Envelopes and Sharp Embeddings of Function Spaces (Research Notes in Mathematics Series)

Envelopes and Sharp Embeddings of Function Spaces

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CHAPMAN & HALL/CRC Research Notes in Mathematics Series Main Editors H. Brezis, Université de Paris R.G. Douglas, Texas A&M University A. Jeffrey, University of Newcastle upon Tyne (Founding Editor) Editorial Board R. Aris, University of Minnesota G.I. Barenblatt, University of California at Berkeley H. Begehr, Freie Universität Berlin P. Bullen, University of British Columbia R.J. Elliott, University of Alberta R.P. Gilbert, University of Delaware D. Jerison, Massachusetts Institute of Technology B. Lawson, State University of New York at Stony Brook

B. Moodie, University of Alberta L.E. Payne, Cornell University D.B. Pearson, University of Hull G.F. Roach, University of Strathclyde I. Stakgold, University of Delaware W.A. Strauss, Brown University J. van der Hoek, University of Adelaide

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Dorothee D. Haroske

Envelopes and Sharp Embeddings of Function Spaces

Boca Raton London New York

Chapman & Hall/CRC is an imprint of the Taylor & Francis Group, an informa business

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Chapman & Hall/CRC Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487‑2742 © 2007 by Taylor & Francis Group, LLC Chapman & Hall/CRC is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid‑free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number‑10: 1‑58488‑750‑8 (Hardcover) International Standard Book Number‑13: 978‑1‑58488‑750‑8 (Hardcover) This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the conse‑ quences of their use. No part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www. copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC) 222 Rosewood Drive, Danvers, MA 01923, 978‑750‑8400. CCC is a not‑for‑profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

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To my family

v

Contents

Preface

ix

I

1

Definition, basic properties, and first examples

1 Introduction 2 Preliminaries, classical function spaces 2.1 Non-increasing rearrangements . . . . . . . . . . . . 2.2 Lebesgue and Lorentz spaces . . . . . . . . . . . . . 2.3 Spaces of continuous functions . . . . . . . . . . . . 2.4 Sobolev spaces, Sobolev’s embedding theorem . . .

3

. . . .

11 11 16 20 26

. . . .

39 39 45 52 55

4 Growth envelopes EG 4.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Examples: Lorentz spaces, Sobolev spaces . . . . . . . . . . .

63 63 66

5 The 5.1 5.2 5.3

continuity envelope function EC Definition and basic properties . . . . . . . . . . . . . . . . . Some lift property . . . . . . . . . . . . . . . . . . . . . . . . Examples: Lipschitz spaces, Sobolev spaces . . . . . . . . . .

75 75 79 84

6 Continuity envelopes EC 6.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Examples: Lipschitz spaces, Sobolev spaces . . . . . . . . . .

91 91 93

II

Results in function spaces, and applications

99

7 Function spaces and embeddings s s 7.1 Spaces of type Bp,q , Fp,q . . . . . . . . . . . . . . . . . . . . 7.2 Embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . .

101 101 106

3 The 3.1 3.2 3.3 3.4

. . . .

growth envelope function EG Definition and basic properties . . . . . . . . . . . . . Examples: Lorentz spaces . . . . . . . . . . . . . . . . Connection with the fundamental function . . . . . . Further examples: Sobolev spaces, weighted Lp -spaces

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

viii

Contents

8 Growth envelopes EG of function spaces Asp,q 8.1 Growth envelopes in the sub-critical case . . . . . . . . . . . 8.2 Growth envelopes in sub-critical borderline cases . . . . . . . 8.3 Growth envelopes in the critical case . . . . . . . . . . . . .

119 119 130 135

9 Continuity envelopes EC of function spaces Asp,q 9.1 Continuity envelopes in the super-critical case . . . . . . . . 9.2 Continuity envelopes in the super-critical borderline case . . 9.3 Continuity envelopes in the critical case . . . . . . . . . . . .

147 147 152 158

10 Envelope functions EG and EC revisited 10.1 Spaces on R+ . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Enveloping functions . . . . . . . . . . . . . . . . . . . . . . 10.3 Global versus local assertions . . . . . . . . . . . . . . . . . .

161 161 167 170

11 Applications 11.1 Hardy inequalities and limiting embeddings . . . . . . . . . . 11.2 Envelopes and lifts . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Compact embeddings . . . . . . . . . . . . . . . . . . . . . .

179 179 187 195

References

211

Symbols

223

Index

225

List of figures

227

Preface We present the new concept of growth envelopes and continuity envelopes in function spaces. This originates from such classical results as the famous Sobolev embedding theorem [Sob38], or, secondly, from the Br´ezis-Wainger result [BW80] on the almost Lipschitz continuity of functions from a Sobolev 1+n/p space Hp (Rn ), 1 < p < ∞. In the first case questions of growth are studied: what can be said about the unboundedness of functions belonging n/p to, say, Hp (Rn ), 1 < p < ∞? We introduce the growth envelope EG (X) = X X (EG (t), uG ) of a function space X ⊂ Lloc 1 , where EGX (t) ∼ sup {f ∗ (t) : kf |Xk ≤ 1} ,

0 < t < 1,

is the growth envelope function of X and uX G ∈ (0, ∞] is some additional index providing an even finer description of the unboundedness of functions belonging to X. Instead of investigating the growth of functions one can also focus on their smoothness, i.e., for X ,→ C it makes sense to replace f ∗ (t) with ω(f,t) t , where ω(f, t) is the modulus of continuity. The continuity envelope function ECX and the continuity envelope EC are introduced completely parallel to EGX and EG , respectively, and similar questions are studied. These concepts are first explained in detail and demonstrated on some classical and rather obvious examples in Part I; in Part II we deal with these s instruments in the context of spaces of Besov and Triebel-Lizorkin type, Bp,q s and Fp,q , respectively. In the end we turn to some applications, e.g., concerning the asymptotic behaviour of approximation numbers of (corresponding) compact embeddings. Further applications are connected with Hardy-type inequalities and limiting embeddings. We discuss the relation between growth and continuity envelopes of a suitable pair of spaces. Problems of global nature are regarded, and we study situations where the envelope function itself belongs to or can be realised in X, respectively. I am especially grateful to Professor Hans Triebel; while he was preparing his book [Tri01] (in which Chapter 2 is devoted to related questions) we could discuss the material at various occasions. This led to the preprint version [Har01], and also became part of [Har02]. But for some reason these results –

ix

x

Preface

though extended, improved, used, cited already – were never published elsewhere. In view of the substantial material, the idea appeared later to collect it in a book rather than a number of papers. This gives me the opportunity for special thanks to Professor David E. Edmunds who offered invaluable mathematical and linguistic comments, and to Professor Ha¨ım Br´ezis who encouraged me in that form of publication. Last but not least I appreciate joint work and exchange of ideas on the subject with many colleagues, in particular Professor Ant´onio M. Caetano and Dr. Susana D. Moura. Finally, I am indebted to my family for their never-ending patience and support.

Jena, March 2006

Dorothee D. Haroske

Part I

Definition, basic properties, and first examples

1

Chapter 1 Introduction

We present our recently developed concept of envelopes in function spaces – a relatively simple tool for the study of classical, and also rather complicated spaces, say, of Besov or Triebel-Lizorkin type, respectively, in so-called “limiting” situations. This subject is still very new, but in our opinion it has grown to such a degree of maturity that it is now worth a coherent account. The topic is studied in two steps: on a more general level, in Part I, where we do not assume any knowledge of the scales of function spaces mentioned above, and subsequently, in Part II, the results are essentially related to spaces of s s . This also explains the main structure of this book. and Fp,q type Bp,q We first describe the common background for both parts and explicate the programme afterwards. In fact, considerable parts of the outcomes were obtained much earlier and already summarised in the long preprint [Har01]; but for some reason they have not yet been published (apart from [Har02], essentially relying on [Har01]). However, we also complemented and extended the presentation [Har01] quite recently. The history of such questions starts in the 1930s with Sobolev’s famous embedding theorem [Sob38] Wpk (Ω) ,→ Lr (Ω),

(1.1)

where Ω ⊂ Rn is a bounded domain with sufficiently smooth boundary, Lr , 1 ≤ r ≤ ∞, stands for the usual Lebesgue space, and Wpk , k ∈ N, 1 ≤ p < ∞, are the classical Sobolev spaces. The latter spaces have been widely accepted as one of the crucial instruments in functional analysis – in particular, in connection with PDEs – and have played a significant role in numerous parts of mathematics for many years. Sobolev’s famous result (1.1) holds for k ∈ N with k < np , and r such that nk − p1 ≥ − 1r (strictly speaking, [Sob38] covers the case nk − p1 > − 1r , whereas the extension to nk − p1 = − 1r was achieved later). In the limiting case, when k = np ∈ N, the inclusion (1.1) does not hold for r = ∞, whereas for all r < ∞, Wpn/p (Ω) ,→ Lr (Ω).

(1.2)

The theory of Sobolev-type embeddings originates in classical inequalities from which integrability properties of a real function can be deduced from

3

4

Envelopes and sharp embeddings of function spaces

those of its derivatives. In that sense (1.2) can be understood simply as the n/p impossibility of specifying integrability conditions of a function f ∈ Wp (Ω) merely by means of Lr conditions. In order to obtain further refinements of the limiting case of (1.1) it becomes necessary to deal with a wider class of function spaces. In the late 1960s Peetre [Pee66], Trudinger [Tru67], Moser [Mos71], and Pohoˇzaev [Poh65] independently found refinements of (1.1) expressed in terms of Orlicz spaces of exponential type, see also Strichartz [Str72]; this was followed by many contributions in the last decades investigating problems related to (1.1) in detail. In 1979 Hansson [Han79] and Br´ezis, Wainger [BW80] showed independently that Wpn/p (Ω) ,→ L∞,p (log L)−1 (Ω),

(1.3)

where 1 < p < ∞, and the spaces Lr,u (log L)a (Ω) appearing in (1.3) provide an even finer tuning, see also Hedberg [Hed72], and sharper results by Maz’ya [Maz72] and [Maz05]. Recently we noticed a revival of interest in limiting embeddings of Sobolev spaces indicated by a considerable number of publications devoted to this subject; let us only mention a series of papers by Edmunds with different co-workers ([EGO96], [EGO97], [EGO00], [EK95], [EKP00]), by Cwikel, Pustylnik [CP98], and – also from the standpoint of applications to spectral theory – the publications [ET95] and [Tri93] by Edmunds and Triebel. This list is by no means complete, but reflects the increased interest in related questions in the last years. There are a lot of different approaches to the modification of (1.1) in order to get – in the adapted framework – appropriately optimal assertions. Apart from the original papers, assertions of this type are indispensable parts in books dealing with Sobolev spaces and related questions, cf. [AF03], [Zie89], [Maz85], [EE87], [EE04], but there is a vast literature devoted to (extensions of) this subject. Returning to the limiting case k = np (1.2), for instance, one can also extend the scale of admitted (source) spaces in another direction, first replacing n/p by the more general fractional Sobolev spaces classical Sobolev spaces Wp n/p s s and Fp,q , respectively. It is well-known, Hp , or even by spaces of type Bp,q n/p

for instance, that Bp,q ,→ L∞ if, and only if, 0 < p < ∞, 0 < q ≤ 1 – n/p but what can be said about the growth of functions f ∈ Bp,q otherwise, i.e., n/p when Bp,q contains essentially unbounded functions ? Edmunds and Triebel proved in [ET99b] that one can characterise such spaces by sharp inequalities involving the non-increasing rearrangement f ∗ of a function f : Let κ be a bounded, continuous, decreasing function on (0, 1] and 1 < p < ∞. Then there is a constant c > 0 such that 1/p  1 ¶p Z µ ∗ ° ° dt f (t)κ(t) ° °   ≤ c °f |Hpn/p ° | log t| t 0

(1.4)

Introduction

5

n/p

for all f ∈ Hp if, and only if, κ is bounded; there are further results related to the case of Hps with 0 < s < np in [Tri99]. s Recall that the scale Fp,q of Triebel-Lizorkin spaces extends the (classical) s Sobolev scale further, whereas Besov spaces Bp,q have been well-known for a long time, either when characterised by differences or – nowadays preferably – in Fourier-analytical terms or via (sub-)atomic decompositions. They appear naturally in signal analysis, in contemporary harmonic analysis, in stochastics, and while studying approximation problems or solving PDEs; thus it is of deep interest to understand these spaces very well – apart from the purely s s functional analytic purpose. The theory of the scales Bp,q and Fp,q has been systematically studied and developed by Triebel in the last decades; his series of monographs [Tri78a], [Tri83], [Tri92], [Tri97], [Tri01], and the forthcoming book [Tri06] can be regarded as the most complete treatment of related questions.

Assertions of type (1.4) are already linked to our concept of envelopes in some sense. We realised that many contributions to the subject of limiting embeddings and sharp inequalities as (partly) mentioned above, share a little disadvantage – beside their unquestioned beauty: as far as characterisations s s of spaces of type Bp,q or Fp,q are concerned, they are usually bound to a certain setting. That is, dealing with such embeddings, one asks, say, for optimality of original or target spaces within a prescribed (fixed) context. We prefer to look for some feature “belonging” to the spaces under consideration only, and, moreover, defined as simply as possible (using classical approaches). This would enable us to gain significantly from the rich history and many forerunners. In view of the above-mentioned papers it was natural to choose the non-increasing rearrangement f ∗ as the basic concept on which our new tool should be built. This led us to the introduction of the growth envelope function of a function space X, EGX (t) :=

sup

f ∗ (t),

0 < t < 1.

(1.5)

kf |Xk≤1

It turns out that in rearrangement-invariant spaces there is a connection between EGX and the fundamental function ϕX ; we ³shall derive´ further properties and give some examples. The pair EG (X) = EGX (t), uX G uX G,

is called the

uX G

≤ ∞, is the infimum of all numbers 0< growth envelope of X, where v satisfying 1/v  ε" #v Z ∗ f (t)  µG (dt) ≤ c kf |Xk (1.6) EGX (t) 0

for some c > 0 and all f ∈ X, and µG is the Borel measure associated with

6

Envelopes and sharp embeddings of function spaces

− log EGX . The result reads for the Lorentz spaces Lp,q as ³ 1 ´ EG (Lp,q ) = t− p , q , and for Sobolev spaces Wpk , ¡ ¢ ³ 1 k ´ EG Wpk = t− p + n , p ,

1 ≤ p < ∞,

k<

n . p

(1.7)

We also deal with some weighted Lp -spaces, illuminating the difference between locally regular weights like (1 + |x|2 )α/2 , α > 0, and corresponding Muckenhoupt Ap weights like |x|α , 0 < α < n(1 − p1 ). In Part II we consider s s characterisations for spaces of type Bp,q and Fp,q , where n( p1 −1)+ ≤ s ≤ np . Returning to our example (1.4) above one proves that ³ ´ ³ ´ 1− 1 EG Hpn/p = |log t| p , p , 1 < p < ∞, (1.8) even in a more general setting. The counterpart for Besov spaces is given by ³ ´ 1 n/p EG (Bp,q ) = | log t|1− q , q , 1 < q ≤ ∞, 0 < p < ∞. (1.9) Unlike [Tri01, Ch. 2] where similar questions have been considered, we also study (some) borderline and weighted cases. In an appropriately modified context it also makes sense to consider embeddings like (1.1) and (1.2) in “super-critical” situations, that is, when k > np . Then by simple monotonicity arguments all distributions are essentially bounded; moreover, one even knows that Wpk ,→ C

(1.10)

in this case, where C stands for the space of all complex-valued bounded uniformly continuous functions on Rn , equipped with the sup-norm. Parallel to the above question of unboundedness it is natural to consider and qualify the continuity of distributions from Wpk in dependence upon k and p. As is well-known, the counterparts of (1.1) and (1.2) yield that for np < k < np + 1, 1 ≤ p < ∞, n (1.11) Wpk ,→ Lipa , 0 < a ≤ k − < 1, p and,

Wp1+n/p ,→ Lipa ,

0 < a < 1,

(1.12)

where Lipa , 0 < a ≤ 1, contains all f ∈ C such that for some c > 0 and all x, h ∈ Rn , |f (x + h) − f (x)| ≤ c |h|a . Similarly to (1.2), the case a = 1 in (1.12) is excluded (unless p = 1 as some special case), i.e., there are 1+n/p functions from Wp that are not Lipschitz-continuous. However, as some

Introduction

7

compensation, one can consider the celebrated result of Br´ezis and Wainger 1+n/p [BW80] in which it was shown that every function u in Hp , 1 < p < ∞, is “almost” Lipschitz-continuous, in the sense that for all x, y ∈ Rn , 0 < |x − y| < 21 , ¯ ¯1− p1 ¯ ¯ ku|Hp1+n/p k . |u(x) − u(y)| ≤ c |x − y| ¯ log |x − y|¯

(1.13)

Here c is a constant independent of x, y and u. In [EH99] we investigated the “sharpness” of this result (concerning the exponent of the log-term), as well as possible extensions to the wider scale of F -spaces and parallel results for B-spaces. Using the classical concept of the modulus of continuity ω(f, t), (1.13) can be reformulated as ω(f, t)

sup 0
1 1− p

t |log t|

° ° ° ° ≤ c °f |Hp1+n/p ° ,

(1.14)

which will be strengthened to 1/p  1 ¸p Z2 · ° ° dt  ω(f, t) ° °  ≤ c °f |Hp1+n/p ° ,   t |log t| t 0

and an assertion similar to (1.4). Consequently, based on observations like (1.14) we shall focus on the smoothness of functions instead of their growth; i.e., when X ,→ C it makes sense to in (1.5) and (1.6), and to introduce the continuity replace f ∗ (t) by ω(f,t) t envelope function X

EC (t) :=

sup kf |Xk≤1

ω(f, t) , t

0 < t < 1,

and the continuity envelope EC in a way completely parallel to that of EGX and EG , respectively. The famous Br´ezis-Wainger result [BW80] then appears as ´ ³ 1 EC (Hp1+n/p ) = | log t|1− p , p , 1 < p < ∞, whereas we obtain for Lipschitz spaces Lipa of order 0 < a < 1, ³ ´ EC (Lipa ) = t−(1−a) , ∞ . The counterpart of (1.7) reads as ´ ¡ ¢ ³ n EC Wpk = t−( p +1−k) , p ,

(1.15)

8

Envelopes and sharp embeddings of function spaces

where 1 ≤ p < ∞, k ∈ N, with np < k < np + 1, refining (1.11). Again the more elaborate results are postponed to Part II. Dealing with Besov spaces, for instance, (1.9) is complemented by ³ ´ ³ ´ 1 1+n/p EC Bp,q = | log t|1− q , q , 1 < q ≤ ∞, 0 < p ≤ ∞. (1.16) Finally, in the last two chapters we return in some sense to the beginning, now discussing different points of view, e.g., whether the envelope functions are indeed (only) envelopes and do not belong to the spaces themselves (posed in a suitable context); in addition, we consider assertions of a more global nature instead of local ones as before. For Sobolev spaces Wpk , k ∈ N0 , 1 ≤ p < ∞, we get Wpk

EG

1

(t) ∼ t− p

for t → ∞,

compared with the local assertion (1.7). Recent projects focus on the underlying domain (reaching as far as spaces defined on fractals, e.g., d-sets or h-sets in [CH06]) and inclusion of further scales of spaces. Another idea is associated with envelopes and interpolation, based on similarly constructed K-envelopes, see [Pus01]. But this is out of the scope of this book. Some obvious applications but with surprisingly sharp results are connected with Hardy inequalities and limiting embeddings, thus coming full circle to our motivation in some sense. In the context of compact embeddings we obtain some new and rather remarkable consequences of our preceding investigations; we combine lift arguments for envelopes, discussed before, with more abstract estimates from approximation theory and obtain results of the type ³ 1´ 1 ak+1 (id : X(U ) −→ C(U )) ≤ c k − n ECX k − n , k ∈ N, for the approximation numbers ak of a compact embedding X(U ) ,→ C(U ), U ⊂ Rn being the unit ball. This describes in some cases the precise asymptotic behaviour of the approximation numbers. In combination with the above-mentioned lift arguments we obtain a counterpart in the context of growth envelopes, too. The book is structured as follows. In Chapter 2 we recall some definitions of (classical) function spaces under consideration, and explain the situation we shall study. In Chapter 3 we introduce the growth envelope function EG and derive some of its properties. This is followed by the introduction of growth envelopes EG in Chapter 4 also providing our main results for Lorentz-Zygmund, Sobolev and weighted (Lebesgue) spaces; Chapters 5 and 6 are parallel to Chapters 3 and 4, replacing the growth envelopes by continuity envelopes. In addition to Sobolev spaces, we explicate spaces of Lipschitz

Introduction

9

type as classical examples. It is our intention that in Part I the reader is not assumed to know more function space theory than the basics on Lebesgue (and Lorentz) spaces, Lipschitz spaces and (classical) Sobolev spaces. We describe ideas and proofs in some detail. In Part II, however, the reader should be more familiar with Fourier theory and fundamentals (like atomic decomposition techniques) of function spaces of Besov and Triebel-Lizorkin type; though we recall important definitions and characterisations as far as needed, we shall not be able to avoid some technicalities and arguments closely linked with the full generality of such scales of spaces. Those who have not yet worked in this field are recommended to special monographs for details and methods, but should be able to understand the results, which are mainly extensions of those in Part I. The last chapters contain material that might be of some interest for all again. In addition to the bibliography, there is a glossary of symbols, an index and a list of figures. The number(s) behind the symbol “I” in the References mark the pages where the corresponding entry is quoted. Each chapter is subdivided into sections. We refer to “Chapter m, Section n” as “Section m.n”. At the moment it seems that those working in this field as well as the publications devoted to the subject are not yet too numerous, but permanently increasing. Though we do not pretend to give a complete survey of all that has been done so far, we shall try to report on closely linked results whenever reasonable and possible.

Chapter 2 Preliminaries, classical function spaces

Let Rn be Euclidean n-space and hxi = (2 + |x|2 )1/2 , x ∈ Rn . Given two (quasi-) Banach spaces X and Y , we write X ,→ Y if X ⊂ Y and the natural embedding of X in Y is continuous. All unimportant positive constants will be denoted by c, occasionally with subscripts. For some κ ∈ R let κ+ = max(κ, 0) and

[κ] = max{k ∈ Z : k ≤ κ} .

(2.1)

¡ ¢ Moreover, for 0 < r ≤ ∞ the number r0 is given by r10 := 1 − 1r + . For non-negative functions f, g : R+ −→ R+ , the symbol f (t) ∼ g(t) will mean that there are positive numbers c1 , c2 such that for all t > 0, c1 f (t) ≤ g(t) ≤ c2 f (t). For convenience, let both dx and | · | stand for the (n-dimensional) Lebesgue measure `n in the sequel. We briefly describe the concept of the non-increasing rearrangement, recall the definitions of some function spaces which will serve us as examples below. This chapter ends with a section devoted to Sobolev’s famous embedding theorem.

2.1

Non-increasing rearrangements

Let [R, µ] be a totally σ-finite measure space, referred to simply as measure space in the sequel. For a function f : R → C, µ-measurable and finite µ-a.e., its distribution function µf : [0, ∞) → [0, ∞] is given by µf (s) := µ ({x ∈ R : |f (x)| > s}) ,

s ≥ 0.

(2.2)

We collect some basic properties and refer to [BS88, Prop. 2.1.3] for a proof.

11

12

Envelopes and sharp embeddings of function spaces

Proposition 2.1 Let f : R → C be a µ-measurable function, finite µ-a.e. (i)

The function µf is non-negative, decreasing, and right-continuous on [0, ∞). For any c 6= 0, µ ¶ s , s ≥ 0. µcf (s) = µf |c|

(ii) Let g : R → C be another µ-measurable function, finite µ-a.e. Then |g| ≤ |f | µ-a.e. implies that µg ≤ µf , and µf +g (s1 + s2 ) ≤ µf (s1 ) + µg (s2 ),

s1 , s2 ≥ 0.

(iii) Let (fn )n be a sequence of µ-measurable functions, finite µ-a.e., such that |f | ≤ lim inf |fn | µ-a.e., n→∞

then µf ≤ lim inf µfn ; n→∞

in particular, |fn | ↑ |f | µ-a.e. implies µfn ↑ µf . We postpone an example and introduce the concept of the non-increasing (or decreasing) rearrangement f ∗ of a function f first. Definition 2.2 For a function f : R → C, µ-measurable and finite µ-a.e., its non-increasing rearrangement f ∗ : [0, ∞) → [0, ∞] by f ∗ (t) = inf {s ≥ 0 : µ ({x ∈ R : |f (x)| > s}) ≤ t} ,

t ≥ 0.

(2.3)

We put inf ∅ = ∞, as usual. Plainly, f ∗ (t) = 0 for t > µ(R). Proposition 2.3 Let f : R → C be a µ-measurable function, finite µ-a.e. (i)

The function f ∗ is non-negative, decreasing, and right-continuous on [0, ∞). For any c 6= 0, ∗

(cf ) (t) = |c|f ∗ (t),

t ≥ 0.

Let 0 < r < ∞, then ∗

r

(|f |r ) (t) = (f ∗ (t)) ,

t ≥ 0.

(ii) Let g : R → C be another µ-measurable function, finite µ-a.e. Then |g| ≤ |f | µ-a.e. implies that g ∗ (t) ≤ f ∗ (t), t ≥ 0, and ∗

(f + g) (s1 + s2 ) ≤ f ∗ (s1 ) + g ∗ (s2 ),

s1 , s2 ≥ 0.

Preliminaries, classical function spaces

13

(iii) Let (fn )n be a sequence of µ-measurable functions, finite µ-a.e., such that |f | ≤ lim inf |fn | µ-a.e., n→∞

then

f ∗ (t) ≤ lim inf fn∗ (t), n→∞

t ≥ 0;

in particular, |fn | ↑ |f | µ-a.e. implies fn∗ ↑ f ∗ . (iv) When µf (s) < ∞, then f ∗ (µf (s)) ≤ s; conversely, if f ∗ (t) < ∞, then µf (f ∗ (t)) ≤ t. (v) f and f ∗ are equi-measurable, i.e., for s ≥ 0, µf (s) = |{t ≥ 0 : f ∗ (t) > s}| = νf ∗ (s),

(2.4)

where ν(·) = | · | stands for the usual Lebesgue measure on R+ . (vi) Let 0 < p < ∞, then Z Z∞ Z∞ p p−1 |f (x)| µ(dx) = p s µf (s)ds = f ∗ (t)p dt, 0

R

(2.5)

0

and for p = ∞, ess sup |f (x)| = inf {s : µf (s) = 0} = f ∗ (0).

(2.6)

x∈R

(vii) Let g : R → C be another µ-measurable function, finite µ-a.e. Then Z Z∞ |f (x)| |g(x)| µ(dx) ≤ f ∗ (t) g ∗ (t) dt. (2.7) R

0

There is plenty of literature on this topic; we refer to [BS88, Ch. 2, Sect. 1], [DL93, Ch. 2, §2], and [EE04, Ch. 3], for instance. Part (vii) is well-known as the Hardy-Littlewood inequality for rearrangements. Example 2.4 We illustrate the above concepts by a few examples. Let m X a1 f (x) = aj χA (x), j a2 j=1 where the Aj , j = 1, . . . , m, are finite µ-measurable subsets of R. Without restriction of generality we may assume that they are pairwise disjoint, Aj ∩ Ak = ∅, j 6= k, and that a1 > a2 > · · · > am > 0.

f (x)

a3 A2

A3

A1 Figure 1

14

Envelopes and sharp embeddings of function spaces

Clearly, µf (s) = 0 for s ≥ a1 , and   k k [ X µf (s) = µ  Aj  = µ(Aj ), j=1

ak+1 ≤ s < ak ,

k = 1, . . . , m,

j=1

where we put am+1 := 0 for convenience. Thus we obtain   k m X X  µf (s) = µ(Aj ) χ[a ,a ) (s), s ≥ 0. k+1

j=1

k=1

k

½ µ(B3 ) µ(A3 )

µf (s)

© µ(B2 )

µ(A2 ) µ(B1 ) µ(A1 ) { a3

s

a2 a1

Figure 2

Using the notation Bk := µf (s) =

k S j=1 m X k=1

µ(Bk ) χ[a

k+1 ,ak )

(s),

f ∗ (t) = ak , ∗

f (t)

µ(B1 )µ(B2 ) µ(B3 ) t

µ (Bk−1 ) ≤ t < µ(Bk ),

for k = 1, . . . , m, where we put µ(B0 ) := 0. Hence, f ∗ (t) =

m X j=1

Figure 3

s ≥ 0.

By (2.3) we can easily compute f ∗ now, since f ∗ (t) = 0, t ≥ µ(Bm ), and

a1 a2

a3

Aj as in Figure 2, this can be written as

aj χ[µ(B

j−1 ),µ(Bj ))

(t)

for t ≥ 0.

Example 2.5 Let [R, µ] = [R+ , ν], where ν = | · | stands for the usual Lebesgue measure on R+ .

Preliminaries, classical function spaces We consider the function g(x) = 1 − e−x ,

15

1

x > 0. g(x) = 1 − e−x

Then ½ µg (s) =

∞, 0 ≤ s < 1 0, s≥1

¾ , 0

hence g ∗ (t) ≡ 1,

1

2

x Figure 4

t ≥ 0,

such that a considerable amount of information is lost. Example 2.6 We finally consider the function ψ on Rn , ( 1 − 1−|x| 2 , |x| < 1, ψ(x) = e 0 , |x| ≥ 1.

(2.8)

ψ(x) xn 0

1

x1

Figure 5 It is well-known that ψ is a compactly supported C ∞ -function in Rn . On the other hand one easily calculates that ( 1 − 1−(t/|ωn |)2/n , t < |ω |, ∗ n ψ (t) ∼ e (2.9) 0 , t ≥ |ωn |. Here |ωn | denotes the (surface) measure of the unit sphere in Rn .

Remark 2.7 For later use we recall the following lemma of Bennett and Sharpley in [BS88, Ch. 2, Lemma 2.5]: Let [R, µ] be a finite non-atomic measure space, and f a real- (or complex-) valued µ-measurable function

16

Envelopes and sharp embeddings of function spaces

that is finite µ-a.e. For any number s with 0 ≤ s ≤ µ(R) there is a measurable set As with µ(As ) = s, such that Z

Zs f ∗ (τ )dτ.

|f |dµ =

(2.10)

0

As

Moreover, the sets As can be constructed so as to increase with s, i.e., 0 ≤ σ ≤ s ≤ µ(R) implies Aσ ⊂ As .

