Theory of function spaces 4 9783030358907, 9783030358914

Table of contents :
Contents......Page 6
Preface......Page 9
Global spaces......Page 11
Hybrid spaces......Page 15
Global spaces......Page 17
Hybrid spaces......Page 20
Relations between global and hybrid spaces......Page 22
Lifts......Page 26
Distinguished representations......Page 27
Fatou property......Page 28
Embeddings in L(Rn) and C(Rn)......Page 30
Embeddings in the space of locally integrable functions......Page 32
Sharp embeddings: Constant differential dimensions......Page 34
Sharp embeddings: Constant smoothness......Page 35
Gagliardo-Nirenberg inequalities......Page 37
The problem......Page 38
The trace theorem......Page 41
Dichotomy: Traces versus density......Page 44
Diffeomorphisms......Page 48
Smooth multipliers......Page 49
Localizations......Page 52
Multiplication algebras......Page 53
Characteristic functions as multipliers......Page 55
Rough multipliers......Page 62
Introduction and distinguished lifts......Page 66
Main assertions......Page 69
The essentials: Why and how......Page 71
Preliminaries and definitions......Page 72
Traces......Page 75
Extensions......Page 77
Embeddings......Page 78
Characteristics of distributions......Page 81
Preliminaries......Page 83
Main assertions......Page 86
Envelopes......Page 91
Positivity......Page 95
Local homogeneity......Page 98
Multipliers, revisited......Page 101
Main assertions......Page 102
Haar wavelets......Page 106
Fubini property......Page 111
Lusin functions......Page 113
Heat kernels......Page 114
Caloric smoothing......Page 117
Introduction and motivation......Page 122
Spaces with negative smoothness......Page 125
Spaces in the distinguished strip......Page 127
Some properties......Page 132
Caloric smoothing......Page 133
Natural habitats......Page 135
The homogeneity rule......Page 136
Classical Sobolev spaces......Page 139
Sobolev spaces......Page 140
Differences......Page 141
Fourier-analytical decompositions and paramultiplication......Page 142
Atoms......Page 143
Quarks......Page 145
Wavelets and paramultiplication......Page 149
Epilogue......Page 152
Bibliography......Page 154
Symbols......Page 164
Index......Page 166

Citation preview

Monographs in Mathematics 107

Hans Triebel

Theory of Function Spaces IV

Monographs in Mathematics Volume 107

Series Editors Herbert Amann, Universität Zürich, Zürich, Switzerland Jean-Pierre Bourguignon, IHES, Bures-sur-Yvette, France William Y. C. Chen, Nankai University, Tianjin, China Associate Editors Huzihiro Araki, Kyoto University, Kyoto, Japan John Ball, Heriot-Watt University, Edinburgh, UK Franco Brezzi, Università degli Studi di Pavia, Pavia, Italy Kung Ching Chang, Peking University, Beijing, China Nigel Hitchin, University of Oxford, Oxford, UK Helmut Hofer, Courant Institute of Mathematical Sciences, New York, USA Horst Knörrer, ETH Zürich, Zürich, Switzerland Don Zagier, Max-Planck-Institut, Bonn, Germany

The foundations of this outstanding book series were laid in 1944. Until the end of the 1970s, a total of 77 volumes appeared, including works of such distinguished mathematicians as Carathéodory, Nevanlinna and Shafarevich, to name a few. The series came to its name and present appearance in the 1980s. In keeping its well-established tradition, only monographs of excellent quality are published in this collection. Comprehensive, in-depth treatments of areas of current interest are presented to a readership ranging from graduate students to professional mathematicians. Concrete examples and applications both within and beyond the immediate domain of mathematics illustrate the import and consequences of the theory under discussion.

More information about this series at http://www.springer.com/series/4843

Hans Triebel

Theory of Function Spaces IV

Hans Triebel Mathematisches Institut Friedrich-Schiller-Universität Jena, Germany

ISSN 1017-0480 ISSN 2296-4886 (electronic) Monographs in Mathematics ISBN 978-3-030-35890-7 ISBN 978-3-030-35891-4 (eBook) https://doi.org/10.1007/978-3-030-35891-4 Mathematics Subject Classification (2010): 46–02, 46E35, 42C40, 42B35 © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Contents Preface 1 Fundamental principles 1.1 Definitions and basic properties . . . . . . . . . . . 1.1.1 Global spaces . . . . . . . . . . . . . . . . . 1.1.2 Hybrid spaces . . . . . . . . . . . . . . . . . 1.2 Wavelet characterizations . . . . . . . . . . . . . . 1.2.1 Global spaces . . . . . . . . . . . . . . . . . 1.2.2 Hybrid spaces . . . . . . . . . . . . . . . . . 1.3 Basic assertions . . . . . . . . . . . . . . . . . . . . 1.3.1 Relations between global and hybrid spaces 1.3.2 Lifts . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Distinguished representations . . . . . . . . 1.3.4 Fatou property . . . . . . . . . . . . . . . .

ix

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

1 1 1 5 7 7 10 12 12 16 17 18

2 The essentials, key theorems 2.1 Embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Embeddings in L∞ (Rn ) and C(Rn ) . . . . . . . . . . . . 2.1.2 Embeddings in the space of locally integrable functions 2.1.3 Sharp embeddings: Constant differential dimensions . . 2.1.4 Sharp embeddings: Constant smoothness . . . . . . . . 2.1.5 Diversity . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.6 Gagliardo-Nirenberg inequalities . . . . . . . . . . . . . 2.2 Traces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 The problem . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 The trace theorem . . . . . . . . . . . . . . . . . . . . . 2.2.3 Dichotomy: Traces versus density . . . . . . . . . . . . . 2.3 Diffeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Multipliers, localizations and multiplication algebras . . . . . . 2.4.1 Smooth multipliers . . . . . . . . . . . . . . . . . . . . . 2.4.2 Localizations . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Multiplication algebras . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

21 21 21 23 25 26 28 28 29 29 32 35 39 40 40 43 44

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

v

vi

Contents

2.5

2.6

2.7

2.4.4 Characteristic functions as multipliers 2.4.5 Rough multipliers . . . . . . . . . . . Extensions . . . . . . . . . . . . . . . . . . . . 2.5.1 Introduction and distinguished lifts . . 2.5.2 Main assertions . . . . . . . . . . . . . Spaces on domains . . . . . . . . . . . . . . . 2.6.1 The essentials: Why and how . . . . . 2.6.2 Preliminaries and definitions . . . . . 2.6.3 Traces . . . . . . . . . . . . . . . . . . 2.6.4 Extensions . . . . . . . . . . . . . . . 2.6.5 Embeddings . . . . . . . . . . . . . . . 2.6.6 Characteristics of distributions . . . . Multipliers: Further properties . . . . . . . . 2.7.1 Preliminaries . . . . . . . . . . . . . . 2.7.2 Main assertions . . . . . . . . . . . . .

3 Further topics 3.1 Envelopes . . . . . . . . . . . . . . 3.2 Positivity . . . . . . . . . . . . . . 3.3 Local homogeneity . . . . . . . . . 3.4 Refined localization spaces . . . . . 3.4.1 Multipliers, revisited . . . . 3.4.2 Main assertions . . . . . . . 3.5 Haar wavelets . . . . . . . . . . . . 3.6 Fubini property . . . . . . . . . . . 3.7 Characterizations in terms of Lusin 3.7.1 Lusin functions . . . . . . . 3.7.2 Heat kernels . . . . . . . . . 3.7.3 Caloric smoothing . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . functions and . . . . . . . . . . . . . . . . . . . . . . . .

4 Complements 4.1 Tempered homogeneous spaces . . . . . . . 4.1.1 Introduction and motivation . . . . 4.1.2 Spaces with negative smoothness . . 4.1.3 Spaces in the distinguished strip . . 4.1.4 Some properties . . . . . . . . . . . 4.1.5 Caloric smoothing . . . . . . . . . . 4.2 Natural habitats and the homogeneity rule 4.2.1 Natural habitats . . . . . . . . . . . 4.2.2 The homogeneity rule . . . . . . . . 4.3 Spaces and tools . . . . . . . . . . . . . . . 4.3.1 Classical Sobolev spaces . . . . . . . 4.3.2 Sobolev spaces . . . . . . . . . . . . 4.3.3 Differences . . . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

46 53 57 57 60 62 62 63 66 68 69 72 74 74 77

. . . . . . . . . . . . . . . . . . . . . . . . heat . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . kernels . . . . . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

83 83 87 90 93 93 94 98 103 105 105 106 109

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

115 115 115 118 120 125 126 128 128 129 132 132 133 134

. . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

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. . . . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

Contents 4.3.4 4.3.5 4.3.6 4.3.7 4.3.8

vii Fourier-analytical decompositions Atoms . . . . . . . . . . . . . . . Quarks . . . . . . . . . . . . . . Wavelets and paramultiplication Epilogue . . . . . . . . . . . . . .

and paramultiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . .

135 136 138 142 145

Bibliography

147

Symbols

157

Index

159

Preface The main motivation to write this book originates from the attempt to incorporate s the nowadays fashionable spaces F∞,q (Rn ), s ∈ R, 0 < q ≤ ∞, into the existing elaborated theory of the spaces Asp,q (Rn ), A ∈ {B, F }, s ∈ R, 0 < p ≤ ∞, 0 < q ≤ ∞ with p < ∞ for the F -spaces. But we decided to extend this task by offering a guided tour through the jungle of these spaces of Sobolev-Besov type. Therefore, this book can also serve as an advanced textbook introducing this topic of Sobolev-Besov-type spaces in a modern and concise way. We concentrate on the essentials, complemented by a kaleidoscope of highlights exploring the grounds of the fascinating mathematical landscape called function spaces. The required hiking equipment is: Fourier analysis, measure theory, functional analysis, and enthusiasm. New proofs rely mostly on wavelet representations. In all other cases, we take over already existing assertions (quite often from our own books), which are carefully commented and adapted. Detailed references are also provided. The table of contents shows which topics are treated in this book, but let us add a few comments at this point. Nowadays, a plethora of function spaces of many types prevails. In this book, however, the aim is not to provide an encyclopaedic survey. Much rather, we deal almost exclusively with the above-mentioned inhomogeneous spaces Asp,q (Rn ) and their counterparts Asp,q (Ω) on domains Ω in Rn . There are two exceptions. Firstly, we will occasionally rely on some assertions for the so-called hybrid spaces LrAsp,q (Rn ), which have the remarkable property s s (Rn ) = F∞,q (Rn ), L0 Fp,q

0 < q ≤ ∞,

s ∈ R,

(0.1)

for all 0 < p < ∞. Secondly, we will deal in the Sections 4.1 and 4.2 with the ∗   tempered homogeneous spaces Asp,q (Rn ) within the dual pairing S(Rn ), S  (Rn ) mostly restricted to their natural habitat 0 < p ≤ ∞,

n

 1 n −1 ≤s≤ . p p

(0.2)

This will be based on heat kernels as discussed in Section 3.7 for the spaces Asp,q (Rn ). Apart from that, we will immediately jump from the Fourier-analytical definition of the spaces Asp,q (Rn ) in Section 1.1 to their wavelet characterization in Section 1.2 without discussing any other means or special cases. This approach

ix

x

Preface

will be compensated to some extent in the final Section 4.3, where we discuss Sobolev and Besov spaces in their more traditional versions based on derivatives and differences of functions, including the related references. Furthermore, we will glance at other means such as atoms and quarks. The fact that this discussion is located at the very end of the book gives us the possibility to make clear how all these special spaces, means and building blocks (including wavelets and heat kernels) are interwoven and how they are connected with the preceding text. Note that this book may be considered both as a supplement to the monographs [T83, T92, T06] with the same title and as an advanced companion of the textbook [HT08]. Throughout the book, we shall use the symbol ∼ (equivalence) as follows: Let I be an arbitrary index set. Then ai ∼ bi

for

i∈I

(equivalence)

(0.3)

for two sets of positive numbers {ai : i ∈ I} and {bi : i ∈ I} means that there are two positive numbers c1 and c2 such that c1 ai ≤ bi ≤ c2 ai

for all

i ∈ I.

(0.4)

Last but not least, I would like to thank Dorothee D. Haroske for producing the figures. Hans Triebel, September 2019

Chapter 1

Fundamental principles 1.1 Definitions and basic properties 1.1.1 Global spaces Throughout the book, we shall use a standard notation. Let N be the collection of all natural numbers and N0 = N ∪ {0}. Let Rn be the Euclidean n-space, where n ∈ N. Furthermore, we set R = R1 , and C is the complex plane. Let S(Rn ) be the Schwartz space of all complex-valued, rapidly decreasing and infinitely differentiable functions on Rn , and let S  (Rn ) be the dual space of all tempered distributions on Rn . Let Lp (Rn ) with 0 < p ≤ ∞ be the standard complex quasiBanach space with respect to the Lebesgue measure in Rn , quasi-normed by  1/p f |Lp (Rn ) = |f (x)|p dx (1.1) Rn

with the natural modification for p = ∞. Similarly, we define Lp (M ) where M is a Lebesgue-measurable subset of Rn . As usual, Z is the collection of all integers; and Zn with n ∈ N denotes the lattice of all points m = (m1 , . . . , mn ) ∈ Rn with mj ∈ Z. If ϕ ∈ S(Rn ) then  ϕ(ξ)  = (F ϕ)(ξ) = (2π)−n/2 e−ixξ ϕ(x) dx, ξ ∈ Rn , (1.2) Rn

denotes the Fourier transform of ϕ. As usual, F −1 ϕ and ϕ∨ stand for the inverse Fourier transform, which n is given by the right-hand side of (1.2) with i instead of −i. Note that xξ = j=1 xj ξj stands for the scalar product in Rn . Both F and F −1 are extended to S  (Rn ) in the standard way. Let ϕ0 ∈ S(Rn ) with and ϕ0 (x) = 0 if |x| ≥ 3/2, (1.3) ϕ0 (x) = 1 if |x| ≤ 1

© Springer Nature Switzerland AG 2020 H. Triebel, Theory of Function Spaces IV, Monographs in Mathematics 107, https://doi.org/10.1007/978-3-030-35891-4_1

1

2

Chapter 1. Fundamental principles

and let

   ϕk (x) = ϕ0 2−k x) − ϕ0 2−k+1 x ,

Since

∞

for

ϕj (x) = 1

x ∈ Rn ,

k ∈ N.

x ∈ Rn ,

(1.4)

(1.5)

j=0

the ϕj form a dyadic resolution of unity. The entire analytic functions (ϕj f)∨ (x) make sense pointwise in Rn for any f ∈ S  (Rn ). Let QJ,M = 2−J M + 2−J (0, 1)n ,

J ∈ Z,

M ∈ Zn ,

(1.6)

be the usual dyadic cubes in Rn , n ∈ N, with sides of length 2−J parallel to the coordinate axes and with 2−J M as the lower left corner. If Q is a cube in Rn and d > 0, then dQ is the cube in Rn that concentrically contains Q and whose side length is d times the side length of Q. Let |Ω| be the Lebesgue measure of the Lebesgue measurable set Ω in Rn . Let a+ = max(a, 0) for a ∈ R. Definition 1.1. Let ϕ = {ϕj }∞ j=0 be the above dyadic resolution of unity. (i) Let 0 < p ≤ ∞, Then

s Bp,q (Rn )

0 < q ≤ ∞, 

s ∈ R.

(1.7)

is the collection of all f ∈ S (R ) such that

s f |Bp,q (Rn )ϕ =

∞ 

n

q 1/q

2jsq (ϕj f)∨ |Lp (Rn )

(1.8)

j=0

is finite (with the usual modification for q = ∞). (ii) Let 0 < p < ∞,

0 < q ≤ ∞,

s ∈ R.

(1.9)

s Then Fp,q (Rn ) is the collection of all f ∈ S  (Rn ) such that ∞



q 1/q  

s Lp (Rn )

f |Fp,q (Rn )ϕ =

2jsq (ϕj f)∨ (·)

(1.10)

j=0

is finite (with the usual modification for q = ∞). s (iii) Let 0 < q ≤ ∞ and s ∈ R. Then F∞,q (Rn ) is the collection of all f ∈ S  (Rn ) such that   q 1/q s f |F∞,q (Rn )ϕ = sup 2Jn/q 2jsq (ϕj f)∨ (x) dx J∈Z,M ∈Zn

QJ,M j≥J +

(1.11) is finite (with the modification for q = ∞ as explained below).

1.1. Definitions and basic properties

3

Remark 1.2. The theory of these global spaces Asp,q (Rn ), A ∈ {B, F }, s ∈ R, 0 < p ≤ ∞ (p < ∞ for F -spaces), 0 < q ≤ ∞ has been developed in many papers and books, including [T83, T92, T06]. In these references, one finds detailed (historical) references and special cases, in particular (fractional) Sobolev spaces, Besov spaces and Hölder-Zygmund spaces. This will not be repeated here. In Section 4.3 at the very end of this book, however, we will collect some related assertions. In addition to the above spaces, we introduce the inhomogeneous space bmo(Rn ) of bounded mean oscillations as the collection of all complex-valued locally Lebesgue-integrable functions f in Rn such that   f |bmo(Rn ) = sup |f (y)| dy + sup |Q|−1 |f (x) − fQ | dx (1.12) |Q|=1

|Q|≤1

Q

Q

is finite, where Q are cubes in Rn and fQ = |Q|−1 Q f (y) dy are the related mean values. It is well known that the spaces in the above definition are independent from the chosen resolution of unity ϕ (equivalent quasi-norms). This justifies our omission of the subscript ϕ in (1.8) and (1.10) in the sequel. As far as the spaces s F∞,q (Rn ) in (1.11) are concerned, we first remark that one can replace J ∈ Z in (1.11) by J ∈ N0 . This follows for q < ∞ from the decomposition of a cube QJ,M with −J ∈ N into 2|J|n cubes Q0,K and a related estimate in (1.11),   ∞ ∞    ∨ q ∨ q 2Jn 2jsq  ϕj f (x) dx ≤ sup 2jsq  ϕj f (x) dx, (1.13) K∈Zn

QJ,M j=0

Q0,K j=0

which results in s (Rn )ϕ ∼ f |F∞,q

sup

J∈N0 ,M ∈Zn

2Jn/q





 ∨ q 1/q 2jsq  ϕj f (x) dx . (1.14)

QJ,M j≥J

s This result coincides with [FrJ90, (12.8), p. 133]. Again, the spaces F∞,q (Rn ) are independent of ϕ (omitted in the sequel). In order to avoid any misunderstandings, we fix that (1.11) with q = ∞ and the related extension of (1.14) to q = ∞ must be understood as  ∨  s f |F∞,∞ (Rn ) = sup sup sup 2js  ϕj f (x) J∈Z,M ∈Zn x∈QJ,M j≥J +

=

J∈N0

=

sup

sup

,M ∈Zn

sup j∈N0 ,x∈Rn

 ∨  sup 2js  ϕj f (x)

x∈QJ,M j≥J js 

2

(1.15)

∨  ϕj f (x)



s (Rn ). = f |B∞,∞ s In other words, the Hölder-Zygmund spaces C s (Rn ) = B∞,∞ (Rn ) can also be incorporated into the F -scale as s s (Rn ) = B∞,∞ (Rn ) = C s (Rn ), F∞,∞

s ∈ R.

(1.16)

4

Chapter 1. Fundamental principles

The well-known equivalent norms in terms of differences for these spaces with s > 0 may be found in (4.94), (4.95). The homogeneous counterpart of the space bmo(Rn ) goes back to [JoN61] where the authors proved some inequalities, mainly in order to support their use in other papers. Also the term bounded mean oscillation has been coined in this paper. Ten years later, it was observed in [Fef71], [FeS72, Theorem 2, p. 145] (again in homogeneous terms) that bmo(Rn ) is the dual of the famous Hardy spaces h1 (Rn ): 0 (Rn ) (1.17) bmo(Rn ) = h1 (Rn ) = F1,2   (in the framework of the dual pairing S(Rn ), S  (Rn ) . Our own attempts to ins corporate p = ∞ into the scale Fp,q (Rn ) in the late 1970s had been motivated by two observations. On the one hand, [Tr78, Section 2.1.4, pp. 33–37], based on [Tri79, Section 4.4, pp. 147–151], had revealed in a tricky explicit construction that it is impossible to extend (1.10) to p = ∞ if one wants the related spaces to be independent of ϕ = {ϕj }∞ j=0 . This had also been mentioned in [T83, Sections 2.3.1, 2.3.2, pp. 45–47], but without any details. Besides, one may also consult the related comments in [FrJ90, Sections 5, 12]. This negative observation is closely related to the failure of some maximal inequalities as recently proved in [Park19, Theorem 1.1, p. 1138]. On the other hand, one can take (1.17) as a motivation for s extending the spaces Fp,q (Rn ) according to Definition 1.1(ii) to p = ∞ by duality. This has been done in [Tr78, Section 2.5.1, pp. 116–122, especially the definition on p. 118], and was essentially repeated in [T83, Section 2.3.4, pp. 50–51]. This fact will be of some use for us later on. We give a description following [T13, p. 36]. Let again ϕ = {ϕj }∞ j=0 be the above dyadic resolution of unity according to (1.3) to (1.5). Let 1 < p ≤ ∞, 1 < q ≤ ∞ and s ∈ R. (1.18)

Then Lsp,q (Rn ) collects all f ∈ S  (Rn ) which can be represented as f=

∞ 

ϕj fj

j=0

∨

with

∞



1/q



2jsq |fj (·)|q |Lp (Rn ) < ∞,

(1.19)

j=0

convergence being in S  (Rn ). Furthermore, ∞



1/q



f |Lsp,q (Rn )ϕ = inf

2jsq |fj (·)|q |Lp (Rn ) < ∞,

(1.20)

j=0 s (Rn ) where the infimum is taken over all admissible representations (1.19). Let Fp,q and bmo(Rn ) be the spaces as introduced in Definition 1.1 and Remark 1.2.

Proposition 1.3. Let p, q, s be as in (1.18) and p1 + p1 = 1q + q1 = 1. Then Lsp,q (Rn ) are Banach spaces. They are independent of ϕ (equivalent norms). Furthermore, s n  (Rn ) = Fp−s Lsp,q (Rn ) = Fp,q  ,q  (R )

(1.21)

1.1. Definitions and basic properties

5

  in the framework of the dual pairing S(Rn ), S  (Rn ) and 0 (Rn ) = L0∞,2 (Rn ). bmo(Rn ) = F∞,2

(1.22)

Remark 1.4. For details, proofs and technical explanations, we refer to [Tr78, Section 2.5.1], [T83, Sections 2.3.4, 2.11.2] and related comments in [FrJ90, Sections 5, 12]. Recall that (1.23) S(Rn ) → Asp,q (Rn ) → S  (Rn ) for all spaces Asp,q (Rn ), A ∈ {B, F }, as introduced in Definition 1.1, where → indicates a continuous (topological) embedding. Duality   must always be understood in the context of the dual pairing S(Rn ), S  (Rn ) where (1.21) is based on the n well-known fact that S(Rn ) is dense in the related spaces Fp−s  ,q  (R ). In particular, the relation −s n  s n F1,q  (R ) = F∞,q (R ),

1 < q ≤ ∞,

s ∈ R,

1 1 +  = 1, q q

(1.24)

will be helpful for us later on. Remark 1.5. The methods resulting in the spaces Ls∞,q (Rn ), 1 < q ≤ ∞, as s (Rn ), 0 < q ≤ ∞, according described in (1.18) to (1.20) on the one hand and F∞,q to (1.11) and (1.14) on the other hand are rather different. The justification of the first equality in (1.21) for p = ∞ and 1 < q ≤ ∞ relies on the independent proofs of the duality according to the related second equalities in (1.21). The definition s of F∞,q (Rn ) in (1.11) applies to all 0 < q ≤ ∞. As mentioned in the beginning of [FrJ90, Section 5], the motivation for relying on the indicated localization by dyadic shrinking mean values goes back to [FeS72]. On the one hand, it is clear s (Rn ) with 0 < q < 1 have no preduals within the dual that the spaces F∞,q   n  n pairing S(R ), S (R ) , since they are no Banach spaces. On the other hand, the s (Rn ) not yet covered by (1.21) can be incorporated remaining Banach spaces F∞,1 ◦

into the duality approach as follows. Let F sp,q (Rn ) be the completion of S(Rn ) in s s (Rn ) where 0 < p, q ≤ ∞ and s ∈ R. Recall that S(Rn ) is dense in Fp,q (Rn ) Fp,q n s for p < ∞, q < ∞. But if max(p, q) = ∞ then S(R ) is not dense in Fp,q (Rn ). According to [Mar87, Theorem 4, p. 87], one has ◦

n F sp,q (Rn ) = Fp−s  ,q  (R ),

1 ≤ p, q ≤ ∞,

s ∈ R,

1 1 1 1 + = +  = 1. (1.25) p p q q

s This incorporates F∞,q (Rn ) with 1 ≤ q ≤ ∞ in (1.21) and (1.24).

1.1.2 Hybrid spaces This book deals with the (inhomogeneous) global spaces Asp,q (Rn ), including, in s (Rn ) as introduced in Definition 1.1. But as already particular, the spaces F∞,q indicated in the Preface, we rely on (0.1) connecting global and hybrid spaces.

6

Chapter 1. Fundamental principles

In this light, it seems reasonable to recall the definition of hybrid spaces and to describe a few relations and properties which are useful in our context. Let ϕ = {ϕj }∞ j=0 be the dyadic resolution of unity as introduced in (1.3)–(1.5) and let QJ,M be the same dyadic cubes as in (1.6). Recall that a+ = max(a, 0), a ∈ R. Definition 1.6. Let ϕ = {ϕj }∞ j=0 be the above resolution of unity. Let 0 < p, q ≤ ∞ s (Rn ) is the (p < ∞ for F -spaces), s ∈ R and −n/p ≤ r < ∞. Then LrBp,q  n collection of all f ∈ S (R ) such that s (Rn )ϕ = f |LrBp,q

sup

J∈Z,M ∈Zn

n

2J(r+ p )



2jsq



q/p     ϕj f ∨ (x)p dx

1/q

QJ,M

j≥J +

(1.26) s (Rn ) is the is finite (with the usual modification for max(p, q) = ∞), and LrFp,q collection of all f ∈ S  (Rn ) such that s (Rn )ϕ = f |LrFp,q

sup

J∈Z,M ∈Zn

n

2J(r+ p )



 QJ,M

 ∨ q p/q 2jsq  ϕj f (x) dx

1/p

j≥J +

(1.27) is finite (with the usual modification for q = ∞). s (Rn ) or Remark 1.7. Again LrAsp,q (Rn ) with A ∈ {B, F } means either LrBp,q r s n L Fp,q (R ). It follows immediately from Definition 1.1(i,ii) that

Asp,q (Rn ) = L−n/pAsp,q (Rn ),

0 < p ≤ ∞,

0 < q ≤ ∞,

s ∈ R,

(1.28)

with p < ∞ for F -spaces. The above inhomogeneous spaces go back to [YSY10], where the related homogeneous spaces studied in [YaY08, YaY10] are modified. These spaces are denoted there as n r s n As,τ p,q (R ) = L Ap,q (R ),

τ=

r 1 + . p n

(1.29)

We refer the reader to [T14, Definition 3.36, pp. 68–69], where one finds in addition n to the above hybrid spaces also the related local spaces LrAsp,q (Rn ) = As,τ p,q (R ) with j ∈ N0 instead of J ∈ Z in (1.26), (1.27). In some sense, the above spaces are in between these local spaces and the global spaces Asp,q (Rn ), which may justify to call them hybrid spaces. The theory of these spaces as well as numerous modifications based on the above Fourier-analytical definitions was developed in [YSY10, Sic12, Sic13], where one also finds related references. Our own approach to local and hybrid spaces is different: We introduced the spaces LrAsp,q (Rn ) and their local versions LrAsp,q (Rn ) in [T13, T14] via approximation procedures near to related wavelet characterizations, as described below. But according to [T14, Theorem 3.38, Corollary 3.39, pp. 69–70] and [YSY13], these two methods result in the same spaces. In this book, we chose the Fourier-analytical definition because we are mainly interested in the spaces Asp,q (Rn ) according to Definition 1.1. As far

1.2. Wavelet characterizations

7

as a systematic study of the spaces LrAsp,q (Rn ) is concerned, we refer the reader to the above-mentioned literature. We rely mainly on [T14]. For the following result, recall again that σ C σ (Rn ) = B∞,∞ (Rn ),

σ ∈ R,

(1.30)

are the Hölder-Zygmund spaces. Proposition 1.8. Let 0 < p, q ≤ ∞ (p < ∞ for F -spaces), s ∈ R and −n/p ≤ r < ∞. Then the spaces LrAsp,q (Rn ) according to Definition 1.6 and their local counterparts LrAsp,q (Rn ) are quasi-Banach spaces. They are independent of the chosen resolution of unity ϕ = {ϕj }∞ j=0 (equivalent quasi-norms) and S(Rn ) → LrAsp,q (Rn ) → LrAsp,q (Rn ) → C s+r (Rn ) → S  (Rn ). Furthermore,

L−n/p Asp,q (Rn ) = Asp,q (Rn )

(1.31) (1.32)

and s s s F∞,q (Rn ) = L0 Fq,q (Rn ) = L0 Bq,q (Rn ),

q < ∞.

(1.33)

Proof. The equality (1.32) which was already mentioned in (1.28) follows from the Definitions 1.1(i,ii) and 1.6. The continuous embeddings in (1.31) are covered by [T14, Theorem 3.20, p. 60], with a reference to [T13, Corollary 1.38, p. 32] as far as the local spaces LrAsp,q (Rn ) are concerned. The already mentioned Fourieranalytical reformulations follow according to [T14, Theorem 3.38, Corollary 3.39, pp. 69–70]. Furthermore, (1.33) follows from (1.11), (1.26) and (1.27) with p = q < ∞. 

1.2 Wavelet characterizations 1.2.1 Global spaces We mainly rely on wavelet characterizations both of the global spaces Asp,q (Rn ) and of their hybrid generalizations LrAsp,q (Rn ). We give a description following [T14, Section 3.2.3, pp. 51–54]. For this, we suppose that the reader is familiar with Daubechies-type wavelets in Rn and the related multiresolution analysis. The standard references are [Dau92, Mal99, Mey92, Woj97]. A short summary of what is needed may also be found in [T06, Section 1.7]. As usual, C u (R) with u ∈ N collects all bounded complex-valued continuous functions on R having continuous bounded derivatives up to order u inclusively. Let ψF ∈ C u (R),

ψM ∈ C u (R),

u ∈ N,

be real compactly supported Daubechies wavelets with  ψM (x) xv dx = 0 for all v ∈ N0 with v < u. R

(1.34)

(1.35)

8

Chapter 1. Fundamental principles

Recall that ψF is called the scaling function (father wavelet) and ψM the associated wavelet (mother wavelet). We extend these wavelets from R to Rn by the usual multiresolution procedure. Let n ∈ N, and let G = (G1 , . . . , Gn ) ∈ G0 = {F, M }n ,

(1.36)

which means that Gr is either F or M . Furthermore, let G = (G1 , . . . , Gn ) ∈ G∗ = Gj = {F, M }n∗ ,

j ∈ N,

(1.37)

which means that Gr is either F or M , where ∗ indicates that at least one of the components of G must be an M . Hence G0 has 2n elements, whereas Gj with j ∈ N and G∗ have 2n − 1 elements. Let j (x) = ψG,m

n 

  ψ G l 2 j x l − ml ,

G ∈ Gj ,

m ∈ Zn ,

x ∈ Rn ,

(1.38)

l=1

where (now) j ∈ N0 . We always assume that ψF and ψM in (1.34) have an L2 -norm of 1. Then  jn/2 j  2 ψG,m : j ∈ N0 , G ∈ Gj , m ∈ Zn (1.39) is an orthonormal basis in L2 (Rn ) (for any u ∈ N) and f=

∞

j λj,G m ψG,m

(1.40)

j=0 G∈Gj m∈Zn

with j,G jn λj,G m = λm (f ) = 2

 Rn

  j j f (x) ψG,m (x) dx = 2jn f, ψG,m

(1.41)

is the corresponding expansion. Let χj,m be the characteristic function of the cube Qj,m = 2−j m + 2−j (0, 1)n according to (1.6). Definition 1.9. Let   j n λ = λj,G . m ∈ C : j ∈ N0 , G ∈ G , m ∈ Z (i) Let 0 < p, q ≤ ∞ (p < ∞ for the f -spaces) and s ∈ R. Then   bsp,q (Rn ) = λ : λ |bsp,q (Rn ) < ∞

(1.42)

(1.43)

with λ |bsp,q (Rn ) =

∞  j=0

n

2j(s− p )q

 G∈Gj

p |λj,G m |

q/p 1/q

(1.44)

m∈Zn

and   s s fp,q (Rn ) = λ : λ |fp,q (Rn ) < ∞

(1.45)

1.2. Wavelet characterizations

9

with



s λ |fp,q (Rn ) =



q 1/q    Lp (Rn )

2jsq λj,G

m χj,m (·)



(1.46)

j∈N0 ,G∈Gj , m∈Zn

(with the usual modifications for max(p, q) = ∞). (ii) Let 0 < q ≤ ∞ and s ∈ R. Then   s s f∞,q (Rn ) = λ : λ |f∞,q (Rn ) < ∞

(1.47)

with s (Rn ) = λ |f∞,q

sup

J∈N0 ,M ∈Zn



q 2jsq−(j−J)n |λj,G m |

1/q (1.48)

j≥J,G∈Gj , m:Qj,m ⊂QJ,M

s (Rn ) = bs∞,∞ (Rn ) for q = ∞). (with f∞,∞

Remark 1.10. The above definition coincides with [T14, Definition 3.10, p. 53] s with exception of f∞,q (Rn ). Recall that we pay special attention to the spaces s n F∞,q (R ). In particular, part (ii) is the sequence counterpart of Definition 1.1(iii) and (1.14). Let n ∈ N and   1  σp(n) = n max , 1 − 1 , p

  1 1  (n) σp,q = n max , , 1 − 1 , p q

(1.49)

where 0 < p, q ≤ ∞. Proposition 1.11.

(i) Let 0 < p ≤ ∞, 0 < q ≤ ∞, s ∈ R and u > max(s, σp(n) − s).

(1.50)

s (Rn ) if, and only if, it can be represented as Let f ∈ S  (Rn ). Then f ∈ Bp,q

f=



j λj,G m ψG,m ,

λ ∈ bsp,q (Rn ),

(1.51)

j∈N0 ,G∈Gj , m∈Zn

the unconditional convergence being in S  (Rn ). The representation (1.51) is unique,   j j,G jn λj,G f, ψG,m , (1.52) m = λm (f ) = 2 and I:

  f → λj,G m (f )

s (Rn ) onto bsp,q (Rn ). is an isomorphic map of Bp,q

(1.53)

10

Chapter 1. Fundamental principles

(ii) Let 0 < p ≤ ∞, 0 < q ≤ ∞, s ∈ R and (n) − s). u > max(s, σp,q

(1.54)

s (Rn ) if, and only if, it can be represented as Let f ∈ S  (Rn ). Then f ∈ Fp,q

f=



j λj,G m ψG,m ,

s λ ∈ fp,q (Rn ),

(1.55)

j

j∈N0 ,G∈G , m∈Zn

the unconditional convergence being in S  (Rn ). The representation (1.55) is unique with (1.52). Furthermore, I in (1.53) is an isomorphic map of s s Fp,q (Rn ) onto fp,q (Rn ). s (Rn ) and Remark 1.12. This is the wavelet characterization of the spaces Bp,q s n Fp,q (R ) as introduced in Definition 1.1. We refer the reader again to [T14, Section 3.2.3, pp. 51–54] where one finds further explanations and references, and to s Corollary 1.21 below as far as the spaces F∞,q (Rn ) according to Definition 1.1(iii) are concerned.

1.2.2 Hybrid spaces Let again χj,m be the characteristic function of the cube Qj,m = 2−j m+2−j (0, 1)n according to (1.6). Let Gj with j ∈ N0 be as in (1.36), (1.37). Let J + = max(J, 0) if J ∈ Z and   PJ,M = j ∈ N0 , j ≥ J + , G ∈ Gj , m ∈ Zn : Qj,m ⊂ QJ,M , J ∈ Z, M ∈ Zn . (1.56) Definition 1.13. Let 0 < p, q ≤ ∞, s ∈ R and r ≥ −n/p. Let   j n . λ = λj,G m ∈ C : j ∈ N0 , G ∈ G , m ∈ Z

(1.57)

Then   Lr bsp,q (Rn ) = λ : λ |Lr bsp,q (Rn ) < ∞

(1.58)

with λ |Lr bsp,q (Rn ) =

sup

J∈Z,M ∈Zn

n

2J(r+ p )

∞ j=J +

n

2j(s− p )q





p |λj,G m |

q/p 1/q

(1.59)

m:(j,m)∈PJ,M , G∈Gj

and   s s (Rn ) = λ : λ |Lrfp,q (Rn ) < ∞ Lrfp,q

(1.60)

1.2. Wavelet characterizations

11

with s λ |Lrfp,q (Rn )

=

sup

J∈Z,M ∈Zn

 n

2J(r+ p )



 q 1/q  n

  L 2jsq λj,G χ (·) (R )

j,m p m



(1.61)

(j,G,m)∈PJ,M

(with the usual modification for max(p, q) = ∞). Remark 1.14. This coincides with [T14, Definition 3.24, p. 63]. It is the hybrid counterpart of Definition 1.9(i). s (Rn ) in Definition 1.9(ii) are concerned, one has the As far as the spaces f∞,q following assertion. s s Proposition 1.15. Let s ∈ R and 0 < p, q < ∞. Let f∞,q (Rn ) and L0 fp,q (Rn ) be the spaces as introduced in the Definitions 1.9(ii) and 1.13. Then s s s L0 fp,q (Rn ) = L0 fq,q (Rn ) = L0 bsq,q (Rn ) = f∞,q (Rn )

(1.62)

(equivalent quasi-norms). Proof. It is one of the crucial observations in [FrJ90, Corollary 5.7, pp. 75, 133] that the sequence spaces in (1.61) with r = 0, 0 < q ≤ ∞ and 0 < p < ∞ are independent of p (equivalent quasi-norms). This justifies the first equality in (1.62). The second equality follows from (1.59) compared with (1.61). It remains to prove the last equality. Terms with J ∈ N0 , r = 0 and p = q in (1.59) coincide with (1.48). Let −J ∈ N. Then a cube QJ,M with side length 2|J| can be decomposed into 2|J|n cubes Q0,K with side length 1, and the related terms in (1.59) with r = 0 and p = q can be estimated from above via the quasi-norm in (1.48).  (n)

(n)

j Let σp and σp,q be as in (1.49). Let r+ = max(r, 0) if r ∈ R and let ψG,m be the same wavelets as in (1.38) based on (1.34).

Proposition 1.16. Let

(i) Let 0 < p ≤ ∞, 0 < q ≤ ∞, s ∈ R and −n/p ≤ r < ∞. u > max(s + r+ , σp(n) − s).

(1.63)

s (Rn ) if, and only if, it can be represented Let f ∈ S  (Rn ). Then f ∈ Lr Bp,q as j f= λj,G λ ∈ Lr bsp,q (Rn ), (1.64) m ψG,m , j∈N0 ,G∈Gj , m∈Zn

the unconditional convergence being in S  (Rn ). The representation (1.64) is unique,   j j,G jn f, ψG,m , (1.65) λj,G m = λm (f ) = 2 and I: is an isomorphic map of L

r

  f → λj,G m (f )

s Bp,q (Rn )

onto

Lr bsp,q (Rn ).

(1.66)

12

Chapter 1. Fundamental principles

(ii) Let 0 < p < ∞, 0 < q ≤ ∞, s ∈ R and −n/p ≤ r < ∞. Let (n) u > max(s + r+ , σp,q − s).

