, q and d will be fixed later on. Recall, p = max{1, p}. Lemma 4.3. Let q ∈ (0, ∞], s ∈ [0, ∞), τ ∈ [0, ∞) and P be a dyadic cube. Then there exist a positive constant C and N ∈ N, independent of P, such that for all p ∈ (0, ∞] and f ∈ Lτp (Rn ), 1 IP ≡ |P|τ



#

∞

∑

2

jsq P

j=( jP ∨0)

1 ≤C sup τ {Q∈Q: P⊂Q} |Q|

| f (x) − Φ j ∗ f (x)| dx

q/p 1/q

p



2(l(Q)∧1)

t

−sq

q/p

#

[at (x)] dx p

Q

0

dt t

1/q + C TN,P ,

and that for all p ∈ (0, ∞) and f ∈ Lτp (Rn ), ⎧   p/q ⎫1/p  ⎬ ∞ 1 ⎨ jsq q JP ≡ 2 | f (x) − Φ ∗ f (x)| dx j ⎭ |P|τ ⎩ P j=(∑ j ∨0) P

1 ≤C sup τ {Q∈Q: P⊂Q} |Q|

 #

2(l(Q)∧1)

t Q

−sq

0

dt [at (x)] t

 p/q

q

1/p + C TN,P .

dx

Proof. For a dyadic cube P we define 1 sup AP ≡ τ {Q∈Q: P⊂Q} |Q|



2(l(Q)∧1)

t

−sq

q/p

#

[at (x)] dx p

0

Q

dt t

1/q ,

and 1 P ≡ A sup τ |Q| {Q∈Q: P⊂Q}

 #

2(l(Q)∧1)

t Q

0

−sq

dt [at (x)] t q

1/p

 p/q dx

.

4.1 Preparations

75

Step 1. Estimate of IP . Using (4.11) and then splitting the integral into the regions |y| ≥ 1 and |y| < 1, we see that 1 IP  |P|τ



 #

∞

∑

2

jsq

1 + τ |P|



|y|<1

P

j=( jP ∨0)

|Δ2M− j y f (x)||ψ (y)| dy

 #

∞

∑

2 jsq P

j=( jP ∨0)

|y|≥1

q/p 1/q

p dx

 p q/p 1/q · · · dx

≡ I1 + I2 . Since ψ ∈ S (Rn ), we have |ψ (y)| ≤ C(N)(1 + |y|)−N for all y ∈ Rn , where N ∈ N is at our disposal and will be chosen later on and C(N) is a positive constant depending on N. Hence ⎧  p q/p ⎫1/q   ⎬ ∞ 1 ⎨ ∞ jsq −mn I1  2 2 a (x) dx −m− j ∑ ∑ 2 ⎭ |P|τ ⎩ j=( j ∨0) P m=0 P

and ⎧  p q/p ⎫1/q   ⎬ ∞ 1 ⎨ ∞ jsq m(n−N) I2  2 2 a (x) dx . m− j+1 ∑ 2 ⎭ |P|τ ⎩ j=(∑ P m=0 j ∨0) P

Step 1.1. Estimate of I1 . Observe that for all x ∈ Rn , l ∈ Z and 2l− j ≤ t ≤ 2l− j+1 , a2l− j (x) ≤ 2n (at (x) + at/2 (x)).

(4.14)

Furthermore, recall the elementary inequality ⎛

$ $q $ ∞ $ $ ⎝ ∑ $ ∑ fm, j $ $ $ $ p j=0 m=0

⎞d/q

∞

⎠

≤

∞



∑ ∑

m=0

L (Ω )

∞

j=0

d/q  fm, j qL p (Ω )

,

(4.15)

valid for all sequences { fm, j }m, j∈N of locally integrable functions, Ω a measurable subset of Rn and with d = min{1, p, q}. This yields 1 (I1 )  |P|τ d d



1 |P|τ d

∞

∑2

 −mnd

m=0 ∞

∑ 2−mnd

m=0

#

∞

∑

j=( jP ∨0)

2

jsq

q/p d/q [a2−m− j (x)] dx p

P

76

4 Several Equivalent Characterizations



∞

∑

×

2

jsq

j=( jP ∨0)



2−m− j

1 ∞ −m(n+s)d ∑2 |P|τ d m=0  −m−( j ∨0)+1 2

×

P

t

q/p

#

−sq

2−( jP ∨0)+1

t

[at (x) + at/2 (x)] dx p

P



[at (x) + at/2 (x)] dx p

P

0

1  |P|τ d

q/p

 2−m− j+1 #

#

−sq

0

P

q/p [at (x)] dx p

dt t

dt t

dt t

d/q

d/q

d/q

 (AP )d . Step 1.2. Estimate of I2 . Applying again (4.14) and (4.15) we find

×

∑

2

∞

∑

(n−N)md

2 jsq q/p

[at (x) + at/2 (x)] dx p

P

∞

∑2

 2m− j+2

j=( jP ∨0)

m=0

#

1  |P|τ d



∞

1 (I2 )  |P|τ d d

(n−N+s)md



dt t

2m− j+1

d/q

2m−( jP ∨0)+2

t

#

−sq

0

m=0

P

q/p [at (x)] dx p

dt t

d/q .

Therefore, for the case jP ≥ 1, it holds (I2 )d ≤ I12 + I22 + I32 , where

I12

I22

I23

1 ≡ |P|τ d 1 ≡ |P|τ d 1 ≡ |P|τ d

∞

∑2

(n−N+s)md



∑2

(n−N+s)md

∞

∑

2



(n−N+s)md

2m− jP +2



q/p

#

2m− jP +2 2l(P)

[at (x)] dx p

P

2l(P)

m=0

m= jP

t

−sq

0

m=0

jP −1

2l(P)

dt ··· t dt ··· t

dt t

d/q ,

d/q , d/q .

Choose N > n + s + nτ . The fact I12  (AP )d is trivial. To estimate the others, let Pm be the dyadic cube containing P and having side length 2m l(P). Then, under the above condition on N, we obtain

4.1 Preparations

I22

77

1  |P|τ d  (AP )d



jP −1

∑2

m+1  2l(Pi )

∑

(n+s−N)md

m=0



jP −1

∑

2(n+s−N)md

m=0

t

l(Pi )

i=1

−sq

q/p

#

[at (x)] dx p

Pi

dt t

d/q

d/q

m+1

∑ 2inτ q

i=0

 (AP )d . Finally, we split I32 once again. Using our abbreviation TN,P we get I32

1  |P|τ d

∞

∑

2

(n+s−N)md



1 + τd |P|  (AP )d

∞

∑

2

(n+s−N)md





∑

P

2m− jP +2

dt ··· t

1

m= jP ∞

[at (x)] dx p

2l(P)

m= jP

q/p

#

1

d/q

d/q

d/q

jP −1

∑ 2inτ q

2(n+s−N)md

m= jP

dt t

+ (TN,P )d

i=1

 (AP )d + (TN,P )d . In the case jP ≤ 0 we argue in a similar way and find 1 (I2 )  |P|τ d d

∞

∑2

(n+s−N)md



∞

t

−sq

#

0

m=0

1 + τd |P|

2

∑2

(n+s−N)md

m=0

P



2m+2 2

q/p [at (x)] dx

dt ··· t

p

dt t

d/q

d/q

 (AP )d + (TN,P )d . Thus, we have proved the claim in case of IP . Step 2. Estimate of JP . As in Step 1 we conclude ⎧  q  p/q ⎫1/p   ⎨ ⎬ ∞ ∞ 1 JP  2 jsq ∑ 2−mn a2−m− j (x) dx ∑ τ ⎭ |P| ⎩ P j=( j ∨0) m=0 P

⎧  q  p/q ⎫1/p   ⎬ ∞ ∞ 1 ⎨ jsq m(n−N) + τ 2 2 a (x) dx m− j+1 ∑ 2 ⎭ |P| ⎩ P j=(∑ m=0 j ∨0) P

≡ J1 + J2 .

78

4 Several Equivalent Characterizations

This time we continue by using $  $d $ ∞ % ∞ %q 1/q $ $ $ $ $ ∑ %% ∑ fm, j %% $ $ $ p $ j=0 m=0

$ 1/q $ $ ∞ $d $ $ q $ $ ≤ ∑ $ ∑ | fm, j | $ $ $ p m=0 j=0 ∞

L (Ω )

,

(4.16)

L (Ω )

valid for all sequences { fm, j }m, j∈N of locally integrable functions, Ω a measurable subset of Rn and with d = min{1, p, q}. Step 2.1. Estimate of J1 . Applying (4.14) in combination with (4.16), we see that 1 (J1 )d  |P|τ d ×

⎧  ⎨

∞

∑ 2−mnd ⎩

 2−m− j+1 2−m− j

1  |P|τ d

P

m=0

∞

∑

2 jsq

j=( jP ∨0)

dt t

(at (x) + at/2 (x))q

∞

⎧  ⎨ 

m=0

P

∑ 2−m(n+s)d ⎩

⎫d/p ⎬

 p/q dx

2−m−( jP ∨0)+1 0

⎭

dt t −sq [at (x)]q t

⎫d/p ⎬

 p/q dx

⎭

P )d .  (A Step 2.2. Estimate of J2 . Again we use (4.16) and find 1 (J2 )d  |P|τ d

⎧  ⎨

∞

∑ 2(n−N)md ⎩

m=0

P

⎫d/p ⎬

 p/q

∞

∑

j=( jP ∨0)

2 jsq (a2m− j+1 (x))q

dx

⎭

.

We split the right-hand side into three terms, i. e., (J2 )d  J12 + J22 + J32 , where 1 J12 ≡ |P|τ d J22 ≡

1 |P|τ d

1 J32 ≡ |P|τ d

⎧  ⎨

∞

∑ 2(n−N)md ⎩

m=0

( jP ∨0)

∑

2(n−N)md

m=0 ∞

∑

m=( jP ∨0)+1

P

∑

j=( jP ∨0)+m+1

⎧  ⎨ ( jP ∨0)+m ⎩

P

2(n−N)md

∑

2 jsq (a2m− j+1 (x))q

···

⎧  ⎨ ( jP ∨0)+m P

∑

j=( jP ∨0)

dx

⎭

⎫d/p ⎬

 p/q dx

j=( jP ∨0)

⎩

⎫d/p ⎬

 p/q

∞

⎭ ⎫d/p ⎬

 p/q ···

dx

⎭

.

P )d . As above the estimate of Similarly to the estimate of I2 , we obtain J12 , J22  (A 3 the third term J2 is more complicated. We split our considerations into the cases jP < 0 and jP ≥ 0. First, let jP < 0. Then (4.14) leads to

4.1 Preparations

79

1 J32  |P|τ d

∞

⎧  ⎨ 

m=1

P

∑ 2(n+s−N)md ⎩

2m+2 2

dt t −sq [at (x)]q t

⎫d/p ⎬

 p/q dx

⎭

.

Hence J32  (TN,P )d . In case jP ≥ 0, similarly, 1 J32  |P|τ d

∞

∑

2(n−N)md

m= jP

1 + τd |P|

∞

∑

⎧  ⎨ ⎩

2(n−N)md

m= jP

P

∑ 2 jsq(a2m− j+1 (x))q

dx

j= jP

⎧  ⎨ m+ jP ⎩

⎫d/p ⎬

 p/q

m

P

∑

⎭ ⎫d/p ⎬

 p/q 2 jsq (a2m− j+1 (x))q

j=m+1

dx

⎭

P )d ,  (TN,P )d + (A  

which completes the proof of Lemma 4.3. Lemma 4.4. Let s ∈ (0, ∞) and P ∈ Q.

(i) Let p ∈ [1, ∞]. Then, for sufficiently large N, there exists a positive constant C, independent of P, such that TN,P + TN,P ≤ C  f Lτp (Rn ) holds for all f with finite right-hand side. (ii) Let p ∈ (0, 1) and τ ∈ [1/p, ∞). Then, for sufficiently large N, there exists a positive constant C, independent of P, such that TN,P + TN,P ≤ C ( f L1τ (Rn ) +  f Lτp (Rn ) ) holds for all f with finite right-hand side. (iii) Let p ∈ (0, 1) and σ p < s0 < s. Then, for sufficiently large N, there exists a positive constant C, independent of P, such that TN,P + TN,P ≤ C

sup

 f Bs0

{P∈Q, |P|≥1}

p,∞ (2P)

|P|τ

holds for all f with finite right-hand side. Proof. Step 1. Proof of (i). Let p ∈ [1, ∞]. We choose N > max{2n + s, 2n + s + nd(τ − 1)}.

80

4 Several Equivalent Characterizations

Minkowski’s inequality then yields (TN,P )

d

∞

∑

≤

2

(n+s−N)md

m=( jP ∨0)

×



2m−( jP ∨0)+2

|ΔhM 1/2≤|h|<2m−( jP ∨0)+2

P

∑

≤

2

(n+s−N)md

m=( jP ∨0)

1 |P|τ d





  f dL p (Rn ) 2( jP ∨0)nτ d τ

p f (x)| dh

2m−( jP ∨0)+2

q/p dx

dt t

d/q

t −sq−nq

1

#

1/2≤|h|<2m−( jP ∨0)+2

t −sq−nq

1

 #

∞

×

1 |P|τ d

P

|ΔhM

∞

∑

q

1/p f (x)| dx p

dh

dt t

d/q

2(n+s−N)md 2(m−( jP ∨0)+2)nd

m=( jP ∨0)

  f dL p (Rn ) , τ

since s > 0. This proves TN,P   f Lτp (Rn ) . We describe the modifications needed in case of TN,P . Since s > 0 we observe ⎧  ⎨  ⎩

P

  

2m−( jP ∨0)+2

1

dt t −sq [at (x)]q t

  1
& # P



dx

p sup

P

|h|<2m−( jP ∨0)+2

#

|h|<2m−( jP ∨0)+2

P

at (x)

⎫1/p ⎬

 qp

1/p dx

|ΔhM f (x)|dh

|ΔhM

⎭

p

'1/p dx

1/p f (x)| dx p

dh .

This implies (TN,P )d   f dL p (Rn ) 2( jP ∨0)nτ τ

if

∞

∑

m=( jP ∨0)

2(n+s−N)md 2(m−( jP ∨0)+2)nd   f dL p (Rn ) τ

N > 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<|h|
|ΔhM (EP f )(x)|dh

p

 qp dx

dt t

 dq .

0 (Rn ) by differences; Next we apply t ≥ 1 and the known characterizations of Bsp,q see [146, Theorem 3.5.3]. It follows that



2m+2

t 1

−sq

#   t −n P

t/2<|h|
|ΔhM (EP f )(x)| dh

p

 qp dx

dt t

 1q

4.1 Preparations



83

sup

t

−s0

#   t −n P

1
t/2<|h|
|ΔhM (EP f )(x)| dh

p

 1p dx

 EP f Bs0

n p,∞ (R )

 f

s0 B p,∞ (2P)

.  

This proves the claim. Finally, we deal with a supplement of Lemma 4.3. Lemma 4.5. Let p, q ∈ (0, ∞] and τ ∈ [1/p, ∞). Then 1 sup τ {Q∈Q: P⊂Q} |Q|



2(l(Q)∧1)

t

−sq

#

q/p [at (x)] dx p

Q

0



1  sup τ |Q| {Q∈Q: |Q|≤1}

2l(Q)

t

−sq

0

# Q

dt t

1/q

q/p [at (x)] dx p

dt t

1/q

and 1 sup τ |Q| {Q∈Q: P⊂Q}

 p/q 1/p dt t −sq [at (x)]q dx t Q 0  #  p/q 1/p 2l(Q) 1 −sq q dt  sup t [at (x)] dx τ t Q 0 {Q∈Q: |Q|≤1} |Q|  #

2(l(Q)∧1)

hold for all f ∈ Lτp (Rn ) and all P ∈ Q. Proof. Let Q be a dyadic cube such that |Q| = 2rn for some r ∈ N. Then the integral with respect to t extends over the interval (0, 2), i. e., is independent of Q itself. In such a situation we may argue as in proof of Lemma 2.2.  

4.1.3 Means of Differences We consider different types of means of differences. We set  f ♣ s, τ B p, q (Rn )

1 ≡ sup τ P∈Q |P|

 f ♣ s, τ Fp, q (Rn )

1 ≡ sup τ P∈Q |P|



2(l(P)∧1)

t

−sq

P

 #

2(l(P)∧1)

t 0

[at (x)] dx p

0

P

q/p

#

−sq

dt [at (x)] t q

dt t

1/q , 1/p

 p/q dx

,

84

4 Several Equivalent Characterizations

1 τ P∈Q |P|

 f ♥ ≡ sup s, τ B (Rn ) p, q



1 ≡ sup τ |P| P∈Q

t −sq

#



0

×  f ♥ s, τ Fp, q (Rn )

2(l(P)∧1)

t

−n

P

  # P

t/2≤|h|
|ΔhM

q/p f (x)| dh dx p

dt t

1/q ,

t −sqt −n

0



dt × |ΔhM f (x)|q dh t t/2≤|h|
1/p

 p/q dx

,

and 

2(l(P)∧1)

#

q/p

 f ♠ s, τ B (Rn )  f ♠ s, τ F (Rn )

⎧   p/q ⎫1/p   2(l(P)∧1) ⎨ ⎬ 1 −sq M q dt ≡ sup t sup | Δ f (x)| dx . h τ ⎭ t 0 P∈Q |P| ⎩ P t/2≤|h|
p, q

p, q

t −sq

0

sup t/2≤|h|
P

|ΔhM f (x)| p dx

dt t

1/q

1 ≡ sup τ |P| P∈Q

,

The following conclusion is an immediate corollary of H¨older’s inequality. Lemma 4.6. Let s ∈ [0, ∞) and τ ∈ [0, ∞). (i) Let p ∈ [1, ∞] and q ∈ (0, ∞]. Then  f ♣ ≤  f ♥ ≤  f ♠ . s, τ s, τ s, τ B (Rn ) B (Rn ) B (Rn ) p, q

p, q

p, q

(ii) Let p ∈ (0, ∞) and q ∈ [1, ∞]. Then  f ♣ ≤  f ♥ ≤  f ♠ . s, τ s, τ s, τ F (Rn ) F (Rn ) F (Rn ) p, q

p, q

p, q

4.2 Characterizations by Wavelets The main aim of this section consists in proving characterizations of our spaces As,p,τq (Rn ) in terms of wavelet coefficients. In addition we prepare the characterization of As,p,τq(Rn ) via differences. To begin with we recall some basics of wavelet theory.

4.2 Characterizations by Wavelets

85

4.2.1 Wavelets and Besov-Triebel-Lizorkin Spaces Wavelet bases in Besov and Triebel-Lizorkin spaces are a well-developed concept. We refer to the monographs of Meyer [99], Wojtasczyk [156] and Triebel [148,149] for the general n-dimensional case (for the one-dimensional case we refer to the books of Hernandez and Weiss [72], Kahane and Lemarie-Rieuseut [82] and the article of Bourdaud [18]). Let φ be an orthonormal scaling function on R with  be the corresponding compact support and of sufficiently high regularity. Let ψ orthonormal wavelet. Then the tensor product ansatz yields a scaling function φ and associated wavelets ψ1 , · · · , ψ2n −1 , all defined now on Rn ; see, e. g., [156, Proposition 5.2]. We suppose

φ ∈ CN1 (Rn ) and

supp φ ⊂ [−N2 , N2 ]n

(4.19)

for certain natural numbers N1 and N2 . This implies

ψi ∈ CN1 (Rn ) and

supp ψi ⊂ [−N3 , N3 ]n , i = 1, · · · , 2n − 1

(4.20)

for some N3 ∈ N. For k ∈ Zn , j ∈ Z+ and i = 1, · · · , 2n − 1, we shall use the standard abbreviations in this context:

φ j,k (x) ≡ 2 jn/2 φ (2 j x − k) and ψi, j,k (x) ≡ 2 jn/2 ψi (2 j x − k), x ∈ Rn . Furthermore, it is well known that  Rn

ψi, j,k (x) xγ dx = 0

if

|γ | ≤ N1

(see [156, Proposition 3.1]) and {φ0,k : k ∈ Zn } ∪ {ψi, j,k : k ∈ Zn , j ∈ Z+ , i = 1, · · · , 2n − 1}

(4.21)

yields an orthonormal basis of L2 (Rn ); see [99, Sect. 3.9] or [148, Sect. 3.1]. Recall that for p, q ∈ (0, ∞], any function f ∈ Bsp, q (Rn ) when s ∈ (σ p , N1 ) or f ∈ Fp,s q(Rn ) when s ∈ (σ p, q , N1 ) admits a representation f=

2n −1

∑n ak φ0,k + ∑ ∑ ∑n ai, j,k ψi, j,k

(4.22)

i=1 j∈Z+ k∈Z

k∈Z

in S  (Rn ), where ak ≡  f , φ0,k  and ai, j,k ≡  f , ψi, j,k  (here it will be sufficient to interpret  · , ·  as scalar product in L2 (Rn ), since the functions φ0,k and ψi, j,k are compactly supported continuous functions and f ∈ L1loc (Rn )). Moreover,   f Bsp, q(Rn ) ∼

1/p

∑n |ak | p

k∈Z

⎡

2n −1

+⎣ ∑



∑

i=1 j∈Z+

2 j(s+n/2)q

∑

k∈Zn

q/p ⎤1/q ⎦ , 2− jn |ai, j,k | p

86

4 Several Equivalent Characterizations

and  f Fp,s q(Rn ) $ q 1/q $ 1/q  n  $ $ 2 −1 $ $ q jsq $ $ Q0k | ∼ $ ∑ |ak χ + ∑ ∑ 2 |ai, j,k χQ jk | ∑ $ $ $ k∈Zn i=1 j∈Z+ k∈Zn

L p (Rn )

in the sense of equivalent quasi-norms; see, e. g., [149, Theorem 1.20]. In fact, even more is true. If, for a function f ∈ Lmax{1, p}(Rn ), the right-hand side is finite, then this function belongs to Asp, q (Rn ). For each j ∈ Z, define a projection W j by setting Wj f ≡

∑  f , φ j,k φ j,k if j > 1

and W j f ≡

k∈Zn

∑  f , φ0,k φ0,k if j ≤ 0.

(4.23)

k∈Zn

Then these functions have a second representation Wj f =

∑

 f , φ0,k φ0,k +

k∈Zn

2n −1 j−1

∑ ∑ ∑  f , ψi,t,k ψi,t,k ,

(4.24)

i=1 t=0 k∈Zn

where, when j < 1, the second summation in the right part of (4.24) is void. For any functions f ∈ Lτp (Rn ) with p ∈ (0, ∞), we have the convergence of the wavelet expansions in the following sense: lim  f − W j f L p (Q) = 0

j→∞

for any dyadic cube Q; see [156, Theorem 8.4].

4.2.2 Estimates of Mean-Values of Differences by Wavelet Coefficients To estimate  f ♣ ,  f ♥ and  f ♠ in terms of wavelet coefficients, s, τ s, τ s, τ A (Rn ) A (Rn ) A (Rn ) p, q

p, q

p, q

we need some more abbreviations. For any P ∈ Q we set  f B♣ (P)

& 2(l(P)∧1) 1 ≡ t −sq |P|τ 0  #   p q/p 1/q dt −n M × |Δh f (x)| dh dx , t t P t/2≤|h|
4.2 Characterizations by Wavelets

 f F ♣ (P) ≡

87

& 

1 |P|τ P #  × t −n

2(l(P)∧1)

t/2≤|h|
1  f B♠ (P) ≡ |P|τ



t −sq

0

2(l(P)∧1)

t

−sq

0

|ΔhM

q f (x)| dh #

sup t/2≤|h|
P

|ΔhM

dt t

1/p

 p/q dx

q/p f (x)| dx p

,

dt t

1/q

and ⎧   p/q ⎫1/p   2(l(P)∧1) ⎬ 1 ⎨ dt −sq M q  f F ♠ (P) ≡ t sup | Δ f (x)| dx . h ⎭ |P|τ ⎩ P 0 t t/2≤|h| 0} , IQ,m ≡ {r ∈ Zn : there exists i ∈ {1, · · · , 2n − 1} such that | supp ψi,m,r ∩ Q| > 0} , where | · | denotes the Lebesgue measure in Rn . Let |JQ | and |IQ,m | denote the cardinalities of these sets. It is easy to check that there exists a positive constant C ≡ C(N2 , N3 ) such that |JQ | ≤ C max(1, |Q|)

and

|IQ,m | ≤ C max(1, 2mn |Q|) .

(4.25)

For Q = Q jk and m ∈ Z+ , we set IQ,m ≡

( |l−k|≤M

IQ jl ,m and JQ ≡

(

JQ jl .

|l−k|≤M

The natural number M will be fixed later on. By taking into account the wavelet characterizations of the preceding subsection it is not difficult to obtain local estimates of the difference f − W j f . Lemma 4.7. Let τ ∈ [0, ∞), p ∈ (0, ∞), q ∈ (0, ∞] and Q ≡ 2− jQ ([0, 1)n + kQ )n . (i) If σ p < s < M ≤ N1 , then there exists a positive constant C such that for all f ∈ Lτp (Rn ),  f − W jQ f B♠ (Q) ⎧ ⎛ ⎞q/p ⎫1/q ⎪ ⎪ ⎨ ⎬ 2n −1 ∞ 1 ω (s+n/2)q −ω n p⎠ ⎝ ≤C τ 2 |ai,ω ,r | . (4.26) 2 ∑ ∑ ∑ ⎪ |Q| ⎪ ⎩ω =( jQ ∨0) ⎭ i=1 r∈IQ,ω

88

4 Several Equivalent Characterizations

The inequality (4.26) remains to hold if  f − W jQ f B♠ (Q) is replaced by either  f − W jQ f B♣ (Q) or  f − W jQ f B♥ (Q) . (ii) If σ p, q < s < M ≤ N1 , then there exists a positive constant C such that for all f ∈ Lτp (Rn ),  f − W jQ f F ♣ (Q) $⎡ ⎤1/q $ $ $ $ $ 2n −1 ∞ 1 $⎣ $ wsq q ⎦  ≤C τ $ 2 |a χ | $ i, ω ,r Q ∑ ∑ ωr $ |Q| $ ω =(∑ jQ ∨0) i=1 r∈IQ,ω $ $

. (4.27)

L p (Q)

The inequality (4.27) remains to hold if  f − W jQ f F ♣ (Q) is replaced by  f − W jQ f F ♥ (Q) . (iii) If n/ min{p, q} < s < M ≤ N1 , then the inequality (4.27) remains to hold if  f − W jQ f F ♣ (Q) is replaced by  f − W jQ f F ♠ (Q) . Proof. Let f ∈ Lτp (Rn ). For convenience we put ai,ω ,r ≡ 0 if ω < 0. Then, by (4.22) and (4.24), we have f (x) − W jQ f (x) =

2n −1

∞

∑ ∑ ∑

ai,ω ,r ψi,ω ,r (x) ,

i=1 w= jQ r∈IQ,ω

valid in L p (Ω ), where Ω ≡ ∪|l−kQ |≤M Q jQ l . For simplicity we denote the function on the right-hand side of the above formula by g(x). Step 1. Proof of (i). By making use of the characterization of Bsp, q(Rn ) by differences (see [146, Theorem 2.6.1]), the wavelet characterization of Bsp, q (Rn ) (see Sect. 4.2.1), and the fact that (4.21) is an orthonormal basis of L2 (Rn ), we find that  f − W jQ f B♣ (Q)  q/p 1/q # 2(l(Q)∧1) 1 dt −sq M p = t sup |Δh g(x)| dx τ |Q| t P 0 t/2≤|h|
4.2 Characterizations by Wavelets

89

⎡  q/p ⎤1/q 2n −1 1 ⎣ 2w(s+n/2)q ∑ ∑ 2−ω n|g, ψi,ω ,k | p ⎦ |Q|τ ω∑ i=1 ∈Z+ k∈Zn

+

⎡ ∼

∞

1 ⎢ 2w(s+n/2)q ⎣ |Q|τ ω =(∑ j ∨0) Q

2n −1

⎛

∑⎝ ∑

i=1

⎞q/p ⎤1/q ⎥ 2−ω n |ai,ω ,r | p ⎠ ⎦ .

r∈IQ,ω

For  f −W jQ f B♣ (Q) and for  f −W jQ f B♥ (Q) , by [146, Theorems 2.6.1/3.5.3], the above argument is also feasible. Step 2. Proof of (ii). Similarly, by the same group of arguments (for the characterization of Fp,s q (Rn ) by differences see [146, Theorem 3.5.3]; for the wavelet characterization of Fp,s q(Rn ) see [149, Theorem 1.20]), we have  f − W jQ f F ♣ (Q)   q  p/q 1/p #  2(l(Q)∧1) dt 1 = t −sq t −n |ΔhM g(x)| dh dx |Q|τ t Q t/2≤|h|
 +

2n −1

∑ ∑

ω ∈Z+ i=1

⎫ ⎪ ⎬

1/q $ $ $ wsq q ∑n 2 |g, ψi,ω ,k χQω k | $$ $ k∈Z

L p (Rn )

$⎡ ⎤1/q $ $ $ $ $ 2n −1 ∞ 1 $⎣ $ wsq q⎦  ∼ 2 |a χ | $ $ i, ω ,r Q ∑ ∑ ∑ ω r $ |Q|τ $ ω =( j ∨0) i=1 r∈I Q,ω Q $ $

⎪ ⎭

.

L p (Q)

By [146, Theorem 2.6.2], we know that the above argument is also feasible for  f − W jQ f F ♥ (Q) , which completes the proof of Lemma 4.7. Step 3. Proof of (iii). We can argue as before but this time the restriction for the characterizations by differences is different; see [146, Theorems 2.6.2/3.5.3]. Step 4. A technical remark. Of course, in all preceding steps we have to argue by starting with the finiteness of the right-hand side for a function f belonging to Lτp (Rn ). This always implies that g ∈ Asp,q (Rn ) as a consequence of the wavelet characterization recalled in Sect. 4.2.1.  

90

4 Several Equivalent Characterizations

It remains to estimate the projections W j f . For having a more compact notation we set  f  s, τ B p, q (Rn )

1 ≡ sup τ {P∈Q: |P|≥1} |P|



1

∑

p

| f , φ0,k |

p

k∈JP

⎧   qp ⎫ 1q ⎨ ⎬ 2n −1 ∞ 1 j(s+n/2)q − jn p + sup 2 2 | f , ψ | i, j,k ∑ ∑ ∑ τ ⎭ P∈Q |P| ⎩ j=( j ∨0) i=1 k∈I P, j

P

and  f  s, τ Fp, q (Rn )

  1p 1 p ≡ sup ∑ | f , φ0,k | τ {P∈Q: |P|≥1} |P| k∈JP $  1q $ $ $ ∞ 2n −1 $ 1 $ jsq q $ $ + sup 2 | f , ψi, j,k  χQ jk | ∑ ∑ ∑ $ τ $ |P| P∈Q $ j=( jP ∨0) i=1 k∈IP, j $

.

L p (P)

We then have the following conclusions. Lemma 4.8. Let τ ∈ [0, ∞), p ∈ (0, ∞), q ∈ (0, ∞], 0 < M ≤ N1 and Q ∈ Q. (i) If σ p < s < {M ∧ (M + n(1/p − τ ))}, then there exists a positive constant C such that for all f ∈ Lτp (Rn ), W jQ f B♣ (Q) + W jQ f B♥ (Q) + W jQ f B♠ (Q) ≤ C  f  . s, τ B (Rn ) p, q

(ii) If σ p, q < s < {M ∧ (M + n(1/p − τ ))}, then there exists a positive constant C such that for all f ∈ Lτp (Rn ), . W jQ f F ♣ (Q) + W jQ f F ♥ (Q) ≤ C f  s, τ F (Rn ) p, q

(iii) If n/ min{p, q} < s < {M ∧ M + n(1/p − τ )}, then there exists a positive constant C such that for all f ∈ Lτp (Rn ), W jQ f F ♠ (Q) ≤ C f  . s, τ F (Rn ) p, q

Proof. We shall concentrate on the terms W jQ f (x)A♠ (Q) . By (4.24), for all f ∈ p

Lτ (Rn ), we have W jQ f (x) =

∑

k∈Zn

ak φ0,k (x) +

2n −1 jQ −1

∑ ∑ ∑

i=1 j=0 k∈IQ, j

ai, j,k ψi, j,k (x) ,

4.2 Characterizations by Wavelets

91

valid in L p (∪|l−kQ |≤M Q jQ l ). Recall, if jQ < 1, then the second summation on the right-hand side of above equality is void. Our main tool will be the elementary inequality sup |∂ α g(y)| , (4.28) |ΔhM g(x)|  |h|M sup |α |=M |x−y|≤M|h|

valid for all functions g ∈ C (R ). We shall apply this inequality with respect to the elements of our wavelet basis which belong to CN1 (Rn ) by assumption. Step 1. Estimate of ∑k∈Zn ak φ0,k . Notice that ΔhM φ0,k (x) = 0 for x ∈ Q and k ∈ / JQ . We put g(x) ≡ ∑ ak φ0,k . M

n

k∈JQ

In analogy to the proof of Lemma 4.7 it follows that $ $ $ $ $ $ $ ∑ ak φ0,k $ $k∈Zn $ ♠ 1 ≤ |Q|τ



B (Q)

2(l(Q)∧1)

t

#

−sq

0

sup Q

t/2≤|h|
|ΔhM g(x)| p dx

q/p

dt t

1/q

1 gBsp, q(Rn ) |Q|τ  1/p 1 p  . ∑ |ak | |Q|τ k∈J ≤

Q

This estimate will be applied for large cubes Q, i. e., |Q| ≥ 1. If |Q| < 1, we argue as follows. The cardinality of JQ is uniformly bounded, for simplicity say 1. Then gB♠(Q)

1 ≤ |Q|τ 



|Q|1/p

|Q|τ  |akQ | ,

2(l(Q)∧1)

t 0

−sq

# sup t/2≤|h|
Q

|ΔhM akQ φ0,kQ (x)| p dx

q/p

dt t

1/q

|akQ |φ CM (Rn ) [l(Q)]M−s

if M + n/p − nτ ≥ s. Summarizing we get $ $ $ $ $ $ $ ∑ ak φ0,k $ $ $k∈Zn

B♠ (Q)

1  sup τ |P| {P∈Q: |P|≥1}



1/p

∑

|ak | p

.

k∈JP

Similarly, for the Triebel-Lizorkin-type spaces, we also have $ $ $ $ $ $ $ ∑ ak φ0,k $ $ $k∈Zn

F ♠ (Q)

1  sup τ |P| {P∈Q: |P|≥1}



1/p

∑

k∈JP

|ak |

p

,

(4.29)

92

4 Several Equivalent Characterizations

if n/ min{p, q} < s < M + n/p − nτ . Replacing  ∑k∈Zn ak φ0,k F ♠ (Q) either by  ∑k∈Zn ak φ0,k F ♣ (Q) or by  ∑k∈Zn ak φ0,k F ♥ (Q) , the restrictions

σ p,q < s < M + n/p − nτ are sufficient. jQ −1 Step 2. Estimate of ∑ j=0 ∑k∈IQ, j ai, j,k ψi, j,k . Let jQ ≥ 1. For all j ∈ {0, · · · , jQ − 1}, i ∈ {1, · · · , 2n − 1} and x ∈ Rn we define

∑

gi j (x) ≡

ai, j,k ψi, j,k (x) .

k∈IQ, j

Recall that |IQ, j |  2−[( jQ − j)∧0]n; see (4.25). By using (4.28) we obtain $ $ $ jQ −1 $ $ $ $ ∑ gi j $ $ j=0 $

B♠ (Q)

⎧ ⎛ ⎞q ⎫1/q  j −1 ⎬ |Q|1/p ⎨ 2l(Q) −sq Mq ⎝ Q jn/2 jM ⎠ dt  t t |a |2 2 i, j,k ∑ ∑ |Q|τ ⎩ 0 t ⎭ j=0 k∈I Q, j

 |Q|(M−s)/n+1/p−τ

jQ −1

∑ ∑

2 j(M+n/2) |ai, j,k |

j=0 k∈IQ, j

 |Q|(M−s)/n+1/p−τ

jQ −1

∑

j=0

 × 

2 j(M−s+n/p−nτ )

sup

sup 2

 j(s+n/2+nτ −n/p)

j∈{0,···, jQ −1} k∈IQ, j

|ai, j,k |

sup 2 j(s+n/2+nτ −n/p)|ai, j,k |,

sup

j∈{0,···, jQ −1} k∈IQ, j

because of M > s + nτ − n/p. Similarly, $ $ $ jQ −1 $ $ $ $ ∑ gi j $ $ j=0 $

F ♣ (Q)



sup

sup

sup 2 j(s+n/2+nτ −n/p)|ai, j,k |.

i∈{1,··· ,2n −1} j∈{0,···, jQ −1} k∈IQ, j

Notice that IQ, j ⊂ IP, j if Q ⊂ P. It is easy to check that sup

sup 2 j(s+n/2+nτ −n/p)|ai, j,k |

sup

i∈{1,···,2n −1} j∈{0,···, jQ −1} k∈IQ, j

⎧  q/p ⎫1/q ⎬ 2n −1 1 ⎨ ∞ j(s+n/2)q − jn p  sup 2 2 |a | i, j,k ∑ ∑ ∑ τ ⎭ P∈Q |P| ⎩ j=( j ∨0) i=1 k∈IP, j P

4.2 Characterizations by Wavelets

93

and i∈{1,··· ,2n −1}

sup 2 j(s+n/2+nτ −n/p)|ai, j,k |

sup

sup

j∈{0,···, jQ −1} k∈IQ, j

⎧   p/q ⎫1/q  ⎬ ∞ 2n −1 1 ⎨ js q   sup (2 |a | χ (x)) dx , Q i, j,k ∑ ∑ ∑ jk τ ⎭ P∈Q |P| ⎩ P j=( j ∨0) i=1 k∈IP, j P

 

which completes the proof of Lemma 4.8.