2.2

Lebesgue and Lorentz spaces

Let [R, µ] be a measure space again, and 0 < p ≤ ∞. Then Lp (R) are the usual (quasi-) Banach spaces consisting of all µ-a.e. finite functions f for which ³Z ´ |f (x)|p µ(dx)

kf |Lp (R)k =

1/p

,

(2.11)

R

(with the usual modification if p = ∞) is finite. A natural refinement of this scale of Lebesgue spaces is provided by the Lorentz (-Zygmund) spaces Lp,q (log L)a . Definition 2.8 Let [R, µ] be some measure space and 0 < p, q ≤ ∞. (i)

The Lorentz space Lp,q = Lp,q (R) consists of all µ-a.e. finite functions f for which the quantity   1/q ∞   Z   i h q dt   1  ∗    , 0 < q < ∞ t p f (t) t kf |Lp,q k = (2.12) 0     1   ∗   , q=∞  sup t p f (t)  0
is finite. (ii) Let a ∈ R. The Lorentz-Zygmund space Lp,q (log L)a = Lp,q (log L)a (R) consists of all µ-measurable µ-a.e. finite functions f for which kf |Lp,q (log L)a k   1/q ∞   Z   h 1 iq dt    a    , q < ∞ t p (1 + | log t|) f ∗ (t) t = 0      sup t p1 (1 + | log t|)a f ∗ (t)   , q = ∞   0
is finite.

(2.13)

Preliminaries, classical function spaces

17

Remark 2.9 These definitions are well-known and can be found, for instance, in [BS88, Ch. 4, Defs. 4.1, 6.13] and [BR80], respectively. Note that (2.12) and (2.13) do not give a norm in any case, not even for p, q ≥ 1. However, replacing the non-increasing rearrangement f ∗ in (2.12) and (2.13) by its maximal function f ∗∗ , given by 1 f (t) = t

Zt f ∗ (s) ds,

∗∗

t > 0,

(2.14)

0

one obtains for 1 < p < ∞, 1 ≤ q ≤ ∞, or p = q = ∞, a norm in that way, see [BS88, Ch. 4, Thm. 4.6]. An essential advantage of the maximal function f ∗∗ is that – unlike f ∗ – it possesses a certain sub-additivity property, (f + g)∗∗ (t) ≤ f ∗∗ (t) + g ∗∗ (t),

t > 0,

(2.15)

cf. [BS88, Ch. 2, (3.10)]. Moreover, for 1 < p ≤ ∞ and 1 ≤ q ≤ ∞, the corresponding expressions (2.12) with f ∗ and f ∗∗ , respectively, are equivalent; cf. [BS88, Ch. 4, Lemma 4.5]. Note that we dealt with an alternative setting of spaces Lp,q (log L)a (Rn ) (or, more generally, when µ(R) = ∞) in [Har98], [Har00a]. We do not repeat this different approach based on some extrapolation procedure here in detail; the resulting spaces Lp,q (log L)a (R) coincide with the spaces defined by (2.13) in case of a finite underlying measure space [R, µ], but differ otherwise (when µ(R) = ∞). By our introductory remarks we had to restrict ourselves in general. However, we may admit in these elementary to spaces X ⊂ Lloc 1 examples the full range of parameters as we stick with the case of “functions” rather than distributions at the moment. Obviously, Lp,p = Lp , and Lp,q (log L)0 = Lp,q . Note that L∞,q , 0 < q < ∞, is trivial; i.e., it contains the zero function only. The same happens for spaces of type Lp,q (log L)a when p = ∞, 0 < q < ∞, and a + 1/q ≥ 0, or p = q = ∞, but a > 0. Thus when p = ∞ we only study spaces Lp,q (log L)a in the sequel, where a + 1/q < 0 for 0 < q < ∞, or a ≤ 0 for q = ∞, respectively. The spaces Lp,q (log L)a are monotonically ordered in q (for fixed p and a) as well as in a (when p, q are fixed). In general, that is, when µ(R) = ∞, there is no monotonicity in p. But for fixed p, there is an interplay between q and a; cf. [BR80, Thms. 9.3, 9.5]. Proposition 2.10 Let 0 < p ≤ ∞, 0 < q, r ≤ ∞, a, b ∈ R, with a + 1q < 0, b + 1r < 0 if r, q < p = ∞, or a ≤ 0 when p = q = ∞, b ≤ 0 if p = r = ∞, respectively. (i)

Then Lp,q (log L)a ,→ Lp,r (log L)b

(2.16)

18

Envelopes and sharp embeddings of function spaces whenever either q ≤ r, or

q > r,

a≥b 1 1 a+ >b+ . q r

(ii) Let 0 < q ≤ r ≤ ∞. Then L∞,q (log L)a ,→ L∞,r (log L)b if a+

(2.17)

1 1 = b+ . q r

a

a Lp,q (log L)a a+

1 q

a+

= const.

1 q

=0

1 q

L∞,q (log L)a

1 q

L∞,r (log L)b

Lp,r (log L)b

p=∞

0
These conditions can, in general, not be relaxed, see [BR80, Rem. 9.4]. According to (ii), spaces L∞,q (log L)a are ordered along the “diagonals” a + 1/q = const., see also Figure 6 where we indicated in the shaded areas admitted parameters ( 1r , b) for target spaces Lp,r (log L)b such that for a fixed source space we have Lp,q (log L)a ,→ Lp,r (log L)b . For finite measure spaces [R, µ], say, µ(R) = 1, and for p = q = ∞ and a ≥ 0, one has L∞,∞ (log L)−a = Lexp,a , where the latter are the Zygmund spaces consisting of all µ-measurable functions f on R for which there is a constant λ = λ(f ) > 0 such that Z 1/a exp (λ|f (x)|) dµ < ∞, (2.18) R

(if a = 0, this is interpreted as f is bounded, i.e., Lexp,0 = L∞ ). In particular, for a ≥ 0, −a kf |Lexp,a k ∼ sup (1 − log t) f ∗ (t), (2.19) 0
Preliminaries, classical function spaces

19

cf. [BS88, Ch. 4, Def. 6.11, Lemma 6.12]. The spaces Lp , 1 ≤ p ≤ ∞, belong to the category of Banach function spaces (or lattices); we briefly recall this notion and follow [BS88, Ch. 1, Sect. 1] closely. Let [R, µ] be a measure space and M+ the cone of µ-measurable functions on R with non-negative values. The characteristic function of a µ-measurable set A is denoted by χA . Definition 2.11 A mapping % : M+ → [0, ∞] is called a Banach function quasi-norm if, for all g, f , fn , n ∈ N, in M+ , for all γ ≥ 0, and for all µ-measurable subsets A of R, the following properties hold: (i)

%(f ) = 0 if, and only if, f = 0

µ-a.e.

(ii) %(γf ) = γ %(f ) ³ (iii) %(f + g) ≤ c%

´ %(f ) + %(g)

for some c% ≥ 1

(iv) 0 ≤ g ≤ f

µ-a.e. implies %(g) ≤ %(f )

(v) 0 ≤ fn ↑ f

µ-a.e. implies %(fn ) ↑ %(f )

(vi) µ(A) < ∞ implies %(χA ) < ∞ Z (vii) µ(A) < ∞ implies

f dµ ≤ cA,% %(f )

for some constant cA,%

A

depending on A and %, but independent of f Note that a mapping % : M+ → [0, ∞] is called a Banach function norm (or simply function norm) if it satisfies conditions (i)-(vii) with c% = 1 in (iii). Definition 2.12 Let % be a function quasi-norm over [R, µ]. The collection X = X(%) of all real or complex valued µ-measurable functions on R, for which % (|f |) < ∞, is called a quasi-Banach function space over [R, µ]. The quasi-norm of a function f ∈ X is given by kf |Xk = % (|f |). If % is a Banach function norm, then k·|Xk is a norm, X is called a Banach function space, and X endowed with the norm k·|Xk is a Banach space. Returning to our above-explained notation X ,→ Y let us remark that for Banach function spaces X and Y over the same measure space [R, µ] the condition X ⊂ Y already implies X ,→ Y ; cf. [BS88, Ch. 1, Thms. 1.6, 1.8].

20

2.3

Envelopes and sharp embeddings of function spaces

Spaces of continuous functions

Let C(Rn ) be the space of all complex-valued bounded uniformly continuous functions on Rn , equipped with the sup-norm as usual. If m ∈ N, we define C m (Rn ) = {f : Dα f ∈ C(Rn ) for all |α| ≤ m}. Here α = (α1 , . . . , αn ) ∈ Nn0 stands for some multi-index, |α| = α1 + · · · + αn ,

α ∈ Nn0 ,

Dα are classical derivatives, Dα =

1 ∂xα 1

∂ |α| , n . . . ∂xα n

α ∈ Nn0 ,

and C m (Rn ) is endowed with the norm X kf |C m (Rn )k = kDα f |C(Rn )k , |α|≤m

where k · |C(Rn )k can obviously be replaced by k · |L∞ (Rn )k in this case. The set of all compactly supported, infinitely often differentiable functions is denoted by C0∞ (Rn ), as usual. n Recall the concept of the difference operator ∆m h , m ∈ N0 , h ∈ R : Let f n be an arbitrary function on R ; then

(∆1h f )(x) = f (x + h) − f (x),

(∆m+1 f )(x) = ∆1h (∆m h f )(x), h

(2.20)

where x, h ∈ Rn . For convenience we may write ∆h instead of ∆1h . Furthermore, the r-th modulus of smoothness of a function f ∈ Lp (Rn ), 1 ≤ p ≤ ∞, r ∈ N, is defined by ωr (f, t)p = sup k∆rh f |Lp (Rn )k,

t > 0,

(2.21)

|h|≤t

see [BS88, Ch. 5, Def. 4.2] or [DL93, Ch. 2, §7]. Note that each modulus ωr (f, t)p , 1 ≤ p ≤ ∞, r ∈ N, is a non-negative, continuous, increasing function of t > 0. Obviously, ωr (f, t)p ≤ 2r kf |Lp (Rn )k ,

r ∈ N,

1 ≤ p ≤ ∞,

t > 0.

Moreover, ωr (f, t)p & ωr (f, 0)p = 0 for t ↓ 0. We shall write ω(f, t)p instead of ω1 (f, t)p and omit the index p = ∞ if there is no danger of confusion, that is, ω(g, t) instead of ω(g, t)∞ . We refer to the literature mentioned above for further details.

Preliminaries, classical function spaces

21

Marchaud’s inequality states the following: let f ∈ Lp (Rn ), 1 ≤ p ≤ ∞, t > 0, and k ∈ N; then Z ∞ k k ωk+1 (f, u)p du , (2.22) ωk (f, t)p ≤ t log 2 uk u t see [BS88, Ch. 5, (4.11)] or [DL93, Ch. 2, Thm. 8.1] (for the one-dimensional case). When s ∈ R+ we consider H¨older-Zygmund spaces C s . For a positive number a ∈ R+ , let a = bac + {a},

bac := max{k ∈ Z : k < a},

0 < {a} ≤ 1.

(2.23)

Definition 2.13 Let s > 0. The H¨older-Zygmund space C s (Rn ) consists of all f ∈ C bsc (Rn ), such that ° 2 α ° ° ° °∆ D f |C(Rn )° X ° h s n bsc n ° (2.24) kf |C (R )k = °f |C (R )° + sup |h|{s} h6=0 |α|=bsc

is finite. It is well-known that C m (Rn ) 6= C m (Rn ), m ∈ N. However, these H¨olderZygmund spaces C s fit precisely in the scale of Besov spaces that we study in Part II. Finally, as for smoothness assertions of order 1, we shall be concerned with Lipschitz spaces, too. Definition 2.14 Let 0 < a ≤ 1. The Lipschitz space Lipa (Rn ) is defined as the set of all f ∈ C(Rn ) such that kf |Lipa (Rn )k := kf |C(Rn ) k +

sup 0
ω(f, t) ta

(2.25)

is finite. Remark 2.15 Note that the restriction 0 < a ≤ 1 is quite natural, as otherwise the spaces contain only constants; when a = 1 one recovers the classical Lipschitz space Lip1 (Rn ), normed by ° ° ω(f, t) ° ° 1 . °f |Lip (Rn )° = kf |C(Rn )k + sup t 0
(2.26)

22

Envelopes and sharp embeddings of function spaces

For later use we shall introduce some refinement of the scale of Lipschitz spaces. This essentially follows the ideas presented in [EH99], [EH00], [Har00b]. Definition 2.16 Let 1 ≤ p ≤ ∞, 0 < q ≤ ∞, α > q = ∞). Then that

n Lip(1,−α) p, q (R )

1 q

(with α ≥ 0 if

is defined as the set of all f ∈ Lp (Rn ) such

° ° ° ° (1,−α) °f |Lip p, q (Rn )° := kf |Lp (Rn )k +

ÃZ 0

1 2

·

ω(f, t)p t | log t|α

¸q

dt t

!1/q (2.27)

(with the usual modification if q = ∞) is finite. Note that Definition 2.16 coincides with [EH00, Def. 4.1] when q = ∞, and in case of p = q = ∞, α ≥ 0, we recover the logarithmic Lipschitz spaces, Lip(1,−α) = Lip(1,−α) ∞, ∞ introduced in [EH99, Def. 1.1]. In particular, for α = 0, p = q = ∞, these are nothing else than the classical Lipschitz spaces Lip1 (Rn ). Convention. As long as there is no danger of confusion we shall omit the “∞”-indices in the classical setting, i.e., we write Lip(1,−α) instead of Lip(1,−α) ∞, ∞ . The restriction α > 1q is quite natural as otherwise we have Lip(1,−α) = {0} p, q only, see [Har00b, Rem. 18]. However, when q = ∞ we may also admit α = 0, whereas Lip(1,−α) would consist only of constants were α allowed to be negative. The somehow unusual notation using −α (instead of α) is simply due to the fact that we want to emphasise that the additional smoothness parameter α acts in such a way that the usual spaces Lip1 (Rn ) are extended: Lip1 (Rn ) ,→ Lip(1,−α) (Rn ) for all α ≥ 0, i.e., the spaces become larger when less smoothness is assumed – as it should be in some reasonable notation. Remark 2.17 The spaces Lip(1,−α) (Rn ), α ≥ 0, can also be obtained as a special case of the more general spaces C 0, σ(t) (Ω), Ω ⊆ Rn , which were introduced by Kufner, John and Fuˇc´ık; see [KJF77, Def. 7.2.12]. Moreover, spaces of type Lip(1,0) p,∞ = Lip(1, Lp ) are considered by DeVore and Lorentz in [DL93, Ch. 2, §9], where Rn is being replaced by some interval [a, b] ⊂ R and 0 < p ≤ ∞. Similarly, spaces Lip(α, p) were studied by Kolyada in [Kol89]. We introduce the Zygmund spaces C (1,−α) (Rn ), α ≥ 0, as refinements of C (Rn ), given by Definition 2.13, and counterparts of the spaces Lip(1,−α) , see also [EH99]. 1

Preliminaries, classical function spaces

23

Definition 2.18 Let α ≥ 0. Then the space C (1,−α) (Rn ) is defined as the set of all f ∈ C(Rn ) such that kf |C (1,−α) (Rn )k = kf |L∞ (Rn )k +

sup

x, h ∈ Rn 0 < |h| < 1/2

|(∆2h f )(x)| α < ∞. |h| |log |h||

(2.28)

Remark 2.19 Though it might not be obvious at first glance there is an essential difference between spaces of type, say, Lip(1,−α) and C (1,−α) , α ≥ 0 – concerning their compatibility with spaces of Besov type as will be clear in Part II, especially in Section 7.2. Remark 2.20 In view of applications, suitably adapted H¨older inequalities are often needed; we give an example for spaces Lip(1,−α) p, ∞ , that can be found in [EH00, Prop. 4.3, Rem. 4.4]. Let 1 ≤ p, q ≤ ∞ such that 0 ≤ 1r = p1 + 1q ≤ 1. Let α, β ≥ 0. Then max(α,β)) Lip(1,−α) · Lip(1,−β) ,→ Lip(1,− ,→ Lip(1,−(α+β)) . p, ∞ q, ∞ r, ∞ r, ∞

(2.29)

The following extrapolation type result for spaces Lip(1,−α) was obtained p, q in [EH00, Prop. 4.2(i)], [Har00b, Prop. 7]; for details about extrapolation techniques we refer to [Mil94]. Proposition 2.21 Let 1 ≤ p ≤ ∞. (i) Let q = ∞, α > 0. Then f ∈ Lip(1,−α) if, and only if, f belongs to p, ∞ Lp and there is some c > 0 such that for all λ, 0 < λ < 1, ω(f, t)p ≤ c λ−α . 1−λ 0
Moreover, we obtain as an equivalent norm in Lip(1,−α) p, ∞ , ° ° ° (1,−α) ° °f |Lip p, ∞ ° ∼ kf |Lp k + sup λα 0<λ<1

ω(f, t)p . 1−λ 0
(2.30)

(ii) Let 0 < q < ∞, α > 1q . Then f ∈ Lip(1,−α) if, and only if, f belongs p, q to Lp and there is some c > 0 such that Z 0

1

Z

1 2

λαq 0

·

ω(f, t)p t1−λ

¸q

dt dλ ≤ c. t λ

24

Envelopes and sharp embeddings of function spaces Moreover, ° ° ° (1,−α) ° °f |Lip p, q ° ∼ kf |Lp k +

ÃZ

Z

1

λ 0

αq 0

1 2

·

ω(f, t)p t1−λ

¸q

dt dλ t λ

!1/q . (2.31)

Remark 2.22 When p = ∞ Proposition 2.21(i) coincides with the result of Krbec and Schmeisser in [KS01, Prop. 2.5] which was also our motivation for the above extension; part (i) was already presented in [EH00, Prop. 4.2(i)]. We give a (sharp) embedding result for logarithmic Lipschitz spaces Lip(1,−α) and Zygmund spaces C (1,−α) in the spirit of Proposition 2.10. Proposition 2.23 Let 1 ≤ p ≤ ∞, 0 < q, r ≤ ∞, α > 1q , β > 1r . Then Lip(1,−α) ,→ Lip(1,−β) p, q p, r

(2.32)

if, and only if, either r ≥ q, or

r < q,

1 ≥α− r 1 β− >α− r

β−

Remark 2.24 The result was proved in [Har00b, Prop. 16]. The similarity to Proposition 2.10 is obvious. Let us again point out the somehow astonishing result that concerning the embedding Lip(1,−α) into p, q (1,−β) Lip p, r one can “compensate” some gain of logarithmic smoothness −β > −α by “paying” with the additional index q, that is, as long as (−β) − (−α) ≤ 1q − 1r , r ≥ q.

1 , q 1 . q α

Lip(1,−β) p, r Lip(1,−α) p, q α=

1 q 1 q

Figure 7

Dealing with Zygmund spaces C (1,−α) , we restrict ourselves to p = q = ∞ at the moment (but will return to this setting in a more general approach in Section 7.2. Recall our convention Lip(1,−α) = Lip(1,−α) ∞, ∞ . We obtained in [EH00, Prop. 2.7] the following embedding result. Proposition 2.25 Let α, β, γ be non-negative real numbers. Then Lip(1,−α) ,→ C (1,−β) ,→ Lip(1,−γ)

(2.33)

Preliminaries, classical function spaces

25

if, and only if, β ≥ α,

and

γ ≥ β + 1.

Finally we consider spaces Lip(a,−α) ∞, q , 0 < a ≤ 1, 0 < q ≤ ∞, α ∈ R, which are “close” to Lipa . For convenience, we deal with p = ∞ exclusively. Definition 2.26 Let 0 < a < 1, 0 < q ≤ ∞, and   α ∈ R α > 1q   α≥0

 0 < a < 1, 0 < q ≤ ∞  

if if

a = 1, 0 < q < ∞

if

a = 1,

q=∞

 

.

(2.34)

n n The space Lip(a,−α) ∞, q (R ) is defined as the set of all f ∈ C(R ) such that

° ° ° ° (a,−α) °f |Lip∞, q (Rn )° := kf |C(Rn ) k +

ÃZ

1 2

·

0

ω(f, t) ta | log t|α

¸q

dt t

!1/q (2.35)

(with the usual modification if q = ∞) is finite.

Remark 2.27 The above spaces first appeared (in this notation) in [Har00b] in connection with limiting embeddings, extending the case a = 1 studied in [EH99] and [EH00]. We obtained different characterisations of spaces Lip(a,−α) in the sense of Proposition 2.21: Let 0 < a ≤ 1, 0 < q ≤ ∞, ∞, q and α ∈ R satisfy (2.34). Then f ∈ Lip(a,−α) ∞, q if, and only if, f belongs to C and there is some c > 0 such that Z

Z

a

λ

1 2

αq

0

·

0

ω(f, t) ta−λ

¸q

dt dλ ≤ c. t λ

Moreover, ° ° ° (a,−α) ° °f |Lip∞, q ° ∼ kf |Ck +

ÃZ

Z

a

λ 0

αq 0

1 2

·

ω(f, t) ta−λ

¸q

dt dλ t λ

!1/q .

(2.36)

Let us finally mention that there are interesting applications connected with Lipschitz spaces Lip(1,−α) ∞, q , see the book [Lio98] by Lions and the paper [Vis98] by Vishik.

26

Envelopes and sharp embeddings of function spaces

2.4

Sobolev spaces, Sobolev’s embedding theorem

Let Wpk (Rn ) be the space of those functions f on Rn (or locally regular n distributions f ∈ Lloc 1 (R )) for which all weak derivatives of order at most α k ∈ N, D f , |α| ≤ k, belong to Lp (Rn ), 1 ≤ p < ∞. It is well-known that the space Wpk (Rn ) is a Banach space equipped with the norm ´1/p ° ° ³ X p °f |Wpk (Rn )° = kDα f |Lp (Rn )k .

(2.37)

|α|≤k

Moreover, the compactly supported smooth functions, C0∞ (Rn ), are dense in Wpk (Rn ). There is a great deal of literature on Sobolev spaces, see [AF03], [Maz85], [EE87], [Zie89]. Dealing with Sobolev spaces on domains, Wpk (Ω), questions of extendability, definition by restriction procedures or – alternatively – by intrinsic characterisations, connected with the geometry of the underlying domain, too, are highly non-trivial and well-studied. In the present book we are, however, essentially interested in spaces on Rn and will thus postpone a more refined discussion of spaces on domains to elsewhere.

k supercritical

k=

n p

Wpk critical 1 subcritical 1

1 p

Figure 8

We recall the famous Sobolev embedding theorem in a version used in the

Preliminaries, classical function spaces

27

sequel. Note, in particular, that we consider here the case Ω = Rn only, though there are many extensions for (bounded) domains Ω ⊂ Rn that satisfy certain conditions (cone condition, segment condition, local Lipschitz condition, etc.). We rely here on the formulation given in [AF03, Thm. 4.12] and [EE87, Ch. V] with Ω = Rn . In the above ( p1 , s)-diagram we indicate the different cases that will be considered later on in detail. Each space Wpk is marked here by its smoothness parameter s = k ∈ N0 and the integrability p ∈ [1, ∞). This s diagram will be enriched and filled when we deal with spaces of type Bp,q and s n Fp,q in Part II. As we deal with spaces on R only, we shall omit the “Rn ” from their notation as long as there is no danger of confusion. Theorem 2.28 Let k ∈ N and 1 ≤ p < ∞. (i)

super-critical case Assume that k >

n or k = n and p = 1; then p

and

Wpk ,→ C,

(2.38)

Wpk+1 ,→ Lip1 .

(2.39)

Wpk ,→ Lr .

(2.40)

Moreover, for p ≤ r ≤ ∞,

If, in addition, k <

n + 1, then p

Wpk ,→ Lipa , and, for k =

0
n < 1, p

(2.41)

n + 1, p Wp1+n/p ,→ Lipa ,

with

W11+n ,→ Lipa ,

0 < a < 1, 0 < a ≤ 1.

(2.42) (2.43)

(ii) critical case Assume that k =

n , then p Wpk ,→ Lr

for p ≤ r < ∞.

(2.44)

28

Envelopes and sharp embeddings of function spaces

(iii) sub-critical case Assume that k <

n , then p Wpk ,→ Lr

for p ≤ r ≤ p∗ =

(2.45)

np . n − kp

P r o o f : As mentioned above, there are many proofs in the literature. For convenience, we give a proof here that essentially follows [AF03, Thm. 4.12] and [EE87, Ch. V] and deal with Ω = Rn only. Step 1. We start with the sub-critical case (iii) and k = 1; thus assume np . First we show that there exists 1 ≤ p < n, and let p∗ be given by p∗ = n−p 1 a constant c > 0 such that for all u ∈ Wp , ° ° ku|Lp∗ k ≤ c °u|Wp1 ° .

(2.46)

By the density of C0∞ in Wpk we may assume by standard arguments that u ∈ C01 ∩ Wp1 . Let α = (α1 , . . . , αn ) ∈ Nn0 , |α| = 1, then α = α(i) = (0, . . . , 1, . . . , 0) with αk = δik , i, k = 1, . . . , n. We write in obvious notation (i) Di for Dα , i = 1, . . . , n. Let x = (x1 , . . . , xn ) ∈ Rn , then Zx1 u(x) =

Z∞ D1 u(ξ1 , x2 , . . . , xn )dξ1 = −

−∞

D1 u(ξ1 , x2 , . . . , xn )dξ1 , x1

hence |u(x)| ≤

1 2

Z∞ |D1 u(ξ1 , x2 , . . . , xn )| dξ1 , −∞

and, dealing with the other coordinates in a similar way,  ∞  Z n Y n  |2u(x)| ≤ |Dj u(x1 , . . . , ξj , . . . , xn )| dξj  . j=1

(2.47)

−∞

1 , integrate it with respect to x1 and apply We raise (2.47) to the power n−1 an iterated version of H¨older’s inequality in the form

Z∞ Ãn−1 Y −∞

k=1

1 ! n−1

gk (t)

dt ≤

n−1 Y k=1



Z∞



1  n−1

gk (t)dt −∞

;

Preliminaries, classical function spaces

29

this gives Z∞

n

|2u(x1 , . . . , xn )| n−1 dx1 −∞



1  n−1

Z∞

≤

|D1 u(x)| dx1 

n Y

 

k=2

−∞

1  n−1

Z∞ Z∞

|Dk u(x)| dxk dx1 

.

−∞ −∞

We proceed with the remaining coordinates in the same way, where every time precisely n − 1 factors are specifically involved in the integration; thus Z |2u(x)|

n n−1

dx ≤

n Y

 

k=1

Rn

1  n−1

Z

|Dk u(x)| dx

,

Rn

which can be reformulated as n ° 1 Y ° 1 ° ° n ° ≤ kDk u|L1 k n . °u|L n−1 2

(2.48)

k=1

The arithmetic-geometric inequality implies ° ° ° ° n ° ≤ °u|L n−1

n ° 1 X 1 ° °u|W11 ° , kDk u|L1 k ≤ 2n 2n

(2.49)

k=1

which is the desired result for p = 1. When p > 1, then we apply (2.49) to v = |u| conditions. Consequently,

(n−1)p n−p

(n−1)p n−p

which satisfies the required differentiability

n−p ° (n−1)p ° ° ° n ° ku|Lp∗ k = °v|L n−1 n−p ! (n−1)p à n X ≤ c1 kDk v|L1 k

k=1

 ≤ c2 

n−p  (n−1)p

n Z X

|u(x)|

k=1 Rn

≤ c3 ku|Lp∗ k

> 1, and

n(p−1) p(n−1)

n(p−1) n−p

Ã

n X

k=1

|Dk u(x)| dx n−p ! (n−1)p

kDk u|Lp k

n(p−1) ° ° n−p ≤ c4 ku|Lp∗ k p(n−1) °u|Wp1 ° (n−1)p ,

30

Envelopes and sharp embeddings of function spaces

where we used H¨older’s inequality again. This leads to (2.46). Step 2. We complete the proof of (iii). Assume k ∈ N, k < np , and u ∈ Wpk . np Consider pj given by pj+1 = n−pjj , j = 0, . . . , k − 1, with p0 = p; thus pk = p∗ . We apply (2.46) and obtain consecutively X ° X ° ° ° °Dα u|Lp ° ≤ c2 °Dβ u|Lp ° k−1 k−2

ku|Lp∗ k ≤ c1

|α|=1

.. . X ≤ ck kDγ u|Lp0 k

|β|=2

≤

° ° c °u|Wpk ° .

(2.50)

|γ|=k

Thus (2.45) is verified for r = p∗ and, simply by definition, for r = p. Let now r be such that p < r < p∗ , and u ∈ Wpk ; then another application of H¨older’s inequality together with (2.50) leads to 

 r1

Z

ku|Lr k = 

|u(x)|

p(p∗ −r) p∗ −p

|u(x)|

p∗ (r−p) p∗ −p

dx

Rn p(p∗ −r)

p∗ (r−p)

≤ ku|Lp k r(p∗ −p) ku|Lp∗ k r(p∗ −p) ° p∗ (r−p) ° ° ° ° p(p∗∗ −r) ° ∗ ≤ c °u|Wpk ° r(p −p) °u|Wpk ° r(p −p) = c °u|Wpk ° . n p . We may assume p > 1, the rest n/p is postponed to Step 4. First let u ∈ Wp with supp u ⊂ Ω, |Ω| < ∞, and p pr ns . Then we have for s := p+r that 1 ≤ s < p and r = n−ks r ≥ p0 = p−1 = s∗ .

Step 3.

We study the limiting case k =

By H¨older’s inequality we obtain that

° ° ° ° °u|Wsk ° ≤ c|Ω| 1s − p1 °u|Wpk ° . Moreover, we may apply (iii) to Wsk and Lr since k = r = s∗ , leading to

pr n p s(p+r)

° ° ° ° 1 ° ° ° ° ku|Lr k ≤ c °u|Wsn/p ° ≤ c0 |Ω| r °u|Wpn/p ° .