(1.67)

s Let f ∈ S  (Rn ). Then f ∈ Lr Fp,q (Rn ) if, and only if, it can be represented as j s f= λj,G λ ∈ Lrfp,q (Rn ), (1.68) m ψG,m , j∈N0 ,G∈Gj , m∈Zn

the unconditional convergence being in S  (Rn ). The representation (1.68) is unique with (1.65). Furthermore, I in (1.66) is an isomorphic map of s s (Rn ) onto Lrfp,q (Rn ). Lr Fp,q Remark 1.17. This result essentially coincides with [T14, Theorem 3.26, p. 64] adapted to the formulation of Proposition 1.11. The proof relies on a counterpart for the local spaces LrAsp,q (Rn ) according to [T13, Theorem 1.32, p. 24]. For r = −n/p, one has (1.32) with p < ∞ for F -spaces. In this case, the related sequence spaces in the Definitions 1.9, 1.13 and also the wavelet representations in s the Propositions 1.11, 1.16 coincide. Using (1.33) one can incorporate F∞,q (Rn ) in Proposition 1.11 based on (1.64) and the related sequence space L0 bsq,q (Rn ) according to (1.59). We will return to this point in Corollary 1.21 below. Otherwise we are looking for a stronger version as already indicated in (0.1).

1.3 Basic assertions 1.3.1 Relations between global and hybrid spaces We do not deal with the hybrid spaces LrAsp,q (Rn ) for their own sake. Compared to [T14], we have nothing new to say about them. But we collect a few properties which show how global and hybrid spaces are related to each other. Let again Asp,q (Rn ) and LrAsp,q (Rn ) with A ∈ {B, F } be the global and hybrid spaces as introduced in the Definitions 1.1, 1.6, complemented by bmo(Rn ) according to (1.12). Let again C s (Rn ) be the same Hölder-Zygmund spaces as in (1.16). Recall that → indicates a continuous (topological) embedding. A strict embedding is a (continuous) embedding where the related spaces do not coincide. Proposition 1.18. s ∈ R. Then

(i) Let 0 < p ≤ ∞ (p < ∞ for F -spaces), 0 < q ≤ ∞ and L−n/p Asp,q (Rn ) = Asp,q (Rn ).

(1.69)

(ii) Let 0 < p ≤ ∞ (p < ∞ for F-spaces), 0 < q ≤ ∞, s ∈ R and r > 0. Then LrAsp,q (Rn ) = C s+r (Rn ).

(1.70)

1.3. Basic assertions

13

(iii) Let 0 < p < ∞, 0 < q ≤ ∞ and s ∈ R. Then s s (Rn ) = F∞,q (Rn ) L0 Fp,q

(1.71)

s s F∞,∞ (Rn ) = B∞,∞ (Rn ) = C s (Rn ).

(1.72)

with (iv) Let 0 < p < ∞. Then 0 0 (Rn ) = L0 Fp,2 (Rn ). bmo(Rn ) = F∞,2

(1.73)

(v) Let 0 < p < ∞ and s ∈ R. Then s (Rn ) = C s (Rn ). L0 Bp,∞

(1.74)

Let in addition 0 < q < ∞. Then s s (Rn ) → L0 Bp,q (Rn ) B∞,q

(1.75)

is a strict embedding. Proof. Step 1. Part (i) coincides with (1.32) and (1.28), respectively (included here for sake of completeness). For r > 0, the embedding (1.31) for all r with r ≥ −n/p can be strengthened by (1.70). This is covered by [T14, Proposition 3.54, Remark 3.55, pp. 92–93] with a reference to [T13, Theorem 2.1, p. 45] and [YaY13, Theorem 2, p. 560]. Step 2. For the proof of part (iii), we use the wavelet characterization according to Proposition 1.16(ii) with r = 0 in (1.68) and (1.61). This yields (1.71) with q < ∞ from (1.33) and the Propositions 1.15, 1.16(ii). Let now q = ∞ and p < ∞. Then it follows from (1.26), (1.27) and (1.8) that s s f |L0 Bp,∞ (Rn ) ≤ f |L0 Fp,∞ (Rn )   s ≤ sup 2js (ϕj f)∨ (x) = f |B∞,∞ (Rn ).

(1.76)

j∈N0 ,x∈Rn

As for the converse, we begin with a preparation. Let ϕ, ψ ∈ S(Rn ),

supp ϕ  ⊂ {ξ : |ξ| ≤ 1},

Then  ∨ (x) = c ϕ(x) = (ϕ ψ)



 = 1 if |ξ| ≤ 1. ψ(ξ)

ϕ(y) ψ(x − y) dy.

(1.77)

(1.78)

Rn

With QJ,M = 2−J M + 2−J (0, 1)n , M ∈ Zn , J ∈ Z as above, one obtains from the rapid decay of ψ that  |ϕ(y)| dy. (1.79) |ϕ(x)| ≤ c sup M ∈Zn

Q0,M

14

Chapter 1. Fundamental principles

If 1 ≤ p < ∞ then one has by Hölder’s inequality  1/p |ϕ(y)|p dy . sup |ϕ(x)| ≤ c sup M ∈Zn

x∈Rn

(1.80)

Q0,M

If 0 < p < 1 then (1.80) is a consequence of sup |ϕ(x)| ≤ c sup |ϕ(x)|1−p · sup

x∈Rn



M ∈Zn

x∈Rn

|ϕ(y)|p dx.

(1.81)

Q0,M

One can extend (1.80) to f ∈ S  (Rn ) with supp f ⊂ {ξ : |ξ| ≤ 1} by using the Paley-Wiener-Schwartz theorem as described in [T83, p. 22]. Let fJ ∈ S  (Rn ) J −J x) with supp f J ⊂ {ξ : |ξ| ≤ 2 }, J ∈ N0 . Then one can apply (1.80) with fJ (2 instead of ϕ, which results in 1/p  Jn/p |fJ (y)|p dy , 0 < p < ∞. (1.82) sup |fJ (x)| ≤ c sup 2 M ∈Zn

x∈Rn

QJ,M

By inserting fJ (x) = (ϕJ f)∨ (x), the converse of (1.76) follows from (1.26). This proves (1.71) with q = ∞, (1.72) using (1.16), and (1.74). Step 3. Part (iv) is covered by (1.22) combined with (1.71). Step 4. The proof of (1.75) with s = 0 and 0 < q < ∞ relies on the wavelet expansions (1.51), (1.52) and (1.64), (1.65) with ∞ 1/q  0 q (Rn ) ∼ sup |λj,G (1.83) f |B∞,q m | n j=0 G∈Gj m∈Z

and 0 (Rn ) f |L0 Bp,q

∼

sup

J∈Z,M ∈Zn

2Jn/p

∞

2−jnq/p

j=J + G∈Gj





p |λj,G m |

q/p 1/q

.

(1.84)

m:(j,m)∈PJ,M

Then the embedding (1.75) follows from  q/p p q |λj,G | ≤ c 2(j−J)qn/p sup |λj,G m m | .

(1.85)

m∈Zn

m:(j,m)∈PJ,M

For any j ∈ N0 and G ∈ Gj , let =1 λj,G 0

and

λj,G m =0

if

m = 0.

(1.86)

Inserted in (1.51), (1.83) and (1.64), (1.84), this leads to 0 (Rn ) f ∈ B∞,q

and

0 f ∈ L0 Bp,q (Rn ).

(1.87)

This shows that the two spaces in (1.75) do not coincide if s = 0. These same arguments can be extended of all s ∈ R, which also follows from the lifting assertions in Theorem 1.22 and (1.96). 

1.3. Basic assertions

15

Remark 1.19. The crucial assertion (1.71) = (0.1) may also be found in [YSY10, Proposition 2.4(iii), p. 41], with a reference to [YaY10] as far as the homogeneous counterpart is concerned. But the indicated reduction to [FrJ90] as in the proof of Proposition 1.15 requires greater care. Additionally, the case q = ∞ must be considered separately. The independence of p in (1.71) and in particular in (1.73) and its homogeneous version BMO(Rn ) has a little history. Firstly, it may be considered as a sequence version of a corresponding independence related to BMOnorms as discussed in [Ste93, Corollary, p. 144] as a generalization of the related assertions in [JoN61]. Secondly, if one relies on [T14, Definition 3.18, p. 58] for 0 (Rn ), the spaces LrAsp,q (Rn ) and the Littlewood-Paley assertion Lp (Rn ) = Fp,2 1 < p < ∞, then one can complement (1.73) by 0 0 bmo(Rn ) = F∞,2 (Rn ) = L0 F∞,2 (Rn ) = L0 Lp (Rn ),

1 < p < ∞,

(1.88)

where bmo(Rn ) = L0 L2 (Rn )

(1.89)

n

(and its homogeneous counterpart for BMO(R )) goes back to [Mey92, Theorem 4, p. 154], where it is proved with the help of wavelet arguments. For further discussions we refer the reader to [T13, pp. 110–111] and [T14, Remark 3.59, p. 94]. As far as assertions of the type (1.74) and the strict embedding in (1.75) are concerned, one may also consult [YaY13] and the references given therein. Remark 1.20. The proof of (1.76) and its inverse show that for s ∈ R and 0 < p < ∞, s s f |L0Bp,∞ (Rn ) ∼ f |L0Fp,∞ (Rn ) s ∼ f |B∞,∞ (Rn )

∼

sup

J∈N0 ;M ∈Zn

J(s+ n p)

2



1/p   (ϕJ f)∨ (x)p dx ,

(1.90)

QJ,M

where again the terms with −J ∈ N can be incorporated as in (1.13), (1.14). This makes clear that (1.72), if based on (1.71) and (1.74), is not a notational agreement (as occasionally used in the literature, including our own papers and books) but an assertion. It shows that the spaces C s (Rn ), s ∈ R, fit in the scale of both the global spaces and the hybrid spaces (and also of the local spaces LrAsp,q (Rn ) defined as in (1.26), (1.27) with J ∈ N0 instead of J ∈ Z), L0 Asp,∞ (Rn ) = L0 Asp,∞ (Rn ) = As∞,∞ (Rn ) = C s (Rn ),

s ∈ R,

0 < p < ∞, (1.91) A ∈ {B, F }. This observation is not new. A modified version of its homogeneous counterpart for p = 1 and related spaces with Muckenhoupt weights may be found in [BuT00, Theorem 3, p. 541]. Furthermore, the converse of (1.76) also follows from (1.31) with r = 0 based on wavelets arguments. We justified (1.91) already in [T13, Remark 2.3, pp. 47–48] and [T14, Remark 3.59, p. 94]. On the other hand, s the strict embedding (1.75) shows that the global spaces B∞,q (Rn ) cannot be incorporated into the scale of the hybrid spaces.

16

Chapter 1. Fundamental principles

s (Rn ) in Finally, we justify the wavelet characterization for the spaces F∞,q s n Proposition 1.11(ii). For that purpose, let f∞,q (R ) with 0 < q ≤ ∞ be the sequence spaces as introduced in Definition 1.9(ii).

Corollary 1.21. Let 0 < q ≤ ∞, s ∈ R and u > max(s, σq(n) − s).

(1.92)

s (Rn ) if, and only if, Let f ∈ S  (Rn ). Then f ∈ F∞,q j s f= λj,G λ ∈ f∞,q (Rn ), m ψG,m ,

(1.93)

j∈N0 ,G∈Gj , m∈Zn

the unconditional convergence being in S  (Rn ). The representation (1.93) is unique s (Rn ) onto with (1.52). Furthermore, I in (1.53) is an isomorphic map of F∞,q s n f∞,q (R ). Proof. The case q = ∞ is covered by (1.72), Definition 1.9(ii) and Proposition s 1.11(i) applied to B∞,∞ (Rn ). If q < ∞ then the above assertion follows from (1.71), Proposition 1.16(ii) with p ≥ q and Proposition 1.15. 

1.3.2 Lifts It is well-known that the classical lifts  ∨ Iδ : f → ξ−δ f with ξ = (1 + |ξ|2 )1/2 ,

ξ ∈ Rn ,

δ ∈ R,

(1.94)

n map the global spaces Asp,q (Rn ) isomorphically onto As+δ p,q (R ) according to Definition 1.1(i,ii) with A ∈ {B, F }, 0 < p ≤ ∞ (p < ∞ for the F -spaces), 0 < q ≤ ∞ and s ∈ R: n Iδ Asp,q (Rn ) = As+δ δ ∈ R, (1.95) p,q (R ),

[T83, Section 2.3.8, pp. 58–59]. We extended this observation in [T14, Theorem 3.72, p. 102] to all hybrid spaces LrAsp,q (Rn ) according to Definition 1.6 with A ∈ {B, F }, 0 < p ≤ ∞ (p < ∞ for F -spaces), 0 < q ≤ ∞, s ∈ R, and −n/p ≤ r < ∞: n Iδ LrAsp,q (Rn ) = LrAs+δ p,q (R ),

δ ∈ R.

(1.96)

This applies in particular to the related spaces in (1.71). Theorem 1.22. Let Asp,q (Rn ) with A ∈ {B, F }, 0 < p ≤ ∞, 0 < q ≤ ∞ and s ∈ R be the global spaces according to Definition 1.1. Then Iδ maps Asp,q (Rn ) n isomorphically onto As+δ p,q (R ): n Iδ Asp,q (Rn ) = As+δ p,q (R ).

(1.97)

s (Rn ) Proof. If one applies (1.96) to (1.71), one can incorporate the spaces F∞,q into the above-mentioned classical assertion. 

1.3. Basic assertions

17

Remark 1.23. As a rule of thumb, assertions known so far for the spaces Asp,q (Rn ) s (Rn ). The with p < ∞ for the F -spaces can now be extended to the spaces F∞,q above theorem is a somewhat simple example. But it is not clear how to extend the s s corresponding proof in [T83, p. 59] for the spaces Fp,q (Rn ) with p < ∞ to F∞,q (Rn ) according to (1.11). The justification of (1.96) in [T14, p. 102] with a reference to [T13, pp. 88–89] relies on already known atomic and wavelet characterizations. The lifted version of (1.73) will be of some interest later on. This justifies the introduction of the notation s bmos (Rn ) = Is bmo(Rn ) = F∞,2 (Rn ),

s ∈ R.

(1.98)

1.3.3 Distinguished representations Let again QJ,M = 2−J M + 2−J (0, 1)n , where J ∈ Z and M ∈ Zn . Let d QJ,M be a cube that concentrically contains QJ,M has a side length d 2−J , d > 0. Let 0 < p < ∞, 0 < q ≤ ∞ and s < 0. Then the first equivalence in s (Rn ) ∼ f |F∞,q

∼

sup

J∈Z,M ∈Zn

s 2Jn/p f |Fp,q (2QJ,M )

sup

J∈N0 ,M ∈Zn

s 2Jn/p f |Fp,q (2QJ,M )

(1.99)

is covered by (1.71) combined with [T14, Theorem 3.64, p. 97] and [T13, Theorem 2.29, p. 75]. For −J ∈ N, [T14, (3.294), p. 94] provides s 2Jn/p f |Fp,q (2QJ,M ) ≤ c

sup

2Q0,L ⊂4QJ,M

s f |Fp,q (2Q0,L ).

(1.100)

This proves the second equivalence in (1.99). Let 0 < p < ∞, 0 < q ≤ ∞, s ∈ R and s < m ∈ N0 . If one combines the above arguments with [T14, Corollary 3.66, p. 98], one obtains s f |F∞,q (Rn ) ∼

sup J∈N0 ,M ∈Zn , 0≤|α|≤m

s−m 2Jn/p Dα f |Fp,q (2QJ,M ).

(1.101)

This observation is interesting in itself. It is in good agreement with a corresponding reduction of J ∈ Z to J ∈ N0 in (1.11), (1.14). But it can also be used to justify the following assertion. Theorem 1.24. Let Asp,q (Rn ) with A ∈ {B, F }, 0 < p ≤ ∞, 0 < q ≤ ∞ and s ∈ R be the global spaces according to Definition 1.1. Let m ∈ N0 . Then f |Asp,q (Rn ) ∼

sup 0≤|α|≤m

(equivalent quasi-norms).

n Dα f |As−m p,q (R )

(1.102)

18

Chapter 1. Fundamental principles

Proof. If p < ∞ for the F -spaces then (1.102) is a very classical assertion for the related spaces Asp,q (Rn ), see [T83, Theorem 2.3.8, pp. 58–59]. It remains to extend s (Rn ). With Iδ as in (1.94), one has this property to F∞,q I δ D α f = D α Iδ f

for

f ∈ S  (Rn ),

δ ∈ R,

α ∈ Nn0 .

(1.103)

Then it follows from Theorem 1.22 as well as from (1.99), (1.101) with s + δ < m that s s+δ f |F∞,q (Rn ) ∼ Iδ f |F∞,q (Rn ) ∼

sup 0≤|α|≤m

∼

s+δ−m Dα Iδ f |F∞,q (Rn )

D f α

sup 0≤|α|≤m

(1.104)

s−m |F∞,q (Rn ).

s (Rn ) as well. This proves (1.102) for the spaces F∞,q



1.3.4 Fatou property Let A(Rn ) be a quasi-normed space in S  (Rn ) with A(Rn ) → S  (Rn ) (continuous embedding). Then A(Rn ) is said to have the Fatou property if there is a positive constant c such that from sup gj |A(Rn ) < ∞

and

gj → g in S  (Rn )

(1.105)

j∈N

it follows that g ∈ A(Rn ) and g |A(Rn ) ≤ c sup gj |A(Rn ).

(1.106)

j∈N

Theorem 1.25. All spaces Asp,q (Rn ) with A ∈ {B, F }, s ∈ R and 0 < p, q ≤ ∞ according to Definition 1.1 have the Fatou property. Proof. Let ψ ∈ S(Rn ). Then   (ψ f)∨ (x) = c f, ψ ∨ (x − ·) ,

f ∈ S  (Rn ),

x ∈ Rn .

(1.107)

This reduces the Fatou property for the Fourier-analytically defined spaces Asp,q (Rn ) according to (1.8), (1.10), (1.11) to the classical measure-theoretical Fa tou property for Lp -spaces. Remark 1.26. The first detailed discussion of the Fatou property was given in [Fra86], including the notation Fatou property in the context of distributions. But this remarkable property had already been used before, at least implicitly, for example in [Pee75, Pee76]. It proved to be very useful, especially in connection with mapping properties of (more or less singular) linear operators. A recent discussion of the Fatou property in some distinguished spaces in the context of tempered distributions – including several applications – can be found in [Tri17c]. Note that s this discussion does not cover the spaces F∞,q (Rn ). But they can be incorporated quite easily.

1.3. Basic assertions

19

Remark 1.27. According to Proposition 1.18, the spaces Asp,q (Rn ) are special cases of the hybrid spaces as introduced in Definition 1.6. Applying the above arguments shows that all spaces LrAsp,q (Rn ) have the Fatou property. This was already observed in [YSY10, Section 2.3, p. 48].

Chapter 2

The essentials, key theorems 2.1 Embeddings 2.1.1 Embeddings in L∞ (Rn ) and C(Rn ) Embeddings between function spaces as introduced in Definition 1.1 and into other distinguished subspaces of S  (Rn ) play a crucial role in the theory of function spaces. In this chapter, we wish to extend the already existing assertions for the s spaces Asp,q (Rn ) with p < ∞ for the F -spaces to F∞,q (Rn ). First we deal with embeddings into L∞ (Rn ) and C(Rn ) as target spaces. We always assume f (x) = 0 if x ∈ Rn is not a Lebesgue point of the locally Lebesgue-integrable function f (x) in Rn . Then f |L∞ (Rn ) = sup |f (x)| (2.1) x∈Rn

is the norm in the usual space L∞ (Rn ) of (essentially) bounded complex-valued functions in Rn . Let C(Rn ) be the space of all continuous bounded complexvalued functions in Rn normed by (2.1). Both L∞ (Rn ) and C(Rn ) are considered as subspaces of S  (Rn ). As usual, → indicates a continuous embedding. We begin with a preparation. Proposition 2.1. Let 0 < p < ∞, 0 < q ≤ ∞ and s ∈ R. Then s+ n

s (Rn ). Bp,∞p (Rn ) → F∞,q

(2.2)

Proof. By Theorem 1.22 we may assume s = 0. We use the wavelet expansion for n/p f ∈ Bp,∞ (Rn ) according to Proposition 1.11(i). Then one has in particular n/p (Rn ) ∼ f |Bp,∞

sup j∈N0 ,G∈Gj



p |λj,G m |

1/p

,

0 < p < ∞.

(2.3)

m∈Zn

© Springer Nature Switzerland AG 2020 H. Triebel, Theory of Function Spaces IV, Monographs in Mathematics 107, https://doi.org/10.1007/978-3-030-35891-4_2

21

22

Chapter 2. The essentials, key theorems

0 (Rn ) from (1.71) and Proposition 1.16(ii) Similarly, one obtains for f ∈ F∞,q 0 (Rn ) ∼ f |F∞,q

sup

J∈Z,M ∈Zn



2Jn/p

Qj,m ⊂QJ,M , j∈N0 ,G∈Gj

 1/q  j,G

λm χj,m (·)q |Lp (Rn ) .

(2.4) We assume 0 < q < p < ∞. For fixed j, the cubes Qj,m are disjoint. Then it follows from the triangle inequality applied to Lp/q (Rn ) that





Qj,m ⊂QJ,M , j∈N0 ,G∈Gj

≤



≤



 1/q  j,G

λm χj,m (·)q |Lp (Rn )





j≥J + ,G∈Gj

m:Qj,m ⊂QJ,M

2−jnq/p





1/q   j,G λm χj,m (·)q |Lp/q (Rn )



 j,G p q/p λm 

(2.5)

1/q

m:Qj,m ⊂QJ,M

j≥J + ,G∈Gj

n/p ≤ c 2−Jn/p f |Bp,∞ (Rn ),

where we used (2.3). By inserting this result into (2.4), one obtains 0 n/p (Rn ) ≤ c f |Bp,∞ (Rn ), f |F∞,q

n/p f ∈ Bp,∞ (Rn ).

This proves (2.2).

(2.6) 

Remark 2.2. The embedding (2.2) goes back to [Mar95, Lemma 16, p. 253], where a corresponding assertion for q ≥ 1 in [Mar87, Corollary 4, p. 88] was extended to 0 < q ≤ ∞. The proof in [Mar95] relies on the Fourier-analytical definitions of the involved spaces according to (1.8) and (1.11). It is even shorter. This can be taken as a counterexample to the rule of thumb that wavelet arguments shorten the proofs based on other means substantially. The above proposition, including (2.4), is interesting in itself. But it can also be used to justify the following assertion. Theorem 2.3. Let Asp,q (Rn ) with A ∈ {B, F }, s ∈ R, 0 < p ≤ ∞ and 0 < q ≤ ∞ be the spaces according to Definition 1.1. Then s (Rn ) → L∞ (Rn ) Bp,q

(2.7)

if, and only if, 

either or

s> s=

n p, n p,

0 < q ≤ ∞, 0 < q ≤ 1,

(2.8)

and s (Rn ) → L∞ (Rn ) Fp,q

(2.9)

2.1. Embeddings

23

if, and only if, 

either or

s> s=

n p, n p,

0 < q ≤ ∞, 0 < p ≤ 1, 0 < q ≤ ∞.

(2.10)

Furthermore, one can replace L∞ (Rn ) in (2.7) and (2.9) by C(Rn ). Proof. Under the additional assumption p < ∞ for the F -spaces, the above assertions are well known. They are covered by [SiT95, Theorem 3.3.1, p. 113] and were repeated in [ET96, pp. 44–45] and [T08, p. 229], including some references. s (Rn ) are monotonous with respect to q. This follows For fixed s, the spaces F∞,q from (1.71) and Proposition 1.16(ii) based on (1.61). Then one has s s F∞,q (Rn ) → B∞,∞ (Rn ) → C(Rn ) → L∞ (Rn ),

s > 0.

(2.11)

If one assumes 0 < q ≤ ∞,

0 F∞,q (Rn ) → L∞ (Rn ),

(2.12)

n/p

one obtains by (2.2) a corresponding embedding for Bp,∞ (Rn ). But this contradicts s (Rn ) are also monotonous with respect to (2.7), (2.8). By (1.14), the spaces F∞,q s – which ensures that there is no embedding (2.9) for s < 0 either. This proves the theorem for both L∞ (Rn ) and C(Rn ) as target space. 

2.1.2 Embeddings in the space of locally integrable functions Let again A(Rn ) be a quasi-Banach space with S(Rn ) → A(Rn ) → S  (Rn ).

(2.13)

n Recall that Lloc 1 (R ) consists of all Lebesgue-measurable (complex-valued) funcn tions in R which are integrable on any bounded domain in Rn . Then n A(Rn ) ⊂ Lloc 1 (R )

(2.14)

means that any f ∈ A(Rn ) can be represented as a locally integrable function such that  f (x) ϕ(x) dx,

f (ϕ) = Rn

Recall that

ϕ ∈ D(Rn ) = C0∞ (Rn ).

  1  σp(n) = n max , 1 − 1 , p

0 < p ≤ ∞.

(2.15)

(2.16)

Theorem 2.4. Let Asp,q (Rn ) with A ∈ {B, F }, s ∈ R, 0 < p ≤ ∞, 0 < q ≤ ∞ be the spaces according Definition 1.1. Then s n Bp,q (Rn ) ⊂ Lloc 1 (R )

(2.17)

24

Chapter 2. The essentials, key theorems

if, and only if, ⎧ ⎪ ⎨either or ⎪ ⎩ or

(n)

0 < p ≤ ∞, s > σp , 0 < q ≤ ∞, (n) 0 < p ≤ 1, s = σp , 0 < q ≤ 1, 1 < p ≤ ∞, s = 0, 0 < q ≤ min(p, 2),

(2.18)

and s n (Rn ) ⊂ Lloc Fp,q 1 (R )

(2.19)

if, and only if, ⎧ ⎪ ⎨either or ⎪ ⎩ or

(n)

0 < p < 1, s ≥ σp , 0 < q ≤ ∞, 1 ≤ p ≤ ∞, s > 0, 0 < q ≤ ∞, 1 ≤ p ≤ ∞, s = 0, 0 < q ≤ 2.

(2.20)

Proof. The above assertion for the spaces Asp,q (Rn ) with the additional assumption p < ∞ for the F -spaces goes back to [SiT95, Theorem 3.3.2, p. 114] and may also be found in [T01, Theorem 11.2, pp. 168–169]. It remains to justify (2.20) for the s (Rn ). From (1.71) as well as the Proposition 1.16 and (1.61), it follows spaces F∞,q that −∞ < s1 < s0 < ∞,

s0 s1 s1 B∞,∞ (Rn ) → B∞,∞ (Rn ), (Rn ) → F∞,q

0 < q ≤ ∞. (2.21)

Using (2.17) with (2.18), it remains to show that 0 n F∞,q (Rn ) ⊂ Lloc 1 (R )

if, and only if, 0 < q ≤ 2.

(2.22)

The if-part follows from (1.12), (1.73) and the monotonicity with respect to q, 0 0 n F∞,q (Rn ) → F∞,2 (Rn ) = bmo(Rn ) ⊂ Lloc 1 (R ),

0 < q ≤ 2.

(2.23)

The only-if-part is a consequence of the last line in (2.18) and of 0 0 B∞,q (Rn ) → F∞,q (Rn ),

0 < q ≤ ∞,

(2.24)

covered by (2.45), (2.46) below. Alternatively, one can prove (2.24) for 1 < q < ∞ (which is sufficient for our purpose) using 0 n 0 n F1,q  (R ) → B1,q  (R ),

1 1 +  = 1, q q

(2.25)

the duality (1.24) and its B-counterpart according to [T83, Theorem 2.11.2, p. 178]. 

2.1. Embeddings

25

2.1.3 Sharp embeddings: Constant differential dimensions Recall that s − np is sometimes called the differential dimension of the space Asp,q (Rn ). This comes from the so-called Sobolev embedding Hps (Rn ) → Lq (Rn ),

s > 0,

1 < p < ∞,

s−

n n =− 0 such that for all f ∈ Fps00,q0 (Rn ), s f |Fp,q (Rn ) ≤ c f |Fps00,q0 (Rn )1−θ · f |Fps11,q1 (Rn )θ .

(2.55)

Proof. If f ∈ Fps00,q0 (Rn ), it follows from (2.32) that f ∈ Fps11,q1 (Rn ) and f ∈ s Fp,q (Rn ). By (2.37), we may assume that q0 = q1 = ∞. For {aj }j∈Z ⊂ C, [BrM01, Lemma 3.7, p. 394] provides ∞ 

2jsq |aj |q

j=−∞

1/q

 1−θ  θ sup 2js1 |aj | . ≤ c sup 2js0 |aj | j∈Z

(2.56)

j∈Z

We insert (2.56) into the wavelet representation for the F -spaces according to Proposition 1.11(ii) with (2.35) for p1 = ∞. Then (2.55) follows from Hölder’s inequality.  Remark 2.12. The proof of (2.56) uses a similar splitting technique as in [Run86, Section 5.1, p. 328] and [RuS96, Section 2.2.5, pp. 37–40], going back to [Jaw77] and [T83, Section 2.7.1, pp. 129–131] in connection with embeddings of the type (2.32). The proof of (2.55) in [BrM01, Lemma 3.1, p. 393] is formulated in terms of s s spaces Fp,q which coincide with Fp,q for p < ∞, but not for the case p = ∞ (which is also mentioned there). As already said, historical and recent references about Gagliardo-Nirenberg inequalities may be found in [Tri14] and [T13, Chapter 4]. They are not repeated here. But as far as related inequalities in terms of so-called ∗

tempered homogeneous function spaces Asp,q (Rn ) are concerned, we additionally refer the reader to [Tri17b]. These spaces will be considered in some detail in Section 4.1 below. Finally, we refer to [HaS17] for dealing with Gagliardo-Nirenberg inequalities in spaces with dominating mixed smoothness.

2.2 Traces 2.2.1 The problem In [T92, Chapter 4] we dealt with the so-called key problems of the theory of function spaces as considered here. These are 1. traces of Asp,q (Rn ) on hyper-planes,

30

Chapter 2. The essentials, key theorems

2. diffeomorphisms of Asp,q (Rn ), 3. extensions of corresponding spaces Asp,q (Rn+ ) to Asp,q (Rn ), where Rn+ = {x ∈ Rn : xn > 0}, and 4. diverse types of pointwise multipliers. This paves the way to an efficient theory of the corresponding spaces Asp,q (Ω) on (bounded smooth) domains Ω in Rn and their numerous applications including, for example, boundary value problems for elliptic PDEs. The first comprehensive satisfactory treatment of these problems for all spaces Asp,q (Rn ), A ∈ {B, F }, s ∈ R, 0 < p ≤ ∞ (p < ∞ for F -spaces), 0 < q ≤ ∞ (2.57) was given in [T92, Chapter 4]. We also refer the reader to Section 2.6.1, where we discuss the crucial role that the above problems have played in the theory of function spaces of the type Asp,q (Rn ) since the late 1970s. Not only in [T92] but also in [T06, T08], we returned to these topics – now using more advanced instruments that had become available, especially atomic and wavelet representations. In this s Section, we wish to incorporate the spaces F∞,q (Rn ) into the existing related assertions for the spaces in (2.57). Let us first deal with traces. In the entire Section 2.2, we always assume n ∈ N and n ≥ 2. Let x = (x , xn ), x ∈ Rn−1 , xn ∈ R. The question arises for which spaces Asp,q (Rn ) the trace tr , tr f = f (x , 0),

x ∈ Rn−1 ,

f ∈ Asp,q (Rn ),

(2.58)

generates a linear and bounded operator into (or better onto) some space Atu,v (Rn−1 ). This question has a long history, going back to the late 1950s. In [T78, Sections 2.9.3, 2.9.4, pp. 218–226], we dealt with the trace problem for the s (Rn ), s > 0, 1 ≤ p ≤ ∞, 1 ≤ q ≤ ∞ and the Sobolev classical Besov spaces Bp,q s n s n spaces Hp (R ) = Fp,2 (R ), s > 0, 1 < p < ∞, including the classical Sobolev spaces Wpk (Rn ) = Hpk (Rn ), k ∈ N, 1 < p < ∞. This work also contains detailed (historical) references. The traditional definitions of these classical spaces are recalled in Section 4.3 below. This theory has later been extended to the spaces Asp,q (Rn ) as follows. Let 0 < p ≤ ∞ with p < ∞ for F -spaces, 0 < q ≤ ∞ and s−

  1  1 > σp(n−1) = (n − 1) max , 1 − 1 . p p

(2.59)

Then s− 1

tr :

s Bp,q (Rn ) → Bp,q p (Rn−1 )

tr :

s Fp,q (Rn ) → Bp,p p (Rn−1 )

(2.60)

and s− 1

(2.61)

2.2. Traces

31

are linear and bounded maps onto the indicated trace spaces, and there are related extension operators ext, tr ◦ ext = id,

s− 1

identity in Bp,q p (Rn−1 )

(2.62)

s− 1

0 < p ≤ ∞,

0 < q ≤ ∞,

(2.63)

s− 1

0 < p < ∞,

0 < q ≤ ∞.

(2.64)

such that ext :

s (Rn ), Bp,q p (Rn−1 ) → Bp,q

ext :

s (Rn ), Bp,p p (Rn−1 ) → Fp,q

and

A somewhat sophisticated Fourier-analytical proof and related references may be found in [T83, Section 2.7.2, pp. 131–139], including the fact that there is a universal extension operator ext. In [T92, Section 4.4, pp. 212–221], we returned to this topic, now based on atoms and local means, and justified again that the trace operators tr in (2.60), (2.61) are linear and bounded maps onto the indicated target spaces. While this shortens some arguments, it does not cover the fact that there are related linear extension operators with (2.63), (2.64). Nowadays one can rely on wavelet expansions. With their help, one can construct linear extension operators satisfying (2.63) and (2.64), respectively, which are common for huge families of spaces. This has been done in [T08, Section 5.1.3, pp. 139–147] with bounded smooth domains Ω and their boundaries Γ = ∂Ω instead of Rn and Rn−1 , respectively. The arguments used therein are local and can also be used, even in a simplified way, for the construction of so-called wavelet-friendly linear extension operators justifying (2.63), (2.64). They are again common for huge families of these spaces. At this point, we are interested in an extension of (2.61), (2.62) and (2.64) to p = ∞. In other words, we wish to justify that tr :

s (Rn ) → C s (Rn−1 ), F∞,q

s > 0,

0 < q ≤ ∞,

(2.65)

s (Rn ), C s (Rn−1 ) → F∞,q

s > 0,

0 < q ≤ ∞,

(2.66)

C s (Rn−1 ),

s > 0.

(2.67)

2 ≤ n ∈ N, and ext :

for suitable linear extension operators such that tr ◦ ext = id,

identity in

Recall that s s (Rn−1 ) = F∞,∞ (Rn−1 ), C s (Rn−1 ) = B∞,∞

s ∈ R,

(2.68)

are the usual Hölder-Zygmund spaces according to (2.35). Some assertions of this type are already known. A Fourier-analytical proof of (2.65) and (2.66) restricted

32

Chapter 2. The essentials, key theorems

to 1 ≤ q ≤ ∞ can be found in [Mar87, Theorem 6, p. 91]. Based on atomic arguments, it was justified in [FrJ90, Theorem 11.2, pp. 128, 134] that tr in (2.65) is a map onto C s (Rn−1 ). But this does not cover the existence of a linear extension operator satisfying (2.66). This shortcoming is quite similar to the above-mentioned atomic argument in [T92]. One may also consult [YSY10, Section 6.3.2, pp. 166– 168] for further discussions.

2.2.2 The trace theorem As already indicated, we wish to incorporate (2.65)–(2.67) into (2.59)–(2.64) using the wavelet arguments according to [T08, Section 5.1.3., pp. 139–147]. For p = ∞, it follows from Theorem 2.3 that As∞,q (Rn ) → C(Rn ), s > 0, and traces make sense pointwise. We always rely on wavelet or atomic expansions with smooth building blocks that again have pointwise traces. If p < ∞ and q < ∞, (2.60) and (2.61) are defined by completion. If p < ∞ and q = ∞, one can additionally use the Fatou property according to Theorem 1.25. The extension operators are constructed by smooth wavelet expansions. This ensures that they are common extensions for families of spaces. We will return to this point in Remark 2.14 below, but do not stress these technicalities otherwise. Theorem 2.13. Let Asp,q (Rn ), 2 ≤ n ∈ N, with A ∈ {B, F }, 0 < p ≤ ∞, 0 < q ≤ ∞ and   1  1 (2.69) s − > σp(n−1) = (n − 1) max , 1 − 1 p p be the corresponding spaces according to Definition 1.1. Then s− 1

tr :

s (Rn ) → Bp,q p (Rn−1 ) Bp,q

tr :

s Fp,q (Rn ) → Bp,p p (Rn−1 ).

and

(2.70)

s− 1

(2.71)

Furthermore, there are linear and bounded extension operators ext with tr ◦ ext = id, such that

identity in

s− 1

Bp,q p (Rn−1 )

s− 1

ext :

s Bp,q p (Rn−1 ) → Bp,q (Rn )

ext :

s Bp,p p (Rn−1 ) → Fp,q (Rn ).

and

(2.72) (2.73)

s− 1

(2.74)

Proof. Step 1. As indicated in Section 2.2.1, the above theorem is known for all involved spaces Asp,q (Rn ) with p < ∞ for the F -spaces. The atomic proof in [T92, Section 4.4, pp. 212–221] fits in the above explanations as to how traces should be understood. Furthermore, (2.38) and (2.70) with p = q = ∞ imply tr :

s s F∞,q (Rn ) → B∞,∞ (Rn−1 ) = C s (Rn−1 ),

0 < q ≤ ∞,

s > 0.

(2.75)

2.2. Traces

33

In other words, it remains to prove (2.66), ext :

0 < q ≤ ∞,

s (Rn ), C s (Rn−1 ) → F∞,q

s > 0,

(2.76)

with (2.67). Step 2. Let x = (x1 , . . . , xn−1 ) with 2 ≤ n ∈ N. We rely on the wavelet expansion of f ∈ C s (Rn−1 ), s > 0, according to Proposition 1.11(i), f=

j  λj,G m ψG,m (x ),



j ψG,m (x ) =

j∈N0 ,G∈Gj , m∈Zn−1

n−1 

ψGl (2j xl − ml ),

(2.77)

l=1

where   j j,G j(n−1) λj,G f, ψG,m , m = λm (f ) = 2

j ∈ N0 ,

m ∈ Zn−1 ,

with G0 = {F, M }n−1 , Gj = {F, M }n−1,∗ for j ∈ N and    sup 2js λj,G f |C s (Rn−1 ) ∼ m (f ) .