Lemma 4.9. Let p, q ∈ (0, ∞], τ ∈ [0, ∞) and s ∈ (0, ∞). Then there exists a positive constant C such that for all f ∈ Lτp (Rn ), 1 Φτ ,p ( f ) ≡ sup τ |P| {P∈Q: |P|≥1}

#

|Φ ∗ f (x)| dx

1/p

p

P

≤ C f  . s, τ A (Rn ) p, q

Proof. By the wavelet expansion (4.22) of f , we see that 1 |P|τ

#

|Φ ∗ f (x)| dx

1/p

p

P

1  |P|τ +

  P

∑ |ak ||Φ ∗ φ0,k (x)|

k∈Zn

p

2n −1 ∞

∑ ∑ ∑ |ai, j,k ||Φ ∗ ψi, j,k (x)|

1/p dx

.

i=1 j=0 k∈Zn

It is easy to check that for all x ∈ P and k ∈ Zn , |Φ ∗ φ0,k (x)| 

1 , (1 + |x − k|)n+δ

where we choose

δ > max {n[τ ∨ (1/p) − 1], 0} . This estimate combined with (2.11) when p ≤ 1 or H¨older’s inequality when p > 1 yields 1 AP ≡ |P|τ 1  |P|τ

  P

  P

p

∑ |ak ||Φ ∗ φ0,k (x)|

1/p dx

k∈Zn

1 ∑n |ak | (1 + |x − k|)n+δ k∈Z

1/p

p dx

94

4 Several Equivalent Characterizations

1  |P|τ



∑ |ak |

 p

k∈Zn

⎡ 1 ⎣ ∞  ∑ |P|τ m=0 k∈J

1 dx P (1 + |x − k|)(n+δ )p

∑

|ak | p



2m P\2m−1 P

1/p

⎤1/p 1 dx⎦ , P (1 + |x − k|)(n+δ )p

where when m = 0, JP\2−1 P is replaced by JP . Notice that for all x ∈ P and k ∈ J2m P\2m−1P , |x − k| ∼ 2m l(P), which implies that 1 AP  |P|τ 



1/p

∞

∑2

−m(n+δ )p

m=0

∑

1−(n+δ )p/n

|ak | |P| p

k∈J2m P

1/p

∞

∑ 2−m(n+δ )p+mnτ p|P|1−(n+δ )p/n



m=0

1  sup τ {P∈Q: |P|≥1} |P|



1/p

∑

|ak |

1 sup τ |P| {P∈Q: |P|≥1}



1/p

∑

|ak | p

k∈JP

.

p

k∈JP

Similarly, we have that for all i ∈ {1, · · · , 2n − 1}, j ∈ Z+ , k ∈ Zn and x ∈ Rn , |Φ ∗ ψi, j,k (x)| 

2− jn/2 , (1 + |x − 2− j k|)n+δ

where we choose

δ > max {0, n(1/p − 1), n(τ − 1), n(τ p − 1)}. This estimate yields

BP

1 ≡ |P|τ 1  |P|τ

  P

  P

p

2n −1 ∞

∑ ∑ ∑ |ai, j,k ||Φ ∗ ψi, j,k (x)|

1/p dx

i=1 j=0 k∈Zn 2n −1 ∞

∑

i=1

2− jn/2 ∑ ∑n |ai, j,k | (1 + |x − 2− jk|)n+δ j=0 k∈Z

1/p

p dx

.

We remark that for all j ≥ 0, k ∈ I2m P\2m−1 P, j and x ∈ P, 1 + |x − 2− j k|  2m l(P). Then for the case p ≤ 1, applying (2.11), we obtain ⎡ n 1 ⎣ ∞ −m(n+δ )p 2 −1 ∞ 2 BP  ∑ ∑ ∑ |P|τ m=0 i=1 j=0 k∈I

⎤1/p

∑

2m P\2m−1 P, j

|ai, j,k | p 2− jnp/2|P|1−(n+δ )p/n⎦

.

4.2 Characterizations by Wavelets

95

If q ≤ p, using (2.11) again yields  BP 

1/q

∞

∑2

−m(n+δ )q+mnτ q

m=0

sup 2

− j[s−n(1/p−1)]q

j∈Z+

⎧  q/p ⎫1/q ⎨ ⎬ ∞ 2n −1 1 j(s+n/2)q − jn p × sup 2 2 |a | i, j,k ∑ ∑ ∑ τ ⎭ {P∈Q: |P|≥1} |P| ⎩ j=0 i=1 k∈I P, j

⎧  q/p ⎫1/q n ⎬ 1 ⎨ ∞ j(s+n/2)q 2 −1 − jn p  sup 2 2 |a | ; i, j,k ∑ ∑ ∑ τ ⎭ {P∈Q: |P|≥1} |P| ⎩ j=0 i=1 k∈IP, j if q < p, using H¨older’s inequality instead of (2.11) yields the same estimate. For the case p > 1, let ε ∈ (2/p − 1 − 2s/n, 2/p −1). Then by H¨older’s inequality, we see that 1 BP  |P|τ



2n −1 ∞

∑∑∑

|ai, j,k |

i=1 j=0 k∈Zn

 p

2− jnε p/2 dx P (1 + |x − 2− j k|)n+δ

⎡ n 1 ⎣ ∞ −m(n+δ ) 2 −1 ∞  2 ∑ ∑∑ |P|τ m=0 i=1 j=0 k∈I

∑

1/p ⎤1/p

|ai, j,k | p 2− jnε p/2|P|−δ /n ⎦

.

2m P\2m−1 P, j

Similarly, applying (2.11) when q ≤ p or H¨older’s inequality when q > p, we also obtain ⎧  q/p ⎫1/q ⎨ ⎬ 2n −1 ∞ 1 sup 2 j(s+n/2)q ∑ 2− jn |ai, j,k | p , BP  ∑ ∑ τ ⎭ {P∈Q: |P|≥1} |P| ⎩ j=0 i=1 k∈I P, j

which together with the estimate of AP yields the desired inequality for Bs,p,τq (Rn ). To prove the desired inequality for Fp,s, qτ (Rn ), notice that when p ≤ q, the desired inequality for Fp,s, qτ (Rn ) is an immediate corollary of the above inequality and Minkowski’s inequality. We only consider the remaining case p > q. Notice that ⎡ ⎛ n  1 ⎣ ⎝ ∞ 2 −1 ∞ BP  ∑ ∑ ∑ |P|τ P m=0 i=1 j=0 k∈I

∑

2m P\2m−1 P, j

⎞ p ⎤1/p 2− jn/2 ⎠ dx⎦ . |ai, j,k | (1 + |x − 2− jk|)n+δ

Then when p ≤ 1, applying (2.11), we have ⎡ ⎛  2n −1 ∞ 1 ⎣ ∞ ⎝ BP  ∑ P ∑∑ |P|τ m=0 i=1 j=0 k∈I

∑

2m P\2m−1 P, j

⎞ p ⎤1/p 2− jn/2 ⎠ dx⎦ , |ai, j,k | (1 + |x − 2− jk|)n+δ

96

4 Several Equivalent Characterizations

and when p > 1, applying Minkowski’s inequality, we obtain

BP

1  |P|τ

⎡

∞

∑⎣

m=0

 P

⎛ ⎝

2n −1 ∞

∑∑

∑

i=1 j=0 k∈I2m P\2m−1 P, j

⎞ p ⎤1/p 2− jn/2 ⎠ dx⎦ . |ai, j,k | (1 + |x − 2− jk|)n+δ

Recall that 1 + |x − 2− j k|  2m l(P) holds for all j ≥ 0, k ∈ I2m P\2m−1P, j and x ∈ P. Let a ∈ (0, min{1, p, q}). Then for all x ∈ P,

∑

k∈I2m P\2m−1 P, j

|ai, j,k |q 2− jnq/2 (1 + |x − 2− jk|)(n+δ )q ⎛

⎞q a

 2−m(n+δ )q [l(P)]−(n+δ )q 2− jnq/2 ⎝

∑

|ai, j,k |a ⎠

k∈I2m P, j

 2−m(n+δ )q [l(P)]−(n+δ )q 2− jnq(2m |P|)q/a ⎛ ⎞q a  1 a a Q jk (y) dy⎠ ×⎝ m |ai, j,k | χ |2 P| 2m P k∈I∑m 2 P, j ⎡ ⎛  2−m(n+δ )q [l(P)]−(n+δ )q 2− jnq(2m |P|)q/a ⎣M ⎝

⎞

∑

k∈I2m P, j

⎤

|ai, j,k |a χQa jk ⎠ (x)⎦ .

Since q < p, using (2.11) and the Fefferman-Stein vector valued maximal inequality, we obtain the desired conclusion, and then complete the proof of Lemma 4.9.  

4.2.3 The Wavelet Characterization of As,p,qτ (Rn ) After these preparations we are now in a position to turn to the wavelet characterization of our classes As,p,τq (Rn ). Let N1 be as in (4.19). If N1 ≥ s + nτ , then for all k ∈ Zn , i = 1, · · · , 2n − 1 and j ∈ Z+ , φ0k and ψi jk are multiples of inhomogeneous smooth analysis As,p,τq (Rn )-molecules, supported respectively near Q0k and Q jk . Then Theorem 3.2(ii) tells us that for all f ∈ As,p,τq (Rn ), ⎧ ⎪ 1 ⎨ sup τ {P∈Q: |P|≥1} |P| ⎪ ⎩

∑

k∈Zn Q0k ⊂P

| f , φ0,k | p

⎫1/p ⎪ ⎬ ⎪ ⎭

  f As,p,τq (Rn ) ,

{| f , ψi, j,k |}l(Q jk )≤1 as,p,τq(Rn )   f As,p,τq (Rn ) ,

i = 1, . . . , 2n − 1 ,

4.2 Characterizations by Wavelets

97

which further implies that  f    f As,p,τq(Rn ) . These estimates allow to close s, τ A p, q (Rn ) the circle which we started with Lemma 4.2. This yields the following wavelet characterization of As,p,τq (Rn ). Theorem 4.1. Let the generators φ and ψ of the wavelet system satisfy the conditions in (4.19), (4.20) with respect to N1 , N2 , N3 ∈ N. Let s ∈ R and p, q ∈ (0, ∞]. (i) We further suppose σ p < s, max{s, s + nτ − n/p} < N1 and 0 ≤ τ < τs,p . Then f ∈ Bs,p,τq (Rn ) if, and only if f is locally integrable and  f  < ∞. Further s, τ B (Rn ) p, q

 f  and  f Bs,p,τq (Rn ) are equivalent. s, τ B (Rn ) p, q

(ii) This time we suppose p ∈ (0, ∞), σ p,q < s, max{s, s+ nτ − n/p} < N1 and 0 ≤ τ < τs,p,q . Then f ∈ Fp,s, qτ (Rn ) if, and only if f is locally integrable and  f  < s, τ F (Rn ) p, q

∞. Further  f  and  f Fp,s, qτ (Rn ) are equivalent. s, τ F (Rn ) p, q

Proof. To finish the proof, it remains to show  f As,p,τq(Rn )   f  . In fact, by s, τ A p, q (Rn ) Lemmas 4.2–4.9, we have +  f Bs,p,τq (Rn )   f  s, τ B (Rn ) p, q



 f  + s, τ B p, q (Rn )

1 τ |P| {P∈Q: l(P)≥1}

#

sup

P

|Φ ∗ f (x)| p dx

1/p

sup TN,P P∈Q

and + sup TN,P .  f Fp,s, τq (Rn )   f  s, τ F (Rn ) p, q

P∈Q

Thus, we need to estimate TN,P (TN,P in the F-case). The first step consists in applying Lemma 4.4. We have to give estimates of  f Lτp (Rn ) ,  f L1τ (Rn ) , and sup {P∈Q, |P|≥1}

 f Bs0

p,∞ (2P)

|P|τ

.

Step 1. Estimate of  f Lτp (Rn ) . Let P be a dyadic cube such that |P| ≥ 1. Using the compact support of the generators φ and ψ we immediately obtain the so-called L p -stability of the corresponding shifts, i. e., % p 1/p   % 1/p % % % % p ∼ ∑ |ak | , % ∑ ak φ0,k % dx % P %k∈J k∈JP P % p 1/p  1/p  % % % % % mn( 12 − 1p ) p ∼2 % ∑ ai,m,k ψi,m,k % dx ∑ |ai,m,k | % P %k∈I k∈I P,m

P,m

and the constants behind ∼ do neither depend on the dyadic cube P nor on m ∈ Z+ . Since s > 0 we conclude

98

4 Several Equivalent Characterizations

 f Lτp (Rn )   f  . s, τ A (Rn )

(4.30)

p, q

s− n +n, τ

Let 0 < p < 1. Since s > σ p we have As,p,τq (Rn ) ⊂ B1,∞p p = 1 we have proved  f L1τ (Rn )   f  . s, τ A (Rn )

(Rn ). Using (4.30) with

p, q

Step 2. Estimate of supP∈Q, |P|≥1  f  f (x) =

s0 B p,∞ (2P)

/|P|τ . Starting point is the identity

2n −1 ∞

∑ ak φ0,k (x) + ∑ ∑ ∑

ai, j,k ψi, j,k (x) ,

i=1 j=0 k∈IP, j

k∈JP

which is valid on P. Taking the right-hand side of this formula we obtain an extension of the restriction of f to P to Rn . Hence  f Bsp,∞ 0 (P)

$ $ $ $ 2n −1 ∞ $ $ ≤ $ ∑ ak φ0,k (x) + ∑ ∑ ∑ ai, j,k ψi, j,k (x)$ $k∈J $ i=1 j=0 k∈I P

P, j

 

1/p

∑ |ak |

+

p



2n −1

∑

sup 2

∑

j(s0 +n/2)

i=1 j∈Z+

k∈JP

s

0 (Rn ) B p,∞

2

− jn

1/p

|ai, j,k |

p

;

k∈IP, j

see Sect. 4.2.1. With the obvious modifications, needed for the replacement of P by 2P, this implies sup  f Bs0 (2P) /|P|τ   f  . s ,τ B 0 (Rn ) P∈Q, |P|≥1

Since  f s0 , τ

B p,∞ (Rn )

p,∞

p,∞

  f  , the proof is complete. As, τ (Rn )

 

p,q

Remark 4.1. When τ = 0, Theorem 4.1 is just the classical result for Asp, q (Rn ); see, e. g., [99, Sect. 2.9], [82, Sect. 6.5], [65], [72, Remark 6.8/8], [149, Theorem 1.20] and [156, Corollary 9.10]. For τ ∈ [1/p, ∞) we can simplify the characterization. Therefore we need a further abbreviation. Let ||| f ||| ≡ sup | f , φ0,k | + s, τ B (Rn ) p, q

k∈Zn

×

⎧ ⎨

∞

∑ ⎩ j= jP

1 τ |P| {P∈Q: |P|≤1}

2 j(s+n/2)q

sup

2n −1

∑

i=1



∑

k∈IP, j

2− jn | f , ψi, j,k | p

q/p ⎫1/q ⎬ ⎭

4.2 Characterizations by Wavelets

99

and ≡ sup | f , φ0,k | + ||| f ||| s, τ F (Rn ) p, q

k∈Zn

1 τ |P| {P∈Q: |P|≤1} sup

$ 1/q $ $ $ ∞ 2n −1 $ $ jsq q $  ×$ 2 | f , ψ  χ | Q jk i, j,k $ $ ∑ ∑ ∑ $ $ j= jP i=1 k∈IP, j

.

L p (P)

Theorem 4.2. Under the same conditions as in Theorem 4.1, but assuming additionally τ ∈ [1/p, ∞), we have the following: all assertions in Theorem 4.1 remain true if  f  is replaced by ||| f ||| . s, τ s, τ A (Rn ) A (Rn ) p, q

p, q

Proof. Instead of Lemma 4.3 we employ Lemma 4.5. The equivalence of 1 sup τ {P∈Q: |P|≥1} |P|



1/p

∑

| f , φ0,k |

p

k∈JP

and supk∈Zn | f , φ0,k | is obvious if τ ∈ [1/p, ∞).

 

We finish this subsection by completing the proof of Proposition 2.2. Proof of Proposition 2.2 (Continued). We have to prove that the spaces As,p,τq11 (Rn ) and As,p,τq02 (Rn ) are incomparable, if s ∈ (σ p , ∞) and 0 ≤ τ0 < τ1 < 1/p. s, τ Step 1. We shall prove As,p,qτ1 (Rn ) ⊂ A p,q0 (Rn ). Let  > nτ . For any dyadic cube Q with |Q| ≥ 1 we define fQ ≡

∑ |k| φ0,k .

k∈JQ

Theorem 4.1 implies that in case τ < 1/p, 

 fQ Bs,p,qτ (Rn ) ∼ |Q| n + p −τ . 1

By Qm we denote the cube with center in xm ≡ (2m+2 + 2m , 0, . . . , 0) and side length 2m . For all α ∈ R, let gα ≡

∞



∑ m−α |Qm |− n − p +τ1 fQm . 1

m=1

Lemma 2.1 in combination with Proposition 2.1(iii) implies that gα ∈ As,p,qτ1 (Rn )

if

α>

1 . min{1, p, q}

100

4 Several Equivalent Characterizations

On the other hand, by the disjointness of the cubes Qm , we obtain 

1 |Qm |τ0

1/p

∑

|gα , φ0,k |

∼ m−α |Qm |τ1 −τ0 .

p

k∈JQm

The right-hand side is unbounded, hence gα ∈ As,p,qτ0 (Rn ) (applying again Proposition 2.1(iii) to avoid the restriction s > σ p,q ). s, τ Step 2. It remains to prove A p,q0 (Rn ) ⊂ As,p,qτ1 (Rn ). We consider the sequence ∞ {ψ1, j,0 } j=0 of functions. Obviously ψ1, j,0  ∼ 2 j(nτ +s+ 2 − p ) . s, τ A (Rn ) n

n

p, q

Let hα ≡

∞

∑ j−α 2− j(nτ0+s+ 2 − p ) ψ1, j,0 . n

n

j=1

As above it follows that hα ∈ As,p,qτ0 (Rn ) \ As,p,qτ1 (Rn ) .  

This finishes the proof of Proposition 2.2.

4.2.4 The Wavelet Characterization of Fs∞,q (Rn ) s (Rn ). By means of Proposition A case of particular importance is given by F∞,q 2.4(iii) and Theorem 4.2 we obtain in case s > n max{0, 1q − 1}: a locally intes,1/p

s (Rn ) if, and only if f belongs to F n grable function f belongs to F∞,q p,q (R ) for some p < ∞, if, and only if

||| f ||| s (Rn ) ≡ sup | f , φ0,k | + F∞,q k∈Zn

1 1/p |P| {P∈Q: |P|≤1} sup

$ 1/q $ $ $ ∞ 2n −1 $ $ jsq q $ $ Q jk | × $ ∑ ∑ ∑ 2 | f , ψi, j,k χ $ $ $ j= jP i=1 k∈IP, j

<∞

L p (P)

n , ∞). Let q ∈ (0, ∞). Then we can choose p = q. Interchanging for some p ∈ ( n+s s,1/q

s,1/q

integration and summation (or use Fq,q (Rn ) = Bq,q (Rn )) we can carry out the integration and obtain the following corollary. Corollary 4.2. Let q ∈ (0, ∞) and s ∈ (n max{0, 1q − 1}, ∞). A tempered distribution s (Rn ) if, and only if f ∈ S  (Rn ) belongs to F∞,q

4.2 Characterizations by Wavelets

101

||| f ||| s (Rn ) ≡ sup | f , φ0,k | + F∞,q k∈Zn



×

1 1/q {P∈Q: |P|≤1} |P| sup

∞ 2n −1

∑ ∑ ∑

2

j(s+n( 21 − 1q ))q

1/q

| f , ψi, j,k |

q

< ∞ . (4.31)

j= jP i=1 k∈IP, j

Furthermore, ||| f ||| Fs

n ∞,q (R )

and  f F∞,q s (Rn ) are equivalent.

Remark 4.2. (i) Using Proposition 5.1 below, we obtain s 0 (Rn ) = I−s (F∞,2 (Rn )) = I−s ( bmo (Rn )) . F∞,2

Wavelet characterizations of the homogeneous counterparts of these spaces have been obtained in [5]. (ii) Interesting limiting cases are bmo (Rn ) and vmo (Rn ); see items (xiii) and (xiv)  be a compactly in Sect. 1.3. Also BMO (Rn ) is of interest, of course. Let ψ supported, continuously differentiable wavelet on R and let ψ1 , . . . , ψ2n −1 be the associated generators for a wavelet basis of L2 (Rn ). Only here we shall use the convention ψi, j,k (x) ≡ 2 jn/2 ψi (2 j x − k) also for j < 0. Then a locally integrable function f belongs to BMO (Rn ) if, and only if 1 sup 1/2 |P| P∈Q



1/2

∞ 2n −1

∑ ∑ ∑

| f , ψi, j,k |

< ∞;

2

j= jP i=1 k∈IP, j

see [99, Sect. 5.6] and [156, Example 8.8]. In the literature sometimes the convention   1/2

n

∑ | f , ψi, j,k |2

| f , ψQ | ≡

i=1

is used with Q = Q j,k . In this language we obtain that a locally integrable function f belongs to BMO (Rn ) if, and only if 

1 sup |P| P∈Q

1/2

∑ | f , ψQ |2

< ∞,

Q⊂P

The formula (4.31) remains to be true for bmo(Rn ), i. e., if s = 0 and q = 2. For this result we refer to [5]. Moreover, there one can also find a wavelet characterization of vmo (Rn ). Defining for ε > 0  Nε ( f ) ≡ sup

l(P)≤ε

1 |P|

1/2

∑

Q⊂P

| f , ψQ |2

102

4 Several Equivalent Characterizations

and N0 ( f ) ≡ lim Nε ( f ) , ε →0

then a function f ∈

bmo(Rn )

belongs to vmo (Rn ) if, and only if N0 ( f ) = 0.

4.3 Characterizations of As,p,τq (Rn ) by Differences Since the restrictions for Besov spaces differ from those for Triebel-Lizorkin spaces, we split the description of our results. Recall, the three expressions  f ♣ , s, τ A (Rn ) p, q

 f ♥ and  f ♠ have been defined in Sect. 4.1.3. In addition we need s, τ s, τ A (Rn ) A (Rn ) p, q

p, q

we denote the quantity obtained by the following local versions. By ||| f |||♣ As, τ (Rn ) p, q

replacing, in the definition of  f ♣ , s, τ A (Rn ) p, q

sup

by

sup {P∈Q: |P|≤1}

P∈Q

.

Similar are the definitions of ||| f |||♥ and ||| f |||♠ . Observe that in all local s, τ s, τ A (Rn ) A (Rn ) p, q

p, q

cases the integral (supremum) with respect to t extends over the interval (0, 2l(P)) instead of 2(l(P) ∧ 1). The following general rule applies to all results within this section. If we can , then we get a characteriprove a characterization of As,p,τq (Rn ) by using ||| f |||♣ s, τ A (Rn ) p, q

zation as well by using  f ♣ ; the same is true for ||| f |||♥ and ||| f |||♠ . s, τ s, τ s, τ A (Rn ) A (Rn ) A (Rn ) p, q

p, q

p, q

However, in such a situation we concentrate on the first possibility. For later applications we also mention that we can always switch in our characterizations from the used annulus {h ∈ Rn : t/2 ≤ |h| < t} to the associated ball {h ∈ Rn : |h| < t} without changing the assertion.

s, τ n 4.3.1 Characterizations of Fp, q (R ) by Differences

We shall distinguish three cases by following the different estimates we got for the remainder TN,P in Lemma 4.4. Most transparent is the situation if p ∈ [1, ∞). Theorem 4.3. Let p ∈ [1, ∞), q ∈ (0, ∞],

σ1,q < s ≤ max{s, s + nτ − n/p} < M with M ∈ N and τ ∈ [0, τs,p,q ).

4.3 Characterizations of As,p,τq (Rn ) by Differences

103

(i) A function f ∈ Fp,s, qτ (Rn ) if, and only if f ∈ Lτ (Rn ) and  f ♣ < ∞. s, τ F (Rn ) p

p, q

and  f Fp,s, qτ (Rn ) are equivalent. Moreover, Further,  f Lτp (Rn ) +  f ♣ s, τ F (Rn ) p, q

when q ∈ [1, ∞], then these assertions remain true if  f ♣ is replaced s, τ F (Rn ) p, q

. by  f ♥ s, τ F (Rn ) p, q

(ii) Let in addition τ ∈ [1/p, ∞). A function f ∈ Fp,s, qτ (Rn ) if, and only if N p ( f ) < ∞ < ∞. Further, N p ( f ) + ||| f |||♣ and  f Fp,s, τq (Rn ) are equivaand ||| f |||♣ s, τ s, τ F (Rn ) F (Rn ) p, q

p, q

lent. Moreover, when q ∈ [1, ∞], then these assertions remain true if ||| f |||♣ s, τ F (Rn ) p, q

is replaced by ||| f |||♥ . s, τ F (Rn ) p, q

. By Minkowski’s inequality, we find Proof. Step 1. We deal with  f ♣ s, τ F (Rn ) p, q

1 |P|τ

# P

|Φ ∗ f (x)| p dx

1/p   f Lτp (Rn )

as long as P ∈ Q and |P| ≥ 1. Moreover, we know Fp,s, qτ (Rn ) ⊂ Lτ (Rn ); see Propositions 2.1 and 2.7. Now we make use of Lemmas 4.2, 4.3 and 4.4(i) to obtain p

  f Fp,s, τq (Rn ) 

 f  + s, τ A p, q (Rn )

1 sup τ {P∈Q: l(P)≥1} |P|

#

|Φ ∗ f (x)| dx

1/p

p

P

+ TN,P +  f Lτp (Rn ) .   f ♣ s, τ F (Rn ) p, q



p  f ♣ s, τ n +  f Lτ (Rn ) . Fp, q (R )

Next we employ Lemmas 4.7, 4.8, combine it with Theorem 4.1(ii), and obtain +  f Lτp (Rn )   f  .  f ♣ s, τ s, τ F (Rn ) F (Rn ) p, q

p, q

. This proves the claim for  f ♣ s, τ F (Rn ) p, q

Step 2. We deal with  f ♥ . In comparison with Step 1 we have to mods, τ F (Rn ) p, q

ify the estimate from below. For this, under the restriction 1 ≤ q ≤ ∞, we refer to Lemma 4.6. Step 3. Proof of (ii). We employ Lemma 4.5 instead of Lemma 4.3,  f ♣ s, τ F (Rn ) p, q

(resp.  f ♥ ) replaced by ||| f |||♣ (resp. ||| f |||♥ ). For the replacement s, τ s, τ s, τ F (Rn ) F (Rn ) F (Rn ) p, q

of  f Lτp (Rn ) by N p ( f ), see (2.5).

p, q

p, q

 

Remark 4.3. For τ = 0 we are back in the classical situation; see [83, 85, 91] and [145, Sect. 2.5.11]. Next we consider the case p ∈ (0, 1) and τ ∈ [1/p, ∞).

104

4 Several Equivalent Characterizations

Theorem 4.4. Let p ∈ (0, 1), q ∈ (0, ∞],

σ p,q < s ≤ max{s, s + nτ − n/p} < M with M ∈ N and τ ∈ [1/p, τs,p,q ). Then f ∈ Fp,s, qτ (Rn ) if, and only if N1 ( f ) < ∞ < ∞. Further, N1 ( f ) + ||| f |||♣ and  f Fp,s, τq (Rn ) are equivalent. and ||| f |||♣ s, τ s, τ F (Rn ) F (Rn ) p, q

p, q

Moreover, when q ∈ [1, ∞], then these assertions remain true if ||| f |||♣ is res, τ F (Rn ) p, q

. placed by ||| f |||♥ s, τ F (Rn ) p, q

Proof. The used arguments are the same as in proof of Theorem 4.3 except the estimate of 1/p # 1 p Φτ ,p ( f ) = sup | Φ ∗ f (x)| dx . τ P {P∈Q: |P|≥1} |P| For the estimate of Φτ ,p ( f ) we shall use Lemma 4.9. This proves that  f Fp,s, qτ (Rn ) and Φτ ,p ( f ) +  f Lτp (Rn ) +  f L1τ (Rn ) +  f ♣ F s, τ (Rn ) p, q

are equivalent. Making use of the argument in (2.5) we obtain for any dyadic cube P with |P| ≥ 1, 1 |P|τ

#

|Φ ∗ f (x)| dx p

P

#

1/p ≤ ≤

sup {Q∈Q: |Q|=1}



sup {Q∈Q: |Q|=1} Q

|Φ ∗ f (x)| dx

1/p

p

Q

|Φ ∗ f (x)| dx  N1 ( f ) ,

where we used in the last step an convolution inequality. Similarly,  f L1τ (Rn ) +  f Lτp (Rn )  N1 ( f ) can be proved. This shows

Φτ ,p ( f ) +  f Lτp (Rn ) +  f L1τ (Rn ) +  f ♣  N1 ( f ) + ||| f |||♣ ; s, τ s, τ F (Rn ) F (Rn ) p, q

p, q

 

see Lemma 4.5. Finally we study the case p ∈ (0, 1) and τ ∈ [0, 1/p). Theorem 4.5. Let p ∈ (0, 1), q ∈ (0, ∞],

σ p,q < s ≤ max{s, s + nτ − n/p} < M

4.3 Characterizations of As,p,τq (Rn ) by Differences

105

with M ∈ N and τ ∈ [0, min{τs,p,q , 1/p}). Let σ p < s0 < s. Then f ∈ Fp,s, qτ (Rn ) if, and only if f ∈ Lτp (Rn ),  f Bs0

p,∞ (2P)

sup

|P|τ

{P∈Q, |P|≥1}

< ∞,

and  f ♣ < ∞. Further, F s, τ (Rn ) p, q

sup {P∈Q, |P|≥1}

 f Bsp,∞ 0 (2P) |P|τ

+  f Lτp (Rn ) +  f ♣ s, τ F (Rn ) p, q

and  f Fp,s, τq (Rn ) are equivalent. Moreover, when q ∈ [1, ∞], then these assertions

remain true if  f ♣ is replaced by  f ♥ . s, τ s, τ F (Rn ) F (Rn ) p, q

p, q

Proof. We only comment on the modifications. The estimate of  f Bs0

p,∞ (2P)

sup {P∈Q, |P|≥1}

|P|τ

from above follows from an argument as in Step 2 of the proof for Theorem 4.1. Further, to estimate of Φτ ,p ( f ) we shall have a closer look to the proof of Lemma 4.9 and combine it with Theorem 4.1 (applied with s0 ). Then we obtain

Φτ ,p ( f ) 

sup {P∈Q, |P|≥1}

 f Bs0

p,∞ (2P)

|P|τ

.  

The proof is complete.

up to now. We summarize our findings Nothing has been said about  f ♠ s, τ F (Rn ) p, q

also with respect to this quantity as follows. Theorem 4.6. We may replace  f ♣ by  f ♠ in Theorems 4.3–4.5 if s, τ s, τ F (Rn ) F (Rn ) p, q

additionally s ∈ (n/ min{p, q}, ∞) is satisfied.

p, q

Remark 4.4. (i) Further characterizations by differences for Fp,s, qτ (Rn ) can be obtained from the s, 1/u−1/p s (Rn ) (see Corollary 3.3(i)) in combination with (Rn ) = E pqu identity Fu, q Corollary 4.11 below. (ii) For τ = 0 we refer to Seeger [130] and Triebel [146, Sect. 3.5.3].