=

n r s p+r

<

n s,

(2.51)

When p < r < p0 , hence 1 < p < 2, then similar to Step 2 we may interpolate 0 (2.51) and Wpk ,→ Lp ; let κ := pp0 −p older’s inequality again −r > 1, then by H¨

Preliminaries, classical function spaces

31

together with (2.51) for r = p0 , Z

p0

p

r

|u(x)| κ |u(x)| κ0 dx

ku|Lr k = Rn

p0

p

≤ ku|Lp k κ ku|Lp0 k κ0 ° ° p00 ° ° κp 1 ° °κ ° ° ≤ c °u|Wpn/p ° |Ω| κ0 °u|Wpn/p ° ° °r r−p ° ° ≤ c |Ω| p0 −p °u|Wpn/p ° , such that finally, 

Z

min( pr−p 0 −p ,1)

|u(x)|r dx ≤ c |Ω|



 pr

X Z

|Dα u(x)|p dx

(2.52)

|α|≤ n p Ω

Ω

for all r, p ≤ r < ∞. In order to remove the dependence on Ω, suppose that we cover Rn with cubes Qm , m ∈ Zn , of side length b > 1, centered at m ∈ Zn , and with sides parallel to the axes of coordinates. Thus |Qm | = bn for all m ∈ Zn , and we have a controlled overlapping, #{l ∈ Zn : Ql ∩Qm 6= ∅} ≤ c, m ∈ Zn ; for a finite set M the number of its elements is denoted by # M , as usual. Let dQ, d > 0, denote the cube with the same centre as Q and side length db. Then, with a suitably adapted partition of unity, (2.52) with Ω = Qm and r ≥ p finally complete the proof of (ii), Z r

m∈Zn

Rn

≤c

Z

X

|u(x)|r dx =

ku|Lr k =

|u(x)|r dx b−1 Qm



X

min( pr−p 0 −p ,1)

|Qm |

 ≤ c0 b

 ≤ c00 



|Dα u(x)|p dx

|α|≤ n p Qm

m∈Zn

n min( pr−p 0 −p ,1)



X

X

 pr

Z

X

Z

 pr

|Dα u(x)|p dx

m∈Zn Q |α|≤ n p m

X Z

 pr

|Dα u(x)|p dx

° °r ° ° = c00 °u|Wpn/p ° .

|α|≤ n p Rn

Step 4. We prove (2.38). Note that this immediately implies (2.39) as well as (2.40) for r = ∞, ° ° ku|L∞ k ≤ c °u|Wpk ° , (2.53)

32

Envelopes and sharp embeddings of function spaces

and for p ≤ r < ∞, Z r

|u(x)|p |u(x)|r−p dx ≤ ku|Ckr−p ku|Lp kp

ku|Lr k = Rn

≤ c ku|Wpk kr−p ku|Wpk kp = c ku|Wpk kr . In view of the above argument it is sufficient to deal with the case of u ∈ C0∞ ∩Wpk such that supp u ⊂ Ω with |Ω| < ∞, and to check the dependence of the constants upon Ω. For simplicity we may even assume from the beginning that Ω is the above cube with edges parallel to the axes of coordinates, and side-length b ≥ 1. Let x ∈ Ω and use the Taylor expansion for u in Ω, X cα Dα u(x)(y − x)α + Rx (y), y ∈ Ω, u(y) = |α|≤k−1

where we may choose the integral representation for the remainder term, Z1

X

Rx (y) =

(1 − t)k−1 Dα u ((1 − t)x + ty)) dt.

α

cα,k (y − x)

|α|=k

0

Thus, we have for any y ∈ Ω, Z |Rx (y)| dx ≤

Z

X

cα,k

|α|=k

Ω

≤c

Z1 |y − x|

(1 − t)k−1 |Dα u ((1 − t)x + ty)) |dt dx 0

Ω

1 X Z

k

Z (1 − t)−n−1

|α|=k 0

|y − z|k |Dα u(z)| dz dt, Ωt

where Ωt = {z ∈ Rn : z = (1 − t)x + ty, x ∈ Ω}. Our special choice of Ω then allows us to estimate further, Z |Rx (y)| dx ≤ c

X Z

1−

0

n

≤c b

(1 − t)−n−1 dt dz

|y − z|k |Dα u(z)|

|α|=k Ω

Ω

|z−y| b

Z

X Z

0 k−n

|y − z|

α

|D u(z)| dz.

(2.54)

|α|=k Ω

Assume k = n, p = 1, then (2.54) leads to Z X |Rx (y)|dx ≤ c bn kDα u|L1 k ≤ c0 bn ku|W1n k , Ω

|α|=k

(2.55)

Preliminaries, classical function spaces whereas for k >

n p,

33

p ≥ 1, we conclude by H¨older’s inequality for any y ∈ Ω, 

Z

X

|Rx (y)|dx ≤ c bn

0

kDα u|Lp k 

|α|=k

Ω

 ° ° ≤ c0 bn °u|Wpk ° 

1/p0

Z

|y − z|(k−n)p dz  Ω

1/p0

Z2b 0

%n−1+(k−n)p d% 0

00

≤c b

n+k− n p

° ° °u|Wpk ° ,

(2.56)

(suitably adapted for p = 1). The last integral converges, since k > np implies n + (k − n)p0 > 0. Concerning the Taylor polynomial we estimate similarly, Z ¯ X ¯ X Z ¯ ¯ cα Dα u(x)(y − x)α ¯dx ≤ c bk−1 |Dα u(x)|dx ¯ |α|≤k−1

Ω

|α|≤k−1 Ω X k−1+ pn0

≤cb

kDα u|Lp k

|α|≤k−1 0

≤c b

n−1+k− n p

° ° °u|Wpk ° ,

(2.57)

for any y ∈ Ω. Finally, since ¶ Z Z µ¯ X ¯ ¯ ¯ bn |u(y)| = |u(y)|dx ≤ cα Dα u(x)(y − x)α ¯ + |Rx (y)| dx, ¯ Ω

Ω

|α|≤k−1

estimates (2.55), (2.56) and (2.57) yield ° n ° |u(y)| ≤ c bk− p °u|Wpk ° for any y ∈ Ω, where either k > np , p ≥ 1, or k = n, p = 1. This yields (2.38) first for u ∈ C0∞ ∩ Wpk with supp u ⊂ Ω; completely parallel to the end of Step 3 this can then be extended to the general situation. Step 5.

It remains to prove

Wpk ,→ Lipa

if

  n n n 0 < a ≤ k − p, p < k < p + 1  0
In view of (2.38) and (2.25) it is sufficient to verify sup Rn

x, y ∈ 0 < |x − y| < 1

|u(x) − u(y)| ≤ c ku|Wpk k |x − y|a

(2.58)

34

Envelopes and sharp embeddings of function spaces

in the cases explicated in (2.58). Moreover, we claim that it is sufficient to deal with the case k = 1, that is, to show X |u(x) − u(y)| ≤ c kDα u|Lr k, (2.59) sup |x − y|a 0<|x−y|<1 |α|≤1

n r.

where n < r ≤ ∞, 0 < a ≤ 1 − Taking (2.59) for granted, we can choose r such that k − np = 1 − nr when np < k < np + 1, and (2.45) applied to Dβ u, |β| = 1, implies the first case in (2.58). When k = np + 1, we choose r such that p < r < ∞ and 0 < 1 − nr < 1; hence (2.44) completes the corresponding argument. Finally, dealing with the third case in (2.58), that is, p = 1, k = n + 1, we choose r = ∞ and use (2.40) together with (2.59). Consequently, it is sufficient to prove (2.59). We deal with the local setting first and consider a unit cube Ω with edges parallel to the axes of coordinates. Let x, y ∈ Ω, with 0 < |x − y| = δ < 1. We denote by Ωδ the cube with faces parallel to those of Ω, side length δ, and such that x, y ∈ Ωδ ⊂ Ω. Let ξ ∈ Ωδ , then Z1 d u(x + t(ξ − x))dt, u(x) = u(ξ) − dt 0

and √ |u(x) − u(ξ)| ≤ n δ

Z1 |∇u(x + t(ξ − x))| dt, 0

as x, ξ ∈ Ωδ . Straightforward calculation then gives, Z ¯ ¯ ¯Z ¯ ¯ ¯ ¯ −n −n ¯ u(ξ)dξ ¯ = δ ¯ (u(x) − u(ξ))dξ ¯ ¯u(x) − δ Ωδ

Ωδ

Z ≤δ

−n

|u(x) − u(ξ)| dξ Ωδ

√ ≤ n δ 1−n

Z Z1 |∇u(x + t(ξ − x))| dt dξ Ωδ 0

√ ≤ n δ 1−n

Z

Z1 t

−n

0

|∇u(η)| dη dt

Ωδ,t

√ ≤ n δ 1−n k∇u|Lr k

Z1 1

t−n |Ωδ,t | r0 dt 0

≤cδ

1− n r

X |α|≤1

Z1 n

α

t− r dt

kD u|Lr k 0

Preliminaries, classical function spaces X n kDα u|Lr k, ≤ c0 δ 1− r

35

|α|≤1

where we applied H¨older’s inequality for r < ∞ and used the assumption r > n; the modification for r = ∞ is obvious. Likewise we have Z ¯ ¯ X n ¯ ¯ kDα u|Lr k, ¯u(y) − δ −n u(ξ)dξ ¯ ≤ c δ 1− r |α|≤1

Ωδ

such that the triangle inequality together with the substitution δ = |x − y| leads to X n kDα u|Lr k, |u(x) − u(y)| ≤ c |x − y|1− r |α|≤1

for all x, y ∈ Ω, x 6= y. We consider a covering of Rn with cubes Qm as described above and centered at m ∈ Zn , then for arbitrary x, y ∈ Rn with |x − y| < 2b there is at least one cube Qm such that x, y ∈ Qm and we can apply the above argument, sup

n

x, y ∈ R 0 < |x − y| < 1

|u(x) − u(y)| n |x − y|1− r

≤ sup m∈Zn

≤ c0

|u(x) − u(y)| + c sup |u(x)| n |x − y|1− r x∈Rn

sup x, y ∈ Qm 0 < |x − y| <

X

b 2

kDα u|Lr k + c ku|Ck

|α|≤1 00

≤c

X

kDα u|Lr k

|α|≤1

using (2.38) with r > n in the final step. This proves (2.59) and finishes the whole proof. Remark 2.29 As already mentioned in the introduction, a lot of work has been done since Sobolev’s pioneering paper [Sob38], dealing with extensions, sharpness, best constants, dependence upon the (geometry of the) underlying domain of the above embeddings or inequalities, respectively. We do not want to repeat this here, but refer to the introduction. The result of Br´ezis-Wainger [BW80], mentioned in the introduction (1.14) obviously complements (2.42). With the notation introduced in Definition 2.26 and taking into consideration that Wpm = Hpm , m ∈ N0 , 1 < p < ∞, (1.14) can be reformulated as 1 1 (1,− p10 ) + = 1. , 1 < p < ∞, (2.60) Wp1+n/p ,→ Lip p p0

36

Envelopes and sharp embeddings of function spaces

Example 2.30 Concerning the sharpness of Theorem 2.28 we restrict ourselves to an example merely, since the main idea of envelopes just relies on those cases where we do not have the corresponding embeddings. We explain this in detail in the next chapters. However, this immediately proves the sharpness of the above assertions, too (and is also known already). Dealing with the critical case (ii) when k = np = 1 and n > 1, see (2.44) and (2.38), we present a family of functions hν that belong to Wn1 , but not to L∞ , as long as 0 < ν < 1 − n1 . For x ∈ Rn , consider the radial functions  ¶ µ 1  − logν 2, 0 < |x| < 1 logν 1 + , hν (x) = |x|   0 , otherwise  

hν (x)

x ∈ Rn .

(2.61)

xn

0

1 x1

Figure 9 Plainly, hν 6∈ L∞ whenever ν > 0, whereas for all 0 < r < ∞, Z r r khν |Lr k = |hν (x)| dx Rn

Z1 ≤ c1 + c2 0

¶ µ 1 d% %n−1 logrν 1 + %

Z1 %n−1−ε d% ≤ c4 < ∞

≤ c1 + c3 0

for all n ∈ N and sufficiently small ε > 0. Using the abbreviation Dk introduced in Step 1 of the proof of Theorem 2.28, straightforward calculations

Preliminaries, classical function spaces

37

lead to the estimates Z n

n

kDk hν |Ln k =

|Dk hν (x)| dx Rn

Z1 %n−1

≤ c1

´ ³ logn(ν−1) 1 + %1 (%2 + %)n

0

Z1 µ ≤ c2 0

≤ c2

% %2 + %

³ ´ ¶n−1 logn(ν−1) 1 + 1 % %2 + %

´ ³ Z1 logn(ν−1) 1 + 1 % %2 + %

0

≤ c3 log

d%

n(ν−1)+1

d%

d%

¶ ¯1 µ 1 ¯¯ ≤ c4 < ∞ 1+ % ¯0

assuming that ν < 1 − n1 . Hence,  1/n X ° ° n °hν |Wn1 ° =  kDα hν |Ln k  |α|≤1

≤ c khν |Ln k + c

n X

kDk hν |Ln k ≤ c0 < ∞

k=1

for 0 < ν < 1 − n1 . Another well-known example following the same idea, i.e., g ∈ Wn1 \ L∞ , is given by   ¶¶ µ µ 1   − log log 2, 0 < |x| < 1 log log 1 + g(x) = , x ∈ Rn . |x|   0 , otherwise

Chapter 3 The growth envelope function EG

We already mentioned in our introductory remarks that characterisations like (1.4) gave reason to study the behaviour of the non-increasing rearrangement f ∗ of a function f , in particular, when these spaces contain essentially unbounded functions. This leads to the concept of growth envelopes. Our results s s for spaces of type Bp,q or Fp,q are postponed to Part II; we start with some simple features and examples in order to give a better feeling for what is really “measured” by growth envelopes. For that reason we test our new envelope tool on rather classical spaces like Lorentz (-Zygmund) spaces first – before arriving at more surprising results in Part II. Moreover, there is also an interesting point at the end of this chapter: the recognition of growth envelope functions in terms of fundamental functions in rearrangement-invariant spaces. Finally, preparing some later discussion of global versus local behaviour in Section 10.3 we already add some “higher-order” and “weighted” examples in Section 3.4.

3.1

Definition and basic properties

Definition 3.1 Let [R, µ] be a measure space and X a quasi-normed function space on R. The growth envelope function EGX : (0, ∞) → [0, ∞] of X is defined by EGX (t) := sup f ∗ (t), t > 0. (3.1) kf |Xk≤1

We adopt the usual convention to put EGX (τ ) := ∞ when {f ∗ (τ ) : kf |Xk ≤ 1} is not bounded from above for some τ > 0. Remark 3.2 Note that (3.1) immediately causes some problem when taking into account that we shall always deal with equivalent (quasi-) norms in the underlying function space (rather than a fixed one). Assume we have two different, but equivalent (quasi-) norms k · |Xk1 and k · |Xk2 in X. Then every function f ∈ X with kf |Xk1 ≤ 1, f 6≡ 0, is connected with some gf := cf , where c = kf |Xk1 /kf |Xk2 , kgf |Xk2 ≤ 1, and gf∗ = cf ∗ , leading 39

40

Envelopes and sharp embeddings of function spaces

to a different, but equivalent expression for EGX . So, strictly speaking, we are concerned with equivalence classes of growth envelope functions, where we choose one representative EGX (t) ∼

sup

f ∗ (t),

t > 0.

kf |Xk≤1

However, we shall not make this difference between equivalence class and representative in the sequel. Furthermore, by (3.1) the growth envelope EGX (t) is defined for all values t > 0, but at the moment we are only interested in local characterisations (singularities) of the spaces referring to small values of t > 0, say, 0 < t < 1, whereas questions of global behaviour (t → ∞) are postponed to Section 10.3. This preference for local studies also implies that we could formally transfer many of our results from spaces on Rn to their counterparts on bounded domains, or, more precisely, from measure spaces [R, µ] with µ(R) = ∞ to finite measure spaces. The necessary modifications in the case of our examples in Section 3.2 are obvious; we shall thus mainly deal with function spaces on Rn in the sequel. We briefly discuss the obvious question whether the growth envelope function EGX is always finite for t > 0 or what necessary/sufficient conditions on X (or the underlying measure space) imply this; recall notation (2.2). Lemma 3.3 Let [R, µ] be a measure space. (i) There are function spaces X on R which do not have a growth envelope function in the sense that EGX (t) is not finite for t > 0. (ii) Let X be a (quasi-) normed function space on R. Then EGX (t) is finite for any t > 0 if, and only if, sup

µf (λ) −→ 0

for

λ → ∞.

(3.2)

kf |Xk≤1

P r o o f : Concerning (i), we take a simple counter-example,¡ obviously such ¢ that X 6,→ L∞ . Let [R, µ] = [R¡n , | · |]¢ and put X := L∞ hxi−1 . Hence f (x) = hxi belongs to X = L∞ hxi−1 , kf |Xk = 1, but f ∗ (t) is not finite for any t > 0. As for (ii), we first prove that EGX (t) < ∞ for any t > 0 implies (3.2). Hence we have for any t > 0 that there is some Mt > 0 such that for all f ∈ X, kf |Xk ≤ 1, we have f ∗ (t) ≤ Mt . Thus there is for any t > 0 some Mt > 0 such that for all f ∈ X, kf |Xk ≤ 1, we know µf (λ) ≤ t for any λ > Mt . In other words, for any t > 0 there is some Mt > 0 such that for all λ > Mt the expression supkf |Xk≤1 µf (λ) is bounded from above by t. But this is nothing else than a reformulation of (3.2). The converse can be shown by the same (standard) argument.

The growth envelope function EG

41

Hence the definition of EGX is non-trivial and reasonable. We now collect a few elementary properties of it. Simplifying technical matters in the sequel we introduce the number τ0 by n o τ0 = τ0G (X) := sup t > 0 : EGX (t) > 0 . (3.3) Note that EGX (t) = 0 for some t > 0 implies f ∗ (t) = 0 for all f ∈ X, kf |Xk ≤ 1; hence – by some scaling argument – g ∗ (t) = 0 for all g ∈ X. But then X contains only functions having a support with finite measure, i.e., µ ({x ∈ R : |g(x)| > 0}) ≤ t for all g ∈ X. This is true, in particular, when µ(R) ≤ t. Proposition 3.4 Let [R, µ] be a measure space and X a quasi-normed function space on R. ³ ´∗ (i) EGX is monotonically decreasing and right-continuous, EGX = EGX . (ii) If R has finite measure, i.e., µ(R) < ∞, then EGX (t) = 0 for t > µ(R) and any function space X on R. (iii) We have X ,→ L∞ if, and only if, EGX (·) is bounded, i.e., sup EGX (t) = lim EGX (t) is finite. In that case,

t>0

t↓0

EGX (0) := lim EGX (t) = kid : X → L∞ k . t↓0

(iv) Let Xi = Xi (R), i = 1, 2, be function spaces on R. Then X1 ,→ X2 implies that there is some positive constant c such that for all t > 0, EGX1 (t) ≤ c EGX2 (t). One may choose c = kid : X1 → X2 k in that case. (v) Let κ : (0, ∞) → [0, ∞) be a non-negative function and EGX (t) < ∞ for t > 0. Then κ(·) is bounded on (0, τ0 ) if, and only if, there exists c > 0 such that for all f ∈ X, kf |Xk ≤ 1, sup 0
κ(t) ∗ f (t) ≤ c . EGX (t)

(3.4)

P r o o f : As for (i), note that f ∗ is monotonically decreasing for any f ∈ X; thus also EGX is monotonically decreasing by standard arguments. This implies ³ ´∗ EGX = EGX , too. Likewise the right-continuity of any f ∗ implies that of EGX . Assertion (ii) is clear, because obviously f ∗ (t) = 0 for any t > µ(R) and any f ∈ X in that case.

42

Envelopes and sharp embeddings of function spaces

We prove (iii). Assume first that X ,→ L∞ , i.e., there exists c > 0 such that for all f ∈ X, kf |L∞ k ≤ c kf |Xk. Then f ∗ (0) = lim f ∗ (t) = kf |L∞ k < ∞, t↓0

which implies EGX (0) := lim EGX (t) = sup t↓0

≤ c

sup

f ∗ (t) =

t>0 kf |Xk≤1

sup

f ∗ (0) =

sup kf |Xk≤1

sup

kf |L∞ k

kf |Xk≤1

kf |Xk = c .

kf |Xk≤1

Conversely, let X 6,→ L∞ . Then by [BS88, Ch. 1, Thm. 1.8] X 6⊂ L∞ for any Banach function space X, and there is nothing else to prove. Otherwise, when X ⊂ L∞ , but id : X → L∞ is not continuous, one finds a sequence (tn )n such that EGX (tn ) > n, n ∈ N, and the monotonicity of EGX leads to sup EGX (t) = ∞. t>0

(iv), °¡ 1let¢ f ∈ °X1 , kf |X1 k1 ≤∗ 1, and¡ 1put ¢ ¢ c := kid : X1 → X2 k. Then ¡Verifying 1 ° f |X2 ° ≤ 1, and f (t) = f ∗ (t) ≤ E X2 (t). Consequently G c f ∈ X2 , c c c for any f ∈ X1 , kf |X1 k ≤ 1, we obtain f ∗ (t) ≤ c EGX2 (t), implying (iv). We first prove the necessity of (3.4) for the boundedness of κ. Thus let κ be a positive and bounded function, i.e., there is some c > 0 such that for all 0 < t < τ0 we have κ(t) ≤ c. On the other hand, by definition EGX (t) ≥ f ∗ (t) for any t > 0 and any f ∈ X, kf |Xk ≤ 1; hence there is some c > 0 such that for all f ∈ X, kf |Xk ≤ 1, and all 0 < t < τ0 , κ(t) ∗ f (t) ≤ c. EGX (t) This implies (3.4). It remains to show the sufficiency of (3.4) for the boundedness of κ. But obviously (3.4) leads to the existence of some c > 0 such that κ(t) ∗ f (t) ≤ c . sup sup X E 0
κ(t) EGX (t)

sup

f ∗ (t) ≤ c ,

kf |Xk≤1

but by the definition of EGX this reduces to bounded.

sup κ(t) ≤ c, i.e., κ is 0
Remark 3.5 We shall see in the next section that some counterpart of (iv) in the sense of (iii), i.e., that some relation of the envelope functions implies some (continuous) embedding for the corresponding spaces, cannot hold in

The growth envelope function EG

43

general; see Remark 3.13. Concerning (v), one may prove even more, namely that in some sense EGX is the only such function with the property described above. Corollary 3.6 Let [R, µ] be a measure space, X a (quasi-) normed function space over R, and ψ : (0, ∞) → (0, ∞] a positive, monotonically decreasing function with the following property: For any non-negative function κ : (0, ∞) → [0, ∞), κ(·) is bounded on (0, τ0 ) if, and only if, there exists c > 0 such that for all f ∈ X, kf |Xk ≤ 1, sup 0
κ(t) ∗ f (t) ≤ c . ψ(t)

(3.5)

Then ψ ∼ EGX , i.e., there are numbers c2 > c1 > 0 such that whenever 0 < t < τ0 , c1 ψ(t) ≤ EGX (t) ≤ c2 ψ(t). (3.6) P r o o f : We have by (3.5) with κ ≡ 1 that there is some c > 0 such that for all f ∈ X, kf |Xk ≤ 1, sup 0
1 f ∗ (t) ≤ c , ψ(t)

i.e., that there exists c > 0 such that for all t ∈ (0, τ0 ) and all f ∈ X, kf |Xk ≤ 1, f ∗ (t) ≤ c ψ(t). Hence by the definition of EGX (t), there exists c > 0 such that for all 0 < t < τ0 , EGX (t) ≤ c ψ(t) .

(3.7)

It remains to show the converse. Let κ be a non-negative function. Then we obtain by (3.4) that κ is bounded if, and only if, there is some c > 0 such that for all f ∈ X, kf |Xk ≤ 1, sup 0
κ(t) ∗ f (t) ≤ c , EGX (t)

i.e.,

sup 0
κ(t) ψ(t) ∗ f (t) ≤ c . ψ(t) EGX (t)

Now we conclude from (3.5) with κ e := κ ψ/EGX that this happens if, and only if, κ e is bounded. Hence we know that κ is bounded if, and only if, κ e = κ ψ/EGX is bounded. Let, in particular, κ ≡ 1; then this provides the inequality converse to (3.7) and ends the proof. Remark 3.7 Triebel discussed earlier the closely related concept of a growth s s function ψ, where [R, µ] = [Rn , `n ] and X = Bp,q or X = Fp,q . In our X notation this means nothing else than ψ ∼ 1/EG .

44

Envelopes and sharp embeddings of function spaces

Later on we shall work with measure spaces [R, µ] and function spaces X such that there is a representative in the equivalence class of EGX which is continuous near 0. In particular, for our purpose it was sufficient to obtain ¡ ¢ ¡ ¢ EGX 2−j ∼ EGX 2−j+1

(3.8)

for some j0 ∈ N and all j ≥ j0 . In the case of [R, µ] = [Rn , `n ] we can, for instance, assume that X additionally satisfies ° ³ 1 ´ ° ° ° °f 2− n · |X ° ≤ c kf |Xk

(3.9)

for some c > 0 and all f ∈ X; this generalises [Tri01, (12.38)]. The monotonicity of EGX , see Proposition 3.4(i), immediately yields “≥” in (3.8), 1 whereas the converse inequality uses functions fn (x) := f (2− n x) built upon ∗ ∗ f ∈ X, say, with kf |Xk ≤ 1. Plainly fn (2t) = f (t), the rest is covered 1 by (3.9). Of course, any number λ0 < 1 replacing 2− n in (3.9) would do, modifying (3.8) appropriately. Lemma 3.8 Let the measure space [R, µ] and the function space X be such that for any f ∈ X there is some fµ ∈ X with fµ∗ (2−j+1 ) = f ∗ (2−j ) for some j0 ∈ N0 and all j ≥ j0 , and kfµ |Xk ≤ c kf |Xk. Then (3.8) holds for all j ≥ j0 , ¡ ¢ ¡ ¢ EGX 2−j ∼ EGX 2−j+1 .

P r o o f : The proof copies the idea of [Rn , `n ] with fµ = fn described above. Proposition 3.4(i) implies “≥” in (3.8), whereas we conclude from ¡ ¢ EGX 2−j =

sup kf |Xk≤1

¡ ¢ f ∗ 2−j ≤ c

sup kfµ |Xk≤1

¡ ¢ ¡ ¢ fµ∗ 2−j+1 ≤ c EGX 2−j+1

the converse estimate.

Remark 3.9 We shall need the above assertion only in connection with the continuity of (a function equivalent to) EGX near 0. Assume, for instance, that the measure µ has the property that µ(τ A) = hµ (τ ) µ(A) for small τ and A in the σ-algebra of R. Then we can put fµ such that fµ (x) := f (τ0 x), if τ0 satisfies hµ (τ0 ) = 21 . (In case of [R, µ] = [Rn , `n ] this refers to 1 hµ (τ ) = τ n and thus τ0 = 2− n .) Consequently we had to pose the restriction on X that kf (τ0 ·) |Xk ≤ c kf |Xk for all f ∈ X, see (3.9).

The growth envelope function EG

3.2

45

Examples: Lorentz spaces

For convenience we adopt the following notation. Here and in the sequel we shall mean by Im(µ) = [0, µ(R)] that the range of µ is the whole interval [0, µ(R)], i.e., that for every number s ∈ [0, µ(R)] there is some As ⊂ R in the σ-algebra of R with µ(As ) = s. We start with a preparatory lemma as the “extremal” functions below will be often used in the sequence. So it appears convenient to extract this argument from the subsequent considerations. In the sequel, let ¯ ¯ © ª Kr (x0 ) = x ∈ Rn : ¯x − x0 ¯ < r (3.10) always stand for the open ball with radius r > 0 centred at x0 ∈ Rn . Lemma 3.10 Let 0 < s < µ(R) and As ⊂ R with µ(As ) = s. (i) Let 0 < r ≤ ∞, and 1 fs = s− r χA . s

Then

1

fs∗ (t) = s− r χ[0,s) (t),

t > 0,

(3.11) (3.12)

and for 0 < p, q ≤ ∞, 1

1

kfs |Lp,q k ∼ s p − r .

(3.13)

(ii) Let [R, µ] = [Rn , | · |], and As = Kcs1/n (0), where c > 0 is suitably chosen such that µ(As ) = |Kcs1/n (0)| = s. Let for 0 < r < ∞, κ > 0, γ ∈ R, fs,κ (x) = s−( r −κ ) |x|−nκ (1 + |log |x||) 1

−γ

χA (x), s

x ∈ Rn .

(3.14)

Then ∗ fs,κ (t) ∼ s−( r −κ ) t−κ (1 + | log t|) 1

−γ

χ[0,s) (t),

t > 0,

(3.15)

and for 0 < p < ∞, 0 < q ≤ ∞, a ∈ R, and κ such that 0 < κ < p1 , 1

1

kfs,κ |Lp,q (log L)a k ∼ s p − r (1 + | log s|)

a−γ

,

s > 0.

(3.16)

(iii) Let [R, µ] = [Rn , | · |], and As = Kcs1/n (0) with µ(As ) = |Kcs1/n (0)| = s for appropriately ³chosen´c > 0. Let for 0 < s < 1, 0 < q < ∞, a, κ ∈ R, with 0 < κ < − a + 1q , fs,κ (x) = (1 + | log s|)

κ

−(a+ q1 +κ)

|log |x||

χA (x), s

x ∈ Rn ;

(3.17)

46

Envelopes and sharp embeddings of function spaces then κ

∗ fs,κ (t) ∼ (1 + | log s|) |log t|

−(a+ q1 +κ)

χ[0,s) (t),

t > 0,

(3.18)

and kfs,κ |L∞,q (log L)a k ∼ 1,

0 < s < 1.

(3.19)

(iv) Assume [R, µ] = [K% (0), | · |] with |K% (0)| = 1, and As = Kcs1/n (0) with µ(As ) = |Kcs1/n (0)| = s for appropriately chosen c > 0 and 0 < s < 1. Let for a ≥ 0, a

hs (x) = |log |x|| then

χA (x),

x ∈ R;

s

a

h∗s (t) ∼ |log t| χ[0,s) (t),

(3.20)

t > 0,

and khs |Lexp,a k ∼ 1.