(2.78)

(2.79)

j∈N0 ,G∈Gj , m∈Zn−1

We now use a simplified version of the wavelet-friendly extension as constructed in [T08, Section 5.1.3, pp. 139–147]. Let χ ∈ D(R) = C0∞ (R),

supp χ ⊂ (−1, 1),

χ(t) = 1 if |t| ≤ 1/2,

and x = (x , xn ), x ∈ Rn−1 , xn ∈ R. Then j  j λj,G g = extf = m (f ) ψG,m (x ) χ(2 xn )

(2.80)

(2.81)

j

j∈N0 ,G∈G , m∈Zn−1

is an atomic expansion for the hybrid spaces LrAsp,q (Rn ) as described in [T14, Theorem 3.33, p. 67] (including all requested moment conditions, if required). By (2.80) and (2.77)–(2.79), one has g(x , 0) = f (x ),

x ∈ Rn−1 ,

(2.82)

as a pointwise trace. Using (1.71) with 0 < p < ∞, (1.61) and the above-mentioned s atomic representation theorem for L0 Fp,q (Rn ), one obtains





q 1/q

 s jsq  j,G n

 g |F∞,q (Rn ) ≤ c sup 2Jn/p

2 χ (·) |L (R ) λ j,m p m

. J∈Z,M ∈Zn

Qj,m ⊂QJ,M , j∈N0 ,G∈Gj

(2.83)

34

Chapter 2. The essentials, key theorems

This construction ensures that only n-dimensional cubes Qj,m near xn = 0 are relevant, typically Qj,m × (−2−j , 2−j ) and QJ,M × (−2−J , 2−J ) with m ∈ Zn−1 and M ∈ Zn−1 . With p = q, one has similarly as in (2.44)

  q 1/q s  g |F∞,q (Rn ) ≤ c sup 2jsq 2−(j−J)n λj,G . (2.84) m j∈Z,M ∈Zn−1

Qj,m ⊂QJ,M , j∈N0 ,G∈Gj

For fixed J ∈ Z and M ∈ Zn−1 , one has ∼ 2(j−J)(n−1) , j ≥ J, j ∈ N0 , relevant cubes Qj,m . The use of (2.79) leads to  1/q s (Rn ) ≤ c f |C s (Rn−1 ) sup 2−j+J g |F∞,q J∈Z (2.85) j≥J ≤ c f |C s (Rn−1 ). Then it follows from (2.78) and (2.82) that g = extf according to (2.81) is a linear extension operator satisfying (2.72), (2.75), (2.76).  Remark 2.14. It is of interest to construct common extension operators for huge families of relevant function spaces. First we remark that g = extf is a simplified version of g in [T08, (5.75), p. 142] used in the proof of [T08, Theorem 5.14, p. 143]. We relied on wavelet expansions as described in Section 1.2.1, where the smoothness u ∈ N in (1.34) is at our disposal. In particular, the construction in s (Rn ) covered by the above theorem with (2.81) applies to all spaces Bp,q 0 < p ≤ ∞,

0 < q ≤ ∞,

1 + σp(n−1) < s < u. p

(2.86)

The situation for the F -spaces is somewhat different. According to [T08, Theorem s (Rn ) with s ∈ R, 0 < p < ∞, 0 < q ≤ ∞ 1.7, p. 5], atomic expansions in Fp,q require the related atoms to have moment conditions of order L with   1 1  (n) (n) (2.87) − s, σp,q = n max , , 1 − 1 . L > σp,q p q By [T14, Theorem 3.3, p. 67], one has the same restrictions for atomic expansions s s (Rn ) = L0 Fp,q (Rn ). If χ in (2.80) additionally satisfies in F∞,q  χ(t) tl dt = 0, l = 0, . . . , L − 1, (2.88) R j (x )χ(2j xn ) ψG,m

in (2.81) have the required moment conditions. then the atoms This applies to all spaces that are covered by the above theorem. In particular, for any ε with 0 < ε < 1 there is a common extension operator as constructed in s s (2.81) for all spaces Bp,q (Rn ) with (2.86) and all spaces Fp,q (Rn ) with 0 < p ≤ ∞,

ε < q ≤ ∞,

1 + σp(n−1) < s < u. p

(2.89)

2.2. Traces

35

The question arises whether one can get rid of these somewhat disturbing moment conditions at least for large s but small q > 0. But this is not possible. More on this issue is discussed in [ScV09, Section 3.2, p. 160] with a reference to [Schn09, Section 3.2, p. 126]. Remark 2.15. In order to ensure that the trace makes sense, we needed s > 0 in (2.75) and more generally (2.69) in (2.70), (2.71). But the construction in (2.81) based on (2.80), (2.88) applies to all s ∈ R. This is similar to the related Fourieranalytical approach in [T83, Section 2.7.2, pp. 131–133]. Remark 2.16. Both in [T83, Section 2.7.2, pp. 131–139] and in [T08, Section 5.1.3, pp. 139–147] (with Rn instead of domains) we dealt not only with the trace of functions, but also of some derivatives,   tr r : f → tr ∂nk f : 0 ≤ k ≤ r , r ∈ N0 , (2.90) with tr g as in (2.58) and ∂nk g = ∂ k g/∂xkn . Instead of (2.69) and (2.86), one now assumes 1 0 < p ≤ ∞, 0 < q ≤ ∞, r + + σp(n−1) < s < u. (2.91) p Then one has to replace (2.70), (2.71) by tr r :

s Bp,q (Rn ) →

r 

s− 1 −k

Bp,q p

(Rn−1 )

(2.92)

(Rn−1 )

(2.93)

k=0

and tr r :

s Fp,q (Rn ) →

r 

s− 1 −k

Bp,p p

k=0

with the interesting special case tr r :

s F∞,q (Rn ) →

r 

C s−k (Rn−1 ).

(2.94)

k=0

One has to imitate the arguments in [T08] for constructing the corresponding extension operators. But this is a technical matter and will not be done here in detail. In particular, tr r in (2.93) and (2.94) are maps onto the related target spaces.

2.2.3 Dichotomy: Traces versus density The following Subsection is inserted for two reasons. Firstly, we wish to provide a better understanding of the restrictions (2.69), including the sinister role played (n−1) by σp . Secondly, we will clarify in Proposition 2.50 the conditions under which the characteristic function χQ of the cube Q = (0, 1)n is an element of Asp,q (Rn ). It

36

Chapter 2. The essentials, key theorems

  will turn out that s = 1/p with 0 < p ≤ ∞ is the critical line in an p1 , s -diagram as illustrated in Figure 3.1, p. 99, in connection with the nearby Haar functions. It is quite obvious that spaces Asp,q (Rn ) such that χQ ∈ Asp,q (Rn ) cannot have traces on (n−1)-dimensional hyper-planes in the context of Theorem 2.13. This is in good agreement with (2.69), at least if p ≥ 1. In order to get a better understanding of what is going on, we will have a closer look at the dichotomy between traces on sets or hyper-planes and the related density assertions. We will be very brief and refer the reader to [T08, Section 6.4, pp. 215–236] based on [Tri08] for further details. Let Asp,q (Rn ) be again the spaces as introduced in Definition 1.1. It is well known that S(Rn ) and D(Rn ) = C0∞ (Rn ) are dense in Asp,q (Rn ) if p < ∞ and q < ∞, [T83, Theorem 2.3.3, p. 48]. Let Γ be a compact d-set in Rn , 0 < d < n, or (as considered here) the d-dimensional hyper-plane   d ∈ N, d < n. (2.95) Γd = x ∈ Rn : x = (x1 , . . . , xd , 0, . . . , 0) , As usual, D(Rn \ Γd ) collects all C ∞ -functions in Rn with compact support in the domain (= open set) Rn \ Γd . The dichotomy studied in [T08] deals with the alternative  either D(Rn \ Γd ) is dense in Asp,q (Rn ), (2.96) or Asp,q (Rn ) has a trace on Γd . So we have  dealt  with traces  in the framework of the dual pairings  far, S(Rn ), S  (Rn ) and S(Rd ), S  (Rd ) with d = n − 1, in the context of the related spaces Asp,q (Rn ) and Asp,q (Rd ), respectively. But this point of view is inadequate for the alternative in (2.96), in particular if one wishes to include spaces with p < 1. The following modification is sufficient for our purpose: We always assume that Γd according to (2.95) is furnished with the usual d-dimensional Lebesgue measure and that Lp (Γd ), 0 < p < ∞, is the related Lebesgue space. With γ ∈ Γd we mean γ = x = (x1 , . . . , xd , 0, . . . , 0) as in (2.95). Definition 2.17. Let s ∈ R and 0 < p, q < ∞ such that for some c > 0, ϕ |Lp (Γd ) ≤ c ϕ |Asp,q (Rn )

for all ϕ ∈ S(Rn ).

(2.97)

Then the trace operator trd , Asp,q (Rn ) → Lp (Γd ) (2.98)   is the completion of the pointwise trace trd ϕ (γ) = ϕ(γ) with ϕ ∈ S(Rn ) and γ ∈ Γd . trd :

Remark 2.18. Note that we modified [T08, Definition 6.61, p. 218], where we mainly dealt with compact d-sets in Rn (which also explains the above notation). For p < 1, the target spaces Lp (Γd ) do not fit into the context of S  (Γd ) = S  (Rd ). This requires some extra measurable considerations, which are outlined in detail in [T08, Remark 6.62, pp. 218–219] and will not be repeated here.

2.2. Traces

37

After these preparations, we can now adapt [T08, Definition 6.66, p. 221]. Recall that the spaces Asp,q (Rn ) for fixed p are monotonous (continuously embedded into each other) with respect to s independently of q, that for fixed s, p the s s spaces Bp,q (Rn ) are monotonous with respect to q, and that the spaces Fp,q (Rn ) are also monotonous with respect to q according to Theorem 2.9. Definition 2.19. Let 2 ≤ n ∈ N and 0 < p < ∞. Let A ∈ {B, F }. Then   Ap (Rn ) = Asp,q (Rn ) : 0 < q < ∞, s ∈ R .

(2.99)

Let Γd with d ∈ N and d < n be the hyper-plane according to (2.95). Let σ ∈ R. Then   with 0 < u < ∞ (2.100) D Ap (Rn ), Lp (Γd ) = (σ, u)   is called the dichotomy of Ap (Rn ), Lp (Γd ) if 

s > σ, s = σ,

0 < q < ∞, 0 < q ≤ u,

(2.101)

s = σ, s < σ,

u < q < ∞, 0 < q < ∞.

(2.102)

Asp,q (Rn ) → Lp (Γd ) exists for

trd : and

 D(R \ Γd ) is dense in n

Asp,q (Rn )

for

Furthermore,   D Ap (Rn ), Lp (Γp ) = (σ, 0) means that  trd exists for s > σ, 0 < q < ∞, D(Rn \ Γd ) is dense in Asp,q (Rn ) for s ≤ σ, 0 < q < ∞,

(2.103)

(2.104)

and   D Ap (Rn ), Lp (Γd ) = (σ, ∞) means that  trd exists for s ≥ σ, 0 < q < ∞, D(Rn \ Γd ) is dense in Asp,q (Rn ) for s < σ, 0 < q < ∞.

(2.105)

(2.106)

Remark 2.20. The above-mentioned monotonicity (continuous embedding into each other) with respect to s and q shows that this definition makes sense. In particular, σ is the breaking point for the smoothness s, whereas u stands for a finer tuning. A discussion of the dichotomy between density and traces according to Definition 2.17 may be found in [T08, Proposition 6.63, p. 219].

38

Chapter 2. The essentials, key theorems

Theorem 2.21. Let 2 ≤ n ∈ N, d ∈ N and d < n. Let 0 < p < ∞. Then  

n



D Bp (R ), Lp (Γd ) =

n−d p , 1  n−d p ,p



if p > 1, if p ≤ 1,

(2.107)

if p > 1, if p ≤ 1.

(2.108)

and  

n



D Fp (R ), Lp (Γd ) =

n−d p ,0   n−d p ,∞



Remark 2.22. This assertion coincides with [T08, Corollary 6.69, p. 225], where it is formulated as a by-product of corresponding assertions for compact d-sets Γd in Rn with 0 < d < n. One also finds detailed (atomic) proofs and further related references there. In addition, we wish to mention [CaH15] which deals with the dichotomy problem in more general function spaces and more general fractal sets and contains many related recent references. Remark 2.23. The above assertion is interesting in itself. But there are two main reasons for us to include these results. Firstly, we wish to comment on the restriction (2.69) in Theorem 2.13. The breaking point s = 1/p is the same as in (2.107) and (2.108)  with d = n − 1. For p < 1, (2.69) comes with the restriction s− p1 > (n−1) p1 −1 as opposed to s = p1 in (2.107), (2.108). The related trace trd with d = n − 1 according to Definition 2.17 exists if s > 1/p, but outside the theory of distributions. In other words, if one  within  the framework of  wishes to stay distributions – here the dual pairings S(Rn ), S  (Rn ) and S(Rn−1 ), S  (Rn−1 )   (n−1) = (n − 1) p1 − 1 > 0 if p < 1. The –, one has to pay a price, namely σp second reason, closely related to this discussion, is the question under which circumstances one has χQ ∈ Asp,q (Rn ), where χQ is the characteristic function of the cube Q = (0, 1)n . The final answer according to Proposition 2.50 makes clear that s = 1/p is again the breaking point independently of n ∈ N. If χQ ∈ Asp,q (Rn ), then there are no traces in which way ever they are defined within the theory of distributions (underlying Theorem 2.13) or according to Definition 2.17. This makes clear that s = 1/p in Proposition 2.50 is also the correct breaking point for spaces Asp,q (Rn ) with p < 1 and not the possible counterpart (2.69) within the theory of distributions. Remark 2.24. We also refer the reader to [T92, Section 4.4.3, pp. 220–221] and [T08, Section 6.4.5, pp. 226–228], both Sections carrying the name Curiosities. There, one can see what may happen if one mingles arguments from the framework of distributions with measure-theoretical aspects, say, related to Lp (Rn ) with p < 1.

2.3. Diffeomorphisms

39

2.3 Diffeomorphisms A continuous one-to-one map of Rn , n ∈ N, onto itself,   y = ψ(x) = ψ1 (x), . . . , ψn (x) , x ∈ Rn ,   y ∈ Rn , x = ψ −1 (y) = ψ1−1 (y), . . . , ψn−1 (y) ,

(2.109) (2.110)

is called a diffeomorphism if all components ψj (x) and ψj−1 (y) are real C ∞ functions on Rn and if for j = 1, . . . , n,   sup |Dα ψj (x)| + |Dα ψj−1 (x)| < ∞ for all α ∈ Nn0 with |α| > 0. (2.111) x∈Rn

  Then ϕ → ϕ ◦ ψ, given by (ϕ ◦ ψ)(x) = ϕ ψ(x) , is a one-to-one map of S(Rn ) onto itself. This can be extended to a one-to-one map of S  (Rn ) onto itself by   (f ◦ ψ)(ϕ) = f, |det ψ∗−1 | ϕ ◦ ψ −1 , (2.112) f ∈ S  (Rn ), ϕ ∈ S(Rn ), as the distributional version of     f ◦ ψ (x) ϕ(x) dx = Rn

Rn

    f (y) ϕ ◦ ψ −1 (y)det ψ∗−1 (y) dy,

(2.113)

(change of variables), where ψ∗−1 is the Jacobian and det ψ∗−1 its determinant. The first proof that Dψ :

Asp,q (Rn ) → Asp,q (Rn ),

Dψ f = f ◦ ψ

(2.114)

is an isomorphic map for all spaces Asp,q (Rn ) with s ∈ R, 0 < p ≤ ∞ (p < ∞ for F -spaces), 0 < q ≤ ∞ goes back to [T83, Section 2.10, pp. 173–176]. But it requires some awkward restrictions for ψ. In [T92, Section 4.3.2, pp. 207–211], we therefore gave a new proof covering all admitted ψ. This new proof relies on so-called local means, but it requires again a lot of efforts. A streamlined version on less than one page can be found in [T19, Section 1.3.8, pp. 66–68] as a preparation for a corresponding assertion for related spaces with dominating mixed smoothness. s One can incorporate the spaces F∞,q (Rn ), s ∈ R, 0 < q ≤ ∞, as follows. Theorem 2.25. Let ψ be the above diffeomorphism of Rn onto itself. Let A ∈ {B, F }, s ∈ R, 0 < p ≤ ∞, 0 < q ≤ ∞. Then Dψ according to (2.114) is an isomorphic map. Proof. Using (1.101) with 0 < p < ∞, one can extend the already known assertion for the spaces Asp,q (Rn ) with A ∈ {B, F }, s ∈ R, 0 < p ≤ ∞ (p < ∞ for F -spaces), s (Rn ).  0 < q ≤ ∞ to F∞,q Remark 2.26. A one-to-one map ψ of Rn onto itself according to (2.109), (2.110) satisfying (2.111) not necessarily for all α ∈ Nn0 but only for 0 < |α| ≤ k ∈ N is called a k-diffeomorphism. One may ask how large k must be such that the related

40

Chapter 2. The essentials, key theorems

k-diffeomorphism Dψ according to (2.114) is still an isomorphic map of a given space Asp,q (Rn ) onto itself. Some corresponding (but not satisfactory) assertions may be found in [T92, Remark 4.3.2/2, p. 211]. But we refer the reader to [Scha13] where problems of this type have been studied in detail. Remark 2.27. An extension of the above theorem to the hybrid spaces LrAsp,q (Rn ) according to Definition 1.6 may be found in [T14, Theorem 3.69, p. 101]. We also refer the reader to [YSY10, Section 6.2, pp. 160–162].

2.4 Multipliers, localizations and multiplication algebras 2.4.1 Smooth multipliers Throughout this book, pointwise multipliers and multiplication properties for the spaces Asp,q (Rn ) will play some role. Let A(Rn ) be a quasi-Banach space in Rn with the continuous embeddings S(Rn ) → A(Rn ) → S  (Rn ).

(2.115)

Then g ∈ L∞ (R ) is said to be a pointwise multiplier for A(R ) if n

n

f → g f

generates a bounded map in A(Rn ).

(2.116)

Of course, one has to say what this multiplication means in this generality. But for this, we can rely on previous considerations. For the spaces A(Rn ) = Asp,q (Rn ) according to Definition 1.1, one finds careful discussions and rather final assertions in [RuS96, Chapter 4], [T83, Section 2.8], [T92, Section 4.2], [T06, Section 2.3] and most recently in [Scha13]. We extended this theory in [T13, Section 2.7.1] to the local spaces LrAsp,q (Rn ) and in [T14, Section 3.6.4] to the hybrid spaces LrAsp,q (Rn ). If g in (2.116) is sufficiently smooth, one can justify the related points (Rn ) in a few lines wise multiplier property for all spaces Asp,q (Rn ) including F∞,q without any technical complications. Recall that C k (Rn ) with k ∈ N0 is the collection of all complex-valued continuous functions g in Rn having classical continuous derivatives up to the order k inclusively with g |C k (Rn ) =

|Dα g(x)| < ∞.

(2.117)

  1 1  (n) σp,q = n max , , 1 − 1 . p q

(2.118)

sup |α|≤k, x∈Rn

Let as before   1  σp(n) = n max , 1 − 1 , p

Theorem 2.28. Let Asp,q (Rn ) with A ∈ {B, F }, s ∈ R and 0 < p, q ≤ ∞ be the spaces according to Definition 1.1. Then there is a number k ∈ N and a constant c > 0 such that gf |Asp,q (Rn ) ≤ c g |C k (Rn ) · f |Asp,q (Rn ) for all g ∈ C k (Rn ) and all f ∈ Asp,q (Rn ).

(2.119)

2.4. Multipliers, localizations and multiplication algebras

41

Proof. Step 1. According to [T08, Theorem 1.7, p. 5], repeated in Section 4.3.5 below, one does not need moment conditions for the atomic expansions in the (n) s s (Rn ) with 0 < p ≤ ∞, 0 < q ≤ ∞, s > σp and in the spaces Fp,q (Rn ) spaces Bp,q (n)

with 0 < p < ∞, 0 < q ≤ ∞, s > σp,q . Multiplication of these expansions with g ∈ C k (Rn ), k > s, proves (2.119). Using (1.71) and the related atomic expansions according to [T14, Theorem 3.33, p. 67], one can extend these arguments to the (n) s (Rn ) with 0 < q ≤ ∞ and s > σq . spaces F∞,q (n)

Step 2. Let s ∈ R and 0 < p, q ≤ ∞. Let l ∈ N0 such that 2l + s > σp for (n) the B-spaces and 2l + s > σp,q for the F -spaces. Then one can apply Step 1 to n As+2l p,q (R ). Theorem 1.22 then yields n s n (id − Δ)l As+2l p,q (R ) = Ap,q (R ). n Applying this result to f = (id − Δ)l h with h ∈ As+2l p,q (R ), one obtains Dα (gα h). gf = (id − Δ)l gh +

(2.120)

(2.121)

|α|≤2l−1

We choose k ∈ N sufficiently large such that g and gα are pointwise multipliers in n As+2l p,q (R ) according to Step 1. Then it follows from Theorem 1.24 that n n gα h |As+2l gf |Asp,q (Rn ) ≤ c gh |As+2l p,q (R ) + c p,q (R ) |α|≤2l−1 n ≤ c g |C k (Rn ) · h |As+2l p,q (R ) 

≤ c g |C (R ) · f k

n

|Asp,q (Rn ).

(2.122) 

Remark 2.29. The proof is surprisingly simple. Step 1 uses atomic expansions without moment conditions, Step 2 relies on lifts that are suitably adapted according to (2.121). For the spaces covered by Step 1, one has the natural restriction k > s. The functions gα in (2.121) have derivatives of g up to order 2l − 1. Then a direct application of Step 1 requires k ≥ s + 4l, which does not look very promising. Next, we discuss sharper versions which rely on more sophisticated arguments. Let again

C (Rn ) = B∞,∞ (Rn ),

be the above Hölder-Zygmund spaces. Let

(n) σp

 ∈ R,

(2.123)

be as in (2.118).

Theorem 2.30. Let with A ∈ {B, F }, s ∈ R and 0 < p, q ≤ ∞ be the spaces according to Definition 1.1. Then   g ∈ C (Rn ) with  > max s, σp(n) − s (2.124) Asp,q (Rn )

is a pointwise multiplier for Asp,q (Rn ). Furthermore, there is a constant c > 0 such that (2.125) g f |Asp,q (Rn ) ≤ c g |C (Rn ) · f |Asp,q (Rn ) for all g ∈ C (Rn ) and all f ∈ Asp,q (Rn ).

42

Chapter 2. The essentials, key theorems

Proof. Step 1. The above assertion is known if, in addition, p < ∞ for the F s (Rn ) goes back to [Fra86, Theorem 1, spaces. The first proof for the spaces Fp,q p. 43], which is an improvement of the corresponding assertions for the B-spaces and the F -spaces in [T83, Theorem 2.8.2, Corollary 2.8.2, pp. 140–143], based on [Tr78, Theorem 2.6.1, p. 128]. We returned to this topic in [T92, Section 4.2.2, pp. 202–206] and proved the above assertions, again with p < ∞ for the F -spaces, using so-called local means. A new approach to these questions, based on modified atomic expansions, can be found in [Scha13]. It remains to incorporate the spaces s (Rn ), F∞,q

s ∈ R,

0 < q ≤ ∞.

(2.126)

Step 2. It follows from the duality (1.24) and Step 1 that any g ∈ C (Rn ) with s (Rn ) with q > 1 satisfying (2.125). For  > |s| is a pointwise multiplier for F∞,q s < 0, one can use (1.99) with 1 < p < ∞ and 0 < q ≤ ∞. Pointwise multipliers s s (Rn ) are uniformly pointwise multipliers for Fp,q (2QJ,M ). Then it follows for Fp,q from Step 1 that g ∈ C (Rn ) with  > |s| is a pointwise multiplier for the spaces s F∞,q (Rn ) with s < 0 and 0 < q ≤ ∞ satisfying (2.125). It remains to extend this s (Rn ) with s ≥ 0 and 0 < q ≤ 1. For this, let 0 < θ < 1, assertion to the spaces F∞,q s1 = −s0 < 0 ≤ s < s0 < ,

0 < q1 < ∞,

and s = (1 − θ)s1 + θs0 ,

1 < q0 < ∞,

1−θ 1 θ = + . q q0 q1

Then the so-called ±-method of interpolation provides  s1  s s0 F∞,q (Rn ) = F∞,q (Rn ), F∞,q (Rn ), θ . 1 0

(2.127)

(2.128)

(2.129)

We refer the reader to the inhomogeneous version of [FrJ90, Theorem 8.5, pp. 98, 134] and [YSY15, Theorem 2.12, Remark 2.13, p. 1843]. By the above assertions, we know that any g ∈ C (Rn ) is a pointwise multiplier in the two spaces on the right-hand side of (2.129). Then it follows from the interpolation property that g s is also a pointwise multiplier in F∞,q (Rn ). If q1 → 0 then q → 0. In other words, s this procedure covers any space F∞,q (Rn ) with s ≥ 0 and 0 < q ≤ 1 in the desired way.  Remark 2.31. The classical real and complex interpolation methods do not work in the above context. The same applies to the complex interpolation method for distinguished quasi-Banach spaces, including Asp,q (Rn ), as developed in [MeM00, KMM07]. We will use this method later on in connection with characteristic functions as pointwise multipliers. It turns out, however, that the method does not apply if both spaces in the related interpolation couple are of the type s F∞,q (Rn ). The (maybe not so well-known) ±-method of interpolation removes this shortcoming. The method itself goes back to [Pee71, GuP77, Nil85]. In addition to [FrJ90], we refer the reader to the comprehensive survey [YSY15] dealing with

2.4. Multipliers, localizations and multiplication algebras

43

diverse interpolation methods for a large variety of spaces, including in particular the hybrid spaces in the τ -version according to (1.29). One may also consult [YYZ13]. Remark 2.32. One may ask to which extent the restrictions for  in (2.124) are sharp or at least natural. For p ≥ 1, this is rather obvious. Then one has  > |s|. But this observation remains valid for all spaces Asp,q (Rn ) with A ∈ {B, F }, s ∈ R and 0 < p, q ≤ ∞. In particular, according to [T83, Corollary 2.8.2, p. 143] there   (n) are functions g ∈ C (Rn ) with  < max s, σp − s which cannot be pointwise s (Rn ) can be incorporated into this negamultipliers in Asp,q (Rn ). The spaces F∞,q tive assertion by interpolation. Remark 2.33. The two Theorems 2.28 and 2.30 are quite different. By accepting atomic representations and lifting (in the somewhat tricky version (2.121)), the proof of Theorem 2.28 is rather straightforward. Theorem 2.30 is sharper, but it relies on heavy machinery altogether. Pointwise multiplier assertions of the type (2.119) can be extended to more general hybrid spaces LrAsp,q (Rn ). We refer the reader to [T14, Theorem 3.69, p. 101] and, based on (1.29), to [YSY10, Section 6.1, pp. 147–160]. Remark 2.34. Let A ∈ {B, F }, s > 0 and 0 < q ≤ ∞. Then it follows from Theorem 2.61 below that g is a pointwise multiplier for As∞,q (Rn ) if, and only if, g ∈ As∞,q (Rn ). This improves the related assertion g ∈ C (Rn ),  > s, in Theorem 2.30.

2.4.2 Localizations Let ψ be a compactly supported C ∞ -function in Rn , n ∈ N, such that ψM (x) = 1, ψM (x) = ψ(x − M ), x ∈ Rn ,

(2.130)

M ∈Zn

is a resolution of unity. Theorem 2.35. Let s ∈ R and 0 < p, q ≤ ∞. Then s f |Fp,q (Rn ) ∼



s ψM f |Fp,q (Rn )p

1/p

(2.131)

M ∈Zn

(with the usual modification for p = ∞) are equivalent quasi-norms. Furthermore, s f |Bp,q (Rn ) ∼



s ψM f |Bp,q (Rn )r

1/r

(2.132)

M ∈Zn

for some r with 0 < r ≤ ∞ (modification in the case of r = ∞) if, and only if, p = q = r.

44

Chapter 2. The essentials, key theorems

Proof. The above assertions coincide with [T92, Theorem 2.4.7, pp. 124–125] under the additional assumption p < ∞ for the F -spaces. In other words, it remains to show that s s (Rn ) ∼ sup ψM f |F∞,q (Rn ) (2.133) f |F∞,q M ∈Zn

(equivalent quasi-norms). The estimate of the right-hand side of (2.133) from above by the left-hand side follows from Theorem 2.28. The opposite estimate is  a consequence of (1.101) if one inserts f = K∈Zn ψK f .  Remark 2.36. It follows again from Theorem 2.28 that one can replace (2.130) in the above theorem by ψM (x) ∼ 1, x ∈ Rn . (2.134) M ∈Zn s Furthermore , if ψM f ∈ Fp,q (Rn ), f ∈ S  (Rn ), for all M ∈ Zn such that the s right-hand side of (2.131) is finite, then f ∈ Fp,q (Rn ). This follows from Theorem 2.30 and the Fatou property according to Theorem 1.25.

2.4.3 Multiplication algebras In Theorem 2.4, we clarified under which circumstances the space Asp,q (Rn ) with A ∈ {B, F }, s ∈ R and 0 < p, q ≤ ∞ consists entirely of regular distributions. In this case, the product of two elements of these spaces makes sense at least pointwise a.e. (almost everywhere). One may ask for additional properties of these products. Of special interest is the question of whether the product of two elements of Asp,q (Rn ) is again an element of this space. Definition 2.37. A quasi-Banach space A(Rn ) on Rn with S(Rn ) → A(Rn ) → S  (Rn ),

n A(Rn ) ⊂ Lloc 1 (R ),

(2.135)

is said to be a multiplication algebra if f1 f2 ∈ A(Rn ) whenever f1 ∈ A(Rn ), f2 ∈ A(Rn ) and if there is a constant c > 0 such that f1 f2 |A(Rn ) ≤ c f1 |A(Rn ) · f2 |A(Rn )

(2.136)

for all f1 ∈ A(Rn ), f2 ∈ A(Rn ). Remark 2.38. We dealt several times with the question of the circumstances under which quasi-Banach spaces A(Rn ) satisfying (2.135) are multiplication algebras. The final answer for the spaces A(Rn ) = Asp,q (Rn ) with p < ∞ for the F -spaces will be quoted and commented in the proof of the Theorem 2.41 below. Related ∗

assertions for the so-called tempered homogeneous spaces Asp,q (Rn ) may be found in [Tri17a, Theorem 3.7, pp. 354–355]. We will return to this point in Theorem 4.13 r A(Rn ) with dominating mixed below. Multiplication algebras for some spaces Sp,q

2.4. Multipliers, localizations and multiplication algebras

45

smoothness are considered in [T19, Section 1.4.2, pp. 77-83]. The corresponding assertions for the hybrid spaces LrAsp,q (Rn ) and their local counterparts LrAsp,q (Rn ) will be quoted in the proof of Theorem 2.41 below. The question of whether a quasiBanach space satisfying (2.135) is a multiplication algebra might be of interest in itself, but it is also of some use in connection with Cauchy problems for nonlinear PDEs, including non-linear heat equations, the Navier-Stokes equations in hydrodynamics, the Keller-Segel equations for chemotaxis and the Lengyel-Epstein systems generating Turing patterns in mathematical biology and chemistry. As far as our own related contributions are concerned, we refer the reader to [T13, T14, T17, Tri19]. Proposition 2.39. Let Asp,q (Rn ) with A ∈ {B, F }, s ∈ R and 0 < p, q ≤ ∞ be the spaces according to Definition 1.1, satisfying in addition (2.135) with A(Rn ) = Asp,q (Rn ). If Asp,q (Rn ) is a multiplication algebra, then Asp,q (Rn ) → L∞ (Rn ).

(2.137)

Remark 2.40. A short proof of this well-known assertion may be found in [T13, Proposition 2.41, p. 90]. In other words, the search for possible multiplication algebras is now shifted from the spaces in Theorem 2.4 to the spaces in Theorem 2.3. Theorem 2.41. Let Asp,q (Rn ) with A ∈ {B, F }, s ∈ R and 0 < p, q ≤ ∞ be the spaces according to Definition 1.1, satisfying in addition (2.135) with A(Rn ) = s Asp,q (Rn ). Then Bp,q (Rn ) is a multiplication algebra if, and only if, 

either or

s > n/p where 0 < p, q ≤ ∞, s = n/p where 0 < p < ∞, 0 < q ≤ 1,

(2.138)

s (Rn ) is a multiplication algebra if, and only if, and Fp,q



either or

s > n/p where 0 < p, q ≤ ∞, s = n/p where 0 < p ≤ 1, 0 < q ≤ ∞.

(2.139)

Proof. The above assertion with p < ∞ for the F -spaces has a substantial history. The final formulation and related references may be found in [T13, Theorem 1.16, Remark 1.17, p. 12]. The conditions (2.8) and (2.138) differ for p = ∞. This somewhat tricky point has again quite a history and was eventually clarified in [SiT95, Remark 3.3.2, p. 114 and Corollary 4.3.2, Remark 4.3.5, p. 120]. According to Proposition 2.39 and (2.9), (2.10), it remains to prove that s (Rn ), A(Rn ) = F∞,q

s > 0,

0 < q ≤ ∞,

(2.140)

is a multiplication algebra. But this is an immediate consequence of [T14, Theorem 3.60(ii), p. 95], based on [T13, Theorem 2.43, p. 91], applied to (1.71). 

46

Chapter 2. The essentials, key theorems

Remark 2.42. The proof looks rather simple. But one should be well aware that it relies on two substantial ingredients. On the one hand, we used related characterizations for the corresponding spaces Asp,q (Rn ) with p < ∞ for the F -spaces; and on the other hand, we used the assertion that the hybrid spaces LrAsp,q (Rn ) are multiplication algebras if s + r > 0, see [T14, Theorem 3.60, p. 95]. Further comments about multiplication algebras can be found in [T14, Section 3.6.2, pp. 95–96], [T13, Section 2.5, pp. 89–95] and [YSY10, Section 6.1.2.4, pp. 156–157]. Finally, let us s (Rn ) with s > 0, mention that it was already stated in [Mar87, p. 90] that F∞,q 1 ≤ q ≤ ∞, is a multiplication algebra. The arguments in this paper rely on the s definition of the space F∞,q (Rn ) = Ls∞,q (Rn ) according to (1.18)–(1.21) extended by (1.15).

2.4.4 Characteristic functions as multipliers Let χ+ be the characteristic function of the half-space Rn+ = {x = (x1 , . . . , xn ) : xn > 0},

n ∈ N,

(2.141)

and let χQ be the characteristic function of the cube Q = (0, 1)n . If χ+ is a pointwise multiplier for a space Asp,q (Rn ), then χQ is also a pointwise multiplier for this space. The converse for the F -spaces follows from the localization property according to Theorem 2.35. This does not apply to the B-spaces. But in this case, one can rely on the arguments in [RuS96, pp. 211–214] which work equally for χ+ and χQ as far as the related only-if-assertions are concerned. We first clarify to which (limiting) spaces χQ belongs. If χ+ is a pointwise multiplier for a space Asp,q (Rn ), then χQ must be an element of this space. In addition, we take the opportunity to complement Theorem 2.3 as to deal with L∞ (Rn ) as source space. Proposition 2.43. Let A0∞,q (Rn ) with A ∈ {B, F } and 0 < q ≤ ∞ be the spaces according to Definition 1.1(i,iii). Then 0 L∞ (Rn ) → B∞,q (Rn )

if, and only if, q = ∞

(2.142)

if, and only if, 2 ≤ q ≤ ∞.

(2.143)

if, and only if, q = ∞

(2.144)

for all 0 < q ≤ ∞.

(2.145)

and 0 (Rn ) L∞ (Rn ) → F∞,q

Furthermore, 0 χQ ∈ B∞,q (Rn )

and 0 χQ ∈ F∞,q (Rn )

Proof. Step 1. Let f ∈ L∞ (R ). Then it follows from Proposition 1.11(i) that n

0 n (Rn ) ∼ sup |λj,G f |B∞,∞ m (f )| ≤ c f |L∞ (R ).

(2.146)

j,G,m

This proves the embedding (2.142) with q = ∞. According to (1.24), the spaces 1 1 0 n n 0 n F1,q  (R ) and L1 (R ) are the preduals of F∞,q (R ), 1 < q ≤ ∞, q + q  = 1 and

2.4. Multipliers, localizations and multiplication algebras

47

0 n n L∞ (Rn ). Furthermore, F1,q  (R ) and L1 (R ) have the same localization property with p = 1 as described in (2.131). Therefore, the corresponding embedding in (2.19), (2.20) is equivalent to

if, and only if, 0 < v ≤ 2.

0 F1,v (Rn ) → L1 (Rn )

(2.147)

With the help of the indicated duality, one obtains the embedding (2.143) for 2 ≤ q ≤ ∞ as well as the assertion that there is no embedding for 1 < q < 2. This can be extended to 0 < q < 2 by the monotonicity of these spaces with respect to q according to Theorem 2.9. Step 2. We prove (2.144). This also covers the only-if-part of (2.142). For that 0 purpose, let ∂ n = ∂1 · · · ∂n with ∂j f = ∂f /∂xj . If χQ ∈ B∞,q (Rn ) then −n (Rn ) ∂ n χQ = δ + + ∈ B∞,q

(2.148)

where δ is the δ-distribution and ++ indicates further δ-distributions with the corners of Q = (0, 1)n as off-points. But it is well-known (and a direct consequence of the Fourier-analytical Definition 1.1(i)) that (2.148) requires q = ∞. This 0 (Rn ) = proves (2.144). The proof of (2.145) relies on Proposition 1.16 and F∞,q 0 0 n 0 0 n L Fq,q (R ) = L Bq,q (R ), 0 < q < ∞, according to (1.71) and (1.33). By (1.65), j one has always |λj,G m (χQ )| ≤ c. But if the support of ψG,m has an empty intersecj,G tion with the faces of Q, then λm (χQ ) = 0, j ∈ N. This shows that one has q (j−J)(n−1) |λj,G (2.149) m | ≤ c2 m:(j,m)∈PJ,M

in (1.59), now with r = s = 0 and 0 < p = q < ∞. From this, it follows that 0 χQ |F∞,q (Rn ) ≤ c

≤c

sup

J∈Z,M ∈Zn

sup

J∈Z,M ∈Zn

2Jn/q



2−jn 2(j−J)(n−1)

1/q

j≥J J/q

2



−j

2

1/q

(2.150) < ∞.

j≥J

This proves (2.145).



Remark 2.44. Both (2.142) and (2.147) are known and may be found in [SiT95, Theorem 3.1.1, p. 112]. Furthermore, (2.144) and (2.145) are also covered by the recent paper [YSY18], where they are proved by similar wavelet arguments as s above in the larger context of the hybrid spaces Lr Bp,q (Rn ). Remark 2.45. We comment on the difference between (2.144) and (2.145) from s the point of view of positive cones in function spaces. Recall that f ∈ Fp,q (Rn ) with s ∈ R and 0 < p, q ≤ ∞ is called a positive distribution if f (ϕ) ≥ 0

for any ϕ ∈ S(Rn ) with ϕ(x) ≥ 0, x ∈ Rn .

(2.151)

48

Chapter 2. The essentials, key theorems

Then it follows from the Radon-Riesz theorem that f is a Radon measure, see +

[Mal95, pp. 61/62, 71, 75]. The positive cone F sp,q (Rn ) is the collection of all pos+

s itive f ∈ Fp,q (Rn ). If 0 < p ≤ ∞ and s < 0, then F sp,q (Rn ) is independent of 0 < q ≤ ∞, +

+

F sp,q1 (Rn ) = F sp,q2 (Rn ),

0 < q1 , q2 ≤ ∞.

(2.152)

This remarkable observation has some history. We refer the reader to [T06, Remark 1.134, p. 84] and [AdH96, Corollary 4.3.9, pp. 103, 126]. The final proof of (2.152) including p = ∞ may be found in [JPW90]. But it is also covered (in a somewhat implicit way) by [Net89]. Note that there is no counterpart of (2.152) for positive s (Rn ). This also follows from the reduction of the only-if-part of (2.144) cones in Bp,q to (2.148) and −n δ ∈ B∞,q (Rn ) if, and only if, q = ∞. (2.153) Obviously, δ is a positive distribution (the Dirac measure). On the other hand, it follows from (2.145) that −n δ ∈ F∞,q (Rn )

for all 0 < q ≤ ∞.