106

4 Several Equivalent Characterizations

4.3.2 Characterizations of Bs,p,τq (Rn ) by Differences We keep the same structure as in Sect. 4.3.1. Theorem 4.7. Let p ∈ [1, ∞], q ∈ (0, ∞], 0 < s ≤ max{s, s + nτ − n/p} < M with M ∈ N and 0 ≤ τ < τs,p =

s 1 + . n p

(i) Then f ∈ Bs,p,τq (Rn ) if, and only if f ∈ Lτp (Rn ) and  f ♣ < ∞. Furthers, τ B (Rn ) p, q

more,  f Lτp (Rn ) +  f ♣ and  f Bs,p,τq(Rn ) are equivalent. Moreover, these s, τ B (Rn ) p, q

is replaced either by  f ♥ or by assertions remain true if  f ♣ s, τ s, τ B (Rn ) B (Rn ) p, q

 f ♠ . s, τ B (Rn )

p, q

p, q

(ii) Let in addition τ ∈ [1/p, ∞). Then f ∈ Bs,p,τq(Rn ) if, and only if N p ( f ) < ∞ < ∞ (or ||| f |||♥ < ∞ or ||| f |||♠ < ∞). Furthermore, and ||| f |||♣ s, τ s, τ s, τ B (Rn ) B (Rn ) B (Rn ) p, q

p, q

p, q

, N p ( f )+ ||| f |||♥ , N p ( f )+ ||| f |||♠ and  f Bs,p,τq(Rn ) N p ( f )+ ||| f |||♣ s, τ s, τ s, τ B (Rn ) B (Rn ) B (Rn ) p, q

p, q

p, q

are equivalent. Proof. The arguments are essentially the same as in the case of Theorem 4.3. Changes are caused by Lemmas 4.7, 4.8 (σ p instead of σ p,q ) and Lemma 4.6.   Most interesting is the case p = q = ∞. Corollary 4.3. Let 0 < s ≤ s + nτ < M with M ∈ N and τ ∈ [0, s/n). Then f ∈ τ (Rn ) if, and only if f ∈ L∞ (Rn ) and Bs,∞,∞ ||| f |||♠ s, τ B

n ∞,∞ (R )

≡

sup {P∈Q: |P|≤1}

Further,  f L∞ (Rn ) + ||| f |||♠ s, τ B

1 |P|τ

n ∞,∞(R )

sup sup |ΔhM f (x)| < ∞ .

t/2≤|h|
τ and  f Bs,∞,∞ (Rn ) are equivalent norms. Moreover,

these assertions remain true if ||| f |||♠ s, τ B ||| f |||♣ τ n . Bs, ∞,∞(R )

t −s

sup 0
n ∞,∞ (R )

is replaced either by ||| f |||♥ s, τ B

n ∞,∞(R )

or by

Remark 4.5. There are many references for the case τ = 0. We refer to [14,109], [9, Theorem 6.2.5], and [145, Sect. 2.5.12]. Next we consider the case p ∈ (0, 1) and τ ∈ [1/p, ∞).

4.3 Characterizations of As,p,τq (Rn ) by Differences

107

Theorem 4.8. Let p ∈ (0, 1), q ∈ (0, ∞],

σ p < s ≤ max{s, s + nτ − n/p} < M with M ∈ N and

1 s ≤ τ < τs,p = 1 + . p n

Then f ∈ Bs,p,τq(Rn ) if, and only if N1 ( f ) < ∞ and ||| f |||♣ < ∞. Further, N1 ( f ) + s, τ B (Rn ) p, q

and  f Bs,p,τq(Rn ) are equivalent. Moreover, these assertions remain true  f ♣ s, τ B (Rn ) p, q

if ||| f |||♣ is replaced by ||| f |||♠ . s, τ s, τ B (Rn ) B (Rn ) p, q

p, q

Finally we study the case p ∈ (0, 1) and τ ∈ [0, 1/p). Theorem 4.9. Let p ∈ (0, 1), q ∈ (0, ∞],

σ p < s ≤ max{s, s + nτ − n/p} < M with M ∈ N and ' & s 1 . 0 ≤ τ < min τs,p , 1/p = min 1 + , n p )

*

Let σ p < s0 < s. Then f ∈ Bs,p,τq (Rn ) if, and only if f ∈ Lτp (Rn ), sup

 f Bs0

{P∈Q, |P|≥1}

p,∞ (2P)

|P|τ

< ∞,

and  f ♣ < ∞. Further, s, τ B (Rn ) p, q

sup {P∈Q, |P|≥1}

 f Bs0

p,∞ (2P)

|P|τ

+  f Lτp (Rn ) +  f ♣ s, τ B (Rn ) p, q

and  f Bs,p,τq(Rn ) are equivalent. Moreover, these assertions remain true if  f ♣ s, τ F (Rn ) is replaced by  f ♠ . s, τ B (Rn )

p, q

p, q

Remark 4.6. τ (i) Further characterizations by differences for Bs,p,∞ (Rn ) can be obtained from the s, 1/u−1/p s (Rn ) (see Corollary 3.3(ii)) in combination with (Rn ) = N p∞u identity Bu, ∞ Corollary 4.13 below. (ii) For τ = 0 we refer to Triebel [146, Sect. 3.5.3].

108

4 Several Equivalent Characterizations

4.3.3 The Classes As,p,τq (Rn ) and Their Relations to Q Spaces In the homogeneous situation the case of first order differences in a symmetrized version has a certain history, connected with Q spaces; see item (xx) in Sect. 1.3. Here we are studying the inhomogeneous counterpart. Let M = 1 and choose p = q. Then the following corollary is a simple conclusion of Theorem 4.7. Recall that s, τ (Rn ) = Bs,p,τp(Rn ). As,p,pτ (Rn ) = Fp,p

Corollary 4.4. Let p ∈ [1, ∞], 0 < s ≤ max{s, s + nτ − n/p} < 1

and

0≤τ <

1 s + . p n

Then f ∈ As,p,pτ (Rn ) if, and only if f ∈ Lτp (Rn ) and  f ♦As, τ (Rn ) ≡ sup p,p

P∈Q

1 |P|τ

& |h|<2(l(P)∧1)

|h|−sp

 P

| f (x + h) − f (x)| p dx

dh |h|n

'1/p

< ∞.

Furthermore,  f Lτp (Rn ) +  f ♦As, τ (Rn ) and  f As,p,pτ (Rn ) are equivalent. If in addition p,p

τ ∈ [1/p, ∞), then it will be enough to extend the supremum in  f ♦As, τ (Rn ) only over p,p

those dyadic cubes P such that |P| ≤ 1.

. In  f ♥ Proof. Our starting point is Theorem 4.7 with respect to  f ♥ Bs, τ (Rn ) Bs, τ (Rn ) p,p

p,p

we may interchange the order of integration and obtain  2(l(P)∧1) 0

∼

t −sp



t −n P



|h|<2(l(P)∧1)

 t/2≤|h|
|h|−sp



P

| f (x + h) − f (x)| p dh dx

| f (x + h) − f (x)| p dx

dt t

dh . |h|n  

The proof is complete. Now it is only a small step to the next conclusion. Corollary 4.5. Let p, s and τ be as in Corollary 4.4. (i) Then f ∈ As,p,pτ (Rn ) if, and only if f ∈ Lτp (Rn ) and ||| f |||♦As, τ (Rn ) ≡ sup p,p

P∈Q

1 |P|τ

&  P

| f (x) − f (y)| p dx dy sp+n P |x − y|

'1/p

< ∞.

Furthermore,  f Lτp (Rn ) + ||| f |||♦As, τ (Rn ) and  f As,p,pτ (Rn ) are equivalent. p,p

(4.32)

4.3 Characterizations of As,p,τq (Rn ) by Differences

109

(ii) If in addition τ ∈ [1/p, ∞) then it will be enough to extend the supremum in (4.32) over those dyadic cubes P such that |P| ≤ 1. Proof. Step 1. Replacing x + h by y in the definition of  f ♦As, τ (Rn ) we obtain p,p

|x − y| < 2(l(P) ∧ 1). Now we distinguish two cases, small cubes and large cubes. Let P be a dyadic cube with l(P) ≤ 1. Then 1 |P|τ

& |h|<2(l(P)∧1)

|h|−sp

 P

| f (x + h) − f (x)| p dx

'1/p | f (x) − f (y)| p dx dy sp+n 4P P |x − y| '1/p &  1 | f (x) − f (y)| p  sup dx dy . τ sp+n Q Q |x − y| Q∈Q |Q| 1 ≤ |P|τ

& 

dh |h|n

'1/p

Now let P be a dyadic cube with l(P) > 1. Then it is obvious that  |h|<1

|h|−sp

 P

| f (x + h) − f (x)| p dx

dh ≤ |h|n



 4P

| f (x) − f (y)| p dx dy . sp+n P |x − y|

Hence  f ♦As, τ (Rn )  ||| f |||♦As, τ (Rn ) follows. p,p

p,p

Step 2. We turn to the converse inequality. It is enough to consider large cubes. For small cubes we can argue as in Step 1. Let P be a dyadic cube with l(P) > 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|<1 P '1/p &  | f (x) − f (y)| p 1 I2 ≡ dx dh . |P|τ |h|sp+n |h|>1 P I1 ≡

1 |P|τ

&



The first term I1 can be estimated by   f ♦As, τ (Rn ) . The second term I2 allows an p,p

estimate by  f Lτp (Rn ) . This finishes the proof of (i). The modifications for proving (ii) are obvious.   Remark 4.7. The above Corollary could be formulated also as follows: The spaces As,2,τ2 (Rn ) and Qα (Rn ) ∩ L2τ (Rn ) coincide in the sense of equivalent norms as far as 0 < s = α < 1 and τ = 12 − α /n ≥ 0.

110

4 Several Equivalent Characterizations

4.3.4 The Characterization of Fs∞, q (Rn ) by Differences We concentrate on a combination of Proposition 2.4, we will use the identity s, 1/q

s, 1/q

s Fq,q (Rn ) = Bq,q (Rn ) = F∞,q (Rn ),

and Theorem 4.7 if q ∈ [1, ∞) or Theorem 4.8 if q ∈ (0, 1). The case q = ∞ has been treated in Corollary 4.3. First we study the case that q ∈ [1, ∞). s (Rn ) if, and Corollary 4.6. Let q ∈ [1, ∞) and s ∈ (0, M) with M ∈ N. Then f ∈ F∞,q only if Nq ( f ) < ∞ and

 ||| f |||♠ s (Rn ) F∞,q

≡

sup {P∈Q: |P|≤1}

1 |P|

Further, Nq ( f ) + ||| f |||♠ Fs

n ∞,q (R )

 2 l(P)

t



−sq

0

sup P

t/2≤|h|
|ΔhM

dt f (x)| dx t

1/q

q

< ∞.

and  f F∞,q s (Rn ) are equivalent.

We continue with q ∈ (0, 1). Corollary 4.7. Let q ∈ (0, 1) and n −n < s < M q s (Rn ) if, and only if N ( f ) < ∞ and ||| f |||♠ with M ∈ N. Then f ∈ F∞,q 1 Fs

Furthermore,

N1 ( f ) + ||| f |||♠ s (Rn ) F∞,q

n ∞,q (R )

< ∞.

and  f F∞,q s (Rn ) are equivalent.

Remark 4.8. s (Rn ) ⊂ L∞ (Rn ) if s > 0, it is not difficult to prove (i) Let q ∈ (0, ∞). Since F∞,q s (Rn ) if, and only if f ∈ L∞ (Rn ) that under the above restrictions, f ∈ F∞,q ♠ and ||| f |||♠ s (Rn ) are s (Rn ) < ∞. Furthermore,  f L∞ (Rn ) + ||| f |||F s (Rn ) and  f F∞,q F∞,q ∞,q equivalent. (ii) Of course, in both cases one could also work with &  2l(P) 1 ||| f |||♣ ≡ sup t −sq s n F∞,q (R ) {P∈Q: |P|≤1} |P| 0 q '1    dt q −n M × |Δh f (x)|dh dx t t P t/2≤|h|
& sup {P∈Q: |P|≤1}

We omit the details.

1 |P|

 2l(P)

t 0

−sq





t P

−n

|ΔhM t/2≤|h|
dt f (x)| dh dx t q

' 1q

.

4.4 Characterizations via Oscillations

111

4.4 Characterizations via Oscillations In this section we establish the oscillation characterization of As,p,τq (Rn ) which leads to the interpretation as local approximation spaces.

4.4.1 Preparations Let M ∈ Z+ . Denote by PM (Rn ) the set of all polynomials of total degree less than or equal to M. For a fixed ball B(x,t) = {y ∈ Rn : |x − y| < t} and u ∈ (0, ∞], define the local oscillation of f by setting, for all x ∈ Rn and t ∈ (0, ∞), oscM u

#  f (x,t) ≡ inf t −n

1/u | f (y) − P(y)| dy u

B(x,t)

,

(4.33)

where the infimum is taken over all polynomials P(y) ∈ PM (Rn ) and suitable modification is made when u = ∞; see, for example, [146, Sect. 1.7.2]. Following Seeger [130], we deduce that differences can be easily dominated by oscillations. Recall, the abbreviations  f B♣ (Q) and  f F ♣ (Q) have been introduced in Sect. 4.2.2. Lemma 4.10. Let M ∈ N, Q ∈ Q, p, q ∈ (0, ∞], τ ∈ [0, ∞), and s > 0. (i) There exists a positive constant C such that for all f ∈ L1loc (Rn ), 1  f B♣ (Q) ≤ C τ |Q|



2(l(Q)∧1)

t

−sq



0

Q

+ M−1 ,p osc1 f (x, Mt) dx

q/p

dt t

1/q

(ii) There exists a positive constant C such that for all f ∈ L1loc (Rn ), 1  f F ♣ (Q) ≤ C τ |Q|

 

2(l(Q)∧1)

t Q

−sq

0

+ M−1 ,q dt osc1 f (x, Mt) t

1/p

 p/q dx

.

Proof. For any x ∈ Rn , let Pt f ∈ PM−1 (Rn ) be the best approximate of f in L1 (B(x, Mt)). It follows that

ΔhM f (x) = ΔhM ( f − Pt f )(x) = (−1) ( f (x) − Pt f (x)) + M

M−1

∑ (−i)

i=0

i

#

 M ( f − Pt f )(x + (M − i)h). i

112

4 Several Equivalent Characterizations

Clearly, if i = 0, . . . , M − 1, then 1 |Q|τ



2(l(Q)∧1)

1  |Q|τ

|h|
Q

0

1  |Q|τ



2(l(Q)∧1)

t

−sq

Q 2(l(Q)∧1)

t −sq

|( f − Pt f )(x + (M − i)h)| dh p

 #  t −n

0



p

 #  −sq t t −n

 Q

0

B(x,Mt)

|( f − Pt f )(y)| dy

+ M−1 ,p osc1 f (x, Mt) dx

 qp

 qp dx

 qp dx

dt t

dt t

 1q

 1q

 1q

dt t

.

Now we turn to the estimate of the term for i = M. Recall that for almost every x ∈ Rn , f (x) = liml→∞ P2−l t f (x) and |Pt f (x)| ≤

1 |B(x,t)|

 B(x,t)

| f (y)| dy;

see [46, (2.3) and (2.7)]. Furthermore, if P ∈ PM−1 (Rn ), then & inf

P∈PM−1 (Rn )

B

'  | f (y) + P(y) − P(y)| dy =

& inf

P∈PM−1 (Rn )

B

' | f (y) − P(y)| dy .

Consequently, | f (x) − Pt f (x)| ≤

∞

∑ |P2−l−1t f (x) − P2−lt f (x)|

l=0 ∞

≤ 2 ∑ |P2−l t ( f − P2−l t f ) (x)| 

l=0 ∞

1

∑ |B(x, 2−l t)|

l=0

 B(x,2−l t)

| f (y) − P2−l t f (y)| dy.

Next we employ (4.15) and obtain with d = min{1, p, q} that 

2(l(Q)∧1)

0



∞

∑

 #  t −sq t −n 2(l(Q)∧1)

t

×

−sq

 # Q

0

l=0

|h|
Q



p |( f − Pt f )(x)| dh

B(x,2−l t)

dx

1 |B(x, 2−l t)| p



 qp

| f (y) − P2−l t f (y)| dy

 qp dx

dt t

 dq

dt t

 dq

4.4 Characterizations via Oscillations ∞

∑ 2−lsd



&

2(l(Q)∧1)

0

l=0

113

ω −sq

 # Q

1 |B(x, ω )|

 dq dω | f (y) − Pω f (y)| dy dx × ω B(x,ω )  dq   qp  2(l(Q)∧1) , + dt p  t −sq f (x, Mt) dx , oscM−1 1 t Q 0 p



 qp

since s > 0. Similarly, for the F-case, we employ (4.16) and proceed as above. This finishes the proof.   Next we recall Whitney’s approximation theorem in a version given in [70, Theorem A.1] but traced to Brudnyi [27] and Nevskii [107]. We refer also to the appendix in [70] for some further remarks concerning the rich history of this result. Proposition 4.1. Let p ∈ (0, ∞], M ∈ N, ρ ∈ (0, 1] and f ∈ L ploc (Rn ). Then there exists a positive constant C = C(M, ρ , p, n) such that for any cube Q with side length a there is a polynomial P ∈ PM−1 (Rn ) satisfying  Q

| f (y) − P(y)| p dy ≤ C a−n



 |h|<ρ a

Q

|ΔhM f (y)| p dydz .

(4.34)

As an immediate consequence of (4.34) we obtain the inequality f (x,t)  oscM−1 p

#  t −2n



|h|<ρ t

#

 t −n/p sup

|h|<ρ t

Q

Q

|ΔhM f (y)| p dy dh |ΔhM f (y)| p dy

1/p

1/p ,

(4.35)

where B(x,t) ⊂ Q and Q has side length 2t. Observe, oscM−1 f (x,t) is monotone with p respect to p. In view of this property and with p ≥ 1, Lemma 4.10 in combination with (4.35) yields characterizations by means of oscillations as long as the space is characterized by  f ♣ and  f ♠ simultaneously. Since we know characterizations of Fp,s, qτ (Rn ) by means of  f ♠ only under s, τ F (Rn ) p, q

very restrictive conditions (see Theorem 4.6), we proceed in a different way. We shall compare oscillations and wavelet coefficients. Most of the arguments will be the same as used in the comparison of differences and wavelet coefficients. We need some more abbreviations. For P ∈ Q and f ∈ L1loc (Rn ) we define 1  f B(P) ≡ |P|τ

 0

2(l(P)∧1)

t −sq

 P

q/p (oscM−1 f (x,t)) p dx 1

dt t

1/q ,

114

4 Several Equivalent Characterizations

and 1  f F(P) ≡ |P|τ

  

2(l(P)∧1)

t P

0

−sq

+ M−1 ,q dt osc1 f (x,t) t

1/p

 p/q

.

dx

Lemma 4.11. (i) Let p, q, s, τ be as in Lemma 4.7(i). Then there exists a positive constant C such that for all Q ∈ Q and f ∈ L1loc (Rn ),  f − W jQ f B(Q) ⎧ ⎛ ⎞q/p ⎫1/q ⎪ ⎪ ⎨ ⎬ 2n −1 ∞ 1 ω (s+n/2)q −ω n p⎠ ⎝ 2 ≤C τ 2 |ai,ω ,r | . ∑ ∑ ∑ ⎪ |Q| ⎪ ⎩ω =( jQ ∨0) ⎭ i=1 r∈IQ,ω (ii) Let p, q, s, τ be as in Lemma 4.7(ii). Then there exists a positive constant C such that for all Q ∈ Q and f ∈ L1loc (Rn ), $⎡ ⎤1/q $ $ $ $ $ ∞ 2n −1 1 $⎣ wsq q Qω r | ⎦ $  f − W jQ f F(Q) ≤ C τ $ 2 |ai,ω ,r χ $ ∑ ∑ ∑ $ |Q| $ ω =( j ∨0) i=1 r∈I Q,ω Q $ $

.

L p (Q)

Proof. We follow the proof of Lemma 4.7 and in particular, employ the same notions as there. Instead the characterization by differences we employ the oscillation characterization of Asp, q (Rn ) (see [146, Theorem 1.7.3]) in combination with the wavelet characterization of Asp, q (Rn ) (see Sect. 4.2.1). As above this yields 1 gFp,s q(Rn ) |Q|τ $⎡ ⎤1/q $ $ $ n $ $ 2 −1 1 $⎣ ∞ $ wsq q ⎦  ∼ 2 |a χ | $ $ i,ω ,r Qω r ∑ ∑ $ |Q|τ $ ω =(∑ jQ ∨0) i=1 r∈IQ,ω $ $

 f − W jQ f F(Q) ≤

,

L p (Q)

and similar estimates are true for the B-case. This finishes the proof of Lemma 4.11.   Now we turn to the counterpart of Lemma 4.8. Lemma 4.12. Let Q ∈ Q and p, q, M, s, τ be as in Lemma 4.8. Then there exists a positive constant C such that for all f ∈ L1loc (Rn ), . W jQ f A(Q) ≤ C f  s, τ A (Rn ) p, q

4.4 Characterizations via Oscillations

115

Proof. We follow the proof of Lemma 4.8. Instead of (4.28) we use the inequality % % % % 1 ∂αg % α% (x)(y − x) %  gCM (Rn ) |y − x|M %g(y) − ∑ α % % α ! ∂y |α |≤M−1 with respect to g = φ0,k and g = ψi, j,k . No further ideas are needed. Details are left to the reader.  

4.4.2 Oscillations and Besov-Type Spaces Most of our arguments are based on the characterization by differences. So we keep the structure of displaying the results as there. We shall use the abbreviations ≡ sup  f A(P) ,  f ♣,os As, τ (Rn ) p, q

P∈Q

 f ♣,osl s, τ A p, q (Rn )

≡

sup {P∈Q: |P|≤1}

 f A(P)

with A ∈ {B, F}. Theorem 4.10. Let p ∈ [1, ∞], q ∈ (0, ∞], 0 < s ≤ max{s, s + nτ − n/p} < M with M ∈ N and 0≤τ <

s 1 + . n p

(i) Then f ∈ Bs,p,τq(Rn ) if, and only if f ∈ Lτp (Rn ) and  f ♣,os < ∞, where s, τ B (Rn ) p, q

≡ sup  f ♣,os s, τ B (Rn ) p, q

P∈Q

1 |P|τ

&

2(l(P)∧1)

t −sq

0

×

 P

+ M−1 ,p osc1 f (x,t) dx

q

p

and  f Bs,p,τq(Rn ) are equivalent. Further,  f Lτp (Rn ) +  f ♣,os s, τ B (Rn ) p, q

dt t

 1q . (4.36)

(ii) Let additionally τ ∈ [1/p, ∞). Then f ∈ Bs,p,τq (Rn ) if, and only if N p ( f ) < ∞ and  f ♣,osl < ∞. Further, N p ( f ) +  f ♣,osl and  f Bs,p,τq(Rn ) are equivalent. s, τ s, τ B (Rn ) B (Rn ) p, q

p, q

Proof. The inequality  f Lτp (Rn ) +  f ♣,os   f Bs,p,τq(Rn ) s, τ B (Rn ) p, q

116

4 Several Equivalent Characterizations

follows from Theorem 4.7 and (4.35). The inequality  f Bs,p,τq (Rn )   f Lτp (Rn ) +  f ♣,os s, τ B (Rn ) p, q

 

is a consequence of Theorem 4.7 and Lemma 4.10. by Remark 4.9. The same proof yields the following: if we replace  f ♣,os s, τ B (Rn ) p, q

 f ♥,os s, τ B p, q (Rn )

1 ≡ sup τ P∈Q |P|



2(l(P)∧1)

t

−sq



0

P

+ M−1 ,p osc p f (x,t) dx

q/p

dt t

1/q ,

then all assertions in Theorem 4.10 remain true. Also in this case we may replace  f ♥,os by  f ♥,osl , defined in analogy to  f ♣,osl , if τ ≥ 1/p. Bs, τ (Rn ) Bs, τ (Rn ) Bs, τ (Rn ) p, q

p, q

p, q

Next we consider the case p ∈ (0, 1) and τ ∈ [1/p, ∞). Theorem 4.11. Let p ∈ (0, 1), q ∈ (0, ∞],

σ p < s ≤ max{s, s + nτ − n/p} < M with M ∈ N and s 1 ≤ τ < 1+ . p n Then f ∈ Bs,p,τq(Rn ) if, and only if N1 ( f ) < ∞ and  f ♣,osl < ∞. Further, N1 ( f ) + s, τ B (Rn ) p, q

 f ♣,osl and  f Bs,p,τq (Rn ) are equivalent. s, τ B (Rn ) p, q

Proof. We combine Theorem 4.8, Lemmas 4.10–4.12 and Theorem 4.1. Finally we study the case p ∈ (0, 1) and τ ∈ [0, 1/p). Theorem 4.12. Let p ∈ (0, 1), q ∈ (0, ∞],

σ p < s ≤ max{s, s + nτ − n/p} < M with M ∈ N and

& ' s 1 0 ≤ τ < min 1 + , . n p

Let σ p < s0 < s. Then f ∈ Bs,p,τq (Rn ) if, and only if Φτ ,p ( f ) < ∞, sup {P∈Q, |P|≥1}

 f Bs0

p,∞ (2P)

|P|τ

< ∞,

 

4.4 Characterizations via Oscillations

117

and  f ♣,os < ∞. Further, s, τ B (Rn ) p, q

 f Bs0

p,∞ (2P)

sup {P∈Q, |P|≥1}

|P|τ

+ Φτ ,p ( f ) +  f Lτp (Rn ) +  f ♣,os s, τ B (Rn ) p, q

and  f Bs,p,τq(Rn ) are equivalent. Proof. We combine Theorem 4.9, Lemmas 4.10–4.12 and Theorem 4.1.

 

Remark 4.10. τ (Rn ) can be obtained from the (i) Further characterizations by oscillations for Bs,p,∞ s, 1/u−1/p n s n (R ) = N p∞u (R ) (see Corollary 3.3(ii)) in combination with identity Bu, ∞ Corollary 4.17 below. (ii) For τ = 0 we refer to Triebel [146, Sect. 3.5.2]. Some more references can be found in [146, Sect. 1.7].

4.4.3 Oscillations and Triebel-Lizorkin-Type Spaces Now we are in the position to formulate the results with respect to Fp,s, qτ (Rn ). Theorem 4.13. Let p ∈ [1, ∞), q ∈ (0, ∞],

σ1,q < s ≤ max{s, s + nτ − n/p} < M with M ∈ N and τ ∈ [0, τs,p,q ). < ∞. Further, (i) Then f ∈ Fp,s, qτ (Rn ) if, and only if f ∈ Lτp (Rn ) and  f ♣,os s, τ F (Rn ) and  f Fp,s, τq (Rn ) are equivalent.  f Lτp (Rn ) +  f ♣,os s, τ F (Rn )

p, q

p, q

(ii) Let additionally τ ∈ [1/p, ∞). Then f ∈ Fp,s, qτ (Rn ) if, and only if N p ( f ) < ∞ and < ∞. Further, N p ( f ) +  f ♣,osl and  f Fp,s, qτ (Rn ) are equivalent.  f ♣,osl s, τ s, τ F (Rn ) F (Rn ) p, q

p, q

Proof. We combine Theorem 4.13, Lemma 4.10, Theorem 4.1(ii) and Lemmas 4.11, 4.12.   Next we consider the case p ∈ (0, 1) and τ ∈ [1/p, ∞). Theorem 4.14. Let p ∈ (0, 1), q ∈ (0, ∞],

σ p,q < s ≤ max{s, s + nτ − n/p} < M with M ∈ N and τ ∈ [1/p, τs,p,q). Then f ∈ Fp,s, qτ (Rn ) if, and only if N1 ( f ) < ∞ and  f ♣,osl < ∞. Further, N1 ( f ) +  f ♣,osl and  f Fp,s, qτ (Rn ) are equivalent. s, τ s, τ F (Rn ) F (Rn ) p, q

p, q

118

4 Several Equivalent Characterizations

Proof. We combine Theorem 4.14, Lemma 4.10, Theorem 4.1(ii) and Lemmas 4.11, 4.12.   Finally we study the case p ∈ (0, 1) and τ ∈ [0, 1/p). Theorem 4.15. Let p ∈ (0, 1), q ∈ (0, ∞],

σ p,q < s ≤ max{s, s + nτ − n/p} < M with M ∈ N and τ ∈ [0, min{τs,p,q , 1/p}). Let σ p < s0 < s. Then f ∈ Fp,s, qτ (Rn ) if, and only if Φτ ,p ( f ) < ∞,  f Bsp,∞ 0 (2P)

sup {P∈Q, |P|≥1}

|P|τ

< ∞,

and  f ♣,os < ∞. Further, F s, τ (Rn ) p, q

sup {P∈Q, |P|≥1}

 f Bs0

p,∞ (2P)

|P|τ

+ Φτ ,p ( f ) +  f Lτp (Rn ) +  f ♣,os s, τ F (Rn ) p, q

and  f Fp,s, qτ (Rn ) are equivalent. Proof. We combine Theorem 4.14, Lemma 4.10, Theorem 4.1(ii), Lemmas 4.11 and 4.12.   Remark 4.11. (i) Further characterizations by oscillations for Fp,s, qτ (Rn ) can be obtained from the s, 1/u−1/p s (Rn ) (see Corollary 3.3(i)) in combination with (Rn ) = E pqu identity Fu, q Corollary 4.16 below. (ii) For τ = 0 we refer to Seeger [130] and Triebel [146, Sect. 3.5.2]. (iii) A technical remark. Let C ≥ 2 be a fixed positive constant. All the theorems in Sects. 4.3 and 4.4 remain true if we replace 2(l(P) ∧ 1) by C(l(P) ∧ 1).

4.4.4 Oscillations and Fs∞, q (Rn ) We concentrate on a combination of Proposition 2.4, we will use the idens, 1/q s, 1/q s (Rn ), and Theorem 4.10 if q ∈ [1, ∞) or tity Fq,q (Rn ) = Bq,q (Rn ) = F∞,q Theorem 4.11 if q ∈ (0, 1). The case q = ∞ is contained in Theorem 4.10. First we study q ∈ [1, ∞). s (Rn ) if, and Corollary 4.8. Let q ∈ [1, ∞) and s ∈ (0, M) with M ∈ N. Then f ∈ F∞,q only if Nq ( f ) < ∞ and

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

 f ♥,osl F s (Rn ) ≡ ∞,q

& sup {P∈Q: |P|≤1}

1 |P|

 2l(P) 0

t −sq (oscM−1 f (x,t))q q

dt t

119

'1/q < ∞.

Further, Nq ( f ) +  f ♥,osl and  f F∞,q s (Rn ) are equivalent. F s (Rn ) ∞,q

We continue with q ∈ (0, 1). Corollary 4.9. Let q ∈ (0, 1) and n −n < s < M q s (Rn ) if, and only if N ( f ) < ∞ and with M ∈ N. Then f ∈ F∞,q 1

 f ♣,osl ≡ F s (Rn ) ∞,q

& sup {P∈Q: |P|≤1}

1 |P|

 2l(P) 0

t −sq (oscM−1 f (x,t))q 1

dt t

'1/q < ∞.

Furthermore, N1 ( f ) +  f ♣,osl and  f F∞,q s (Rn ) are equivalent. F s (Rn ) ∞,q

s (Rn ) ⊂ L∞ (Rn ) if s > 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

4 Several Equivalent Characterizations

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

4 Several Equivalent Characterizations

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 < u ≤ p ≤ ∞ such that & ' 1 1 1 s > 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 ⎠

<∞.

4.5 The Hedberg-Netrusov Approach to Spaces of Besov-Triebel-Lizorkin Type  s (Rn ) and  f  The quasi-norms  f N pqu Ns

n pqu (R )

123

are equivalent.

As above this can be reformulated a bit. Corollary 4.13. Let u, p, q, v, r, s be as in Corollary 4.12. 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 (Rn ) pqu

≡ sup P∈Q

  

1 1

1

|P| u − p

+ sup P∈Q

P

1

B(x,1)



1

1

t

1

|P| u − p

1/u

u/v | f (y)| dy v

−sq

q/u



 s (Rn ) and  f  The quasi-norms  f N pqu Ns

n pqu (R )

|bv,t (x)| dx u

P

0

dx dt t

1/q < ∞.

are equivalent.

s, 1/u−1/p

s (Rn ) (see Corollary Remark 4.14. In view of the identity Bu, ∞ (Rn ) = N p∞u 3.3(i)), the above corollary supplements the results of Sect. 4.3.2.

4.5.3 Characterizations by Oscillations Corollary 4.10 and 4.12 have direct counterparts in [70] which we now recall. Corollary 4.14. Let v, u, r, p, s be as in Corollary 4.10. 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 #E pqu s (Rn )

< ∞, where

$ 1/q $ $ $ ∞ $ $ / 0 jsq −j q $ $ osc f (·, 2 ) p n +$ ∑ 2 v $ Mu (R ) $ $ j=1

$ $ ≡ $  f Lv (B(·,1)) $

.

p Mu (Rn )

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

n pqu (R )

are equivalent.

Corollary 4.15. Let v, u, r, p, s be as in Corollary 4.12. Then the following assertions are equivalent for functions in Lrloc (Rn ): s (Rn ); (i) f ∈ N pqu

124

4 Several Equivalent Characterizations

(ii) f ∈ Lvloc (Rn ) and  f #N s

n pqu (R )

 f #N pqu s (Rn )

< ∞, where 

$ $ ≡ $  f Lv (B(·,1)) $

∑2

+

p

Mu (Rn )

1/q

∞

jsq

 oscv f (·, 2

−j

j=1

# The quasi-norms  f N pqu s (Rn ) and  f  Ns

q )M p (Rn ) u

.

are equivalent.

n pqu (R )

The quantities oscu f (x,t) are not monotone in general. However, there exists a positive constant C such that for all f ∈ Lvloc (Rn ) and all x ∈ Rn , oscv f (x,t) ≤ C oscv f (x, t¯)

t < t¯ < 2t .

with

This is sufficient for establishing the counterparts of Corollaries 4.11 and 4.13. Corollary 4.16. Let v, u, r, p, s be as in Corollary 4.10. Then the following assertions are equivalent for functions in Lrloc (Rn ): s (i) f ∈ E pqu (Rn ); v (ii) f ∈ L loc (Rn ) and

||| f |||#E pqu s (Rn )

 

1

≡ sup

1/u

u/v

| f (y)| dy 1 1 P B(x,1) |P| u − p $ # 1/q $ $ $ 1 $ $ q dt −sq +$ t [oscv f (x,t)] $ $ $ 0 t v

dx

P∈Q

< ∞. p

Mu (Rn )

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

n pqu (R )

are equivalent.

s, 1/u−1/p

s (Rn ) (see Corollary Remark 4.15. In view of the identity Fu, q (Rn ) = E pqu 3.3(i)), the above corollary supplements the results of Sect. 4.4.3.

Corollary 4.17. Let v, u, r, p, s be as in Corollary 4.12. Then the following assertions are equivalent for functions in Lrloc (Rn ): s (i) f ∈ N pqu (Rn ); (ii) f ∈ Lvloc (Rn ) and

||| f |||#N pqu s (Rn )

≡ sup P∈Q

+

  

1 1

1

|P| u − p

#

0

1

P

B(x,1)

| f (y)| dy

t −sq  oscv f (x,t)qM p (Rn ) u

# The quasi-norms  f N pqu s (Rn ) and ||| f ||| Ns

n pqu (R )

1/u

u/v v

dt t

are equivalent.

dx

1/q

< ∞.