(3.21)

P r o o f : The first assertion in (i) is obvious, see Example 2.4; for the second one note that  q1  q1  s  s Z Z h i q 1 1 1 1 q dt 1  = s− r  t p −1 dt ∼ s p − r , s > 0. kfs |Lp,q k ∼  t p s− r t 0

0

We turn to (ii). Again, straightforward calculation leads to (3.15), so that we can further conclude by (2.13), kfs,κ |Lp,q (log L)a k ∼ s−(

1 r −κ

 s  q1 Z 1 )  t( p −κ )q−1 (1 + | log t|)(a−γ)q dt 0

∼ s−(

1 r −κ

) s(

1 p −κ−ε

 s  q1 Z ) (1 + | log s|)(a−γ)  tεq−1 dt 0

∼s

1 1 p−r

(1 + | log s|)

(a−γ)

,

s > 0,

for 0 < ε < p1 − κ and with obvious modification for q = ∞. Assertion (3.18) follows directly from our assumptions on the parameters, and again by (2.13) for small s, 1 Zs h iq dt q 1 a −(a+ q +κ) κ  (1 + | log t|) |log t| kfs,κ |L∞,q (log L)a k ∼ (1 + | log s|)  t 

0

κ

∼ (1 + | log s|) (1 + | log s|)

−κ

∼ 1.

The growth envelope function EG

47

It remains to check (iv); recall (2.18), (2.19). Observe that Z Z 1/a 1/a exp (λ|hs (x)|) µ(dx) = |x|−λ µ(dx) + µ(R \ As ) R

As 1/n cs Z 1/a

≤ c0

rn−1−λ

dr + 1 − s ≤ C

0 a

for any λ < na , i.e., hs ∈ Lexp,a . Moreover, h∗s (t) ∼ |log t| χ[0,s) (t) and thus by (2.19), khs |Lexp,a k ∼ sup (1 − log t)−a |log t|

a

∼ 1.

0
Remark 3.11 In general, dealing with function spaces X (of distributions) we shall stick with the assumption X ⊂ Lloc 1 as mentioned above. However, in case of Lorentz (-Zygmund) spaces as given by Definition 2.8 we may incorporate parameters p, q ≤ 1, always keeping in mind this slight abuse of notation. Proposition 3.12 Let [R, µ] be a σ-finite measure space with Im(µ) = [0, µ(R)] or a finite non-atomic measure space. Then we obtain for Lp,q = Lp,q (R), 0 < p, q ≤ ∞ (with q = ∞ when p = ∞), 1

L

EG p,q (t) ∼ t− p ,

0 < t < µ(R).

(3.22)

P r o o f : Step 1. We first prove the assertion for 0 < p < ∞, 0 < q ≤ ∞. Assume f ∈ Lp,q , kf |Lp,q k ≤ 1, and let τ > 0 be such that τ < µ(R); then by (2.12), 1/q  τ 1/q Z Zτ h iq dt q 1  ≥ f ∗ (τ )  t p −1 dt 1 ≥ kf |Lp,q k ≥  t p f ∗ (t) t 

µ = f ∗ (τ )

0

0

p pq τ q

¶1/q , 1

and this implies f ∗ (τ ) ≤ cpq τ − p for any τ , 0 < τ < µ(R). Thus 1 L EG p,q (t) ≤ C t− p , 0 < p < ∞, 0 < q ≤ ∞. Step 2. We show the converse inequality for 0 < p < ∞, 0 < q ≤ ∞. Let 0 < s < µ(R) and As ⊂ R with µ(As ) = s. The existence of such an As is a consequence of our assumption: while using the first alternative (concerning the range of µ) it is obvious, otherwise, since µ(R) < ∞ and

48

Envelopes and sharp embeddings of function spaces

Definitions 2.11(vi) and 2.12 then imply that χR belongs to the µ-measurable functions finite µ-a.e., Remark 2.7 with f = χR yields the existence of such a set As for every s, 0 ≤ s ≤ µ(R). We apply Lemma 3.10(i) with r = p; then fs ∈ Lp,q with kfs |Lp,q k ∼ 1, and 1

L

1

EG p,q (t) ≥ sup fs∗ (t) = sup fs∗ (t) ∼ sup s− p ∼ t− p , s>t

s>0

0 < t < µ(R).

s>t

Step 3. It remains to verify (3.22) for p = q = ∞. Obviously (3.22) with p = q = ∞ is to be understood in the sense that there are constants c2 > c1 > 0 such that for all t > 0 c1 ≤ EGL∞ (t) ≤ c2 . It is clear by Definition 3.1 and Proposition 3.4(iii), that EGL∞ (t) ≤ c. Conversely, we apply (3.11) with r = ∞; then fs = χA ∈ L∞ , s > 0, s

kfs |L∞ k = 1, and for t > 0, EGL∞ (t) ≥ sup fs∗ (t) = sup 1 = 1. s>0

s>t

Remark 3.13 We return to Remark 3.5. One can easily calculate that for, say, 0 < p, q < ∞, µ ¶ q1 1 q L t− p EG p,q (t) = p with fixed k·|Lp,q k now as given in (2.12). Consider spaces Lp,q and Lp,r , where 0 < p, q, r < ∞. Assuming that some counterpart of Proposition 3.4(iii) was true, i.e., that the existence of some positive c > 0 such L L that EG p,q (t) ≤ c EG p,r (t) for small t > 0 led to Lp,q ,→ Lp,r , we thus had to verify that there is some c > 0 such that for all 0 < p, q, r < ∞ satisfying 1/q 1/r (q/p) ≤ c (r/p) it follows r ≥ q; see also Figure 10.

1 f (x) =

0

2

4

³ ´1/x x p

6

8

, p=2

10

Figure 10 1/q

The converse, however, is true: for all c > 0 there are p, q, r with (q/p) ≤ 1/r c (r/p) and r < q (given some c > 0, choose p with c > e−1/pe , r = pe, and q > pe sufficiently large).

The growth envelope function EG

³ Let Lp,q

49 h i [0, ∞) be the Lorentz space with respect to [R, µ] = [0, ∞), | · | . ´

Corollary 3.14 Let 0 < p, q ≤ ∞ (with q = ∞ if p = ∞). Then ³ ´ L EG p,q (·) ∈ Lp,q [0, ∞) if, and only if, q = ∞.

(3.23)

P r o o f : This follows immediately from (3.22), Definition (2.12), and Proposition 3.4(i). L

We denote by EGp,q;a = EG p,q convenience.

(log L)a (R)

, 0 < p < ∞, 0 < q ≤ ∞, a ∈ R, for

Proposition 3.15 Let [R, µ] be a σ-finite measure space with [0, µ(R)] or a finite non-atomic measure space. Then 1

−a

EGp,q;a (t) ∼ t− p (1 + | log t|)

,

0 < t < µ(R),

Im(µ) =

(3.24)

for 0 < p < ∞, 0 < q ≤ ∞, a ∈ R. Proof:

Step 1. First let q = ∞; thus kf |Lp,∞ (log L)a k ≤ 1 implies 1

−a

f ∗ (t) ≤ c t− p (1 + | log t|) for any 0 < t < µ(R). Consequently, 1 −a p,∞;a −p EG (t) ≤ c t (1 + | log t|) . On the other hand, Lp,q (log L)a ,→ Lp,∞ (log L)a , 0 < q ≤ ∞, which together with Proposition 3.4(iv) leads to 1 −a EGp,q;a (t) ≤ c t− p (1 + | log t|) , 0 < t < µ(R). Step 2. For simplicity we shall first describe the setting in [R, µ] = [Ω, | · |], where Ω ⊆ Rn is such that it contains As = Kcs1/n (0). We use the construction (3.14) with r = p and 0 < κ < p1 ; then by Lemma 3.10(ii) fs,κ ∈ Lp,q (log L)a , kfs,κ |Lp,q (log L)a k ∼ 1, and ∗ EGp,q;a (t) ≥ sup fs,κ (t) ∼ t−κ (1 + | log t|) s>0

∼t

1 −p

−a

(1 + | log t|)

−a

sup s−( p −κ ) 1

s>t

.

This proves (3.24) when [R, µ] = [Ω, | · |]. Step 3. In the general case [R, µ] we can construct the counterpart of fs,κ in (3.14) by some limit procedure arising from simple functions. For 0 < s < µ(R), m ∈ N, let gm (x) :=

m X k=1

ak χAm,s (x), k

50

Envelopes and sharp embeddings of function spaces

where we assume that the coefficients satisfy a1 > a2 > · · · > am > 0, and m S that the sets Am,s are pairwise disjoint subsets of R with Am,s = As k k and s =

m P k=1

k=1

µ (Am,s k ).

In particular, put −κ

ak ∼ [kµ (Am,s k )]

−a

|log (kµ (Am,s k ))|

s and µ (Am,s k ) ∼ m , k = 1, . . . , m. For the monotonicity of {ak } one might have to choose κ properly and s sufficiently small. Then

mk =

k X

µ (Am,s ) ∼ k i

i=1

s ∼ kµ (Am,s k ), m

and we obtain by Example 2.4 that ∗ (t) gm

=

m X k=1

ak χ[m

k−1 ,mk )

(t),

that is, for t ∼ kµ (Am,s k ), −κ

∗ (t) ∼ ak ∼ [kµ (Am,s gm k )]

−a

|log (kµ (Am,s k ))|

∼ t−κ |log t|

−a

.

Now a limit procedure m → ∞ leads to the function g(x) = lim gm (x) m→∞ on As , and finally fs,κ (x) := s−( p −κ ) g(x)χA (x), 1

s

x ∈ R,

is the desired counterpart of (3.14). ³ ´ Let Lp,q (log L)a [0, ε) be the Lorentz-Zygmund space with respect to h i [R, µ] = [0, ε), | · | . Then (3.24), Definition (2.13) and Proposition 3.4(i) imply the following counterpart of Corollary 3.14. Corollary 3.16 Let 0 < p < ∞, 0 < q ≤ ∞, a ∈ R. Then ³ ´ EGp,q;a (·) ∈ Lp,q (log L)a [0, ε) if, and only if, q = ∞.

(3.25)

We deal with the case p = ∞ now. Proposition 3.17 Let [R, µ] be a σ-finite measure space with Im(µ) = [0, µ(R)] or a finite non-atomic measure space. Then we obtain for a ∈ R, 0 < q < ∞, with a + 1/q < 0, and L∞,q (log L)a = L∞,q (log L)a (R), −(a+ q1 )

EG∞,q;a (t) ∼ (1 + | log t|) where ε ≤ min (1, µ(R)).

,

0 < t < ε,

(3.26)

The growth envelope function EG

51

P r o o f : Step 1. Let f ∈ L∞,q (log L)a with kf |L∞,q (log L)a k ≤ 1 . By (2.13) and the monotonicity of f ∗ this implies for any number τ , 0 < τ < µ(R), that 1/q  τ 1/q  τ Z Z dt dt q aq a   ≤  [(1 + | log t|) f ∗ (t)] f ∗ (τ )  (1 + | log t|) t t 0



0

1/q

Z∞ a

≤

q

[(1 + | log t|) f ∗ (t)] 0

dt  t

≤ 1,

i.e.,, since a + 1/q < 0, −1/q  τ Z dt −(a+ q1 ) aq  ≤ c (1 + | log τ |) . f ∗ (τ ) ≤  (1 + | log t|) t 0

Step 2. Let 0 < s < 1, 0 < κ < −(a + 1q ); for simplicity we only describe the setting in [R, µ] = [Rn , | · |]. Then by Lemma 3.10(iii), fs,κ ∈ L∞,q (log L)a , κ

∗ with fs,κ (t) ∼ (1 + | log s|) |log t| ∗ EG∞,q;a (t) ≥ sup fs,κ (t)

kfs,κ |L∞,q (log L)a k ∼ 1, −(a+ q1 +κ)

χ[0,s) (t). Hence,

−(a+ q1 +κ)

∼ |log t|

= |log t|

−(a+ q1 +κ)

sup (1 + | log s|)

κ

t<s<1

0<s<1

κ

(1 + | log t|)

∼

(1 + | log t|)

−(a+ q1 )

for 0 < t < 1. Recall L∞,∞ (log L)−a = Lexp,a for a ≥ 0 and µ(R) < ∞. Proposition 3.18 Let [R, µ] be a non-atomic finite measure space, µ(R) = 1, and a ≥ 0. Then L

a

EG exp,a (t) ∼ (1 − log t) ,

0 < t < 1.

(3.27)

P r o o f : Note that the case a = 0 is covered by Proposition 3.12; thus we assume a > 0 now. Moreover, by (2.19) we see that kf |Lexp,a k ≤ 1 implies L a a f ∗ (t) ≤ c (1 − log t) . Hence EG exp,a (t) ≤ c (1 − log t) . Conversely, consider the functions hs given by Lemma 3.10(iv); thus for 0 < t < 1, L

EG exp,a (t) ≥

a

a

sup h∗s (t) ≥ c sup |log t| ∼ |log t| ,

0<s<1

s>t

52

Envelopes and sharp embeddings of function spaces

as desired. ³ ´ h i Let Lexp,a [0, 1] be the exponential space with respect to [R, µ] = [0, 1], |·| . Corollary 3.19 Let a ≥ 0. Then ³ ´ L EG exp,a (·) ∈ Lexp,a [0, 1] .

(3.28)

P r o o f : This is an immediate consequence of (3.27), (2.19) and Proposition 3.4(i).

3.3

Connection with the fundamental function

In rearrangement-invariant function spaces X one has the concept of the “fundamental function” ϕX ; we now investigate its connection with the growth envelope function EGX as defined above. For convenience we assume in this section that all function spaces are considered over the measure space [Rn , `n ], i.e., Rn equipped with the (n-dimensional) Lebesgue measure `n . We closely follow the presentation in [BS88, Ch. 2, §5]. Recall the notion of a Banach function quasi-norm as presented in Definition 2.11. A function quasi-norm % over a measure space [R, µ] is said to be rearrangement-invariant, if %(f ) = %(g) for every pair of equi-measurable functions f and g in M+ that are finite µ-a.e., i.e., if µf (λ) = µg (λ) for all λ ≥ 0 implies %(f ) = %(g). A (quasi-) Banach function space X generated by a rearrangement-invariant (quasi-) norm % is called a rearrangementinvariant (quasi-) Banach function space or simply a rearrangement-invariant space. Note that by Definitions 2.11(vi) and 2.12 we have χA ∈ X for all A ⊂ R with µ(A) < ∞. Definition 3.20 Let X be a rearrangement-invariant Banach function space over [Rn , `n ]. For each t > 0, let At ⊂ Rn be such that `n (At ) = t, and let ° ¯ ° ° ° ϕX (t) = °χA ¯X ° . (3.29) t

The function ϕX so defined is called the fundamental function of X. Note that the particular choice of the set At with `n (At ) = t is immaterial since if Bt is any other subset Bt ⊂ Rn with `n (Bt ) = t, then χA and χB are t

t

The growth envelope function EG

53

equi-measurable, and so kχA |Xk = kχB |Xk because of the rearrangementt t invariance of X. Hence ϕX is well-defined. We start with some well-known examples. Let 1 ≤ p ≤ ∞, and Lp = Lp (Rn ); then ( ) 1 ,1≤p<∞ tp ϕLp (t) = , t ≥ 0, χ(0,∞) (t), p=∞ cf. [BS88, p. 65]. Moreover, when 1 ≤ q ≤ p < ∞ or p = q = ∞, then Lp,q 1 is rearrangement-invariant and ϕLp,q (t) = t p , see [BS88, Ch. 4, Thm. 4.3]. (In view of Remark 2.9 one can further prove that Lp,q is a rearrangementinvariant Banach space for 1 < p < ∞, 1 ≤ q ≤ ∞, or p = q = ∞, when f ∗ in (2.12) is replaced by f ∗∗ ; cf. [BS88, Ch. 4, Thm. 4.6].) Likewise, let Ω ⊂ Rn have finite measure, say, `n (Ω) = 1. Then it is known that L1 (log L)1 (Ω) and Lexp,1 (Ω) are rearrangement-invariant with fundamental functions ϕL1 (log L)1 (t) = t (1 + | log t|) , and ϕLexp,1 (t) = (1 + | log t|)

−1

,

0 < t < 1, 0 < t < 1,

see [BS88, Ch. 4, Thm. 6.4]. So in view of Propositions 3.12, 3.15 and 3.18 the following assertion seems quite reasonable. Proposition 3.21 Let X be a rearrangement-invariant Banach function space over [Rn , `n ], and ϕX the corresponding fundamental function. Then EGX (t) =

1 , ϕX (t)

t > 0.

(3.30)

P r o o f : Step 1. Let t > 0 and At ⊂ Rn be such that `n (At ) = t. Then (by our remarks above) gt := χA /ϕX (t) ∈ X, kgt |Xk = 1, and t

gt∗ (s)

=

(χA )∗ (s) t

ϕX (t)

=

χ[0,t) (s) ϕX (t)

.

Note that by [BS88, Ch. 2, Cor. 5.3] we have ϕX (t) = 0 if, and only if, t = 0. Moreover, ϕX is continuous and increasing; thus EGX (s) =

sup kf |Xk≤1

f ∗ (s) ≥ sup gt∗ (s) = sup t>0

t>0

χ[0,t) (s) ϕX (t)

= sup t>s

1 1 = . ϕX (t) ϕX (s)

Step 2. It remains to prove the converse inequality. The rearrangementinvariance of X implies that for any g ∈ X 0 , X 0 being the associate space to X, its norm is given by Z∞ 0

kg|X k =

f ∗ (s)g ∗ (s)ds,

sup kf |Xk≤1

0

54

Envelopes and sharp embeddings of function spaces

see [BS88, Ch. 2, Cor. 4.4]. Let again t > 0 and At ⊂ Rn be such that `n (At ) = t. Hence for g = χA ∈ X 0 (note that X 0 is also rearrangementt invariant with that norm and thus χA ∈ X 0 , too, for any At ⊂ Rn ) we t obtain  t  Z ° ° ¯ ° ° (3.31) ϕX 0 (t) = °χA ¯X 0 ° = sup  f ∗ (s)ds ≥ t sup f ∗ (t), t

kf |Xk≤1

kf |Xk≤1

0

where we used the monotonicity of f ∗ (t) again. Thus (3.31) implies ϕX 0 (t) ≥ t EGX (t). On the other hand, [BS88, Ch. 2, Thm. 5.2] provides ϕX (t) ϕX 0 (t) = t, leading to EGX (t) ≤ ϕ 1(t) for all t > 0. X

Remark 3.22 One can prove a counterpart of Proposition 3.21 when the underlying measure space [R, µ] = [Rn , `n ] is replaced by some non-atomic finite measure space [R, µ]. Carro, Pick, Soria and Stepanov studied related questions in [CPSS01]; in particular, [CPSS01, Rem. 2.5(ii)] essentially coincides with (3.30), where their function %X (t) corresponds to EGX (t). Moreover, when X is a rearrangementinvariant Banach function space, there is a counterpart of Proposition 3.4(iii) in [CPSS01, Thm. 2.8(iii)]: X ,→ Lq,∞

⇐⇒

1

sup t q EGX (t) < ∞,

0 < q < ∞.

t>0

A further property of the fundamental function ϕX is its quasi-concavity by which the following is meant: A non-negative function ϕ defined on R+ is called quasi-concave, if ϕ(t) is increasing on (0, ∞), ϕ(t) = 0 if, and is decreasing on (0, ∞); see [BS88, Ch. 2, Def. 5.6]. only if, t = 0, and ϕ(t) t Observe that every non-negative concave function on R+ , that vanishes only at the origin, is quasi-concave; the converse, however, is not true. However, any quasi-concave function ϕ is equivalent to its least concave majorant ϕ, e cf. [BS88, Ch. 2, Prop. 5.10]. Corollary 3.23 Let X be a rearrangement-invariant Banach function space over [Rn , `n ], put ψG (t) = t EGX (t), t > 0. (3.32) (i)

The function ψG (t) is monotonically increasing in t > 0.

(ii) If lim ψG (t) = 0, then ψG (t) is equivalent to some concave function t↓0

for t > 0. (iii) The growth envelope function EGX (t) is equivalent to some convex function for t > 0.

The growth envelope function EG Proof:

55

Part (i) follows immediately from Proposition 3.21 and a correϕ (t)

is decreassponding result for the fundamental function, stating that Xt ing, cf. [BS88, Ch. 2, Cor. 5.3]. As for (iii), we know that ϕ is quasi-concave; thus by the above-mentioned result it is equivalent to some concave function: hence application of Proposition 3.21 yields that 1/EGX (t) is equivalent to some concave function for t > 0. One verifies that EGX (t) is equivalent to some convex function on (0, ∞) then. Finally, (ii) is a consequence of (i), Proposition 3.4(i) and the general statements on concave and quasi-concave functions as repeated above.

Remark 3.24 The question naturally arises whether the rearrangement-invariance of X is really necessary or to what extent this assumption can be weakened (not to mention extensions of [R, µ] = [Rn , `n ] at the moment). In all the cases we studied, i.e., spaces of type Lp,q (log L)a and Asp,q (postponed to Part II), respectively, we obtain the above-described behaviour of EGX and is satisfied (incorporating in a slight abuse of ψG whenever X ⊂ Lloc 1 notation the case of constant functions ψG in (i), too; then also X = L1 with EGX (t) ∼ t−1 and thus ψG (t) ∼ 1 is covered): functions of type µ

EGX (t) ∼ t−κ |log t| ,

t>0

small,

with 0 < κ < 1, µ ∈ R, or κ = 0, µ > 0, lead to functions ψG (t) clearly satisfying Corollary 3.23 (with the above-mentioned extension to κ = 1, µ ≤ 0). On the other hand, as we did not observe a direct application of (an extended version of) Corollary 3.23 so far, we postpone the study of what happens when X is not rearrangement-invariant.

3.4

Further examples: spaces

Sobolev spaces, weighted Lp -

Returning to our starting point, Sobolev’s famous embedding result Theorem 2.28, we study (classical) Sobolev spaces Wpk , 1 ≤ p < ∞, k ∈ N, now. Moreover, as we intend to compare local and global behaviour of EGX (t), i.e., for 0 < t < 1, or t → ∞, respectively, in Section 10.3, we prepare this a little and briefly deal with some weighted spaces Lp (w). For convenience we retain the setting [R, µ] = [Rn , `n ] from the last section and shall always assume Ω = Rn unless otherwise stated.

56

Envelopes and sharp embeddings of function spaces

In view of Theorem 2.28(i) we have Wpk ,→ L∞ for k > np , 1 ≤ p < ∞, or k = n and p = 1, such that by Proposition 3.4(iii) the corresponding growth envelope function is bounded. Hence we are left to study the following cases now: ( ) n , 1 ≤ p k < < ∞ p Wpk 6,→ L∞ if . (3.33) k = np , 1 < p < ∞ We start with the sub-critical case k <

n p.

Proposition 3.25 Let 1 ≤ p < ∞, n ≥ 2, k ∈ N0 , with k < Wpk

EG

k

1

(t) ∼ t− p + n ,

n p.

Then

0 < t < 1.

(3.34)

P r o o f : By Wp0 = Lp the case k = 0 is covered by Proposition 3.12. Moreover, Sobolev’s famous embedding result (2.45) immediately yields Wpk

EG

(t) ≤ c EG p (t) ≤ c0 t−( p − n ) , L

1

∗

k

0 < t < 1,

with p1∗ = p1 − nk , applying Propositions 3.4(iv) and 3.12. Hence it remains to prove the converse inequality. Let R > 0 and consider functions ¡ ¢ n fR (x) = Rk− p ψ R−1 x , x ∈ Rn , (3.35) where ψ(x) is some compactly supported C ∞ -function in Rn , e.g., as given by Example 2.6, ( 1 − 1−|x| 2 , |x| < 1, ψ(x) = e (3.36) 0 , |x| ≥ 1. At the moment we may restrict ourselves to small R, 0 < R < 1. Clearly, by the above construction, ¡ ¢ n Dα fR (x) = Rk− p R−|α| (Dα ψ) R−1 x , α ∈ Nn0 , (3.37) and thus ° ° ¡ ¢ n kDα fR |Lp k = Rk− p −|α| °Dα ψ R−1 · |Lp ° = Rk−|α| kDα ψ|Lp k .

(3.38)

Consequently, for 0 < R < 1, ³ X ´1/p ° ° ° ° p °fR |Wpk ° ≤ R(k−|α|)p kDα ψ|Lp k ≤ °ψ|Wpk ° , |α|≤k

° ° and gR := kψ|Wpk k−1 fR ∈ Wpk , °gR |Wpk ° ≤ 1. On the other hand, by (2.9) and Proposition 2.3, n

∗ gR (t)

¡ ¢ Rk− p ψ ∗ R−n t . = kψ|Wpk k

(3.39)

The growth envelope function EG

57

Let 0 < t < 1 and choose R0 = R0 (t) = d t1/n such that i.e.,

R0−n t

³

t |ωn |

´1/n

< R0 < 1,

< |ωn | for appropriate d > 0. This finally leads to Wpk

EG

(t) ≥

sup 0
k− n p

∗ ∗ gR (t) ≥ gR (t) ≥ c R0 0

k

1

≥ c0 t n − p ,

completing the proof.

Remark 3.26 In Part II when we deal with spaces of Besov and Triebels s Lizorkin type, Bp,q and Fp,q , respectively, we shall stress arguments similar to those above and “extremal” functions of type (3.35) will recur and emerge as so-called “atoms” in the corresponding spaces. Moreover, slightly adapted functions of type (3.35) will be used to investigate global assertions, see Section 10.3, then for large values of R, R → ∞. Obviously, we have f ∈ Wpk ,→ Lp and hence, by Propositions 3.4(iv) and 1

3.12 immediately f ∗ (t) ≤ c t− p , t > 0. This results in a worse upper bound than (3.34) for small t (local singularities). In other words, Sobolev’s famous embedding theorem (2.45) reads in this context as the conclusion that the increased smoothness assertions imposed on f ∈ Wpk (compared with f ∈ Lp simply) lead to the reduction of admitted local singularities (unboundedness). We shall see in Section 10.3 that this is different from the global behaviour. Moreover, by Sobolev’s embedding (2.45) we only know Wpk ,→ Lr , 1r = p1 − nk , Wk

but the corresponding growth envelope functions EG p and EGLr even coincide – unlike the underlying spaces. We return to this observation in more general context in Remark 8.2. We deal with the case k =

n p,

1 < p < ∞ now. Clearly, by (2.44) and

Wpn/p

Proposition 3.4(iii), EG (t) cannot be bounded for t ↓ 0. On the other hand, results of Trudinger [Tru67], Yudovich [Yud61], Pohoˇzaev [Poh65], Moser [Mos71], Strichartz [Str72], Maz’ya [Maz72] yield (in our notation) Wpn/p (Ω) ,→ Lexp, 10 (Ω), p

1 < p < ∞,

(3.40)

for a bounded domain Ω ⊂ Rn , say, with |Ω| ≤ 1. This leads to the following result. Proposition 3.27 Let 1 < p < ∞ be such that Wpn/p

EG

1

(t) ∼ |log t| p0 ,

n p

= k ∈ N. Then

0
1 . 2

(3.41)

58

Envelopes and sharp embeddings of function spaces n/p

(Rn ). Then we have locally always ° ° 1 1 ° ° f ∗ (t) ≤ c °f |Wpn/p ° |log t| p0 , 0 < t < , 2

Proof:

Let f ∈ Wp

(3.42)

cf. [BW80, p. 787], so that the upper estimate immediately follows, Wpn/p

EG

1

(t) ≤ c | log t| p0 ,

1 < p < ∞,

0
1 . 2

(3.43)

For the converse, we need a refined version of (3.35), that is, fm (x) =

m−1 X

¡ ¢ ψ 2j x ,

m ∈ N,

(3.44)

j=0

and ψ given by (3.36). Then by (3.37) and k = (Dα fm ) (x) =

m−1 X

n p,

¡ ¢ 2j|α| (Dα ψ) 2j x ,

m ∈ N,

α ∈ Nn0 ,

j=0

so that for α ∈ Nn0 , |α| ≤ k,  kDα fm |Lp k ≤ 

m−1 X

1/p p 2j|α|p−jn kDα ψ|Lp k 

j=0

 ≤ c kDα ψ|Lp k 

m−1 X

1/p 2

jp(k− n p )

j=0

= c m1/p kDα ψ|Lp k . Consequently, ° ° ° ° ° ° ° ° °fm |Wpn/p ° ≤ c0 m1/p °ψ|Wpn/p ° ≤ C m1/p ,

m ∈ N.

(3.45)

On the other hand, ( ∗ fm (t)

∼

m

,

t ≤ 2−mn ,

(3.46)

|log t|, 2−mn ≤ t ≤ 1,

so that finally, for 0 < t < 21 , Wpn/p

EG

−1/p

∗ (t) ≥ c sup m−1/p fm (t) ≥ c m0 m∈N

1

∗ fm (t) ≥ c0 | log t|− p +1 0

where we have chosen m0 ∈ N such that t ∼ 2 2−m0 n , i.e., m0 ∼ | log t|.

The growth envelope function EG

59

Remark 3.28 In Part II, more precisely, in Theorem 8.16(i) we shall prove the counterpart of Proposition 3.27 in a more general setting. We come to weighted Lp -spaces now. There are essentially two reasonable ways to explain Lp (w) for some positive weight function w ∈ Lloc 1 , i.e., f ∈ Lp (w) ⇐⇒ wf ∈ Lp , or, alternatively, f ∈ Lp (w) ⇐⇒ w1/p f ∈ Lp . We shall stick to the first possibility, mainly because of historical reasons in connection with weighted spaces of Besov or Triebel-Lizorkin type, see Remark 3.29 below. Hence, ³Z ´1/p kf |Lp (w)k = kwf |Lp k = |f (x)|p w(x)p dx . (3.47) Rn

We start with some “admissible” weights of at most polynomial growth. More precisely, we shall mean by this the collection of all positive C ∞ functions w on Rn with the following properties: (i) for all γ ∈ Nn0 there exists a positive constant cγ with |Dγ w(x)| ≤ cγ w(x) for all

x ∈ Rn ,

(ii) there exist two constants c > 0 and α ≥ 0 such that for all x, y ∈ Rn , 0 < w(x) ≤ c w(y) hx − yiα . As a prototype we may take first wα (x) = hxiα ,

α ∈ R,

x ∈ Rn .