(2.154)

This may be considered as a simple illustration of (2.152). Corollary 2.46. Let Asp,q (Rn ) with A ∈ {B, F }, s ∈ R and 0 < p, q ≤ ∞ be the spaces according to Definition 1.1. Then L∞ (Rn ) → Asp,q (Rn )

(2.155)

if, and only if, either p = ∞, s < 0 or as in (2.142), (2.143). Proof. Let f (x) = 1 for x ∈ Rn . Then one has by (1.8) and (2.38) that (2.155) requires p = ∞. The above assertions now follow from Proposition 2.43 and the s (Rn ) with respect to s and q, combined with the embedding monotonicity of B∞,q in Theorem 2.9.  Remark 2.47. By similar arguments, it follows that χQ ∈ As∞,q (Rn ) if, and only if, either s < 0 or as in (2.144), (2.145). Next, we ask for which spaces Asp,q (Rn ) according to Definition 1.1 the characteristic function χ+ of the half space Rn+ in (2.141) is a pointwise multiplier. As far as technicalities are concerned, one may consult the references mentioned at the beginning of Section 2.4.1. But one may also first deal with χ+ ϕ, ϕ ∈ S(Rn ), combined with completion (using the fact that S(Rn ) is dense in Asp,q (Rn ) if p < ∞, q < ∞) or a Fatou argument based on Theorem 1.25. This will not be discussed in detail here. Apart from that, we take over related known assertions for the spaces Asp,q (Rn ) with p < ∞ for the F -spaces and concentrate on the spaces As∞,q (Rn ). As mentioned above, if χ+ is a pointwise multiplier for As∞,q (Rn ), then χQ ∈ As∞,q (Rn ), and only the related spaces covered by Remark 2.47 are possible candidates.

2.4. Multipliers, localizations and multiplication algebras

49

Theorem 2.48. Let Asp,q (Rn ) with A ∈ {B, F }, s ∈ R and 0 < p, q ≤ ∞ be the 0 −1 (Rn ) and for F∞,q (Rn ). spaces according to Definition 1.1 with q ≥ 1 for F∞,q s n Then χ+ is a pointwise multiplier for Ap,q (R ) if, and only if,   1  1 1 (2.156) max n − 1 , − 1 < s < . p p p Proof. Step 1. The above assertion with p < ∞ for the F -spaces coincides with the formulation in [T08, p. 169]. A corresponding complete proof may be found in [RuS96, Section 4.6.3, pp. 207–221]. But this property has a long history. For related references, we refer the reader to [RuS96, p. 258] and [T06, Remark 2.32, s (Rn ). p. 146]. It remains to prove the corresponding assertions for the spaces F∞,q Step 2. Let 1 < p < ∞ and p1 − 1 < s < 0. Then it follows from Step 1 that χ+ is also a pointwise multiplier for the spaces covered by (1.99). This shows that χ+ is a pointwise multiplier for the spaces −1 < s < 0,

s (Rn ), F∞,q

0 < q ≤ ∞.

(2.157)

Step 3. According to the above comments, χ+ is not a pointwise multiplier for the s spaces F∞,q (Rn ) with s > 0. Furthermore, it follows from Step 1 that χ+ is not a s s (Rn ) = B∞,∞ (Rn ) if s < −1. Let pointwise multiplier for the spaces F∞,∞ s0 < −1, 0 < q0 < ∞

and

1 < p1 < ∞,

1 1 − 1 < s1 < . p1 p1

(2.158)

1−θ 1 θ = + . q q0 p1

(2.159)

Let 0 < θ < 1 and s = (1 − θ)s0 + θs1 ,

θ 1 = , p p1

Then [KMM07, Theorems 5.2, 9.1, pp. 137, 157] provides   s0 s (Rn ) F∞,q0 (Rn ), Fps11,p1 (Rn ) θ = Fp,q

(2.160)

where [·, ·]θ stands for the complex interpolation method for some quasi-Banach spaces as developed there (covering the spaces in (2.160)). If χ+ was a pointwise s0 multiplier for F∞,q (Rn ), it would follow from Step 1 and the interpolation prop0 s (Rn ). But if θ > 0 is small, one erty that χ+ is also a pointwise multiplier for Fp,q has s < −1 and gets a contradiction to the only-if-part of (2.156) covered by Step 1. The following spaces remain: s (Rn ) F∞,q

Let q > If χ+ is

with

s = 0, s = −1 and q ≥ 1.

−s 1 1 n 1. By (1.24), the space F1,q  (R ) , q + q = s n a pointwise multiplier for F∞,q (R ), then

1, is the predual of

   −s n s (χ+ f, g) : g |F∞,q χ+ f |F1,q (Rn ) ≤ 1  (R ) = sup    s = sup (f, χ+ g) : g |F∞,q (Rn ) ≤ 1 ≤ cf

−s n |F1,q  (R )

(2.161) s F∞,q (Rn ).

(2.162)

50

Chapter 2. The essentials, key theorems

−s n shows that χ+ is also a pointwise multiplier for F1,q  (R ). But this contradicts the only-if-part of (2.156) with p = 1 and s = 0 or s = 1. Using (1.25) and the Fatou s (Rn ) according to Theorem 1.25, one can extend this argument property of F1,∞ to q = 1. 

Remark 2.49. We could not remove the ugly assumption q ≥ 1 for the spaces 0 −1 (Rn ) and F∞,q (Rn ). One would expect that χ+ is not a pointwise multiplier F∞,q 0 −1 for all spaces F∞,q (Rn ) and F∞,q (Rn ) with 0 < q ≤ ∞. But the above duality argument does not work for q < 1. One could try to elaborate the wavelet technique as used in Step 2 of the proof of Proposition 2.43 in order to disprove that χ+ 0 −1 is a pointwise multiplier in the spaces F∞,q (Rn ) and F∞,q (Rn ), 0 < q < ∞. This, however, requires a more detailed examination of the coefficients λj,G m (χQ ) used there. In any case, it turns out that χ+ is a pointwise multiplier for the distinguished spaces bmos (Rn ) according to (1.98) if, and only if, −1 < s < 0. This excludes the spaces bmo(Rn ) = bmo0 (Rn ) and bmo−1 (Rn ), which play an important role in the recent theory of nonlinear PDEs, including Navier-Stokes equations. The main motivation for writing this book is to naturally incorporate the s (Rn ) into already existing assertions for the spaces Asp,q (Rn ) with spaces F∞,q p < ∞ for the F -spaces. Theorem 2.48 is a typical example where one can replace χ+ by χQ as mentioned in the beginning of this Section 2.4.4. As a complement, one may try to characterize the spaces Asp,q (Rn ) such that χQ ∈ Asp,q (Rn ), which is simpler than asking for which spaces χQ is a pointwise multiplier. According to Remark 2.47, we have a final answer for the spaces As∞,q (Rn ). Related assertions for the spaces Asp,q (Rn ) with p < ∞ are more or less known, but we have no convincing reference which could be simply quoted. This may justify why we fix the outcome and give a short proof. Proposition 2.50. Let Asp,q (Rn ) with A ∈ {B, F }, s ∈ R and 0 < p, q ≤ ∞ be the spaces according to Definition 1.1. Let χQ be the characteristic function of the cube Q = (0, 1)n . Then 1/p χQ ∈ Bp,q (Rn )

if, and only if, q = ∞

(2.163)

1/p (Rn ) χQ ∈ Fp,q

if, and only if, p = ∞.

(2.164)

and Furthermore, χQ ∈ Asp,q (Rn ) if, and only if, either s < 1/p or as in (2.163), (2.164). Proof. Step 1. The last assertion follows from (2.163) and elementary embeddings. The case p = ∞ is covered by Proposition 2.43. In other words, we may assume p < ∞ and s = 1/p in what follows. Step 2. Let n = 1 and let χI be the characteristic function of the interval I = (0, 1). j j,M By (1.52) and (1.35), one has |λj,G m (χI )| ≤ c with λm (χI ) = 0 if supp ψM,m ⊂

2.4. Multipliers, localizations and multiplication algebras

51

1/p

R \ {0, 1} and j ∈ N. Then χI ∈ Bp,∞ (R) follows from Proposition 1.11(i). If 1 1/p p −1 (R), where δ is the δ-distribution and δ its χI ∈ Bp,q (R) then χ = δ − δ ∈ Bp,q I

1

−1

p (R) requires q = ∞, which follows counterpart with 1 as off-point. But δ ∈ Bp,q 1 p −1

from the Fourier-analytical definition of Bp,q (R) according to (1.8). This proves (2.163) for n = 1. By (2.28), we find 0 < p < p1 < ∞,

1/p 1 Fp,q (R) → Bp1/p (R), 1 ,p

0 < q ≤ ∞.

(2.165)

1/p

In particular, χI is not an element of Fp,q (R), p < ∞. Step 3. We reduce the proof of (2.163), (2.164) with p < ∞ and n ≥ 2 to the corresponding one-dimensional case. We rely again on the wavelet expansion according to Proposition 1.11 for f = χQ . In addition to (1.35), one has R ψF (t) dt = 1, [T06, p. 31] based on the references given in Section 1.2.1. Then it follows from j the product structure of both χQ and the wavelets ψG,m in (1.38) that only coj,G efficients λm (χQ ) according to (1.41) near the faces of Q, say xn = 0, are of relevance, typically of the type  j jn χQ (x) ψG,m (x) dx λj,G m (χQ ) = 2 Rn  (2.166) j j,M = 2j χI (xn ) ψM,m (x ) dx = λ (χ ) n n I mn n R

where G = (F, . . . , F, M ) and m = (m1 , . . . , mn ). This applies for fixed mn to ∼ 2j(n−1) coefficients λj,G m (χQ ), j ∈ N, complemented by estimates for similar terms referring to edges and corners and to j = 0. Inserted in (1.44), this gives λ(χQ ) |bsp,q (Rn ) ∼ λ(χI ) |bsp,q (R).

(2.167)

Then (2.163) follows from Proposition 1.11(i) with s = 1/p and Step 2. For the proof of (2.164), we may assume 0 < p < ∞ and q = ∞. Inserting (2.166) in (1.46), where Lp (Rn ) can be replaced by Lp (Q), one obtains s s λ(χQ ) |fp,∞ (Rn ) ∼ λ(χI ) |fp,∞ (R).

(2.168)

Then (2.164) follows from Proposition 1.11(ii) with 0 < p < ∞, s = 1/p and Step 2. Note that we did not care about what happens at the lower-dimensional faces of Q. While this is not really necessary, it may justify the modification of the proof in the following remark such that this question does not arise.  Remark 2.51. We modify the above proof as follows. Let n ∈ N. For j ∈ N0 , there are at most ∼ 2j(n−1) non-zero coefficients λj,G m (χQ ) according to (2.166). 1/p They are uniformly bounded. Then χQ ∈ Bp,∞ (Rn ) is an immediate consequence 1/p of Proposition 1.11 and bp,∞ (Rn ) according to (1.44). Let n ≥ 2 and p < ∞. As

52

Chapter 2. The essentials, key theorems

above, we reduce the corresponding assertion in Proposition 2.50 to n = 1. Then (2.164) follows in modification of (2.168) from 1/p 1/p (Rn ) ≥ c λ(χI ) |fp,∞ (R) = ∞ λ(χQ ) |fp,∞

(2.169)

for some c > 0 and what we already know. A similar result is obtained for the B-spaces in modification of (2.167) with s = 1/p and q < ∞. In other words, there is no need to discuss what happens at lower-dimensional faces of Q. Remark 2.52. If χQ ∈ Asp,q (Rn ) then it is quite clear that Asp,q (Rn ) cannot have traces on Rn−1 in the understanding of Theorem 2.13. The breaking point in Proposition 2.50 is s = 1/p for all 0 < p ≤ ∞. For p ≥ 1, this is in good agreement with (2.69), while the situation seems to be different for p < 1. We discussed this point in some detail in Section 2.2.3, especially in Remark 2.23, and the outcome is that Proposition 2.50 and traces on (n − 1)-dimensional hyper-planes are still in good agreement if one admits traces on Rn−1 in the context of all spaces Lp (Rn−1 ) with 0 < p < ∞. Remark 2.53. Let again 0 < p, q ≤ ∞ and   1 1  (n) σp,q = n max , , 1 − 1 . p q

  1  σp(n) = n max , 1 − 1 , p

(2.170) (n)

According to Theorem 2.4, the spaces Asp,q (Rn ) with A ∈ {B, F }, s > σp 0 < p, q ≤ ∞ consist entirely of regular distributions. Then j

xj → f x (xj ) = f (x),

xj = (x1 , . . . , xj−1 , xj+1 , . . . , xn )

and

(2.171)

with f ∈ Asp,q (Rn ) makes sense a.e. (almost everywhere), and one can ask whether these spaces have the Fubini property, f |Asp,q (Rn ) ∼

n

xj s

f |Ap,q (R) |Lp (Rn−1 )

(2.172)

j=1

(equivalent quasi-norms), 2 ≤ n ∈ N. According to Theorem 3.25 below, based on [T01, Theorem 4.4, p. 36], the spaces s Fp,q (Rn )

with

0 < p < ∞,

0 < q ≤ ∞,

(n) s > σp,q

(2.173)

s have the Fubini property, whereas the spaces Bp,q (Rn ) have the Fubini property if, and only if, 0 < p = q ≤ ∞. In any case, one can reduce (2.168) from Rn to R based on 1/p 1/p χQ |Fp,∞ (Rn ) ∼ χI |Fp,∞ (R)

if

n−1 < p < ∞. n

(2.174)

2.4. Multipliers, localizations and multiplication algebras

53

Remark 2.54. The conditions for Asp,q (Rn ) in Theorem 2.48 and in Proposition 2.50 1/p

are different at s = 1/p. This is well known, at least for Bp,q (Rn ), and occasionally mentioned in the literature as a preparation for the deeper assertions in Theorem 2.48 for the spaces Asp,q (Rn ) with p < ∞ for the F -spaces. We proved the onedimensional case of Proposition 2.50 for the spaces Asp,q (R) (p < ∞ for F -spaces) r A(Rn ) with dominating in [T10, p. 263] in the larger context of the spaces Sp,q mixed smoothness. We do not deal here with these spaces, but refer to [T10, T19] for definitions and details. Still, we formulate the outcome, which is the same as in Proposition 2.50: r Let Sp,q A(Rn ) with A ∈ {B, F }, r ∈ R and 0 < p, q ≤ ∞ (with p < ∞ for the F -spaces) be the spaces with dominating mixed smoothness according to the Fourier-analytical definition in [T10, pp. 23, 33]. Then 1/p B(Rn ) χQ ∈ Sp,q

if, and only if,

q = ∞.

(2.175)

r A(Rn ) if, and only if, either r < 1/p or as in (2.175). Furthermore, χQ ∈ Sp,q We refer the reader to [T10, Proposition 6.3, pp. 250–251] for a Fourieranalytical proof (where 1 + p1 in [T10, (6.20), (6.21), p. 251] must be replaced by 1 p − 1). In any case, the conditions are the same as in Proposition 2.50 (p < ∞ for F -spaces). One may try to remove the restriction p < ∞ for F -spaces also in the context of the spaces with dominating mixed smoothness and to introduce the r F (Rn ) appropriately. But so far, this has not yet been done. spaces S∞,q

2.4.5 Rough multipliers So far, we dealt with smooth pointwise multipliers in Section 2.4.1 and asked under which circumstances characteristic functions of cubes and half-spaces are pointwise multipliers in Section 2.4.4. Now, we are interested in general properties of pointwise multipliers in a given space A(Rn ). We adopt the same point of view as in the beginning of Section 2.4.3, where we dealt with multiplication algebras. Definition 2.55. Let A(Rn ) be a quasi-Banach space in Rn with S(Rn ) → A(Rn ) → S  (Rn ),

n A(Rn ) ⊂ Lloc 1 (R ).

(2.176)

n n n Then m ∈ Lloc 1 (R ) is said to be a pointwise multiplier for A(R ) if mf ∈ A(R ) n for any f ∈ A(R ) (a.e., almost everywhere) and if there is a constant c > 0 such that mf |A(Rn ) ≤ c f |A(Rn ), f ∈ A(Rn ). (2.177)     n Let M A(R ) be the collection of all pointwise multipliers and let m |M A(Rn )  be the related operator quasi-norm,

    m |M A(Rn )  = sup mf |A(Rn ) : f |A(Rn ) ≤ 1 .

(2.178)

54

Chapter 2. The essentials, key theorems

Remark 2.56. This is the counterpart of Definition 2.37. Let A(Rn ) be a quasiBanach space satisfying (2.176). Then one has the continuous embedding S(Rn ) → A(Rn ) → L1 (Ω) → S  (Rn )

(2.179)

for any bounded domain Ω in Rn . Here, L1 (Ω) is the usual space of Lebesgueintegrable functions in Ω, interpreted as a subspace of S  (Rn ) (extended by zero outside of Ω). This assertion follows from the closed graph theorem, see [Woj91, p. 4], [Rud91, 2.15, p. 51].   Let us collect some well-known properties of M A(Rn ) . Relations between pointwise multiplier spaces and multiplication algebras as introduced in Definition 2.37 are of particular interest for us.   Proposition 2.57. The collection M A(Rn ) of all pointwise multipliers of a translation-invariant quasi-Banach space A(Rn ) satisfying (2.176) is a translationinvariant quasi-Banach space and a multiplication algebra with   (2.180) M A(Rn ) → L∞ (Rn ). Remark 2.58. This result essentially coincides with [T19, Proposition 1.46, p. 42]. The proof given there shows that one has     w ∈ M A(Rn ) , (2.181) w |L∞ (Rn ) ≤ w |M A(Rn ) , with a factor 1. Further comments and references may be found in [T06, Section 2.3.1, pp. 136–137]. Definition 2.59. Let ψ = {ψM }M ∈Zn be a resolution of unity according to (2.130). Let Asp,q (Rn ) with A ∈ {B, F }, s ∈ R and 0 < p, q ≤ ∞ be the spaces according to Definition 1.1. Then Asp,q,unif (Rn ) is the collection of all f ∈ S  (Rn ) such that f |Asp,q,unif (Rn )ψ = sup ψM f |Asp,q (Rn ) M ∈Zn

(2.182)

is finite. Remark 2.60. It follows from Theorem 2.28 that these spaces are independent of the chosen resolution of unity ψ. This justifies our omission of the subscript ψ in (2.182) in the sequel. By the same argument, one can also replace (2.130) by (2.134), ψM (x) ∼ 1, x ∈ Rn . (2.183) M ∈Zn

It follows by standard arguments that Asp,q,unif (Rn ) are quasi-Banach spaces. Furthermore, we have Asp,q (Rn ) → Asp,q,unif (Rn ). (2.184)

2.4. Multipliers, localizations and multiplication algebras

55

Next, we wish to clarify the relations between Asp,q (Rn ), Asp,q,unif (Rn ) and  M Asp,q (Rn ) . By Theorem 2.35, we have so far 

s ∈ R,

s s (Rn ) = F∞,q,unif (Rn ), F∞,q

0 < q ≤ ∞.

(2.185)

It follows from the same theorem that there is no counterpart for the spaces  s (Rn ) with q < ∞. According to Proposition 2.57, all spaces M Asp,q (Rn ) are B∞,q multiplication algebras. n Theorem 2.61. (i) Let Asp,q (Rn ) ⊂ Lloc 1 (R ) according to Theorem 2.4. Then  s  n M Ap,q (R ) is a multiplication algebra and

  M Asp,q (Rn ) → L∞ (Rn ) ∩ Asp,q,unif (Rn ).

(2.186)

  M Asp,q (Rn ) = Asp,q (Rn )

(2.187)

Furthermore, if, and only if, p = ∞, 0 < q ≤ ∞ and s > 0. s (Rn ) be a multiplication algebra according to Theorem 2.41. Then (ii) Let Fp,q

 s  s M Fp,q (Rn ) = Fp,q,unif (Rn ).

(2.188)

Proof. Step 1. We prove part (i). Proposition 2.57 and (2.177) with f = ψM (combined with the translation-invariance of Asp,q (Rn )) justify the first assertions of part (i) including (2.186). Next, we prove (2.187). As remarked in the proof of Theorem 2.10, the function f (x) = 1, x ∈ Rn , does not belong to Asp,q (Rn )   if p < ∞, but to M Asp,q (Rn ) . This reduces (2.187) to p = ∞. In addition, As∞,q (Rn ) must be a multiplication algebra. Then it follows from Theorem 2.41 that s > 0. On the other hand, any m ∈ As∞,q (Rn ) is a pointwise multiplier. As for the converse, we remark that f (x) = 1, x ∈ Rn , is an element of As∞,q (Rn ). This follows from (1.8) and (1.11). Inserted in (2.177), this shows that m ∈ As∞,q (Rn ) for any pointwise multiplier m in As∞,q (Rn ). Step 2. We prove part (ii). The case p = ∞ is covered by (2.187) and (2.185). s (Rn ) is a Let now p < ∞. By (2.186), it remains to prove that any m ∈ Fp,q,unif s n 2 pointwise multiplier for Fp,q (R ). We use Theorem 2.35 with ψM instead of ψM , based on (2.134). Then it follows from s 2 s (Rn )p ∼ ψM mf |Fp,q (Rn )p mf |Fp,q m∈Zn



≤c

s s ψM f |Fp,q (Rn )p · sup ψK m |Fp,q (Rn )p (2.189) K∈Zn

M ∈Zn

≤c



s m |Fp,q,unif (Rn )p

s · f |Fp,q (Rn )p

s that m is a pointwise multiplier in Fp,q (Rn ).



56

Chapter 2. The essentials, key theorems

Remark 2.62. Let us add a technical comment. One first proves (2.189) for mχj with χj (x) = χ(2−j x), χ ∈ D(Rn ), χ(x) = 1 for |x| ≤ 1, and then uses the Fatou property according to Theorem 1.25. If Asp,q (Rn ) is a multiplication algebra, one can complement (2.137) and (2.186) by   Asp,q (Rn ) → M Asp,q (Rn ) → Asp,q,unif (Rn ). (2.190) The question arises how these three spaces are related to each other. So far, we s (Rn ) which are of special interest have (2.185) and (2.188) for the spaces F∞,q in this book. But nowadays, the following (almost) final answers prevail for all multiplication algebras Asp,q (Rn ). Recall that the embedding A0 → A1 is called strict if A0 and A1 do not coincide. Corollary 2.63. Let Asp,q (Rn ) be a multiplication algebra according to Theorem 2.41. Then (2.190) can be strengthened as follows:  s  s s F∞,q (Rn ) = M F∞,q (Rn ) = F∞,q,unif (Rn ) if 0 < q ≤ ∞, s > 0; (2.191)  s  s s (Rn ) = M B∞,q (Rn ) → B∞,q,unif (Rn ) if 0 < q < ∞, s > 0; (2.192) B∞,q  s  s s Fp,q (Rn ) → M Fp,q (Rn ) = Fp,q,unif (Rn )  either s > np , 0 < p < ∞, 0 < q ≤ ∞, if or s = np , 0 < p ≤ 1, 0 < q ≤ ∞;  s  s s Bp,q (Rn ) → M Bp,q (Rn ) = Bp,q,unif (Rn )

if

(2.193)

n ; p (2.194)

0 < p < ∞, p ≤ q ≤ ∞, s >

and  s  s s Bp,q (Rn ) → M Bp,q (Rn ) → Bp,q,unif (Rn )

if

0 < q < p < ∞, s >

n . (2.195) p

All embeddings → in (2.192) to (2.195) are strict. Proof. The equalities in (2.191) follow from (2.185) and (2.188) with p = ∞. The assertions in (2.192) are covered by (2.187), Theorem 2.35 and (2.190). The function f (x) = 1, x ∈ Rn , belongs to the second spaces in (2.193)–(2.195), but not to the first ones. The equality in (2.193) comes from (2.188). The remaining equality in (2.194) and the second strict embedding in (2.195) are the main results of the remarkable paper [NgS18].  Remark 2.64. In [T06, Section 2.3, pp. 136–146], we dealt with pointwise multipliers for the spaces Asp,q (Rn ) preferably with 0 < p ≤ ∞ (p < ∞ for F -spaces),     (n) 0 < q ≤ ∞ and s > σp = n max p1 , 1 − 1 . This applies in particular to the

2.5. Extensions

57

related assertions in Theorem 2.61. Roughly speaking, we extended the consideras (Rn ) which are of particular interest in this book. tions in [T06] to the spaces F∞,q References to the related literature may be found in [T06] and [NgS18], which will not be repeated here. The first equality in (2.191) may also be found in [YSY10, Theorem 6.4, p. 157] with some restrictions for the parameters. Remark 2.65. Compared with Theorem 2.41, the Corollary 2.63 covers all muln/p tiplication algebras Asp,q (Rn ) except the limiting cases Bp,q (Rn ), 0 < p < ∞, 0 < q ≤ 1. By the above arguments, the first strict embeddings in (2.194), (2.195) also apply to the cases  n/p n  n/p Bp,q (Rn ) → M Bp,q (R ) if 0 < p < ∞, 0 < q ≤ 1. (2.196) According to [NgS18, Theorem 1.9, p. 209], one has again the second strict embedding in (2.195) 0 < q < p ≤ 1 and s = n/p. In [NgS18], one also finds a  if s description of M Bp,q (Rn ) for the spaces covered by (2.195), including the just mentioned limiting case.

2.5 Extensions 2.5.1 Introduction and distinguished lifts So far, we dealt with three of the four key problems listed in the beginning of Section 2.2.1: traces (Section 2.2), diffeomorphisms (Section 2.3), and pointwise multipliers (Section 2.4). Now we have a closer look at the extension problem. Let again Rn+ = {x = (x1 , . . . , xn ) ∈ Rn : xn > 0} (2.197) and Rn− = {x = (x1 , . . . , xn ) ∈ Rn : xn < 0}.

(2.198)

The closure Rn+ of Rn+ consists of all x ∈ Rn with xn ≥ 0. As usual, D (Rn+ ) denotes the set of all distributions on Rn+ . Furthermore, g|Rn+ ∈ D (Rn+ ) stands for the restriction of g ∈ S  (Rn ) to Rn+ . Definition 2.66. Let Asp,q (Rn ) with A ∈ {B, F }, s ∈ R and 0 < p, q ≤ ∞ be the spaces according to Definition 1.1. Then   Asp,q (Rn+ ) = f ∈ D (Rn+ ) : f = g|Rn+ for some g ∈ Asp,q (Rn ) , (2.199) f |Asp,q (Rn+ ) = inf g |Asp,q (Rn ),

(2.200)

where the infimum is taken over all g ∈ Asp,q (Rn ) with g|Rn+ = f . Furthermore,   sp,q (Rn ) = f ∈ Asp,q (Rn ) : supp f ⊂ Rn . A + +

(2.201)

58

Chapter 2. The essentials, key theorems

Remark 2.67. These notations will be extended in Definition 2.74 to arbitrary domains (= open sets) Ω in Rn . It follows from standard arguments that Asp,q (Rn+ ) is a quasi-Banach space (and a Banach space for p ≥ 1, q ≥ 1), continuously s (Rn ) embedded in D (Rn+ ) or the restriction of S  (Rn ) to Rn+ . Furthermore, A p,q + is a closed subspace of Asp,q (Rn ). The restriction operator re , S  (Rn ) → D (Rn+ )

re f = f |Rn+ :

(2.202)

generates a linear and bounded map from Asp,q (Rn ) onto Asp,q (Rn+ ). We ask for a linear and bounded extension operator ext, ext :

Asp,q (Rn+ ) → Asp,q (Rn )

(2.203)

such that re ◦ ext = id,

identity in Asp,q (Rn+ ).

(2.204) Asp,q (Rn+ ).

The Of interest are common extension operators for families of spaces (more or less explicit) construction of extension operators is a central topic in the theory of function spaces. There are several methods, for which we will give some references in Remark 2.73 later on. We are again interested in incorporating s (Rn ) into already existing assertions for the spaces Asp,q (Rn ) with the spaces F∞,q p < ∞ for the F -spaces. But it is not so clear to which extent the available methods s for the spaces Asp,q (Rn ) with p < ∞ for F -spaces can be extended to F∞,q (Rn ). We rely here on characteristic functions as pointwise multipliers, combined with distinguished lifts. It follows from Theorem 2.48 that ext,  f (x) if x ∈ Rn+ , extf = (2.205) 0 otherwise, is an extension operator according to (2.203) for all spaces Asp,q (Rn ) with p, s as s (Rn ) with −1 < s < 0 in (2.156) and 0 < q ≤ ∞. This applies in particular to F∞,q and 0 < q ≤ ∞. We wish to transfer this assertion by lifting it to other spaces s F∞,q (Rn ). So far, we have the lift Iδ in (1.94) and Theorem 1.22. But it is quite clear that one needs some modifications which apply simultaneously to Asp,q (Rn ) and Asp,q (Rn+ ). For this purpose, we introduced in [T78, Section 2.10.3, pp. 231–233] the operators Iδ+ , Iδ− , Iδ+ f =



Iδ− f =



iξn + ξ  

−δ

iξn − ξ  

−δ

f f

∨

∨

,

f ∈ S  (Rn ),

(2.206)

,

f ∈ S  (Rn ),

(2.207)

1/2  n−1 (with the obvious modification for n = 1), δ ∈ R. where ξ   = 1 + l=1 |ξl |2 Combining sharp Marcinkiewicz-Fourier multiplier assertions for Lp , 1 < p
0 and σ > 0 a common extension operator for all spaces s s Bp,q (Rn ) with ε < p ≤ ∞, 0 < q ≤ ∞, |s| < σ and all spaces Fp,q (Rn ) with δ ε < p < ∞, ε < q ≤ ∞, |s| < σ. The extension operator ext according to (2.217) applies to the spaces Asp,q (Rn ) satisfying (2.219). This excludes spaces with small p > 0 in Rn , n ≥ 2. The interest in common extension operators comes from applications, for example interpolation. In any case, one can combine the related assertions according to the Steps 1 and 2 for this purpose. Apart from that, interpolation will not play a central role in this book. Remark 2.73. Extensions for spaces on Rn+ or on domains Ω in Rn to their counterparts in Rn play a decisive role in the theory of function spaces of the type Asp,q (Rn ) and their recent offsprings. In [T92, Theorem 4.5.5, pp. 229–235], we used an extended Hestenes method which covers all spaces Asp,q (Rn ) (p < ∞ for F -spaces) simultaneously. This approach avoids lifting, but requires elaborated properties of these spaces. It basically goes back to [T78, Section 2.9.3, pp. 218–219], where s s these constructions are used for the spaces Bp,q (Rn ) and Hps (Rn ) = Fp,2 (Rn ), s ∈ R, 1 < p < ∞, 1 ≤ q ≤ ∞. In this case, one does not need lifting as described in Proposition 2.68. If the classical Hestenes method is used, one first needs the (n) (n) (n) restrictions s > σp for the B-spaces and s > σp,q for the F -spaces with σp , (n) σp,q as in (1.49). This technically simpler approach may be found in [T92, Section 4.5.2, pp. 223–226]. As indicated in [T92, Corollary 4.5.2, p. 225], one can extend these assertions to all spaces Asp,q (Rn ) with p < ∞ for F -spaces. In this context, one may also consult [T83, Section 2.9, pp. 166–173] and the related improvements in [Fra86]. An illustrated description for how the original Hestenes method works for the classical Sobolev spaces Wpk (Rn ), 1 < p < ∞, k ∈ N, may be found in [HT08, Section 3.4]. Finally, we refer the reader to the remarkable construction

62

Chapter 2. The essentials, key theorems

of universal extension operators in [Ryc99a], with [Ryc98] as a forerunner for all spaces Asp,q (Rn ) with p < ∞ for F -spaces. But again, these constructions require rather specific properties of these spaces. The question arises to which extent the above-outlined methods can be carried over to the more general hybrid spaces s (Rn ) as special LrAsp,q (Rn ) as introduced in Definition 1.6, with the spaces F∞,q cases according to Proposition 1.18(iii). We have no references that the sophisticated methods in [Ryc99a] as well as in [T92, Section 4.5.5, pp. 228–235] (extended Hestenes approach) have been applied to the spaces LrAsp,q (Rn ), or at least to s F∞,q (Rn ) – in which case one would not need the lifts as described in Proposition 2.68. The related arguments in [YSY10, Section 6.4.1, pp. 168–172] rely on the classical Hestenes method and the above lifts, basically extending the approach in [T92, Section 4.5.2, pp. 223–226] from some spaces Asp,q (Rn ) to corresponding s spaces LrAsp,q (Rn ). This covers in particular the spaces F∞,q (Rn ). Our method as suggested here is different. Instead of the classical Hestenes method, we used that the characteristic function of the half-space Rn+ is a pointwise multiplier in the spaces Asp,q (Rn ) satisfying (2.216) (and again the lifts as described in Proposition 2.68).

2.6 Spaces on domains 2.6.1 The essentials: Why and how The proof of the pudding is in the eating, says an English proverb. In other words, one should clarify whether a new mathematical development is worth being studied. This might be a matter of non-trivial challenging tasks if one additionally asks for desirable relations to neighbouring fields. We apply this point of view to s s (Rn ) and Fp,q (Rn ) the function spaces as considered in this book. The spaces Bp,q with 0 < p, q ≤ ∞ (p < ∞ for the F -spaces) and s ∈ R according to Definition 1.1 were introduced in the decade from the mid 1960s to the mid 1970s. Related references may be found in Section 4.3.4 below. This Fourier-analytical approach fits very well to problems related to differential equations or pseudodifferential operators (with constant coefficients) on Rn . Furthermore, these spaces unify the already existing (classical) Sobolev spaces and Besov spaces, as will be recalled in the Sections 4.3.1–4.3.3. On the other hand, the Sobolev spaces on (bounded) domains in Rn had been introduced in [Sob50] (based on his papers in the late 1930s) as a tool to deal with partial differential operators in domains, including boundary value problems for elliptic equations. One can take this as a guide and s s ask for spaces Bp,q (Ω) and Fp,q (Ω) in domains Ω in Rn which admit a convincing substantial theory for elliptic boundary value problems on Ω and its boundary ∂Ω. s s First, one has to say what is meant by Bp,q (Ω) and Fp,q (Ω). In (4.87), we recall the intrinsic definition of the Sobolev spaces as introduced in [Sob50]. But this s s does not fit to the more general spaces Bp,q (Rn ) and Fp,q (Rn ) from above. Theres s fore, the related spaces Bp,q (Ω) and Fp,q (Ω) will be defined by a restriction of the

2.6. Spaces on domains

63

s s (Rn ) and Fp,q (Rn ) to Ω. For this task, it is decisive to transfer crucial spaces Bp,q properties for the spaces on Rn to their counterparts on Ω. This was the situation at the end of the 1970s, and it was solved step by step over several years. The outcome is a substantial part of [Tr78, T83] including boundary value problems for regular elliptic differential equations in smooth bounded domains Ω in Rn , [T83, Chapter 4]. As far as the underlying spaces on domains are concerned, one can reduce this task to the so-called key problems as already listed in the beginning of Section 2.2.1:

1. traces on hyper-planes (Section 2.2), 2. diffeomorphisms (Section 2.3), 3. pointwise multipliers (especially Section 2.4.1), 4. extensions (Section 2.5). In addition to these key problems, the essentials cover some more advanced multiplier problems (Sections 2.4.2–2.4.5), embeddings (Section 2.1) and the present Section 2.6 as the proof of the pudding. Recall that we are always s s (Rn ) and F∞,q (Ω) in already existing interested in incorporating the spaces F∞,q assertions for the spaces Asp,q (Rn ) and Asp,q (Ω) with p < ∞ for the F -spaces.

2.6.2 Preliminaries and definitions First, let us fix some notation. Let Ω be an arbitrary domain in Rn , where domain means a non-empty open set. Then Lp (Ω) with 0 < p ≤ ∞ is the standard quasiBanach space of all complex-valued Lebesgue measurable functions f on Ω such that  1/p |f (x)|p dx (2.222) f |Lp (Ω) = Ω

(with the natural modification for p = ∞) is finite. As usual, D(Ω) = C0∞ (Ω) stands for the collection of all complex-valued infinitely differentiable functions on Rn with compact support in Ω. Let D (Ω) be the dual space of all distributions in Ω. Let g ∈ S  (Rn ). We denote by g|Ω its restriction to Ω, g|Ω ∈ D (Ω) :

(g|Ω)(ϕ) = g(ϕ)

for ϕ ∈ D(Ω).

(2.223)

Definition 2.74. Let Ω be an arbitrary domain in Rn with Ω = Rn and let Asp,q (Rn ) with A ∈ {B, F }, s ∈ R, and 0 < p, q ≤ ∞ be the spaces according to Definition 1.1. (i) Then Asp,q (Ω) = {f ∈ D (Ω) : f = g|Ω for some g ∈ Asp,q (Rn )}, f |Asp,q (Ω) = inf g |Asp,q (Rn ), where the infimum is taken over all g ∈

Asp,q (Rn )

with g|Ω = f .

(2.224) (2.225)

64

Chapter 2. The essentials, key theorems

(ii) Let sp,q (Ω) = {f ∈ Asp,q (Rn ) : supp f ⊂ Ω}. A

(2.226)

Then s (Ω)}, sp,q (Ω) = {f ∈ D (Ω) : f = g|Ω for some g ∈ A A p,q s (Ω) = inf g |As (Rn ), f |A p,q p,q

(2.227) (2.228)

s (Ω) with g|Ω = f . where the infimum is taken over all g ∈ A p,q s (Ω) Remark 2.75. This is the extension of Definition 2.66 from Rn+ to Ω. While A p,q s n s s  is a closed subspace of Ap,q (R ), the spaces Ap,q (Ω) and Ap,q (Ω) must be consid  ered in the context of the dual pairing D(Ω), D (Ω) . It follows from standard arguments that they are quasi-Banach spaces (Banach spaces if p ≥ 1, q ≥ 1). The formulations coincide more or less with [T08, Section 2.1.1, pp. 28–30], now exs (Rn ) and its counterpart on Ω. But in contrast to [T08], we do not tended to F∞,q deal with the spaces Asp,q (Ω) in rough domains and their peculiarities. Instead, we assume that Ω is a bounded smooth domain. Then the study of the spaces Asp,q (Ω) can be reduced to the corresponding assertions for the spaces Asp,q (Rn ) and Asp,q (Rn+ ) based on the key problems as listed in the beginning of Section 2.2.1 and also discussed in the preceding Section 2.6.1: traces (Theorem 2.13), diffeomorphisms (Theorem 2.25), extensions (Theorem 2.71) and smooth pointwise multipliers (Theorem 2.28). In all these theorems, we naturally incorporated the s spaces F∞,q (Rn ) in already existing assertions for the spaces Asp,q (Rn ) with p < ∞ for the F -spaces. The reduction of assertions for the spaces on Ω to corresponding ones on Rn and Rn+ is standard. We have nothing essentially new to say, the exception being that the previous restriction p < ∞ for F -spaces is now deleted. But it is reasonable to fix the underlying definitions as well as the outcome. Let Ω be an (arbitrary) domain in Rn . Let m ∈ N0 . Then C m (Ω) is the collection of all (complex-valued) continuous functions in Ω having classical continuous derivatives Dα f with |α| ≤ m, which can be extended continuously to Ω such that f |C m (Ω) = sup |Dα f (x)| < ∞. (2.229) |α|≤m

Let C(Ω) = C 0 (Ω) and C ∞ (Ω) =

x∈Ω

∞ !

C m (Ω).

(2.230)

m=0

Definition 2.76. (i) Let n ∈ N, n ≥ 2 and k ∈ N. Then a special C k -domain in Rn is the collection of all points x = (x , xn ) ∈ Rn with x ∈ Rn−1 such that h(x ) < xn < ∞, where h ∈ C k (Rn−1 ).