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

125

Remark 4.16. (i) In view of the identity s, 1/u−1/p

Bu, ∞

s (Rn ) = N p∞u (Rn )

(see Corollary 3.3(i)), the above corollary supplements the results of Sect. 4.4.2. s s (Rn ) and N pqu (Rn ) in terms of atoms (ii) In [70] also the characterizations of E pqu are given.

τ n 4.6 A Characterization of As, p, q (R ) via a Localization s n of Ap, q (R )

In this section we always assume that 0<s<1

and

0 < s + nτ −

n < 1. p

(4.39)

Our main tool will be the characterization of As,p,τq (Rn ) by means of first order differences; see Sect. 4.3. Under these restrictions we establish a characterization of As,p,τq (Rn ) via a localization of Asp, q (Rn ), an idea which we picked up from [5], where s (Rn ). Restricted to this section, we such characterizations are investigated for F∞,2 define  f B ≡

 f ♣ F

⎧ ⎨ ⎩

0

2

⎫1/q dt ⎬

$# 1/p $  $ $q $ −sq $ −n p t $ t | f (· + h) − f ( · )| dh $ $ $ p t/2≤|h|
L (Rn )

t ⎭

,

$& q '1/q $ #  $ $ 2 dt $ $ −sq −n ≡$ t | f (· + h) − f ( · )| dh t $ $ $ 0 t t/2≤|h|
L p (Rn )

and $& '1/q $  $ $ 2 dt $ $  f ♥ t −sqt −n | f (· + h) − f ( · )|q dh $ F ≡$ $ $ 0 t t/2≤|h|
.

L p (Rn )

Then [145, Theorem 2.5.9] implies that  f L p (Rn ) +  f A and  f Asp, q(Rn ) are equivalent quasi-norms in Asp, q (Rn ) (s > σ p,q if Asp, q (Rn )=Fp,s q (Rn )). Further, recall that by Proposition 2.6, As,p,τq(Rn ) ⊂ Z s+nτ −n/p(Rn ) = Cs+nτ −n/p(Rn ) since 0 < s + nτ − n/p < 1. Thus, we only deal with H¨older continuous functions.

126

4 Several Equivalent Characterizations

4.6.1 A Characterization of Bs,p,τq (Rn ) We need some preparations. Lemma 4.13. Let p, q ∈ (0, ∞], τ ∈ [0, ∞) and s be as in (4.39). Let

Ψ ∈ Bsp, q(Rn ) ∩ L∞ (Rn ) be a function such that 

supp Ψ ⊂ [−1, 1]n and

Rn

Ψ (y) dy ≡ CΨ > 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|


p

t/2≤|h|


' 1q | f (x + h) − fQ | |ΨQ (x + h)| dh · · · p

t/2≤|h|
p

and I4 ≡

&

2

2l(Q)



' 1q | f (x) − fQ | |ΨQ (x)| dh · · · , p

t/2≤|h|
p

where when l(Q) ≥ 1, I3 and I4 are void. Then it follows that ( f − fQ )ΨQ B  I1 + I2 + I3 + I4 .

(4.47)

Obviously, I1 ≤ |Q|−1/2 Ψ L∞ (Rn )    2(l(Q)∧1) × t −sq t −n 0

PQ

t/2≤|h|
 |Q|τ −1/2 Ψ L∞ (Rn )  f ♥ . s, τ B (Rn ) p, q

|Δh1 f (x)| p dh dx

q/p

dt t

1/q

(4.48)

Next we turn to I2 . Since |h| < t < 2(l(Q) ∧ 1), for either x ∈ PQ or x + h ∈ PQ , we conclude that in any case x, x + h ∈ 3PQ . Hence we may employ (4.44) and find

128

4 Several Equivalent Characterizations

I2  |Q|(s+nτ −n/p)/n f Cs+nτ −n/p (Rn ) Ψ L∞ (Rn )    2(l(Q)∧1)

×

t −sq

Rn

0

t −n

t/2≤|h|
|Δh1ΨQ (x)| p dh dx

q/p

dt t

1/q

∼ |Q|(s+nτ −n/p)/n f Cs+nτ −n/p (Rn ) Ψ L∞ (Rn ) J , where J≡



2(l(Q)∧1)

t

−sq

 Rn

0

t



−n

t/2≤|h|
|Δh1ΨQ (x)| p dh dx

q/p

dt t

1/q .

Let Q ≡ Q j,k . Using a transformation of coordinates we can estimate J 

J = |Q|

2(l(Q)∧1)

1/p−1/2

t

−sq

 Rn

0

= |Q|1/p−1/2

&

2(l(Q)∧1)

0



= |Q|

1/p−1/2 js

t

2

 t/2≤|h|
· · · (2 j t)−n

2 j+1 (l(Q)∧1)

−n

|Δ21j hΨ (x)| p dh dx

 2 j t/2≤|h|<2 j t

#  · · · t −n

t/2≤|h|
0

q/p

dt t

1/q

'1/q |Δh1Ψ (x)| p dh · · · |Δh1Ψ (x)| dh

1/q

p

···

≤ |Q|1/p−1/22 js Ψ Bsp, q(Rn ) . Inserting this estimate in our previous one we get I2  |Q|τ −1/2  f Cs+nτ −n/p (Rn ) Ψ Bsp, q(Rn ) Ψ L∞ (Rn ) .

(4.49)

. Observe To estimate I3 we will have an advantage from working with  f ♥ Bs, τ (Rn ) p, q

that we can apply (4.44). Consequently, I3  |Q|(s+nτ −n/p)/n f Cs+nτ −n/p (Rn )    2

×

t −sq

Rn

2l(Q)

t −n

t/2≤|h|
|ΨQ (x + h)| p dhdx

 |Q|(s+nτ −n/p)/n f Cs+nτ −n/p (Rn ) ΨQ L p (Rn ) (s+nτ −n/p)/n

 |Q|

|Q|

1−1 p 2

|Q|

 |Q|τ − 2  f Cs+nτ −n/p (Rn ) . 1

In the same way one derives

−s/n

&

q/p

2 2l(Q)

t −sq

dt t dt t

1/q '1/q

 f Cs+nτ −n/p (Rn ) (4.50)

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

129

I4  |Q|τ − 2  f Cs+nτ −n/p (Rn ) .

(4.51)

1

Combining (4.48)–(4.51) with (4.47) we get the estimate (4.41), which completes the proof of Lemma 4.13.   Remark 4.17. If supp Ψ ⊂ [−N, N]n for some N, then the assertion (4.41) remains true, probably with a different constant, but this is not of relevance for us. be defined Lemma 4.14. Let p, q ∈ (0, ∞], τ ∈ [0, ∞) and s ∈ (0, 1). Let  · ♥ s, τ A (Rn ) p, q

as in Sect. 4.1.3 with M = 1. Let Ψ be a locally integrable function satisfying √ √ Ψ (x) = 1 if x ∈ [−3 n, 3 n]n .

(4.52)

Then there exists a positive constant C such that for all f ∈ L1loc (Rn ), &

|Q|1/2 ≤ C sup τ Q∈Q |Q|

 f ♥ s, τ B p, q (Rn )

× t −n

2(l(Q)∧1)

t

−sq

 Q

0

 t/2≤|h|
|Δh1 (( f − fQ )ΨQ )(x)| p dh dx

q/p

dt t

1/q

and  f ♥ s, τ Fp, q (Rn )

|Q|1/2 ≤ C sup τ Q∈Q |Q| × t −n

&  Q

2(l(Q)∧1)

t −sq

0



dt |Δh1 (( f − fQ )ΨQ )(x)|q dh t t/2≤|h|
1/p

 p/q dx

.

A similar estimate holds for  f ♣ . F s, τ (Rn ) p, q

Proof. Since

ΨQ (x) = ΨQ (x + h) = |Q|−1/2 , we then have that for all x ∈ Q and t < 2(l(Q) ∧ 1),  t/2≤|h|
|Δh1 f (x)| dh =

 t/2≤|h|
= |Q|1/2

|Δh1 ( f − fQ )(x)| dh



t/2≤|h|
which completes the proof of Lemma 4.14.

|Δh1 (( f − fQ )ΨQ )(x)| dh,  

From Theorem 4.7, Lemmas 4.13, 4.14 and Remark 4.17, we deduce the following conclusion.

130

4 Several Equivalent Characterizations

Theorem 4.16. Let s ∈ R, p ∈ [1, ∞), q ∈ (0, ∞] and τ ∈ [0, ∞) such that ' & 1 <s<1 n max 0, τ − p

and

0 < s + nτ −

n < 1. p

(4.53)

Let Ψ ∈ Bsp, q (Rn ) ∩ L∞ (Rn ) be a function satisfying CΨ > 0 (see (4.40)) and (4.52). (i) Then f ∈ Bs,p,τq(Rn ) if, and only if f ∈ Lτp (Rn ) and  f  ≡ sup |Q|1/2−τ ( f − fQ )ΨQ Bsp, q(Rn ) < ∞. s, τ B (Rn ) p, q

(4.54)

Q∈Q

Furthermore,  f Lτp (Rn ) +  f  and  f Bs,p,τq (Rn ) are equivalent. Bs, τ (Rn ) p, q

(ii) Let in addition τ ∈ [1/p, ∞). Then f ∈ Bs,p,τq(Rn ) if, and only if N p ( f ) < ∞ and ≡ ||| f ||| s, τ B (Rn ) p, q

sup {Q∈Q: |Q|≤1}

|Q|1/2−τ ( f − fQ )ΨQ Bsp, q(Rn ) < ∞.

(4.55)

Furthermore, N p ( f ) + ||| f ||| and  f Bs,p,τq (Rn ) are equivalent. s, τ B (Rn ) p, q

Proof. Step 1. Let f ∈ Bs,p,τq (Rn ). By Propositions 2.7 and 2.1, we know that Bs,p,τq (Rn ) ⊂ Lτp (Rn ). From Proposition 2.6 we derive that Bs,p,τq (Rn ) ⊂ Cs+nτ −n/p(Rn ). Next we employ Lemma 4.13, Remark 4.17 in combination with Theorem 4.7(i) and obtain ( f − fQ )ΨQ B  |Q|τ −1/2  f Bs,p,τq(Rn ) . Recall that the quasi-norms  f L p (Rn ) +  f B and  f Bsp, q(Rn ) are equivalent in Bsp, q (Rn ). Thus, we only need to estimate ( f − fQ )ΨQ L p (Rn ) . We shall use the notation from the proof of Lemma 4.13. In fact, since supp ΨQ ⊂ PQ , we see that ( f − fQ )ΨQ L p (Rn ) ≤ |Q|−1/2 Ψ L∞ (Rn )

# PQ

1/p | f (x) − fQ | p dx

.

When |Q| ≤ 1, by (4.44) and s > 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 H¨older’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

4 Several Equivalent Characterizations

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|
L p (Rn )

,

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

133

$& 2(l(Q)∧1) #  $ I2 ≡ $ · · · t −n | f (x + h) − fQ | $ 0 t/2≤|h|
L p (Rn )

and

$& #  q ' 1q $ $ $ 2 $ $ −n I4 ≡ $ ··· t | f (x) − fQ | |ΨQ (x)| dh · · · $ $ $ 2l(Q) t/2≤|h|
,

L p (Rn )

where, as above, when l(Q) ≥ 1, then I3 and I4 are void. Again we have ( f − fQ )ΨQ ♣ F  I1 + I2 + I3 + I4 . Concerning the estimates of I1 and I2 we can argue as in the proof of Lemma 4.13 and obtain I1  |Q|τ −1/2 Ψ L∞ (Rn )  f ♣ s, τ F (Rn ) p, q

as well as

I2  |Q|τ −1/2  f Cs+nτ −n/p (Rn ) Ψ Fp,s q(Rn ) Ψ L∞ (Rn ) .

To estimate I3 we will use the Hardy-Littlewood maximal function and obtain by using (4.44) that I3  |Q|(s+nτ −n/p)/n f Cs+nτ −n/p (Rn ) $& #  q '1/q $ $  2 $ dt $ $ −sq −n t ×$ t |ΨQ (x + h)| dh $ $ 2l(Q) $ p n t t/2≤|h|
L p (Rn )

τ − 12

 Ψ L p (Rn ) |Q|

 f Cs+nτ −n/p (Rn ) ,

because of 1 < p < ∞. In the same way one derives I4  |Q|τ − 2  f Cs+nτ −n/p (Rn ) . 1

Collecting the estimates of I1 − I4 we have proved (4.57).

 

Remark 4.18. As in the B-case, if supp Ψ ⊂ [−N, N]n for some N, then the assertion (4.57) remains true, probably with a different constant.

134

4 Several Equivalent Characterizations

From Lemmas 4.15, 4.14 and Remark 4.18, we now deduce the counterpart of Theorem 4.16. Theorem 4.17. Let s ∈ R, p ∈ (1, ∞), q ∈ (0, ∞] and τ ∈ [0, ∞) such that & ' 1 1 n max 0, − 1, τ − <s<1 q p

and

0 < s + nτ −

n < 1. p

(4.58)

Let Ψ ∈ Fp,s q (Rn ) ∩ L∞ (Rn ) be a function satisfying CΨ > 0 (see (4.40)) and (4.52). (i) Then f ∈ Fp,s, qτ (Rn ) if, and only if f ∈ Lτp (Rn ) and  f  ≡ sup |Q|1/2−τ ( f − fQ )ΨQ Fp,s q(Rn ) < ∞. s, τ F (Rn ) p, q

Q∈Q

Furthermore,  f Lτp (Rn ) +  f  and  f Fp,s, qτ (Rn ) are equivalent. F s, τ (Rn ) p, q

(ii) Let in addition τ ∈ [1/p, ∞). Then f ∈ Fp,s, qτ (Rn ) if, and only if N p ( f ) < ∞ and ||| f ||| ≡ F s, τ (Rn ) p, q

sup {Q∈Q: |Q|≤1}

|Q|1/2−τ ( f − fQ )ΨQ Fp,s q(Rn ) < ∞.

Furthermore, N p ( f ) + ||| f ||| and  f Fp,s, qτ (Rn ) are equivalent. s, τ F (Rn ) p, q

Proof. The only difference in comparison with the proof of Theorem 4.16 consists in the fact that we use Theorem 4.3 instead of Theorem 4.7.   It is well known that X ∈ Fp,s q (Rn ) if, and only if either s < 1/p and q is arbitrary; see [119, Lemma 2.3.1/3]. Therefore we obtain the counterpart of Corollary 4.18. Corollary 4.19. Let p ∈ (1, ∞), q ∈ (0, ∞], and s and τ be as in (4.58). Let in addition s < 1/p. Then f ∈ Fp,s, qτ (Rn ) if, and only if f ∈ Lτp (Rn ) and $# $   $ $ 1  1/2−τ $ $  f F s, τ (Rn ) ≡ sup |P| $ f − |P| P f (y) dy XP $ s n < ∞. p, q ∗ P∈Q Fp, q (R ) Furthermore,  f Lτp (Rn ) +  f  and  f Fp,s, qτ (Rn ) are equivalent. s, τ F (Rn ) p, q

s (Rn ) 4.6.3 A Characterization of F∞,q

We employ Proposition 2.4 and Theorem 4.17(ii). Theorem 4.18. Let p ∈ (1, ∞), q ∈ (0, ∞) and ' & 1 n max 0, − 1 < s < 1 . q

(4.59)

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

135

s Let Ψ ∈ F∞,q (Rn ) be a function satisfying CΨ > 0 (see (4.40)) and (4.52). Then s n f ∈ F∞,q (R ) if, and only if N p ( f ) < ∞ and

 f  Fs

n ∞,q (R )

≡

sup {Q∈Q: |Q|≤1}

Furthermore, N p ( f ) +  f  Fs

n ∞,q (R )

|Q|1/2−τ ( f − fQ )ΨQ Fp,s q(Rn ) < ∞.

s (Rn ) are equivalent. and  f F∞,q

Remark 4.19. s (Rn ) can be characterized by quantities, using (i) It is a bit surprising that F∞,q s n Fp,q (R ), and p can be chosen as we want (within (1, ∞)). (ii) As mentioned at the beginning of this section we have taken over the idea for those characterizations from the paper [5]. There the homogeneous situation with p = q = 2 is investigated.

Chapter 5

Pseudo-Differential Operators

The main purpose of this chapter is to obtain the boundedness on As,p,τq (Rn ) of all pseudo-differential operators of type (1,1) with inhomogeneous symbols. The smooth molecular decomposition characterizations of As,p,τq(Rn ) play an important role in this chapter.

μ

5.1 Pseudo-Differential Operators of Class S 1,1 (Rn ) We begin with recalling the following class of inhomogeneous symbols, which is a special case of H¨ormander class of symbols; see, for example, [73, 74] and [146, Chap. 6]. Definition 5.1. Let μ ∈ R. A smooth function a defined on Rn × Rn is called to μ belong to the class S1,1 (Rn ), if a satisfies the following differential inequalities that n for all α , β ∈ Z+ , β

sup (1 + |ξ |)−μ −|α |+|β ||∂xα ∂ξ a(x, ξ )| < ∞.

x, ξ ∈Rn

The main result of this section is the following. Theorem 5.1. Let s ∈ R and p, q ∈ (0, ∞]. Further we assume τ ∈ [0, τs,p,q ) if As,p,τq (Rn ) = Fp,s, qτ (Rn ) and τ ∈ [0, τs,p ) if As,p,τq (Rn ) = Bs,p,τq (Rn ). Let μ ∈ R, a ∈ μ S1,1 (Rn ) and a(x, D) be the corresponding pseudo-differential operator such that a(x, D)( f )(x) ≡ s+ μ , τ

for all smooth molecules f for A p, q

 Rn

eixξ a(x, ξ ) f(ξ ) d ξ

(Rn ) and x ∈ Rn .

(i) If s > σ p,q (s > σ p if As,p,τq (Rn ) = Bs,p,τq (Rn )), then a(x, D) is a continuous linear s+ μ ,τ mapping from A p,q (Rn ) to As,p,τq (Rn ).

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 5, c Springer-Verlag Berlin Heidelberg 2010 

137

138

5 Pseudo-Differential Operators

(ii) If s ≤ σ p,q (s ≤ σ p if As,p,τq (Rn ) = Bs,p,τq (Rn )), assume further that its formal adjoint a(x, D)∗ satisfies a(x, D)∗ (xβ ) ∈ P(Rn )

(5.1)

for all β ∈ Zn+ with |β | ≤ N, where N is as in Definition 3.2. Then a(x, D) is a s+ μ , τ continuous linear mapping from A p, q (Rn ) to As,p,τq (Rn ). Proof. For simplicity, in what follows, we write T ≡ a(x, D). s+ μ , τ Let f ∈ A p, q (Rn ), Φ , Ψ , ϕ and ψ be as in Lemma 2.3. Then by the Calder´on reproducing formula, we have f=

∑

∞

 f , ΦQ  ΨQ + ∑

∑

j=1 l(Q)=2− j

l(Q)=1

 f , ϕQ  ψQ

in S  (Rn ). Set tQ ≡  f , ΦQ  if l(Q) = 1 and tQ ≡  f , ϕQ  if l(Q) < 1. Then from s+ μ , τ the ϕ -transform characterization of A p, q (Rn ) obtained in Theorem 2.1, we deduce s+ μ , τ that {tQ }l(Q)≤1 ∈ a p, q (Rn ) with {tQ }l(Q)≤1 as+μ , τ (Rn )  f As+μ , τ (Rn ) , p, q

p, q

which further implies that {|Q|−μ /ntQ }l(Q)≤1 as,p,τq(Rn )  f As+μ , τ (Rn ) . p, q

(5.2)

We claim that T(f) ≡

∑

tQ T (ψQ )

l(Q)≤1

in S  (Rn ) and satisfies T ( f ) As,p,τq (Rn )  f As+μ , τ (Rn ) , p, q

where and in what follows, when l(Q) = 1, ψ is replaced by Ψ . To see this, by (5.2) and the smooth molecular decomposition characterizations of As,p,τq (Rn ) (see Theorem 3.2), it suffices to prove that for all Q ∈ Q with l(Q) ≤ 1, |Q|μ /n T (ψQ ) is a multiple of a smooth synthesis molecule for As,p,τq(Rn ) supported near Q. The proof is similar to that in [69]. For the reader’s convenience, we give some details. For all dyadic cubes Q = Q jk and x ∈ Rn , we set TQ (ψ )(x) ≡

 Rn

 (ξ ) d ξ . eixξ a(2− j (x + k), 2 j ξ )ψ

Then an argument, using a change of variables, yields that for all x ∈ Rn , T (ψQ )(x) =

 Rn

Q (ξ ) d ξ = 2 jn/2 TQ (ψ )(2 j x − k). eixξ a(x, ξ )ψ

μ

5.1 Pseudo-Differential Operators of Class S1,1 (Rn )

139

Recall that J = n/ min{1, p, q} if As,p,τq (Rn ) means Fp,s, qτ (Rn ), and J = n/ min{1, p} if As,p,τq (Rn ) means Bs,p,τq (Rn ). Fix a multi-index γ and M ∈ N such that M > max{J, J − s}/2. Then for all x ∈ Rn ,

∂ γ TQ (ψ )(x) =

 Rn

eixξ

∑ Cδ (iξ )δ ∂xγ −δ [a(2− j (x + k), 2 j ξ )]ψ (ξ ) d ξ

δ ≤γ

for certain constants Cδ , where δ ≤ γ means that δi ≤ γi for all i ∈ {1, · · · , n}. By (I − Δξ )M (eixξ ) = (1 + |x|2 )M eixξ and an integration by parts, we obtain that for all x ∈ Rn ,

∂ γ TQ (ψ )(x) =



Rn

eixξ

(I − Δξ )M (1 + |x|2 )M

∑ Cδ (iξ )δ ∂xγ −δ a(2− j (x + k), 2 j ξ )ψ (ξ ) d ξ .

(5.3)

δ ≤γ

Leibniz’s rule and Definition 5.1 yield that for all ξ ∈ Rn ,     (ξ )] (I − Δξ )M ∂xγ −δ [(iξ )δ a(2− j (x + k), 2 j ξ )ψ     β γ −δ  α  −j j δ  ∂ ∂ [a(2 (x + k), 2 ξ )] ∂ [(i ξ ) ψ ( ξ )]     ∑ ξ ξ x |α +β |≤2M



∑

|α +β |≤2M

    (ξ )] . 2 j|β | 2− j(|γ |−|δ |) (1 + |2 j ξ |)μ +|γ |−|δ |−|β | ∂ξα [(iξ )δ ψ

If l(Q) = 1, then j = 0. The above estimate and (2.1) imply that the right-hand side of (5.3) is pointwise bounded by C(γ )(1 + |x|2 )−M for some positive constant C(γ ). If l(Q) < 1, then j ≥ 1 and for all 1/2 ≤ |ξ | ≤ 2, 1 + |2 j ξ | ∼ |2 j ξ |. This fact together  also yields that there exists with the above estimate and the support condition of ψ a positive constant C(γ ) such that the right-hand side of (5.3) is pointwise bounded by C(γ )2 j μ (1 + |x|2)−M . Thus, for all x ∈ Rn , we have |∂ γ TQ (ψ )(x)| ≤ C(γ )2 j μ (1 + |x|2)−M for all Q ∈ Q with l(Q) ≤ 1. Using dilation and translation then deduces that for all x ∈ Rn , |∂ γ T (ψQ )(x)| ≤ C(γ )2 j μ 2 j|γ | 2 jn (1 + |2 j x − k|2 )−M ≤ C(γ )|Q|−μ /n−1/2−|γ |/n(1 + l(Q)−1 |x − xQ |)−2M . Therefore, a constant multiple of |Q|μ /n T (ψQ ) satisfies (3.4) through (3.7). Notice that if s > J − n, we allow us not to postulate the vanishing moment condition (3.3). Then |Q|μ /n T (ψQ ) is a multiple of a smooth synthesis molecule for As,p,τq (Rn ) supported near Q. If s ≤ J − n, we must check the vanishing moment

140

5 Pseudo-Differential Operators

condition for T (ψQ ) when l(Q) < 1. In fact, by (2.2) and the hypothesis (5.1), we see that for all β ∈ Zn+ with |β | ≤ N,  Rn

xβ T (ψQ )(x) dx = xβ , T (ψQ ) = T ∗ (xβ ), ψQ  = 0.

Thus, in the case when s ≤ J − n, |Q|μ /n T (ψQ ) is also a multiple of a smooth synthesis molecule for As,p,τq (Rn ) supported near Q, which completes the proof of Theorem 5.1.

Remark 5.1. (i) Since we only need to control a finite number of derivatives of molecules we obtain the estimate a(x, D)|As,p,τq (Rn ) → As,p,τq (Rn ) β

 max sup (1 + |ξ |)−μ −|α |+|β | |∂xα ∂ξ a(x, ξ )| |α |,|β |≤M x,ξ

(5.4)

for some M ≡ M(s, p, q, τ ), where a(x, D)|As,p,τq (Rn ) → As,p,τq (Rn ) denotes the operator norm of a(x, D) from As,p,τq (Rn ) to As,p,τq (Rn ). (ii) The counterpart of Theorem 5.1 for the homogeneous Besov-type space B˙ s,p,τq (Rn ) and Triebel-Lizorkin-type space F˙p,s, qτ (Rn ) were already obtained in [127, Theorem 1.5]. (iii) The proof given above uses ideas of Grafakos and Torres [69], which itself has been based on [61, 66, 140, 141]. (iv) The boundedness of pseudo-differential operators of the “exotic” class μ S1,1 (Rn ) has its own history. Here we only mention the contributions of Meyer [98] (boundedness on H ps (Rn ), s > 0, 1 < p < ∞), Bourdaud [16] (boundedness on Bsp,q(Rn ), s > 0, 1 ≤ p, q ≤ ∞), Runst [117] and Torres [140]. The last two authors have dealt with the general case of Besov-Triebel-Lizorkin spaces including values of p and q less than 1. As an immediate consequence of Theorem 5.1, we have the following conclusion. Corollary 5.1. Let γ ∈ Zn+ and s, p, q and τ be as in Theorem 5.1. Then the operator s+|γ |, τ ∂ γ : A p, q (Rn ) → As,p,τq (Rn ) is continuous. Form Theorem 5.1 and the smooth atomic decomposition characterization of As,p,τq (Rn ), we also deduce the following result. Corollary 5.2. Let s, p, q and τ be as in Theorem 5.1. Assume that l ∈ Z+ such that s + 2l > σ p,g if As,p,τq (Rn ) = Fp,s, qτ (Rn ), and s + 2l > σ p if As,p,τq (Rn ) = Bs,p,τq(Rn ). Then τ n any f ∈ As,p,τq (Rn ) can be represented as f = (I + (−Δ )l )h with h ∈ As+2l, p,q (R ) and C−1 f As,p,τq(Rn ) ≤ h As+2l, τ (Rn ) ≤ C f As,p,τq(Rn ) , p, q

where C is a positive constant independent of f and h.

μ

5.1 Pseudo-Differential Operators of Class S1,1 (Rn )

141

τ n Proof. We first show that the operator I + (−Δ )l is continuous from As+2l, p, q (R ) τ n to As,p,τq (Rn ). Let h ∈ As+2l, p, q (R ). By Theorem 3.3 and Remark 3.1, there exist a sequence t ≡ {tQ }l(Q)≤1 ⊂ C satisfying

t as+2l, τ (Rn )  h As+2l, τ (Rn ) p, q

p, q

τ n and a family {aQ }l(Q)≤1 of smooth atoms for As+2l, p, q (R ) such that h = ∑l(Q)≤1 tQ aQ  n in S (R ), where the smooth atom aQ has the regularity condition that

∂ β aQ L∞ (Rn ) ≤ |Q|−1/2−|β |/n 

 and the moment condition that n xβ aQ (x) dx = 0 if |β | ≤ N,  where if |β | ≤ K, R  ≥ max{s + nτ + 1, 0} + 2l and N  ≥ max{J − n − s, −1} + 2l. In view of the K actual construction in [64, p. 132], we see that tQ aQ is obtained canonically for all τ n h ∈ As+2l, p, q (R ). We now claim that f ≡ (I + (−Δ )l )h ≡

∑

tQ (I + (−Δ )l )aQ

l(Q)≤1

converges in S  (Rn ) and satisfies f As,p,τq(Rn )  h As+2l, τ (Rn ) . p, q

To this end, by the inequality that 2l

{|Q|− n tQ }l(Q)≤1 as,p,τq(Rn ) = t as+2l, τ (Rn )  h As+2l, τ (Rn ) p, q

p, q

and Theorem 3.3 again, it suffices to prove that for each Q ∈ Q with l(Q) ≤ 1, 2l bQ ≡ |Q| n (I + (−Δ )l )aQ is a constant multiple of a smooth atom for As,p,τq(Rn ) supported near Q. Obviously, bQ satisfies the support condition (3.13). On the other hand, since for  all β ∈ Zn+ with |β | ≤ K, ∂ β aQ L∞ (Rn ) ≤ |Q|−1/2−|β |/n,  − 2l and Q ∈ Q with l(Q) ≤ 1, then for all γ ∈ Zn+ with |β | ≤ K ∂ γ bQ L∞ (Rn ) = |Q| n ∂ γ (I + (−Δ )l )aQ L∞ (Rn )  2l ≤ |Q| n |Q|−1/2−|γ |/n + |Q|−1/2−|γ |/n−2l/n 2l

 |Q|−1/2−|γ |/n. Similarly, by the moment condition of aQ , we obtain that  Rn

 − 2l. xγ bQ (x) dx = 0 if |γ | ≤ N

142

5 Pseudo-Differential Operators

Thus, a constant multiple of bQ satisfies the regularity condition (3.15) and the moment condition (3.14), which proves the previous claim and further implies that τ s, τ n n I + (−Δ )l is continuous from As+2l, p, q (R ) to A p, q (R ). To finish the proof of Corollary 5.2, we need to show that I + (−Δ )l is a surjective operator. Let f ∈ As,p,τq (Rn ) and set a(x, ξ ) ≡ (1 + |ξ |2l )−1 −2l for all x, ξ ∈ Rn . It is easy to see that a ∈ S1,1 (Rn ). By s + 2l > J − n and Theorem 5.1, the corresponding operator a(x, D) is a continuous linear mapping from τ s+2l, τ n As,p,τq (Rn ) to As+2l, (Rn ) and p, q (R ). Set h ≡ a(x, D) f . Then h ∈ A p, q

f ≡ (I + (−Δ )l )h,



which completes the proof of Corollary 5.2. Remark 5.2. Corollaries 5.1 and 5.2 will be of certain use in the next chapter.

In addition, by Theorem 5.1, we also obtain the so-called lifting properties for the spaces As,p,τq(Rn ). Let σ ∈ R. Recall that the lifting operator Iσ is defined by

Iσ f ≡ (1 + | · |2)σ /2 f,

f ∈ S  (Rn );

see, for example, [145, p. 58]. It is well known that Iσ is a one-to-one mapping from S  (Rn ) onto itself. Notice that σ (Rn ). a(x, ξ ) ≡ (1 + |ξ |2)σ /2 ∈ S1,1 Applying Theorem 5.1, we have the following result. Proposition 5.1. Let σ ∈ R and s, p, q and τ be as in Theorem 5.1. Then the operσ,τ n ator Iσ maps As,p,τq (Rn ) isomorphically onto As− p, q (R ). We remark that Proposition 5.1 when τ = 0 generalizes the classic conclusion in [145, Theorem 2.3.8].

5.2 Composition of Functions in As,p,τq (Rn ) Let f : R → R be a smooth function such that f (0) = 0. Then there is a well-known connection between mapping properties of the nonlinear composition operator Tf : g → f ◦ g ,

g ∈ As,p,τq (Rn ) ,

0 (Rn ). We and the boundedness of pseudo-differential operators from the class S1,1 follow [98] and [100, Sect. 16.2].

5.2 Composition of Functions in As,p,τq (Rn )

143

Let ψ , {ψ j } j∈Z+ and ϕ j be defined as in (1.1) and (1.2). Observe M

∑ ψ j (x) = ψ (2−M x) → 1

if

M → ∞.

j=0

For g ∈ C(Rn ) we define

Δ jg ≡ ϕ j ∗ g

S j g ≡ F −1 [ψ (2− j ξ ) F g(ξ )] .

and

Then the composition f ◦ g can be written as f ◦ g = f ◦ S0g + ( f ◦ S1g − f ◦ S0 g) + . . . + ( f ◦ S j+1g − f ◦ S j g) + . . . . The convergence of the latter telescopic series follows from the inequality ⎞ ⎛ | f (u) − f (v)| ⎠ |g(x) − g j (x)| | f ◦ g(x) − f ◦ S j g(x)| ≤ ⎝ sup |u − v| |u|,|v|≤ g ∞ u=v

combined with the uniform convergence of S j g to g. With m j (x) ≡

 1 0

f  (S j g(x) + t Δ j g(x)) dt,

we can rewrite f ◦ g as ∞

f ◦ g(x) = f ◦ S0 g + ∑ m j (x) Δ j g(x) .

(5.5)

j=0

Lemma 5.1. Let g ∈ C(Rn ). Then the linear operator L g(x) ≡

∞

∑ m j (x)Δ j g(x)

j=0

with symbol a(x, ξ ) ≡

∞

∑ m j (x) ψ j (ξ )

j=0 0 (Rn ). belongs to S1,1

Proof. Let j ∈ N and let 2 j−1 ≤ |ξ | ≤ 2 j . Then only ψ j−1 (ξ ) and ψ j (ξ ) can be different from 0. Thus, a(x, ξ ) is finite and β |∂xα ∂ξ a(x, ξ )|

   j    β  α =  ∑ ∂x m (x) ∂ξ ψ (ξ ) = j−1        2− j|β | max ∂xα m (x) ∂ β ψ (2− ξ ) − 2|β | ∂ β ψ (2−+1ξ )  . j−1≤≤ j

144

5 Pseudo-Differential Operators

Let

b j (x,t) ≡ S j g(x) + t Δ j g(x) .

Then

|∂ γ b j (x,t)|  2 j|γ | g L∞ (Rn )

with constants behind  independent of x ∈ Rn , t ∈ (0, 1), j ∈ N and g ∈ C(Rn ). Faa die Bruno’s formula yields |∂xα m j (x)| 

|α |

∑ | f (k+1) (b j (x,t))|

k=1

∑

∂ γ b j (x,t) . . . ∂ γ b j (x,t) 1

k

γ 1 +...+γ k =α

 2 j|α | f C|α |+1 (B(0, g

L∞ (Rn )

  |α | g . ∞ n + g ∞ n L (R ) )) L (R )

With an obvious modification for |ξ | ≤ 1 we have found the estimate β

|∂xα ∂ξ a(x, ξ )|  f C|α |+1 (B(0, g ∞ n )) L (R )   |α | × g L∞ (Rn ) + g L∞(Rn ) (1 + |ξ |)|α |−|β | .

(5.6)

This proves the lemma.