(3.48)

Remark 3.29 The study of such special weights has a little history, in particular in connection with spaces of Besov or Triebel-Lizorkin type as performed in Part II in detail. In [HT94a], [HT94b], [Har95], [Har97] we studied characterisations and compact embeddings of corresponding spaces, see also [ET96, Ch. 4] and [ST87] for a more general approach. Quite recently, there was a renewed interest in this topic leading to the series of papers [KLSS06a],[KLSS06b], [KLSSxx] and [HT05], [Skr05]. Applications are described in [HT94b]. Note that for α < −

n p

the spaces Lp (wα ) do not possess a growth envelope L (w )

function in the sense of Lemma 3.3(i), i.e., EG p α (t) is not finite for (small) t > 0. This can be seen by taking fβ (x) = hxiβ ∈ Lp (wα ), 0 < β < −α − np , ∗ such that (fβ ) (t) does not exist for t > 0. Plainly, when Rn is replaced by some bounded domain Ω ⊂ Rn , we have Lp (Ω, wα ) = Lp (Ω),

Ω ⊂ Rn , Ω bounded,

α ∈ R,

60

Envelopes and sharp embeddings of function spaces

and all considerations from the unweighted case apply. However, as already mentioned a few times, we are essentially interested in the situation when Ω = Rn at this moment. Proposition 3.30 Let 0 < p < ∞, α ≥ 0, then L (wα )

EG p Proof:

1

(t) ∼ t− p ,

0 < t < 1.

(3.49)

Note that wα (x) ≥ 1, x ∈ Rn , hence Lp (wα ) ,→ Lp ,

α ≥ 0,

(3.50) 1

L (w )

and Propositions 3.4(iv), 3.12 imply EG p α (t) ≤ t− p , t > 0. Conversely, using Lemma 3.10(i) with r = p, it is sufficient to show that kfs |Lp (wα )k ∼ 1 for small s, Z Z p kfs |Lp (wα )k = s−1 wα (x)p dx = s−1 hxiαp dx As

As

1/n

cs Z −1

=s

αp ¡ ¢ αp 1 + |x|2 2 dx ≤ c00 s−1 (1 + c0 ) 2 |As | ≤ C

0

as 0 < s < 1. In the same way as in the proof of Proposition 3.12 this leads 1 L (w ) to EG p α (t) ≥ t− p , 0 < t < 1, thus completing the proof. Remark 3.31 It is immediately clear that assertion (3.49) remains unchanged when wα is replaced by an arbitrary admissible weight w that is bounded from below, w(x) ≥ c > 0, x ∈ Rn ; one can either adapt the above proof appropriately or, even simpler, conclude that due to the admissibility of w, in particular (ii), there are constants c0 > c > 0 and α ≥ 0 such that c ≤ w(x) ≤ c0 hxiα ,

x ∈ Rn .

Thus Lp (wα ) ,→ Lp (w) ,→ Lp ,

(3.51)

and Propositions 3.4(iv), 3.12, 3.30 complete the argument. Finally, we illuminate another famous class of weights, the Muckenhoupt Ap weights. Recall that a function w ∈ Lloc 1 , w > 0 a.e., satisfies the Ap condition, 1 < p < ∞, if there is some constant A > 0 such that for all balls B in Rn ,  p/p0 Z Z 0 p 1 1 w(x)dx  w(x)− p dx ≤ A < ∞. |B| |B| B

B

The growth envelope function EG

61

For p = 1 the condition reads as w ∈ A1 if there is some A > 0 such that for all balls B in Rn and for a.e. x ∈ B, Z 1 w(x)dx ≤ A w(x), |B| B

and, finally, A∞ :=

[

Ap .

p>1

Remark 3.32 The class of such weights has been intensively studied in the past; we refer to [Muc72], [Muc74], and the monographs [GCR85], [Ste93, Ch. V] for a detailed account. Additionally we mention that in [Ryc01] there was introduced a more general concept of weight functions, Aloc p , containing both the above admissible weights as well as Ap weights. Again we content ourselves with an example and consider the counterpart to wα , wα (x) = |x|α , α ∈ R. (3.52) It is well-known that wα ∈ Ap , 1 < p < ∞, if, and only if, − np < α < cf. [Ste93, Ch. V, §6.4].

n p0 ,

Lemma 3.33 Let 1 ≤ u ≤ ∞, 1 < p < ∞, 0 ≤ α < pn0 , and p0 be given by 1 α 1 α p0 = p + n . Let w stand for w or wα , respectively. Then Lp,u (w) ,→ Lp0 ,u ,

(3.53)

Lp (w) ,→ Lp0 ,p .

(3.54)

and, in particular,

P r o o f : We combine H¨older’s inequality and real interpolation arguments for Lorentz spaces, (Lr0 ,q0 , Lr1 ,q1 )θ,p = Lr,p , (3.55) where 0 < θ < 1, 0 < r0 , r1 < ∞, r0 6= r1 , 0 < q0 , q1 , p ≤ ∞, and 1−θ θ 1 = + ; r r0 r1

(3.56)

this is the very classical interpolation result for Lorentz spaces, cf. [BL76, Thm. 5.3.1] and [Tri78a, Thm. 1.18.6/2]. In that way one can show that if 1 < p, q < ∞ with 0 < 1r = p1 + 1q < 1, and 1 ≤ u ≤ ∞, then Lp,u · Lq,∞ ⊂ Lr,u ,

(3.57)

62

Envelopes and sharp embeddings of function spaces

in the sense that whenever f ∈ Lp,u and g ∈ Lq,∞ , then f g ∈ Lr,p ; for a short proof see [Har98, Lemma 2.12]. Furthermore, w−1 ∈ Ln/α,∞ . Thus n , and thus r = p0 given by p10 = p1 + α with q = α n ; (3.57) implies ° ° ° ° kf |Lp ,u k = °f ww−1 |Lp ,u ° ≤ c kf w|Lp,u k °w−1 |L n ,∞ ° ≤ c0 kf |Lp,u (w)k , 0

0

α

i.e., Lp,u (w) ,→ Lp0 ,u . With u = p, (3.54) immediately follows. Remark 3.34 In Corollary 11.7 we complement the above embedding assertions by sufficient conditions for (3.54). Proposition 3.35 Let 1 < p < ∞, 0 ≤ α < L (wα )

EG p

α

n p0 ;

1

(t) ∼ t− n − p ,

then t > 0.

(3.58)

P r o o f : The upper estimate is an immediate consequence of (3.54) with w = wα and Propositions 3.4(iv), 3.12. For the lower one consider extremal functions of type (3.11) with r = p0 ; thus p

kfs |Lp (wα )k = s

− pp 0

Z wα (x)p dx ∼ s

− pp 0

1/n cs Z

³ 1 ´αp+n p ∼ c0 , |x|αp dx ∼ s− p0 s n

0

As

i.e., (up to possible normalising factors) we have kfs |Lp (wα )k ≤ 1. Hence, L (wα )

EG p

1

α

1

(t) ≥ sup fs∗ (t) ≥ c sup s− p0 ∼ t− n − p , s>0

s>t

t > 0,

where we additionally used (3.12). Remark 3.36 Note that the argument for the lower bound works for all α > − np . Similarly, by (3.54) we had the counterpart of the upper estimate for wα , too. But for small t > 0 this obviously leads to a weaker estimate than (3.49); however, for large numbers t it is not difficult to predict that – as in the case of wα in (3.58) – the number p0 (and thus α) will determine the L (w ) behaviour of EG p α (t), t → ∞. This discussion is postponed to Section 10.3. Comparing Propositions 3.30 and 3.35 it is clear that the influences of the locally regular weight wα and the Ap -weight wα , concerning local singularities in the underlying spaces, are different. As mentioned above, parameters α ≤ − np are not admitted for wα or wα , respectively, whereas the case − np < α < 0 is reasonable to consider. This will be done elsewhere.

Chapter 4 Growth envelopes EG

We shall need a finer characterisation than that provided by the growth envelope functions only. By Proposition 3.15 it is obvious, for instance, that EGX alone cannot distinguish between different spaces like Lp,q1 (log L)a and Lp,q2 (log L)a . So it is desirable to complement EGX by some expression, naturally belonging to EGX , but yielding – as a test – the number q (or a related quantity) in case of Lp,q (log L)a spaces. Again a more substantial justification for complementing EGX by this additional expression results from more coms s ) than Lp,q (log L)a ; but in these classical and Fp,q plicated spaces (like Bp,q cases the outcome can be checked immediately. The missing link is obtained by the introduction of some “characteristic” index uX G , which gives a finer measure of the (local) integrability of functions belonging to X. Moreover, the definition below is also motivated by (sharp) inequalities of type (1.4) with κ ≡ 1.

4.1

Definition

We start with some preliminaries. Let ψ be a real continuous monotonically increasing function on the interval [0, ε] for some small ε > 0. Assume ψ(0) = 0 and ψ(t) > 0 if 0 < t ≤ ε. Let µlog ψ be the associated Borel measure with respect to the distribution function log ψ; if, in addition, ψ is continuously differentiable in (0, ε) then µlog ψ (dt) =

ψ 0 (t) dt ψ(t)

(4.1)

in (0, ε); cf. [Lan93, p. 285] or [Hal74, §15(9)]. The following result of Triebel [Tri01, Prop. 12.2] is essential for our argument below.

Proposition 4.1 (i)

Let ψ and µlog ψ be as above, and 0 < r0 ≤ r1 < ∞. Then there are 63

64

Envelopes and sharp embeddings of function spaces numbers c2 > c1 > 0 such that  ε 1/r1 Z r sup ψ(t) g(t) ≤ c1  [ψ(t) g(t)] 1 µlog ψ (dt)

0
0

 ε 1/r0 Z r ≤ c2  [ψ(t) g(t)] 0 µlog ψ (dt)

(4.2)

0

for all functions g(t) ≥ 0, which are monotonically decreasing. (ii) Let ψ1 , ψ2 be two equivalent functions as above and µlog ψ1 , µlog ψ2 the corresponding measures. Assume 0 < r ≤ ∞. Then  ε 1/r  ε 1/r Z Z  [ψ1 (t)g(t)]r µlog ψ1 (dt) ∼  [ψ2 (t)g(t)]r µlog ψ2 (dt) (4.3) 0

0

(usual modification if r = ∞) for all functions g(t) ≥ 0, which are monotonically decreasing. In a slight abuse of notation we shall mean by µG the Borel measure associated with a function ψ (as described above and) equivalent to 1/EGX , where X is some function space satisfying (3.9) and X 6,→ L∞ ; that is, ψ(t) ∼ 1/EGX (t), 0 < t < ε. Note that the equivalence class of growth envelope functions EGX of a space X satisfying the assumptions of Lemma 3.8 contains a continuous representative. If EGX is continuously differentiable, then

µG (dt) ∼ −

³ ´0 EGX (t) EGX (t)

dt

(4.4)

for small t > 0. This approach coincides with that presented by Triebel in [Tri01, Sect. 12.1] and [Tri01, Sect. 12.8]. Recall our notation τ0 in (3.3). Definition 4.2 Let [R, µ] be a measure space and X 6,→ L∞ some (quasi-) normed function space on R satisfying the assumptions of Lemma 3.8, and let EGX be the corresponding growth envelope function. Assume 0 < ε < τ0 . X The index uX G , 0 < uG ≤ ∞, is defined as the infimum of all numbers v, 0 < v ≤ ∞, such that 1/v  ε" #v Z ∗ f (t)  µG (dt) ≤ c kf |Xk EGX (t) 0

(4.5)

Growth envelopes EG

65

(with the usual modification if v = ∞) holds for some c > 0 and all f ∈ X. Then ´ ¡ ¢ ³ EG X = EGX (·), uX (4.6) G is called the growth envelope for the function space X. Remark 4.3 It is clear by Proposition 3.4(v) (with κ ≡ 1) that (4.5) holds with v = ∞ in any case. Thus the question arises whether (depending upon the underlying function space X) there is some smaller v such that (4.5) is still satisfied. Moreover, it is reasonable to ask for the smallest parameter v satisfying (4.5) as the corresponding expressions on the left-hand side are monotonically ordered in v by Proposition 4.1(i) with g = f ∗ and ψ ∼ 1/EGX . The number uX G in Definition 4.2 is defined as the infimum of all numbers v satisfying (4.5); however, it is not clear at the moment, whether this infimum is in fact always a minimum. More precisely, one can study the question what assumptions (on the function space X and the underlying measure space) imply that uX G satisfies (4.5), too. So far we only know that all cases we studied (as presented below) are examples for the latter case (when uX G happens to be a minimum), but lack a general answer. Remark 4.4 Note that we explicitly excluded the case X ,→ L∞ (in particular, X = L∞ ) in Definition 4.2 above. The problem obviously arises from our notation (4.4). One may, however, adopt the (reasonable) opinion that – in case of bounded growth functions EGX (that is, according to Proposition 3.4(iii), when X ,→ L∞ ) – (4.5) is replaced by sup f ∗ (t) ≤ c kf |Xk , 0
for some c > 0 and all f ∈ X; thus uX G := ∞. The following assertion is not very complicated to prove but quite effective in application later on. Proposition 4.5 Let [R, µ] be a measure space, and Xi , i = 1, 2, (quasi-) normed function spaces on R with X1 ,→ X2 . Assume for their growth envelope functions EGX1 (t) ∼ EGX2 (t), 0 < t < ε. (4.7) i Then we get for the corresponding indices uX G , i = 1, 2, that 1 2 uX ≤ uX G G .

(4.8)

66

Envelopes and sharp embeddings of function spaces

P r o o f : In view of Remark 4.4 we need not assume Xi 6,→ L∞ , i = 1, 2, explicitly; when X2 ,→ L∞ , then X1 ,→ X2 in connection with Remark 4.4 X2 1 implies uX G = uG = ∞ which satisfies (4.8), too. Otherwise, when X1 ,→ L∞ , then by Proposition 3.4(iii) EGX1 (t) is bounded; hence by (4.7) EGX2 (t) is bounded, too, and another application of Proposition 3.4(iii) yields X2 ,→ L∞ resulting in the preceding argument. Let f ∈ X1 ,→ X2 ; we conclude by (4.7) together with Proposition 4.1 with g = f ∗ that à Zε " 0

f ∗ (t) EGX1 (t)

2 #u X G

2 !1/uX G 1 µX G (dt)

à Zε " ∼ 0

f ∗ (t) EGX2 (t)

2 #u X G

2 !1/uX G 2 µX G (dt)

≤ c kf |X2 k ≤ c0 kf |X1 k 1 for all f ∈ X1 . Thus, by definition of uX we obtain (4.8). The modificaG tions for the general case (i.e., when the infimum in (4.5) cannot be replaced by a minimum) are obvious.

Remark 4.6 We give another interpretation of the meaning of (4.5) in terms µ of sharp embeddings. Assume that EGX (t) ∼ t−α |log t| with α > 0, µ ∈ R, X or α = 0, µ > 0 (recall the monotonicity of EG near 0). Then µG (dt) ∼

dt t

if α > 0,

and

µG (dt) ∼

dt t | log t|

if α = 0,

and (4.5) can be reformulated as follows: What is the smallest space of type L α1 ,v (log L)−µ (α > 0) or of type L∞,v (log L)−(µ+ 1 ) (α = 0), respectively, v such that X can be embedded into it continuously? Having this idea in mind the results in Section 4.2 are not very astonishing. However, this is only some interpretation of (4.5); the definition itself is independent of any scale of Lorentz spaces as target spaces.

4.2

Examples: Lorentz spaces, Sobolev spaces

We give here our main result for those examples already considered in Sections 3.2 and 3.4. Theorem 4.7 Let [R, µ] be a σ-finite measure space with Im(µ) = [0, µ(R)] or a finite non-atomic measure space. (i)

Let 0 < p, q ≤ ∞ (with q = ∞ when p = ∞). Then ³ ´ ³ 1 ´ EG Lp,q = t− p , q .

(4.9)

Growth envelopes EG

67

(ii) Let 0 < p < ∞, 0 < q ≤ ∞, and a ∈ R. Then ³ ´ ³ 1 ´ −a EG Lp,q (log L)a = t− p |log t| , q . (iii) Let 0 < q < ∞, a ∈ R, with a +

1 q

< 0. Then

³ ´ ³ ´ −(a+ q1 ) , q . EG L∞,q (log L)a = |log t| Proof:

(4.10)

(4.11)

Step 1. We begin with (i). Assume first p < ∞. We know by 1

L

Proposition 3.12 that EG p,q (t) ∼ t− p , t > 0. Thus it remains to verify the index q according to (4.5); that is, we look for the smallest possible number v such that for some c > 0 and all f ∈ Lp,q , 1/v  ε Z h iv dt 1   ≤ c kf |Lp,q k . t p f ∗ (t) t

(4.12)

0

1

L

in that Note that by (4.4) and EG p,q (t) ∼ t− p we have µG (dt) ∼ dt t case. In view of (2.12) this question can be reformulated as follows: we ask for the smallest possible number v such that Lp,q ,→ Lp,v (at least locally). But here it follows immediately that v ≥ q, i.e., the least admitted number v is q. It remains to deal with the case p = ∞, but we may now refer to Remark 4.4. Step 2.

We care about (ii). Proposition 3.15 yields 1

−a

EGp,q;a (t) ∼ t− p (1 + | log t|)

1

∼ t− p |log t|

−a

,

0 < t < ε.

Together with (4.4) this implies µG (dt) ∼ dt t again. We thus look for possible numbers v such that 1/v  ε Z h i v 1 dt  a  ≤ c kf |Lp,q (log L)a k t p (1 + | log t|) f ∗ (t) t 0

for some c > 0 and all f ∈ Lp,q (log L)a . Again we may understand this in the sense that Lp,q (log L)a ,→ Lp,v (log L)a (locally) for some number v. However, this implies v ≥ q again, and (4.10) is proved. Step 3.

It remains to deal with (iii). We obtain from Proposition 3.17 that

EG∞,q;a (t) ∼ (1 + | log t|)

−(a+ q1 )

∼ |log t|

µG (dt) ∼

−(a+ q1 )

, 0 < t < ε. Consequently

1 dt 1 + | log t| t

68

Envelopes and sharp embeddings of function spaces

now. Hence we have to determine the smallest number v such that 1/v  ε Z h iv dt 1 1 a+ −   ≤ c kf |L∞,q (log L)a k (1 + | log t|) q v f ∗ (t) t 0

holds for some c > 0 and all f ∈ L∞,q (log L)a . We apply Proposition 2.10 with r = v, b = a + 1q − v1 , and get v ≥ q; this proves (4.11). Remark 4.8 As already announced in Remark 4.6, the above results were to be expected in view of the reformulation of (4.5). We thus obtained what we wanted – a method to recover the fine index q in case of Lorentz (-Zygmund) spaces Lp,q (log L)a . Moreover, with the help of Lemma 3.10 one could immediately construct counter-examples to disprove the existence of some v < p satisfying (4.12); similarly for the other cases. For the cases p, q ≤ 1 we refer to Remark 3.11. In view of Section 3.3 the question arises naturally whether we can also identify uX G as some quantity, known for a long time (and in possibly another context) in Banach space theory. By Theorem 4.7 one has to look for expressions only which take the value q when X = Lp,q (log L)a . Recall L∞,∞ (log L)−a = Lexp,a for a ≥ 0 and µ(R) < ∞. Proposition 4.9 Let [R, µ] be a non-atomic finite measure space, µ(R) = 1, and a ≥ 0. Then ³ ´ ³ ´ a EG Lexp,a = |log t| , ∞ . (4.13) L

a

P r o o f : We know by Proposition 3.18 that EG exp,a (t) ∼ (1 + | log t|) , 0 < t < 1. For which numbers v is there some c > 0 such that for all f ∈ Lexp,a , 1/v  ε Z h iv dt 1 −(a+ v ) ∗ −a   ≤ c sup (1 + | log t|) f ∗ (t) f (t) (1 + | log t|) t 0
is the question that remains. But obviously this leads to v = ∞ only, recall Proposition 2.10. Proposition 4.10 Let 1 ≤ p < ∞, n ≥ 2, k ∈ N0 , with k < ´ ¡ ¢ ³ 1 k EG Wpk = t− p + n , p .

n p.

Then

Growth envelopes EG Proof: that

69

In view of Proposition 3.25 and Definition 4.2 it remains to prove 1/v Zε h iv dt ° ° k 1   ≤ c °f |Wpk ° t p − n f ∗ (t) t 

(4.14)

0

holds for some c > 0 and all f ∈ Wpk if, and only if, v ≥ p. First we show that there cannot be some number v < p that guarantees (4.14) for all f ∈ Wpk ; this is done by contradiction using a refined construction of “extremal” functions based upon (3.35) with R = 2−j . ¯ Let ¯ {bj }j∈N be a 0 n ¯ 0¯ sequence of non-negative numbers, and let x ∈ R , x > 4, such that ¡ ¢ ¡ ¢ supp ψ 2j · −x0 ∩ supp ψ 2r · −x0 = ∅ for j 6= r, j, r ∈ N0 , and ψ given by (3.36). Then fb (x) :=

∞ X

¡ ¢ n 2−j(k− p ) bj ψ 2j x − x0 ,

x ∈ Rn ,

(4.15)

j=1

belongs to Wpk for b ∈ `p , and due to the disjointness of the supports, 1/p  ∞ X ° ° °fb |Wpk ° ≤ c  = c kb|`p k . bpj 

(4.16)

j=1

¡ ¢ n On the other hand, fb∗ c 2−jn ≥ c0 bj 2−j(k− p ) , j ∈ N0 . For convenience we may assume b1 = · · · = bJ−1 = 0, where J is suitably chosen such that 2−J ∼ ε, given by (4.14). Then by monotonicity arguments, 1/v   ε 1/v Z h ∞ h iv dt iv X ¡ ¢ k k 1 1   ∼ 2−jn( p − n ) fb∗ c 2−jn  t p − n fb∗ (t) t 0

j=J

1/v ∞ h iv X k 1 1 k ≥c  2−jn( p − n ) bj 2−jn( n − p )  

j=J

 ∼

∞ X

1/v bvj 

.

j=J

Together with (4.16) this implies kb|`p k ≥ c kb|`v k for arbitrary sequences of non-negative numbers. This obviously requires v ≥ p. For the converse we may use real interpolation and Sobolev’s embedding: by (2.45) we know Wpki ,→ Lqi ,

1 k 1 = − , i = 0, 1; qi pi n

70

Envelopes and sharp embeddings of function spaces

on the other hand, for 1 < p < ∞, ¡ k ¢ Wp0 , Wpk1 θ,p = Wpk

and

(Lq0 , Lq1 )θ,p = Lq,p ,

where 0 < θ < 1, 1 < q0 < q1 < ∞, 1 < p0 < p1 < ∞, k ∈ N, and 1−θ θ 1 = + q q0 q1

1 1−θ θ = + . p p0 p1

and

(4.17)

The Wpk -interpolation coincides with [Tri78a, 2.4.2/(10)]; the L-part is the very classical interpolation result for Lebesgue spaces, cf. [BL76, Thm. 5.3.1] and [Tri78a, Thm. 1.18.6/2]. Thus Wpk ,→ Lp∗ ,p ,

1 k 1 = − , ∗ p p n

1 < p < ∞,

(4.18)

see also [Pee66]. The case p = 1 can be incorporated using results by [Alv77b], [Alv77a],  r1  r1 ∞ Z h Z∞ h i i r r 1 1 1 dt  dt   ≤ c t p |∇u|∗ (t) , t p − n u∗ (t) t t 

(4.19)

0

0

for 1 ≤ r ≤ p < n and sufficiently smooth u ∈ Wp1 , see also [Tal94, Thm. 4B]. With p = r = 1 this corresponds to the case k = 1 < n, Z∞ 1

t1− n u∗ (t) 0

dt ≤ c t

Z∞

° ° |∇u|∗ (t)dt = k|∇u| |L1 k ≤ c °u|W11 °

0

and covers (4.14) with v = p = 1 and k = 1. We indicate how this result can be iterated to include k ∈ N, k ≥ 2. Assume u ∈ Wp2 , k = 2 < np , u sufficiently smooth; then |∇u| ∈ Wp1 , 1 ≤ p < n2 . Moreover, by (an iterated version of) Sobolev’s embedding theorem (2.45) we know that u ∈ Wp2 ,→ Wp1∗ with np . By our assumptions, 1 < p∗ < n such that we can apply (4.19) to p∗ := n−p u ∈ Wp1∗ (with p replaced by p∗ ) and obtain for 1 ≤ r ≤ p∗ < n, 1 1 ∞ Z h Z∞ h ir dt r ir dt r 1 1 1  ≤ c  ,  t p∗ |∇u|∗ (t) t p∗ − n u∗ (t) t t 

0

0

i.e., with r = 1, Z∞ t 0

2 1 p−n

dt u (t) ≤ c t

Z∞ 1

1

t p − n |∇u|∗ (t)

∗

0

dt , t

(4.20)

Growth envelopes EG

71

np < n. On the other hand, (4.19) (with r = 1) applied to for 1 ≤ n−p g = |∇u| ∈ Wp1 leads to

Z∞ t

1 1 p−n

0

dt ≤ c |∇u| (t) t

Z∞ 1

∗

t p |∇g|∗ (t) 0

dt , t

(4.21)

for 1 ≤ p < n. Consequently, (4.20) and (4.21) with p = 1 yield Z∞ t 0

2 1− n

dt u (t) ≤ c t

Z∞

∗

° ° ° ° |∇g|∗ (t) dt ≤ c0 °g|W11 ° ≤ c00 °u|Wp2 ° ;

0

This gives (4.14) with v = p = 1 and k = 2. Obviously this process can be iterated so as to cover (4.14) with v = p = 1 and all k ∈ N. Alternatively, one may use [KPxx, Thm. 4.2]. A combination of Theorem 4.7(i) and Proposition 4.5 finishes the proof. We continue Proposition 3.27 dealing with the case k =

n p.

Proposition 4.11 Let 1 < p < ∞ be such that np = k ∈ N. Then ´ ³ ´ ³ 1 EG Wpn/p = |log t| p0 , p .

Proof:

In view of Proposition 3.27 and µG (dt) ∼

1 dt , | log t| t

0
1 , 2

we have to show that 1/v  ε Z h ° ° iv dt 1 1 ° ° −( + )   ≤ c °f |Wpn/p ° , |log t| p0 v f ∗ (t) t

f ∈ Wpn/p ,

(4.22)

0

holds if, and only if, v ≥ p. Note that by a result of Hansson [Han79, (3.13)] we have in that case (locally) 1/p  ε Z · ∗ ¸p ° ° dt f (t) ° °   ≤ c °f |Wpn/p ° , (4.23) |log t| t 0

so that v ≤ p; see also Maz’ya [Maz72]. For the converse inequality we proceed by contradiction. Assume that (4.22) holds for some v < p and consider slightly modified functions of type (3.44), i.e., fm,b (x) =

m−1 X j=0

¡ ¢ bj ψ 2j x ,

m ∈ N,

(4.24)

72

Envelopes and sharp embeddings of function spaces

where ψ is given by (3.36) and bj ≥ 0, j = 0, . . . , m − 1. Then just as in (3.45),  1/p ° ° ° ° m−1 X X p ° ° ° ° bj  kDα fm,b |Lp k ≤ c °ψ|Wpn/p °  °fm,b |Wpn/p ° ≤ j=0

|α|≤k



m−1 X

≤ c0 

1/p bpj 

.

j=0

If we choose 1

bj = (j + 1)− p (1 + log(j + 1)) then

 m ° ° X ° ° °fm,b |Wpn/p ° ≤ c  j=1

− v1

,

j = 0, . . . , m − 1, 1/p

1 p/v



j (1 + log j)

≤ c0

(4.25)

uniformly in m ∈ N because p > v. On the other hand, for r = 1, . . . , m, r−1 X ¡ −rn ¢ 1 −1 ∗ fm,b c2 ≥ c bj ≥ c r br−1 = c r p0 (1 + log r) v ,

(4.26)

j=0

so that finally (4.22) implies for sufficiently large m ∈ N,  ε" #v  v1 #v ! v1 Ãm " Z ∗ ∗ ° ° X fm,b (c 2−rn ) f (t) dt ° m,b n/p °  ≥ c2 °fm,b |Wp ° ≥ c1  1 1 + v1 +1 t p0 |log t| r p0 v r=1 0 Ãm ! v1 X 1 ≥ c3 r (1 + log r) r=1 which does not converge for m → ∞ in contrast to the left-hand side. From this contradiction we conclude v ≥ p and the proof is complete.

Remark 4.12 This result reappears as a special case from Theorem 8.16(i) below. Finally we deal with the weighted Lp -spaces as introduced in Section 3.4. Proposition 4.13 Let 1 < p < ∞, 0 < α <

n p0 .

Then

³ 1 ´ EG (Lp (wα )) = t− p , p

(4.27)

Growth envelopes EG and

73

³ 1 α ´ EG (Lp (wα )) = t− p − n , p .

(4.28)

P r o o f : In view of Propositions 3.30 and 3.35 we only have to deal with the L (w ) L (wα ) indices uG p α and uG p . Moreover, (3.50), (3.54), Proposition 4.5, and Theorem 4.7(i) imply L (wα )

uG p

≤ p,

L (wα )

uG p

≤ p.