(2.231)

2.6. Spaces on domains

65

(ii) Let n ∈ N, n ≥ 2 and k ∈ N. Then a bounded C k -domain in Rn is a bounded connected domain in Rn whose boundary ∂Ω can be covered by finitely many open balls Kj in Rn , j = 1, . . . , J , centered at ∂Ω such that Kj ∩ Ω = Kj ∩ Ωj ,

with

j = 1, . . . , J,

(2.232)

where Ωj are rotations of suitable special C k -domains in Rn . (iii) Let n ∈ N, n ≥ 2. If Ω is a bounded C k -domain for every k ∈ N, then it is called a bounded C ∞ -domain. (iv) For n = 1, the term bounded C ∞ -domain simply means an open bounded interval. Remark 2.77. This definition basically coincides with [HT08, Section A.2, pp. 246– 247], where one also finds an illustration and the usual definition of what is meant by the outer normal ν(σ), σ ∈ ∂Ω, that will be needed in the sequel. The above definition ensures that the boundary Γ = ∂Ω of a bounded C ∞ domain in Rn , n ≥ 2, is a compact d-dimensional C ∞ -manifold with d = n − 1. In particular, there are an atlas {Vm , ψm }M m=1 , M ∈ N, consisting of open sets Vm in "M Γ such that m=1 Vm = Γ and homeomorphic maps ψm , ψm :

Vm ⇐⇒ Um = ψm (Vm ) ⊂ Rd ,

m = 1, . . . , M,

(2.233)

d

of Vm onto connected bounded open sets Um in R such that     −1 if Vm ∩ Vk = ∅ : ψm Vm ∩ Vk ⇐⇒ ψk Vm ∩ Vk ψk ◦ ψ m

(2.234)

are diffeomorphic C ∞ -maps with positive Jacobians. In this context, (2.234) are the usual compatibility conditions for overlapping open sets Vm . We furnish this ddimensional compact C ∞ -manifolds with the d-dimensional Hausdorff measure μ. The image measure ψm [μ] restricted to Vm is of course equivalent to the Lebesgue measure in Um . In this approach, we followed [T08, Section 5.1.2, pp. 132–133], where one also finds related references, in particular to [Mat95, Section 4]. Let Lp (Γ) with 0 < p ≤ ∞ be the collection of all complex-valued μ-measurable functions on Γ such that  1/p |f (γ)|p μ(d γ) (2.235) f |Lp (Γ) = Γ

is finite (with the usual modification for p = ∞). Additionally, one can lift func−1 . tions, distributions and function spaces from Rd to Γ via the above mappings ψm This applies in particular to the space of test functions D(Γ) = C ∞ (Γ) and its dual D (Γ), the space of distributions on Γ. Let {χm }M m=1 ⊂ D(Γ) be a resolution of unity such that supp χm ⊂ Vm

and

M m=1

χm (γ) = 1

if γ ∈ Γ.

(2.236)

66

Chapter 2. The essentials, key theorems

If f ∈ D (Γ) then −1 ∈ D (Um ) ∩ S  (Rd ), (χm f ) ◦ ψm

m = 1, . . . , M,

(2.237)

(extended by zero outside of Um and appropriately interpreted). Definition 2.78. Let Γ be a compact d-dimensional C ∞ -manifold, d ∈ N. Let Asp,q (Rd ) with A ∈ {B, F }, s ∈ R and 0 < p, q ≤ ∞ be the spaces according to Definition 1.1. Then   −1 Asp,q (Γ) = f ∈ D (Γ) : (χm f ) ◦ ψm ∈ Asp,q (Rd ), m = 1, . . . , M (2.238) and f |Asp,q (Γ) =

M

−1

(χm f ) ◦ ψm |Asp,q (Rd ).

(2.239)

m=1

Remark 2.79. We followed again [T08, Section 5.1.2, pp. 132–139] where we dealt with wavelet expansions for these spaces. This will not be repeated here. Now, we s (Γ). It follows by standard arguments, based on diffeomorphic incorporated F∞,q maps and pointwise multipliers according to the Theorems 2.25 and 2.28, that these spaces are independent of the compatible atlases (local charts) and related resolutions of unity (equivalent quasi-norms).

2.6.3 Traces Let Ω be a bounded C ∞ -domain in Rn , n ≥ 2, as described in Definition 2.76 and let Asp,q (Ω) and Asp,q (Γ) with Γ = ∂Ω be the related spaces according to the Definitions 2.74 and 2.78. We are interested in the counterpart of Theorem 2.13 on the trace. Similar to there, let s−

  1  1 > σp(n−1) = (n − 1) max , 1 − 1 . p p

(2.240)

Theorem 2.4 and the above localization ensure that s− 1

Bp,q p (Γ) → L1 (Γ)

(2.241)

for all possible trace spaces. Let S(Ω) = S(Rn )|Ω be the pointwise restriction of S(Rn ) to Ω and let tr Γ ϕ = ϕ(γ),

γ ∈ Γ,

ϕ ∈ S(Ω),

(2.242)

be the related pointwise trace on Γ = ∂Ω. Then it makes sense to ask for a positive constant such that tr Γ ϕ |L1 (Γ) ≤ c ϕ |Asp,q (Ω)

for all ϕ ∈ S(Ω).

(2.243)

2.6. Spaces on domains

67

The above localization shows that we are otherwise in the same situation as in the beginning of Section 2.2.2, including Fatou arguments. We are interested in a counterpart of Theorem 2.13, asking again for the trace space tr Γ Asp,q (Ω) and a related linear and bounded extension operator extΓ from this trace space to Asp,q (Ω) such that tr Γ ◦ extΓ = id,

identity in tr Γ Asp,q (Ω).

(2.244)

Theorem 2.80. Let Ω be a bounded C ∞ -domain in Rn , n ≥ 2. Let Asp,q (Ω) be the above spaces with A ∈ {B, F }, 0 < p ≤ ∞, 0 < q ≤ ∞ and   1  1 (2.245) s − > σp(n−1) = (n − 1) max , 1 − 1 . p p s− 1

Let Bp,q p (Γ) with Γ = ∂Ω be the corresponding spaces according to the Definitions 2.74 and 2.78. Then s− 1 s (Ω) → Bp,q p (Γ) (2.246) tr Γ : Bp,q and tr Γ :

s− 1

s Fp,q (Ω) → Bp,p p (Γ).

(2.247)

Furthermore, there are linear and bounded extension operators extΓ with s− 1

tr Γ ◦ extΓ = id, such that

identity in Bp,q p (Γ), s− 1

extΓ :

s Bp,q p (Γ) → Bp,q (Ω)

extΓ :

s Bp,p p (Γ) → Fp,q (Ω).

and

s− 1

(2.248) (2.249) (2.250)

Proof. This is the counterpart of Theorem 2.13 and also a special case of [T08, Theorem 5.14, p. 143] based on [T08, Theorem 5.9, pp. 136–137]. There, the arguments rely on the same constructions as used in Definition 2.78 and can now also s s (Ω) and their trace spaces C s (Γ) = B∞,∞ (Γ) with be applied to the spaces F∞,q s > 0.  Remark 2.81. Similarly, one can transfer the discussion in Remark 2.14 about common extension operators to the above situation. This also applies to Remark 2.16. In the same way as there, one can now ask not only for traces tr Γ f of functions but also for the counterpart of (2.90), # $ ∂j f : 0≤j≤r , r ∈ N, (2.251) tr rΓ : f → tr Γ j ∂ν where ν indicates the outer normal as defined in [HT08, p. 247]. This extends [T08, Theorem 5.14, p. 143] to tr rΓ :

s F∞,q (Ω) →

r  j=0

C s−j (Γ),

s > r ∈ N0 .

(2.252)

68

Chapter 2. The essentials, key theorems

But this will not be done here in detail.

2.6.4 Extensions Let Ω be a bounded C ∞ -domain in Rn as described in Definition 2.76 (respectively a bounded interval for n = 1) and let Asp,q (Ω) be the related spaces introduced in Definition 2.74 by restriction. Then the restriction operator re Ω , re Ω (f ) = f |Ω :

S  (Ω) → D (Ω)

(2.253)

generates a linear and bounded map from Asp,q (Rn ) onto Asp,q (Ω). It is the counterpart of re in (2.202). Similar to there, one may ask for linear and bounded extension operators extΩ , extΩ :

Asp,q (Ω) → Asp,q (Rn )

(2.254)

such that re Ω ◦ extΩ = id,

identity in Asp,q (Ω).

(2.255)

Asp,q (Ω)

has the extension property if there exists an exWe say that the space tension operator extΩ satisfying (2.254) and (2.255). As far as common extension operators for families of spaces are concerned, we add a comment in Remark 2.83 below. Theorem 2.82. Let Ω be a bounded C ∞ -domain in Rn . Then all spaces Asp,q (Ω) with A ∈ {B, F }, s ∈ R and 0 < p, q ≤ ∞ according to Definition 2.74(i) have the extension property. Proof. This follows from Theorem 2.71 and the localizations as described in Section 2.6.2 in the same way as in [T92, Section 5.1.3, pp. 238–240].  Remark 2.83. The comments about common extension operators in Remark 2.72 can be transferred from Rn+ to bounded C ∞ -domains Ω. s -spaces Remark 2.84. It is one of the main aims of this book to incorporate F∞,q s with s ∈ R, 0 < q ≤ ∞ into already existing assertions for Ap,q -spaces with s ∈ R, 0 < p ≤ ∞ (p < ∞ for F -spaces), 0 < q ≤ ∞. As far as extensions of corresponding spaces on Rn+ and on bounded C ∞ -domains to corresponding spaces on Rn are concerned, the Theorems 2.71 and 2.82, complemented by the Remarks 2.72 and 2.83, give first satisfactory answers. But we relied on rather specific arguments, such as (2.215) based on characteristic functions as pointwise multipliers, distinguished lifts in (2.217), and the standard localization technique for smooth domains as described in Section 2.6.2. In Remark 2.73, we already discussed several extension methods and wondered to which extent they can be used for the above purposes. But this is not so clear. For any bounded Lipschitz domain Ω in Rn , there are universal extension operators for all spaces Asp,q (Ω) with A ∈ {B, F }, s ∈ R, 0 < p ≤ ∞ (p < ∞ for F -spaces) and 0 < q ≤ ∞. This is

2.6. Spaces on domains

69

the main result of [Ryc99a]. Further references and discussions may be found in [T06, Section 1.11.5, pp. 64–66]. It would be of interest to incorporate the spaces s (Ω), s ∈ R, 0 < q ≤ ∞ into this assertion. But the sophisticated properties F∞,q s used in [Ryc99a] are apparently not (yet) available for the spaces F∞,q (Ω). Finally, we refer the reader to [T08, Chapter 4] where we dealt with the extension problem for spaces Asp,q (Ω) (p < ∞ for F -spaces) in rough domains. Again, one can ask s (Ω) can be incorporated into the related assertions. But whether the spaces F∞,q in [T08], we used sophisticated wavelet frames in rough domains, and there has s been no counterpart for the spaces F∞,q (Ω) so far. An examination of the details shows, however, that there is a good chance for incorporating the related spaces s F∞,q (Ω) at least as far as they are covered by Theorem 3.9 (local homogeneity) and Theorem 3.15 (refined localization spaces) below. It would be of interest to check this proposal. More generally, it seems to be possible to extend [T08, Sections 4.1, 4.2, pp. 101–108], based on [T08, Section 3.2, pp. 77–91], from the related global spaces Asp,q (Ω) to the corresponding hybrid spaces LrAsp,q (Ω). This would complement [T13, T14, YSY10] in a decisive way.

2.6.5 Embeddings According to Definition 2.74, the spaces Asp,q (Ω) on domains Ω in Rn are restrictions of Asp,q (Rn ) to Ω. Therefore, many properties of the spaces Asp,q (Rn ) can be immediately transferred to the spaces Asp,q (Ω). This also applies to embeddings and pointwise multipliers, including multiplication algebras, as considered in the Sections 2.1 and 2.4. But if Ω is bounded, some new aspects arise. For example, one can ask which continuous embeddings between spaces on Ω are compact and how the degree of compactness can be measured in terms of, say, entropy numbers or approximation numbers. This has been done in many books and papers, including, for example, [ET96] and [T06]. The Ω-version of (2.45), (2.46) shows that nothing new can be expected once the corresponding assertions are extended s (Ω). But we wish to illuminate the differences between spaces on Rn and to F∞,q on bounded domains Ω by having a closer look at limiting embeddings, again s including F∞,q (Ω) naturally. Theorem 2.85. (i) Let s0 ∈ R, s1 ∈ R and 0 < p0 , p1 , q0 , q1 ≤ ∞. Let independently Asp00 ,q0 (Rn ) be either Bps00 ,q0 (Rn ) or Fps00,q0 (Rn ) and Asp11 ,q1 (Rn ) be either Bps11 ,q1 (Rn ) or Fps11,q1 (Rn ) such that Asp00 ,q0 (Rn ) → Asp11 ,q1 (Rn ).

(2.256)

Then p0 ≤ p1 . (ii) Let Ω be a bounded C ∞ -domain in Rn . Let s ∈ R, 0 < p1 ≤ p0 ≤ ∞ and 0 < q0 , q1 ≤ ∞. Then Bps0 ,q0 (Ω) → Bps1 ,q1 (Ω)

if, and only if, q0 ≤ q1 ,

(2.257)

70

Chapter 2. The essentials, key theorems and Fps0 ,q0 (Ω) → Fps1 ,q1 (Ω)

if, and only if, q0 ≤ q1 .

(2.258)

Proof. Step 1. We prove part (i). From (1.8) and (2.38), it follows that f (x) = 1, x ∈ Rn , is an element of all spaces As∞,q (Rn ), s ∈ R, 0 < q ≤ ∞, but does not belong to any space Asp,q (Rn ) with p < ∞. This proves part (i) for p0 = ∞. Let now 0 < p1 < p0 < ∞. By (2.38), it is sufficient to deal with the B-spaces. 0,G We rely on Proposition 1.11(i) assuming λj,G m = 0 with the exception of λm , n G = (M, . . . , M ), m ∈ Z , such that {λ0,G m }m∈Zn ∈ p0 ,

{λ0,G m }m∈Zn ∈ p1 .

(2.259)

This disproves (2.256) if p1 < p0 . Step 2. We prove part (ii). According to Theorem 2.30, one can multiply the extension operator in Theorem 2.82 with a smooth compactly supported cut-off function that has the value 1 in Ω. Then it follows by standard arguments that it is sufficient to deal with functions f ∈ Asp,q (Rn ) having wavelet expansions j j λj,G supp ψG,m ⊂ ω, (2.260) f= m ψG,m , j∈N0 ,G∈Gj , m∈Zn

in the related spaces for some bounded domain ω, based on the Propositions 1.11, 1.16 and (1.71). Let us first justify the if-assertions. For this, let f ∈ Fps0 ,q0 (Rn ) with f as in (2.260) and p0 < ∞. One can replace Lp0 (Rn ) in (1.46) by Lp0 (ω). Then (2.258) follows from Hölder’s inequality and the monotonicity of the q spaces. If p0 = ∞ and p1 < ∞, then one may choose p = p1 in (1.71), (1.46) and (1.61) with the same outcome. For p0 = p1 = ∞, one can again rely on (1.61) for some p, 0 < p < ∞, and the monotonicity of the q -spaces. Let f ∈ Bps0 ,q0 (Rn ) with f as in (2.260). Then one can use either (1.44) directly or convert the p -sum multiplied with 2−jn/p into an Lp -integral over ω and use the same arguments as above. Step 3. It remains to prove the only-if-assertions in part (ii). Let f=

∞ j=0

λj



j ψG,m ,

G = (M, . . . , M ),

(2.261)

0≤ml ≤2j

be a special case of (2.260). By (1.44), one has s (Rn ) ∼ f |Bp,q

∞ 

2jsq |λj |q

1/q

(2.262)

j=0

(with the usual modification for q = ∞). If q1 < q0 then there are numbers λj > 0 such that and f∈ / Bps1 ,q1 (Rn ). (2.263) f ∈ Bps0 ,q0 (Rn )

2.6. Spaces on domains

71

This proves the only-if-part of (2.257). Let p < ∞. Inserting (2.261) in (1.46), one obtains

 ∞

1/q



s jsq q n

f |Fp,q (Rn ) ∼

2 |λ | χ (·) |L (R ) j j,m p



j=0 0≤ml ≤2j

∞

 1/q



jsq q ∼

2 |λj | |Lp (Q)

∼

j=0 ∞ 

2jsq |λj |q

(2.264)

1/q

j=0

where Q is a suitably chosen cube in Rn . This is the same result as in (2.262). It proves the only-if-part of (2.258) for p0 < ∞. Let now p0 = ∞ and p1 < ∞. Then the assumed embedding in (2.258) and (2.38) requires that s s B∞,q (Rn ) → F∞,q (Rn ) → Fps1 ,q1 (Rn ). 0 0

(2.265)

Now q0 ≤ q1 follows from (2.262) and (2.264). The remaining case p0 = p1 = ∞ is covered by (2.43) and (2.44).  Remark 2.86. Part (ii) with p0 < ∞ for the F -spaces is essentially covered by [T19, Theorem 1.50, Corollary 1.51, Remark 1.52, pp. 45–47 ]. This also applies to the construction (2.261) to (2.264). There, one finds further comments and references, as well as related counterparts for spaces with dominating mixed smoothness. Remark 2.87. One can combine the above assertions with Theorem 2.5. Let n ∈ N, s ∈ R and 0 < p ≤ ∞. Let a+ = max(a, 0) for a ∈ R. Let # 1 Dn1 (p, s) = (r, σ) : p ≤ r ≤ ∞, σ < s − n p #  1 Dn2 (p, s) = (r, σ) : p ≤ r ≤ ∞, σ ≤ s − n p # 1 Dn3 (p, s) = (r, σ) : 0 < r ≤ ∞, σ < s − n p # 1 Dn4 (p, s) = (r, σ) : 0 < r ≤ ∞, σ ≤ s − n p

1 $ , r 1 $ − , r 1 $ − + , r 1 $ − + , r −

(2.266) (2.267) (2.268) (2.269)

as illustrated in Figure 2.1, p. 72. Let s0 ∈ R, s1 ∈ R and 0 < p0 , p1 , q0 , q1 ≤ ∞. Let independently Asp00 ,q0 (Rn ) be either Bps00 ,q0 (Rn ) or Fps00,q0 (Rn ) and Asp11 ,q1 (Rn ) be either Bps11 ,q1 (Rn ) or Fps11,q1 (Rn ). Then Asp00 ,q0 (Rn ) → Asp11 ,q1 (Rn )

if

(p1 , s1 ) ∈ Dn1 (p0 , s0 ).

(2.270)

72

Chapter 2. The essentials, key theorems

σ

σ ( p1 , s)

s

s−

s

s−

n p

1 p

( p1 , s)

n p

1 r

1 p

Dn1

1 r

Dn3

Figure 2.1: Embeddings If (p1 , s1 ) ∈ / Dn2 (p0 , s0 ) then there is no embedding of Asp00 ,q0 (Rn ) into Asp11 ,q1 (Rn ). Let Ω be a bounded C ∞ -domain in Rn . Then Asp00 ,q0 (Ω) → Asp11 ,q1 (Ω)

if

(p1 , s1 ) ∈ Dn3 (p0 , s0 ).

(2.271)

If (p1 , s1 ) ∈ / Dn4 (p0 , q0 ) then there is no embedding of Asp00 ,q0 (Ω) into Asp11 ,q1 (Ω). This follows from the Theorems 2.5 and 2.9, their local versions with Ω instead of Rn as explicitly stated in [T06, p. 60] and Theorem 2.85. These assertions show the typical differences between spaces on Rn and spaces on bounded domains. The above-mentioned theorems indicate what can be expected in the limiting cases (p1 , s1 ) ∈ Dn2 (p0 , s0 )\Dn1 (p0 , s0 )

and

(p1 , s1 ) ∈ Dn4 (p0 , s0 )\Dn3 (p0 , s0 ). (2.272)

s s (Ω) are always naturally included. (Rn ) and F∞,q The spaces F∞,q

2.6.6 Characteristics of distributions s (Rn ) with s ∈ R and 0 < p ≤ ∞ has a compact support, then it follows If f ∈ Bp,∞ from Corollary 2.7 and Theorem 2.85(ii) that σ f ∈ Br,∞ (Rn ),

0 < r ≤ ∞,

σ ≤s−

1 1 − . p r +

(2.273)

2.6. Spaces on domains

73

s

sf (t)

( p1 , s) s

s−

n p

1 p

t=

1 p

Figure 2.2: Characteristics In other words, f belongs to all spaces covered by the region Dn4 (p, s) according s to (2.269). But for an individual compactly supported element f ∈ Bp,∞ (Rn ), the situation might be much better. This question is of no interest if f ∈ D(Rn ) = C0∞ (Rn ) is a compactly supported C ∞ -function. In order to exclude this case, one introduces the so-called singular support of f ∈ S  (Rn ) given by   sing supp f = x ∈ Rn : f |B(x, τ ) = C ∞ for any τ > 0 , (2.274) where again B(x, τ ) is the open ball centered at x ∈ Rn and of radius τ , and f |Ω ∈ D (Ω) is the restriction of f to Ω. Definition 2.88. Let f ∈ S  (Rn ) with supp f compact

and

sing supp f = ∅.

(2.275)

0 ≤ t = 1/p < ∞,

(2.276)

Then   s sf (t) = sup s : f ∈ Bp,∞ (Rn ) , is called the characteristic of f . Remark 2.89. If f ∈ Bpsll ,∞ (Rn ) with l = 0, 1, then one has by (1.8) and Hölder’s inequality that s (Rn ), f ∈ Bp,∞

1−θ 1 θ = + , p p0 p1

s = (1 − θ)s0 + θs1 ,

(2.277)

0 < θ < 1. In combination with the discussions in Remark 2.87, it follows that sf (t), 0 ≤ t = 1/p < ∞ is a concave increasing (= not decreasing) continuous

74

Chapter 2. The essentials, key theorems

function with a slope of at most n, as shown in Figure 2.2, p. 73. There is a converse, which will be discussed in the following theorem. Let again |M | be the Lebesgue measure of a Lebesgue measurable set M in Rn . Theorem 2.90. Let s(t) for 0 ≤ t < ∞ be a real continuous increasing (= not decreasing) concave function with a slope that is smaller than or equal to n. Then there is an f ∈ S  (Rn ) with (2.275) and sf (t) as in (2.276) such that sf (t) = s(t), Furthermore,   sing supp f  = 0

and

0 ≤ t < ∞.

s(t) (Rn ), f ∈ Bp,∞

(2.278)

p = 1/t.

(2.279)

Remark 2.91. This assertion is covered by [T06, Theorem 7.48, p. 324]. Its proof relies on compactly supported fractal Radon measures in Rn , interpreted as elements of S  (Rn ), where both assumptions in (2.275) play a decisive role. But according to [Ved07], there is an alternative way based on characteristics sf (t) for C ∞ -functions in Rn without support conditions. This is a global matter. In addition to the above curves s(t), there are now some other possibilities. These s observations have been extended to the weighted spaces Bp,q (Rn , wα ) collecting  n all f ∈ S (R ) such that s s (Rn , wα ) = wα f |Bp,q (Rn ) f |Bp,q

(2.280)

is finite, where wα (x) = (1 + |x|2 )α/2 , x ∈ Rn , α ∈ R, s ∈ R and 0 < p, q ≤ ∞. For fixed p and q, one has according to [Kab08] that S(Rn ) =

!

s Bp,q (Rn , wα ),

α,s∈R

S  (Rn ) =

%

s Bp,q (Rn , wα ).

(2.281)

α,s∈R

This shows that the assertions in [Ved07] cover all C ∞ -functions f ∈ S  (Rn ). The proofs in [T06, Ved07] rely on wavelet arguments.

2.7 Multipliers: Further properties 2.7.1 Preliminaries   Let M Asp,q (Rn ) be the collection of all pointwise multipliers for the spaces Asp,q (Rn ) as introduced in Definition 2.55. In Corollary 2.63 and Theorem 2.30, we specified the basic assertion   (2.282) M Asp,q (Rn ) → L∞ (Rn ) ∩ Asp,q,unif (Rn )

2.7. Multipliers: Further properties

75

according to Theorem 2.61. In particular, if A ∈ {B, F } and 0 < p, q ≤ ∞ then C (Rn ) · Asp,q (Rn ) → Asp,q (Rn ) (n)

where again σp

if

 > s > σp(n) ,

(2.283)

  = n max( p1 , 1) − 1 . Recall that σ σ C σ (Rn ) = B∞,∞ (Rn ) = F∞,∞ (Rn ),

σ ∈ R,

(2.284)

see (1.16). We now wish to employ some of the assertions of the preceding sections for a further discussion of pointwise multipliers. In particular, it follows from (2.282) and the local embeddings as described in Remark 2.87 that (2.283) cannot be extended to  < s. As far as the limiting case  = s is concerned, one has the following assertion. Proposition 2.92. Let A ∈ {B, F }, 0 < p ≤ ∞ and s > n/p. Then C s (Rn ) · Asp,q (Rn ) → Asp,q (Rn )

(2.285)

if, and only if, q = ∞. Proof. According to Theorem 2.85(ii), one has for bounded C ∞ -domains Ω that C s (Ω) → Asp,q (Ω)

if, and only if, q = ∞.

(2.286)

Then (2.282) in combination with standard arguments disproves (2.285) for the case q < ∞. On the other hand, (2.284), (2.286) and Corollary 2.63 show that C s (Rn ) · Asp,∞ (Rn ) → Asp,∞ (Rn ),

0 < p ≤ ∞,

s > n/p.

(2.287) 

Remark 2.93. In addition to (2.191)–(2.193), we relied on the substantial assertion (2.194) where the above restriction s > n/p comes from. In the case of the F s spaces, one can replace s > n/p by the assumption that Fp,q (Rn ) is a multiplication algebra, which coincides with (2.191), (2.193). The main aim of this Section 2.7 is to ask for counterparts of (2.287) with q instead of ∞, extending (2.285) to Asp1 ,q (Rn ) · Asp0 ,q (Rn ) → Asp0 ,q (Rn ),

(2.288)

where 0 < p0 ≤ p1 ≤ ∞ and 0 < q ≤ ∞. Of course, (2.288) must always be interpreted as f0 f1 |Asp0 ,q (Rn ) ≤ c f0 |Asp0 ,q (Rn ) · f1 |Asp1 ,q (Rn )

(2.289)

for some c > 0 and all f0 ∈ Asp0 ,q (Rn ), f1 ∈ Asp1 ,q (Rn ). If p0 = p1 = p then (2.288) means that Asp,q (Rn ) is a multiplication algebra. Then according to Theorem 2.41, one has a final answer. As will be briefly discussed below, the case p1 < p0 does not

76

Chapter 2. The essentials, key theorems

make sense. In other words, only assertions of the type (2.288) with p0 < p1 ≤ ∞ are of interest, as also suggested by the local version of (2.288) and the local embeddings in Theorem 2.85 with s > n/p0 . But the step from bounded domains to Rn seems to be a substantial task. The only related assertion known to us may be found in [T83, Remark 2.8.2/1, p. 143]: Let s > 0, 1 ≤ p ≤ ∞ and 0 < q ≤ ∞. Then s s s B∞,q (Rn ) · Bp,q (Rn ) → Bp,q (Rn ). (2.290) This result is essentially a by-product of assertions of the type (2.283) proved in [T83, Section 2.8.2, pp. 140–145], relying on the heavy machinery of paramultiplication as described in Section 4.3.4 below. We are mainly interested in bilinear maps of the type (2.288). In other words, using (2.282) we ask for conditions that ensure   (2.291) Asp1 ,q (Rn ) → M Asp0 ,q (Rn ) → L∞ (Rn ) ∩ Asp0 ,q,unif (Rn ), 0 < p0 ≤ p1 ≤ ∞ and 0 < q ≤ ∞. If p1 < p0 then it follows from Remark 2.87 that there is no local embedding of Asp1 ,q (Rn ) into Asp0 ,q (Rn ), and nothing like (2.291) can be expected. Proposition 2.39 and the Theorems 2.3, 2.41 suggest that multiplication algebras are the first candidates for this task. Proposition 2.92 with q = ∞ may be considered as a further example. In order to provide a better understanding, it seems reasonable to discuss multiplication properties of the type (2.288) in a somewhat larger context. We concentrate on F -spaces for reasons which will be explained in the following Section 2.7.2. Let s (Rn ) ≤ c f0 |Fps0 ,q (Rn ) · f1 |Fps1 ,q (Rn ) f0 f1 |Fp,q

(2.292)

for all f0 ∈ Fps0 ,q (Rn ) and f1 ∈ Fps1 ,q (Rn ), where 0 < p ≤ p0 ≤ p1 < ∞

and

s > σp(n)

(2.293)

s (Rn )ϕ be the homogeneous counterpart of (1.10) as as in (2.283). Let f |F˙p,q described in (4.7) below. According to (4.2), one has the global homogeneity n s s (Rn ) ∼ λs− p f |F˙ p,q (Rn ), f (λ·) |F˙ p,q

λ > 0,

(2.294)

where the equivalence constants are independent of f and λ. Furthermore, if 0 < (n) p < ∞, 0 < q ≤ ∞ and s > σp , then s s (Rn ) ∼ f |Lp (Rn ) + f |F˙ p,q (Rn ) f |Fp,q

(2.295)

are equivalent quasi-norms, see [T92, Theorem 2.3.3, p. 98]. By inserting f0 (λ·) and f1 (λ·) in (2.292), one obtains from (2.294) and (2.295) that s (Rn ) λ− p f0 f1 |Lp (Rn ) + λs− p f0 f1 |F˙ p,q 1    −n s− n ∼ λ pj fj |Lpj (Rn ) + λ pj fj |F˙ psj ,q (Rn ) . n

n

j=0

(2.296)

2.7. Multipliers: Further properties

77

Then

n  1 n 1 1 n  ≤ s− ≤ + s− and + p p0 p1 p p0 p1 follow from λ → ∞ and λ → 0 in (2.296). This can be rewritten as s−

1 1 1 + − p0 p1 p

s≥n

and

1 1 1 ≤ + . p p0 p1

(2.297)

(2.298)

In particular, we have  1 1 n + − 1 ≥ s − + n > 0. p0 p1 p

2s − n

(2.299)

This shows that (2.298) recovers the necessary conditions ensuring (2.292) with (2.293) according to [SiT95, Theorem 4.1.1, p. 116] and [RuS96, Section 4.8.1, p. 232]. These conditions, with the possible exceptions of limiting cases, are also sufficient.     (n) Proposition 2.94. Let 0 < p < p0 ≤ p1 < ∞ and s > σp = n max p1 , 1 − 1 . Let 0 < q ≤ ∞, s>n

1 1 1 + − p0 p1 p

and

1 1 1 + . < p p0 p1

(2.300)

Then s (Rn ). Fps1 ,q (Rn ) · Fps0 ,q (Rn ) → Fp,q

(2.301)

Proof. This is a special case of [SiT95, Theorem 4.4.1, p. 120] and [RuS96, Theorem 4.8.2/2, p. 239].  Remark 2.95. The above references also cover Proposition 2.94 with B instead of F , where additionally p < p0 ≤ p1 = ∞ can be admitted. But our main aim is to ask for assertions of the type (2.288) and (2.291). In addition to multiplication algebras, we are interested in conditions that ensure (2.291) for p0 < p1 ≤ ∞, extending (2.301) (and its B-counterpart) to p = p0 < p1 ≤ ∞. Proposition 2.92 with q = ∞ may be considered a first example. Apart from that, it seems to be a rather tricky task. The following Section 2.7.2 deals with this question mainly in the context of F -spaces. The above discussions can be seen as an introduction to this topic.

2.7.2 Main assertions In this Section, we wish to extend (2.301) to 0 < p = p0 ≤ ∞ and all p1 with p0 ≤ p1 ≤ ∞. Then Fps0 ,q (Rn ) must be a multiplication algebra and one has (2.288), (2.289) and (2.291) with A = F . Recall that Fps0 ,q (Rn ) is a multiplication algebra according to Definition 2.37 and Theorem 2.41 if, and only if,  either s > n/p0 where 0 < p0 , q ≤ ∞, (2.302) or s = n/p0 where 0 < p0 ≤ 1, 0 < q ≤ ∞.

78

Chapter 2. The essentials, key theorems

s (Rn ), as always in this In contrast to Proposition 2.94, we now incorporate F∞,q book.

Theorem 2.96. Let 0 < p0 ≤ p1 ≤ ∞ and 0 < q ≤ ∞. Let s be as in (2.302). Then Fps1 ,q (Rn ) · Fps0 ,q (Rn ) → Fps0 ,q (Rn ).

(2.303)

Proof. So far, we know that Fps0 ,q (Rn ) is a multiplication algebra, Fps0 ,q (Rn ) · Fps0 ,q (Rn ) → Fps0 ,q (Rn ).

(2.304)

According to Theorem 2.35 and Remark 2.36, we may choose ψM in (2.130) such that  1/p0 f |Fps0 ,q (Rn ) ∼ ψM f |Fps0 ,q (Rn )p0 M ∈Zn

∼



2 ψM f |Fps0 ,q (Rn )p0

1/p0

(2.305) .

M ∈Zn

Let f0 ∈ Fps0 ,q (Rn ) and

Fps0 ,q (Rn )

and f1 ∈

Fps1 ,q (Rn ).

Then Theorem 2.85(ii) ensures ψM f1 ∈

ψM f1 |Fps0 ,q (Rn ) ≤ c ψM f1 |Fps1 ,q (Rn ),

(2.306)

where c > 0 is independent of M ∈ Z . Using (2.304) and Theorem 2.28, one obtains n

2 ψM f0 f1 |Fps0 ,q (Rn ) ≤ c ψM f0 |Fps0 ,q (Rn ) · ψM f1 |Fps0 ,q (Rn )

≤ c ψM f0 |Fps0 ,q (Rn ) · f1 |Fps1 ,q (Rn ).

(2.307)

Summing over M ∈ Zn and (2.305) show that f0 f1 |Fps0 ,q (Rn ) ≤ c f0 |Fps0 ,q (Rn ) · f1 |Fps1 ,q (Rn ),

(2.308)

where we used that the right-hand sides in (2.305) are characterizing quasi-norms according to Remark 2.36.  Remark 2.97. As mentioned after (2.291), the restriction p0 ≤ p1 is necessary. Otherwise the proof is surprisingly short. But one must be aware that we did not only use the embedding Theorem 2.85(ii) valid for both B-spaces and F -spaces, but also the localization property for F -spaces according to Theorem 2.35, which has no counterpart for B-spaces. In particular, the above arguments cannot be extended to B-spaces in Rn . This does not mean, however, that there are no assertions of this type for B-spaces. We shall return to this point at the end of this Section 2.7.2, namely in Proposition 2.101. Remark 2.98. Let s (Rn ), Hps (Rn ) = Fp,2

s ∈ R,

1 < p < ∞,

(2.309)

2.7. Multipliers: Further properties

79

be the Sobolev spaces as recalled in Section 4.3.2 below, with the classical Sobolev spaces k ∈ N, 1 < p < ∞, (2.310) Wpk (Rn ) = Hpk (Rn ), as special cases, see the Sections 4.3.1 and 4.3.2. Let s bmos (Rn ) = F∞,2 (Rn ),

s ∈ R,

(2.311)

be as in (1.98). Then the above theorem implies Hps1 (Rn ) · Hps0 (Rn ) → Hps0 (Rn ),

1 < p0 ≤ p1 < ∞,

bmos (Rn ) · Hps0 (Rn ) → Hps0 (Rn ),

1 < p0 < ∞,

s> s>

n , p0

n , p0

(2.312) (2.313)

with the special cases Wpk1 (Rn ) · Wpk0 (Rn ) → Wpk0 (Rn ),

1 < p0 ≤ p1 < ∞,

bmok (Rn ) · Wpk0 (Rn ) → Wpk0 (Rn ),

1 < p0 < ∞,

n < k ∈ N, (2.314) p0 n < k ∈ N. p0

(2.315)

Although at least (2.314) looks very classical, we have no immediate reference at hand. The location property for F -spaces as used in (2.305) is rather obvious for the classical Sobolev spaces. Nevertheless it seems reasonable to give a direct proof of (2.314), using only Sobolev embeddings and Hölder inequalities for Lp -spaces on a global scale. For this, let k ∈ N and 1 < p < ∞. Then 1/p  Dα f |Lp (Rn )p f |Wpk (Rn ) = α∈Nn 0, 0≤|α|≤k

(2.316)

∼ f |Lp (R ) + n

D f |Lp (R ) α

n

α∈Nn 0 ,|α|=k

are equivalent norms, see Section 4.3.1. Let f0 ∈ Wpk0 (Rn ) and f1 ∈ Wpk1 (Rn ), where 0 < pn1 ≤ pn0 < k ∈ N. Then Theorem 2.3 implies f0 f1 |Lp0 (Rn ) ≤ c f1 |L∞ (Rn ) · f0 |Lp0 (Rn ) ≤ c f1 |Wpk1 (Rn ) · f0 |Wpk0 (Rn ).

(2.317)

Let α ∈ Nn0 , |α| = k. Then Dα (f0 f1 )(x) =



cα0 ,α1 Dα0 f0 (x) Dα1 f1 (x)

(2.318)

α0 +α1 =α

for some cα0 ,α1 ∈ R. One has 0| Dα0 f0 ∈ Wpk−|α (Rn ) 0

and

1| Dα1 f1 ∈ Wpk−|α (Rn ). 1

(2.319)

80

Chapter 2. The essentials, key theorems

For

n p1

< k − |α1 |, it follows as in (2.317) that Dα0 f0 · Dα1 f1 |Lp0 (Rn ) ≤ c f0 |Wpk0 (Rn ) · f1 |Wpk1 (Rn ).

It remains to examine the cases with ⎧   ⎪ − n < min k − |α0 | − pn0 , 0 = − pn0 ≤ 0 ⎪ ⎨ p0 α0 and ⎪ ⎪ ⎩− n < k − |α1 | − n = − n1 ≤ 0. p1 p1 p

(2.320)

(2.321)

α1

For these cases, it follows from (2.319) and the classical Sobolev embedding as covered by Remark 2.87 that Dα0 f0 ∈ Lp0 (Rn ), p0 ≤ p0 < p0α0

and

Dα1 f1 ∈ Lp1 (Rn ), p1 ≤ p1 < p1α1 . (2.322)

By (2.321) and |α0 | + |α1 | = k, one has 1 1 1 1 1 k + 1 ≤ + − < . p0α0 pα 1 p0 p1 n p0

(2.323)

Due to (2.322), one may chose p0 and p1 such that p10 + p11 = p10 . Then Hölder’s inequality, (2.319) and (2.322) show that one has again (2.320). The bilinear embedding (2.314) is now a consequence of (2.317), (2.318) and (2.320) based on (2.316). This elementary (but not obvious) proof of (2.314) shows the role played by the assumption 0 < pn1 ≤ pn0 < k ∈ N in combination with classical Sobolev embeddings. The proof of Theorem 2.96 relies on the local embedding (2.306), for which one has a B-counterpart, and the global property (2.305), for which one has no B-counterpart. In other words, there is a counterpart of Theorem 2.96 for bounded domains in Rn for both F -spaces and B-spaces. Although the related assertions are quite obvious, it might be worthwhile to fix the outcome. The B-counterpart of (2.302) is given by  either s > n/p0 where 0 < p0 , q ≤ ∞, (2.324) or s = n/p0 where 0 < p0 < ∞, 0 < q ≤ 1, characterizing those spaces Bps0 ,q (Rn ) which are multiplication algebras according to Definition 2.37 and Theorem 2.41. In Definition 2.74, we introduced the spaces Asp,q (Ω), A ∈ {B, F }, in arbitrary domains Ω in Rn by restriction. Then Asp,q (Ω) is a multiplication algebra if Asp,q (Rn ) is a multiplication algebra. In particular, this means that Asp,q (Ω) → L1 (Ω) if Ω is bounded. In order to be consistent with our previous considerations, we assume that Ω is a bounded C ∞ -domain. (What follows, however, remains valid for any bounded domain Ω in Rn .)

2.7. Multipliers: Further properties

81

Corollary 2.99. Let Ω be a bounded C ∞ -domain in Rn . Let 0 < p0 ≤ p1 ≤ ∞ and 0 < q ≤ ∞. (i) Let s be as in (2.324). Then Bps1 ,q (Ω) · Bps0 ,q (Ω) → Bps0 ,q (Ω).