Remark 5.3. Lemma 5.1 is in principle known, we refer to [100, Lemma 16.2/1,2]. However, we repeated the proof since we missed a reference for the estimate (5.6), which we need for the proof of the next theorem. We shall work with functions f which are infinitely differentiable, i. e. f ∈ C∞ (R). Theorem 5.2. Assume that p, q ∈ (0, ∞]. Let either s ∈ (σ p,q , ∞) and τ ∈ [0, τs,p,q ) if As,p,τq (Rn ) = Fp,s, qτ (Rn ) or let s ∈ (σ p , ∞) and τ ∈ [0, τs,p ) if As,p,τq (Rn ) = Bs,p,τq(Rn ). Let f ∈ C∞ (R) and f (0) = 0. Then, for all real-valued functions g ∈ As,p,τq (Rn ) ∩C(Rn ), the function f ◦ g also belongs to As,p,τq (Rn ) ∩C(Rn ). The associated operator T f : As,p,τq (Rn ) ∩C(Rn ) → As,p,τq (Rn ) ∩C(Rn ) is bounded. Proof. From Lemma 5.1, Theorem 5.1, (5.4) and (5.6), we deduce that   L g As,p,τq(Rn )  f CM+1 (B(0, g L∞(Rn ) )) g L∞(Rn ) + g M L∞ (Rn ) g As,p,τq(Rn ) for some M ≡ M(s, p, q, τ ) ∈ N. By (5.5), it remains to show that f ◦ S0 g ∈ As,p,τq(Rn ) and to estimate f ◦ S0 g As,p,τq(Rn ) . Of course, f ◦ S0 g ∈ C∞ (Rn ). Since g is bounded, also S0 g is bounded and we have the obvious estimate f ◦ S0g C(Rn ) ≤ f C(B(0, S0g L∞ (Rn ) )) ≤ f C(B(0,c g L∞(Rn ) )) ,

5.2 Composition of Functions in As,p,τq (Rn )

145

where c is a positive constant independent of f and g. To estimate f ◦ S0 g As,p,τq(Rn ) we shall apply the characterizations by differences; see Sect. 4.3. By the elementary embeddings in Proposition 2.1 it will enough to derive an estimate of f ◦ S0 g Bs1 , τ (Rn ) for some s1 > s. Now we make use of an argument which we p,∞ have applied also in the proof of Lemma 4.4. Let f and g be fixed. From the regu1 (P) for any dyadic cube P. For the larity of f and S0 g it is clear that f ◦ S0 g ∈ Bsp,∞ s1 same reasons we also have S0 g ∈ B p,∞ (P). We associate to P an extension EP (S0 g) of the restriction of S0 g to P such that EP (S0 g) Bsp,∞ 1 (Rn ) ≤ 2 S0 g Bs1 (P) . p,∞ Of course, f ◦ EP (S0 g) is an extension of the restriction of ( f ◦ (S0 g)) to P. To have a more precise notation we shall write at ( f ) instead of at ; see (4.12). Then it follows from known estimates of composition operators on Besov spaces (see [119, Theorem 5.3.4/2]) that 1 |P|τ

sup

t

−s1

1/p

 P

0
[at ( f ◦ EP(S0 g))(x)] dx p

1 f ◦ EP (S0 g) Bsp,∞ 1 (Rn ) |P|τ 1  f Cs2 (B(0,c g L∞(Rn ) )) τ EP (S0 g) Bs1 (Rn ) EP (S0 g) sL2∞−1 (Rn ) p,∞ |P| 1  f Cs2 (B(0,c g L∞(Rn ) )) g sL2∞−1 S0 g Bsp,∞ 1 (P) (Rn ) |P|τ 

 f Cs2 (B(0,c g L∞(Rn ) )) g sL2∞−1 S0 g Bs1 , τ (Rn ) (Rn ) p,∞

for some s2 > max{s1 , 1}. Since S0 g Bs1 , τ (Rn )  S0g Bs,p,∞τ (Rn )  g Bs,p,∞τ (Rn ) , p,∞

we finally have obtained f ◦ S0 g ♣s1 , τ

B p,∞ (Rn )

 f Cs2 (B(0,c g L∞(Rn ) )) g sL2∞−1 g Bs,p,∞τ (Rn ) . (Rn )

Having a look at Theorems 4.7, 4.8 and 4.9, we need to estimate also f ◦S0 g Lτp (Rn ) , N1 ( f ◦ S0 g), and sup {P∈Q: |P|≥1}

f ◦ S0g Bs0

p,∞ (2P)

/|P|τ ,

146

5 Pseudo-Differential Operators

where σ p < s0 < s. The estimate of the last term follows from an argument as above. Finally, the estimates of f ◦ S0 g Lτp (Rn ) and N1 ( f ◦ S0g) are obtained by using the local Lipschitz continuity of f and f (0) = 0. We have f ◦ S0 g − f (0) Lτp(Rn ) ≤ f C1 (B(0,c g L∞(Rn ) )) S0 g Lτp (Rn )  f C1 (B(0,c g L∞(Rn ) )) S0 g As,p,τq(Rn )  f C1 (B(0,c g L∞(Rn ) )) g As,p,τq(Rn ) , where we used Propositions 2.7, 2.1 inbetween. In the same way one can estimate N1 ( f ◦ S0 g). This proves f ◦ g ∈ As,p,τq(Rn ) for all g ∈ As,p,τq(Rn ) and at the same time, that bounded subsets of As,p,τq (Rn ) are mapped into bounded subsets of As,p,τq(Rn ).

Remark 5.4. (i) As we have mentioned above, we followed Meyer [98] and Meyer and Coifman [100, Sect. 16.2], where the case of Bessel potential spaces s,0 H ps (Rn ) = Fp,2 (Rn )

with p ∈ (1, ∞), was treated. (ii) In case τ = 0 much more is known about the properties of those composition operators. Surveys can be found in the monographs, Appell and Zabrejko [6] (first order Sobolev spaces, L p -spaces), [118], [119, Chap. 5] and in the paper of Bourdaud [17]. For the latest progress we refer to [20, 21] and [22].

Chapter 6

Key Theorems

In this chapter we focus on some crucial problems including pointwise multipliers, diffeomorphisms and traces, which govern the theory of the spaces As,p,τq(Rn ) to a large extent. These problems are of vital importance both for function spaces treated for their own sake and for applications to partial differential equations. Following Triebel’s monograph [146], we call these assertions key theorems, since these theorems are the basis for the definitions of Besov-type spaces and Triebel-Lizorkin-type spaces on domains. An important tool used in this chapter is the smooth atomic decomposition characterization of As,p,τq (Rn ) in Theorem 3.3.

6.1 Pointwise Multipliers Pointwise multiplication in Besov and Triebel-Lizorkin spaces has been studied extensively in the last 30 years; see, for example, [95, 114, 119, 145, 146] and [96]. The two monographs [95, 96] by Maz’ya and Shaposnikova are completely devoted to this subject. However, the authors restrict their interest essentially to the s (Rn ), 1 < p < ∞, and the Slobodeckij spaces Bsp,p (Rn ), Bessel-potential spaces Fp,2 1 ≤ p ≤ ∞. Let X and Y be two quasi-Banach spaces of functions (distributions). Then the basic question consists in descriptions of the associated multiplier space M(X,Y ) given by M(X,Y ) ≡ { f : f · g ∈ Y for all g ∈ X} . This space is equipped with the induced quasi-norm  f M(X,Y ) ≡ sup  f · gY . gX ≤1

Here, in this lecture note, we will be concerned with the easier problem of proving embeddings into M(X) ≡ M(X, X) with X ≡ As,p,τq (Rn ). We shall present three different approaches. The first one uses atoms and the lifting operator Iσ and is the most general, i. e., it can be applied for all p, q, s and some τ (depending on s and p). The second one uses the Hardy-Littlewood maximal 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 6, c Springer-Verlag Berlin Heidelberg 2010 

147

148

6 Key Theorems

function. As a consequence we have to restrict us to spaces X = Fp,s, qτ (Rn ) with τ < 1/p. However, it leads to a very natural extension of some classical assertions, known for many years in case τ = 0. Our third approach uses characterizations by differences. Here we concentrate on τ = 1/p leaving possible further applications aside.

6.1.1 Smooth Functions are Pointwise Multipliers for As,p,τq (Rn ) We shall prove embeddings of the type Cm (Rn ) ⊂ M(As,p,τq (Rn ))

if

m = m(s, p, q, τ , n)

is large enough. Here Cm (Rn ) is defined in item (iii) in Sect. 1.3. Theorem 6.1. Let s, p, q and τ be as in Theorem 5.1. If m ∈ N is sufficiently large, then there exists a positive constant C(m) such that for all g ∈ Cm (Rn ) and all f ∈ As,p,τq (Rn ),  g f As,p,τq(Rn ) ≤ C(m)



∑

|α |≤m

∂ α gL∞ (Rn )  f As,p,τq(Rn ) .

(6.1)

Proof. First we prove (6.1) under the assumption s > J − n. Let f ∈ As,p,τq (Rn ) and g ∈ Cm (Rn ) with m ≥ max{s + nτ + 1 , 0}. By Theorem 3.3, we can write f as f=

∑

tQ aQ

l(Q)≤1

in S (Rn ), where each aQ is a smooth atom for As,p,τq(Rn ) supported near Q and the sequence t ≡ {tQ }l(Q)≤1 ⊂ C satisfies tas,p,τq(Rn )   f As,p,τq(Rn ) . Set bQ ≡ gaQ , then supp bQ ⊂ 3Q. To show g f ∈ As,p,τq (Rn ), by Theorem 3.3 again, it suffices to prove that each bQ is a constant multiple of a smooth atom for As,p,τq (Rn ) supported near Q. By the assumption s > J − n, there is no need to postulate any moment condition on bQ . Thus, we focus on the regularity condition of bQ . Indeed, since l(Q) ≤ 1 and m ≥ max{s + nτ + 1 , 0}, for all γ ∈ Zn+ with |γ | ≤ max{s + nτ + 1 , 0}, we have ∂ γ bQ L∞ (Rn ) = ∂ γ (gaQ )L∞ (Rn ) ≤

∑ ∂ α gL∞(Rn ) ∂ γ −α aQ L∞ (Rn )

α ≤γ

6.1 Pointwise Multipliers

149

∑ ∂ α gL∞(Rn ) |Q|− 2 − 1

≤

α ≤γ





∑

≤

|γ |−|α | n

α

|α |≤m

1

|γ |

∂ gL∞ (Rn ) |Q|− 2 − n ,

which implies that (6.1) holds provided s > J − n. Now we consider the case when s ≤ J − n. Fix l ∈ N such that s + 2l > J − n. Then by Corollary 5.2, any f ∈ As,p,τq(Rn ) can be represented as f ≡ (I + (−Δ )l )h τ (Rn ) and hAs+2l,τ (Rn ) ∼  f As,p,τq (Rn ) . Similarly to the argument in with h ∈ As+2l, p,q p,q [146, p. 204], we have g f = (I + (−Δ )l )(gh) +

∑

|α |<2l

∂ α (gα h),

where each gα is the summation of terms of type ∂ β g with |β | ≤ 2l. Then by Corollaries 5.1, 5.2 and Proposition 2.1(ii), we obtain g f As,p,τq(Rn )  Let m ∈ N such that

∑

|α |≤2l

gα hAs+2l,τ (Rn ) .

(6.2)

p,q

m − 2l ≥ s + 2l + nτ + 1 .

Then applying the previous proved result to the right-hand side of (6.2) yields that for all |α | ≤ 2l,  gα hAs+2l,τ (Rn )  p,q



∑

|β |≤m−2l

 

β

∑

|β |≤m

∂ gα L∞ (Rn ) hAs+2l,τ (Rn ) p,q



∂ β gL∞ (Rn )  f As,p,τq(Rn ) ,

which together with (6.2) implies (6.1), and then completes the proof of Theorem 6.1.

Remark 6.1. Theorem 6.1 generalizes some classical results on Besov spaces and Triebel-Lizorkin spaces by taking τ = 0; see, for example, [146, Theorem 4.2.2].

6.1.2 Pointwise Multipliers and Paramultiplication In this part we shall improve Theorem 6.1 for certain parameter constellations. We follow a well-known strategy. The product of f and g will be decomposed into three parts (involving paraproducts). These three parts can be estimated by applying some

150

6 Key Theorems

tools from Fourier analysis, in particular Marschall’s inequality in combination with the dyadic ball criterion. The tools rely on the boundedness of the Hardy-Littlewood maximal function. As a consequence we investigate principally the multipliers of the s (Rn ). However, based on their identification with F s,τ (Rn ), τ = 1/u − spaces E pqu u,q 1/p, this leads to corresponding results for M(Fp,s, qτ (Rn )). This procedure causes the restriction to τ < 1/p.

6.1.2.1 Marschall’s Inequality and the Dyadic Ball Criterion This subsection contains some preliminaries. Inspired by Marschall’s paper [94], we shall give a version of his pointwise estimate of pseudo-differential operators b(x, D), that is suitable for us. In Marschall’s inequality the symbol is estimated via the norm of a homogeneous Besov space B˙ sp,q (Rn ) ≡ B˙ s,p,q0 (Rn ). We refer to Chap. 8 for the definitions of these spaces. It will be convenient for us to work with a homogeneous dyadic partition of unity. Let ψ and ψ 1 be as in (1.1) and define

φ1 ≡ ψ 1

φk ≡ φ1 (2−k+1 ·) ,

and

Then it follows that 1=

∞

∑

φk

on

k ∈ Z.

Rn \ {0} .

k=−∞

Proposition 6.1. Let a symbol b ∈ C0∞ (Rn ) and a function f ∈ C∞ (Rn ) be given such that, for A > 0 and R ≥ 1, supp F f ⊂ B(0, AR) and

supp b ⊂ B(0, A) .

(6.3)

Let t ∈ (0, 1]. Then there exists a positive constant C such that 

n

|b(D)u(x)| ≤ C(RA) t −n  b  ˙ n/t B1,t

(Rn )

1/t M(| f |t ) (x).

(6.4)

Here C can be taken as a function of t only. Proof. Since convolutions in S (Rn ) ∗ S (Rn ) are mapped to products by the Fourier transformation, b(D)u(x) = F −1 (bF u)(x) =

1 (2π )n/2

 Rn

F −1 b(x − y)u(y) dy .

With x fixed, y → F −1 b(x − y)u(y) has its spectrum in B(0, A) + B(0, RA) ⊂ B(0, (R + 1)A).

6.1 Pointwise Multipliers

151

Let us recall the Nikol’skij inequality in a form stated in [145, Sect. 1.3.2]. Let 0 < p ≤ q ≤ ∞, b ∈ (0, ∞) and ϕ ∈ S (Rn ) satisfying supp ϕ ⊂ B(0, b), then  ϕ Lq (Rn )  bn(1/p−1/q)  ϕ L p (Rn ) with a constant independent on ϕ and b. Applying this inequality with q = 1 and 0 < t ≤ 1 we obtain that |b(D)u(x)| ≤

 Rn

|F −1 b(x − y)u(y)| dy n

 (RA) t −n  F −1 b(x − ·) u Lt (Rn )   (RA)

n −n t

1/t

∑  φk (x − ·) F −1 b(x − ·) u tLt (Rn )

.

(6.5)

k∈Z

By the obvious estimate sup |φk (y) F

−1

y∈Rn

b(y)| ≤

 Rn

  −1  F (φk F −1 b)(η )  d η ≡ bk ,

one finds  B(x,2k+1 )

| φk (x − y) F −1 b(x − y) f (y)|t dy  btk 2kn M(| f |t )(x).

Inserting this into (6.5) we obtain the desired inequality, since

∑



2kn/t  F −1 [φk F b] L1 (Rn )

k∈Z

t

=  b t˙ n/t

B1,t (Rn )

,



which completes the proof.

Remark 6.2. Proposition 6.1 is a simplified version of an inequality proved in [81]. Now we turn to the dyadic ball criterion for the spaces Fp,s, qτ (Rn ). Proposition 6.2. Let p ∈ (0, ∞), q ∈ (0, ∞], τ ∈ [0, 1/p) and s ∈ (σ p,q , ∞). Suppose {u j }∞j=0 ⊂ S (Rn ) such that supp F u j ⊂ B(0, 2 j+2) , and

j ∈ Z+ ,

⎡   p/q ⎤1/p  ∞ 1 ⎣ dx⎦ < ∞ . A ≡ sup ∑ 2 jsq|u j (x)|q τ P j=0 P∈Q |P|

(6.6)

(6.7)

152

6 Key Theorems

Then u ≡ ∑∞j=0 u j converges in S (Rn ) and its limit u belongs to Fp,s, qτ (Rn ). Furthermore, (6.8) uFp,s, τq (Rn ) ≤ C A , where C is a positive constant independent of {u j }∞j=0 and A. s (Rn ), 1/r = 1/p− τ ; Proof. The proof will be based on the identity Fp,s, qτ (Rn ) = Er,q,p see Corollary 3.3. Step 1. Temporarily we assume that the sequence {u j }∞j=0 is finite, i. e., u j = 0 if j > N for some N ∈ N. Since u j is a smooth function it belongs to Lr (P) for all P ∈ Q and all r ∈ (0, ∞]. Let {ϕk }∞ k=0 be the system defined in (1.2). Since supp F ϕk ⊂ B(0, 2k+1 ), Marschall’s inequality yields

|ϕk ∗ u j+k−3(x)|  F −1 ϕk  ˙ n/t

B1,t (Rn )

 1/t n (2 j+k ) t −n M(|u j+k−3 |t ) (x)

 1/t n  2 j( t −n) M(|u j+k−3 |t ) (x) , where the constants behind  do not depend on x, j and k. Let u j ≡ 0, if j < 0. Next we observe that, using (6.6), ∞

∑

ϕk ∗ u =

j=(k−3)∨0

ϕk ∗ u j =

∞

∑ ϕk ∗ u j+k−3 .

j=0

Let d = min{1, p, q}. Thus, the vector-valued maximal inequality of Tang and Xu (see (4.38)) implies that udEr,q,p s (Rn ) 

∞

⎡ 1

sup ∑ P∈Q |P|τ d

j=0



∞

∑2



∑2 ∞

P

∑ 2ksq |ϕk ∗ u j+k−3(x)|q

 p/q

⎤d/q dx⎦

k=0

⎡   p/q ⎤d/q  ∞   1 ⎣ q/t sup ∑ 2ksq M(|u j+k−3|t ) (x) dx⎦ τd P k=0 P∈Q |P|

j( nt −n)d

⎡   p/q ⎤d/q  ∞ 1 ⎣ sup dx⎦ ∑ 2ksq|u j+k−3|q τd P k=0 P∈Q |P|

j=0



⎣

∞

j( nt −n)d

j=0 ∞





∑ 2 j( t −n)d 2− jsd Ad . n

(6.9)

j=0

For t approaching min{1, p, q} the condition s > σ p,q becomes sufficient for s (Rn ) by C A. Because of the mentioned coincidence we obtain the estimating uEr,q,p s (Rn ) replaced by u s, τ same conclusion with uEr,q,p Fp, q (Rn ) .

6.1 Pointwise Multipliers

153

Step 2. We remove the restriction to finite sequences. Let q < ∞. Then, by applying the methods of Step 1, we get    ∞    →0 if N → ∞ .  ∑ u j  j=N  s, τ n Fp, q (R )

Thus, {∑Lj=0 u j }L∈N is convergent in Fp,s, qτ (Rn ). Now, let q = ∞. Then, by repeating these arguments with s replaced by s − ε , ε > 0, and q replaced by 1, we s−ε , τ obtain the convergence of {∑Lj=0 u j }L∈N in Fp,1 (Rn ) and therefore in S (Rn ); see Proposition 2.3. Furthermore, Step 1 combined with the Fatou property of Fp,s, qτ (Rn ) (see Proposition 2.8) yields u ∈ Fp,s, qτ (Rn ) and at the same time (6.8).

Later on we also need a supplement dealing with dyadic annuli instead of dyadic balls. Proposition 6.3. Let p ∈ (0, ∞), q ∈ (0, ∞], τ ∈ [0, 1/p) and s ∈ R. Suppose that the sequence {u j }∞j=0 ⊂ S (Rn ) such that supp F u0 ⊂ B(0, 4), supp F u j ⊂ B(0, 2 j+2 ) \ B(0, 2 j−3) , and

⎡





1 ⎣ τ |P| P∈Q

A ≡ sup

P

j ∈ N,

 p/q

∞

∑ 2 jsq|u j (x)|q

(6.10)

⎤1/p dx⎦

< ∞.

(6.11)

j=0

Then u ≡ ∑∞j=0 u j converges in S (Rn ) and its limit u belongs to Fp,s, qτ (Rn ). Furthermore, (6.12) uFp,s, τq (Rn ) ≤ C A , where C is a positive constant independent of {u j } j and A. Proof. Observe that

ϕk ∗ u =

k+2

∑

ϕk ∗ u j ,

k ∈ Z+ .

j=max{k−2, 0}

Based on this identity we can proceed as in the proof of Proposition 6.2. Since the sum with respect to j in the last line of the estimate (6.9) has always less than 6 summands, there is no need for a restriction with respect to s.



6.1.2.2 The Decomposition of the Product Let ψ , {ψ j } j∈Z+ and {ϕ j } j∈Z+ be as in (1.1) and (1.2), respectively. Let ϕ−1 ≡ 0. For f ∈ S (Rn ) we put j

S j f (x) ≡

∑ (ϕ ∗ f )(x) = F −1 [ψ (2− j ξ ) F f (ξ )](x) .

=0

154

6 Key Theorems

Using these smooth approximations with respect to f and g, we define the product of these distributions as f · g ≡ lim S j f · S j g , j→∞



whenever this limit exists in S (R ). For a further discussion of this definition we refer to [80] and [119, Chap. 4]. Related to this definition we introduce the following operators:

Π1 ( f , g) =

n

∞

∑ Sk−2 f · (ϕk ∗ g) ,

(6.13)

k=2

Π2 ( f , g) =

∞

∑ [(ϕk−1 ∗ f ) + (ϕk ∗ f ) + (ϕk+1 ∗ f )] · (ϕk ∗ g) ,

(6.14)

k=0

and

Π3 ( f , g) =

∞

∑ (ϕk ∗ f ) · Sk−2g = Π1(g, f ) .

(6.15)

k=2

It follows that

f · g = Π1 ( f , g) + Π2 ( f , g) + Π3 ( f , g) ,

whenever these three limits exist in consists in

S (Rn ).

The advantage of this decomposition

supp F (Sk−2 f · (ϕk ∗ g)) ⊂ { ξ : 2k−3 ≤ |ξ | ≤ 2k+1 } , and supp F



k+1

∑

=k−1

(6.16)

k = 2, 3, . . .

(6.17)

 (ϕ ∗ f ) · (ϕk ∗ g) ⊂ { ξ : |ξ | ≤ 5 · 2k } ,

k = 0, 1, . . . . (6.18)

This means, we can apply either the dyadic ball criterion or Proposition 6.3 in connection with these operators. Remark 6.3. The splitting technique from formula (6.16) has been invented independently by Peetre [114] and Triebel [143]. 6.1.2.3 Multiplication by H¨older Continuous Functions Theorem 6.2. Let s ∈ R, p ∈ (0, ∞), q ∈ (0, ∞] and τ ∈ [0, 1/p). Suppose that   n ρ > max 0, |s|, − n − s . p Then the embedding Cρ (Rn ) ⊂ M(Fp,s, qτ (Rn )) holds.

(6.19)

6.1 Pointwise Multipliers

155

Proof. It will be enough to estimate Π1 ( f , g), Π2 ( f , g) and Π3 ( f , g) for f ∈ Cρ (Rn ) s (Rn ) and and g ∈ Fp,s, qτ (Rn ). Again we shall employ the identity Fp,s, qτ (Rn ) = Er,q,p 1/r = 1/p − τ ; see Corollary 3.3. Step 1. Estimate of Π1 . Recall that the convolution inequality sup sup |Sk f (x)|   f L∞ (Rn ) .

k∈Z+ x∈Rn

Applying this convolution inequality and Proposition 6.3 with u0 = u1 = 0 and uk+2 ≡ Sk−2 f · (ϕk ∗ g), k ≥ 0, we find Π1 ( f , g)Fp,s, qτ (Rn )

⎡   p/q ⎤1/p  ∞ 1 ⎣  sup dx⎦ ∑ 2ksq |uk (x)|q τ P k=2 P∈Q |P| ⎡   p/q ⎤1/p  ∞ 1 ⎣   f L∞ (Rn ) sup ∑ 2ksq |(ϕk ∗ g)(x)|q dx⎦ τ P k=2 P∈Q |P| s (Rn )   f L∞ (Rn ) gEr,q,p

  f L∞ (Rn ) gFp,s, τq (Rn ) .

(6.20) s+ρ , τ

s, τ Step 2. Estimate of Π2 . Recall that the embedding Fp,∞ (Rn ) ⊂ Fp,q (Rn ); see Proposition 2.1. This time we have to use the dyadic ball criterion. For simplicity we put uk ≡ (ϕk ∗ f ) · (ϕk ∗ g), k ∈ Z+ .

Then we obtain Π2 ( f , g)Fp,s, qτ (Rn )  Π2 ( f , g)F s+ρ , τ (Rn ) p,∞  p 1/p   1 k(s+ρ )  sup |(ϕk ∗ f )(x) · (ϕk ∗ g)(x)| dx sup 2 τ P k∈Z+ P∈Q |P|   p 1/p    1 kρ ks  sup 2 ϕk ∗ f L∞ (Rn ) sup sup 2 |(ϕk ∗ g)(x)| dx τ P k∈Z+ k∈Z+ P∈Q |P| s, τ   f Bρ∞,∞(Rn ) gFp,∞ (Rn ) ,

since

  1 s + ρ > n max 0, − 1 . p

156

6 Key Theorems

Step 3. Estimate of Π3 . Let ρ < ρ be a number which also satisfies (6.19). Again we can apply Proposition 6.3. This yields Π3 ( f , g)Fp,s, qτ (Rn ) ⎡   p/q ⎤1/p  ∞ 1 ⎣  sup dx⎦ ∑ 2ksq |(ϕk ∗ f )(x) · Sk−2g(x)|q τ P k=2 P∈Q |P| ⎡   p/q ⎤1/p ∞  f Bρ∞,∞(Rn )  ⎣  sup 2k(s−ρ )q|Sk−2 g(x)|q dx⎦ ∑ τ |P| P P∈Q k=2  sup P∈Q

   f Bρ∞,∞(Rn )  |P|τ

  f Bρ∞,∞(Rn ) g

P s−ρ , τ

Fp,1

sup 2 k=2,3,...

k(s−ρ )

k−2

p

∑ |(ϕ j ∗ g)(x)|

1/p dx

j=0

(Rn )

s, τ   f Bρ∞,∞(Rn ) gFp,q (Rn )

because of ρ > 0. Summarizing Step 1 through Step 3 we have proved the embedding Bρ∞,∞ (Rn ) ⊂ M(Fp,s, qτ (Rn )). ρ

However, because of Cρ (Rn ) ⊂ B∞,∞ (Rn ) this is sufficient.



Remark 6.4. For τ = 0 this is a well-known result; we refer to [145, Corollary 2.8.2], [64] and [119, Sect. 4.7.1].

6.1.2.4 Multiplication Algebras This time we study the question under which conditions do we have the embedding X ⊂ M(X). Essentially the same methods as used in the proof of Theorem 6.2 apply. Theorem 6.3. Let p ∈ (0, ∞), q ∈ (0, ∞], τ ∈ [0, 1/p) and s ∈ (σ p,q , ∞). Then there exists a positive constant C such that for all f , g ∈ Fp,s, qτ (Rn ) ∩ L∞ (Rn ),

 f · gFp,s, τq (Rn ) ≤ C  f L∞ (Rn ) gFp,s, τq (Rn ) + gL∞(Rn )  f Fp,s, τq (Rn ) . (6.21) Proof. The estimate of Π1 ( f , g), given in (6.20), is totally sufficient for our purpose. Since Π3 ( f , g) = Π1 (g, f ) we also get the estimate of Π3 on this way. Finally, we have to deal with Π2 . Similarly as in Step 2 of the proof of Theorem 6.2 we find Π2 ( f , g)Fp,s, qτ (Rn ) ⎛   p/q ⎞1/p  ∞ 1 ⎝  sup ∑ 2ksq|(ϕk ∗ f )(x) · (ϕk ∗ g)(x)|q dx⎠ τ P k=0 P∈Q |P|

6.1 Pointwise Multipliers

157

 

 sup ϕk ∗ f L∞ (Rn )

k=0,1,...

⎛   p/q ⎞1/p  ∞ 1 ⎝ × sup ∑ 2ksq |(ϕk ∗ g)(x)|q dx⎠ τ P k=0 P∈Q |P| s, τ   f B0∞,∞(Rn ) gFp,∞ (Rn ) ,

where we could apply Proposition 6.2 since s > σ p,q . The proof is completed by taking into account the embedding L∞ (Rn ) ⊂ B0∞,∞ (Rn ).

Remark 6.5. (i) The estimate (6.21) implies that the spaces Fp,s, qτ (Rn )∩ L∞ (Rn ) are algebras with respect to pointwise multiplication. (ii) For τ = 0 we refer to [119, Theorem 4.6.4/2]. Combining Theorem 6.3 with Proposition 2.6 we get the following conclusion concerning the algebra properties of Fp,s, qτ (Rn ). Corollary 6.1. Let s ∈ R, p ∈ (0, ∞), q ∈ (0, ∞] and τ ∈ [0, 1/p) such that  s > n max

 1 1 − τ, − 1 . p q

Then Fp,s, qτ (Rn ) is an algebra with respect to pointwise multiplication. Remark 6.6. (i) For τ = 0 this question had some history. For the Bessel potential spaces s, 0 H ps (Rn ) = Fp,2 (Rn ), p ∈ (1, ∞), it was settled by Strichartz [135]. This was extended by Triebel in [144, Sect. 2.6.2], Kalyabin [84,86] and Franke [59]; see also [119, Theorem 4.6.4/1]. (ii) Characterizations of M(Wpm (Rn )), H ps (Rn ) and M(Bsp,p (Rn )) can be found in s (Rn )), s > n/p, we refer to Franke [95, 96]. For a characterization of M(Fp,q [59] and [119, Theorem 4.9.1/1].

s (Rn )) 6.1.3 A Characterization of M(F∞,q

The methods used in the previous subsection do partly not apply to the spaces s (Rn ), since we always require τ < 1/p; see Proposition 2.4. However, it is quite F∞,q easy to prove the following. Theorem 6.4. Let q ∈ (0, ∞] and s ∈ (σ1,q , ∞). Then s s (Rn )) = F∞,q (Rn ) M(F∞,q

in the sense of equivalent quasi-norms.

158

6 Key Theorems

Proof. Step 1. We shall prove s s (Rn ) ⊂ M(F∞,q (Rn )). F∞,q

There is an elementary approach based on Corollaries 4.6, 4.7 if q < ∞, and Theorem 4.7 (q = ∞). We employ the formula

ΔhM ( f · g)(x) =

M

∑ ck (Δhk f )(x) (ΔhM−k g)(x + kh) ,

k=0

where ck = ck (M) are certain constants depending only on M. Choosing M such that M n −n < s < q 2 then either k > s > n/q − n or M − k > s > n/q − n. Let k0 ∈ Z+ be chosen such that k0 ≤ s < k0 + 1. Thus, if 0 < q < ∞, 

sup t/2≤|h|


|ΔhM ( f · g)(x)|q dx 

k0

∑  f L∞ (Rn )

k=0

+

M

∑

k=k0 +1

sup t/2≤|h|
gL∞(Rn )

|ΔhM−k g(x + kh)|qdx 

sup t/2≤|h|
|Δhk f (x)|q dx .

For t < l(P) ≤ 1 we have 

sup t/2≤|h|
|ΔhM−k g(x + kh)|qdx 



sup

t/2≤|h|
|ΔhM−k g(x)|q dx .

Altogether, in view of Corollaries 4.6, 4.7, this proves ||| f · g|||♠ Fs

n ∞,q(R )

  f L∞ (Rn ) |||g|||♠ Fs

n ∞,q(R )

+ gL∞(Rn ) ||| f |||♠ Fs

n ∞,q(R )

.

The estimate of Nq ( f ) (or N1 ( f )) and the modifications, needed for q = ∞, are obvious. s (Rn ). Thus, if f · g ∈ F s (Rn ) for all Step 2. The function g ≡ 1 belongs to F∞,q ∞,q s n s g ∈ F∞,q (R ), necessarily we get f ∈ F∞,q (Rn ). In addition we have  f M(F∞,q s (Rn )) ≥ This proves the theorem.

 f F∞,q s (Rn ) 1F∞,q s (Rn )

.



6.1 Pointwise Multipliers

159

Remark 6.7. (i) The assertions of the Theorem do not extend to s = 0. E.g, if q = 2, the correct description of M( bmo ) was found by Janson [75]. For a description of M(B0∞,∞ (Rn )) we refer to [87]. s (Rn ) are algebras with respect to point(ii) Theorem 6.4 implies that the spaces F∞,q wise multiplication, at least, if s > σ1,q . For q ≥ 1 a different proof of this fact can be found in [93].

6.1.4 A Characterization of M(Fsp,q (Rn )), s < n/p As said above, in case τ = 0 much more is known; see, for example, [131]. Of s (Rn )), p ∈ (0, 1), certain relevance for this lecture note is the description of M(Fp,q s ∈ (σ p,q , n/p), given by Netrusov [106]. s (Rn )) Theorem 6.5. Let p ∈ (0, 1], q ∈ (0, ∞] and s ∈ (σ p,q , n/p). Then f ∈ M(Fp,q if, and only if f ∈ L∞ (Rn ), f can be represented in S (Rn ) in the form

f=

∞

∑ fj ,

supp F f j ⊂ B(0, 2 j+1 ) \ B(0, 2 j−1) ,

j=0

such that ⎛ sup sup 2

j( np −s)

⎝





j∈Z+ x∈Rn

∞

∑ 2ksq| fk (x)|q

B(x,2− j )

 p/q

⎞1/p dx⎠

< ∞.

k= j

Remark 6.8. Netrusov [106] did not publish a proof of this remarkable result. A proof under more restrictive conditions can be found in [132]. For p = q = 1 some more simple characterizations have been found by Maz’ya and Shaposnikova; see [95, Sect. 3.4.2] Theorem 6.6. Let s = m + σ , where m is a nonnegative integer and σ is a real s (Rn )) if, and only if f ∈ L∞ (Rn ) and number with σ ∈ (0, 1). Then f ∈ M(F1,1 sup sup rs−n 0
∑

 B(x,r)

|α |≤m

+

|∂ α f (y)| dy



 |∂ α f (y) − ∂ α f (x)| dy dx < ∞ . (6.22) |y − x|n+σ B(x,r)

 B(x,r)

We would like to reformulate Theorem 6.6. For this reason we recall the defis, τ n nition of the space Fp,q, unif (R ). Let ψ be as in (1.1). A distribution f belongs to s,τ n Fp,q, unif (R ) if

160

6 Key Theorems

 f F s,τ

n p,q, unif (R )

s,τ ≡ sup  f ψ ( · − λ )Fp,q (Rn ) < ∞ .