It is thus sufficient to verify the converse inequalities. We start with (4.27). Assume that there is some v < p such that for some c > 0 and all f ∈ Lp (wα ) 1/v  ε Z h iv dt 1   ≤ c kf |Lp (wα )k (4.29) t p f ∗ (t) t 0

By Lemma 3.10(ii), in particular (3.14), (3.15) with r = p, κ = p1 , γ ∈ R, we know that for arbitrary s > 0, and As as in Lemma 3.10(ii), n

fs,γ (x) = |x|− p (1 + |log |x||) with

1

∗ (t) ∼ t− p (1 + | log t|) fs,γ

−γ

−γ

χA (x), s

χ[0,s) (t),

x ∈ Rn , t > 0,

does not belong to Lp,v locally for γ ≤ v1 , as  v1   v1  ε min(ε,s) Z Z h i v dt 1 −γv dt  ∗  ∼  (t) (1 + | log t|) t p fs,γ   t t 0

0

diverges for γv ≤ 1. On the other hand, fs,γ ∈ Lp (wα ) for γ > 0 < s < 1, Z p −γp kfs,γ |Lp (wα )k = |x|−n (1 + |log |x||) hxiαp dx

1 p

and all

As

Z1 −γp

≤c

(1 + log r) 0

dr ≤ c0 . r

Thus choosing p1 < γ ≤ v1 and 0 < s < 1, fs,γ disprove (4.29) for v < p. Similarly we proceed in case of wα where (4.29) is replaced now by 1/v  ε Z h iv dt α 1 + ∗   ≤ c kf |Lp (wα )k . (4.30) t p n f (t) t 0

74

Envelopes and sharp embeddings of function spaces

Accordingly we take as counter-examples n

−γ

gs,γ (x) = |x|− p −α (1 + |log |x||) with

1

α

∗ gs,γ (t) ∼ t− p − n (1 + | log t|)

χA (x), s

−γ

χ[0,s) (t),

x ∈ Rn ,

t > 0,

due to Lemma 3.10(ii), that is (3.14), (3.15) with r = p, κ = In the same way as above we find that

1 p

+

α n,

γ ∈ R.

 v1   v1 min(ε,s) Z Zε h i v α 1 dt   −γv dt  ∗  (t) ∼ (1 + | log t|) t p + n gs,γ  t t 

0

0

diverges for γ ≤ v1 , whereas for small s, 0 < s < 1, and γ > p1 , Z p

−γp

kgs,γ |Lp (wα )k =

|x|−n−αp (1 + |log |x||)

|x|αp dx

As

Z1 −γp

(1 + log r)

≤c 0

dr ≤ c0 , r

so that the gs,γ violate condition (4.30) for appropriately chosen numbers s and γ when v < p.

Remark 4.14 In view of Remark 3.31 and our arguments above for wα (x) which can be immediately extended to arbitrary admissible weights that are bounded from below, w(x) ≥ c > 0, x ∈ Rn , we are led to ³ 1 ´ EG (Lp (w)) = t− p , p for all such weights w and 0 < p < ∞. Note that (3.51), Proposition 4.5, Theorem 4.7(i), Proposition 4.13 directly imply this result. In other words, the local regularity of such weights implies that the (local) growth envelopes (i.e., the characterisation of local singularities) remain unchanged compared with an unweighted Lp -space (w ≡ 1). This is essentially different from the corresponding situation for Ap -weights where we only treated wα (x) as an example so far; however, the situation also changes when regarding global assertions instead of local ones, see Section 10.3 below.

Chapter 5 The continuity envelope function EC

As in the case of growth envelopes we first introduce the continuity envelope X function EC , derive some elementary properties, and discuss examples afterwards. In addition, we prove a certain lift property that will be essential in later applications. In the sequel we deal with [R, µ] = [Rn , `n ] only and regard (quasi-) Banach spaces X of functions on Rn .

5.1

Definition and basic properties

This approach is based on the concept of the modulus of continuity and related to questions of Lipschitz continuity; we refer to Section 2.3 for notation and basics. Definition 5.1 Let X ,→ C be a function space on Rn . The continuity X envelope function EC : (0, ∞) → [0, ∞) is defined by X

EC (t) :=

sup kf |Xk≤1

ω(f, t) , t

t > 0.

(5.1)

Remark 5.2 An adapted version of Remark 3.2 holds here, too, concerning the equivalence classes of continuity envelope functions as well as the question of local (instead of global) behaviour of functions, implying our restriction to function spaces on Rn rather than function spaces on domains. We do not want to repeat the arguments in detail. X

X

First we collect a few elementary properties of EC (t). Note that EC (t) X cannot be too small as t ↓ 0, for EC (t) & 0 as t ↓ 0 implies that any element of X is constant. Furthermore, one introduces a number τ0C – parallel to (3.3) – by n o X τ0C = τ0C (X) := sup t > 0 : EC (t) > 0 . (5.2) 75

76

Envelopes and sharp embeddings of function spaces X

However, as EC (t) = 0 for some t > 0 means ω(f, t) = 0 for all f ∈ X (i.e., elements of X are constant) we are mainly interested in spaces X with τ0C (X) = ∞; investigating the local behaviour (small t > 0) at the moment, X it was even sufficient to assume, say, sup{0 < t < 1 : EC (t) > 0} = 1. Proposition 5.3 Let X ,→ C be a function space on Rn . X (i) EC is right-continuous and “essentially monotonically decreasing”, that X is, EC is equivalent to some monotonically decreasing function. X

X

(ii) We have X ,→ Lip1 if, and only if, EC (·) is bounded, i.e., sup EC (t) = t>0

X

lim sup EC (t) is finite. In that case it holds t↓0

° ° ° ° X X EC (0) := lim sup EC (t) = °id : X → Lip1 ° . t↓0

(iii) Let Xi ,→ C, i = 1, 2, be function spaces on Rn . Then X1 ,→ X2 implies that there is a positive constant c such that for all t > 0, X1

X2

EC (t) ≤ c EC (t). One may choose c = kid : X1 → X2 k in that case. (iv) Let X ,→ C be non-trivial, i.e., τ0C (X) = ∞. Let κ : (0, ∞) → [0, ∞) be some non-negative function. Then κ(·) is bounded if, and only if, there is some c > 0 such that for all f ∈ X, kf |Xk ≤ 1, sup t>0

κ(t) ω(f, t) ≤ c. X t E (t)

(5.3)

C

X

P r o o f : Step 1. The continuity of EC follows immediately from the corresponding feature of ω(f, t), see [DL93, Ch. 2, §6]. Similarly we gain from a result of DeVore and Lorentz which provides that 1 ω(f, t) ≤ ω(f, t) ≤ ω(f, t), 2

t > 0,

(5.4)

for any f ∈ C; cf. [DL93, Ch. 2, Lemma 6.1]. Here ω(f, t) is the least concave majorant of ω(f, t) (being itself a modulus of continuity). Consequently, is monotonically decreasing in t > 0. But this as ω(f, t) is concave, ω(f,t) t implies (i). Step 2. We prove (ii). Assume first X ,→ Lip1 . Then there is some c > 0 such that for all f ∈ X, kf |Lip1 k ≤ c kf |Xk. Thus for all f ∈ X, kf |Xk ≤ 1,

The continuity envelope function EC

77

we obtain k 1c f |Lip1 k ≤ 1. Hence, with g := 1c f , f ∈ X, kf |Xk ≤ 1 implies g ∈ Lip1 , kg|Lip1 k ≤ 1 and ω(f, t) = c ω(g, t). Consequently, X

EC (t) =

sup kf |Xk≤1

ω(g, t) ω(f, t) ≤ c sup ≤ c sup kg|Lip1 k ≤ c. 1 t t kg|Lip k≤1 kg|Lip1 k≤1

X

X

X

Thus EC (t) is bounded, and by (i), sup EC (t) ∼ lim sup EC (t) is finite. t>0

t↓0

X

Conversely, assume that there exists C > 0 such that sup EC (t) ≤ C. Then t>0

X

sup EC (t) = sup t>0

sup

t>0 kf |Xk≤1

ω(f, t) = t

sup

sup

kf |Xk≤1 t>0

ω(f, t) ≤ C, t

≤ C. which implies that for any f ∈ X, kf |Xk ≤ 1, we obtain sup ω(f,t) t t>0

Together with our assumption X ,→ C we can conclude that f ∈ Lip1 for any f ∈ X, kf |Xk ≤ 1; but by some scaling argument this implies X ,→ Lip1 , as desired. The proof of (iii) is similar. Step 3.

The restriction in (iv) that X contains essentially more than conX

stants only implies that EC (t) > 0 for any t > 0. So the left-hand side of (5.3) is well-defined. Now one may proceed analogously to the proof of Proposition 3.4(v). Corollary 5.4 Let X ,→ C be non-trivial, ϕ : (0, ∞) → (0, ∞) a positive, monotonically decreasing function with the following property: For any nonnegative function κ : (0, ∞) → [0, ∞) it is the case, that κ(·) is bounded if, and only if, there is some c > 0 such that for all f ∈ X, kf |Xk ≤ 1, sup t>0

κ(t) ω(f, t) ≤ c. ϕ(t) t

(5.5)

X

Then ϕ ∼ EC , i.e., there are numbers c2 > c1 > 0 such that for all t > 0, X

c1 ϕ(t) ≤ EC (t) ≤ c2 ϕ(t).

Proof: here.

(5.6)

The proof is similar to that of Corollary 3.6 and will not be repeated

Remark 5.5 In analogy to Remark 3.7 we mention the continuity function s ψ considered by Triebel with the additional assumptions that X = Bp,q or X s X = Fp,q , it behaves like ψ ∼ 1/EC .

78

Envelopes and sharp embeddings of function spaces

Remark 5.6 In view of Section 3.1, in particular Lemma 3.3, one may ask whether any space X of the above type possesses a continuity envelope X X function EC , that is, whether in any admissible situation EC (t) is finite for X any t > 0. In contrast to EG , see Lemma 3.3, our assumption X ,→ C already implies X

EC (t) =

sup kf |Xk≤1

2 kf |Ck 1 ω(f, t) ≤ sup ≤ 2 kid : X → Ck , t t t kf |Xk≤1

t > 0,

X

i.e., there is some c > 0 such that for all t > 0 , EC (t) ≤ ct . In that sense X any space X ,→ C has a continuity envelope function EC . Corollary 5.7 Let X ,→ C be a function space over [Rn , `n ]; put X

ψC (t) = t EC (t),

t > 0.

(5.7)

(i) The function ψC (t) is monotonically increasing in t > 0, lim ψC (t) = 0. t↓0

(ii) The function ψC (t) is equivalent to some concave function for t > 0. P r o o f : Note at first, that when X is trivial in the sense, that it contains X constants only, we always have EC ≡ 0 and there is nothing to prove. We come to the more interesting case now, when X is not trivial in the above sense. Part (i) follows immediately from the fact that ω(f, t) is monotonically increasing in t > 0 for any f ∈ X, and lim ω(f, t) = ω(f, 0) = 0 for t↓0

any f ∈ X. Moreover, ψC (t) = 0 if, and only if, t = 0. Together with (i) and Proposition 5.3(i) we thus obtain that ψC (t) is equivalent to some quasi-concave function; recall our remarks in front of Corollary 3.23. Another application of [BS88, Ch. 2, Prop. 5.10] finishes the proof. Remark 5.8 The coincidences as well as differences between Corollaries 3.23 and 5.7 are obvious. Note that in all cases we studied we have the counterpart X of Corollary 3.23(iii), too, i.e., EC is (equivalent to) a convex function. More important from our point of view, however, is the observation that obviX ously the (different) envelope functions EGX and EC show similar behaviour; we merely take it as some kind of (delayed) justification that the definition of the two envelope functions – arising in completely different problems when measuring smoothness or unboundedness, respectively – led to parallel concepts, though each one of them separately was initially motivated by suitable classical settings only. As already mentioned for growth envelopes, we shall work on function spaces X such that there is a continuous representative in the equivalence class of X EC , see Lemma 3.8. In particular, for our purpose it was sufficient to obtain the counterpart of (3.8). We give some sufficient condition.

The continuity envelope function EC

79

Lemma 5.9 Assume that X additionally satisfies ° ¡ −1 ¢ ° °f 2 · |X ° ≤ c kf |Xk

(5.8)

for some c > 0 and all f ∈ X. Then ¢ ¢ X¡ X¡ EC 2−j ∼ EC 2−j+1

(5.9)

for some j0 ∈ N and all j ≥ j0 . P r o o f : This generalises [Tri01, (12.78)]; see (5.4) and the similar argument following Proposition 3.4.

5.2

Some lift property

Before we present first examples we shall prepare later considerations by a “lifting” assertion that turns out to be an essential key in the sequel. It provides a relation between the modulus of continuity of a (sufficiently smooth) function and the non-increasing rearrangement of its gradient. The idea is to gain from results obtained in spaces of (sub-) critical type (and hence in terms of growth envelopes) when dealing with (super-) critical spaces (and continuity envelopes). Roughly speaking, we want to “lift” our (sub-) critical results by smoothness 1 to (super-) critical ones. This is at least partly possible. We return to this 11.2 and discuss it in more detail. Recall ³ point later in Section ´ ∂f ∂f (∇f )(x) = ∂x1 (x), . . . , ∂xn (x) , x ∈ Rn , with à n ¯ ¯2 !1/2 ¯ n ¯ X ¯ ∂f X ¯ ¯ ¯ ∂f ¯ ¯ ¯ ¯ |∇f (x)| = ∼ ¯ ∂xl (x)¯ ¯ ∂xl (x)¯ . l=1

(5.10)

l=1

Proposition 5.10 (i) There exists c > 0 such that for all t > 0 and all f ∈ C 1 , Zt ω(f, t) ≤ c

n

Zt s

1 n −1

∗

∗

|∇f | (σ n ) dσ.

|∇f | (s)ds ∼

(5.11)

0

0

(ii) Let 0 < r ≤ ∞, u > 1r , and 0 < ε < 1. Then there is a number c > 0 such that ¸r ¸r Zε · Zε · ∗ dt |∇f | (t) dt ω(f, t) ≤ c (5.12) t | log t|u t | log t|u t 0

0

(with the obvious modification when r = ∞) for all f ∈ C 1 .

80

Envelopes and sharp embeddings of function spaces

(iii) Let 0 < r ≤ ∞, 0 < κ < 1, and 0 < ε < 1. Then there is a number c > 0 such that Zε · 0

ω(f, t) tκ

¸r

dt ≤ c t

Zε h

1

∗

t n (1−κ) |∇f | (t)

0

ir dt t

(5.13)

(with the obvious modification when r = ∞) for all f ∈ C 1 . P r o o f : Step 1. We prove (i). We thank Professor V. Kolyada for the idea for this estimate (5.11). There is a similar observation in [DS84]. Let x, h ∈ Rn ; denote by Q a (closed) cube containing x and x + h with side-length ` = |h| and sides parallel to the coordinate planes. We estimate |f (x) − f (x + h)| ≤ |f (x) − fQ | + |fQ − f (x + h)|, where

1 fQ = |Q|

(5.14)

Z f (y)dy. Q

It is sufficient now to deal with the first term on the right-hand side of (5.14), |f (x) − fQ | as the second one can be handled similarly; thus Z Z 1 1 |f (x) − f (y)|dy = |f (x) − f (x + ξ)| dξ, |f (x) − fQ | ≤ |Q| |Q0 | Q

Q0

where Q0 is simply the translation of Q such that one corner is allocated in the origin, |Q0 | = |Q|. Furthermore, ¯ Z1 ¯ Z1 ¯ ¯ |f (x) − f (x + ξ)| = ¯¯ ∇f (x + τ ξ) · ξdτ ¯¯ ≤ |ξ| |(∇f ) (x + τ ξ)| dτ 0

0

Z1 ≤ `

|(∇f ) (x + τ ξ)| dτ. 0

Taking into account that |Q0 | = |Q| = `n we thus arrive at Z1 |f (x) − fQ | ≤ c `

−(n−1)

Z dτ

0

|(∇f ) (x + τ ξ)| dξ. Q0

A change of variable z = τ ξ in the latter integral yields Z1 |f (x) − fQ | ≤ c `

−(n−1)

τ

−n

Z dτ

0

|(∇f ) (x + z)| dz, Qτ

The continuity envelope function EC

81

where Qτ denotes the dilated cube with side length τ `. In view of |Qτ | = τ n `n and Z 1 ∗∗ |(∇f ) (x + z)| dz ∼ |∇f | (τ n `n ) |Qτ | Qτ

we obtain n

Z1

Z`

|f (x) − fQ | ≤ c `

|∇f |

∗∗

1

s n −1 |∇f |

(τ n `n ) dτ = c0

0

∗∗

(s)ds.

0

By our introductory remarks and standard arguments this implies Zt

1

s n −1 |∇f |

ω(f, t) = sup sup |f (x) − f (x + h)| ≤ C x∈Rn

|h|≤t

n

∗∗

(s)ds. (5.15)

0 ∗∗

In view of (5.11) it remains to replace |∇f | by |∇f |∗ in (5.15). Assume first n > 1; then we can apply Hardy’s inequality (see, for instance, [BS88, Ch. 3, Lemma 3.9]) to the right-hand side of (5.15) and obtain (5.11). In case of n = 1 we prove it directly. Let f ∈ C 1 (R); for simplicity we assume f (0) = 0. Let t > 0, and 0 < x ≤ t; then Zx

Zx 0

|f (x)| ≤

|f 0 |∗ (s)ds = x|f 0 |∗∗ (x) ≤ t|f 0 |∗∗ (t)

|f (τ )|dτ ≤ 0

0

for any x ≤ t; for the second estimate see, for instance, [BS88, Ch. 2, Thm. 2.2]. Thus ω(f, t) ∗∗ ≤ c |f 0 | (t), t

0 < t < ε,

f ∈ C 1 (R),

(5.16)

the modification in the general case (f (0) 6= 0) being obvious. But this is (5.11) in the case n = 1. Step 2. show

We have to verify (5.12) and start with the case r = ∞, that is we ∗

sup 0
|∇f | (t) ω(f, t) ≤ c sup u t | log t|u 0
(5.17)

for u > 0. By (5.11) we have c ω(f, t) ≤ t | log t|u t | log t|u

Zt

n 1

∗

s n −1 |∇f | (s)ds 0 n

Rt ∗

≤ c sup 0<s<ε

|∇f | (s) | log s|u

1

s n −1 | log s|u ds

0

t | log t|u

∗

≤ c0 sup 0<s<ε

|∇f | (s) | log s|u

82

Envelopes and sharp embeddings of function spaces

for small t > 0. Thus (5.17) is shown. Step 3. Assume now r < ∞, u > 1r ; we verify (5.12). Application of (5.11) and a change of variable s = σ n leads to Zε · 0

ω(f, t) t | log t|u

¸r

dt ≤ c t

Zε 0

 1  t | log t|u

r

Zt ∗

|∇f | (σ n ) dσ  0

dt . t

(5.18)

Assume that ε ∼ 2−J , put t = 2−j , j ≥ J, and σ = 2−` , ` ≥ j. Then in view of (5.18) and (5.12) it is sufficient to prove ∞ X

 2j j −u

j=J

∞ X

r ∞ X £ −u ¢¤r ¡ ¢ ∗¡ ∗ k |∇f | 2−kn . (5.19) 2−` |∇f | 2−`n  ≤ C k=J

`=j

The left-hand side can be manipulated into  r "∞ #r ∞ ∞ ∞ X X X X ¡ ¢ ∗ ∗ −(`−j) −u −`n −i −u −(i+j)n   = 2 j |∇f | 2 2 j |∇f | (2 ) j=J

j=J

`=j

i=0

With % < r, we can estimate further  r ∞ ∞ ∞ X ∞ h ir X X X ¡ ¢ ∗ ∗  2−(`−j) j −u |∇f | 2−`n  ≤ c 2−i% j −ur |∇f | (2−(i+j)n ) ; j=J

j=J i=0

`=j

in case of r ≤ 1 this is simply monotonicity, otherwise we need % < r and apply H¨older’s inequality. Changing indices as well as the order of summation provides ∞ X ∞ X

h ir ∗ 2−i% j −ur |∇f | (2−(i+j)n )

j=J i=0

=

µ ∞ X X £ ¢¤r −ur k−J ∗¡ |∇f | 2−kn k 2−ν% ν=0

k=J

k k−ν

¶ur .

k ν = 1 + k−ν ≤ 1 + ν because ν ≤ k − J ≤ k − 1. Thus the Note that k−ν inner sum converges independently of k, k−J X ν=0

µ 2−ν%

k k−ν

¶ur

and we arrive at (5.19) as desired.

≤

k−J X ν=0

ur

2−ν% (1 + ν)

≤ c0 ,

The continuity envelope function EC

83

Step 4. It remains to check (5.13). First let r < ∞; we apply (5.11) and make use of Hardy’s inequality once more, Zε · 0

ω(f, t) tκ

¸r

dt ≤ c1 t

Zε



∗

t−κ 0

Zε ≤ c2 0

r

Zt

|∇f | (σ n )dσ  0

dt t

£ 1−κ ¤r dt ∗ t |∇f | (tn ) t

Zε h ir dτ 1 ∗ . ≤ c3 τ n (1−κ) |∇f | (τ ) τ 0

Finally, when r = ∞, we can estimate in a similar way for 0 < t < ε, and find ω(f, t) ≤ c t−κ tκ

Zt

n 1

∗

s n −1 |∇f | (s)ds 0

¶ Ztn

µ ≤ ct 0

≤c

−κ

sup

0<s
sup s

1 n (1−κ)

s

1 n (1−κ)

κ

∗

s n −1 ds

|∇f | (s) 0

∗

|∇f | (s).

0<s<ε

Remark 5.11 Note that Triebel obtained in [Tri01, Prop. 12.16] assertion (5.12), too, but based on a different estimate replacing (5.11) by ¢ 1 ω(f, t) ∗∗ ¡ ≤ c |∇f | t2n−1 + 3 sup τ − 2 ω(f, τ ) t 0<τ ≤t2

(5.20)

for small ε > 0, all t ∈ (0, ε) and all f ∈ C 1 , cf. [Tri01, Prop. 12.16]. We discuss these results in Section 11.2 below; in particular, the above results also imply estimates for corresponding envelope functions, see Corollary 11.11 below. The exponent 2n−1 (instead of n) in the first term on the right-hand side of (5.20) prevented a result like (5.13) in that case, in contrast to (5.12) where the log-term takes no notice of exponents. Neves derived an assertion similar to (5.13) from (5.20), see [Nev01b, Prop. 4.2.28]. Let us also mention that (5.12) can be derived from (5.18) directly using an extended version of Hardy’s inequality obtained by Bennett and Rudnick in [BR80, Thm. 6.4].

84

5.3

Envelopes and sharp embeddings of function spaces

Examples: Lipschitz spaces, Sobolev spaces

All function spaces are defined on Rn unless otherwise stated. In analogy to Section 3.2 we start with some elementary examples. Proposition 5.12 Let 0 < a ≤ 1. Then Lipa

EC

(t) ∼ t

−(1−a)

,

0 < t < 1.

(5.21)

P r o o f : Step 1. Let 0 < a ≤ 1, f ∈ Lipa with kf |Lipa k ≤ 1. Then by (2.25) we conclude that ω(f, t) ≤ ta for all 0 < t < 1. Thus ω(f, t) −(1−a) ≤ t t Lipa

for all 0 < t < 1 and all f ∈ Lipa , kf |Lipa k ≤ 1. This implies EC t

−(1−a)

(t) ≤

.

Step 2. It remains to show the converse inequality. Let f1 be a continuous function on Rn which can be described as f1 (x) = |x|a χ[0,1) (|x|) + ψ(x)χ[1,∞) (|x|),

x ∈ Rn ,

(5.22)

where the continuous, non-negative, monotonically decreasing function ψ may be chosen so that f1 is continuous, kf1 |Ck ∼ 1, and ½ a |h| , |h| ≤ 1, sup |∆h f1 (x)| ∼ 1 , |h| ≥ 1. n x∈R a

Then ω(f1 , t) ∼ [min(t, 1)] = ta for 0 < t < 1, and kf1 |Lipa k ∼ 1. Consequently, for any 0 < t < 1, Lipa

EC

(t) ≥

ta ω(f1 , t) −(1−a) ∼ = t . t t

This proves (5.21).

Proposition 5.13 Let 0 < q ≤ ∞, α > Lip(1,−α) ∞, q

EC

(t) ∼ |log t|

1 q

α− q1

(with α ≥ 0 if q = ∞). Then ,

0
1 . 2

(5.23)

The continuity envelope function EC

85

P r o o f : Step 1. First let q = ∞; then the corresponding upper bound in (5.23) follows from (2.35) by straightforward calculation. When q < ∞, we (up to constants) and conclude that make use of the monotonicity of ω(f,t) t (1,−α) 1 for any τ , 0 < τ < 2 , and any f ∈ Lip ∞, q , kf |Lip(1,−α) ∞, q k ≤ 1, 1/q   1 1/q ¸q ¸q Zτ · Z2 · dt  ω(f, dt ω(f, t) t)   ≥ 1≥  t | log t|α t t | log t|α t 0

0

 τ 1/q Z dt ω(f, τ ) ω(f, τ )  −(α− q1 )  ≥ c0 |log τ | . ≥ c αq τ t | log t| τ 0

We benefit from αq > 1. Hence there is some c > 0 such that for any τ > 0 (1,−α) and any f ∈ Lip(1,−α) ∞, q , kf |Lip ∞, q k ≤ 1, ω(f, τ ) α− 1 ≤ c |log τ | q , τ yielding the upper estimate in (5.23) for q < ∞. Step 2. We verify the estimate from below in (5.23). We slightly modify the function f1 from (5.22) and adapt it to our setting: for 0 < s < 21 , 0 < q ≤ ∞ and α > 1q let fs (x) = |log s|

α− q1

x ∈ Rn ,

|x| χ[0,s) (|x|) + ψ(x)χ[s,∞) (|x|),

α− q1

and ψ is chosen as above. Then ω(fs , t) ∼ min(t, s) |log s| α− 1 kf |Ck ∼ s |log s| q ≤ cα,q for 0 < s < 21 . On the other hand, Ã Z21 · 0

ω(fs , t) t | log t|α

¸q

dt t

!1/q ≤ c |log s|

α− q1

à Z21 ·

min(s, t) t | log t|α

0

α− q1

à Zs ·

≤ c1 |log s|

0

+ c2 s |log s|

α− q1

α− q1

0 α− q1

à Z21 ·

−(α− q1 )

|log s|

dt t

¸q

Thus

!1/q

dt t

!1/q

1 t | log t|α

dt t | log t|αq

≤ c3 |log s|

≤ c5 |log s|

t t | log t|α

s

à Zs

¸q

.

¸q

dt t

!1/q

!1/q + c4

+ c4 ≤ C,

(5.24)

86

Envelopes and sharp embeddings of function spaces

(with obvious modifications when q = ∞) so that kfs |Lip(1,−α) ∞, q k ∼ 1. On the other hand, Lip(1,−α) ∞, q

EC

(t) ≥ sup fs

fs ∈ Lip(1,−α) ∞, q

and

ω(fs , t) α− 1 min(t, s) α− 1 ∼ sup |log s| q ≥ sup |log s| q t t s>t s>0 α− q1

,

∼ |log t|

where the last estimate results from 0 < t < s < 21 . Proposition 5.14 Let 0 < a < 1, 0 < q ≤ ∞, and α ∈ R. Then Lip(a,−α) ∞, q

EC

α

(t) ∼ t−(1−a) |log t| ,

0
1 . 2

P r o o f : The proof is simply a combination of those for Propositions 5.12 and 5.13, where the extremal functions fs can now be chosen as fs (x) = |log s|

α

sa−1 |x| χ[0,s) (|x|) + ψ(x)χ[s,∞) (|x|),

x ∈ Rn ,

s > 0 small; the modifications are clear otherwise. Having dealt in Proposition 5.13 with the modifications Lip(1,−α) of the ∞, q 1 “upper borderline case” Lip , we finally concentrate on the basic space C. Proposition 5.15 Let C be as above. Then C

EC (t)

∼

t−1 ,

0 < t < 1.

(5.25)

P r o o f : The upper estimate simply comes from ω(f, t) ≤ 2 kf |Ck for all t > 0 and all f ∈ C, see Remark 5.6. Concerning the lower bound we use functions fn (x), n ∈ N, defined by ½ 1 − n|x|, |x| ≤ n1 , (5.26) fn (x) = 0 , otherwise. Clearly, kfn |Ck = 1 and ω(fn , t) = 1 for t ≥ some m ∈ N with m ≥ t10 ; then C

EC (t0 ) = completing the proof.

sup kf |Ck≤1

1 n.

Thus choose for t0 > 0

ω(fm , t0 ) 1 ω(f, t0 ) ≥ = , t0 t0 t0

The continuity envelope function EC

87

In view of Theorem 2.28(i) we have Wpk ,→ C for k > np , 1 ≤ p < ∞, or k = n and p = 1, so that the concept of continuity envelopes makes sense. On the other hand, by (2.39), (2.43), combined with Proposition 5.3(ii) we conclude that unbounded continuity envelope functions appear in the following settings: n  n   p < k < p + 1, 1 ≤ p < ∞   k = np + 1, 1 < p < ∞ Wpk 6,→ Lip1 if (5.27)     k=n , p=1 We start with the super-critical strip

n p


Proposition 5.16 Let 1 ≤ p < ∞, k ∈ N, with Wpk

EC

Proof:

(t) ∼ t−( p +1−k) , n

n p

+ 1, 1 ≤ p < ∞.

n p


n p

+ 1. Then

0 < t < 1.

(5.28)

In view of Theorem 2.28(i), in particular, (2.41), Wpk ,→ Lipa ,

a=k−

n , p

(5.29)

see also [Zie89, Thm. 2.4.4]. Thus Propositions 5.3(iii) and 5.12 imply Wpk

EC

(t) ≤ c t−( p +1−k) , n

0 < t < 1.

Conversely, we return to our examples already used in (3.35), ¡ ¢ n fR (x) = Rk− p ψ R−1 x , x ∈ Rn , 0 < R < 1, where ψ(x) is given by (3.36). Recall that with gR := kψ|Wpk k−1 fR ∈ Wpk we obtain ° ° °fR |Wpk ° ° ° k °gR |Wp ° = ≤ 1, 0 < R < 1. kψ|Wpk k On the other hand, by construction, n

n Rk− p ∼ Rk− p , ω(gR , R) = kψ|Wpk k e

that is, Wpk

EC

(t) ≥

0 < R < 1,

n ω(gt , t) ω(gR , t) ≥ ≥ c tk− p −1 , t t 0
sup

for 0 < t < 1. We consider the limiting case k =

n p

+ 1, 1 < p < ∞, now.