(2.325)

(ii) Let s be as in (2.302). Then Fps1 ,q (Ω) · Fps0 ,q (Ω) → Fps0 ,q (Ω).

(2.326)

Proof. The proof is covered by the preceding comments and Theorem 2.85(ii). (Note that the role of p0 and p1 is changed.)  Remark 2.100. As discussed after (2.291), the restriction p0 ≤ p1 is necessary. Otherwise we have the same special cases as in (2.312)–(2.315) with Ω instead of Rn . For bounded domains Ω in Rn , the assertions (2.325), (2.326) are satisfactory for both B-spaces and F -spaces. For corresponding spaces on Rn , the situation is different. In this case, we have (2.303) but no related B-counterpart. But with the help of (2.290), one can say at least something. Proposition 2.101. Let 1 ≤ p0 ≤ p1 ≤ ∞ and 0 < q < ∞. Let  either s > n/p0 where 1 ≤ p0 ≤ ∞, 0 < q < ∞, or s = n/p0 where 1 ≤ p0 < ∞, 0 < q ≤ 1.

(2.327)

Then Bps1 ,q (Rn ) · Bps0 ,q (Rn ) → Bps0 ,q (Rn ).

(2.328)

Proof. According to Definition 2.37 and Theorem 2.41, the restrictions (2.327) ensure that Bps0 ,q (Rn ) is a multiplication algebra, Bps0 ,q (Rn ) · Bps0 ,q (Rn ) → Bps0 ,q (Rn ).

(2.329)

This particularly covers the case p0 = p1 = ∞. Let 1 ≤ p0 < ∞, 0 < θ < 1 and 1 1−θ p1 = p0 . We rely on the complex interpolation   s s (Rn ) θ = Bps1 ,q (Rn ) (2.330) Bp0 ,q (Rn ), B∞,q according to [KMM07, Theorems 5.2, 9.1, pp. 137, 157], where the above restriction q < ∞ comes from. Then (2.328) follows from (2.290) and (2.329).  Remark 2.102. The restriction p0 ≥ 1 comes from (2.290). This restriction as well as some other ingredients that we used above rely on paramultiplication. Nowadays, one may ask for direct shorter wavelet proofs of the assertions in Theorem 2.96, Proposition 2.101 and possible generalizations. A first step in this direction may be found in [T19, pp. 78–79].

Chapter 3

Further topics 3.1 Envelopes Let Asp,q (Rn ) be a space that entirely consists of regular distributions as characterized in Theorem 2.4, but is not continuously embedded in L∞ (Rn ) according to Theorem 2.3. Then it is interesting to have a closer look at singularities of related unbounded functions belonging to these spaces. Questions of this type have a long history and finally culminated in the theory of envelopes. In this book, we do not deal systematically with this topic, but rather collect a few more or less known assertions illuminating our previous considerations both for spaces on Rn and on bounded domains in Rn . Special attention will be paid to spaces with p = ∞. Let Ω be either Rn , n ∈ N, or a bounded domain in Rn with the unit ball n K = {x ∈ Rn : |x| < 1} as the prototype. Let f ∈ Lloc 1 (Ω) if Ω = R or f ∈ L1 (Ω) if Ω = K. Then the singularity behaviour of f can be expressed in terms of the distribution function μf and the non-increasing rearrangement f ∗ of f given by   λ ≥ 0, (3.1) μf (λ) = {x ∈ Ω : |f (x)| > λ}, where |M | is the (finite or infinite) Lebesgue measure of the measurable set M in Rn , and   f ∗ (t) = inf λ : μf (λ) ≤ t , 0 ≤ t < ∞. (3.2) It is reasonable to admit f ∗ (t) = ∞ if there is no λ ≥ 0 with μf (λ) ≤ t. For (essentially) unbounded (locally) integrable f , one has f ∗ (t) ↑ ∞ if t ↓ 0 in any case. The rearrangement of functions attracted a lot of attention, especially in connection with function spaces. Basic information may be found in [BeS88, DeL93, EdE04]. The related theory of (growth) envelopes goes back to [Har01, Har02] and was developed in greater detail in [T01, Chapter II] and, in particular, in [Har07]. A short description may also be found in [T06, Section 1.9, pp. 39– 55]. Here, we are only interested in some special aspects and adapt our notation

© Springer Nature Switzerland AG 2020 H. Triebel, Theory of Function Spaces IV, Monographs in Mathematics 107, https://doi.org/10.1007/978-3-030-35891-4_3

83

84

Chapter 3. Further topics

appropriately. Let Asp,q (Rn ) be the spaces as described above. Then E ∞Asp,q (t) = sup{f ∗ (t) : f |Asp,q (Rn ) ≤ 1},

0 < t < ε,

(3.3)

for some ε, 0 < ε < 1, is the related envelope function for the space Asp,q (Rn ). Similarly, E 1Asp,q (t) = sup{f ∗ (t) : f |Asp,q (K) ≤ 1},

0 < t < ε,

(3.4)

is the envelope function for the space Asp,q (K) as introduced in Definition 2.74 by restriction. Nowadays, one has final descriptions of the envelope functions for all spaces n Asp,q (Rn ) ⊂ Lloc (3.5) 1 (R ), which are not continuously embedded in L∞ (Rn ), as characterized in the Theorems 2.4, 2.3, and their counterparts Asp,q (K) → L1 (K)

(3.6)

with the same parameters as for the spaces on Rn . For this purpose, one divides the region of the admitted parameters s, p, q into four cases (subcritical, critical, borderline, singular) with the following outcome. Recall that   1  0 < p ≤ ∞. (3.7) σp(n) = n max , 1 − 1 , p Theorem 3.1. Let 0 < t < ε < 1. (i) Subcritical case. If s > 0, 0 < q ≤ ∞ then

and

s−

n n = − with 1 < r < ∞ p r

E 1Asp,q (t) ∼ E ∞Asp,q (t) ∼ t−1/r .

(3.8)

(3.9)

(ii) Critical case. If 0 < p < ∞, then

1 0. In particular, f ∈ B∞,q 0 (Rn ) and we have μf (λ) > 0. By some modification, we may assume f ∈ B∞,q supp f ⊂ K. Let Kl with l ∈ N be congruent disjoint balls in Rn centered at some xl ∈ Rn . Then it follows from the wavelet representation in Proposition 1.11 that g=

∞

0 f (x − xl ) ∈ B∞,q (Rn ),

0 0 g |B∞,q (Rn ) = f |B∞,q (Rn ).

(3.20)

l=1

In particular, we have μg (λ) = ∞ for any λ > 0. Then g ∗ (t) = ∞. This proves the second assertion in (3.18). The second assertion in (3.19) can be proved in the same way with the help of the localization Theorem 2.35.  Remark 3.2. As already mentioned, the above list covers all cases of interest. In s s this book, we pay special attention to the spaces F∞,q (Rn ) and F∞,q (Ω). But again, it turns out that these spaces can be naturally incorporated into the theory of the spaces Asp,q (Rn ) and Asp,q (Ω) with p < ∞ for F -spaces. The peculiarities of part (iv) of the above theorem equally apply to the B-spaces and F -spaces with p = ∞. Remark 3.3. We focused on the envelope functions E 1Asp,q (t) and E ∞Asp,q (t). But there is a finer tuning asking for integrability properties of f ∗ (t)/E ∞Asp,q (t) and f ∗ (t)/E 1Asp,q (t). In the subcritical case, this results in the question for which u, 0 < u ≤ ∞, one has sup t1/r f ∗ (t) ≤ c1

0 0 such that for all f ∈ Fp,q  ε  u dt 1/u s κ(t) t1/r f ∗ (t) ≤ c f |Fp,q (Rn ) (3.23) t 0 if, and only if, simultaneously p ≤ u ≤ ∞ and κ is bounded. There are counterparts for the spaces covered by the critical case. These formulations may be found in [T06, Theorems 1.78, 1.84, pp. 46, 51], based on the above references to [Har01, Har02, Har07, T01]. As far as the borderline and singular cases are concerned, we refer the reader to [Har07, SeT09, Vyb10].

3.2 Positivity Any f ∈ Lp (Rn ), 1 < p < ∞, can be decomposed as f = f 1 − f 2 + if 3 − if 4 , with f |Lp (Rn ) ∼

4

f l (x) ≥ 0,

f l |Lp (Rn ).

x ∈ Rn ,

(3.24)

(3.25)

l=1

Any element f of the Sobolev space Hps (Rn ) with s > 0 and 1 < p < ∞ according to (4.88)–(4.91) can be represented as      Gs (x − y) I−s f (y) dy, x ∈ Rn , (3.26) f (x) = Is I−s f (x) = Rn

where I−s f ∈ Lp (Rn ) can be decomposed as in (3.24)–(3.25) and Gs (y) are the positive Bessel potential kernels, see [AdH96, Sections 1.2.4, 1.2.5, pp. 10–13] or [Gra04, Proposition 6.1.5, p. 418]. Then it follows that also f ∈ Hps (Rn ), s ≥ 0, 1 < p < ∞ can be represented as in (3.24) with 0 ≤ f l ∈ Hps (Rn ) such that f |Hps (Rn ) ∼

4

f l |Hps (Rn ).

(3.27)

l=1

The question arises for which other spaces Asp,q (Rn ) a similar decomposition into non-negative components exists. We dealt with this problem in [T06, Section 3.3.2, pp. 190–192], based on [Tri03]. As always in the present book, we wish to incors porate the spaces F∞,q (Rn ) into the already existing assertions for the spaces Asp,q (Rn ) with p < ∞ for the F -spaces. For this case, it will be described in Remark 3.8 below how this can be done very easily. But since not all details and arguments from [Tri03] have been taken over in [T06], we wish to present a more detailed account. In addition, one can demonstrate once again how other properties of the spaces Asp,q (Rn ) are related to this question.

88

Chapter 3. Further topics

Definition 3.4. Let Asp,q (Rn ) with A ∈ {B, F }, s ∈ R and 0 < p, q ≤ ∞ be the spaces according to Definition 1.1. Then Asp,q (Rn ) is said to have the positivity property if any f ∈ Asp,q (Rn ) can be decomposed as f = f 1 − f 2 + if 3 − if 4

with

and f |Asp,q (Rn ) ∼

f l ≥ 0,

4

f l ∈ Asp,q (Rn ),

f l |Asp,q (Rn )

(3.28)

(3.29)

l=1

with equivalence constants which are independent of f . Remark 3.5. Similarly as in Remark 2.45, an element f ∈ Asp,q (Rn ) is called a positive distribution, f ≥ 0, if f (ϕ) ≥ 0

for any real ϕ ∈ S(Rn ) with ϕ(x) ≥ 0, x ∈ Rn .

(3.30)

It follows from the same references as there, namely [Mal95, pp. 61/62, 71, 75], that any positive distribution can be represented as  ϕ(x) μ(dx), ϕ ∈ S(Rn ), (3.31) f (ϕ) = Rn

where μ is a uniquely determined (positive) Radon measure. Let again   1  σp(n) = n max , 1 − 1 p

and

  1 1  (n) σp,q = n max , , 1 − 1 , p q

(3.32)

where 0 < p, q ≤ ∞ and n ∈ N. Theorem 3.6. Let Asp,q (Rn ) with A ∈ {B, F }, s ∈ R and 0 < p, q ≤ ∞ be the spaces according to Definition 1.1. (n)

s (Rn ) with 0 < p, q ≤ ∞ and s > σp (i) The spaces Bp,q property.

have the positivity

s (ii) The spaces Fp,q (Rn ) with

⎧ ⎪ ⎨either or ⎪ ⎩ or

(n)

0 < p, q ≤ ∞ and s > σp,q , 1 < p ≤ ∞, 0 < q ≤ ∞ and s > n/p, 0 < p ≤ 1, 0 < q ≤ ∞ and s ≥ n/p,

(3.33)

have the positivity property. (n)

(iii) The spaces Asp,q (Rn ) with 0 < p, q ≤ ∞ and s < σp property.

do not have the positivity

3.2. Positivity

89

s (Rn ) Proof. Step 1. It follows from [T06, Theorem 3.48, p. 191] that the spaces Bp,q (n)

with 0 < p ≤ ∞, 0 < q ≤ ∞, s > σp

s and Fp,q (Rn ) with 0 < p < ∞, 0
decompositions. For this purpose, one modifies the related elementary building blocks as described in (4.121), (4.122) such that β 2−J (x − x0 ) k(x) ≥ 0,



x ∈ Rn ,

β ∈ Nn0 ,

(3.34)

and extends the corresponding expansions in (4.135)–(4.137) to all spaces (n) (n) s s (Rn ), s > σp , and Fp,q (Rn ), s > σp,q (with p < ∞ for the F -spaces). Bp,q β Afterwards, the modified counterparts of the building blocks kj,m in (4.136) are β non-negative. Then the decomposition of the complex numbers λj,m gives the desired result. We refer the reader to [T06, p. 191] and the underlying quarkonial decompositions. Step 2. As far as the remaining cases in part (ii) are concerned, it follows from s (Rn ) with Theorem 2.3 that one has to justify that all spaces Fp,q s (Rn ) → C(Rn ) Fp,q

(3.35)

s (Rn ) is real. We rely have the positivity property. We may assume that f ∈ Fp,q on the localization property according to Theorem 2.35 with ψM as in (2.130). We may assume ψM (x) ≥ 0, x ∈ Rn . Let fM ψM (x) with fM = max |f (y)|. (3.36) f 1 (x) = y∈supp ψM

M ∈Zn

The local version of (3.35) ensures   s (Rn ) + + + fM ≤ c ψM f |Fp,q

(3.37)

where + + + indicates N neighbouring terms with N ∈ N being independent of M ∈ Zn . From (2.131) applied to f and f 1 , one obtains that s s (Rn ) ≤ cf |Fp,q (Rn ). f 1 |Fp,q

(3.38)

Then f = f 1 − f 2 with f 2 (x) =



 fM − f (x) ψM (x) ≥ 0,

x ∈ Rn ,

(3.39)

M ∈Zn

is the desired decomposition. Step 3. For the proof of part (iii), we assume that any real, compactly supported f ∈ Asp,q (Rn ) can be decomposed according to (3.28), (3.29) by f = f 1 − f 2 , f 1 ≥ 0, f 2 ≥ 0. Recall that f being real means that f (ϕ) ∈ R for any real

90

Chapter 3. Further topics

ϕ ∈ S(Rn ). From (3.31) with g = f 1 or g = f 2 , it follows that there is a finite, positive, compactly supported Radon measure μ such that   ϕ(x) μ(dx), μ(Rn ) = μ(dx) < ∞, (3.40) g(ϕ) = Rn

Rn

0 ϕ ∈ S(R ) real. We wish to prove that μ is an element of B1,∞ (Rn ). For that purpose, let {ϕj } be the dyadic resolution of unity according to (1.3)–(1.5). Then one has with ϕ(x) = ϕ0 (x) − ϕ0 (2x)    ∨   ∨ jn ϕj μ  (x) = c ϕj (x − y) μ(dy) = c 2 ϕ∨ 2j x − 2j y μ(dy), (3.41) n

Rn

Rn

j ∈ N, x ∈ R . Taking the L1 -norm, one obtains n

(ϕj μ )∨ |L1 (Rn ) ≤ c μ(Rn ), 1

j ∈ N0 ,

(3.42)

2

0 and (1.8) then implies that f and f are elements of B1,∞ (Rn ). In particular, s n 0 Ap,q (R ) is at least locally continuously embedded in B1,∞ (Rn ). But this contra(n)  dicts Remark 2.87 for s < σp . s Remark 3.7. With the exception of F∞,q (Rn ), the above theorem is essentially covered by [Tri03] and at least partly reproduced in [T06, Section 3.3.2, pp. 190– s 192]. The theorem covers in particular (3.24), (3.27) with Hps (Rn ) = Fp,2 (Rn ), 1 < p < ∞, s > 0. Additionally, it follows that this assertion cannot be extended to Hps (Rn ) with 1 < p < ∞, s < 0.

Remark 3.8. If one accepts all the assertions of the theorem with the exception s (Rn ), 0 < q ≤ ∞, s > 0, has the positivity property, then one can that F∞,q s argue more directly. Namely, if f ∈ F∞,q (Rn ) with 0 < q ≤ ∞ and s > 0, then it immediately follows from (2.133) applied to f 1 = f |C(Rn ) and (3.35) that s s (Rn ) ≤ c f |C(Rn ) ≤ c f |F∞,q (Rn ). f 1 |F∞,q

(3.43)

Then f = f 1 − (f 1 − f ) for a real f is the (rather obvious) desired decomposition. s This type of argument also applies to B∞,q (Rn ), 0 < q ≤ ∞, s > 0, where one can rely on (4.97) with p = ∞.

3.3 Local homogeneity Let   Uλ = x = (x1 , . . . , xn ) ∈ Rn : |xj | < λ ,

λ > 0,

(3.44)

be cubes in Rn , n ∈ N, centered at the origin. Let U = U1 , and let Asp,q (Uλ ) and s (Uλ ) with A ∈ {B, F }, s ∈ R and 0 < p, q ≤ ∞ be the corresponding spaces A p,q as introduced in Definition 2.74. With   1  0 < p ≤ ∞, (3.45) σp(n) = n max , 1 − 1 , p

3.3. Local homogeneity

91

as before, let ⎧ s (Uλ ) ⎪ ⎨A p,q s Ap,q (Uλ ) = A0p,q (Uλ ) ⎪ ⎩ s Ap,q (Uλ )

(n)

if 0 < p ≤ ∞, 0 < q ≤ ∞, s > σp , if 1 < p < ∞, 0 < q ≤ ∞, s = 0, if 0 < p ≤ ∞, 0 < q ≤ ∞, s < 0.

(3.46)

Let us add two comments. First, according to Theorem 2.4, the spaces Asp,q (Rn ) (n)

with s > σp consist entirely of regular distributions. Then |∂Uλ | = 0 and the s (Uλ ) = A s (Uλ ) as a closed Definition 2.74 justifies our interpretation of A p,q p,q s n subspace of Ap,q (R ). Secondly, if 1 < p < ∞ and s = 0 then it follows from Theorem 2.48 that the characteristic function of Uλ is a pointwise multiplier for A0p,q (Rn ), with the related multiplier norm being independent of λ. This shows that the identification 0 (Uλ ) = A 0 (Uλ ), A0p,q (Uλ ) = A p,q p,q

1 < p < ∞,

0 < q ≤ ∞,

λ > 0, (3.47)

makes sense. Now we ask under which circumstances one has n

f (λ·) |Asp,q (U ) ∼ λs− p f |Asp,q (Uλ ),

(3.48)

where the equivalence constants are independent of λ with 0 < λ ≤ 1 and of f ∈ Asp,q (Uλ ). We dealt several times with this local homogeneity, in particular in [T01, Corollary 5.16, p. 66], [T08, Theorem 2.11, p. 34] (repeated in [T13, Theorem 1.14, p. 11]) and [T15, Corollary 3.55, p. 116]. This property plays a crucial role in the theory of the isotropic spaces Asp,q in (rough) domains, but also in connection with local and hybrid spaces as considered in [T08, T13, T14]. So far, (3.48) has been obtained for the spaces ⎧  s (Uλ ) if 0 < p ≤ ∞, 0 < q ≤ ∞, s > σp(n) , ⎪ ⎨B p,q s 0 B p,q (Uλ ) = Bp,q (3.49) (Uλ ) if 1 < p < ∞, 0 < q ≤ ∞, s = 0, ⎪ ⎩ s Bp,q (Uλ ) if 0 < p ≤ ∞, 0 < q ≤ ∞, s < 0, and ⎧ s ⎪ ⎨F p,q (Uλ ) s 0 F p,q (Uλ ) = Fp,q (Uλ ) ⎪ ⎩ s Fp,q (Uλ )

(n)

if 0 < p < ∞, 0 < q ≤ ∞, s > σp , if 1 < p < ∞, 1 ≤ q < ∞, s = 0, if 0 < p < ∞, 0 < q ≤ ∞, s < 0.

(3.50)

This is the final result from [T15, Section 3.21, pp. 115–118]. Now, we wish to s (Uλ ). incorporate the spaces F∞,q Theorem 3.9. Let Asp,q (Uλ ) be either ⎧  s (Uλ ) if 0 < p ≤ ∞, 0 < q ≤ ∞, s > σp(n) , ⎪ ⎨B p,q s 0 B p,q (Uλ ) = Bp,q (Uλ ) if 1 < p < ∞, 0 < q ≤ ∞, s = 0, ⎪ ⎩ s Bp,q (Uλ ) if 0 < p ≤ ∞, 0 < q ≤ ∞, s < 0,

(3.51)

92 or

Chapter 3. Further topics

⎧ s ⎪ ⎨Fp,q (Uλ ) s 0 F p,q (Uλ ) = Fp,q (Uλ ) ⎪ ⎩ s Fp,q (Uλ )

(n)

if 0 < p ≤ ∞, 0 < q ≤ ∞, s > σp , if 1 < p < ∞, 1 ≤ q < ∞, s = 0, if 0 < p ≤ ∞, 0 < q ≤ ∞, s < 0,

(3.52)

s (Uλ ) if s > 0. Then with q > 1 for F∞,q n

f (λ·) |Asp,q (U ) ∼ λs− p f |Asp,q (Uλ ),

(3.53)

where the equivalence constants are independent of λ with 0 < λ ≤ 1 and of f ∈ Asp,q (Uλ ). Proof. Step 1. Compared with what is already known, we have to justify (3.53) s for the spaces F∞,q (Rn ) with s = 0 and q > 1 if s > 0. By (1.72), we may assume s (Rn ), s < 0, 0 < q < ∞ and rely on q < ∞. First, we deal with the spaces F∞,q the second equivalence in (1.99). We may assume λ = 2−K , K ∈ N0 . Some related arguments may be found in [T08, Step 1, p. 92]. Let QK = U2−K , K ∈ N0 . Then it follows from (1.99) and the distinguished pointwise multiplier property according to [T08, Theorem 2.13, p. 36] (which is also covered by Theorem 3.11 below) that s (QK ) ∼ f |F∞,q

sup

J≥K,QJ,M ⊂2QK

s 2Jn/p f |Fp,q (2QJ,M ),

(3.54)

s s 0 < p < ∞. Let f ∈ F∞,q (QK ). Then one obtains from (3.48), with Fp,q applied twice, s s (QJ ) ∼ 2−K(s− p ) f |Fp,q (QJ+K ), f (2−K ·) |Fp,q n

J ∈ N0 .

(3.55)

These terms correspond to (1.99) with M = 0. Together with suitable translations for the terms with M = 0, it follows from (1.99) and its modification (3.54) that s s (Q0 ) ∼ 2−Ks f |F∞,q (QK ). f (2−K ·) |F∞,q

(3.56)

s As already mentioned, this is sufficient to justify (3.53) for Asp,q (U ) = F∞,q (U ), U = Q0 , s < 0, < q < ∞. s Step 2. We prove (3.53) for the spaces F∞,q (Uλ ) with s > 0 and 1 < q < ∞. Recall that the case q = ∞ is already covered by the B-spaces. We rely on the duality (1.24), namely −s n  s n F1,q  (R ) = F∞,q (R ),

1 < q < ∞,

s > 0,

1 1 +  = 1. q q

(3.57)

s s (Uλ ) with F∞,q (Uλ ). Let f ∈ By the comments after (3.46), we can identify F∞,q s n F∞,q (R ) with supp f ⊂ Uλ . Then it follows from (3.57) that  #  $   −s s n f (x) g(x) dx : g |F1,q (3.58) f |F∞,q (R ) = sup   (Uλ ) ≤ 1 , Rn

3.4. Refined localization spaces

93

s (Rn ) → L∞ (Rn ). In addition, we may assume where we used the known fact F∞,q that the admitted g are smooth. In particular, the above integral makes sense. Furthermore, we already know that −s −s s+n g(λ·) |F1,q g |F1,q  (Uλ ) ∼ λ  (U ),

0 < λ ≤ 1.

(3.59)

By inserting 

f (x) g(x) dx = λ−s



Rn

f (λx) λs+n g(λx) dx

(3.60)

Rn

into (3.58), it follows that s s (Uλ ) ∼ λ−s f (λ·) |F∞,q (U ), f |F∞,q

(3.61)

s s where again we used the above identifications of F∞,q (Uλ ) with F∞,q (Uλ ) and s s s F∞,q (U ) with F∞,q (U ) (as subspaces of F∞,q (Rn )).  s with s > 0, the above duality argument restricts Remark 3.10. For the spaces F∞,q the equivalence (3.53) to q > 1. It would be of interest to extend (3.53) to all spaces s F∞,q , s > 0, 0 < q ≤ ∞. At least, the above approach covers the spaces s (Rn ), bmos (Rn ) = F∞,2

s = 0,

(3.62)

according to (1.98). For s = 0, one has (3.47) for the spaces A0p,q with 1 < p < ∞. An extension to p = ∞ is at least questionable. Then one has two possible, maybe 0 (Uλ ), 0 < q ≤ ∞. different, candidates A0∞,q (Uλ ) and A ∞,q

3.4 Refined localization spaces 3.4.1 Multipliers, revisited In [T08, Section 2.2.3, pp. 36–40], we used homogeneity properties of the type (3.53) to introduce so-called refined localization spaces. For this purpose, one needs some specific pointwise multiplier assertions which are not covered by our preceding considerations. So far, we have the smooth pointwise multipliers according to the Theorems 2.28, 2.30 and the more sophisticated versions (rough pointwise multipliers) covered by Theorem 2.61. This will be now complemented as follows. Let sup |Dα g(x)| if  ∈ N0 (3.63) g |C (Rn ) = |α|≤ , x∈Rn

as in (2.117), and g |C (Rn ) = g |C [ ] (Rn ) + sup

|Dα g(x) − Dα g(y)| |x − y|{ }

(3.64)

94

Chapter 3. Further topics

for 0 <  = [] + {} with [] ∈ N0 and 0 < {} < 1, where the latter supremum is taken over all x ∈ Rn , y ∈ Rn with 0 < |x − y| < 1, and α ∈ Nn0 such that |α| = []. Let C (Rn ),  > 0, be the corresponding space of complex-valued functions such that the related norm is finite. In particular, we have

(Rn ), C (Rn ) = C (Rn ) = B∞,∞

0   (n) max s, σp − s . Then gf |Asp,q (Uλ ) ≤ c g(λ·) |C (Rn ) · f |Asp,q (Uλ ),

(3.66)

where c is a positive constant which is independent of f ∈ Asp,q (Uλ ), of λ with 0 < λ ≤ 1 and of g ∈ C (Rn ), supp g ⊂ U2λ . (3.67) Proof. By (3.53) and (2.125), one has gf |Asp,q (Uλ ) ∼ λ p −s g(λ·)f (λ·) |Asp,q (U ) n

≤ c λ p −s g(λ·) |C (Rn ) · f (λ·) |Asp,q (U ), n

(3.68)

where we used some standard properties of the related spaces in Uκ , κ = 1 and κ = 2, such as extension and multiplication with cut-off functions. Retransforming (3.68) yields (3.66).  Remark 3.12. Compared with [T08, Theorem 2.13, p. 36], we now have (3.66) for the spaces s (Uλ ), 0 < p < ∞, 0 < q ≤ ∞, s > σp(n) , (3.69) Fp,q (n)

instead of the restriction s > σp,q as in (2.118). This comes from the improved homogeneity (3.50). But it might be more interesting to incorporate the spaces s (Rn ) F∞,q

with either s < 0, 0 < q ≤ ∞, or s > 0, q > 1,

(3.70)

s (Rn ) with s = 0 according to (1.98). covering in particular bmos (Rn ) = F∞,2

3.4.2 Main assertions In Definition 2.74, we introduced the spaces Asp,q (Ω) with A ∈ {B, F }, s ∈ R and 0 < p, q ≤ ∞ in arbitrary domains Ω in Rn by restriction. Afterwards, namely in Section 2.6, we dealt with these spaces on bounded C ∞ -domains Ω by reducing them to their counterparts on Rn and Rn+ according to (2.197), based on smooth pointwise multipliers and diffeomorphisms. This method also works for

3.4. Refined localization spaces

95

C k -domains as introduced in Definition 2.76 if k ∈ N is chosen sufficiently large in dependence on the parameters s, p, q. At least some parts of this theory, for example the extension property as considered in Section 2.6.4, remain valid for (bounded) Lipschitz domains as mentioned in Remark 2.84. It was one of the main aims of [T08] to study function spaces in rough domains. In this case, one needs other methods. One can take the localization property according to Theorem 2.35 as a guide when asking for counterparts on domains Ω in Rn , based on suitable resolutions of unity in Ω. Theorem 2.35 suggests to concentrate on s,rloc F -spaces. This results in the so-called refined localization spaces Fp,q (Ω). Under some restrictions for the (rough) domains Ω and the parameters, one has   s,rloc s s (Ω) = Fp,q (Ω) = f ∈ Fp,q (Rn ) : supp f ⊂ Ω , Fp,q 0 < p < ∞,

0 < q ≤ ∞,

  1 1  (n) = n max , , 1 − 1 s > σp,q p q

(3.71) (3.72)

s and p = q = ∞, s > 0, where Fp,q (Ω) are the same spaces as in Definition 2.74. For the admitted domains Ω, one has |∂Ω| = 0 such that the comments after (3.46) s s s justify Fp,q (Ω) = Fp,q (Ω) and the interpretation of Fp,q (Ω) as a closed subspace s n s (Ω), of Fp,q (R ). One can ask for an extension of this theory to the spaces F∞,q 0 < q ≤ ∞. As far as related refined localization spaces are concerned, one can rely on Theorem 3.11. But it is not so clear whether (3.71) remains valid for corresponding spaces with p = ∞ and 0 < q < ∞. We shall comment on this point in Remark 3.17 below. Let Ω be an arbitrary domain in Rn with Ω = Rn . Let ψ = {ψl,r } be a resolution of unity, ∞ ψl,r (x) = 1 if x ∈ Ω, (3.73) l=0

r

as described in [T08, p. 36] based on the open Whitney cubes Q0l,r ⊂ Q1l,r centered at 2−l mr for some mr ∈ Zn with respective side-lengths 2−l and 2−l+1 such that the cubes Q0l,r are pairwise disjoint, Ω=

%

Q0l,r ,

  dist Q1l,r , ∂Ω ∼ 2−l

if l ∈ N

(3.74)

l,r

(with a suitable modification for l = 0). In particular, we have supp ψl,r ⊂ Q1l,r ,

|Dγ ψl,r (x)| ≤ cγ 2l|γ| ,

x ∈ Ω,

γ ∈ Nn0 .

(3.75)

Definition 3.13. Let Ω be an arbitrary domain in Rn with Ω = Rn . Let ψ = {ψl,r } be a resolution of unity as described above. Then   s,rloc s,rloc (Ω) = f ∈ D (Ω) : f |Fp,q (Ω)ψ < ∞ (3.76) Fp,q

96

Chapter 3. Further topics

with

∞ 

s,rloc f |Fp,q (Ω)ψ =

l=0

s ψl,r f |Fp,q (Rn )p

1/p

(3.77)

r

when 0 < p ≤ ∞,

0 < q ≤ ∞,

  1  s > σp(n) = n max , 1 − 1 p

(3.78)

(q > 1 if p = ∞) and s,rloc f |Fp,q (Ω)ψ =

∞  l=0

when



either or

s ψl,r f |Fp,q (Q1l,r )p

1/p

(3.79)

r

1 < p < ∞, 1 ≤ q < ∞, s = 0, 0 < p ≤ ∞, 0 < q ≤ ∞, s < 0.

(3.80)

Remark 3.14. Compared with [T08, Definition 2.14, pp. 36–37], we now have s > (n) (n) (n) σp in (3.78) instead of s > σp,q , with σp,q as in (3.72). Furthermore, we now incorporated the spaces s (Rn ), F∞,q

s ∈ R,

s = 0,

0 < q < ∞,

(3.81)

with q > 1 if s > 0. Obviously, ψl,r f with f ∈ D (Ω) and ψl,r as in (3.75) is extended outside Q1l,r by zero. Then both (3.77) and (3.79) make sense. According s s to the discussion after (3.46), one can replace Fp,q (Rn ) in (3.77) by Fp,q (Q1l,r ), s,rloc (Ω)ψ = f |Fp,q

∞  l=0

s ψl,r f |Fp,q (Q1l,r )p

1/p

.

(3.82)

r

s (Q1l,r ) On the other hand, despite supp ψl,r f ⊂ Q1l,r it is not possible to replace Fp,q s n in (3.79), (3.80) by Fp,q (R ). This somewhat delicate point is discussed in [T08, Remark 2.22, p. 40].

Theorem 3.15. Let Ω be an arbitrary domain in Rn with Ω = Rn . Then the spaces s,rloc Fp,q (Ω) according to Definition 3.13 are quasi-Banach spaces (Banach spaces if p ≥ 1, q ≥ 1). They are independent of ψ = {ψl,r } (equivalent quasi-norms). Proof. Recall that (3.77) can be replaced by (3.82). Then the independence of s,rloc Fp,q (Ω) of ψ = {ψl,r } follows from Theorem 3.11 applied to g = ψl,r with (3.75) and a possible second resolution of unity. Afterwards, one finds by standard s,rloc (Ω) is a quasi-Banach space.  arguments that Fp,q Remark 3.16. The above theorem extends [T08, Theorem 2.16, p. 37] in particular to the spaces in (3.81). This also applies to its proof. In [T08, Section 2.2.3, pp. 36– 40], we discussed further possibilities and some properties.

3.4. Refined localization spaces

97

Remark 3.17. According to [T01, Theorem 5.14, pp. 60–61], one has (3.71) for 0 < p ≤ ∞,

0 < q ≤ ∞,

  1 1  (n) = n max , , 1 − 1 s > σp,q p q

(3.83)

(q = ∞ if p = ∞) if Ω is a bounded C ∞ -domain in Rn . In [T06, Proposition 4.20, Remark 4.21, footnote, pp. 208–209], we extended this assertion to bounded s Lipschitz domains Ω in Rn . The proofs are based on characterizations of Fp,q (Rn ) in terms of differences and so-called ball means of differences, as will be described in (4.98)–(4.100). But according to [ChS06], this type of characterization of the (n) s spaces Fp,q (Rn ) is no longer possible for s ≤ σp,q . In [T08, Proposition 3.10, pp. 77–78, Theorem 3.28, p. 97], we proved (3.71) for (3.83) (again with q = ∞ if p = ∞) under the weaker assumption that Ω is a so-called E-thick domain (covering Lipschitz domains, but also snowflake domains in R2 ). The arguments are based on representations by atoms without moment conditions, as will be (n) described in (4.117) and (4.118). But again, s > σp,q is indispensable. According to [ScV09, Section 3.2, p. 160] with a reference to [Schn09, Section 3.2, p. 126], one (n) s needs moment conditions for atomic representations for Fp,q -spaces with s ≤ σp,q . s In other words, despite the extension of (3.53) for the spaces Fp,q with 0 < p < ∞, (n)

(n)

(n)

0 < q ≤ ∞ from s > σp,q to σp = σp,p according to Theorem 3.9, it is not clear s with 0 < p < ∞, 0 < q ≤ ∞ and whether (3.71) remains valid for all spaces Fp,q (n)

s > σp . Furthermore, based on Theorem 3.9 it makes sense to ask if s,rloc s F∞,q (Ω) = F∞,q (Ω),

s > 0,

1 < q < ∞.

(3.84)

According to (1.71) and [T14, Theorem 3.33, p. 67], one has representations by atoms without moment conditions in the cases of interest, although the arguments in [T08, p. 78] for ensuring (3.71) with (3.83) (p < ∞) require some sophisticated s in terms of local means. But we have no characterizations of the related spaces Fp,q reference of a sufficiently strong assertion for the spaces in (1.71). In other words, a proof of (3.84) remains an open problem. Let Ω be again an E-thick domain in Rn . In addition to (3.71) with (3.83), [T08, Theorem 3.28, p. 97] also provides the assertion s,rloc s (Ω) = Fp,q (Ω) Fp,q

if 0 < p ≤ ∞, 0 < q ≤ ∞, s < 0,

(3.85)

(with q = ∞ if p = ∞). One may ask for an extension of this assertion to s,rloc s (Ω) = F∞,q (Ω) F∞,q

if 0 < q < ∞, s < 0.

(3.86)

According to Theorem 3.9, one has the required homogeneity (3.53). On the other hand, the proof of (3.85) relies on sophisticated wavelet representations for the related spaces. It remains to be checked whether the corresponding wavelet representations for the spaces in (1.71) are strong enough to ensure (3.86).

98

Chapter 3. Further topics

3.5 Haar wavelets The requested smoothness of the underlying wavelets in the Propositions 1.11 and 1.16 is natural. But it excludes Haar wavelets which have attracted a lot of attention in the theory of function spaces. In [T10, Chapter 2], we dealt in detail with Haar wavelets in some global spaces Asp,q (Rn ), p < ∞ for F -spaces, and we extended these considerations in [T14, Section 3.4.4, pp. 70–78] to related hybrid spaces LrAsp,q (Rn ) as introduced in Definition 1.6. It follows from Proposition 1.18 that the corresponding assertions cover all relevant global spaces Asp,q (Rn ) s including F∞,q (Rn ). Nevertheless, it seems to be reasonable to fix this specification and to add some comments. First we recall some basic definitions. Let y ∈ R and ⎧ ⎪ if 0 < y < 1/2, ⎨1 (3.87) hM (y) = −1 if 1/2 ≤ y < 1, ⎪ ⎩ 0 if y ∈ (0, 1). Let hF (y) = |hM (y)| be the characteristic function of the unit interval (0, 1). This is the counterpart to (1.34), (1.35). We now use the same construction as in (1.36)–(1.41). Let again n ∈ N, and let G = (G1 , . . . , Gn ) ∈ G0 = {F, M }n

(3.88)

which means that Gr is either F or M . Let G = (G1 , . . . , Gn ) ∈ G∗ = Gj ∈ {F, M }n∗ ,

j ∈ N,

(3.89)

which means that Gr is either F or M , where ∗ indicates that at least one of the components of G must be an M . The counterpart of (1.38) is given by hjG,m (x) =

n 

  h G l 2 j x l − ml ,

G ∈ Gj ,

m ∈ Zn ,

(3.90)

l=1

x ∈ Rn , where (now) j ∈ N0 . It is well known that  jn/2 j  2 hG,m : j ∈ N0 , G ∈ Gj , m ∈ Zn

(3.91)

n

is an orthonormal basis in L2 (R ) and that j f= λj,G m hG,m

(3.92)

j∈N0 G∈Gj m∈Zn

with j,G jn λj,G m = λm (f ) = 2

 Rn

  f (x) hjG,m (x) dx = 2jn f, hjG,m

(3.93)

is the corresponding expansion. Let asp,q (Rn ) with a ∈ {b, f } be the sequence s (Rn ). Let again Asp,q (Rn ) with spaces according to Definition 1.9, including f∞,q A ∈ {B, F }.