λ ∈Zn

Let m = 0 in Theorem 6.6. Then, as an immediate conclusion of Corollary 4.5, we obtain the following. s (Rn )) if, and only if Corollary 6.2. Let s ∈ (0, 1). Then f ∈ M(F1,1 s,τ n f ∈ L∞ (Rn ) ∩ F1,1, unif (R ),

where τ = 1 − s/n. Remark 6.9. We conjecture that s,τ s n (Rn )) = L∞ (Rn ) ∩ Fp,q, M(Fp,q unif (R ) ,

τ=

1 s − , p n

holds under the restrictions of Theorem 6.5.

6.2 Diffeomorphisms Let m ∈ N and BC(Rn ) denote the collection of all complex-valued bounded and continuous functions in Rn . We begin with recalling the notion of diffeomorphisms; see, for example, [146, p. 206]. A one-to-one mapping y = ψ (x) of Rn onto Rn is called an m-diffeomorphism if the components ψ j of ψ = (ψ1 , · · · , ψn ) have classical derivatives up to order m with ∂ α ψ j ∈ BC(Rn ) if 0 < |α | ≤ m, and | det ψ∗ (x)| ≥ c > 0 for some positive constant c and all x ∈ Rn , where ψ∗ stands for the Jacobian matrix of ψ . We remark that if ψ is an m-diffeomorphism, then its inverse ψ −1 is also an m-diffeomorphism. The mapping ψ is called a diffeomorphism if it is an m-diffeomorphism for any m ∈ N. Let ψ be a diffeomorphism. It is well known that for any f ∈ S (Rn ), f ◦ ψ ≡ f (ψ (·)) makes sense. If ψ is only an m-diffeomorphism and f ∈ As,p,τq (Rn ), we define f ◦ ψ via smooth atoms for As,p,τq (Rn ). We wish to know that whether the mapping Dψ ( f ) ≡ f ◦ ψ is a linear and bounded operator from As,p,τq (Rn ) onto itself. Based on the smooth atomic decomposition characterization of As,p,τq (Rn ) in Theorem 3.3, we have the following conclusion.

6.2 Diffeomorphisms

161

Theorem 6.7. Let m ∈ N, ψ be an m-diffeomorphism and s, p, q and τ be as in Theorem 5.1. If m ∈ N is sufficiently large, then Dψ is an isomorphic mapping of As,p,τq (Rn ) onto itself. Proof. Notice that ψ is an m-diffeomorphism if and only if its inverse ψ −1 is an m-diffeomorphism. To show Dψ is an isomorphic mapping of As,p,τq (Rn ) onto itself, we only need to prove that Dψ is a linear and continuous operator. We first consider the case when s > J − n. Let f ∈ As,p,τq (Rn ). By Theorem 3.3, we can write f as f = ∑ tQ aQ l(Q)≤1

in S (Rn ), where each aQ is a smooth atom for As,p,τq(Rn ) supported near Q and the sequence t ≡ {tQ }l(Q)≤1 ⊂ C satisfies tas,p,τq(Rn )   f As,p,τq(Rn ) . For all l(Q) ≤ 1, set mQ ≡ aQ (ψ (·)). We claim that Dψ ( f ) ≡

∑

tQ aQ (ψ (·)) =

l(Q)≤1

∑

tQ mQ

l(Q)≤1

converges in S (Rn ) and satisfies Dψ ( f )As,p,τq (Rn )   f As,p,τq(Rn ) . By Theorem 3.3 again, it suffices to prove that each mQ is also a multiple of a smooth  with |Q|  ∼ |Q|. atom for As,p,τq (Rn ) supported near a new dyadic cube Q By the assumption s > J − n, no moment conditions (3.3) are necessary. Since  ∈ Q such that |Q|  ∼ |Q| | det ψ∗ (x)| ≥ c > 0, it is easy to check that there exists a Q  Thus, and ψ −1 (3Q) ⊂ 3Q.  supp mQ = supp (aQ (ψ (·))) ⊂ 3Q. Let m ≥ max{s + nτ + 1 , 0}. By the regularity condition of aQ and l(Q) ≤ 1, we see that for all |γ | ≤ max{s + nτ + 1 , 0} ∂ γ (aQ (ψ (·)))L∞ (Rn ) ≤ ≤

∑ C(α , ψ )∂ α aQ L∞ (Rn )

α ≤γ

∑ C(α , ψ )|Q|−1/2−|α |/n

α ≤γ

≤ C(γ , ψ )|Q|−1/2−|γ |/n . Therefore, each mQ is a multiple of a smooth atom for As,p,τq(Rn ), which further implies that Dψ is a linear and continuous operator from As,p,τq (Rn ) to itself when s > J − n.

162

6 Key Theorems

Now we consider the case when s ≤ J − n. In this case, we use an argument similar to that used in Step 3 of the proof of [146, Theorem 4.3.2]. For convenience of the reader, we give the details. Indeed, if we fix l ∈ N such that s + 2l > J − n, as in the proof of Theorem 6.1, we represent f ∈ As,p,τq (Rn ) as f ≡ (I + (−Δ )l )h τ with h ∈ As+2l, (Rn ) and p,q

hAs+2l,τ (Rn ) ∼  f As,p,τq(Rn ) . p,q

Then, if m ≥ 2l, f (x) = (I + (−Δ )l )h(ψ ◦ ψ −1(x)) =

∑

|α |≤2l

Cα (x)(∂ α h ◦ ψ )(ψ −1(x))

for some bounded and continuous functions Cα . Then, if we choose m as in the proof of Theorem 6.1 in the case s ≤ J − n, by Theorem 6.1, Corollary 5.1 and Proposition 2.1(ii), we have  f ◦ ψ As,p,τq(Rn )  

∑

Cα ∂ α h ◦ ψ As,p,τq(Rn )

∑

∂ α h ◦ ψ As,p,τq(Rn )

|α |≤2l |α |≤2l

 h ◦ ψ As+2l,τ (Rn ) , p,q

which together with the previous proved result when s > J − n and the fact hAs+2l,τ (Rn ) ∼  f As,p,τq(Rn ) p,q

yields the desired result, and then completes the proof of Theorem 6.7.



Theorem 6.7 generalizes the classical results on Besov spaces and TriebelLizorkin spaces by taking τ = 0; see, for example, [146, Proposition 4.3.1, Remark 4.3.1 and Theorem 4.3.2].

6.3 Traces The trace theorem is of crucial interest for boundary value problems of elliptic differential operators. Let x = (x1 , · · · , xn ) ∈ Rn and x ≡ (x1 , · · · , xn−1 ) ∈ Rn−1 . We are interested in properties of the trace operator Tr :

f (x) → f (x , 0).

(6.23)

6.3 Traces

163

For τ = 0 such problems have been treated extensively; see, for example, [145, Sect. 2.7.2] and [62,64]. In this section, we deal with the corresponding problem for the spaces As,p,τq (Rn ). It is easy to see that (6.23) makes sense for all smooth atoms f for As,p,τq(Rn ). We follow the approach of Frazier and Jawerth [62, 64]. They showed the usefulness of atomic characterizations in connection with the trace problem.

6.3.1 Traces of Functions in As,p,τq (Rn ) In this section, to emphasize the dimension n, we denote by Q(Rn ) the collection of all dyadic cubes in Rn and by Q j (Rn ) the collection of all Q ∈ Q(Rn ) with l(Q) = 2− j for all j ∈ Z. The main result of this section is the following theorem. The proof is similar to those given for [127, Theorems 1.3 and 1.4]. Theorem 6.8. Let n ≥ 2, p, q ∈ (0, ∞], s ∈ (1/p + (n − 1)[1/ min{1, p} − 1], ∞), J be as in Definition 3.2 and τ ∈ [0, 1/p + (s + n − J)/n). Then Tr is a linear, continnτ s− 1 , n−1

uous and surjective operator from Bs,p,τq (Rn ) to B p,qp nτ s− 1p , n−1

to Fp,p

(Rn−1 ) and from Fp,s, qτ (Rn )

(Rn−1 ).

Our proof of Theorem 6.8 will take full advantage of the smooth atomic des− 1 , nτ

composition characterizations of As,p,τq(Rn ) and A p,qp n−1 (Rn−1 ). Neither of which requires any moment condition because of the assumption on s. Proof of Theorem 6.8. By similarity, we only consider the Besov-type spaces. Let f ∈ Bs,p,τq (Rn ). By Theorem 3.3, we write f=

∑

tQ aQ

{Q∈Q(Rn ): l(Q)≤1}

in S (Rn ), where each aQ is a smooth atom for Bs,p,τq(Rn ) supported near Q and the coefficient sequence t ≡ {tQ }{Q∈Q(Rn ): l(Q)≤1} ⊂ C satisfies tbs,p,τq(Rn )   f Bs,p,τq(Rn ) . Precisely, the smooth function aQ satisfies the support condition (3.13) and the regularity conditions (3.15) for all |γ | ≤ max{s + nτ + 1 , 0}. Since s ∈ (1/p + (n − 1)(1/ min{1, p} − 1), ∞), the moment condition (3.14) is an empty condition.

164

6 Key Theorems

Recall that tQ aQ is obtained canonically for f ∈ Bs,p,τq (Rn ). Then the definition of Tr( f ) can be rephrased as

∑

Tr( f )(x , 0) ≡

tQ Tr(aQ )(x , 0).

{Q∈Q(Rn ): l(Q)≤1}

We now verify that the summation in the right-hand side of the above equality converges in S (Rn−1 ) and satisfies Tr( f )

nτ s− 1p , n−1

B p,q

(Rn−1 )

  f Bs,p,τq(Rn ) .

/ {0, 1, 2}, Since supp aQ ⊂ 3Q, then if i ∈ aQ ×[(i−1)l(Q ),il(Q )) (· , 0) ≡ 0. Thus, the summation

∑ n

tQ Tr(aQ )(· , 0)

{Q∈Q(R ): l(Q)≤1}

can be re-written as 2

∑

∑

i=0 Q ∈Q(Rn−1 )

tQ ×[(i−1)l(Q ),il(Q )) aQ ×[(i−1)l(Q ),il(Q )) (· , 0).

(6.24)

l(Q )≤1

To show that (6.24) converges in S (Rn−1 ), by Theorem 3.3 again, it is sufficient to prove that each bQ ≡ [l(Q )]1/2 aQ ×[(i−1)l(Q ),il(Q )) (· , 0) s− 1 , nτ

is a smooth atom for B p,qp n−1 (Rn−1 ) supported near Q

    [l(Q )]−1/2tQ ×[(i−1)l(Q ),il(Q )) 

n−1

{Q ∈Q(R

and for all i ∈ / {0, 1, 2},    1 nτ < ∞.  s− ,

): l(Q )≤1} b p n−1 (Rn−1 ) p,q

By similarity, we only consider the case when i = 1. It immediately deduces from the corresponding properties of aQ that bQ satisfies (3.13) and (3.15), namely, bQ

s− 1 , nτ

is a smooth atom for B p,qp n−1 (Rn−1 ) supported near Q . On the other hand,       [l(Q )]−1/2tQ ×[0,l(Q ))  1 nτ   s− ,

n−1

{Q ∈Q(R

=

sup

P ∈Q(Rn−1 )

1 |P |

nτ n−1

⎧ ⎪ ⎪ ⎪ ⎨

): l(Q )≤1} b p n−1 (Rn−1 ) p,q

⎛ ∞

⎪ ∑

⎪ j= j ⎪ ⎩ P

⎜ ⎜ ⎝

∑

Q ⊂P , l(Q )≤1 Q ∈Q(Rn−1 )

⎞q/p ⎫1/q ⎪ ⎪ ⎪ [l(Q )]−p/2 |tQ ×[0,l(Q )) | p ⎟ ⎬ ⎟ [l(Q )] ps−1+(n−1)(p/2−1) ⎠ ⎪ ⎪ ⎪ ⎭

6.3 Traces

=

165

⎧ ⎪ ⎪ ⎪ ⎨

1

sup

P ∈Q(Rn−1 ) |P |

nτ n−1

⎛ ∞

⎜ ⎜ ⎝

⎪ ∑

⎪ j= j ⎪ ⎩ P

∑

Q ⊂P , l(Q )≤1 Q ∈Q(Rn−1 )

⎞q/p ⎫1/q ⎪ ⎪ ⎪ |tQ ×[0,l(Q )) | p ⎟ ⎬ ⎟ [l(Q )] ps+pn/2−n ⎠ ⎪ ⎪ ⎪ ⎭

≤ tbs,p,τq(Rn ) . Therefore, by Theorem 3.3, we obtain that (6.24) converges in S (Rn−1 ) and Tr( f )

nτ s− 1p , n−1

B p,q

(Rn−1 )

 tbs,p,τq(Rn )   f Bs,p,τq(Rn ) .

We now show that Tr is surjective. To this end, by Theorem 3.3 again, any f ∈

s− 1 , nτ B p,qp n−1 (Rn−1 )

can be represented as

∑

f=

λQ aQ

{Q ∈Q(Rn−1 ): l(Q )≤1} nτ s− 1 , n−1

in S (Rn−1 ), where each aQ is a smooth atom for B p,qp Q and the coefficient sequence

(Rn−1 ) supported near

λ ≡ {λQ }{Q ∈Q(Rn−1 ): l(Q )≤1} ⊂ C satisfies λ 

1

nτ

s− p , b p,q n−1 (Rn−1 )

 f

1

nτ

s− p , B p,q n−1 (Rn−1 )

.

Let ϕ ∈ Cc∞ (R) with supp ϕ ⊂ (− 12 , 12 ) and ϕ (0) = 1. For all Q ∈ Q(Rn−1 ) and all

x ∈ R, set ϕQ (x) ≡ ϕ (2− log2 l(Q ) x) and F≡

∑

λQ aQ ⊗ ϕQ .

{Q ∈Q(Rn−1 ): l(Q )≤1}

It is easy to check that each [l(Q )]−1/2 aQ ⊗ ϕQ is a smooth atom for Bs,p,τq(Rn ) supported near Q × [0, l(Q )). Moreover, 1

{[l(Q )] 2 λQ }{Q ∈Q(Rn−1 ): l(Q )≤1} bs,p,τq (Rn )   f 

nτ s− 1p , n−1

B p,q

(Rn−1 )

.

Then Theorem 3.3 implies that F ∈ Bs,p,τq(Rn ) and FBs,p,τq(Rn )   f 

nτ s− 1p , n−1

B p,q

(Rn−1 )

;

furthermore, Tr(F) = f , which shows that Tr is surjective, and then, completes the proof of Theorem 6.8.



166

6 Key Theorems

Remark 6.10. (i) We would like to mention that Theorem 6.8 generalizes the classical trace theorems for Besov spaces and Triebel-Lizorkin spaces by taking τ = 0; see, for example, [13, 77, 108, 109], [145, Sect. 2.7.2] or [146, Sect. 4.4]. (ii) The counterpart of Theorem 6.8 for the homogeneous Besov-type space B˙ s,p,τq (Rn ) and Triebel-Lizorkin-type space F˙p,s, qτ (Rn ) were already obtained in [127]. (iii) Limiting situations for τ = 0, i. e. s = 1p + (n − 1) max{0, 1/p − 1}, are investigated in [28, 64, 112], [146, Sect. 4.4.3] and [56].

s (Rn ) and Some Consequences 6.3.2 Traces of Functions in F∞,q

In view of Proposition 2.4, Theorem 6.8 yields that Tr is a linear, continuous and surjective operator from s, 1/p

1 s−1/p, 1p + p(n−1)

s F∞,q (Rn ) = Fp,q (Rn ) → Fp,p

(Rn−1 ) ,

as long as s > σ p,q (then τ can be chosen to be 1/p ) and s>

  1 1 + (n − 1) max 0, − 1 . p p

s (Rn ) under Tr are well known. We refer to However, the range spaces of F∞,q Marschall [93] and Frazier and Jawerth [64, Theorem 11.2] (in combination with the comments at the end of Sect. 12 in [64]). There it is proved that s s (Rn )) = F∞,∞ (Rn−1 ) = Bs∞,∞ (Rn−1 ) = Z s (Rn−1 ) . Tr(F∞,q

Thus, we got two different characterizations. This is stated as a supplement to Proposition 2.4. Lemma 6.1. Let s ∈ R and p ∈ (0, ∞) such that   1 1 s > + n max 0, − 1 . p p Then 1 s−1/p, 1p + pn

Fp,p

(Rn ) = Z s (Rn )

in the sense of equivalent quasi-norms. This procedure can be iterated by taking into account Tr(Z s (Rn )) = Z s (Rn−1 ). Furthermore, it can be combined with Proposition 2.6.

6.3 Traces

167

Theorem 6.9. Let 0 < p < p0 < ∞, k ∈ N and   k 1 s > + n max 0, − 1 . p p (i) Then s−k/p, 1p n+k n

Fp,q

(Rn ) = Z s (Rn )

if

p ≤ q ≤ ∞,

(6.25)

and n 1 n+k s− k+n p +p , p n

(Rn ) = Z s (Rn )

0

Fp0 ,q

if

0 < q ≤ ∞,

(6.26)

in the sense of equivalent quasi-norms. (ii) We have s−k/p, 1p n+k n

B p,q

(Rn ) = Z s (Rn )

if

p ≤ q ≤ ∞,

(6.27)

and n 1 n+k s− k+n p + p0 , p n

B p0 ,q

(Rn ) = Z s (Rn )

if

p ≤ q ≤ ∞,

(6.28)

in the sense of equivalent quasi-norms. Proof. The iteration of the trace argument yields, in case p ∈ (0, ∞), k ∈ N and   1 k s > + n max 0, − 1 , p p the coincidence of the spaces s−k/p, 1p n+k n

Fp,p

(Rn ) = Z s (Rn )

in the sense of equivalent quasi-norms. Let 0 < p ≤ p1 < ∞. Now we apply Propositions 2.1, 2.6 and Corollary 2.2 and obtain s−k/p, 1p n+k n

Z s (Rn ) = Fp,p ⊂B

s−k/p, 1p n+k n

(Rn ) ⊂ B p,∞

n 1 n+k s− k+n p + p1 , p n p1 ,∞

(Rn )

(Rn ) ⊂ Z s (Rn ) ,

  n n k+n + = s. + nτ − s− p p1 p1

since

Taking p1 = p we have proved (6.25) and (6.27). Now let p1 = p0 > p. Because of s−k/p, 1p n+k n

Fp,p

n 1 n+k s− k+n p + p0 , p n

(Rn ) ⊂ Fp0 ,q

(Rn ) ⊂ Z s (Rn )

168

6 Key Theorems

(see Corollary 2.2), also (6.26) is proved. Finally, (6.28) is a consequence of s−k/p, 1p n+k n

Fp,p

s−k/p, 1p n+k n

(Rn ) = B p,p

n 1 n+k s− k+n p + p0 , p n

(Rn ) ⊂ B p0 ,p

(Rn ) ⊂ Z s (Rn ) ;

see again Corollary 2.2.



Remark 6.11. (i) We believe that Theorem 6.9 is not the final answer to the questions: (α) For which set of parameters p, q, τ , we have Fp,s, qτ (Rn ) = Z s (Rn ); and (β) For which set of parameters p, q, τ , we have Bs,p,τq (Rn ) = Z s (Rn ). (ii) Differently from [146], wherein Triebel established the mapping properties of pointwise multipliers, trace properties and the theorem on diffeomorphisms for Besov spaces and Triebel-Lizorkin spaces via the local mean characterizations of these spaces, in this chapter, we establish these key properties for Bs,p,τq(Rn ) and Fp,s, qτ (Rn ) via the smooth atomic decomposition characterizations of these spaces. λ ,s (Rn ), for all Recently, Drihem [52] independently introduced the spaces L p,q s ∈ R, λ ∈ [0, ∞) and p, q ∈ (0, ∞), and obtained their maximal function and local mean characterizations. As in Triebel [146], these characterizations provide another possible way to obtain the key properties for these spaces. Recall that the spaces λ ,s L p,q (Rn ) when p, q ∈ [1, ∞) were originally introduced by El Baraka [49, 50]. λ ,s (Rn ) for all s ∈ R, λ ∈ [0, ∞) and p, q ∈ (0, ∞) It is easy to see that the spaces L p,q s,λ /(nq) (Rn ). We also point out that the maximal are just the Besov-type spaces B p,q function and local mean characterizations for homogeneous spaces B˙ s,p,τq (Rn ) and F˙p,s, qτ (Rn ) were established in [167].

6.4 Spaces on Rn+ and Smooth Domains In this section, we introduce the Besov-type spaces and Triebel-Lizorkin-type spaces on Rn+ and Ω , where Rn+ ≡ {x = (x , xn ) : x ∈ Rn−1 , xn > 0} and Ω stands for a bounded C∞ domain in Rn ; see, for example, [145, Sect. 3.2.1]. We remark that domain always stands for an open set.

6.4.1 Spaces on Rn+ Let D(Rn+ ) be the set of all C∞ (Rn+ ) functions with compact supports in Rn+ and denoted by D (Rn+ ) its topological dual. The spaces on Rn+ are defined as follows.

6.4 Spaces on Rn+ and Smooth Domains

169

Definition 6.1. Let s ∈ R, τ ∈ [0, ∞) and p, q ∈ (0, ∞]. The space As,p,τq (Rn+ ) is defined to be the restriction of As,p,τq(Rn ) on Rn+ , quasi-normed by  f As,p,τq(Rn ) ≡ inf gAs,p,τq(Rn ) , +

where the infimum is taken over all g ∈ As,p,τq (Rn ) with g|Rn+ = f in the sense of D (Rn+ ). From Definition 6.1, we deduce that if ∞

∑  fm As,p,τq(Rn+ )

min{1, p, q}

< ∞,

m=1

then

∞

∑

fm ∈ As,p,τq (Rn+ ),

m=1

which further yields that the space As,p,τq (Rn+ ) is a quasi-Banach space. In particular, As,p,τq (Rn+ ) is a Banach space if p, q ∈ [1, ∞]. Let m ∈ N. Denoted by Cm (Rn+ ) the set of all functions f on Rn+ such that f = g|Rn+ for some functions g ∈ Cm (Rn ) such that ∂ α gL∞ (Rn ) ∼ ∂ α f L∞ (Rn+ ) for all |α | ≤ m. We also have the following pointwise multiplication assertion for As,p,τq (Rn+ ), which is an immediate corollary of Definition 6.1 and Theorem 6.1. We omit the details. Theorem 6.10. Let m ∈ N and 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 ∈ Cm (Rn+ ) and all f ∈ As,p,τq (Rn+ ), g f As,p,τq(Rn ) ≤ C(m) +

∑

|α |≤m

∂ α gL∞ (Rn+ )  f As,p,τq(Rn ) . +

Next we establish the lifting property for As,p,τq (Rn+ ). Recall that Franke and Runst [60] (see also [119, Proposition 2.4.3]) constructed a family {Jσ }σ ∈R of isomorσ n phisms mapping Asp, q (Rn ) to As− p, q (R ) such that (i) Jσ and J−σ are inverse to each other; (ii) If f ∈ S (Rn ) is supported in Rn−1 × (−∞, 0], so is Jσ f . Notice that the classical lifting operator Iσ does not satisfy the above condition (ii). Following Sawano [124], for ε ∈ [0, ∞), we define a holomorphic function ψε on C by setting, for all z ∈ C,

ψε (z) ≡

 0 −∞

η (t)e−iε tz dt − iz,

170

6 Key Theorems

where η ∈ S (R) is a positive real-valued function supported in (−∞, 0) with integral 2. Let H ≡ {z ∈ C : Im(z) > 0} and H ≡ {z ∈ C : Im(z) ≥ 0}. Define a function φ σ : Rn−1 × H → C by σ

2 σ /2



φ (x , zn ) ≡ (1 + |x | )

&



ψε

zn (1 + |x |2 )σ /2

'σ

.

It was proved in [124, Lemma 4.3] that for all α ∈ Zn+ , there exists a positive constant C(α ) such that for all (x , zn ) ∈ Rn−1 × H, |∂ α φ 1 (x , zn )| ≤ C(α )[(1 + |x |2 )1/2 + |zn |]1−|α | . Especially, when |α | = 0, |φ 1 (x , zn )| ∼ (1 + |x |2 )1/2 + |zn |. Denote again by φ σ the restriction of φ σ to Rn . The above observations imply that σ (Rn ). Define J by setting, for all ξ ∈ Rn , φ σ ∈ S1,1 σ σ  J( σ f (ξ ) ≡ φ (ξ ) f (ξ ).

Then from Theorem 5.1, we deduce that Proposition 5.1 is still true with Iσ replaced by Jσ . Proposition 6.4. Let σ ∈ R, s, p, q and τ be as in Theorem 5.1. Then the operator σ,τ n Jσ maps As,p,τq (Rn ) isomorphically onto As− p, q (R ). Recall that if f ∈ S (Rn ) is supported in Rn−1 × (−∞, 0], so is Jσ f ; see [60] or [124, Proposition 4.6]. We then have the following result, which is an immediate corollary of Proposition 6.4. Proposition 6.5. Let σ ∈ R, s, p, q and τ be as in Theorem 5.1. Let f ∈ As,p,τq (Rn+ ). Then Jσ f ≡ Jσ g|Rn+ does not depend on the choice of the representative g ∈ As,p,τq(Rn ) σ,τ n of f and Jσ maps As,p,τq (Rn+ ) isomorphically onto As− p, q (R+ ). The restriction operator re is a linear and bounded operator from As,p,τq(Rn ) onto s, τ A p, q (Rn+ ). It is natural to ask whether there exists a linear and bounded operator ext from As,p,τq (Rn+ ) into As,p,τq(Rn ) such that re◦ ext is the identity in As,p,τq (Rn+ ). Extension problems for Besov spaces and Triebel-Lizorkin spaces have been studied in depth by Triebel; see, for example, [145, Sect. 2.9] and [146, Sect. 4.5]. Let M ∈ Z+ be large enough, 0 < λ0 < λ1 < · · · < λM and a0 , · · · , aM be real numbers such that for all l ∈ {0, · · · , M}, M

∑ ak (−λk )l = 0.

k=0

6.4 Spaces on Rn+ and Smooth Domains

171

As in [146, Sect. 4.5.2], we define extM by setting, for all functions f on Rn+ and x = (x , xn ) ∈ Rn ,

extM f (x) ≡

⎧ ⎪ ⎨ f (x),

if x ∈ Rn+ ;

⎪ ⎩ ∑ ak f (x , −λk xn ),

if xn ≤ 0.

M

(6.29)

k=0

Then we have the following extension theorem. Similarly to [146, Sect. 4.5.2], its proof relies on the oscillation characterization in Theorems 4.10 and 4.13. Theorem 6.11. Let p ∈ [1, ∞), q ∈ (0, ∞], s ∈ R and τ ∈ [0, ∞). There exists a linear and bounded operator ext from As,p,τq (Rn+ ) into As,p,τq (Rn ) such that re ◦ ext is the identity in As,p,τq (Rn+ ). Proof. We first consider the case when s > 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.

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 7, c Springer-Verlag Berlin Heidelberg 2010 

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|
Set

n+1 Rn+1 Z+ ≡ (x, a) ∈ R+ : − log2 a ∈ Z+ .

n In what follows, for all functions F on Rn+1 Z+ , x ∈ R and j ∈ Z+ , write

F j (x) ≡ F(x, 2− j ). We then introduce the following tent spaces. Definition 7.2. Let s ∈ R, p ∈ (1, ∞), q ∈ [1, ∞) and τ ∈ (0,

1 (p∨q) ].

s,τ n+1 (i) The tent space BTp,q (Rn+1 Z+ ) is defined to be the set of all functions f on RZ+ j such that { f } j∈Z+ are Lebesgue measurable and

 s,τ f BTp,q (Rn+1 ) ≡ inf Z+

ω

∞

∑2

j=0

1/q jsq

f

j

[ω j ]−1 qL p (Rn )

< ∞,

7.1 Tent Spaces

179

where the infimum is taken over all nonnegative Borel measurable functions ω on Rn+1 + with 



Rn

(∞)

[N ω (x)](p∨q) dΛnτ (p∨q) (x) ≤ 1,

(7.3)

and with the restriction that ω is allowed to vanish only where f vanishes. s,τ (ii) The tent space FTp,q (Rn+1 Z+ ) (q = 1) is defined to be the set of all functions f n+1 j on RZ+ such that { f } j∈Z+ are Lebesgue measurable and  1/q   ∞    jsq j q j −q   s,τ ≡ inf  ∑ 2 | f | [ω ] f FTp,q  (Rn+1 ) ω  Z+  j=0

< ∞,

L p (Rn )

where the infimum is taken over all nonnegative Borel measurable functions ω on Rn+1 + with the same restrictions as in (i). These tent spaces are applied later to determine the predual spaces of Bs,p,τq(Rn ) s,τ s,τ s,τ n+1 n+1 and Fp,s, qτ (Rn ). We use ATp,q (Rn+1 Z+ ) to denote either BTp,q (RZ+ ) or FTp,q (RZ+ ). Remark 7.1. s,τ (i) The notion of ATp,q (Rn+1 Z+ ) can be extended to τ = 0. In this case, (7.3) implies that ω has upper bound. In fact, for all nonnegative integers k, set 

Ek ≡ {x ∈ Rn : [N ω (x)](p∨q) /q > 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

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

 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

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

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

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

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

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

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 H¨older’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 H¨older’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

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

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

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



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)

,  

which completes the proof of Lemma 7.4.

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

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

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

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

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|

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

≤ C < ε.

∑

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

∑

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

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 ,

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

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

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

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

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



∞

∑ ∑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 Calder´on 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.  

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

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|<β t

In Definition 7.2, with (7.3) replaced by 

(

Rn

)(p∨q) (∞) dΛnτ (p∨q) (x) ≤ 1, Nβ ω (x)

we obtain another tent space, denoted by ATp,s, qτ (β , Rn+1 Z+ ). Then, obviously, s, τ n+1 ATp,s, qτ (1, Rn+1 Z+ ) = ATp, q (RZ+ ).

Generally, we have the following result. Lemma 7.8. Let β ∈ [1, ∞) and s, τ , p, q be as in Definition 7.2. Then s, τ n+1 ATp,s,qτ (β , Rn+1 Z+ ) = ATp, q (RZ+ )

(7.11)

202

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

and there exists a positive constant C such that for all functions F ∈ ATp,s, qτ (Rn+1 Z+ ) and β ∈ [1, ∞), 

F ATp,s, qτ (Rn+1 ) ≤ F ATp,s, qτ (β , Rn+1 ) ≤ Cβ n((p∨q) +2)/q F ATp,s, qτ (Rn+1 ) . Z+

Z+

Z+

Proof. The first inequality is trivial, so that we only need to verify the second one. Again we only consider the space FTp,s,qτ (Rn+1 Z+ ). s, τ n+1 Let F ∈ FTp, q (RZ+ ) and ω be a nonnegative Borel measurable function on Rn+1 + satisfying (7.3) such that  1/q    ∞     ∑ 2ksq |F k |q [ω k ]−q     k=0

≤ 2 F Fp,s, τq (Rn ) .

L p (Rn )

For any β ∈ [1, ∞), obviously, ⎛

⎞



B(x, β t) ⊂ ⎝

B(y,t/5)⎠ .

y∈B(x,β t) 5n (a+1)n 

By [71, p. 2, Theorem 1.2], there exists a set {a j } j=1 ⎛ B(x, β t) ⊂ ⎝

5n (a+1)n 



⊂ B(0, β t) such that

⎞ B(x + a j ,t)⎠ .

j=1

From this, it follows that for all x ∈ Rn , Nβ ω (x) ≤

5n (β +1)n 

∑

sup |y−x−a j |
j=1

ω (y,t) =

5n (β +1)n 

∑

N ω (x + a j ),

j=1

which together with the geometry property of the Hausdorff capacity and (7.3) yields that 

( Rn

)(p∨q) (∞) Nβ ω (x) dΛnτ (p∨q) (x) (p∨q) +1

 (5 (β + 1) ) n

n

5n (β +1)n  

∑

j=1

Rn





 (5n (β + 1)n )(p∨q) +2 . Set

(∞)

[N ω (x + a j )](p∨q) dΛnτ (p∨q) (x)



 ≡ κω /(5n(β + 1)n)(p∨q) +2 , ω

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

203

 satisfies (7.3). Then we obtain where the positive constant κ is chosen such that ω

F FTp,s, qτ (β , Rn+1 ) Z+

 1/q    ∞   ksq k q  k −q   ≤  ∑ 2 |F | [ω ]    k=0

L p (Rn )

((p∨q) +2)/q

 (5n (β + 1)n)

F FTp,s, qτ (Rn+1 ) , Z+

 

which completes the proof of Lemma 7.8. Proposition 7.2. Let s ∈ R, p ∈ (1, ∞), q ∈ [1, ∞) and τ ∈ [0,

s, τ n AH p, q (R )

is independent of the choices of Φ and ϕ .

1 ]. Then the space (p∨q)

s, τ n Proof. We only consider FH p, q (R ) and τ > 0. Let Φ and ϕ satisfy, respectively, (2.1) and (2.2). Then there exist Ψ and ψ satisfying, respectively, (2.1) and (2.2) s,τ such that (2.6) holds. To emphasize ϕ and ψ , in this proof, we use · FHp,q (ϕ ,Rn ) s,τ s, τ and · FHp,q to replace · . By symmetry, it suffices to show that (ψ ,Rn ) FHp, q (Rn ) s,τ s,τ f FHp,q (ψ ,Rn )  f FHp,q (ϕ ,Rn )

s,τ for all f ∈ FH p,q (ϕ , Rn ).  be a nonnegative Borel measurable function satisfying (7.3) and Let ω

 1/q   ∞    j −q  ∑ 2 jsq |ϕ j ∗ f |q [ω   ]    j=0 

s,τ ≤ 2 f FHp,q (ϕ ,Rn ) .

(7.12)

L p (Rn )

 j ≡ 0 when |k − j| > 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|<2−(k∧ j)

|ϕ j ∗ f (y)| dy

 2l−(k∧ j)−1 ≤|x−y|<2l−(k∧ j)

 |ϕ j ∗ f (y)| dy ,

204

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

Thus by Lemma 2.3 and Remark 7.1(ii), we see that s,τ f FHp,q (ψ ,Rn )    ∞ k+1  ksq  inf  ∑ 2 ∑ 2−|k− j|M 2(k∧ j)n ω  k=0 j=(k−1)∨0 q 1/q     k −1 × |ϕ j ∗ f (y)|[ω (·)] dy   p n |·−y|<2−(k∧ j) L (R )    ∞ ∞ k+1  + ∑ 2−lM+l ρ inf  ∑ 2ksq ∑ 2−|k− j|M 2−ln2(k∧ j)n ω  l=1 k=0 j=(k−1)∨0 q 1/q     × |ϕ j ∗ f (y)|[ω k (·)]−1 dy  l−(k∧ j)  p n |·−y|<2

∞

L (R )

≡ J0 + ∑ 2−lM+l ρ Jl , l=1

where ρ is same as in (7.4). For all l ∈ Z+ and (x, t) ∈ Rn+1 + , set . / κ  (y, s) : |y − x| < 2l+3 s and t/4 ≤ s ≤ 4t , ωl (x, t) ≡ (l+3)n((p∨q)+2) sup ω 2 where κ ∈ (0, ∞) will be determined later. Then, for all x ∈ Rn , / . κ t  (z, s) : |y − z| < 2l+3 s and ≤ s ≤ 4t N ωl (x) = (l+3)n((p∨q)+2) sup sup ω 4 2 |y−x|
7.2 Besov-Hausdorff Spaces and Triebel-Lizorkin-Hausdorff Spaces

205

   ∞ k+1 ln((p∨q) +2)  ksq 2  ∑2 ∑ 2−|k− j|M 2−ln 2(k∧ j)n  k=0 j=(k−1)∨0 q 1/q     j −1/q  × |ϕ j ∗ f (y)|[w  (y)] dy  −(k∧ j) l |·−y|<2 2  p n

L (R )

2

ln((p∨q) +2)

2

ln((p∨q) +2)

 q 1/q    ∞  k+1   1 0 ksq −|k− j|M − js js j −1  × 2 2 2 M 2 | ϕ ∗ f |[ w  ] j ∑  ∑   k=0  p n j=(k−1)∨0 L (R )   1/q   ∞   ( 0 js 1)q    2l((p∨q) +2)   j ]−1  ∑ M 2 |ϕ j ∗ f |[w   j=0  p n L (R )   1/q   ∞    jsq q j −q   2ln((p∨q) +2)  ]  ∑ 2 |ϕ j ∗ f | [w   j=0  p n L (R )

s,τ f FHp,q (ϕ ,Rn ) .