88

Envelopes and sharp embeddings of function spaces

Proposition 5.17 Let 1 < p < ∞ be such that Wp1+n/p

EC

1

(t) ∼ | log t| p0 ,

n p

+ 1 = k ∈ N. Then

0
1 . 2

(5.30)

P r o o f : We make use of our earlier results on growth envelopes, in particular, the result by Hansson [Han79, (3.13)] recalled in (4.23), and a more general statement in Proposition 5.10(ii). Obviously, by density arguments it is sufficient to prove (5.30) for all sufficiently smooth functions. On the other 1+n/p n/p hand, f ∈ Wp implies |∇f | ∈ Wp , and we may apply Proposition 4.11, in particular, (4.23) to |∇f |. Together with (5.12) with 1 < r = p < ∞, u = 1 > p10 , and 0 < ε < 1 this yields the existence of c > 0 such that for all f ∈ C 1, 1/p  ε 1/p  ε ¸p ¸p Z · Z · ∗ dt  |∇f | (t) dt  ω(f, t)  ≤ c t | log t| t | log t| t 0 °0 ° ° ° ≤ c0 °|∇f | |Wpn/p ° ° ° ° ° ≤ c00 °f |Wp1+n/p ° . (5.31) On the other hand, for arbitrary τ , 0 < τ < ε, the left-hand side of (5.31) can be estimated from below by 1/p  τ 1/p  τ ¸p Z Z · dt  ω(f, τ )  dt ω(f, t)   ≥c t | log t| t τ t | log t|p 0

0

ω(f, τ ) −1 | log τ | p0 ≥ c0 τ due to the monotonicity of

ω(f,t) t ,

see (5.4) or [DL93,°Ch. 2, Lemma 6.1]. ° ° 1+n/p 1+n/p ° 1 Thus we have by (5.31) for all f ∈ C ∩ Wp with °f |Wp ° ≤ 1 and all τ , 0 < τ < ε, 1 ω(f, τ ) ≤ c | log τ | p0 , τ i.e., Wp1+n/p

EC

1

(t) ≤ c | log t| p0 ,

0 < t < ε.

For the converse we modify an argument from the proof of Proposition 3.27. In particular, we adapt the construction (3.44) by hm (x) =

m−1 X j=0

¡ ¢ 2−j ψ 2j x ,

m ∈ N,

(5.32)

The continuity envelope function EC

89

and ψ given by (3.36). Then we obtain in the same way as described above, ° ° ° ° ° ° ° ° (5.33) °hm |Wp1+n/p ° ≤ c0 m1/p °ψ|Wp1+n/p ° ≤ C m1/p , m ∈ N. On the other hand, we have for k = 1, . . . , m, k−1 k−1 X X X ¡ ¢ m−1 ¡ ¢ ω hm , 2−k ≥ 2−j ψ(0) − 2−j ψ 2j−k θ ≥ c 2−j+j−k j=0

j=0

j=0

−k

≥ck2

(5.34)

where θ ∈ Cn , |θ| = 1. Let t, 0 < t < 21 , be given and choose m0 ∈ N such that 2−(m0 +1) < t ≤ 2−m0 , i.e., m0 ∼ | log t|. Then by (5.33), (5.34) with k = m0 , Wp1+n/p

EC

1

(t) ≥ c sup m− p m∈N

− 1 ω(hm0 , t) ω(hm , t) ≥ c m0 p t t

1 − 1 +1 ω(hm0 , 2−m0 ) ≥ c00 m0 p ≥ C | log t|− p +1 . −m 0 2

1 −p

≥ c0 m0

This completes the proof.

Remark 5.18 Note that one part of the estimate (5.30) could have been obtained by (2.60) in connection with Propositions 5.3(iii) and 5.13. This would, W 1+n/p

however, not cover the proof for the fine index uC p , see Proposition 6.8 below. Therefore we stressed an alternative line of argument. In Part II, more precisely, in Theorem 9.4(i) we obtain the counterpart of (5.30) in a more general setting. It remains to deal with the case k = n, p = 1. Proposition 5.19 We have W1n

EC

(t) ∼ t−1 ,

0
1 . 2

(5.35)

P r o o f : Clearly, Proposition 5.15 together with the embedding (2.38) and Proposition 5.3(iii) imply the corresponding upper estimate. Conversely, note that the construction from Proposition 5.16 remains valid in case of k = n, p = 1, thus leading for ¡ ¢ ψ R−1 x , 0 < R < 1, gR (x) = kψ|W1n k

90

Envelopes and sharp embeddings of function spaces

to kgR |W1n k

¶ µ R ∼ 1, ω gR , 2

≤ 1,

that is, W1n

EC 1 for 0 < t < . 2

(t) ≥

sup 0
0 < R < 1,

ω(g2t , t) ω(gR , t) ≥ ≥ c t−1 t t

Chapter 6 Continuity envelopes EC

We do not want to repeat our introductory remarks as presented in Chapter 4. Obviously the continuity envelope function as given in Chapter 5 is a suitable counterpart of the growth envelope function handled in Chapter 3; so a parallel X argument may serve as motivation when complementing EC by the index X uC , too.

6.1

Definition

Analogous to the situation described at the beginning of Section 4.1 we shall introduce the Borel measure µC associated with the function ψ as described X in Section 4.1, and equivalent to 1/EC for some function space X with (5.8) X X and X 6,→ Lip1 , ψ(t) ∼ 1/EC (t), 0 < t < ε. Then (granted that EC was continuously differentiable) we obtain ³ ´0 X EC (t) dt (6.1) µC (dt) ∼ − X EC (t) for small t > 0. Definition 6.1 Let X ,→ C be a function space on Rn with (5.8), X 6,→ X Lip1 and continuity envelope function EC . Assume ε > 0. The index uX , C X 0 < uC ≤ ∞, is defined as the infimum of all numbers v, 0 < v ≤ ∞, such that #v !1/v à Zε " ω(f, t) µC (dt) ≤ c kf |Xk (6.2) X t EC (t) 0

(with the usual modification if v = ∞) holds for some c > 0 and all f ∈ X. Then ´ ¡ ¢ ³ X EC X = EC (·), uX (6.3) C is called the continuity envelope for the function space X.

91

92

Envelopes and sharp embeddings of function spaces

Remark 6.2 Proposition 5.3(iv) (with κ ≡ 1) implies that (6.2) holds with v = ∞ in any case; but – depending upon the underlying function space X – there might be some smaller v such that (6.2) is still satisfied. As Proposition 4.1(i) can be applied to the above case, that is, ψ ∼ 1/EC and g(t) ∼ ω(f,t) t , without any difficulties, we have the monotonicity of (6.2) in v. The question posed in Section 4.1, that is, under which assumptions uX = inf {v : 0 < v ≤ ∞, v C

satisfies

(6.2)}

(6.4)

is in fact a minimum, makes sense in that context, too, but is likewise open in general. Again, all the examples studied below are such (possibly special) cases where uX satisfies (6.2). C Remark 6.3 In analogy to Remark 4.4 we handle the case when X ,→ Lip1 separately. Parallel to Remark 4.4 we can include this situation by putting X uX := ∞ as for bounded EC , that is, by Proposition 5.3(ii), when X ,→ Lip1 , C (6.2) can be replaced by ω(f, t) ≤ c kf |Xk , t 0
for some c > 0 and all f ∈ X. We give the counterpart of Proposition 4.5 in terms of continuity envelopes. Proposition 6.4 Let Xi ,→ C, i = 1, 2, be some function spaces on Rn with X1 ,→ X2 . Assume for their continuity envelope functions X1

X2

EC (t) ∼ EC (t),

0 < t < ε.

(6.5)

i Then for the corresponding indices uX C , i = 1, 2, we have 1 2 uX ≤ uX . C C

Proof:

(6.6)

The proof copies that of Proposition 4.5.

Analogous to Remark 4.6 we could interpret the meaning of (6.2) in terms of sharp embeddings in, say, target spaces of type Lip(1,−α) ∞, q . We shall not perform this here in detail. Remark 6.5 In analogy to Remark 5.11 let us mention that Proposition 5.10 leads to an assertion about related fine indices, see Corollary 11.11 below.

Continuity envelopes EC

6.2

93

Examples: Lipschitz spaces, Sobolev spaces

We return to our examples from Section 5.3. Theorem 6.6 (i)

Let 0 < a ≤ 1. Then ³ ´ ³ −(1−a) ´ EC Lipa = t , ∞ .

(ii) Let 0 < q ≤ ∞, α >

1 q

(6.7)

(with α ≥ 0 if q = ∞). Then

³ ´ ´ ³ α− 1 EC Lip(1,−α) = |log t| q , q . ∞, q

(6.8)

(iii) Let 0 < a < 1, 0 < q ≤ ∞, and α ∈ R. Then ³ ´ ³ ´ α −(1−a) EC Lip(a,−α) = t |log t| , q . ∞, q (iv) We have

Proof:

(6.9)

´ ¡ ¢ ³ −1 EC C = t , ∞ .

(6.10)

Step 1. We prove (i). Let first a < 1. Note that Proposition 5.12

Lipa EC (t)

−(1−a)

yields ∼ t , 0 < t < 1. Thus, taking (6.1) and (6.2) into account, we have to check for which numbers v there exists c > 0 such that for all f ∈ Lipa , 1/v  ε ¸v Z · dt ω(f, t)   ≤ c kf |Lipa k . ta t 0

In view of (2.25) this leads to v = ∞ only. Finally, the case a = 1 is covered by Remark 6.3. Step 2.

Lip(1,−α) ∞, q

We come to (ii). In Proposition 5.13 we obtained EC

α− q1

Lip(1,−α) uC ∞, q

1 2.

,0
ω(f, t) t | log t|α−1/q+1/v

¸v

(t) ∼

in (6.8) is the least

1/v ° ° dt  ° ° ≤ c °f |Lip(1,−α) ∞, q ° t

94

Envelopes and sharp embeddings of function spaces

for all f ∈ Lip(1,−α) ∞, q , recall (6.1). In view of (2.35), for which numbers v do we obtain (at least locally) Lip(1,−α) ∞, q

(1,−(α− q1 + v1 ))

,→

Lip ∞, v

we are led to ask. We apply Proposition 2.23 with p = ∞, r = v, and β = α − 1q + v1 to obtain v ≥ q. Step 3. We verify (iii). By Proposition 5.14 we look for the smallest number v such that 1/v  ε ¸v Z · ° ° dt  ω(f, t) ° (a,−α) °  ≤ c |Lip (6.11) °f ∞, q ° ta | log t|α t 0

for some positive number c and all f ∈ Lip(a,−α) ∞, q . Plainly, (2.35) yields Lip(a,−α)

uC ∞, q ≤ q. In order to show the converse inequality we proceed by contradiction and modify our argument presented in [Har00b, Prop. 16] concerning Lip(1,−α) slightly. Let fκ ∈ C be such that ∞, q κ

ω(fκ , t) ∼ ta |log t|

for small t > 0. Clearly, fκ belongs to Lip(a,−α) if, and only if, κ < α − 1q . ∞, q Assume now that (6.11) holds for some number v < q and all f ∈ Lip(a,−α) ∞, q . Then we can choose κ such that α − v1 ≤ κ < α − 1q and arrive at a contradiction in (6.11) immediately. Step 4. It remains to show (iv). In view of Remark 6.2 it is sufficient to prove uC ≥ ∞. We proceed by contradiction, i.e., we disprove that there is C a number v < ∞ such that 1/v  ε Z dt  [ω(f, t)]v  ≤ c kf |Ck (6.12) t 0

holds for all f ∈ C, kf |C k ≤ 1. We make use of the functions fn given by (5.26), n ∈ N. Recall kfn |Ck = 1 and ω(fn , t) = 1 for t ≥ n1 , then (6.12) implies that there exists c > 0 such that for all n ∈ N  ε 1/v Z 1 dt v  [ω(fn , t)] 1 = kfn |Ck ≥ c t 0

 1/v Zε 1 1 v dt  ≥ c0 |log n| v . ≥  [ω(fn , t)]  c t 1 n

Continuity envelopes EC

95

However, this is impossible for all n ∈ N, and so uC = ∞. C Proposition 6.7 Let 1 ≤ p < ∞, k ∈ N, with np < k < ³ n ´ ¡ ¢ EC Wpk = t−( p +1−k) , p . Proof: that

n p

+ 1. Then (6.13)

In view of Proposition 5.16 and Definition 6.1 it remains to prove 

Zε ·

ω(f, t) n tk− p

 0

¸v

1/v ° ° dt  ≤ c °f |Wpk ° t

(6.14)

holds for some c > 0 and all f ∈ Wpk if, and only if, v ≥ p. First we show that there cannot be a number v < p that guarantees (6.14) for all f ∈ Wpk ; this is done by contradiction using a refined construction of “extremal” functions based upon (3.35) with R = 2−j . Let ¯ ¯b = {bj }j∈N be 0 n ¯ 0¯ a sequence of non-negative numbers, and let x ∈ R , x > 4, such that ¡ ¢ ¡ ¢ supp ψ 2j · −x0 ∩ supp ψ 2r · −x0 = ∅ for j 6= r, j, r ∈ N0 , and ψ given by (3.36). Then fb (x) :=

∞ X

¡ ¢ n 2−j(k− p ) bj ψ 2j x − x0 ,

x ∈ Rn ,

(6.15)

j=1

° ° belongs to Wpk for b ∈ `p , and °fb |Wpk ° ≤ c kb|`p k, see (4.16). On the other hand, by construction ¡ ¢ n ω fb , 2−j ≥ c 2−j(k− p ) bj , j ∈ N0 . For convenience we may assume b1 = · · · = bJ−1 = 0, where J is suitably chosen such that 2−J ∼ ε given by (6.14). Then by monotonicity arguments, 1/v  1/v  ε ¸v Z · ∞ h iv X ¡ ¢ n dt ω(f , t) b   ∼ 2−j ( p −k) ω fb , 2−j  n t tk− p j=J

0

1/v ∞ h iv X n n ∼ kb|`v k . ≥c  2−j ( p −k) bj 2−j(k− p )  

j=J

° ° Together with °fb |Wpk ° ≤ c kb|`p k our assumption (6.14) thus implies kb|`p k ≥ c kb|`v k for arbitrary sequences of non-negative numbers. This obviously requires v ≥ p. It remains to verify the converse, i.e., 1/p  ε ¸p Z · ° ° dt ω(f, t)   ≤ c °f |Wpk ° . (6.16) k− n t t p 0

96

Envelopes and sharp embeddings of function spaces

Here we gain from a more general estimate and our previous observations for growth envelopes. In particular, we use Proposition 5.10(iii), which gives us for 1 ≤ p < ∞, 0 < κ < 1, and 0 < ε < 1 the existence of c > 0 such that for all f ∈ C 1 , Zε · 0

ω(f, t) tκ

¸p

dt ≤ c t

Zε h ip dt 1 ∗ t n (1−κ) |∇f | (t) t

(6.17)

0

Obviously, by density arguments it is sufficient to prove (6.16) for all sufficiently smooth functions. On the other hand, f ∈ Wpk with np < k < np − 1, implies |∇f | ∈ Wpk−1 , 0 ≤ k − 1 < np , and we may apply Proposition 4.10, in particular, (4.14) to |∇f |. Together with (6.17) and κ = k − np this completes the proof, 

Zε ·

 0

ω(f, t) n tk− p

¸p

1/p  ε 1/p Z h ip dt k−1 1 dt  ∗ −  ≤ c t p n |∇f | (t) t t 0 ° ° ° ° 0° ≤ c |∇f | |Wpk−1 ° ≤ c00 °f |Wpk ° .

We continue Proposition 5.17 dealing with the case k = Proposition 6.8 Let 1 < p < ∞ be such that

n p

n p

+ 1.

+ 1 = k ∈ N. Then

´ ³ ´ ³ 1 EC Wp1+n/p = |log t| p0 , p .

Proof:

Wp1+n/p

In view of EC

µC (dt) ∼

0

(t) ∼ | log t|1/p by (5.30), and 1 dt , | log t| t

0
1 , 2

we have to show that 1/v  ε ¸v Z · ° ° dt ω(f, t) ° 1+n/p °   ≤ c f |W ° ° 0 p t t | log t|1/p +1/v

(6.18)

0

1+n/p

holds for all f ∈ Wp

W 1+n/p uC p

W 1+n/p

if, and only if, v ≥ p, i.e., uC p

= p. Obviously

(5.31) gives ≤ p and it suffices to prove that (6.18) fails for v < p. We stress an argument parallel to the proof of Proposition 4.11 and proceed by

Continuity envelopes EC

97

contradiction. Assume that (6.18) holds for some v < p and consider slightly modified functions of type (5.32), i.e., hm,b (x) =

m−1 X

¡ ¢ 2−j bj ψ 2j x ,

m ∈ N,

(6.19)

j=0

where ψ is given by (3.36) and bj ≥ 0, j = 0, . . . , m − 1. Then just as in (5.33), ° ° ° ° °hm,b |Wp1+n/p ° ≤

 1/p ° ° m−1 X p ° ° kDα hm,b |Lp k ≤ c °ψ|Wp1+n/p °  bj 

X |α|≤1+ n p



m−1 X

≤c  0

j=0

1/p bpj 

.

j=0

If we choose 1

bj = (j + 1)− p (1 + log(j + 1)) then

 m ° ° X ° ° °hm,b |Wp1+n/p ° ≤ c  j=1

− v1

,

j = 0, . . . , m − 1, 1/p 1 p/v



j (1 + log j)

≤ c0

(6.20)

uniformly in m ∈ N because p > v. On the other hand, parallel to (5.34) one obtains for r = 1, . . . , m, r−1 X ¡ ¢ 1 −1 ω hm,b , 2−r ≥ c 2−r bj ≥ c 2−r r br−1 = c 2−r r p0 (1 + log r) v , (6.21) j=0

so that finally (6.18) implies for sufficiently large m ∈ N,  ε Z · ° ° ° ° 1+n/p  °hm,b |Wp ° ≥ c1 0

ω(hm,b , t) t | log t|1/p0 +1/v

¸v

1/v dt  t

!1/v Ãm · X ω(hm,b , 2−r ) ¸v ≥ c2 2−r r1/p0 +1/v r=1 Ãm !1/v X 1 ≥ c3 r (1 + log r) r=1 which does not converge for m → ∞ in contrast to the left-hand side. Hence by this contradiction we conclude v ≥ p and the proof is complete.

98

Envelopes and sharp embeddings of function spaces

Remark 6.9 The above result reappears as a special case of Theorem 9.4(i) below. We finally return to the case k = n, p = 1, but have not yet a complete result. Lemma 6.10 We have W1n

EC

(t) ∼ t−1 ,

0 < t < ε,

and

Wn

uC 1 ≥ 1.

(6.22)

P r o o f : In view of Proposition 5.19 we have to show the existence of some c > 0 such that for all f ∈ W1n 

Zε

1/v dt  ≤ c kf |W1n k [ω(f, t)] t v

 0 Wn

(6.23)

implies v ≥ 1, i.e., uC 1 ≥ 1. Plainly, a slightly adapted version of the argument used in the proof of Proposition 6.8 immediately covers this case.

Part II

Results in function spaces, and applications

99

Chapter 7 Function spaces and embeddings

s s Function spaces of Besov or Triebel-Lizorkin type, Bp,q and Fp,q , respectively, will serve as outstanding examples in the sequel. Thus we recall briefly the basic ingredients needed for their introduction. In addition, we collect wellknown characterisations and embedding results. Note that all spaces are defined on Rn unless otherwise stated; so we shall omit the “Rn ” in the sequel.

7.1

s s Spaces of type Bp,q , Fp,q

The Schwartz space S and its dual S 0 of all complex-valued tempered distributions have their usual meaning here. Let ϕ0 = ϕ ∈ S be such that supp ϕ ⊂ {y ∈ Rn : |y| < 2}

and

ϕ(x) = 1

if

|x| ≤ 1,

(7.1)

and for each j ∈ N let ϕj (x) = ϕ(2−j x)−ϕ(2−j+1 x). Then {ϕj }∞ j=0 forms a smooth dyadic resolution of unity. Given any f ∈ S 0 , we denote by Ff and F −1 f its Fourier transform and its inverse Fourier transform, respectively. Then F −1 ϕj Ff is an analytic function on Rn . Definition 7.1 Let s ∈ R, 0 < q ≤ ∞, and let {ϕj } be a smooth dyadic resolution of unity. s (i) Let 0 < p ≤ ∞. The space Bp,q is the collection of all f ∈ S 0 such that ∞ ³X ° °q ´1/q s kf |Bp,q k= 2jsq °F −1 ϕj Ff |Lp ° j=0

(with the usual modification if q = ∞) is finite. s (ii) Let 0 < p < ∞. The space Fp,q is the collection of all f ∈ S 0 such that ° ³ ∞ ´1/q ° ° ° ° X jsq −1 ° s ° °f |Fp,q =° |Lp ° 2 |F ϕj Ff (·)|q j=0

(with the usual modification if q = ∞) is finite.

101

102

Envelopes and sharp embeddings of function spaces

Let us introduce the number µ σp = n

¶

1 −1 p

,

0 < p ≤ ∞.

(7.2)

+

s s Remark 7.2 The theory of the spaces Bp,q and Fp,q as given above has been developed in detail in [Tri83] and [Tri92] (and continued and extended in the more recent monographs [Tri97], [Tri01], [Tri06]), but has a longer history already including many contributors; we do not want to discuss this here. s s Let us mention instead that these two scales Bp,q and Fp,q cover (fractional) Sobolev spaces, H¨older-Zygmund spaces, local Hardy spaces, and classical Besov spaces – characterised via derivatives and differences: Let 0 < p ≤ ∞, s > σp , 0 < q ≤ ∞, and r ∈ N with r > s. Then with ωr (f, t)p given by (2.21),

ÃZ s kf |Bp,q k

∼

1 2

kf |Lp k + 0

£ −s ¤q dt t ωr (f, t)p t

!1/q (7.3)

(with the usual modification if q = ∞), see [BS88, Ch. 5, Def. 4.3], [DL93, Ch. 2, §10] (where the Besov spaces are defined in that way) for the Banach case, and [Tri83, Thm. 2.5.12] for what concerns the equivalence of Definition 7.1(i) and characterisation (7.3). In particular, with p = q = ∞, one recovers H¨older-Zygmund spaces C s , s B∞,∞ = Cs,

s > 0.

(7.4)

see Definition 2.13. Let, say, 0 < s < 1; then (in the sense of equivalent norms), ° ° s °f |B∞,∞ ° ∼ kf |Ck + sup ω(f, t) . (7.5) ts 0
Iσ f = F −1 hξiσ Ff,

f ∈ S 0,

σ ∈ R,

(7.6)

is the lift operator; in particular, in case of classical Sobolev spaces Wpk we have for k ∈ N0 , 1 < p < ∞, k Fp,2 = Wpk ,

0 i.e., Fp,2 = Lp .

(7.7)

Convention. In the sequel we shall sometimes write Asp,q , when both s s scales of spaces – either Asp,q = Bp,q or Asp,q = Fp,q – are concerned

Function spaces and embeddings

103

simultaneously and the particular choice does not matter. The lift operator Iσ , given by (7.6), maps Asp,q isomorphically onto As−σ p,q , 0 < p ≤ ∞ (with p < ∞ in F -case), 0 < q ≤ ∞, s, σ ∈ R, ° ° ° ° °Iσ f |As−σ ° ∼ °f |Asp,q ° . (7.8) p,q Another useful characterisation is given in terms of a Sobolev-type norm, ° ° n ° m ¯ ° X ° ° ° ° ° ° X ∂ f ° ¯ ° s α s−m s−m s−m °f |Ap,q ° ∼ °D f |Ap,q ° ∼ °f |Ap,q ° + ° m ¯Ap,q ° , (7.9) ° ∂xj ° j=1

|α|≤m

where s ∈ R, 0 < p, q ≤ ∞ (with p < ∞ in F -case), and m ∈ N; see [Tri83, Thm. 2.3.8]. Remark 7.3 Note that 0 hp = Fp,2 ,

0 < p < ∞,

(7.10)

cf. [Tri83, Thm. 2.5.8/1]; here the local (non-homogeneous) Hardy spaces hp , 0 < p < ∞, are defined in the sense of Goldberg [Gol79a], [Gol79b], see also 0 [Tri83, Sect. 2.2.2]. We shall also need the dual space of h1 , bmo = (h1 ) , see [Gol79b]: the local (non-homogeneous) space of functions of bounded mean satisfying oscillation, bmo , consists of all locally integrable functions f ∈ Lloc 1 that Z Z 1 1 |f (x) − fQ |dx + sup |f (x)|dx (7.11) kf |bmo k = sup |Q|≤1 |Q| |Q|>1 |Q| Q

Q

n is finite, where Q are cubes R in R , and fQ is the mean value of f with 1 respect to Q, fQ = |Q| Q f (x)dx. This definition coincides with [Tri83, 2.2.2(viii)]; see also [BS88, Ch. 5, Def. 7.6, (7.15)].

Remark 7.4 Occasionally we shall refer in what follows to function spaces of generalised smoothness: by this we essentially mean in this context spaces of type Asp,q where the main smoothness, characterised by the parameter s ∈ R, is “disturbed” by some slowly varying function Ψ (in Karamata’s sense), i.e., a positive and measurable function defined on (0, 1] such that lim

t→0

Ψ(st) = 1, Ψ(t)

s ∈ (0, 1].

(7.12)

In particular, a so-called admissible function, that is, a positive monotone function defined on (0, 1] such that Ψ(2−2j ) ∼ Ψ(2−j ), j ∈ N, is up to equivalence a slowly varying function. The standard example one should bear in

104

Envelopes and sharp embeddings of function spaces b

mind is Ψb (t) = (1 + | log t|) , t ∈ (0, 1], b ∈ R. The corresponding function (s,Ψ) (s,Ψ) spaces of type Bp,q and Fp,q are introduced completely parallel to Definition 7.1 simply replacing the term 2js by 2js Ψ(2−j ). These spaces were studied by Edmunds and Triebel in [ET98], [ET99a] and also considered by Moura in [Mou01], [Mou02] when Ψ is an admissible function. For further basic properties, like the independence of the spaces from the chosen dyadic resolution of unity (in the sense of equivalent norms), we refer to [FL06] in a more general setting; the extensive Russian literature can be found in the survey by Kalyabin and Lizorkin [KL87] and the appendix [Liz86]. Our intention to allude to these spaces results from connected envelope results obtained by, and partly in joint work with Caetano and Moura. This more general setting surprisingly offered deeper insight in our results; we shall explain it in more detail below. Example 7.5 If b ∈ R, then b

Ψb (t) = (1 + | log t|) ,

t ∈ (0, 1],

b ∈ R,

(7.13)

s,b is an admissible function. With this particular choice we obtain spaces Bp,q 0 consisting of those f ∈ S for which ∞ ° ° ³X ° °q ´1/q s,b ° °f | Bp,q = 2jsq (1 + j)bq °F −1 ϕj Ff |Lp °

(7.14)

j=0

is finite (usual modification for q = ∞). These spaces were studied by Leopold in [Leo98], [Leo00]. It turns out that the following atomic characterisation of function spaces s s of type Bp,q (or Fp,q ) is sometimes preferred (compared with the above Fourier-analytical approach), e.g., when arguments for entropy numbers of embeddings between such function spaces can thus be transferred to related questions of embeddings in (well-adapted) sequence spaces; we closely follow the presentation in [Tri97, Sect. 13]. Definition 7.6 Let 0 < p ≤ ∞, 0 < q ≤ ∞, and λ = {λνm ∈ C : ν ∈ N0 , m ∈ Zn }. Then ½ bpq =

λ : kλ | bpq k =

µX ∞ ³ X ν=0

|λνm |p

´q/p ¶1/q

¾ < ∞

m∈Zn

(with the usual modification if p = ∞ and/or q = ∞). This definition is a modification of the related one in [FJ90] and coincides with [Tri97, Def. 13.5].

Function spaces and embeddings

105

Let Qνm , ν ∈ N0 , m ∈ Zn , denote a cube in Rn with sides parallel to the axes of coordinates, centred at 2−ν m, and with side length 2−ν . For a cube Q in Rn and r > 0 we shall mean by rQ the cube in Rn concentric with Q and with side length r times the side length of Q. Definition 7.7 (i) Let K ∈ N0 and d > 1. A K times differentiable complex-valued function a on Rn (continuous if K = 0) is called a 1K -atom if supp a ⊂ dQ0m and

for some

|Dα a(x)| ≤ 1

for

m ∈ Zn ,

|α| ≤ K.

(7.15) (7.16)

(ii) Let s ∈ R, 0 < p ≤ ∞, K ∈ N0 , L + 1 ∈ N0 , and d > 1. A K times differentiable complex-valued function a on Rn (continuous if K = 0) is called an (s, p)K,L - atom if for some ν ∈ N0 supp a ⊂ d Qνm

|Dα a(x)| ≤ 2−ν(s−n/p)+|α|ν and

m ∈ Zn ,

for some for

|α| ≤ K,

(7.17) (7.18)

Z xβ a(x) dx = 0

if

|β| ≤ L.

(7.19)

Rn

This definition coincides with [Tri97, Def. 13.3]. It is convenient to write aνm (x) instead of a(x) if this atom is located at Qνm according to (7.15) and (7.17). The number d in (7.15) and (7.17) is unimportant in so far as it simply makes clear that at the level ν some controlled overlapping of the supports of aνm must be allowed. Assumption (7.19) is called a moment condition, where L = −1 means that there are no moment conditions. s is given by the The atomic characterisation of function spaces of type Bp,q following result [Tri97, Thm. 13.8]. Theorem 7.8 Let 0 < p ≤ ∞, 0 < q ≤ ∞, and s ∈ R. Let K ∈ N0 and L + 1 ∈ N0 with K ≥ (1 + [s])+

and

L ≥ max(−1, [σp − s])

(7.20)

s be fixed. Then f ∈ S 0 belongs to Bp,q if, and only if, it can be represented as ∞ X X λνm aνm (x), convergence being in S 0 , (7.21) f = ν=0 m∈Zn

where the aνm are 1K -atoms (ν = 0) or (s, p)K,L -atoms (ν ∈ N) with supp aνm ⊂ d Qνm ,

ν ∈ N0 , m ∈ Zn , d > 1,

(7.22)

106

Envelopes and sharp embeddings of function spaces

and λ ∈ bpq . Furthermore inf kλ | bpq k, where the infimum is taken over all admissible representations (7.21), is an s equivalent quasi-norm in Bp,q . For the proof, historical comments, as well as further remarks and conses quences we refer to [Tri97]; there is also a counterpart for spaces of type Fp,q , but this will not be needed in our context. Remark 7.9 Dealing with F -spaces, the case p = ∞ is usually excluded; however they were introduced already in [Tri78b, 2.5.1] for 1 < q ≤ ∞, see also [Tri83, Sect. 2.3.4]. This definition was modified and extended to 0 < q ≤ ∞ by Frazier and Jawerth in [FJ90, Sect. 5]: f ∈ S 0 (Rn ) belongs s to F∞,q , s ∈ R, 0 < q ≤ ∞, if µ ¶ ¶ q1 Z µX ∞ ¯¡ ¯q ¢ s < ∞, (7.23) k = sup 2νn 2ksq ¯ F −1 ϕk Ff (x)¯ dx kf |F∞,q Qνm

Qνm

k=ν

where the supremum is taken over all dyadic cubes Qνm , m ∈ Zn , ν ∈ N0 . Finally, in the case of a bounded domain Ω ⊂ Rn , the spaces Asp,q (Ω) are defined by restriction, Asp,q (Ω) = {f ∈ D0 (Ω) : ∃ g ∈ Asp,q (Rn ), g|Ω = f },

(7.24)

with kf |Asp,q (Ω)k = inf kg|Asp,q (Rn )k, and the infimum is taken over all g ∈ Asp,q (Rn ), g|Ω = f .