3.5. Haar wavelets

99

s

s

1

s = n( p1 − 1)

1 1+ n1

1 p

−1

s = n( p1 − 1)

1 2

− 12

1 2

1 1 1+ 2n

1 p

−1

B-spaces

H-spaces Figure 3.1: Haar bases, n ∈ N

Theorem 3.18. Let 0 < p, q ≤ ∞,

 1  1  1 max n − 1), − 1 < s < min , 1 p p p

for the B-spaces, see left side of Figure 3.1, p. 99, and ⎧   1 1  1 1 ⎪ ⎪ ⎨0 < p < ∞, 0 < q < ∞, n max p , q , 1) − 1 < s < min p , q , 1 , 1 < p < ∞, 1 < q < ∞, ⎪ ⎪ ⎩1 < p ≤ ∞, 1 < q ≤ ∞,

s = 0,   max p1 , 1q − 1 < s < 0,

(3.94)

(3.95)

s (Rn ). Let for the F -spaces, see right side of Figure 3.1, p. 99, for Hps (Rn ) = Fp,2  n s n f ∈ S (R ). Then f ∈ Ap,q (R ) if, and only if, it can be represented as j λj,G λ ∈ asp,q (Rn ), (3.96) f= m hG,m , j∈N0 ,G∈Gj , m∈Zn

the unconditional convergence being in S  (Rn ). The representation (3.96) is unique,   j,G jn f, hjG,m (3.97) λj,G m = λm (f ) = 2 and I:

  f → λj,G m (f )

is an isomorphic map of Asp,q (Rn ) onto asp,q (Rn ),





f |Asp,q (Rn ) ∼ λ(f ) |asp,q (Rn ) .

(3.98)

(3.99)

Proof. The above assertions essentially coincide (up to the normalization) with [T10, Theorem 2.21, pp. 92–93] under the restriction p < ∞ for the F -spaces.

100

Chapter 3. Further topics

In [T14, Theorem 3.41, p. 74], they were extended to the spaces LrAsp,q (Rn ) with s (Rn ) with 0 < q < ∞ and −n/p ≤ r < −s, including in particular the spaces F∞,q s < 0 (q = ∞ can be incorporated into the B-scale). Let q > 1 and 1q − 1 < s < 0. Then the desired assertions follow from [T14, Theorem 3.41, p. 74] applied to s s (Rn ) = F∞,q (Rn ) with 1 < q < p < ∞ and Proposition 1.15.  L0 Fp,q Remark 3.19. For detailed explanations, (historical) references and Figure 3.1, p. 99, we refer the reader to [T10, Chapter 2]. Similar to there, let s Hps (Rn ) = Fp,2 (Rn ),

0 < p ≤ ∞,

s ∈ R,

(3.100)

s now extended to p = ∞ with bmos (Rn ) = F∞,2 (Rn ) as in (1.98). If both p < ∞ and q < ∞ in (3.94) or (3.95), then

2−j(s− p ) hjG,m : j ∈ N0 , G ∈ Gj , m ∈ Zn



n



(3.101)

is an unconditional (normalized) basis in Asp,q (Rn ). The spaces Asp,q (Rn ) with max(p, q) = ∞ are not separable. In particular, they do not have a (Schauder) basis. Nevertheless, one has the above representations. But the related series in (3.96) converge unconditionally in S  (Rn ) or locally in spaces Aσp,q (Rn ) with σ < s. s This applies in particular to F∞,q (Rn ) with 1 < q ≤ ∞ and 1q − 1 < s < 0. Remark 3.20. The Haar functions go back to [Haar10]. They have attracted a lot of attention since then. In [T10, Section 2.1, pp. 63–71] (classical theory and historical comments), we discussed Haar systems and Faber systems, including advantages and disadvantages compared to trigonometrical systems. There, one also finds related (historical) references, which will not be repeated here. The s first step to deal with Haar bases in the Besov spaces Bp,q (Q), Q = (0, 1)n , 1 < 1 1 p, q < ∞, −1 + p < s < p , goes back to [Tri73a] and was basically repeated in [T78, Section 4.9.4, pp. 338–344]. These findings were extended in [Tri78] and essentially repeated in [T06, Theorem 1.58, p. 29] with the following outcome: The s (Rn ) if Haar system (3.101) is a basis in Bp,q   1  1 1  max n − 1 , − 1 < s < min , 1 , p p p 1  and it is not a basis if p , s does not belong to the set 0 < p, q < ∞,

(3.102)

 1   1 1  max n − 1 , − 1 ≤ s ≤ min , 1 (3.103) p p p   1  1 as indicated in Figure 3.1, p. 99, in the p , s : 0 ≤ p < ∞, s ∈ R halfplane, the natural habitat for the spaces Asp,q (Rn ) according to Section 4.2.1 below. Furthermore, it follows from Proposition 2.50 that the Haar system cannot be a 1/p basis in Bp,q (Rn ), 1 ≤ p < ∞, 0 < q ≤ ∞. The proofs in [Tri73a, T78, Tri78] 0 < p ≤ ∞,

3.5. Haar wavelets

101

are different from the proofs underlying Theorem 3.18. There, one asks whether the projections on subspaces spanned by suitably ordered finite Haar functions are uniformly bounded. If this is the case and if the linear combinations of these functions are dense in the considered space, one can expect that the Haar system is a basis. This covers (3.102), while it does not ensure that this basis is unconditional or that there is an isomorphic map as in (3.97)–(3.99). There is little hope that the Haar system is an unconditional basis in the remaining limiting cases in (3.103) with 0 < p, q < ∞, including isomorphic maps as in (3.97)–(3.99). This would ensure that the characteristic function of a half-space is a pointwise multiplier in the related limiting spaces, thus contradicting Theorem 2.48 and the comments in the beginning of Section 2.4.4. This follows from the well-known assertion that the related projections are uniformly bounded, see [AlK06, Proposition 1.1.4, p. 4] or [Woj91, Proposition 6, p. 37]. As mentioned above, the Haar system cannot be a 1/p (unconditional or conditional) basis in the limiting spaces Bp,q (Rn ), 1 ≤ p < ∞, 0 < q < ∞. New substantial results about these problems have been obtained only very recently. In particular, [GSU19a, Theorem 1.1] found that the above Haar s system is an unconditional basis in Bp,q (Rn ) if, and only if, p, q, s are restricted by (3.102). This paper also contains a detailed discussion of what happens in the s (Rn ) given by the difference of the two sets in (3.103) and limiting cases of Bp,q   (3.102) with p < ∞, q < ∞ in the above p1 , s) : 0 ≤ p1 < ∞, s ∈ R half-plane. There, one asks for conditions ensuring that the above Haar system, if suitably enumerated, is a conditional (i.e., not unconditional) basis in some of these limiting spaces. In [GSU19a], the type of allowed enumeration is restricted to so-called admissible and strongly admissible enumerations of the above Haar system: The admitted enumerations are related to the location and the size of the supports of the Haar systems in a prescribed interplay. In particular, [GSU19a, Theorem 1.3] clarifies under which conditions every strongly admissible enumeration of the s above Haar system is a basis in Bp,q (Rn ). One has an affirmative answer for the s (Rn ) limiting spaces Bp,q with

1  n < p ≤ 1, s = n − 1 n+1 p

if, and only if, q = p.

(3.104)

If one replaces Rn by the n-torus Tn or by Q = (0, 1)n , the situation is different (in contrast to the unconditional basis in (3.102)). Then q = p in (3.104) must be replaced by 0 < q ≤ p. We refer again to [GSU19a] and related assertions in [Osw18], based on [Osw81]. s (Rn ) is even more involved. The Remark 3.21. The situation for the spaces Fp,q s n assertion for Fp,q (R ) with p < ∞ in Theorem 3.18 goes back to [T10, Sections 2.2, 2.3, pp. 71–98]. The use of complex interpolation for quasi-Banach spaces according to [KMM07] caused the q-dependence in (3.95). At that time, we did not know whether this outcome is an artefact of the method on which we relied or not. But it turned out quite recently that the conditions in (3.95) are natural. Let s 1 < p, q < ∞. Then the above Haar system is an unconditional basis in Fp,q (R) if,

102

Chapter 3. Further topics

and only if, max

1 1 1 1 , − 1 < s < min , . p q p q

(3.105)

This is the main result of [SeU17a, SeU17b], based on sophisticated arguments. The final assertion came out quite recently. According to [GSU19b, Theorem 1.1], s the above Haar system is an unconditional basis in Fp,q (Rn ) if, and only if, ⎧   1 1 1 1  ⎪ ⎪ ⎨0 < p < ∞, 0 < q < ∞, n max p , q , 1) − 1 < s < min p , q , 1 , (3.106) 1 < p < ∞, 1 < q < ∞, s = 0, ⎪ ⎪ ⎩1 < p < ∞, 1 < q < ∞, max  1 , 1  − 1 < s < 0. p q This coincides with (3.95), excluding in addition p = ∞ and q = ∞. (Recall that s the spaces Fp,q (Rn ) with max(p, q) = ∞ are not separable.) As already mentioned, s (Rn ) if, and only if, p, q, s the above Haar system is an unconditional basis in Bp,q are restricted by (3.102). This shows the striking difference between the B-spaces and the F -spaces as far as unconditional Haar bases are concerned. One may ask what happens in the context of F -spaces if (p, q, s) is covered by (3.102) but not by (3.106). This problem has been studied in [GSU18, GSU17, GSU19b] in the framework of the already mentioned admissible and strongly admissible enumeration of the above Haar system with the following outcome: According to [GSU18, Theorem 1.1], any admissible enumeration of the Haar system (3.90), (3.91) is s a basis in Fp,q (Rn ) with p, q, s as in (3.102). In particular, if (p, q, s) belongs to (3.102) but not to (3.106), then this basis must be conditional. These observas tions are complemented in [GSU19b, Theorem 1.3], where the spaces Fp,q (Rn ) are characterized such that every strongly admissible enumeration of the above Haar system is a basis in this space. This includes positive and negative assertions for s the limiting spaces Fp,q (Rn ) given by the difference of the two sets in (3.103) and  1  (3.102) in the p , s) : 0 ≤ p1 < ∞, s ∈ R half-plane with p < ∞, q < ∞. Remark 3.22. Theorem 3.18 on the one hand and the above-mentioned papers on the other hand suggest to complement (at least notationally) unconditional bases and conditional bases by weak*-bases. One must be aware that all considerations in the quoted literature take place in the framework of the dual pairing  S(Rn ), S  (Rn ) , and the corresponding duality arguments are occasionally used for proving or disproving that the Haar system is a (unconditional or conditional) basis in the related space Asp,q (Rn ). One could call the Haar system (or any other wavelet system) a weak*-basis in Asp,q (Rn ) if any f ∈ Asp,q (Rn ) can be uniquely represented as j f= λj,G (3.107) m hG,m j∈N0 ,G∈Gj , m∈Zn

without the a-priori-assumption λ ∈ asp,q (Rn ) as in (3.96), converging (unconditionally or conditionally) in S  (Rn ). This point of view is supported by the Fatou

3.6. Fubini property

103

property for all spaces Asp,q (Rn ) according to Theorem 1.25. All spaces covered by Theorem 3.18 have unconditional weak*-Haar-bases, including the related nonseparable spaces Asp,q (Rn ) with max(p, q) = ∞, in particular the corresponding s spaces F∞,q (Rn ). In [GSU19b], one also finds some discussions about the Haar s (Rn ) and in the completion of S(Rn ) in these spaces. system in F∞,q

3.6 Fubini property Let Wpk (Rn ), 1 < p < ∞, k ∈ N, be the classical Sobolev spaces as recalled in Section 4.3.1 below. Let n ≥ 2, x = (x1 , . . . , xn ) ∈ Rn

xj = (x1 , .., xj−1 , xj+1 , .., xn ) ∈ Rn−1 .

and

(3.108)

Let f ∈ Wpk (Rn ) and j

xj → f x (xj ) = f (x),

x ∈ Rn ,

(3.109)

considered as a function on R for any fixed xj ∈ Rn−1 . It is well known and an easy consequence of (4.84) below that Wpk (Rn ) has the so-called Fubini property f |Wpk (Rn ) ∼

n



 j



f x |Wpk (R) Lp (Rn−1 ) ,

(3.110)

j=1

where (in the usual measure-theoretical interpretation) the inner norm is taken with respect to xj ∈ R and where Lp (Rn−1 ) applies to xj ∈ Rn−1 . This assertion was extended in [Str67, Str68] to the Sobolev spaces Hps (Rn ) with 1 < p < ∞, s (Rn ), s > 0, see (4.88)–(4.92), and to the special Besov spaces Bps (Rn ) = Bp,p 1 ≤ p ≤ ∞, s > 0. In [T83, Section 2.5.13, pp. 114–117], we dealt with this problem, and in [T01, Chapter 4, pp. 34–40], we obtained a satisfactory assertion for some spaces Asp,q (Rn ) with p < ∞ for the F -spaces. Let us now return to this s (Rn ) can be incorporated. Let problem and ask to which extent the spaces F∞,q again   1  σp(n) = n max , 1 − 1 p

and

  1 1  (n) σp,q = n max , , 1 − 1 . p q (n)

(3.111)

If f ∈ Asp,q (Rn ) with 0 < p, q ≤ ∞ and s > σp , then it follows from Theorem 2.4 that (3.109) makes sense in the usual a.e. (almost everywhere) measure-theoretical interpretation. If p = ∞ and s > 0, then the corresponding spaces As∞,q (Rn ), 0 < q ≤ ∞, are continuously embedded in the space of bounded continuous functions, see Theorem 2.3, and the traces trL on lines L in Rn ,   (3.112) L = x = (x1 , . . . , xn ) : xj = aj + tbj , t ∈ R ,

104

Chapter 3. Further topics

a = (a1 , . . . , an ) ∈ Rn , b = (b1 , . . . , bn ) ∈ Rn , can be taken pointwise. An iterated application of Theorem 2.13 shows that s s (Rn ) = B∞,q (L), trL B∞,q

and

s trL F∞,q (Rn ) = C s (L),

(3.113)

s s (L) = F∞,∞ (L) as in (2.68) and s > 0, 0 < q ≤ ∞. with (again) C s (L) = B∞,∞

Definition 3.23. Let 2 ≤ n ∈ N. Let Asp,q (Rn ) with A ∈ {B, F }, 0 < p ≤ ∞, (n)

0 < q ≤ ∞ and s > σp

be the global spaces according to Definition 1.1.

(i) Let 0 < p < ∞. The space Asp,q (Rn ) is said to have the Fubini property if for j any f ∈ Asp,q (Rn ) the function f x according to (3.109) belongs to Asp,q (R) for almost all xj ∈ Rn−1 and f |Asp,q (Rn ) ∼

n



 j



f x |Asp,q (R) Lp (Rn−1 )

(3.114)

j=1

(equivalent quasi-norms). (ii) The space As∞,q (Rn ) is said to have the Fubini property if for any f ∈ j As∞,q (Rn ) the function f x according to (3.109) belongs to As∞,q (R) for all xj ∈ Rn−1 and f |As∞,q (Rn ) ∼

n

sup

j n−1 j=1 x ∈R

j

f x |As∞,q (R)

(3.115)

(equivalent quasi-norms). Remark 3.24. The modification in part (ii) compared to part (i) is not obvious, but justified by the pointwise trace according to (3.113). Part (i) imitates the usual formulation of the Fubini theorem for functions belonging to Lp (Rn ) with p < ∞. Theorem 3.25. Let n ≥ 2. s (i) The spaces Fp,q (Rn ) with

0 < p < ∞,

0 σp,q

and

(3.116)

have the Fubini property. (n)

s (ii) The spaces Bp,q (Rn ) with 0 < p, q ≤ ∞ and s > σp if, and only if, p = q.

have the Fubini property

s (iii) The spaces F∞,q (Rn ) with s > 0 and q < ∞ do not have the Fubini poperty.

Proof. The parts (i) and (ii) basically coincide with [T01, Theorem 4.4, p. 36]. There is, however, a minor difference: In contrast to part (ii) of the above definition, s we incorporated B∞,q (Rn ), s > 0, 0 < q ≤ ∞ in [T01, Definition 4.2, p. 35] into

3.7. Characterizations in terms of Lusin functions and heat kernels

105

part (i) of the Definition 3.23 based on (3.113). But this does not influence the arguments in [T01]. It remains to justify part (iii) of the above theorem. According s (L) → C s (L), s > 0, 0 < q < ∞ to Theorem 2.10 and (2.45), the embedding F∞,q s is strict. Then it follows from (3.113) that there are functions f ∈ F∞,q (Rn ) such s (L). This proves that their pointwise restriction to a line L does not belong to F∞,q part (iii).  Remark 3.26. It is reasonable to extend part (i) of Definition 3.23 to the spaces As∞,q (Rn ), s > 0, 0 < q ≤ ∞. However, we could not prove whether part (iii) of Theorem 3.25 remains valid under these modified circumstances. Remark 3.27. The Fubini property is of some interest in itself. But occasionally, it is also of great service. In this context, we refer to [T06, p. 290] and in particular to [T10, p. 89], where we used it in a decisive way in connection with Haar wavelets. A further example is the equivalence in (2.174).

3.7 Characterizations in terms of Lusin functions and heat kernels 3.7.1 Lusin functions s s The definitions of F∞,q (Rn ) according to the modification in (1.14) and of Fp,q (Rn )  ∨  with p < ∞ in (1.10) differ in two points. Firstly, one replaces  ϕJ f (x) by related means over the cubes QJ,M . And secondly, one has an additional summation over j ≥ J inside the integral over QJ,M . But as it turns out, only the second  ∨  modification is essential. With the following result, one can replace  ϕJ f (x) s also in the spaces Fp,q (Rn ) with p < ∞ by related means, called Lusin functions.

Theorem 3.28. Let ϕ = {ϕJ }∞ J=0 be the same dyadic resolution of unity as in s (Rn ) is the collection of all Definition 1.1. Let 0 < p, q < ∞ and s ∈ R. Then Fp,q f ∈ S  (Rn ) such that f

s |Fp,q (Rn )∗ϕ

 ∞



Jn =

2 J=0

{y:|·−y|≤2−J }

 ∨ q 1/q  Lp (Rn )

2Jsq  ϕJ f (y) dy

(3.117)

is finite (equivalent quasi-norms). Remark 3.29. Of course, {y : |x − y| ≤ 2−J } refers to a ball of radius 2−J centered at x ∈ Rn . If one replaces Lp (Rn ) in (3.117) by L∞ (Rn ), one essentially arrives at (1.14), but without the additional summation over j ≥ J. The above assertion goes back to [Pai82] and was mentioned in [T83, Section 2.12.1, pp. 182–183] without proof but with further references. We returned to this topic in [T92, Section 2.4.5, pp. 120–121] in greater detail based on [Tri88, pp. 179–180], where one finds further references and discussions.

106

Chapter 3. Further topics

3.7.2 Heat kernels Thermic (caloric, in terms of Gauss-Weierstrass semi-groups) as well as harmonic (in terms of Cauchy-Poisson semi-groups) characterizations of the spaces Asp,q (Rn ) with A ∈ {B, F }, s ∈ R and 0 < p, q ≤ ∞ have a long history. The beginning of this theory may be found in [T78, Sections 2.5.2, 2.5.3, pp. 190–196], including the related (historical) literature. In [T15, Section 2.4, pp. 13–17], we described more recent assertions and referred the reader to the underlying papers. This will not be repeated here. In this Section, we are mainly interested in showing how s corresponding characterizations for the spaces Fp,q (Rn ) with p < ∞ on the one hand and p = ∞ on the other hand look like. This is essentially (but maybe not totally) covered by the already existing literature. First, we describe the background and recall the well-known characterizations for the spaces Asp,q (Rn ), A ∈ {B, F }, s ∈ R, 0 < p ≤ ∞ (p < ∞ for F -spaces) and 0 < q ≤ ∞ in terms of heat kernels. Let w ∈ S  (Rn ), n ∈ N. Then  |x−y|2 1 Wt w(x) = e− 4t w(y) dy (3.118) n/2 (4πt) Rn   |x−·|2 1 w, e− 4t , t > 0, x ∈ Rn , = n/2 (4πt) is the well-known Gauss-Weierstrass semi-group which can be written in Fourier terms as −t|ξ|2  w(ξ),  ξ ∈ Rn , t > 0, (3.119) W t w(ξ) = e where the Fourier transform according to (1.2) is taken with respect to the space variables x ∈ Rn . Of course, both (3.118) and (3.119) must be interpreted in the context of S  (Rn ). But we recall that (3.118) makes sense pointwise: It is the |y|2

convolution of w ∈ S  (Rn ) and gt (y) = (4πt)−n/2 e− 4t ∈ S(Rn ). In particular,  N/2 |(w ∗ gt )(x)| ≤ ct 1 + |x|2 , x ∈ Rn , (3.120) w ∗ gt ∈ C ∞ (Rn ), for some ct > 0 and some N ∈ N. Some more details and references may be found in [T78, Section 2.2.1, p. 152] and [T14, Section 4.1, pp. 112–114]. Let ∂t = ∂/∂t and ∂tm = ∂ m /∂tm , m ∈ N0 , with ∂t0 f = f . According to [T92, Theorem 2.6.4, p. 152], the following characterization holds. For this, let W1 f = Wt f with t = 1. Proposition 3.30. Let 0 < p ≤ ∞ (p < ∞ for F -spaces), 0 < q ≤ ∞, s ∈ R and s (Rn ) is the collection of all f ∈ S  (Rn ) such that s/2 < m ∈ N0 . Then Bp,q  1

q dt 1/q

s s f |Bp,q (Rn )(m) = W1 f |Lp (Rn ) + t(m− 2 )q ∂tm Wt f |Lp (Rn )

t 0 (3.121) s (Rn ) is the collection of all f ∈ S  (Rn ) such that is finite, and Fp,q s f |Fp,q (Rn )(m) = W1 f |Lp (Rn )

  1 q dt 1/q  s



t(m− 2 )q ∂tm Wt f (·) |Lp (Rn )

+

t 0

(3.122)

3.7. Characterizations in terms of Lusin functions and heat kernels

107

is finite (with the usual modification for q = ∞). Remark 3.31. In [T92, Theorem 2.6.4, p. 152], we formulated the above assertion with (ϕ0 f)∨ instead of W1 f , where ϕ0 is a compactly supported C ∞ -function in Rn with ϕ0 (0) = 0. But the assertion remains valid if ϕ0 ∈ S(Rn ) with ϕ0 (0) = 2 0. This particularly applies to ϕ0 (ξ) = e−|ξ| with (ϕ0 f)∨ (x) = W1 f (x). The advantage of this formulation is that the terms with W1 f (x) can be incorporated into the second terms on the right-hand sides of (3.121) and (3.122) if s < 0 and m = 0. In particular, if s < 0 and 0 < p, q ≤ ∞ (with p < ∞ for the F -spaces), s then Bp,q (Rn ) is the collection of all f ∈ S  (Rn ) such that  1 dt 1/q s (Rn ) = t−sq/2 Wt f |Lp (Rn )q (3.123) f |Bp,q t 0 s is finite, and Fp,q (Rn ) is the collection of all f ∈ S  (Rn ) such that

  1

q dt 1/q 



s (Rn ) =

t−sq/2 Wt f (·) |Lp (Rn )

f |Fp,q t 0

(3.124)

is finite (with the usual modification for q = ∞). Of special interest is C s (Rn ) = s s (Rn ) = F∞,∞ (Rn ) according to (1.72) with s < 0. Then B∞,∞   sup t−s/2 Wt f (x), s < 0, (3.125) f |C s (Rn ) = x∈Rn ,00,x∈Rn

∗

in C s (Rn ), where 0 < p ≤ ∞ (with (4.19) for p = ∞). This follows again from the validity of the mean value property according to (3.132) for all t > 0 and from (3.133). The above equivalent norms are a little bit simpler than their counterparts in [T15, p. 47–48] and [Tri17a, p. 338]. Now, we used the fact that not only the first right-hand side in (3.132), but also the second one generates equivalent norms. Let us now take (4.18) as the extension of Definition 4.1(iii) to q = ∞. For that, let again Asp,q (Rn ) with A ∈ {B, F }, s ∈ R and 0 < p, q ≤ ∞ be the inhomogeneous spaces as introduced in Definition 1.1. ∗

Theorem 4.3. Let Asp,q (Rn ) with A ∈ {B, F }, s < 0 and 0 < p, q ≤ ∞ be the spaces according to Definition 4.1 complemented by (4.18). (i) They are quasi-Banach spaces (Banach spaces if p ≥ 1, q ≥ 1) with ∗

Asp,q (Rn ) → Asp,q (Rn ) → S  (Rn ) and

∗

∗

n

f (λ·) |Asp,q (Rn ) = λs− p f |Asp,q (Rn ), (ii) Furthermore,

(4.21) λ > 0.

∗

S(Rn ) → B sp,q (Rn )

(4.22)

(4.23)

if, and only if,  1 < p ≤ ∞,

  s > n p1 − 1 ,   s = n p1 − 1 ,

and

0 < q ≤ ∞, q = ∞,

∗

S(Rn ) → F sp,q (Rn )

(4.24)

(4.25)

if, and only if, 1 < p ≤ ∞,

s>n

 1 −1 , p

0 < q ≤ ∞.

(4.26)

These assertions remain valid if one replaces S(Rn ) by D(Rn ) = C0∞ (Rn ).

120

Chapter 4. Complements

Remark 4.4. The above assertions are covered by [T15, Theorems 3.3, 3.5, pp. 48– 49, 52] and [Tri17a, Theorem 2.6, p. 340], where detailed proofs can be found. The requested homogeneity (4.12) and the second embedding in (4.13) are satisfied for ∗

all admitted spaces Asp,q (Rn ) with s < 0 and 0 < p, q ≤ ∞. But the first embedding in (4.13) requires that the parameters p, q, s satisfy (4.24) for the B-spaces  and  (4.26) for the F -spaces. This corresponds to the lower borderline s = n p1 − 1 < 0 in (4.14) and Figure 4.1, p. 129. Remark 4.5. We refer the reader to [T15, Tri17a, Tri17b] for further properties of these spaces. In particular, ∗

s f |B˙ p,q (Rn )ϕ ,

f ∈ B sp,q (Rn ),

s f |F˙ p,q (Rn )ϕ ,

f ∈ F sp,q (Rn ),

s0

  t−r/2 Wt f (x),

(4.29)

(4.30)

4.1. Tempered homogeneous spaces

121

be the same spaces as in (4.18), (4.19) with ∗

S(Rn ) → C r (Rn ),

−n ≤ r < 0,

(4.31)

∗

according to Theorem 4.3(ii). We use C r (Rn ) as an endpoint or anchor space for ∗

the spaces Asp,q (Rn ) having the same differential dimension s − np = r. We basically follow [Tri17a, Definition 2.8, pp. 343–344], where we included some limiting cases compared with [T15, Definition 3.22, p. 78]. In addition, we now pay special ∗

attention to F s∞,q (Rn ). We always assume n ∈ N. Definition 4.6.

(i) Let 0 < p ≤ ∞ (p < ∞ for F -spaces), 0 < q ≤ ∞, 1  n −1 0 ∗

0 ∗

q  dτ 1/q (m− 2s )q  m  τ W f (y) dy ∂ τ τ √ τ {y:|x−y|≤ t} (4.35)

if finite. Let F s∞,∞ (Rn ) = B s∞,∞ (Rn ) according to part (i). ∗   (iii) Let 0 < p ≤ ∞, s = n p1 − 1 and s/2 < m ∈ N0 . Then B sp,∞ (Rn ) is the collection of all f ∈ S  (Rn ) such that ∗ ∗

s

f |B sp,∞ (Rn )(m) = f |C −n (Rn ) + sup tm− 2 ∂tm Wt f |Lp (Rn )

t>0

(4.36)

122

Chapter 4. Complements is finite. ∗

(iv) Let 0 < p < ∞, 0 < q ≤ 1, s = n/p and s/2 < m ∈ N. Then B sp,q (Rn ) is the collection of all f ∈ S  (Rn ) such that ∗

f |B sp,q (Rn )(m) = sup |(f, ϕ)|  ∞

q dt 1/q s + t(m− 2 )q ∂tm Wt f |Lp (Rn )

t 0

(4.37)

is finite, where the supremum is taken over all ϕ ∈ S(Rn ) such that ϕ |L1 (Rn ) ≤ 1. ∗

(v) Let 0 < p ≤ 1, 0 < q ≤ ∞, s = n/p and s/2 < m ∈ N. Then F sp,q (Rn ) is the collection of all f ∈ S  (Rn ) such that ∗   f |F sp,q (Rn )(m) = sup (f, ϕ) 

 ∞

(4.38) q dt 1/q   s

Lp (Rn )

+

t(m− 2 )q ∂tm Wt f (·)

t 0 is finite, where the supremum is taken over all ϕ ∈ S(Rn ) such that ϕ |L1 (Rn ) ≤ 1. Remark 4.7. Part (i) essentially coincides with [T15, Definition 3.22, p. 78]. This definition was complemented in [Tri17a, Definition 2.8, pp. 343–344] by the limiting cases in the parts (iii) to (v). Now we added part (ii). They are the homogeneous counterparts of characterizations of the corresponding inhomogeneous spaces in terms of heat kernels according to Proposition 3.30 and Theorem 3.32(ii) with (4.19) = (4.30) instead of W1 f |Lp (Rn ) as a starting term. As far as the first terms on the right-hand sides of (4.37) and (4.38) are concerned, one has   (4.39) sup |(f, ϕ)| : ϕ ∈ S(Rn ), ϕ |L1 (Rn ) ≤ 1 = f |L∞ (Rn ). But we prefer the above version. Then it can be tested for any f ∈ S  (Rn ) whether it belongs to the corresponding space or not. Let us now make a few comments on the relations for the parameters. The differential dimensions r for the spaces ∗

C r (Rn ), −n ≤ r < 0 and zero for L∞ (Rn ), are the same as in the corresponding second terms on the right-hand sides of (4.33)–(4.38), r = s − np (zero for the spaces spaces in (iv) and (v)). Using the explicit calculations in [T15, p. 61], it turns out that in all cases ∗

n

∗

f (λ·) |Asp,q (Rn ) = λs− p f |Asp,q (Rn ),

λ > 0,

(4.40)

as requested in (4.12). In order to provide a better understanding for the restrictions of the parameters, we recall the corresponding assertions for their inhomogeneous counterparts. According to the Theorems 2.5 and 2.9, one has n r (4.41) (Rn ), r =s− , Asp,q (Rn ) → C r (Rn ) = B∞,∞ p

4.1. Tempered homogeneous spaces

123

for all spaces Asp,q (Rn ) with A ∈ {B, F }, s ∈ R and 0 < p, q ≤ ∞. This justifies ∗

why one can rely on C r (Rn ) with −n ≤ r < 0 as the anchor space on the right-hand sides of (4.33)–(4.36). For s = n/p, it follows from Theorem 2.3 that n/p (Rn ) → L∞ (Rn ) Bp,q

if, and only if, 0 < p ≤ ∞, 0 < q ≤ 1,

(4.42)

n/p (Rn ) → L∞ (Rn ) Fp,q

if, and only if, 0 < p ≤ 1, 0 < q ≤ ∞.

(4.43)

and

This explains the corresponding restrictions in the parts (iv) and (v) of Definition ∗

4.6. However, some arguments in [Tri17a] do not apply to possible spaces B 0∞,q (Rn ) with q ≤ 1 covered by Theorem 2.3. This is the reason for the additional restriction p < ∞ in part (iv). As far as the request ∗

S(Rn ) → Asp,q (Rn )

(4.44)

  in (4.13) is concerned, it follows from Theorem 4.3 that the restrictions s > n p1 −1 ∗   in the parts (i), (ii) and s = n p1 − 1 , q = ∞ for B sp,q (Rn ) in part (iii) are natural. In [T15] and the subsequent papers [Tri17a, Tri17b], we developed the theory ∗

∗

of the spaces Asp,q (Rn ). Now, we would like to incorporate F s∞,q (Rn ). But this can be done by the same arguments and by what has already been said about s their inhomogeneous counterparts F∞,q (Rn ). Therefore, we will not stress this point. Instead, we will restrict ourselves to the formulation of some basic assertions complemented in Section 4.1.4 by a few distinguished properties. For that purpose, let again n ∈ N. ∗

Theorem 4.8. The spaces Asp,q (Rn ),

A=B

with

⎧ ⎪ ⎨0 < p ≤ ∞, 0 < q ≤ ∞, 0 < p ≤ ∞, q = ∞, ⎪ ⎩ 0 < p < ∞, 0 < q ≤ 1,

  n p1 − 1 < s <  1 s= p −1 , s = np ,

n p,



  n p1 − 1 < s < s = np ,

n p,

(4.45)

and A=F

with

0 < p ≤ ∞, 0 < q ≤ ∞, 0 < p ≤ 1, 0 < q ≤ ∞,

(4.46)

as introduced in Definition 4.6 are quasi-Banach spaces (Banach spaces if p ≥ 1, ∗

q ≥ 1). Let s/2 < m ∈ N0 . Then f |B sp,q (Rn )(m) according to (4.33), (4.36),

124

Chapter 4. Complements ∗

∗

(4.37) are equivalent quasi-norms in B sp,q (Rn ), and f |F sp,q (Rn )(m) according to ∗

(4.34), (4.35), (4.38) are equivalent quasi-norms in F sp,q (Rn ). Furthermore, ∗

S(Rn ) → Asp,q (Rn ) → S  (Rn ) and

∗

n

∗

f (λ·) |Asp,q (Rn ) = λs− p f |Asp,q (Rn ),

(4.47)

λ > 0.

(4.48)

Proof. As   far as the spaces with p = ∞, s < 0, are concerned, the assertions with n p1 − 1 < s < np are covered by [T15, Theorem 3.24, pp. 79–80] complemented by [T15, Theorem 3.5, p. 52]. This also applies to (4.35) with m = 0. But it can be extended by the same arguments and references as there to any m ∈ N0 . It is the homogeneous counterpart of (3.127). Detailed proofs for the limiting cases according to Definition 4.6(iii)-(v) may be found in [Tri17a, Theorem 2.10, pp. 345–348].  Remark 4.9. All spaces covered by Definition 4.6 satisfy the homogeneity (4.12) = (4.48) and the continuous embedding (4.13) = (4.47). This justifies their name tempered homogeneous 4.3 shows that the lower borderline 0 <   spaces. Theorem   p ≤ ∞, s = n p1 − 1 in the p1 , s -diagram in Figure 4.1, p. 129, is a natural barrier for the left-hand side of (4.47). From the discussions in [T15, Section 3.19, 109–111], it follows that the upper borderline 0 < p ≤ ∞, s = n/p in this  pp. 1 p , s -diagram is a natural barrier for the right-hand side of (4.47). As far as the limiting spaces with s = n/p are concerned, we took the embeddings in (4.42) and (4.43) as a guide. Definition 4.6 and Theorem 4.8 cover all related homogeneous ∗

spaces except possible spaces B 0∞,q (Rn ), 0 < q ≤ 1, for which our arguments do not work. The situation is similar as for multiplication algebras, where (2.137) and (2.138) coincide with the corresponding conditions in Theorem 2.3 except for 0 B∞,q (Rn ), 0 < q ≤ 1. Remark 4.10. In [T15, Tri17a, Tri17b], we developed the theory of the above tem∗

pered homogeneous spaces Asp,q (Rn ) in detail. This will not be repeated here, with a few exceptions. Recall that the interest for homogeneous spaces within the distinguished strip (4.14) arose from some famous nonlinear evolutionary equations in hydrodynamics and chemotaxis and the related so-called critical spaces according to (4.10) and (4.11). Furthermore, one wishes to extend Remark 4.5 to all spaces that were introduced in Definition 4.6. This can be done, with the outcome that s f |B˙ p,q (Rn )ϕ ,

∗

f ∈ B sp,q (Rn ),

(4.49)

according to (4.6) is an equivalent quasi-norm in all related spaces covered by Definition 4.6(i,iii,iv), and that s (Rn )ϕ , f |F˙ p,q

∗

f ∈ F sp,q (Rn ),

(4.50)

4.1. Tempered homogeneous spaces

125

according to (4.7) and (4.8) is an equivalent quasi-norm in all related spaces covered by Definition 4.6(i,ii,v). In other words, as long as one restricts the calculations to the corresponding spaces or to related families of spaces, one can work with these equivalent quasi-norms without any problems modulo polynomials. In [T15], quasi-norms of this type are called domestic. In Theorem 4.3, we described how homogeneous and inhomogeneous spaces with s < 0 are related to each other. This can be complemented as follows. Recall that   1  0 < p ≤ ∞, (4.51) σp(n) = n max , 1 − 1 , p and that r = s −

n p

as before. Then ∗

∗

S(Rn ) → Asp,q (Rn ) → Asp,q (Rn ) → C r (Rn ) → S  (Rn )

(4.52)

∗

for all spaces Asp,q (Rn ),

A=B

with

⎧ (n) ⎪ ⎨0 < p < ∞, 0 < q ≤ ∞, σp < s < (n) 0 < p < ∞, q = ∞, σp = s, ⎪ ⎩ 0 < p < ∞, 0 < q ≤ 1, s = np ,

and

 A=F

(n)

n p,

0 < p < ∞, 0 < q ≤ ∞, σp < s < 0 < p ≤ 1, 0 < q ≤ ∞, s = np ,

with

(4.53)

n p,

(4.54)

∗

with L∞ (Rn ) instead of C r (Rn ) if s = n/p. For the proofs, we refer the reader again to [T15, Theorem 3.24, pp. 79–81] and [Tri17a, Theorem 2.10, pp. 345–346]. The embeddings between homogeneous and inhomogeneous spaces with s < 0 in (4.21) and with s > 0 in (4.52) are different. But this does not come as a surprise, since some duality can be expected.

4.1.4 Some properties The theory of tempered homogeneous spaces was developed in [T15, Tri17a, Tri17b]. This will not be repeated here. But it seems reasonable to mention a few properties which are directly related to the preceding considerations. ∗

So far, we have (4.52) for the spaces Asp,q (Rn ) in (4.53), (4.54) of positive smoothness s. As the following result shows, there is a converse. We use the pre(n) vious notation. In particular, σp has the same meaning as in (4.51). Theorem 4.11. Let Ω be a bounded domain in Rn , n ∈ N. Let Asp,q (Rn ) and ∗

Asp,q (Rn ) be the spaces as introduced in the Definitions 1.1 and 4.6 restricted to  (n) 0 < p < ∞, 0 < q ≤ ∞, σp < s < np , A = B with (4.55) 0 < p < ∞, 0 < q ≤ 1, s = np ,

126

Chapter 4. Complements

and

 with

A=F

0 < p < ∞, 0 < q ≤ ∞, 0 < p ≤ 1, 0 < q ≤ ∞,

(n)

σp < s < s = np .

n p,

(4.56)

Then ∗     f ∈ Asp,q (Rn ) : supp f ⊂ Ω = f ∈ Asp,q (Rn ) : supp f ⊂ Ω .

(4.57)

Proof. We mentioned this property in [T15, (3.232), p. 85] for the spaces with (n) σp < s < np , where we used {x : |x| < ε} instead of Ω. But the arguments given there apply to any bounded domain Ω as well as to the spaces (4.55), (4.56) with s = n/p.  Remark 4.12. In [T15, Theorem 3.27, pp. 84–87], we used this observation to justify ∗

the expansion of some spaces Asp,q (Rn ) in terms of homogeneous Haar systems. This is the homogeneous counterpart of Theorem 3.18. As a second application, we improved in [T15, Corollary 3.55, p. 116] earlier versions of the local homogeneity for some inhomogeneous spaces Asp,q (Rn ), which eventually results in Theorem 3.9. Multiplication algebras of spaces A(Rn ) in S  (Rn ) as introduced in Definition 2.37 are very useful in the theory of nonlinear PDEs. Related references may be found in Remark 2.38. Theorem 2.41 gives a final answer under which conditions the inhomogeneous spaces Asp,q (Rn ) are multiplication algebras. The ∗

following counterpart exists for the homogeneous spaces Asp,q (Rn ). ∗

Theorem 4.13. A space Asp,q (Rn ) as introduced in Definition 4.6 is a multiplication algebra if, and only if, either ∗

∗

∗

∗

Asp,q (Rn ) = B sp,q (Rn ),

0 < p < ∞,

0 < q ≤ 1,

s = n/p,

(4.58)

0 < p ≤ 1,

0 < q ≤ ∞,

s = n/p.