Therefore, choosing M > n((p ∨ q)  + 2) + ρ , we obtain ∞

−lM+l ρ s,τ f FHp,q Jl (ψ ,Rn )  J0 + ∑ 2 l=1



∞



s,τ ∑ 2−lM+lρ 2ln((p∨q) +2) f FHp,q (ϕ ,Rn )

l=0

s,τ  f FHp,q (ϕ ,Rn ) ,

 

which completes the proof of Proposition 7.2. Lemma 7.9. Let s ∈ R, p ∈ (1, ∞), q ∈ [1, ∞) and τ ∈ [0,

1 (p∨q) ].

Then

s, τ n  n S (Rn ) ⊂ AH p, q (R ) ⊂ S (R ) s, τ n and S (Rn ) is dense in AH p, q (R ). s,0 Proof. By similarity and AH p,q (Rn ) = Asp,q (Rn ), we only give the proof of the space s, τ FH p, q (Rn ) in the case when τ > 0. s, τ n  n We first show that FH p, q (R ) ⊂ S (R ), namely, there exists an M ∈ N such s, τ that for all f ∈ FH p, q (Rn ) and φ ∈ S (Rn ),

| f , φ |  f FHp,s, τq (Rn ) φ SM+1 .

206

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

 be a nonnegative Borel measurable function Let Φ , ϕ , Ψ and ψ be as in (2.6). Let ω satisfying (7.3) and (7.12). Then by Lemma 2.3, Lemma 2.4 and Remark 7.1(iii), we obtain | f , φ | ≤

∞

∑



n j=0 R

|ϕ j ∗ φ (x)||ψ j ∗ f (x)| dx

 φ SM+1  φ SM+1

∞

∑ 2− jM+ jnτ

j=0 ∞

 Rn

1  j (x)]−1 dx |ψ j ∗ f (x)|[ω (1 + |x|)n+M

∞

∑ 2− jM+ jnτ ∑ 2−l(n+M)

j=0



 j (x)]−1 dx. |ψ j ∗ f (x)|[ω

|x|<2l

l=0

Choosing M > nτ − s, by H¨older’s inequality, we have | f , φ |  φ SM+1 ×

% |x|<2l

∞

∞

j=0

l=0

∑ 2− jM+ jnτ − js ∑ 2−l(n+M)2ln(1−1/p) 2

jsp

−p

|ψ j ∗ f (x)| [ω (x)] p

 f FHp,s, τq (Rn ) φ SM+1

j

&1/p dx

∞

∞

j=0

l=0

∑ 2− jM+ jnτ − js ∑ 2−l(n+M)2ln(1−1/p)

 f FHp,s, τq (Rn ) φ SM+1 . s, τ n  n Thus, FH p, q (R ) ⊂ S (R ). s, τ n n Next we prove that S (Rn ) ⊂ FH p, q (R ). For all φ ∈ S (R ), by Lemma 2.4, we have  1/q   ∞  ksq −kMq   2 2 k −q  [ ω ] φ FHp,s, τq (Rn )  φ SM+1 inf  ∑   nq+Mq ω   k=0 (1 + | · |) L p (Rn )

≡ φ SM+1 J, where M ∈ N will be determined later. Let ε be a positive real number such that nτ + ε > nτ (p ∨ q) . For all l ∈ Z+ , set 

ωl (x, t) ≡ κ 2

 −lnτ (p∨q)

min 1,



2l

 |x|2 + t 2

nτ +ε 1/(p∨q) ,

where the positive constant κ is same as in the proof of Lemma 7.1. Then, by an argument similar to that used in the proof of Lemma 7.1, we see that ωl satisfies (7.3), and hence

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

J = inf ω

⎧ ⎨ ⎩

+∑

2ksq 2−kMq ∑ (1 + |x|)nq+Mq [ω k (x)]−q k=0

|x|<1

 2l−1 ≤|x|<2l

l=1

ω

∞



∞

 inf



⎧ ⎨ ⎩



+ ∑ 2l ρ inf ω

l=1

2

 p/q dx

2

∑ (1 + |x|)nq+Mq [ω k (x)]−q

dx

k=0

⎧ ⎨

 2l−1 ≤|x|<2l

⎭

⎫1/p ⎬

 p/q

2ksq 2−kMq

dx

k=0

⎩

⎫1/p ⎬

 p/q

ksq −kMq

∑ (1 + |x|)nq+Mq [ω k (x)]−q

|x|<1

∞

∞

∞

207

⎭

∞

2ksq 2−kMq ∑ (1 + |x|)nq+Mq [ω k (x)]−q k=0

⎫1/p ⎬

 p/q dx

⎭

.

Notice that ωl ∼ 2−lnτ in T (B(0, 2l )). If we choose M > max {s, nτ + ρ − n(1 − 1/p)}, then J 

⎧ ⎨ ⎩

|x|<1 ∞



∞

2ksq 2−kMq ∑ nq+Mq k=0 (1 + |x|)

+ ∑ 2lnτ +l ρ l=1



⎧ ⎨ ⎩



2l−1 ≤|x|<2l

∑ 2lnτ +lρ 2−l(n+M) ⎩

l=0

dx

⎭

∞

2ksq 2−kMq ∑ (1 + |x|)nq+Mq k=0

⎧ ⎨

∞

⎫1/p ⎬

 p/q



|x|<2l

∞

∑ 2ksq 2−kMq

k=0

⎫1/p ⎬

 p/q dx

⎭

⎫1/p ⎬

 p/q dx

⎭

< ∞. s, τ n We then obtain φ FHp,s, τq (Rn )  φ SM+1 , and hence S (Rn ) ⊂ FH p, q (R ).

s, τ s, τ n n To show S (Rn ) is dense in FH p, q (R ), let f ∈ FH p, q (R ), Φ , ϕ , Ψ and ψ be as in (2.6). Then by Lemma 2.3,

f=

∞

∑ ψ j ∗ ϕ j ∗ f

j=0

in S  (Rn ). For all m, s ∈ N, set fm, s ≡

∑

0≤ j≤m

 j ∗ [(ϕ j ∗ f )γs ], ψ

208

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

where γs ∈ C∞ (Rn ) satisfying 0 ≤ γs ≤ 1, supp γs ⊂ B(0, 2s) and γs ≡ 1 on B(0, s). Then, by [134, p. 23, Theorem 3.13], fm, s ∈ S (Rn ) and fm, s → f in S  (Rn ) as s, m → ∞. s, τ n  satisfies (7.3) and (7.12). To prove fm, s → f in FH p, q (R ) as m, s → ∞, assume ω Then, using Lemma 2.4 and the vector-valued inequality of Fefferman and Stein, similarly to the proof of Proposition 7.2, we see that      j ∗ ϕ j ∗ f   f − ∑ ψ  s, τ n  0≤ j≤m FHp, q (R )    ∞  = inf  ∑ 2ksq ∑ χ{ j∈Z: k−1≤ j≤k+1} ( j) ω  j>m k=0 ×

q

 Rn

 j (· − y)||ϕ j ∗ f (y)| dy |ϕk ∗ ψ

 1q    k −q  [ω (·)]  

L p (Rn )

 1   q   ( ) q js j −1    ] )   ∑ M(2 |ϕ j ∗ f |[ω   p n  j>m L (R )   1   q   jsq q  j −1   ]   ∑ 2 |ϕ j ∗ f | [ω  p n  j>m L (R )

→0 as m → 0. Thus, for any ε > 0, there exists an mε ∈ N such that for all m ≥ mε ,      j ∗ ϕ j ∗ f  f − ∑ ψ    0≤ j≤m

s, τ

< ε /2ρ +1 .

FHp, q (Rn )

Fix m ∈ N. Repeating the argument in the proof of Proposition 7.2 again, by the Lebesgue dominated convergence theorem and lims→∞ γs (x) = 1 for all x ∈ Rn , we see that if s → ∞, then       j ∗ [(ϕ j ∗ f )γs ] − ∑ ψ j ∗ ϕ j ∗ f    ∑ ψ  s, τ n 0≤ j≤m 0≤ j≤m FHp, q (R )  1   q   jsq q q  j −q   2 | ϕ ∗ f | | γ − 1| [ ω ] j s  ∑   0≤ j≤m  p n L (R )

→ 0.

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

209

Thus, there exists an sm, ε ∈ N such that for all s ≥ sm, ε ,       j ∗ [(ϕ j ∗ f )γs ] − ∑ ψ j ∗ ϕ j ∗ f   ∑ ψ  0≤ j≤m  0≤ j≤m

s, τ FHp, q (Rn )

< ε /2ρ +1.

Thus, for any ε > 0, choosing m ≥ mε and s ≥ sm, ε , we have f − fm, s FHp,s, τq (Rn )     ρ j ∗ ϕ j ∗ f  ≤ 2 f − ∑ ψ    s, τ n 0≤ j≤m FHp, q (R )       j ∗ [(ϕ j ∗ f )γs ] − ∑ ψ j ∗ ϕ j ∗ f  +2ρ  ∑ ψ  0≤ j≤m  0≤ j≤m

s, τ

FHp, q (Rn )

< ε,

s, τ n which implies S (Rn ) is dense in FH p, q (R ), and hence completes the proof of Lemma 7.9.   s, τ s, τ n n To establish the dual relation between AH p, q (R ) and A p, q (R ), we need the following result. 1 Lemma 7.10. Let s ∈ R, p ∈ (1, ∞), q ∈ [1, ∞) and τ ∈ [0, (p∨q)  ], Φ and ϕ satisfy, respectively, (2.1) and (2.2). Then πϕ is a bounded linear operator from s,τ s, τ n ATp,q (Rn+1 Z+ ) to AH p, q (R ).

Proof. Let ω be a nonnegative Borel measurable function satisfy (7.3) and  1/q   ∞     ∑ 2 jsq |F j |q [ω j ]−q     j=0 

s,τ ≤ 2 F FTp,q (Rn+1 ) . Z+

L p (Rn )

By Lemma 2.4 and Remark 7.1(iii), similarly to the proof of Lemma 7.7, we obtain s,τ that (7.8) holds in S  (Rn ) for all F ∈ ATp,q (Rn+1 Z+ ). Repeating the argument in the proof of Proposition 7.2 with ϕ j ∗ f replaced by s,τ (Rn+1 F j , we obtain that for all F ∈ ATp,q Z+ ), s,τ πϕ (F) AHp,s, τq (Rn )  F ATp,q (Rn+1 ) , Z+

s,τ s, τ n which implies that πϕ is a bounded linear operator from ATp,q (Rn+1 Z+ ) to AH p, q (R ), and hence completes the proof of Lemma 7.10.  

We have the following dual theorem. The proof is similar to that of its homogeneous counterpart in [164, 165].

210

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

s, τ n Theorem 7.3. Let p, q, s, τ be as in Definition 7.5. The dual space of AH p, q (R ) is −s, τ −s, τ A p , q (Rn ) in the following sense: if g ∈ A p , q (Rn ), then the linear functional

L( f ) =

 Rn

f (x)g(x) dx,

(7.13)

s, τ n defined initially for all f ∈ S (Rn ), has a bounded extension to AH p, q (R ). s, τ n Conversely, if L is a bounded linear functional on AH p, q (R ), then there exists τ n g ∈ A−s, p , q (R ) so that g A−s, τ (Rn )  L p , q

and L can be written in the form (7.13) for all f ∈ S (Rn ). s, τ n Proof. We only consider FH p, q (R ). Let Φ , ϕ , Ψ and ψ be as in (2.6). For all −s, τ f ∈ S (Rn ) and g ∈ Fp , q (Rn ), applying the Calder´on reproducing formula in Lemmas 2.3, 7.7 and 7.10, and Theorem 7.1, we obtain 2 3   ∞  k ∗ ϕk ∗ g  |Lg ( f )| = | f , g| ≡  f , ∑ ψ   k=0     ∞   (ψk ∗ f )(x)(ϕk ∗ g)(x) dx = ∑   Rn k=0 s,τ  ρψ ( f ) FTp,q (Rn+1 ) ρϕ (g) FW −s,τ (Rn+1 ) p ,q

Z+

Z+

∼ f FHp,s, τq (Rn ) g F −s,τ (Rn ) . p,q



τp Thus each g ∈ Fp−s, (Rn ) induces a bounded linear functional Lg on the space  ,q n S (R ) with

Lg  g F −s,τ (Rn ) . p,q

s, τ n By Lemma 7.9, S (Rn ) is dense in FH p, q (R ), so that Lg can be extended to a s, τ n bounded linear functional on FH p, q (R ) with

Lg  g F −s,τ (Rn ) . p,q

s, τ n Conversely, let L be a bounded linear functional on FH p, L ≡ L ◦ πψ . q (R ). Set  s,τ (Rn+1 Obviously,  L is linear. By Lemma 7.10, we know that for all F ∈ FTp,q Z+ ), then s, τ n πψ (F) ∈ FH p, q (R ) and

| L(F)| = |L(πψ (F))| ≤ L πψ (F) FHp,s, τq (Rn )  L F FTp,q s,τ (Rn+1 ) . Z+

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

211

s,τ  Thus,  L becomes a bounded linear functional on FTp,q (Rn+1 Z+ ) with L  L . By −s,τ n+1 Theorem 7.1, there exists a function G ∈ FWp ,q (RZ+ ) with

G FW −s,τ (Rn+1 )  L p ,q

Z+

s,τ such that for all F ∈ FTp,q (Rn+1 Z+ ),

 L(F) =



∞

∑ F k (x)Gk (x) dx. Rn k=0

Notice that by Lemma 2.3, the composite operator πψ ρϕ is the identity on S (Rn ). For all f ∈ S (Rn ), we have L( f ) = L(πψ (ρϕ ( f )))  ρϕ ( f )) = L( = = =



∞

∑ ϕk ∗ f (y)Gk (y) dy Rn



Rn



Rn

k=0

∞

f (x) ∑ (ϕk ∗ Gk )(x) dx k=0

f (x)g(x) dx,

τ n where g ≡ πϕ (G). By Lemma 7.7, g ∈ Fp−s,  ,q (R ) and

g F −s,τ (Rn )  G FW −s,τ (Rn+1 )  L , p,q

p ,q

Z+

which completes the proof of Theorem 7.3.

 

Remark 7.4. When τ = 0, then Theorem 7.3 is the classical dual result of inhomogeneous Besov spaces and Triebel-Lizorkin spaces; see [145, Theorem 2.11.2]. s, τ n It turns out that the spaces AH p, q (R ) satisfy many properties similar to those of We begin with their ϕ -transform characterizations. We define the corres, τ n sponding sequence spaces to AH p, q (R ) as follows. Recall that Q denotes the set of n all dyadic cubes in R and

As,p,τq (Rn ).

Q j ≡ Q ∈ Q : l(Q) = 2− j for j ∈ Z. Definition 7.6. Let p ∈ (1, ∞) and s ∈ R. s, τ 1 n (i) If q ∈ [1, ∞) and τ ∈ [0, (p∨q)  ], the sequence space bH p, q (R ) is then defined to be the set of all t ≡ {tQ }l(Q)≤1 ⊂ C such that

212

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

⎧ ⎨

 q   jsq  − j −1 Q [ω (·, 2 )]  t bHp,s, τq(Rn ) ≡ inf ∑ 2  ∑ |tQ |χ  ω ⎩ Q∈Q  j=0 ∞

L p (Rn )

j

⎫1 ⎬q ⎭

< ∞,

where the infimum is taken over all nonnegative Borel measurable functions ω on Rn+1 + such that ω satisfies (7.3) and with the restriction that for any j ∈ Z+ , ω (·, 2− j ) is allowed to vanish only where ∑Q∈Q j |tQ |χQ vanishes. s, τ 1 n (ii) If q ∈ (1, ∞) and τ ∈ [0, (p∨q)  ], the sequence space f H p, q (R ) is then defined to be the set of all t ≡ {tQ }l(Q)≤1 ⊂ C such that   q  1q   ∞    jsq − j −1   t f Hp,s, τq (Rn ) ≡ inf  ∑ 2 ∑ |tQ |χQ [ω (·, 2 )]  ω   j=0 Q∈Q j

< ∞,

L p (Rn )

where the infimum is taken over all nonnegative Borel measurable functions ω on Rn+1 + with the same restrictions as in (i). s, τ s, τ s, τ n n n We also use aH p, q (R ) to denote either bH p, q (R ) or f H p, q (R ). Similarly to the proof of (7.4), we see that · aHp,s, τq (Rn ) is a quasi-norm, namely, there exists a s, τ n nonnegative constant ρ ∈ [0, 1] such that for all t1 , t2 ∈ aH p, q (R ),

t1 + t2 aHp,s, τq (Rn ) ≤ 2ρ ( t1 aHp,s, τq (Rn ) + t2 aHp,s, τq (Rn ) ).

(7.14)

s, τ s, τ n n ˙ p, The homogeneous counterpart of aH p, q (R ), denoted by aH q (R ), was already s, τ n introduced in [168]. Similarly to a p, q(R ), we define s, τ n ˙ s, τ n V : aH p, q (R ) → aH p, q (R )

by setting (V t)Q ≡ tQ if l(Q) ≤ 1, and (Vt)Q ≡ 0 otherwise. Then V is an isometric s, τ s, τ n n ˙ p, embedding of aH p, q (R ) in aH q (R ). Define s, τ n s, τ n W : aH˙ p, q (R ) → aH p, q (R )

by setting (W t)Q ≡ tQ if l(Q) ≤ 1. Then W is continuous and W ◦ V is the identity s, τ n on aH p, q (R ). s, τ n Next we show that the inverse ϕ -transform is well defined on aH p, q (R ). 1 Lemma 7.11. Let p ∈ (1, ∞), q ∈ [1, ∞), s ∈ R and τ ∈ [0, (p∨q)  ]. Then for all t ∈

s, τ n aH p, q (R ),

Tψ t ≡

∑

tQ ψQ

l(Q)≤1 s, τ n  n converges in S  (Rn ); moreover, Tψ : aH p, q (R ) → S (R ) is continuous.

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

213

s, τ n Proof. By similarity, we only consider the space bH p, q (R ). Let s, τ n t ≡ {tQ }l(Q)≤1 ∈ bH p, q (R ).

We need to show that there exists an M ∈ Z+ such that for all φ ∈ S (Rn ),

∑

|tQ ||ψQ , φ |  φ SM .

l(Q)≤1

Choose a Borel function ω on Rn+1 + satisfying (7.3) as well as ⎧ ⎨

⎫1 ⎬q

 q    jsq  − j −1  2 |t | χ [ ω (·, 2 )]   Q Q ∑ ∑  ⎩ j=0 Q∈Q ∞

L p (Rn )

j

⎭

≤ 2 t bHp,s, τq(Rn ) .

(7.15)

−nτ . Then for all Q ∈ Q , by By Remark 7.1(iii), for all (x, s) ∈ Rn+1 j + , ω (x, s)  s H¨older’s inequality and (7.15), we have

|tQ | ≤ |Q|−τ − p |tQ | 1

% Q

[ω (x, 2− j )]−p dx

& 1p

 |Q| n + 2 −τ − p t bHp,s, τq(Rn ) . s

1

1

(7.16)

Recall that as a special case of [24, Lemma 2.11], there exists a positive constant L0 such that for all j ∈ Z+ ,

∑

(1 + |xQ|n )−L0  2n j .

(7.17)

Q∈Q j

Furthermore, since φ ∈ S (Rn ) and ψ satisfies either (2.1) or (2.2), a standard computation gives us that if L > max{1/p + 1/2 − s/n − τ , 1/p + 3/2 + s/n + τ , L0}, then there exists an M ∈ Z+ such that for all Q ∈ Q j , |ψQ , φ |  φ SM (1 + |xQ|n )−L 2− jnL ;

(7.18)

see also [165, p. 10] and [24, (3.18)]. Using (7.16), (7.18) and (7.17), we conclude that

∑

l(Q)≤1

|tQ ||ψQ , φ |  t bHp,s, τq(Rn ) φ SM ×

∑

|Q| n + 2 −τ − p (1 + |xQ|n )−L 2−nL jQ s

1

1

l(Q)≤1

 t bHp,s, τq(Rn ) φ SM , which completes the proof of Lemma 7.11.

 

214

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

Now we present our main result of this section, which also implies that the spaces s, τ n AH p, q (R ) are independent of the choices of ϕ and Φ . 1 Theorem 7.4. Let p ∈ (1, ∞), q ∈ [1, ∞), s ∈ R, τ ∈ [0, (p∨q)  ], ϕ and ψ be as in Definition 2.1. Then s, τ n s, τ n Sϕ : AH p, q (R ) → aH p, q (R )

and

s, τ n s, τ n Tψ : aH p, q (R ) → AH p, q (R )

s, τ n are bounded; moreover, Tψ ◦ Sϕ is the identity on AH p, q (R ).

s, τ n Theorem 7.4 is the ϕ -transform characterization of AH p, q (R ). We remark that the homogeneous counterpart of Theorem 7.4 was already obtained in [168, Theorem 2.1]. To prove Theorem 7.4, we need to establish the counterparts of s, τ n Lemmas 2.6, 2.8 and 2.9 on AH p, q (R ). Let γ be a fixed integer. Replacing ϕ j by ϕ j−γ (ϕ0 by Φ−γ ) in Definition 7.5, we s, τ n ∗ obtain a quasi-norm in AH p, q (R ), which is denoted by f AH s, τ (Rn ) . As a counterp, q part of Lemma 2.6, we have the following conclusion. Its proof is similar to those of Lemma 2.6 and [64, Lemma 12.1].

Lemma 7.12. Let s, p, q, τ be as in Theorem 7.4. The quasi-norms f ∗AH s, τ (Rn ) p, q

and f AHp,s, τq (Rn ) are equivalent.

s, τ n Proof. By similarity, we only consider FH p, q (R ) and the case when γ > 0. First we prove f FHp,s, τq (Rn )  f ∗FH s, τ (Rn ) . p, q

It suffices to prove that I ≡ inf ω

 Rn

−p

|Φ ∗ f (x)| [ω (x, 1)] p

1/p dx

 f ∗FH s, τ (Rn ) . p, q

To this end, let ω satisfy (7.3) and ⎧  q/p ⎫1/p ⎨ ⎬ ∞ 2 jsq |ϕ j−γ ∗ f (x)|q [ω (x, 2− j )]−q dx  f ∗FH s, τ (Rn ) . ∑ p, q ⎩ Rn j=0 ⎭ For each m ∈ N and i ∈ {−γ , −γ + 1, · · · , 1}, define 

ωm,i (x,t) ≡ 2−mn((p∨q) +2) sup{ω (y, s) : |x − y| ≤ 2m , s = 2it}. Similarly to the proof of Proposition 7.2, we see that there exists a positive constant C(γ ) such that C(γ ) ωm,i satisfies (7.3). Moreover, [ωm,i (x, 1)]−1 ω (x − y, 2−i )  1 for all |y| ≤ 2m .

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

215

Similarly to the proof of Lemma 2.6, we choose ηi ∈ S (Rn ), i = −γ , · · · , 1, such that

Φ ∗ f ≡ η−γ ∗ Φ−γ ∗ f +

1

∑

i=−γ +1

ηi ∗ ϕi ∗ f .

On the other hand, let χ0 be the characteristic function on unit ball B(0, 1) and χ0 be the characteristic function on B(0, 2m ) \ B(0, 2m−1 ) for all m ∈ N. Then by Minkowski’s inequality and the fact that ηi ∈ S (Rn ), choosing M > ρ + n((p ∨ q) + 2), we have I   



1

∑

inf

1

∞

i=−γ

ω

∑ ∑

Rn

2mρ

|ηi ∗ ϕi ∗ f (x)| p [ω (x, 1)]−p dx

 Rn

i=−γ m=0

|(ηi χm ) ∗ ϕi ∗ f (x)| p [ωm,i (x, 1)]−p dx

∞

1

1/p 1/p



∑ ∑ 2mρ 2mn((p∨q) +2)

i=−γ m=0

×

 Rn

|ηi (y)|χm (y)

 f ∗FH s, τ (Rn ) p, q

 Rn ∞

1

|ϕi ∗ f (x − y)| p[ω (x − y, 2−i )]−p dx 

∑ ∑ 2mρ 2mn((p∨q) +2)



i=−γ m=0 ∗  f FH s, τ (Rn ) , p, q

which implies that

Rn

1/p dy

χm (y) dy (1 + |y|)n+M

f FHp,s, τq (Rn )  f ∗FH s, τ (Rn ) . p, q

The converse inequality follows from a similar argument. This finishes the proof of Lemma 7.12.   Lemma 7.13. Let s, p, q, τ be as in Theorem 7.4 and λ ∈ (n, ∞) be sufficiently s, τ n large. Then there exists a positive constant C such that for all t ∈ aH p, q (R ), ∗ s, τ t aHp,s, τq(Rn ) ≤ t p∧q, n ≤ C t aH s, τ (Rn ) , λ aHp, q (R ) p, q ∗ where t p∧q, λ is as in (2.17).

Proof. The inequality ∗ s, τ t aHp,s, τq(Rn ) ≤ t p∧q, n λ aHp, q (R )

216

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

being trivial, we only need to concentrate on ∗ s, τ t p∧q, n  t aH s, τ (Rn ) . λ aHp, q (R ) p, q s, τ n Also, by similarity, we only consider the space bH p, q (R ). Let s, τ n t ≡ {tQ }Q∈Q ∈ bH p, q (R ).

We choose a Borel function ω as in the proof of Lemma 7.11. For all cubes Q ∈ Q j and m ∈ N, we set A0 (Q) ≡ {P ∈ Q j : 2 j |xP − xQ | ≤ 1} and Am (Q) ≡ {P ∈ Q j : 2m−1 < 2 j |xP − xQ | ≤ 2m }. The triangle inequality that |x − y| ≤ |x − xQ | + |xQ − xP | + |xP − y| gives us that

√ |x − y| ≤ 3 n2m− j

provided x ∈ Q, y ∈ P and P ∈ Am (Q). For all m ∈ Z+ and (x, s) ∈ Rn+1 + , we set 

ωm (x, s) ≡ 2−mn((p∨q) +2) sup{ω (y, s) : y ∈ Rn , |y − x| <

√ m+2 n2 s}.

By the argument in the proof of Lemma 7.1, we know that ωm still satisfies (7.3) modulo multiplicative constants independent of m. Also it follows from the definition of ωm that for all x ∈ Q with Q ∈ Q j and y ∈ P with P ∈ Am (Q), 

ω (y, 2− j )  2mn((p∨q) +2) ωm (x, 2− j ). Then for all r ∈ (0, ∞) and a ∈ (0, r), using this estimate and the monotonicity of a/r , we obtain that for all x ∈ Q, |tP |r ∑ (1 + 2 j |xQ − xP|)λ [ωm (x, 2− j )]−r P∈Am (Q)  r/a  2−mλ

∑

|tP |a [ωm (x, 2− j )]−a

P∈Am (Q)

2

−mλ + jnr/a



r/a

∑

Rn P∈A (Q) m

− j −a

|tP | χP (y)[ωm (x, 2 )] a

dy

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

r/a



2

−mλ + jnr/a+mnr((p∨q)+2)

2

−mλ +mnr/a+mnr((p∨q) +2)

∑

− j −a

|tP | χP (y)[ω (y, 2 )] a

Rn P∈A (Q) m



217



∑

M

dy

 − j −a

|tP | χP [ω (·, 2 )] a

r/a (x)

.

P∈Am (Q)

Recall that M denotes the Hardy-Littlewood maximal operator on Rn . ∗,m For all m ∈ Z+ , set tr,∗,m λ ≡ {(tr,λ )Q }Q∈Q with  (tr,∗,m λ )Q

|tP |r ∑ (1 + l(P)−1|xP − xQ |)λ P∈Am (Q)

≡

 1r .

In what follows, choose a ∈ (0, p ∧ q) and

λ > n(p ∧ q)/a + n(p ∧ q){(p ∨ q) + 2} + (p ∧ q)ρ , where ρ is a nonnegative constant as in (7.14). By (7.14), the previous pointwise p estimate and the L a (Rn )-boundedness of M, we obtain ∗ s, τ t p∧q, n λ bHp, q (R )



∞

∗,m s, τ ∑ 2ρ m t p∧q, λ bHp, q (Rn )

m=0



∞

∑2

ρm

m=0

⎧ ⎨

⎡

∑ ⎩ j=0





∞

2

jsq ⎣

∑

Rn Q∈Q

j

|tP | p∧q ∑ (1 + l(P)−1|xP − xQ|)λ P∈Am (Q)



p p∧q

⎤ qp ⎫ 1q ⎪ ⎬ χQ (x) p ⎦ × dx ⎪ [ωm (x, 2− j )] p ⎭ 

∞



∑ 2− p∧q {−λ +n(p∧q)/a+n(p∧q){(p∨q) +2}+(p∧q)ρ } m

m=0

⎡

∞

⎢ × ⎣ ∑ 2 jsq j=0

⎧ ⎨ ⎩

 Rn

 M

P )a (|tP |χ −j a P∈Q j [ω (·, 2 )]

∑



 ap (x)

⎫ q ⎤ 1q ⎬p ⎥ dx ⎦ ⎭

 t bHp,s, τq(Rn ) , which completes the proof of Lemma 7.13.

 

With Lemma 2.6 replaced by Lemma 7.12, similarly to the proof of Lemma 2.9, we have the following conclusion.

218

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

Lemma 7.14. Let s, p, q, τ be as in Theorem 7.4 and γ ∈ Z+ be sufficiently large. s, τ n Then there exists a constant C ∈ [1, ∞) such that for all f ∈ AH p, q (R ), C−1 infγ ( f ) aHp,s, τq (Rn ) ≤ f AHp,s, τq (Rn ) ≤ sup( f ) aHp,s, τq (Rn ) ≤ C infγ ( f ) aHp,s, τq (Rn ) . With Lemmas 7.12, 7.13 and 7.14, the proof of Theorem 7.4 is similar to that of Theorem 2.1. We omit the details. Applying Theorem 7.4, we have the following Sobolev-type embedding props, τ n erties of AH p, q (R ), whose homogeneous counterparts were already established in [168, Proposition 2.2]. Proposition 7.3. Let 1 < p0 < p1 < ∞ and −∞ < s1 < s0 < ∞. Assume in addition that s0 − n/p0 = s1 − n/p1. (i) If q ∈ [1, ∞) and

  τ ∈ 0, min

1 1 , (p0 ∨ q) (p1 ∨ q)



such that τ (p0 ∨ q) = τ (p1 ∨ q) , then BH ps00,, τq (Rn ) → BH ps11,, τq (Rn ). (ii) If q, r ∈ (1, ∞) and

  τ ∈ 0, min

1 1 , (p0 ∨ r) (p1 ∨ q)



such that τ (p0 ∨ r) ≤ τ (p1 ∨ q) , then FH ps00,, τr (Rn ) → FH ps11,, τq (Rn ). Proof. By Theorem 7.4 and similarity, it suffices to consider the corresponding ses, τ n quence spaces f H p, q (R ), that is, to prove that t f H s1 , τ (Rn )  t f H s0 , τ (Rn ) p1 , q

p0 , r

f H ps00,, τr (Rn ).

for all t ∈ It was already proved in [168, Proposition 2.2] that t f H˙ s1 , τ (Rn )  t f H˙ s0 , τ (Rn ) p1 , q

p0 , r

s ,τ s, τ n for all t ∈ f H˙ p00 , r (Rn ). Recall that V is an isometric embedding of aH p, q (R ) in s , τ s, τ aH˙ p, q(Rn ). Then for all t ∈ f H p00 , r (Rn ), we have

t f H s1 , τ (Rn ) ∼ Vt f H˙ s1 , τ (Rn )  Vt f H˙ s0 , τ (Rn ) ∼ t f H s0 , τ (Rn ) , p1 , q

p1 , q

which completes the proof of Proposition 7.3.

p0 , r

p0 , r

 

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

219

Remark 7.5. When τ = 0, Proposition 7.3 recovers the corresponding results on Besov spaces and Triebel-Lizorkin spaces in [145, p. 129]. We also remark that the restriction that τ (p0 ∨q) = τ (p1 ∨q) in Proposition 7.3(i) is necessary, and sharp in this sense; see [168, Proposition 2.3]. However, it is still unclear that if the restriction that τ (p0 ∨ r) ≤ τ (p1 ∨ q) in Proposition 7.3(ii) is sharp. Corresponding to Theorem 7.3, we also have the dual result for sequence spaces. The homogeneous counterpart of the following conclusion was obtained in [168, Proposition 2.1]. Proposition 7.4. Let s, p, q, τ be as in Theorem 7.4. Then −s,τ s, τ n ∗ n (aH p, q (R )) = a p ,q (R )

τ n in the following sense: if t ≡ {tQ }l(Q)≤1 ∈ a−s, p ,q (R ), then the map

λ = {λQ }l(Q)≤1 → λ ,t ≡

∑

λQtQ

l(Q)≤1 s, τ n defines a continuous linear functional on aH p, q (R ) with operator norm no more s, τ n ∗ than a constant multiple of t a−s,τ (Rn ) . Conversely, every L ∈ (aH p, q (R )) is of

this form for a certain t ∈

p ,q

τ n a−s, p ,q (R )

and t a−s,τ (Rn ) is no more than a constant p ,q

multiple of the operator norm L .

s, τ s, τ n n Proof. We only consider the spaces bH p, q (R ) because the assertion for f H p, q (R ) can be proved similarly. τ s, τ n n For t ≡ {tQ }l(Q)≤1 ∈ b−s, p ,q (R ) and λ ≡ {λQ }l(Q)≤1 ∈ bH p, q (R ), let F and G be n functions on Rn+1 Z+ defined by setting, for all x ∈ R and j ∈ Z+ ,

F(x, 2− j ) ≡

∑

Q |λQ |χ

∑

P . |tP |χ

Q∈Q j

and G(x, 2− j ) ≡

P∈Q j

Since s, τ F BTp,q s,τ (Rn+1 ) ∼ λ bHp, q (Rn ) Z+

and G BW −s,τ (Rn+1 ) ∼ t b−s,τ (Rn ) , p ,q

Z+

p ,q

220

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

by Theorem 7.1, we have        ∑ λQ t Q  ≤ l(Q)≤1  =

∑



∑ ∑

n j∈Z+ R Q∈Q j P∈Q j

∑

j∈Z+



Rn

Q (x)|tP |χ P (x) dx |λQ |χ

F(x, 2− j )G(x, 2− j ) dx

s,τ  F BTp,q (Rn+1 ) G BW −s,τ (Rn+1 ) p ,q

Z+

Z+

∼ λ bHp,s, τq (Rn ) t b−s,τ (Rn ) , p ,q

τ s, τ n n ∗ which implies that b−s, p ,q (R ) → (bH p, q (R )) . Conversely, since the subspace consisting of all sequences with finite nons, τ s, τ n n ∗ vanishing elements are dense in bH p, q (R ), we know that every L ∈ (bH p, q (R )) is of the form λ → ∑l(Q)≤1 λQt Q for a certain t ≡ {tQ }l(Q)≤1 ⊂ C. It remains to show that t b−s,τ (Rn )  L (bHp,s, τq (Rn ))∗ . p ,q

Fix P ∈ Q and a ∈ R. For j ≥ ( jP ∨ 0), let X j be the set of all Q ∈ Q j satisfying Q ⊂ P and let μ be a measure on X j such that the μ -measure of the “point” Q is |Q|/|P|τ a . Also, let qP denote the set of all {a j } j≥( jP ∨0) ⊂ C with  {a j } j≥( jP ∨0) q ≡ P

∞

∑

1/q |a j |

q

j=( jP ∨0)

and P ( p (X j , d μ )) denote the set of all {aQ, j }Q∈Q j , Q⊂P, j≥( jP ∨0) ⊂ C with q

{aQ, j }Q∈Q j (Rn ), Q⊂P, j≥( jP ∨0) q ( p (X j ,d μ )) P ⎛   q ⎞1/q ∞

∑

≡⎝

∑

p

|aQ, j | p |Q||P|−τ a

⎠

.

j=( jP ∨0) Q∈Q j , Q⊂P q



It is well known that the dual space of P ( p (X j , d μ )) is P ( p (X j , d μ )); see, for example, [145, p. 177]. Via this observation and the already proved conclusion of this proposition, we see that q

⎧ ⎫1  q ⎪ q  ⎪ ⎨  p ⎬

∞ p s 1 1 ∑ ∑ |Q|− n − 2 |tQ | |Q| ⎪ |P|τ ⎪ ⎩ j=( j ∨0) Q∈Q , Q⊂P ⎭ j

P

s

1

= {|Q|− n − 2 |tQ |}Q∈Q j , Q⊂P, j≥( jP ∨0)

q



P ( p (X j ,d μ ))

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

=

sup

{λQ }Q∈Q , Q⊂P, j≥( j ∨0) q ( p (X ,d μ )) j P j P

≤

sup

221

   ∞ |Q|   − ns − 21 |tQ | τ p   ∑ ∑ λQ |Q| |P|  ≤1  j=( j ∨0) Q∈Q , Q⊂P

{λQ }Q∈Q , Q⊂P, j≥( j ∨0) q ( p (X ,d μ )) ≤1 j P j

j

P

L (bHp,s, τq (Rn ))∗

  s 1    × {λQ |Q|− n − 2 |Q|/|P|τ p }Q∈Q j , Q⊂P, j≥( jP ∨0)  P

s, τ n bHp, q (R )

.