(4.59)

or Asp,q (Rn ) = F sp,q (Rn ),

Remark 4.14. This coincides with [Tri17a, Theorem 3.7, pp. 354–355]. The proof given there relies on the global homogeneity according to (4.48), which makes clear that only spaces with s = n/p can be multiplication algebras, and on Theorem 4.11 combined with Theorem 2.41.

4.1.5 Caloric smoothing In Theorem 3.35, we dealt with caloric smoothing in the context of the inhomogeneous spaces Asp,q (Rn ). The same arguments can be used for justifying corresponding assertions for the related homogeneous spaces.

4.1. Tempered homogeneous spaces

127

∗

Proposition 4.15. Let d ≥ 0. Let Asp,q (Rn ) with A ∈ {B, F } be the spaces according to Definition 4.1, restricted by 0 < p, q ≤ ∞ and s + d + 2q < 0. Then there is a ∗

constant c > 0 such that for all t with t > 0 and all w ∈ Asp,q (Rn ), ∗

∗

n s n td/2 Wt w |As+d p,q (R ) ≤ c w |Ap,q (R ).

(4.60)

Proof. If one replaces 0 < t < 1 in the quasi-norms in (3.123), (3.124) and (3.127) with m = 0 by 0 < t < ∞, one obtains the corresponding quasi-norms in Definition ∗

∗

n s n 4.1 both for As+d p,q (R ) and Ap,q (R ). This shows that one can apply the arguments in Step 1 of the proof of Theorem 3.35, based on the boundedness of (t, τ ) in (3.139). Instead of the embedding (2.45) in connection with (3.143), one now uses ∗

∗

F s∞,q (Rn ) → C s (Rn ),

0 < q ≤ ∞,

s < 0,

(4.61)

which is covered by [T15, Proposition 3.7, p. 56].



There is the temptation to extend (4.60) similarly as in Step 2 of the proof of Theorem 3.35 by lifting to all homogeneous spaces. This is supported by the related quasi-norms in (4.6)–(4.8) and the counterpart I˙δ Wt I˙−δ = Wt , t > 0, δ ∈ R, (4.62) of (3.144), based on the lifts I˙δ :

 ∨ f → |ξ|−δ f ,

ξ ∈ Rn ,

δ ∈ R,

(4.63)

see [T15, Proposition 2.18, p. 23]. But one must be well aware that one gets into troubled waters, especially if one wishes to stay within the dual pairing   S(Rn ), S  (Rn ) as we suggested above. Note, however, that there is a somewhat more complicated yet direct way, which we already prepared in Remark 3.38 above. ∗

Theorem 4.16. Let d ≥ 0. Let A ∈ {B, F } and p, q, s such that both Asp,q (Rn ) and ∗

n As+d p,q (R ) are covered by Definition 4.6. Then there is a constant c > 0 such that ∗

for all t with t > 0 and all w ∈ Asp,q (Rn ), ∗

∗

n s n td/2 Wt w |As+d p,q (R ) ≤ c w |Ap,q (R ).

(4.64)

Proof. If r + d < 0, Proposition 4.15 implies ∗

∗

td/2 Wt w |C r+d (Rn ) ≤ c w |C r (Rn ),

0 < t < ∞.

(4.65)

0 < t < ∞.

(4.66)

Furthermore, it follows from (4.30) that ∗

td/2 Wt w |L∞ (Rn ) ≤ w |C −d (Rn ),

d > 0,

This covers all first terms in the quasi-norms (4.33)–(4.38), where we used (4.39). The second terms in these quasi-norms can be treated as in Remark 3.38, where (3.150) is covered by (4.33) with p = q = ∞. 

128

Chapter 4. Complements

4.2 Natural habitats and the homogeneity rule 4.2.1 Natural habitats   The natural habitat within the framework of the dual pairing S(Rn ), S  (Rn ) occupied by the inhomogeneous spaces Asp,q (Rn ), n ∈ N, as introduced in Definition 1.1 is the closed half-space # 1  $ , s : 0 < p ≤ ∞, s ∈ R (4.67) p in the related p1 -s-diagram, complemented by 0 < q ≤ ∞, see Figure 4.1, p. 129. The corresponding natural habitat within the framework of the dual pairing ∗   S(Rn ), S  (Rn ) occupied by the homogeneous spaces Asp,q (Rn ), n ∈ N, as introduced in Definition 4.6 is the distinguished strip # 1  1  n$ , s : 0 < p ≤ ∞, n − 1 ≤ s ≤ (4.68) p p p in the related p1 -s-diagram, complemented by 0 < q ≤ ∞ with the indicated re  strictions for p and q at the borderlines s = n p1 − 1 and s = np . Recall that any ∗

admitted homogeneous space Asp,q (Rn ) should satisfy not only the global homogeneity (4.12) = (4.48) but also the topological embedding (4.13) = (4.47). s Remark 4.17. After the spaces F∞,q (Rn ) have been incorporated into the theory of s n the inhomogeneous spaces Ap,q (R ), it is rather obvious to call (4.67) the natural ∗

habitat for these spaces. For the homogeneous spaces Asp,q (Rn ), however, the situ∗

ation is a little bit different. The theory of the spaces Asp,q (Rn ) with 0 < p, q ≤ ∞   and s < 0 as described in Section 4.1.2 within the dual pairing S(Rn ), S  (Rn ) looks reasonable and avoids the possible ambiguity modulo polynomials. This also ∗

applies to the corresponding spaces Asp,q (Rn ) with positive smoothness, 0 < p, q ≤ ∞

and

σp(n) < s
n max , , p q r r (4.99) Let 0 < u ≤ r and s < m ∈ N. Then

  1

1/q 

s q dt Lp (Rn )

(Rn )m,u = f |Lp (Rn ) +

t−sq dm f |Fp,q

(4.100) t,u f (·) t 0 s (Rn ), see (with the usual modification for q = ∞) are equivalent quasi-norms in Fp,q [T92, Theorem 3.5.3, p. 194]. In any case, if p, q, s are restricted according to (4.96)

4.3. Spaces and tools

135

s s (Rn ) and Fp,q (Rn ) have equivalent quasior (4.99), then the related spaces Bp,q norms in terms of differences and ball means of differences. It follows from Theorem 2.4 that the restriction (4.96) for the B-spaces is natural. But this also applies to the restriction (4.99) with r = u = 1 for the F -spaces, see [ChS06]. Characteriza  (n) s tions of the spaces F∞,q (Rn ) with 0 < q < ∞ and s > σp = n max( 1q , 1) − 1 in terms of the above ball means may be found in [YSY10, Section 4.3.4, p. 110] and in greater detail in the recent paper [Park19a], based on [Park19]. In some contrast to our previous books [T78, T83, T92, T06], we did not use descriptions in terms of differences in the above text. Instead, we mainly relied on wavelets, complemented by Fourier-analytical arguments and heat kernels. But one may compare the above formulations with their heat kernel counterparts according to Proposis s (Rn ), the related restrictions Bp,q (Ω) to domains Ω in tion 3.30. The spaces Bp,q Rn and their intrinsically defined modifications with s > 0 and 1 ≤ p, q ≤ ∞ have been studied by the Russian school since the 1960s in great detail. Differences, means of differences and related integral representations play a central role up to our time. The standard references are [Nik77] (first edition 1969) and [BIN75].

4.3.4 Fourier-analytical decompositions and paramultiplication s s The Fourier-analytical introduction of the spaces Bp,q (Rn ) and Fp,q (Rn ) with 0 < p, q ≤ ∞ (with p < ∞ for the F -spaces) and s ∈ R according to Definition 1.1(i,ii) goes back to [Pee67, Pee73, Pee75, Tri73b, Tri73c] and was afterwards elaborated in [Pee76, T78, Tr77, Tr78, T83]. It turned out rather quickly that cross multiplication on the Fourier side is an effective tool for studying products of elements belonging to these spaces as well as related operators. This was used in [Pee76, pp. 144–146] and [Tri77]. At this point, we give a brief description. For that purpose, let {ϕk }∞ k=0 be the resolution of unity according to (1.3)–(1.5). Then g ∈ S  (Rn ) and f ∈ S  (Rn ) can be represented as

g=

∞

bj (x),

f=

j=0

∞

 ∨ bj (x) = ϕj g (x),

cj (x),

 ∨ cj (x) = ϕj f (x).

j=0

   ∨ (x) of the product gf , Of interest are the building blocks ϕk gf



 ϕk gf

∨

 (x) = Rn

ϕ∨ 1 (y)

∞

bj (x − 2−k y) cl (x − 2−k y) dy,

(4.101)

(4.102)

j,l=0

k ∈ N, x ∈ Rn . A closer examination of the supports of the factors in the products shows that many of them are zero. The remaining terms can be subordinated to

136

Chapter 4. Complements

the three model cases 

k (x)

=

k  j=0



k (x) =

 k

(x) =

Rn

k  l=0 ∞  l=k

Rn

Rn

−k ϕ∨ y) ck (x − 2−k y) dy, 1 (y) bj (x − 2

(4.103)

−k ϕ∨ y) cl (x − 2−k y) dy, 1 (y) bk (x − 2

(4.104)

−k ϕ∨ y) cl (x − 2−k y) dy. 1 (y) bl (x − 2

(4.105)

This multiplication procedure coincides with [Tri77, (7)–(9), (13)–(15)]. It was repeated in [Tr78, Section 2.6.1, pp. 127–133] and [T83, Section 2.8.2, pp. 140–145]. There, one also finds further technical explanations. Together with several types of maximal functions, it is an effective instrument for dealing with the multiplication of elements of the spaces Asp,q (Rn ) in the framework of the dual pairing   S(Rn ), S  (Rn ) . Later on, this method was also used in [ET96, SiT95, RuS96] and in many other papers and books up to our time, including [BCD11, Saw18]. The same proposal was made a few years later in [Bony81], where it was called paramultiplication. Today, this notation is commonly used. In [Bony81], it was used for studying nonlinear PDEs and so-called paradifferential operators. An elaborated version of this application may be found in [MeC97, Chapter 16]. Nowadays, there is the possibility to shift paramultiplication from the Fourier side to the space side based on wavelet decompositions. We will return to this point in Section 4.3.7 below.

4.3.5 Atoms In Definition 1.1, we introduced the spaces Asp,q (Rn ) in the traditional way, based on Fourier-analytical decompositions. But in contrast to most of the papers and books mentioned in the Sections 4.3.1 to 4.3.4 (including our own ones), we did not rely on Fourier-analytical arguments in the proofs of the assertions in the above text. Instead, we worked almost exclusively with wavelet characterizations as described in Section 1.2.1. But all devices that are commonly used, such as derivatives and differences, Fourier-analytical decompositions, atoms, quarks and wavelets, have their advantages and disadvantages. In particular, it is reasonable to compare the most distinguished building blocks such as atoms, quarks, and wavelets. Atoms in the spaces Asp,q (Rn ) have a long history going back to the 1980s, see [FrJ85, FrJ90, Tor91]. More detailed (historical) references may be found in [T92, Section 1.9]. In Step 1 of the proof of Theorem 2.28, we already used atoms in a very typical situation. Now, we recall some details of the underlying reference [T08, Theorem 1.7, p. 5]. Let again χj,m be the characteristic function of the cube Qj,m = 2−j m + (p) 2−j (0, 1)n according to (1.6), n ∈ N. Let 0 < p, q ≤ ∞ and χj,m = 2jn/p χj,m .

4.3. Spaces and tools

137

Then bp,q (Rn ) is the collection of all sequences   λ = λj,m ∈ C : j ∈ N0 , m ∈ Zn such that λ |bp,q (Rn ) =

∞ 

|λj,m |p

q/p 1/q

(4.106)

< ∞,

(4.107)

m∈Zn

j=0

and fp,q (Rn ) is the collection of all these sequences such that



λ |fp,q (Rn ) =



(p)

|λj,m χj,m (·)|q

j∈N0 ,m∈Zn

1/q  Lp (Rn )

< ∞,

(4.108)

with the usual modification for max(p, q) = ∞. This is the atomic counterpart of Definition 1.9 with a different normalization. If Q is a cube in Rn and d > 0, then d Q is the cube in Rn which concentrically contains Q and has a side-length of d times the side-length of Q. Let xβ =

n 

β

xj j ,

x = (x1 , . . . , xn ) ∈ Rn ,

β = (β1 , . . . , βn ) ∈ Nn0 .

(4.109)

j=1

Let s ∈ R, 0 < p ≤ ∞, K ∈ N0 , L ∈ N0 and d > 1. Then the L∞ -functions aj,m : Rn → C with j ∈ N0 , m ∈ Zn , are called (s, p)K,L,d -atoms if supp aj,m ⊂ d Qj,m ,

j ∈ N0 ,

m ∈ Zn ,

(4.110)

if all (classical) derivatives Dα aj,m with |α| ≤ K exist such that |Dα aj,m (x)| ≤ 2−j(s− p )+j|α| , n

|α| ≤ K,

j ∈ N0 ,

m ∈ Zn ,

(4.111)

and if 

xβ aj,m (x) dx = 0,

|β| < L,

j ∈ N,

m ∈ Zn .

(4.112)

Rn

For a0,m , no cancellation (4.112) is required. For L = 0, (4.112) is empty (no condition). If K = 0, then (4.111) means that aj,m ∈ L∞ (Rn ) and |aj,m (x)| ≤ n 2−j(s− p ) . This basically coincides with [T08, Definition 1.5, p. 4]. A slightly more general version (adapted to Faber bases) may be found in [T10, Definition 1.5, p. 4]. Let again   1  σp(n) = n max , 1 − 1 , p

  1 1  (n) σp,q = n max , , 1 − 1 . p q

Then one has the following atomic representation theorem.

(4.113)

138

Chapter 4. Complements

(i) Let 0 < p, q ≤ ∞ and s ∈ R. Let K ∈ N0 , L ∈ N0 and d > 1 with K > s,

L > σp(n) − s,

(4.114)

s be fixed. Then f ∈ S  (Rn ) belongs to Bp,q (Rn ) if, and only if, it can be represented as ∞ f= λj,m aj,m , (4.115) j=0 m∈Zn

where the aj,m are (s, p)K,L,d -atoms and λ ∈ bp,q (Rn ). Furthermore, s (Rn ) ∼ inf λ |bp,q (Rn ) f |Bp,q

(4.116)

are equivalent quasi-norms, where the infimum is taken over all admissible representations (4.115). (ii) Let 0 < p < ∞, 0 < q ≤ ∞ and s ∈ R. Let K ∈ N0 , L ∈ N0 and d > 1 with K > s,

(n) − s, L > σp,q

(4.117)

s be fixed. Then f ∈ S  (Rn ) belongs to Fp,q (Rn ) if, and only if, it can be represented by (4.115), where the aj,m are (s, p)K,L,d -atoms and λ ∈ fp,q (Rn ). Furthermore, s (Rn ) ∼ inf λ |fp,q (Rn ) f |Fp,q

(4.118)

are equivalent quasi-norms, where the infimum is taken over all admissible representations (4.115). This result essentially coincides with [T08, Theorem 1.7, p. 5], with a reference to [T06, Section 1.5.1] for some technical explanations. A corresponding atomic s (Rn ), 0 < q < ∞, is covered by (1.71) and the decomposition for the spaces F∞,q related assertion in [T14, Theorem 3.33, p. 67]. One can compare the above atomic representation theorem, complemented by s its F∞,q (Rn )-counterpart, with its wavelet counterparts according to Proposition 1.11. If p < ∞, q < ∞, then the wavelet system (1.39) generates unconditional bases and one always has isomorphic maps onto the indicated sequence spaces. In the above text, we used this advantage quite often. The atoms are more qualitative. The representations (4.115) with λ ∈ bp,q (Rn ) or λ ∈ fp,q (Rn ) are not unique, and instead of isomorphic maps onto related sequence spaces, one has the rights (Rn )-counterpart. But this hand sides of (4.116), (4.118) and a corresponding F∞,q qualitative nature of the atoms makes them more flexible. A typical example for the application of this property are both steps of the proof of Theorem 2.28.

4.3.6 Quarks The above text mainly relies on wavelets, occasionally complemented by Fourieranalytical and atomic arguments. These different devices have their own advantages and disadvantages. While on the one hand wavelets seem to be very rigid,

4.3. Spaces and tools

139

it remains unclear on the other hand how to find optimal coefficients in the more flexible atomic representations of the spaces Asp,q (Rn ) as described in (4.114)– (4.118). At least, this was the situation some 20 years ago, when building blocks were sought which are simpler than wavelets but more constructive than the vague atomic decompositions. This search resulted in the decomposition of atoms into more handsome elementary constructive building blocks called quarks (in analogy to physics). We developed the respective theory of quarkonial decompositions in [T97] and applied the outcome to function spaces related to rough structures, including fractals. A summary of this theory can be found in [T06, Section 1.6, pp. 19–25]. Later on, however, it turned out that wavelets in Rn can be manipulated such that related modified versions remain bases in spaces of the type Asp,q (Ω) where Ω is a (rough) domain in Rn . The corresponding theory was developed in [T08], based in a decisive way on the local homogeneity for all spaces of interest as discussed in Section 3.3. This knowledge was simply not available in the 1990s, and it needed ten more years to be fully elaborated. But the simplicity of the quarkonial building blocks remained useful for numerical purposes, see [DOR13, DKR17]. Furthermore, there is a second aspect. Instead of derivatives, differences and Fourier-analytical decompositions, there are several proposals in the literature to introduce function spaces in a more abstract or axiomatic way, or based on more fundamental building blocks. In this context, we refer to [HeN07] in particular, which used related aspects from [AdH96], where spaces of the type Asp,q (Rn ) (and diverse variations of them) are introduced axiomatically. A somewhat different approach are proposals to construct function spaces via elementary building blocks that are simpler than the related Fourier-analytical decompositions or the qualitative definitions in terms of derivatives or differences. This was just one of our main aims in [T01] (as the title already indicates), where we further elaborated what had been developed before in [T97]. We do not repeat this theory in detail, but only describe the typical constructions following [T06, Section 3.2]. Let k be a non-negative C ∞ -function in Rn with   supp k ⊂ y ∈ Rn : |y| < 2J (4.119) for some J ∈ N such that

k(x − m) = 1

where

x ∈ Rn .

(4.120)

m∈Zn

Let xβ with x = (x1 , . . . , xn ) ∈ Rn and β ∈ Nn0 be as in (4.109) and let ax = (ax1 , . . . , axn ), a ∈ R. Then  β k β (x) = 2−J x k(x), x ∈ Rn , β ∈ Nn0 , (4.121) and   β kj,m (x) = k β 2j x − m ,

j ∈ N0 ,

x ∈ Rn .

(4.122)

These are the elementary building blocks, called quarks, that we are looking for. For any fixed β ∈ Nn0 , they are L∞ -normalized atoms as introduced in (4.110),

140

Chapter 4. Complements

(4.111) without the moment conditions (4.112). On the other hand, we wish to have a representation which is comparable to the wavelet expansion in Proposition 1.11. For this purpose, we need a second system of functions taking over the role of the kernels in (1.52), which is (somewhat surprisingly at the first glance) β in (4.122). If β = (β1 , . . . , βn ) ∈ Nn0 then totally independent of the quarks kj,m &n n |β| = j=1 βj and β! = j=1 βj ! (with 0! = 1). Let xβ be as in (4.121). Let ω be a C ∞ -function in Rn with supp ω ⊂ (−π, π)n

and

|x| ≤ 2.

if

ω(x) = 1

(4.123)

Let for some J ∈ N as in (4.119) ω β (x) =

i|β| 2J|β| β x ω(x), (2π)n β!

and Ωβ (x) =



x ∈ Rn ,

(ω β )∨ (m) e−imx ,

β ∈ Nn0

x ∈ Rn .

(4.124)

(4.125)

m∈Zn

Let ϕ0 be a C ∞ -function in Rn with ϕ0 (x) = 1 if |x| ≤ 1 and

ϕ0 (x) = 0 if |x| ≥ 3/2

(4.126)

as in (1.3) and let ϕ(x) = ϕ0 (x) − ϕ0 (2x), x ∈ Rn . Then the counterparts ΦβF and ΦβM of the scaling function ψF (father wavelet) and the wavelet ψM (mother wavelet) in (1.34) are given on the Fourier side by (ΦβF )∨ (ξ) = ϕ0 (ξ) Ωβ (ξ),

β ∈ Nn0 ,

ξ ∈ Rn ,

(4.127)

(ΦβM )∨ (ξ)

β∈

ξ∈R .

(4.128)

β

= ϕ(ξ) Ω (ξ),

Nn0 ,

n

The quarkonial version of (1.38), used as kernels in (1.41), is now given by  ΦβF (x − m) if j = 0, β Φj,m (x) = (4.129) ΦβM (2j x − m) if j ∈ N, where again x ∈ Rn , β ∈ Nn0 and m ∈ Zn . Instead of G ∈ Gj in (1.38), one now has β ∈ Nn0 . Furthermore, we have Φβj,m ∈ S(Rn ). One can preserve the product structure of (1.38) by choosing ω in (4.123) as a product of adequate functions in R. Then at least ω β (x) and Ωβ (x) in (4.124), (4.125) have a suitable product structure. (We will return to this point in Remark 4.21 below.) We do not deal with the spaces Asp,q (Rn ), A ∈ {B, F }, in full generality. Just on the contrary, we concentrate on   1  s Bps (Rn ) = Bp,p (Rn ), 0 < p ≤ ∞, s > σp(n) = n max , 1 − 1 . (4.130) p

4.3. Spaces and tools

141

This restriction ensures that no moment conditions for the building blocks are required, like for the atomic expansions according to (4.114)–(4.116). In addition, we need related sequence spaces that adapt the wavelet version (1.34)–(1.44) properly, but restricted to Bps (Rn ) in (4.130). Let 0 < p ≤ ∞,

s ∈ R,

 ∈ R,

(4.131)

and   λ = λβj,m ∈ C : j ∈ N0 , β ∈ Nn0 , m ∈ Zn .

(4.132)

  n s,

n bs,

p (R ) = λ : λ |bp (R ) < ∞

(4.133)

Then with n λ |bs,

p (R ) =

∞ 

n

2 |β|p+j(s− p )p |λβj,m |p

1/p

(4.134)

n β∈Nn 0 j=0 m∈Z

(with the usual modification for p = ∞). After these preparations, one can formulate the counterparts of both Proposition 1.11(i) and (4.114)–(4.116), restricted to (4.130). Let (4.135) 0 < p ≤ ∞, s > σp(n) and  ≥ 0. Then f ∈ S  (Rn ) belongs to Bps (Rn ) if, and only if, it can be represented as f=



β λβj,m kj,m ,

n λ ∈ bs,

p (R ),

(4.136)

β∈Nn 0 ,j∈N0 , n m∈Z

the unconditional convergence being in S  (Rn ). Furthermore, n f |Bps (Rn ) ∼ inf λ |bs,

p (R ),

(4.137)

where the infimum is taken over all admissible representations (4.136). Let   λβj,m (f ) = 2jn f, Φβj,m , (4.138) f ∈ S  (Rn ), with Φβj,m as in (4.129), and let   λ(f ) = λβj,m (f ) : j ∈ N0 , β ∈ Nn0 , m ∈ Zn .

(4.139)

Then f ∈ Bps (Rn ) can be represented as f=

β∈Nn 0 ,j∈N0 , m∈Zn

β λβj,m (f ) kj,m ,

(4.140)

142

Chapter 4. Complements

the unconditional convergence being in S  (Rn ), and n f |Bps (Rn ) ∼ λ(f ) |bs,

p (R )

(4.141)

(equivalent quasi-norms). For technical explanations and some calculations about the rather specific functions in (4.123)–(4.129), we refer the reader to [T06, Section 3.2.1, pp. 161–163]. The above assertion essentially coincides with [T06, Theorem 3.21, p. 165], where also a detailed proof can be found. In [T06, Sections 3.2, 3.3, pp. 161–192], we extended this theory into several directions, including some applications. In particular, the justification of the positivity property according to Theorem 3.6 partly relies on modified quarkonial decompositions, which is mentioned in Step 1 of the corresponding proof. Note that the above assertion is somewhat between the the wavelet expansions in (1.51), (1.52) and the atomic representations according to (4.115), (4.116). We have no longer uniqueness as in (1.51) and (1.52), but we still have the optimal so-called frame representations (4.140), (4.141) in terms of the constructive linear continuous functionals λβj,m (f ) according to (4.138). Remark 4.21. As mentioned above, if one chooses the basic functions k in (4.119) and ω in (4.123) as products of corresponding functions on R, then also k β and ω β are product functions. While this might not be overly useful in the framework of the inhomogeneous spaces Asp,q (Rn ), the situation is different for the spaces r A(Rn ), A ∈ {B, F }, with dominating mixed smoothness. These spaces have Sp,q a long history which may be found in [ST87]. But also quite recently, they attracted a lot of attention, especially in connection with numerical problems. The corresponding theory and related references may be found in [T10] and especially in [T19]. The product structure of the above basic functions paves the way to r quarkonial representations of the spaces Sp,q A(Rn ). But this has not yet been done.

4.3.7 Wavelets and paramultiplication Compactly supported wavelets were introduced in Section 1.2.1. After the Fourieranalytical introduction of the spaces Asp,q (Rn ), A ∈ {B, F }, s ∈ R and 0 < p, q ≤ ∞, in Definition 1.1, we immediately jumped to their wavelet characterizations according to Proposition 1.11. These characterizations being the basis for most of the arguments that followed afterwards. At this point, we would like to add only a few comments. Among the outstanding building blocks that are commonly used these days, including atoms (Section 4.3.5) and quarks (Section 4.3.6), the wavelet representations according to Proposition 1.11 have the advantage that they pave the way to shift problems for the spaces Asp,q (Rn ), A ∈ {B, F }, s ∈ R and 0 < p, q ≤ ∞ to their sequence counterparts. We took this for granted on the basis of related references. But one may ask for a turn of the tide, namely one s defines the spaces Asp,q (Rn ) in their full generality, including F∞,q (Rn ), s ∈ R

4.3. Spaces and tools

143

and 0 < q ≤∞, by the related wavelet characterizations within the topological  dual pairing S(Rn ), S  (Rn ) and then tries to reach the safe harbour of Fourieranalytical descriptions. No doubt that this can be done. One only has to reverse the already existing chain of arguments as evolved over years and decades. But as another important task, one should try to find a convincing and powerful way on only a few pages. This is also interesting from the point of view of the structure of the underlying decomposition spaces. As already mentioned in Section 4.3.6, we made such an attempt in [T01] (under the title ’The structure of functions’), but based on quarks and not on wavelets. Nevertheless, one can take the arguments given there as a guide for stepping from decomposition spaces to their Fourieranalytical versions. It might be even reasonable to start with a concise introduction to wavelets, relying on, e.g., [Woj97] or a condensed version as in [Lem02, Chapter 8]. There is a second point which might be more promising for future research. In (4.101)–(4.105), we described paramultiplication on the Fourier side as a powerful tool for studying products of elements that belong to some functions spaces. But there is an equally effective paramultiplication on the space (or original) side, at least as far as multiplications of elements of function spaces are concerned. This approach can be based on the wavelet expansions according to Proposition 1.11. As a typical example, one may ask for a wavelet proof of Theorem 2.41 characterizing those function spaces Asp,q (Rn ) which are multiplication algebras. The related proofs for Asp,q (Rn ) with p < ∞ for F -spaces in [Tr78, T83, SiT95, RuS96] rely extensively on the Fourier-analytical paramultiplication as described in Secs tion 4.3.4. We incorporated the spaces F∞,q (Rn ) according to (2.140) as a special case of [T14, Theorem 3.60(ii), p. 95] with a reference to [T13, Theorem 2.43(ii), p. 91]. The (somewhat sketchy) proof given there can be described as paramultiplication for wavelets on the space side (as already indicated in [T13, Remark 1.18, p. 12–13]). We returned to this type of arguments in [T19, Section 1.4.2, pp. 77– 83], where we asked which of the special spaces Spr B(Rn ) with dominating mixed smoothness are multiplications algebras. For this purpose, one first needs a proof r (R) is a multiplication algebra if that Bpr (R) = Bp,p  either 1 < p ≤ ∞, r > 1/p, (4.142) or 0 < p ≤ 1, r ≥ 1/p, which can then be extended from one dimension to higher dimensions by standard tensor product arguments that are typical for spaces Spr B(Rn ) with dominating mixed smoothness. This has been done in [T19, pp. 82–83]. It illustrates what we wish to call paramultiplication on the space side. In what follows, we outline the main ideas of this concept. First, we adapt the wavelet expansions as described in Section 1.2.1 to our specific needs. Let again ψF ∈ C u (R) and ψM ∈ C u (R), u ∈ N, be the real scaling function (father wavelet) and the real wavelet (mother wavelet) as in (1.34), (1.35) with L2 -norm 1. We modify (1.36)–(1.41) in the one-dimensional case as follows.

144 Let ψ−1,m (t) =

Chapter 4. Complements

√

2 ψF (t − m)

ψk,m (t) = ψM (2k t − m),

and

t ∈ R,

with k ∈ N0 and m ∈ Z. Let N−1 = N0 ∪ {−1}. Then   k/2 2 ψk,m (t) : k ∈ N−1 , m ∈ Z

(4.143)

(4.144)

is an orthonormal basis in L2 (R). Let u > r > σp(1) = σp = max

1  , 1 − 1, p

0 < p ≤ ∞.

(4.145)

According to Theorem 2.4, any f ∈ Bpr (R) is locally integrable. In modification of Proposition 1.11, any f ∈ Bpr (R) can be uniquely represented as 1 f= λk,m 2−k(r− p ) ψk,m (4.146) k∈N−1 ,m∈Z

with 1

λk,m = λk,m (f ) = 2k(r− p +1)

 f (y) ψk,m (y) dy

(4.147)

R

and f |Bpr (R) ∼ λ | p ,

f ∈ Bpr (R),

λ = {λk,m }.

(4.148)

Let f1 , f2 ∈ Bpr (R)

with

0 < p ≤ ∞,

r > σp .

(4.149)

Bpr (R)

The question of whether f = f1 f2 belongs to can be reduced to (4.146)– (4.148). In particular, inserting the related expansions for f1 and f2 in (4.147) yields  1 λk,m (f ) = 2k(r− p +1) f1 (y)f2 (y)ψk,m (y) dy R 1 2 1 1 k(r− p +1) λk1 ,m1 (f1 )λk2 ,m2 (f2 ) 2−(k +k )(r− p ) Ikk,m =2 1 ,k 2 ,m1 ,m2 k1 ,k2 ∈N−1 , m1 ,m2 ∈Z

(4.150) where Ikk,m 1 ,k 2 ,m1 ,m2 =

 ψk,m (y) ψk1 ,m1 (y) ψk2 ,m2 (y) dy.

(4.151)

R

Recall that (4.148) is an isomorphic map of Bpr (R) onto p . This shows that the question of whether (4.152) {μ1k1 ,m1 } × {μ2k2 ,m2 } → {μk,m } with 1



μk,m = 2k(r− p +1)

k ,k ∈N−1 , m1 ,m2 ∈Z 1

2

μ1k1 ,m1 μ2k2 ,m2 2−(k

1

1 +k2 )(r− p )

Ikk,m 1 ,k 2 ,m1 ,m2

(4.153)

4.3. Spaces and tools

145

is a bounded bilinear map Mp :

p × p → p ,

0 < p ≤ ∞,

(4.154)

is equivalent to the problem of whether Bpr (R) is a multiplication algebra. We adopted the above formulation from [T19, pp. 78–79]. It follows from [T19, Proposition 1.73, p. 79] that Mp with p as in (4.142) is a bounded bilinear map in p according to (4.154). Then Bpr (R) with p and r as in (4.142) is a multiplication algebra. This is a wavelet proof of the related special case of Theorem 2.41. The estimates of the integrals in (4.151) can be reduced to the cases k 2 ≤ k 1 ≤ k and k < k 1 , k 2 ≤ k 1 by using our knowledge of the supports of the involved functions. This is quite similar to the paramultiplication on the Fourier side based on (4.103)–(4.105) and may be called paramultiplication on the space side. As mentioned above, we needed this type of argument for asking under which circumstances the special spaces Spr B(Rn ) with dominating mixed smoothness are multiplication algebras, see [T19, Theorem 1.75, p. 82]. But it might be a challenging problem to give a new proof of Theorem 2.41 by extending the above paramultiplication on the space side. As first steps in this direction, one may consider the related arguments in [T13, Section 2.5.2] and [T14, Section 3.6.2] in the context of the so-called local and hybrid spaces.

4.3.8 Epilogue The main aim of this book was to present the theory of the inhomogeneous function spaces Asp,q (Rn ) and their counterparts Asp,q (Ω) on domains Ω in Rn in full generality, A ∈ {B, F }, s ∈ R and 0 < p, q ≤ ∞, as it stands now – the result of crucial (historical) developments over decades. We concentrated on the essentials (key theorems) and a few selected highlights. But the main motivation to write this book comes from the attempt to incorporate the nowadays fashionable s (Rn ), s ∈ R, 0 < q ≤ ∞, into the already existing theory of the spaces F∞,q s spaces Ap,q (Rn ) with p < ∞ for the F -spaces. Otherwise this book would not exist. Nowadays, there is a plethora of function spaces of diverse types. Many of them are related to the Sobolev spaces and the classical Besov spaces, as mentioned in the Sections 4.3.1 to 4.3.3 including related references. But it was not our aim to discuss these modifications and generalizations. Just on the contrary. We discussed to some extent only those types of function spaces from which we believe that they are intimately related to the spaces Asp,q (Rn ) and Asp,q (Ω). This applies to the hybrid spaces LrAsp,q (Rn ) as introduced in Section 1.1.2, mainly bes s (Rn ) can be identified with F∞,q (Rn ) according to (1.71). Secondly, cause L0 Fp,q ∗

we discussed in Section 4.1 the homogeneous spaces Asp,q (Rn ), and we advocated in Section 4.2 that they should be considered in their natural habitat  (4.68) where  they can be treated within the framework of the dual pairing S(Rn ), S  (Rn ) . r A(Rn ) with domiFinally, we mentioned more or less in passing the spaces Sp,q nating mixed smoothness, with references to [ST87, T10, T19]. These spaces have

146

Chapter 4. Complements

remarkable properties, and one could take the above text as a guide for studying them in greater detail than has already been done.

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Symbols

157

Symbols Sets C, 1 dQ, 2 Dnl (p, s), 71 Γd , 36 G∗ , Gj , 8 μf (λ), 83 N, N0 , 1 PJ,M , 10 QJ,M , 2 Rn , R, 1 Rn+ , 46, 57 Rn− , 57 sing supp , 73 Uλ , 90 Z, Zn , 1

sp,q (Ω), 64 A s (Ω), 64 A p,q Asp,q (Γ), 66 Asp,q (Uλ ), 91 s (Rn ), 57 A p,q

+

LrAsp,q (Rn ), 6 s (Rn ), 2 Bp,q s (Rn ), 116 B˙ p,q ∗

B sp,q (Rn ), 118, 121 s LrBp,q (Rn ), 6 n C(R ), 21 C k (Rn ), 40 C (Rn ), 93 C m (Ω), C ∞ (Ω), 64 C s (Rn ), 3, 31 ∗

Spaces bp,q (Rn ), 137 bsp,q (Rn ), 8 n bs,

p (R ), 141 bmo(Rn ), 3, 15 bmos (Rn ), 17, 79 Lr bsp,q (Rn ), 10 fp,q (Rn ), 137 s (Rn ), 8 fp,q s f∞,q (Rn ), 9, 11 r s L fp,q (Rn ), 10 Asp,q (Rn ), 3 Asp,q (Rn+ ), 57 Asp,q (Ω), 63 n As,τ p,q (R ), 6 Asp,q,unif (Rn ), 54

C s (Rn ), 118 D(Ω), 63 D(Γ) = C ∞ (Γ), 65 D(Rn \ Γd ), 36 s Fp,q (Rn ), 2 s (Rn ), 2, 3 F∞,q F˙ s (Rn ), 116 p,q ∗ F sp,q (Rn ),

118, 121

+

F sp,q (Rn ), 48 s,rloc (Ω), 95 Fp,q r s L Fp,q (Rn ), 6 Hps (Rn ), 59, 133 Lp (Rn ), Lp (M ), 1 Lp (Ω), 63, 132 Lp (Γ), 65 L∞ (Rn ), 21

© Springer Nature Switzerland AG 2020 H. Triebel, Theory of Function Spaces IV, Monographs in Mathematics 107, https://doi.org/10.1007/978-3-030-35891-4

158 n Lloc 1 (R ), 23 Lsp,q (Rn ), 4   M A(Rn ) , 53 S(Rn ), S  (Rn ), 1 ˙ n ), 116 S(R Wpk (Rn ), 132

Wpk (Ω), 132 Wpk [Ω], 133

Operators ϕ,  F ϕ, 1 ϕ∨ , F −1 ϕ, 1 Dψ , 39 ext, 31, 58 extΓ , 67 extΩ , 68 extδ , 60 Iδ , 16 Iδ+ , Iδ− , 58 Jδ+ , Jδ− , 59 re , 58 re Ω , 68 tr , 30 trd , 36 trL , 104 tr Γ , 66 tr r , 35 tr rΓ , 67

Functions, functionals Dα , 132 dm t,u f , 134 E 1 Asp,q (t), 84 E ∞ Asp,q (t), 84

Symbols f ∗ (t), 83 hM (y), hF (y), 98 hjG,m , 98 sf (t), 73 Wt w, 106 Wtα w, 112 xβ , 137 ∂t , ∂tm , 106 ∂j , ∂jm , 132 Δlh , 134 ψ F , ψM , 7 j ,8 ψG,m χQ , 46 χ+ , 46 χj,m , 8

Numbers, relations ∼, x →, 5 a+ = max(a, 0), 6 (n)

(n)

σp , σp,q , 9, 34, 52   D Ap (Rn ), Lp (Γd ) , 37

Index

159

Index atom, 137 characteristic, distribution, 73 counter–example, 22 dichotomy, 36 diffeomorphism, 39 differences, 134 differences, ball means, 134 differential dimension, 25, 121 distribution, positive, 47, 88 diversity, 28 domain = open set domain, C k , 65 domain, special C k , 64 domain, C ∞ , 65 embedding, Jawerth–Franke, 25 embedding, sharp, 25, 26 embedding, strict, 12 envelope, 86 envelope, function, 84 envelope, number, 86 extension operator, 31

Figure 2.1, 72 Figure 2.2, 73 Figure 3.1, 99 Figure 4.1, 129 Fourier transform, 1 function, Lusin, 105

habitat, natural, 128 heat kernel, 106 Hilbert, 117 homogeneity, global, 117, 124 homogeneity, local, 91 inequality, Gagliardo–Nirenberg, 28 interpolation, complex, 49, 128 interpolation, ±–method, 42 key problems, 29, 63 lift, 16 localization, 43 manifold, C ∞ , 65 multiplication algebra, 44, 126 multiplier, pointwise, 40, 53 Nature, Wisdom, 117 paramultiplication, 136, 145 positivity, 88 property, extension, 60, 68 property, Fatou, 18 property, Fubini, 52, 104 pudding, 62 quarks, 138 representation, distinguished, 17 resolution of unity, 2 restriction, 63

© Springer Nature Switzerland AG 2020 H. Triebel, Theory of Function Spaces IV, Monographs in Mathematics 107, https://doi.org/10.1007/978-3-030-35891-4

160 rule, homogeneity, 129, 131 scaling heuristics, 115 semi–group, Gauss–Weierstrass, 106, 110 smoothing, caloric, 109, 126 Sobolev embedding, 25 space, classical Sobolev, 132 spaces, critical, supercritical, 117 space, Hölder–Zygmund, 3, 7, 41, 134 space, hybrid, 5, 6 space, local, 6 space, refined localization, 95 space, Sobolev, 133 space, tempered homogeneous, 115, 124 strip, distinguished, 120, 128 traces, 29, 30 wavelets, 7 wavelets, Haar, 98

Index