To finish the proof of this proposition, it suffices to show that   s 1    {λQ |Q|− n − 2 |Q|/|P|τ p }Q∈Q j , Q⊂P, j≥( jP ∨0) 

s, τ

bHp, q (Rn )

1

for all sequences λ satisfying {λQ }Q∈Q j , Q⊂P, j≥( jP ∨0) q ( p (X j ,d μ )) ≤ 1. P

√

In fact, let B ≡ B(cP , nl(P)) ⊂ Rn and ω be as in the proof of Lemma 7.7 associated with B, then ω satisfies (7.3) and for all x ∈ P and j ≥ ( jP ∨ 0), [ω (x, 2− j )]−1 ∼ [l(P)]nτ . We then obtain that   s 1    {λQ |Q|− n − 2 |Q|/|P|τ p }Q∈Q j , Q⊂P, j≥( jP ∨0)  

⎧ ⎨

s, τ n bHp, q (R )

∞



⎩ j=(∑ j ∨0)

2 jsq

⎧ ⎪ ⎨

⎡

P

 qp ⎫ 1q % &p  ⎬ s |Q| [ω (x, 2− j )]−p dx |λQ ||Q|− n −1 τ p ∑ ⎭ |P| Q Q∈Q j , Q⊂P

⎤ q ⎫ 1q p⎪ ⎬ ∞ p τ p ⎦ ⎣ ∼ | λ | |Q|/|P| Q ∑ ∑ ⎪ ⎪ ⎩ j=( jP ∨0) Q∈Q j (Rn ), Q⊂P ⎭ ∼ {λQ }Q∈Q j , Q⊂P, j≥( jP ∨0) q ( p (X j ,d μ )) P

 1, which completes the proof of Proposition 7.4.

 

Remark 7.6. By Proposition 7.4 and the ϕ -transform characterizations of As,p,τq(Rn ) s, τ s, τ −s, τ n n ∗ n and AH p, q (R ), we also obtain the duality that (AH p, q (R )) = A p, q (R ). This gives another proof of this conclusion, which is different from that of Theorem 7.3. Next we establish the smooth atomic and molecular decomposition characters, τ s, τ n n izations of AH p, q (R ). We first obtain the boundedness on aH p, q (R ) of almost diagonal operators in Definition 3.1.

222

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

1 Theorem 7.5. Let p ∈ (1, ∞), q ∈ [1, ∞), s ∈ R, ε ∈ (0, ∞) and τ ∈ [0, (p∨q)  ]. Then s,τ n all the ε -almost diagonal operators are bounded on aH p,q (R ) if τ > 2nτ .

To prove this theorem, we need some technical lemmas established in [168]. For the reader’s convenient, we give their proofs. Lemma 7.15. Let d ∈ (0, n] and Ω be an open set in Rn such that Ω = ∪∞j=1 B j , where {B j }∞j=1 ≡ {B(X j , R j )}∞j=1 is a countable collection of balls. Define (∞)

Λd (Ω , {B j }∞j=1 )  ≡ inf

∞

∑

rkd

k=1

:Ω⊂



∞ 

B(xk , rk ), B(xk , rk ) ⊃ B j if B j ∩ B(xk , rk ) = 0/ .

k=1

Then there exists a positive constant C, independent of Ω , {B j }∞j=1 and d, such that (∞)

(∞)

(∞)

Λd (Ω ) ≤ Λd (Ω , {B j }∞j=1 ) ≤ C(46)d Λd (Ω ). Proof. The first inequality is trivial. We only need to prove the second one. Without loss of generality, we may assume sup j∈N R j < ∞. By the well-known (5r)-covering lemma (see, for example, [53, Theorem 2.19]), there exists a subset J ∗ of N such that ∞ 

(3B j ) ⊂

j=1



j∈J∗

(15B j )

and χ j∈J∗ χ(3B j ) ≤ 1. Furthermore, by its construction, if B j , j ∈ N, intersects B j for some j ∈ J ∗ , we have that (3B j ) ⊂ (15B j ). Let {B(xk , rk )}k∈N be a collection of balls such that Ω ⊂ ∪∞ k=1 B(xk , rk ) and ∞ d ≤ 2Λ (∞) (Ω ). Set r ∑k=1 k d K1 ≡ {k ∈ N : When B(xk , 45rk ) ∩ B j = 0/ for any j ∈ N, then rk ≥ 135R j } and J1 ≡ { j ∈ N : B j ∩ B(xk , 45rk ) = 0/ for some k ∈ K1 }. Also define J2 ≡ (N \ J1 ) and K2 ≡ (N \ K1 ). We remark that if k ∈ K2 , then there exists j ∈ J2 such that B j ∩ B(xk , 45rk ) = 0/ and 135R j > rk . Notice that  Bj ⊂ Ω ⊂

∞ 

 B(xk , rk ) .

k=1

Thus, for each j ∈ J2 , we have Bj ⊂

 k∈K2 , B(xk ,rk )∩B j =0/

B(xk , rk ),

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

223

d

and then, by d ≤ n and the monotonicity of  n , we see that

∑ rkd

∑ |B(xk , rk )| n d

∼

k∈K2

k∈K2

∑∗



∑

d

j∈J ∩J2 k∈K2 , B j ∩B(xk ,45rk ) =0/

|B(xk , rk )| n

⎛



∑

⎝

∑∗

Rdj ,

j∈J∗ ∩J2



j∈J ∩J2

⎞d

n

∑

|B(xk , rk )|⎠

k∈K2 , B j ∩B(xk ,45rk ) =0/

which further yields that

∑ rkd + ∑∗

k∈K1

Rdj 

j∈J ∩J2

∑ rkd .

k∈K

On the other hand, we have

Ω⊂

∞  j=1

Bj ⊂

 j∈J∗

 (15B j ) =



 j∈J∗ ∩J1

(15B j )





 j∈J∗ ∩J2

 (15B j )

and hence 

Ω⊂







B(xk , 46rk )



 j∈J∗ ∩J2

k∈K1

 (15B j ) .

Notice that for k ∈ K1 , B(xk , 45rk ) meets B j for some j ∈ N, which gives us rk ≥ 135R j and further implies that B(xk , 46rk ) ⊃ B j . Also, for j ∈ J ∗ and j ∈ N, if B j ∩ B j = 0, / then (15B j ) ⊃ B j . As a result, we conclude that {B(xk , 46rk )}k∈K1



{15B j } j∈J∗ ∩J2

is the desired covering of Ω , and hence (∞)

Λd (Ω , {B j }∞j=1 ) ≤

∑ (46rk )d + ∑∗

k∈K1

j∈J ∩J2

(∞)

(15R j )d  (46)d Λd (Ω ),

which completes the proof of Lemma 7.15.

 

Lemma 7.16. Let β ∈ [1, ∞), λ ∈ (0, ∞) and ω be a nonnegative Borel measurable function on Rn+1 + . Then there exists a positive constant C, independent of β , ω and λ , such that (∞) 0

Λd

1 (∞) {x ∈ Rn : Nβ ω (x) > λ } ≤ Cβ d Λd ({x ∈ Rn : N ω (x) > λ }) ,

224

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

where Nβ ω (x) ≡

sup ω (y,t).

|y−x|<β t

Proof. Observe that {x ∈ Rn : N ω (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 Linder¨of 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 )

.

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

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

 

which completes the proof of Lemma 7.16.

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)
By Definition 3.1, we see that for Q ∈ Q, |(A0t)Q | 

∑

{P∈Q: l(Q)≤l(P)}

%

l(Q) l(P)

& n+ε 2

|tP | −1 (1 + l(P) |xQ − xP |)n+ε

.

Thus, we have A0t f H 0,τ (Rn ) p,q

   ∞ j n+ε  − 2q  inf  ∑ ∑ |Q| χQ ∑ ∑ 2(i− j) 2 ω  i=−∞ P∈Qi j=0 Q∈Q j q  1q   − j −1  |tP |[ω (·, 2 )]  × . n+ ε  (1 + 2i|xQ − xP |)  L p (Rn )

226

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

Let ω be a nonnegative Borel measurable function satisfying (7.3) and  1   ∞ q   − j −q  ∑ ∑ |tQ |q [χ   ω (·, 2 )] Q    j=0 Q∈Q j 

 t f H 0,τ (Rn ) . p,q

L p (Rn )

Let A0,i (Q) ≡ {P ∈ Qi : 2i |xP − xQ | ≤ and

√ n/2}

√ √ Am,i (Q) ≡ {P ∈ Qi : 2m−1 n/2 < 2i |xP − xQ | ≤ 2m n/2}

for all i ∈ Z and m ∈ Z+ . Define

ωm (x,t) ≡ 2−mnτ

sup

√ y∈B(x, n2m+1t)

ω (y,t)

−mnτ N√ for all (x,t) ∈ Rn+1 + . Then N ωm  2 n2m+2 ω and

[ωm (x, 2− j )]−1 ω (y, 2−i )  2mnτ for m ∈ Z+ , x ∈ Q with Q ∈ Q j , y ∈ P with P ∈ Am,i (Q) and i ≤ j. Moreover, using Lemma 7.16, we see that a constant multiple of ωm also satisfies (7.3). Similarly to the proof of Lemma 2.4, we have that for all x ∈ Q, ⎞ ⎛ |tP |[ωm (x, 2− j )]−1 ∑ (1 + 2i|x − x |)n+ε  2−mε +mnτ M ⎝ ∑ |tP |χP[ω (·, 2−i )]−1 ⎠ (x). P Q P∈Am,i (Q) P∈Am,i (Q) Thus, choosing ε > 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 )

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

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

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

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|

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

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

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

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 ).

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

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

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

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

 

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

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

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

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 |
 (x ) ≤ N ω (x , 0) for all x ∈ Rn−1 . where C is a positive constant chosen so that N ω    (x )](p∨q) > 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 ).

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

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

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

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

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

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 )

7.3 A ( vmo , h1 )-Type Duality Result

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

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

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 H¨older’s inequality, we obtain λ a0,τ (Rn )  G AW 0,τ (Rn+1 )  1, p,p

p,p

Z+

7.3 A ( vmo , h1 )-Type Duality Result

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

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

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 ))

7.3 A ( vmo , h1 )-Type Duality Result

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

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

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 < p . Thus, the claim is true. From the above claim, it follows that if τ > 0 and p = q, only when 1 ≤ q < p < ∞, the conclusion of Theorem 7.12 may be true for the spaces 0 Bs,p,τq(Rn ) τ n and BH p−s,  ,q (R ), which is unclear so far to us; see also Remark 7.2. (ii) Similarly, we claim that when τ > 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) .

7.3 A ( vmo , h1 )-Type Duality Result

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 < p < q < ∞, then by the embedding s−n/p+n/q,τ

Fp,s, qτ (Rn ) ⊂ Fq,r

(Rn )

in Corollary 2.2 with r > 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

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

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

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

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 H¨older’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 Calder´on 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 H¨older’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

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

Let t1 ∈ (s1 , s). By H¨older’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.

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 8, c Springer-Verlag Berlin Heidelberg 2010 

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

8 Homogeneous Spaces

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 Calder´on 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

8 Homogeneous Spaces

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

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

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 Calder´on 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

8 Homogeneous Spaces

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



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

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)<|x−2− j k|≤2i l(P)

ϕ j ∗ f (2− j k) = 2 jn/2 f , ϕQ ,

then by Theorem 8.1, we see that ⎡ −(n+δ ) ⎢

|II(x)|  [l(P)]

⎤1 ∞

⎣∑

p

∞

∑

j= jP i=0

∑

 [l(P)]

p −(i+ j)(n+δ )p⎥

|ϕ j ∗ f (2 k)| 2

k∈Zn |x−2− j k|≤2i l(P)



−(n+δ )

−j

τ  f B˙ s,p,∞ (Rn )

∞

∑

∞

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

1

∑ |2 P| 2

j= jP i=0

⎦

i

τ p − jsp jn −(i+ j)(n+δ )p

2 2

p

260

8 Homogeneous Spaces

where we choose δ > 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

8 Homogeneous Spaces

 (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|<2−m jP −1  ,  2n −1 ∞ ∞ 1 msq mn  2 2 ∑ ∑ ∑ ∑ |ai, j,k | |P|τ P m= j −1 i=1 j=m+1 k∈IP, j P q - p/q ⎫1/p 1/p   ⎬ m 1 M × |Δh ψi, j,k (x)| dx dx + τ ··· ∑ ··· ⎭ |P| 2−m−2 ≤|h|<2−m j=−∞ ≡ I + II.

It suffices to show that both I and II are dominated by  f  . s, τ F˙ (Rn ) p, q

For I, by the support condition of ψi (see (4.20)), we see that for all |h| < 2−m and l ∈ {0, · · · , M}, supp ψi (2 j (· + (M − l)h) − k) ⊂ 2− j (k + [−N3 , N3 ]n ) + 2−m M(−1, 1)n ≡ EM,m, j . Thus, M

1 I ∑ τ l=0 |P|

 , P

×



∞

∑

2

msq

2mn

m= jP −1

2n −1

∞

∑ ∑

2 jn/2

i=1 j=m+1

∑

|ai, j,k |χEM,m, j (x)

k∈IP, j

⎫1/p ⎬

q - p/q

 2−m−2 ≤|h|<2−m

|ψi (2 j (x + (M − l)h) − k)| dh

dx

⎭

.

If q ∈ (0, 1], since ψi ∈ CN1 (Rn ) and has compact support, by (2.11), we have ⎧ , - p/q ⎫1/p  ⎨ ⎬ 2n −1 ∞ ∞ 1 I  2msq ∑ ∑ 2 jnq/2 ∑ |ai, j,k |q χEM,m, j (x) dx . ∑ τ ⎭ |P| ⎩ P m= j −1 i=1 j=m+1 k∈I P

P, j

Since s > 0, for any fix x ∈ P, ∞

∑

2msq

m= jP −1

= 2−sq

∞

∑

2 jnq/2

j=m+1 ∞

∑ ∑

j= jP k∈IP, j

∑

|ai, j,k |q χEM,m, j (x)

k∈IP, j

2 j(s+n/2)q|ai, j,k |q χEM, j−1, j (x),

264

8 Homogeneous Spaces

which together with the fact that EM, j−1, j is covered by a union of finite number . If q ∈ (1, ∞], letting ε ∈ (0, s), by dyadic cubes in Q j implies that I   f  s, τ F˙p, q (Rn ) H¨older’s inequality, we see that ⎧ ⎛ ⎪ n ⎪ M ∞ 2 −1 ∞ 1 ⎨ ⎜ msq ⎜ ∑ 2 I ∑ τ ∑ ∑ 2 jn/2 ⎝ ⎪ |P| P ⎪ m= jP −1 i=1 j=m+1 l=0 ⎩ ,

-1/q

∑

×

|ai, j,k | χEM,m, j (x) q

k∈IP, j

⎤q

⎫1/p ⎪ ⎪ ⎪ ⎬

⎞ p/q

⎟ ⎦ ⎟ ⎠

dx

⎪ ⎪ ⎪ ⎭

⎧ ,  2n −1 ∞ ∞ 1 ⎨ ∑ τ 2m(s−ε )q ∑ ∑ 2 jnq/2+ jε q ∑ ⎩ P m= j −1 i=1 j=m+1 l=0 |P| P M

⎫1/p ⎬

- p/q

∑

×

|ai, j,k |q χEM,m, j (x)

dx

k∈IP, j

⎭

  f  . s, τ F˙ (Rn ) p, q

The estimate of II is similar to that of Lemma 4.8. In fact, by (4.28), we have ⎧ ,    ∞ 2n −1 m 1 ⎨ msq mn 2 2 II  ∑ ∑ ∑ 2 jn/2 |P|τ ⎩ P m= jP −1 2−m−2 ≤|h|<2−m i=1 j=−∞ ×

∑

q - p/q

|ai, j,k |χEM,m, j (x)|h|M 2 jM

k∈IP, j

(1 + |2 j (x + θ1 h + · · · + θM h) − k|)n+δ

dh

dx

⎫1/p ⎬ ⎭

,

where δ ∈ (0, ∞) can be sufficiently large. Since j ≤ m and |h| < 2−m , we see that |2 j (x + θ1 h + · · · + θM h) − k| ≥ |2 j x − k| − M2 j |h| ≥ |2 j x − k| − M, which further deduces that ⎧ , n  ∞ 2 −1 m 1 ⎨ msq II  2 ∑ ∑ 2 jn/22( j−m)M |P|τ ⎩ P m=∑ j −1 i=1 j=−∞ P

×

∑

k∈IP, j

|ai, j,k |χEM,m, j (x) (1 + |2 j x − k|)n+δ

⎫1/p ⎬

q - p/q dx

⎭

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

265

⎧ ⎛ ⎡ ⎪ m 1 ⎨ ⎝ ∞ msq ⎣  2 2− js 2( j−m)M 2 jn/p ∑ ∑ |P|τ ⎪ ⎩ P m= j −1 j=−∞ P

, ×

∑ ∑

2 jsp2 jnp/22− jn |ai, j,k | p

⎫1/p ⎪ ⎬

⎤q ⎞ p/q

-1/p

2n −1

χEM,m, j (x)⎦ ⎠

dx

i=1 k∈IP, j

⎪ ⎭

,

where the last inequality follows from (2.11) when p ≤ 1 or H¨older’s inequality when p > 1. Since j ≤ m, a finite union of dyadic cubes in Q j covers EM,m, j . Then -1/p

,

2n −1

∑ ∑

2

jsp jnp/2 − jn

2

2

|ai, j,k |

i=1 k∈IP, j

p

 2−( j∧ jP )nτ  f  s, τ F˙ (Rn ) p, q

and hence, by s < {M ∧ M + n(1/p − τ )}, we have 1 II  |P|τ

 , P



∞

∑

2

msq

m= jP −1

m

∑

2− js 2( j−m)M 2 jn/p2−( j∧ jP )nτ

j=−∞

⎫1/p ⎬

q - p/q × χEM,m, j (x)

dx

⎭

 f  s, τ F˙ (Rn ) p, q

  f  , s, τ F˙ (Rn ) p, q

which completes the proof of Lemma 8.4.

 

By Lemmas 8.3 and 8.4, to prove Theorems 8.3 and 8.4, we only need to prove that  f    f A˙ s,p,τq(Rn ) . s, τ A˙ (Rn ) p, q

In fact, letting N1 ≥ s + nτ , by the properties of wavelets (see [99, p. 108]), we know that each ψi, j,k is a constant multiple of a homogeneous smooth analysis molecule for A˙ s,p,τq (Rn ) introduced in [165, Definition 4.2]. From the smooth molecular decomposition characterization of A˙ s,p,τq(Rn ) in [165, Theorem 4.2], we deduce that { f , ψi, j,k } j∈Z, k∈Zn a˙s,p,τq(Rn )   f A˙ s,p,τq(Rn ) , which further implies that  f    f A˙ s,p,τq(Rn ) s, τ A˙ (Rn ) p, q

and then yields Theorems 8.3 and 8.4.

266

8 Homogeneous Spaces

Remark 8.4. For the wavelet characterizations on B˙ sp, q (Rn ) and F˙p,s q (Rn ), we refer to, e. g., [65, 99]. For the difference characterization on B˙ sp, q(Rn ) with p, q ∈ [1, ∞], we refer to [9, p. 147, Theorem 6.3.1] (see also [145, Sect. 5.2.3]); while for the difference characterization on F˙p,s q(Rn ) with p, q ∈ [1, ∞], see [22]. Corresponding to Sect. 4.4, we now establish the oscillation characterization of A˙ s,p,τq (Rn ). For all f ∈ L1loc (Rn ), define  f B˙ s, τ (Rn ) p, q

1 ≡ sup τ P∈Q |P|



2l(P)

t

−sq

(

0

0 P

1p oscM−1 f (x, Mt) 1

)q/p dx

dt t

1/q

and  f F˙ s, τ (Rn ) p, q

1 ≡ sup τ P∈Q |P|

 (

2l(P)

t P

−sq

0

1q dt 0 M−1 osc1 f (x, Mt) t

1/p

) p/q dx

.

Then we have the following conclusion. Theorem 8.5. Let s, p, q, τ , N1 and M be as in Theorem 8.3. If f ∈ A˙ s,p,τq (Rn ), then there exists a function g ∈ L ploc (Rn ) such that f = g in S∞ (Rn ) and gA˙ s, τ (Rn ) ≤ C f A˙ s,p,τq (Rn ) ; p, q

if g ∈ L ploc (Rn ) ∩ S∞ (Rn ) satisfies that gA˙ s, τ (Rn ) < ∞, then g ∈ A˙ s,p,τq(Rn ) and p, q

gA˙ s,p,τq(Rn ) ≤ CgA˙ s, τ (Rn ) , p, q

where C is a positive constant independent of f and g. Proof. We only consider Triebel-Lizorkin spaces. If g ∈ L ploc (Rn ) ∩ S∞ (Rn ) satisfies that gA˙ s, τ (Rn ) < ∞, similarly to the proof of Lemma 4.10, we obtain that for p, q

all g ∈ L ploc (Rn ) ∩ S∞ (Rn ),

g♣  gA˙ s, τ (Rn ) , A˙ s, τ (Rn ) p, q

p, q

which together with Theorem 8.4 yields that g ∈ A˙ s,p,τq (Rn ) and gA˙ s,p,τq(Rn )  gA˙ s, τ (Rn ) . p, q

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

267

p On the other hand, for f ∈ A˙ s,p,τq (Rn ), by Proposition 8.2, there exists g ∈ L loc (Rn )  n such that f = g in S∞ (R ). It remains to show that

gA˙ s, τ (Rn )  gA˙ s,p,τq(Rn ) . p, q

To this end, we need to estimate  (

1 IR ≡ |R|τ

2l(R)

t R

0 −sq

0

1q dt g(x, Mt) oscM−1 1 t

 1p

) qp

,

dx

where R is dyadic cube. It is easy to see that   R



2l(R)

t −sq

(

0

⎧  ⎨ ⎩

R

t −n

B(x,Mt)

(

∞

∑

inf

P∈PM−1 (Rn )

)q



2ksq

k= jR −1

inf

P∈PM−1 (Rn )

Recall that g=

2kn

|g(y) − P(y)| dy

B(x,M2−k )

∑∑ ∑

 1p

qp dx

⎫1 ⎬p

)q  qp



2n −1

dt t

|g(y) − P(y)| dy

dx

⎭

.

ai, j,m ψi, j,m

i=1 j∈Z m∈Zn

holds in L ploc (Rn ) when p ∈ (0, ∞), where ai, j,m ≡ g, ψi, j,m  (see the proof of Lemma 8.4). Then for any ε > 0, there exists an N ≡ N(x, k, R) ∈ N such that N > 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

8 Homogeneous Spaces

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

<∞

s,τ s,τ and when AH˙ p,q (Rn ) = F H˙ p,q (Rn ) (q = 1),

$ 1 $ $ q$ $ $ 2 2 q jsq − j −1 $ $ 2 2 s,τ  f F H˙ p,q 2 ϕ j ∗ f [ω (·, 2 )] ∑ (Rn ) ≡ inf $ $ ω $ $ j∈Z

< ∞,

L p (Rn )

and the function ω runs over all nonnegative Borel measurable functions on Rn+1 + satisfying (7.3) and with the restriction that for any j ∈ Z, ω (·, 2− j ) is allowed to vanish only where ϕ j ∗ f vanishes. s, τ s, τ n n ˙ p, The spaces BH˙ p, q (R ) and F H q (R ) were original introduced, respectively, in [164, Sect. 5] and [165, Sect. 6] and proved therein to be the predual spaces of τ n ˙ −s, τ n ˙ s, τ n ˙ s, τ n B˙ −s, p , q (R ) and Fp , q (R ). We also recall that the spaces BH p, q (R ) and F H p, q (R ) cover the Hardy-Hausdorff space HH−1 α (Rn ), which was introduced in [44] and proved therein to be the predual of the space Qα (Rn ); see [164, Remark 5.1]. We remark that all results in Chap. 7 have counterparts for homogeneous case. Indeed, in [168], we obtained the ϕ -transform characterization, Sobolev-type embedding properties, smooth atomic and molecular decomposition characterizations s, τ s, τ n n ˙ p, for BH˙ p, q (R ) and F H q (R ), which further deduced the trace theorem, the boundedness of pseudo-differential operators with homogeneous symbols and the lifting s, τ s, τ n n ˙ p, property of BH˙ p, q (R ) and F H q (R ). The dual properties in Sect. 7.3 have homogeneous counterparts (see [166]) and the real interpolation properties in Sect. 7.4 are s, τ s, τ n n ˙ p, true for the homogeneous spaces BH˙ p, q (R ) and F H q (R ) by a similar proof. We also point out that in [167], the maximal function and local mean characterizations s, τ s, τ n n ˙ p, of BH˙ p, q (R ) and F H q (R ) were established.

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Index

s,τ ATp,q (Rn+1 Z+ )-atom, 181 s, τ A p, q (Ω ), 173 As,p,τq (Rn ), 22 As,p,τq (Rn+ ), 169 s,τ BTp,q (Rn+1 Z+ ), 178 s, τ B p, q (Rn ), 12, 22 C(Rn ), 7 Cm (Rn ), 7 Cm (Rn+ ), 169 Csp (Rn ), 10 Cs (Rn ), 7 Cc∞ (Rn ), 18 D(Rn+ ), 168 E loc , 19 E unif , 19 s,τ FTp,q (Rn+1 Z+ ), 179 s F∞, q (Rn ), 11 Fp,s, qτ (Rn ), 12, 22 Hps (Rn ), 11 K-functional, 246 L1loc (Rn ), 73 L∞ (Rn ), 6 L p (Rn ), 6 Lτp (Rn ), 7, 46 N0 , 102 Nε , 102 Qα (Rn ), 10 Sϕ , 31 Tψ , 31 TN,P , 74 Wps (Rn ), 8 (∞)

Λd , 177 Θ (Rn , L), 65 s, τ n AHp, q (R ), 201 s, τ BHp, q (Rn ), 13, 201 BMO (Rn ), 8 L p,λ (Rn ), 9 P(Rn ), 251

S  (Rn )/P(Rn ), 251 S (Rn ), 18 μ S1,1 (Rn ), 137 S∞ (Rn ), 251 s, τ n AH˙ p, q (R ), 269 s, τ n ˙ BHp, q (R ), 269 B˙ s,p,τq (Rn ), 13, 253 s, τ n F H˙ p, q (R ), 269 s, τ n ˙ Fp, q (R ), 13, 253 b˙ s,p,τq (Rn ), 254 s, τ n f˙p, q (R ), 254 sq (Mup (Rn )), 120 s, τ n FHp, q (R ), 13, 201 n Lip s(R ), 7 s (Rn ), 12 E pqu IQ,m , 87 JQ , 87 λ ,s L p,q (Rn ), 168 Mup (Rn ), 9 s (Rn ), 12 N pqu PM (Rn ), 9, 111 Q, 18  · ♣,osl , 115 As, τ (Rn ) p, q

 · ♣,os , 115 s, τ A (Rn ) p, q

Rn+1 Z+ , 178 s, τ n aHp, q (R ), 212 s, τ n ), 211 bHp, (R q bmo (Rn ), 9 cmo (Rn ), 9 s, τ n f Hp, q (R ), 212 σ p , 18 σ p, q , 18 vmo (Rn ), 9 τs,p,q , 18 τs,p , 18 ϕ -transform, 31, 32, 138, 214  (Rn , L), 261 Θ 279

280

Index

TN,P , 74 (∞) Λd , 178 Z+ , 18 as,p,τq (Rn ), 30 bs,p,τq (Rn ), 30 s, τ n f p, q (R ), 30 p h (Rn ), 11 p ∨ q, 18 p ∧ q, 18 Tps (Rn ), 10 Mup (Rn )(sq ), 120 Z m (Rn ), 7 CMO (Rn ), 9 VMO(Rn ), 8 oscM u f , 111 ||| · ||| Fs

∞,q(R

||| · |||♣ Fs

 · ♣,osl F s (Rn ) , 119 ∞,q

 · ♣ , 261 B˙ s, τ (Rn ) p, q

 · ♣ , 261 F˙ s, τ (Rn ) p, q

 · ♦As, τ (Rn ) , 108 p,p

 · ♥,osl F s (Rn ) , 119 ∞,q

, 116  · ♥,osl As, τ (Rn ) p, q

 · ♥,os , 116 As, τ (Rn ) p, q

 · B˙s, τ (Rn ) , 266 p, q

 · F˙ s, τ (Rn ) , 266 p, q

 ·  , 72 Bs, τ (Rn ) p, q

 · SM , 18 , 72  ·  F s, τ (Rn ) p, q

n)

, 100

n , 110 ∞,q(R )  ||| · |||Bs, τ (Rn ) , 130 p, q ||| · ||| s, τ n , 134 Fp, q (R )  ||| · |||Bs, τ (Rn ) , 98 p, q ||| · ||| s, τ n , 99 Fp, q (R ) ♣ ||| · |||As, τ (Rn ) , 102 p, q ||| · |||♦As, τ (Rn ) , 108 p,p , 102 ||| · |||♥ As,p,τq(Rn ) ♥ ||| · |||Fs (Rn ) , 110 ∞,q , 102 ||| · |||♠ As,p,τq(Rn ) ♠ ||| · |||Fs (Rn ) , 110 ∞,q  ·  , 130 s, τ B p, q (Rn )  ·  , 90 s, τ B p, q (Rn ) , 90  ·  Fp,s, qτ (Rn ) ♣  · Bs, τ (Rn ) , 83 p, q  · ♣ , 83 Fp,s, qτ (Rn ) ♥  · Bs, τ (Rn ) , 84 p, q  · ♥ , 84 s, τ Fp, q (Rn ) ♠  · Bs, τ (Rn ) , 84 p, q  · ♠ , 84 Fp,s, qτ (Rn )   · F s, τ (Rn ) , 134 p, q  ·  , 132 Bs,p,τq (Rn )   · F s, τ (Rn ) , 134 p, q , 260  ·  A˙ s,p,τq (Rn )

almost diagonal operator, 50 Besov space, 8, 11 Besov-Hausdorff space, 13, 201, 268 Besov-Morrey space, 12 Besov-type space, 12, 13, 22, 253 Bessel-potential space, 11 Calder´on reproducing formula, 24, 254 Campanato space, 9 Choquet integral, 178 diffeomorphism, 160 diffeomorphism theorem, 160 difference, 73 dyadic cube, 18 dyadic Hausdorff capacity, 178 Fatou property, 48 H¨older space, 7 H¨older-Zygmund space, 45 Hausdorff capacity, 177 Hedberg-Netrusov approach, 119 interpolation, 246 lifting operator, 142, 169 lifting property, 142, 170

Index Lipschitz space, 7 Local approximation space, 9 Local Hardy space, 11

281 smooth atom, 58, 59, 160, 163, 229 smooth molecule, 56, 138 smooth synthesis molecule, 54, 228 Sobolev space, 7 Sobolev-type embedding, 44, 218

Morrey space, 9

nontangential maximal function, 178

oscillation, 9, 111

tent space, 178, 187 trace, 162 Triebel-Lizorkin space, 11 Triebel-Lizorkin-Hausdorff space, 13, 201, 268 Triebel-Lizorkin-Morrey space, 12 Triebel-Lizorkin-type space, 12, 13, 22, 253

pseudo-differential operator, 137 wavelet, 85, 97, 256 Schwartz function, 18 Slobodeckij space, 8 smooth analysis molecule, 54, 228

Zygmund space, 7

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