7.2

Embeddings

For convenience, we briefly collect some well-known facts about so-called (non-) limiting embeddings. As already explained in Part I, Sobolev’s famous embedding result, Theorem 2.28, led to a large number of further embedding results in more general function spaces, say, of type Asp,q . Inasmuch as there is no difference we shall again omit Ω or Rn in the formulation below. Let s s Asp,q stand for Bp,q or Fp,q , respectively. Proposition 7.10 Let s ∈ R, 0 < p ≤ ∞ (with p < ∞ for F -spaces), and 0 < q ≤ ∞. (i) Then Asp,q ,→ Asp,r for q ≤ r ≤ ∞. (7.25)

Function spaces and embeddings

107

(ii) Let 0 < r ≤ ∞ and ε > 0, then s As+ε p,q ,→ Ap,r .

(7.26)

s s s Bp,min(p,q) ,→ Fp,q ,→ Bp,max(p,q) .

(7.27)

(iii) Assume 0 < p < ∞, then

P r o o f : This result is well-known and can be found, for instance, in [Tri83, Prop. 2.3.2/2]. We include a short proof here for the convenience of nonspecialists, to demonstrate the interplay of the different Lp - and `q -norms involved in the definition of spaces Asp,q . In view of Definition 7.1, embedding (7.25) is a direct consequence of the monotonicity of `u -spaces, i.e., `q ,→ `r whenever q ≤ r ≤ ∞. Similarly, for arbitrary 0 < r ≤ ∞ and ε > 0 we can argue ∞ ° ° ³X ° °r ´1/r s ° °f |Bp,r = 2jsr °F −1 ϕj Ff |Lp ° j=0 ∞ ³X

° ° ≤ sup 2j(s+ε) °F −1 ϕj Ff |Lp ° j

2−jεr

´1/r

j=0

° ° s+ε ° ≤ c °f |Bp,∞ ,

which together with (i) finishes the proof in the B-case, with obvious modifications for r = ∞. The argument for the F -case works in a parallel way. s Concerning (iii), we first consider q < p; thus (7.27) reduces to Bp,q ,→ s s since the right-hand embedding already follows from Fp,p = Bp,p . Let s f ∈ Bp,q , then

s Fp,q ,

° ° s ° °f |Fp,q =

µ Z µX ∞ Rn

=

µ Z µ X ∞

µ X ∞ j=0

|F

ϕj Ff (x)|

q

¶1/p dx

¶u 2jsq |F −1 ϕj Ff (x)|q

¶1/(uq) dx

j=0

µ X ∞ µ Z j=0

=

2

−1

j=0

Rn

≤

¶p/q jsq

¶1/u ¶1/q jsqu

2

|F

−1

ϕj Ff (x)|

Rn

2

jsp

qu

° −1 ° °F ϕj Ff |Lp °q

¶1/q

dx ° ° s ° = °f |Bp,q ,

108

Envelopes and sharp embeddings of function spaces

where we applied, in addition, the generalised triangle inequality to u = pq > 1, cf. [HLP52, Thm. 202]. Conversely, when p < q, similar arguments lead to µ X µ Z ¶q/p ¶1/q ∞ ° ° s ° jsq −1 p °f |Bp,q = 2 |F ϕj Ff (x)| dx j=0

=

j=0

≤

¶v ¶1/(vp) 2jsp |F −1 ϕj Ff (x)|p dx

Rn

µ Z µ X ∞ µ Z µX ∞

q p

¶p/q jsq

2

|F

−1

q

ϕj Ff (x)|

¶1/p ° ° s ° dx = °f |Fp,q ,

j=0

Rn

where we used v = proof.

¶1/v ¶1/p 2jspv |F −1 ϕj Ff (x)|pv dx

j=0

Rn

=

Rn

µ X ∞ µ Z

s s > 1. Due to (7.27) and Bp,p = Fp,p this completes the

Dealing with classical spaces such as Lp , 1 ≤ p ≤ ∞, or C, one can 0 , complement (7.7) by the following elementary considerations. Let f ∈ Bp,1 then by the properties of a smooth dyadic resolution of unity (ϕk )k , ¯p ¶1/p µ Z ¯X ¯ ∞ −1 ¯ ¯ kf |Lp k = F ϕj Ff (x)¯¯ dx ¯ ≤

Rn j=0 ∞ Xµ Z

¯ −1 ¯ ¯F ϕj Ff (x)¯p dx

j=0

¶1/p

° ° 0 ° = °f |Bp,1 ,

Rn

with obvious modifications if p = ∞. Moreover, every F −1 ϕj Ff (x), j ∈ N0 , is bounded and uniformly continuous on Rn , such that the above argument implies f ∈ C, too. Conversely, for 1 ≤ p ≤ ∞, we can rewrite F −1 ϕj Ff (x), j ∈ N, using the properties of the (inverse) Fourier transform, Z ¡ −1 ¢ F −1 ϕj Ff (x) = F ϕj (y)f (x − y) dy, x ∈ Rn , j ∈ N, Rn

such that the generalised triangle inequality (for integrals), cf. [HLP52, Thm. 202], implies Z ° −1 ° ¯¡ ¢ ¯ °F ϕj Ff |Lp ° ≤ kf |Lp k ¯ F −1 ϕj (y)¯ dy ≤ c kf |Lp k Rn

uniformly for ¡ all j ∈ ¢ N, where we applied the special structure of (7.1), i.e., ϕj (y) = ϕ1 2−j+1y , j ∈ N, y ∈ Rn . Thus the above calculations lead to 0 0 Bp,1 ,→ Lp ,→ Bp,∞ ,

1 ≤ p ≤ ∞,

(7.28)

Function spaces and embeddings

109

and 0 0 B∞,1 ,→ C ,→ B∞,∞ .

(7.29)

In view of (7.9) one obtains the following “lifted” embeddings, see [Tri83, Prop. 2.5.7, (2.5.7/10,11)]. Proposition 7.11 Let m ∈ N0 , 1 ≤ p < ∞, then m m Bp,1 ,→ Wpm ,→ Bp,∞ ,

(7.30)

m m ,→ C m ,→ B∞,∞ . B∞,1

(7.31)

and

Dealing with spaces with different metrics, we have for 0 < p1 < p2 < ∞, 0 < q1 , q2 , q ≤ ∞ and s1 − pn1 = s2 − pn2 , Bps11 ,q ,→ Bps22 ,q

and

Fps11,q1 ,→ Fps22,q2 .

(7.32)

Concerning B-spaces this is essentially due to some Plancherel-Polya-Nikolskij inequality (cf. [Tri83, (1.3.2/5), Rem. 1.4.1/4]), that gives in our case ” “ ° ° ° −1 ° 1 1 °F ϕj Ff |Lp ° ≤ c 2jn p1 − p2 °F −1 ϕj Ff |Lp1 ° , 2

j ∈ N0 ,

(7.33)

whereas the argument in the F -case needs some more care; we refer to [Tri83, Thm. 2.7.1]. Let us introduce the notation ¶ µ ¶ µ n n − s2 − . (7.34) δ = s1 − p1 p2 Note that we thus have Asp11 ,q1 ,→ Asp22 ,q2

(7.35)

for all admitted parameters 0 < q1 , q2 ≤ ∞, assuming that s1 > s2 , 0 < p1 ≤ p2 ≤ ∞ (with p2 < ∞ in the F -case), and δ > 0, whereas this is not true for δ = 0 and all q-parameters in the B-case, see (7.32). We return to this point in Section 11.1. This is some reason to call δ = 0 as a limiting case referring back to Sobolev’s famous result. In the sequel we shall need a counterpart of (7.32) concerning the case when both B- as well as F -spaces are involved (as source or target spaces, respectively). Having different smoothness parameters si in the spaces under consideration, then the situation (7.27) is improved as follows; we gain from a result of Sickel and Triebel in [ST95, Thm. 3.2.1].

110

Envelopes and sharp embeddings of function spaces

Proposition 7.12 Let 0 < p0 < p < p1 ≤ ∞, s, s0 , s1 ∈ R, with s0 −

n n n = s− = s1 − , p0 p p1

(7.36)

and 0 < q, u, v ≤ ∞. Then s Bps00 ,u ,→ Fp,q ,→ Bps11 ,v

(7.37)

0 < u ≤ p ≤ v ≤ ∞.

(7.38)

if, and only if,

For a proof we refer to [ST95, Sect. 5.2]. Note that the “if”-part of the right-hand embedding is due to Jawerth [Jaw77], whereas the “if”-part of the left-hand embedding was proved by Franke [Fra86]. In particular, s Bps00 ,p ,→ Fp,q ,→ Bps11 ,p

for 0 < p0 < p < p1 ≤ ∞, s ∈ R, s0 −

(7.39)

n n n = s − = s1 − , and 0 < q ≤ ∞. p0 p p1

Our main goal in the next sections is to study growth envelopes or continuity envelopes of spaces of type Asp,q , respectively. Hence we are only interested in spaces of “functions” (i.e., regular distributions), such that Asp,q 6,→ L∞ (in case of growth envelopes), or satisfying Asp,q ,→ C and Asp,q 6,→ Lip1 , respectively. We discuss necessary and sufficient conditions for both cases now. Proposition 7.13 Let 0 < p ≤ ∞ (with p < ∞ for F -spaces), and 0 < q ≤ ∞. Then, n/p

if, and only if,

0 < p ≤ 1,

0 < q ≤ ∞,

(7.40)

n/p

if, and only if,

0 < p ≤ ∞,

0 < q ≤ 1,

(7.41)

F p,q ,→ L∞ and B p,q ,→ L∞

where L∞ in (7.40) and (7.41) can be replaced by C. For a proof we refer to [ET96, 2.3.3(iii)]. Obviously, (7.40) extends Theorem 2.28(ii) due to (7.7). Moreover, by (7.26), we obtain Asp,q ,→ L∞ ,

s>

n , p

0 < p, q ≤ ∞,

(7.42)

(with p < ∞ in F -case), where L∞ can be replaced by C, too. In view of Proposition 3.4(iii) it is clear that the spaces given by (7.40), (7.41), and

Function spaces and embeddings

111

(7.42), respectively, are of no further interest in our context, because the corresponding (growth) envelope functions are bounded. In the usual ( p1 , s)-diagram, where any space of the above type is characterised by its parameters s and p (neglecting q for the moment), that is Asp,q ↔ ( p1 , s), these embeddings correspond to embeddings along lines with slope n, i.e., s − np ≡ const. Note that this is the enriched version of Figure 8.

s=

s s=

n p

n p

+1 supercritical

1

critical ³ s=n

1 p

´ −1

subcritical 1

1 p

Figure 11 In view of the historical background and our above considerations, that is, the question whether a space contains essentially unbounded functions, it is reasonable to call embeddings (or spaces) of type Asp,q with s − np = 0 “critical”, whereas situations with s − np > 0 and s − np < 0 are regarded as “super-critical” or “sub-critical”, respectively. Moreover, as indicated in Figure 11, we shall merely study spaces where σp ≤ s ≤ np + 1. The lower bound is connected with the assumption Asp,q ⊂ Lloc 1 that we have to impose, i.e., that we deal with locally integrable functions. This implies that we have to assume s ≥ σp ; a complete treatment of this problem Asp,q ⊂ Lloc can be found in [ST95, Thm. 3.3.2], where Sickel and 1 Triebel obtained the following result:   either 0 < p < 1 , s ≥ σp , 0 < q ≤ ∞, s or 1 ≤ p < ∞, s > σp , 0 < q ≤ ∞, Fp,q ⊂ Lloc ⇐⇒ (7.43) 1  or 1 ≤ p < ∞, s = σp , 0 < q ≤ 2 .

112

Envelopes and sharp embeddings of function spaces

The parallel assertion for B-spaces reads as   either 0 < p ≤ ∞, s > σp , 0 < q ≤ ∞, s or 0 < p ≤ 1 , s = σp , 0 < q ≤ 1, Bp,q ⊂ Lloc ⇐⇒ (7.44) 1  or 1 < p ≤ ∞, s = σp , 0 < q ≤ min(p, 2) . Summarising our above observations we thus concentrate on spaces Asp,q ,

σp ≤ s ≤

n , p

0 < p, q ≤ ∞,

in connection with growth envelopes. As for the study of continuity envelopes we already know by Proposition 7.13 and (7.42) that we shall assume s ≥ np in general. On the other hand, spaces with s > np + 1 are not very interesting in our context due to the following result. Proposition 7.14 Let 0 < p ≤ ∞ (with p < ∞ for F -spaces), and 0 < q ≤ ∞. Then 1+n/p Fp,q ,→ Lip1

if, and only if,

0
if, and only if,

0
and

0 < q ≤ ∞,

(7.45)

0 < q ≤ 1.

(7.46)

and 1+n/p Bp,q ,→ Lip1

and

This is the (lifted) counterpart of (7.40) and (7.41). For a proof we refer to [Tri01, Thm. 11.4] and [EH99, Thm. 2.1]. Again, (7.26) implies Asp,q ,→ Lip1 ,

s>

n + 1, p

0 < p, q ≤ ∞,

(7.47)

(with p < ∞ in F -case). Hence, in view of Proposition 5.3(ii) it is clear that spaces given by (7.45), (7.46), and (7.47), respectively, are of no further interest in our context, because the corresponding envelope functions are bounded; we refer back to Section 5.1. Thus we shall rely on the notation as indicated in Figure 11, where both the super-critical and the sub-critical case are represented by the corresponding strips in the diagram. We shall briefly dwell upon sharp embeddings of spaces Asp,q into spaces introduced in of Lipschitz type, including the more general scale Lip(1,−α) p, q Section 2.3. Since we have already reserved the expression “limiting” (in connection with embeddings) for situations described by (7.34), we shall adopt the saying “sharp embedding” now when – at least for one parameter – there cannot be chosen any “better” (smaller or larger, respectively) value such

Function spaces and embeddings

113

that the embedding still holds. For instance, returning to the famous result of Br´ezis and Wainger [BW80], see (1.13) and rewritten now as 0

Hp1+n/p ,→ Lip(1,−1/p ) ,

(7.48)

see also (1.14), (2.60), one can ask whether the embedding (7.48) is sharp in the sense that Hp1+n/p 6,→ Lip(1,−α) if α < p10 (by the monotonicity of spaces Lip(1,−α) in α one clearly looks for the smallest value of α). We formulate our result in terms of spaces Lip(1,−α) introduced in Definip, q tion 2.16. Proposition 7.15 Let 0 < q, v ≤ ∞, α >

1 v

(with α ≥ 0 if v = ∞).

(i) Let 0 < p ≤ ∞. Then 1+n/p Bp,q ,→ Lip(1,−α) ∞, v

if, and only if,

α≥

1 1 + . v q0

(7.49)

1 . q0

(7.50)

1 1 + . v p0

(7.51)

In particular, for v = ∞, (1,−α) 1+n/p Bp,q ,→ Lip(1,−α) ∞, ∞ = Lip

if, and only if,

α≥

(ii) Let 0 < p < ∞. Then 1+n/p Fp,q ,→ Lip(1,−α) ∞, v

if, and only if,

α≥

In particular, for v = ∞, (1,−α) 1+n/p Fp,q ,→ Lip(1,−α) ∞, ∞ = Lip

if, and only if,

α≥

1 . p0

(7.52)

We refer to [EH99, Thm. 2.1] for the case v = ∞ and to [Har02, Prop. 3.3.5] otherwise. Clearly, (7.50) and (7.52) coincide with Proposition 7.14 when α = 0. In particular, (7.52) implies that for 1 < p < ∞ and 0 < q ≤ ∞ there is some c > 0 such that for all x, y ∈ Rn , 0 < |x − y| < 21 , and all 1+n/p f ∈ Fp,q , ¯ ¯1/p0 ° ° ¯ ¯ ° 1+n/p ° |f (x) − f (y)| ≤ c |x − y| ¯ log |x − y|¯ °f |Fp,q °,

(7.53)

114

Envelopes and sharp embeddings of function spaces

where the exponent p10 is sharp. Similarly, for 0 < p ≤ ∞ and 1 < q ≤ ∞ there is some c > 0 such that for all x, y ∈ Rn , 0 < |x − y| < 21 , and all 1+n/p f ∈ Bp,q , ¯ ¯1/q0 ° ° ¯ ¯ ° 1+n/p ° |f (x) − f (y)| ≤ c |x − y| ¯ log |x − y|¯ °f |Bp,q °,

(7.54)

s = Hps , s ∈ R, where the exponent q10 is sharp, see (7.50). Using Fp,2 1 < p < ∞, (7.51) generalises (1.13). For other works on sharpness of related embeddings see [EGO97], [EGO00] and [EK95], see also our discussion in [EH99, Rem. 2.5]. 1 Example 7.16 Let p = q = ∞, such that B∞,∞ = C 1 in view (7.4). Then, according to (7.54), there is some c > 0 such that for all f ∈ C 1 , ¯ ¯ ¯ ¯ |f (x) − f (0)| ≤ c |x| ¯ log |x|¯ kf |C 1 k, (7.55)

for all x, 0 < |x| < 21 . The exponent 1 of | log |x|| in (7.55) is sharp. This was first observed by Zygmund [Zyg45]. Remark 7.17 The sharpness assertion essentially relies on results on extremal functions as presented in [EH99, Prop. 2.2]: Let 1 < p < ∞ and σ > p1 . There is a function gpσ with 1+n/p , gpσ ∈ Bp,p

gpσ (0) = 0,

¯ ¯1/p0 ³ ¯ ¯´−σ ¯ ¯ ¯ ¯ |gpσ (x)| ≥ c |x| ¯ log |x|¯ log ¯ log ε|x|¯ for some c > 0, small ε > 0 and x = (x1 , 0, . . . , 0), 0 < x1 < δ, δ > 0 small. This is some “lifted” version of an example given by Triebel in [Tri93, Thms. 3.1.2, 4.2.2]; see also [ET96, Thm. 2.7.1]. within the scale of Besov We want to locate the Lipschitz spaces Lip(1,−α) p, q s,b spaces, refined by the logarithmic spaces Bp,q introduced in Example 7.5, see (7.14). We start with the classical setting b = 0. Proposition 7.18 Let 1 ≤ p < ∞, 0 < q, v ≤ ∞, α > v = ∞). Then ( 1 Bp,q

,→

Lip(1,−α) p, v

if

α≥ α>

1 q0 1 1 v + q0

1 v

(with α ≥ 0 if

,

v = ∞,

,

v < ∞.

(7.56)

Function spaces and embeddings

115

This result is proved and discussed in [Har00b, Prop. 11] for all p, 1 ≤ p ≤ ∞, but the case p = ∞ is now replaced by the better result Proposition 7.15. Furthermore, for p < ∞ and v = ∞ (7.56) is covered by [EH00, Prop. 4.2(ii)] already. Comparing (7.56) and (7.49) the question naturally arises 1 whether Bp,q ,→ Lip(1,−α) remains true for α = v1 + q10 and v < ∞, p, v p < ∞. This is not so clear at the moment. However, when p = ∞ [Har00b, Cor. 20] implies that there cannot be an embedding like (7.56) for α < v1 + q10 . Otherwise, for 1 ≤ p < ∞, there is an improved version of (7.56) by Neves in [Nev01a, Prop. 5.2] based upon Timan’s inequality [DL93, Ch. 2, Thm. 8.4] instead of Marchaud’s (2.22). As already mentioned, we are interested in extensions of (7.4) and 1 1 B∞,1 ,→ Lip1 ,→ B∞,∞ ,

see [Tri83, (2.5.7/2), (2.5.7/11)], to spaces of type Lip(1,−α) , C (1,−α) . In [EH99, Props. 4.2, 4.4] we proved such a result. Proposition 7.19 Let α ≥ 0. Then 1,−α 1,−α . B∞,1 ,→ Lip(1,−α) ,→ C (1,−α) = B∞,∞

(7.57)

Moreover, 1,−α B∞,q ,→ Lip(1,−α)

if, and only if,

0 < q ≤ 1.

s,b Our so far final embedding results related to spaces Bp,q and Lip(1,−α) is p, q the following, collecting [Har00b, Prop. 23, Cors. 25, 26]. We shall use it later on to derive envelope assertions for spaces Lip(1,−α) p, q .

Corollary 7.20 Let 1 ≤ p ≤ ∞, 0 < q ≤ ∞, α > 1q . (i) Then ( 1,−β Bp,1

Moreover,

,→

Lip(1,−α) p, q

1,−α+ 1

if

q Bp,min(q,1)

β <α− β≤α

,→

1 q

,

0 < q < ∞,

,

q = ∞.

(7.58)

Lip(1,−α) p, q .

(7.59)

. Lip(1,−α) p, q

(7.60)

(ii) Let 1 ≤ q ≤ ∞, α > 1. Then 1,−α+1 Bp,q

,→

116

Envelopes and sharp embeddings of function spaces

(iii) Let 0 < v ≤ ∞, β > v1 . Then Lip(1,−α) p, q

1,−β Bp,v

,→

if

 β −

1 v

≥α−

1 q

,

v ≥ q,

β −

1 v

>α−

1 q

,

v < q.

(7.61)

Concerning the question “where” the Lipschitz spaces Lip(1,−α) can be p, q s,b found within the scale of Besov spaces Bp,q , we arrived in [Har00b, Sect. 4] at the following diagram, where q ∗ := min(q, 1), 1,−α+ q1∗

1,−α+ 1∗ Bp,q∗ q

% & {z

| ≡

1,−α+ q1 , Bp,q

Bp,q

& %

1,−α+ q1

Bp,q∗

Lip(1,−α) p, q

}

0
% &

1,−α Bp,q 1,−α+ 1

& 1,−α . B % p,∞

Bp,∞ q {z } | 1,−α ≡ Bp,∞ ,q=∞ 1,−α+ 1

q . These We have the same diagram with Lip(1,−α) replaced by Bp,q p, q spaces, however, are not comparable (in the above sense) when 1 < q < ∞. There are a lot of further related approaches to spaces of Lipschitz type; we refer to [Har00b] for more details.

Remark 7.21 As mentioned in Remark 7.4 above, we shall occasionally refer (s,Ψ) to function spaces of generalised smoothness Ap,q in the sequel, where Ψ is a slowly varying function or an admissible function. Note that many of the above embeddings have their immediate counterpart in this context; in particular, this covers embeddings (7.25), (7.26), (7.27), (7.32), (7.39), always assuming that the same function Ψ is involved both in source and target space; cf. [Mou02, Prop. 1.1.13], [BM03], [HM04]. In addition, one has for all admitted parameters and all ε > 0, (s,Ψ) As+ε ,→ As−ε p,q ,→ Ap,q p,q .

(7.62)

Concerning Propositions 7.13, 7.14 there are the following counterparts: Let 0 < p, q ≤ ∞ (with p < ∞ for F -spaces), and Ψ as above, then (n/p,Ψ) Bp,q ,→ C

if, and only if,

³ ¡ ¢−1 ´ Ψ 2−j

∈ `q0

(7.63)

(n/p,Ψ) Fp,q ,→ C

if, and only if,

³ ¡ ¢−1 ´ Ψ 2−j

∈ `p0 ,

(7.64)

j∈N

and j∈N

Function spaces and embeddings

117

where C may be replaced by L∞ . We refer to [CM04a, Prop. 3.11] for a proof; a forerunner related to the case 1 < p, q < ∞ was obtained in [Kal82]. As for the embeddings in Lipschitz spaces, we obtained in [CH05, Prop. 2.2] ³ ¡ ¢−1 ´ (1+n/p,Ψ) Bp,q ,→ Lip1 if, and only if, Ψ 2−j ∈ `q0 , (7.65) j∈N

and (1+n/p,Ψ) Fp,q ,→ Lip1

³ ¡ ¢−1 ´ Ψ 2−j

if, and only if,

j∈N

∈ `p0 .

(7.66)

Due to (7.43), (7.44), and (7.62), the assumption ¶ µ 1 −1 s > σp = n p + (s,Ψ)

already implies Ap,q

⊂ Lloc 1 .

s,b Example 7.22 We return to our Example 7.5 concerning spaces Bp,q . In view of Propositions 7.13, 7.14, (7.62), (7.63), and (7.65) we thus have s,b Bp,q ,→ L∞

whenever s > case,

n p,

and b ∈ R, 0 < q ≤ ∞ arbitrary, whereas in the limiting ½

n/p,b Bp,q ,→ L∞

if, and only if,

Otherwise, assuming s >

n p,

b > q10 , 1 < q ≤ ∞ b≥0 ,0
¾ .

we obtain s,b Bp,q ,→ Lip1

if s >

n p

+ 1, and b ∈ R, 0 < q ≤ ∞ arbitrary, or, in the limiting case, ½

1+n/p,b Bp,q ,→ Lip1

if, and only if,

b > q10 , 1 < q ≤ ∞ b≥0 ,0
¾ .

(7.67)

Chapter 8 Growth envelopes EG of function spaces Asp,q

We study growth envelopes, introduced in Chapters 3 and 4, in the context of function spaces of type Asp,q , where σp ≤ s ≤ np .

8.1

Growth envelopes in the sub-critical case

We first deal with spaces of type Asp,q where 0 < p < ∞, 0 < q ≤ ∞, and σp ≤ s < np – called sub-critical according to our notation in Figure 11 (and the explanations given there). The borderline case s = σp needs some additional care; this refers to the thick lines in Figure 12. This situation is studied separately, but postponed to the end of this section.

s

³

s

s=

1 r

´

1 p, s

³

n p

s=n

1

1 p

1 p

´ −1

1 p

Figure 12

First we consider the “sub-critical strip” where σp < s < np , 0 < p < ∞ and 0 < q ≤ ∞. Let 1 < r < ∞, then all spaces on the line with slope n

119

120

Envelopes and sharp embeddings of function spaces

and “foot-point” 1r (see Figure 12) belong to this sub-critical area. Moreover, as all spaces of type Asp,q (with such parameters) can be embedded in, say, suitable Lebesgue spaces Lu , it makes sense to study their growth envelopes. Theorem 8.1 Let 0 < q ≤ ∞, 0 < p < ∞, and σp < s <

and

n p.

Then

³ ´ ³ 1 s ´ s EG Fp,q = t− p + n , p

(8.1)

³ ´ ³ 1 s ´ s EG Bp,q = t− p + n , q .

(8.2)

P r o o f : This result is known, see [Tri01, Thm. 15.2]; we give here a proof for later reference when we deal with borderline cases and indicate necessary modifications. Let 1 < r < ∞ be given by np − s = nr , as indicated in Figure 12. Bs Fs As 1 Thus the above assertions read as EG p,q (t) ∼ t− r , with uG p,q = q, uGp,q = p. As

1

Step 1. We first show EG p,q (t) ∼ t− r under the above assumptions. Note that by (7.7) and (7.32) we have s 0 Fp,q ,→ Fr,2 = Lr ,

1
(8.3) Fs

Now Proposition 3.4(iv) and Theorem 4.7(i) immediately imply EG p,q (t) ≤ 1 c t− r . Concerning the corresponding estimate for B-spaces, note that by a real interpolation argument ¡ s ¢ s Fp0 ,q , Fps1 ,q θ,p = Fp,q and (Lr0 , Lr1 )θ,p = Lr,p , (8.4) where 0 < θ < 1, 1 < r0 < r1 < ∞, 0 < p0 < p1 < ∞, s > 0, and 0 < q ≤ ∞. Here 1−θ θ 1 1−θ θ 1 = + and = + . (8.5) r r0 r1 p p0 p1 The F -interpolation in (8.4) restricted to 1 < pi < ∞ and 1 < q < ∞ coincides with [Tri78a, 2.4.2/(6)], whereas the general case is due to Frazier and Jawerth, [FJ90, Cor. 6.7 and §12]; the L-part in (8.4) is the very classical interpolation result for Lebesgue spaces, see (3.55) with r0 = q0 , r1 = q1 , and cf. [BL76, Thm. 5.3.1] and [Tri78a, Thm. 1.18.6/2]. Thus (8.3) together with (8.4) results in s Fp,q ,→ Lr,p . (8.6) s s The B-case now follows from another real interpolation (with Bp,p = Fp,p ) or (8.6) with (7.39), s Bp,q ,→ Lr,q ; (8.7)

Growth envelopes EG of function spaces Asp,q

121

for the case p, q ≥ 1 see also [BS88, Ch. 5, Cor. 4.20]. This finally leads to 1 Bs EG p,q (t) ≤ c t− r , using Proposition 3.4(iv) and Theorem 4.7(i) again. Step 2. It remains to show the converse inequalities. We use the example given in [Tri99, 3.2] and return to the construction introduced in Part I in (3.35). Let ψ(x) be the compactly supported C ∞ -function in Rn given by (3.36); then for j ∈ N, the functions ¡ ¢ n fj (x) := 2j r ψ 2j x , x ∈ Rn , (8.8) s are atoms in Bp,q in the sub-critical case: one has (up to a normalising factor © ª which can be neglected) supp fj ⊂ x ∈ Rn : |x| ≤ 2−j , and n

n

|Dγ fj (x)| ≤ 2−j(s− p )+j|γ| = 2j r +j|γ| ,

|γ| ≤ K,

where K ∈ N can be chosen arbitrarily large. Moreover, dealing with the sub-critical B-setting one does not need any moment conditions, as ¶ µ ¶ µ 1 n n 1 −1 −s=n −1 − + < 0 σp − s = n (8.9) p p p r + + for any p, 0°< p < °∞. In particular, the functions fj , j ∈ N, belong to s s ° Bp,q with °fj |Bp,q ∼ 1 (independent of j ∈ N). This can be seen by Theorem 7.8. On the other hand one calculates ¡ ¢ n fj∗ 2−jn ∼ 2j r , j ∈ N, implying

s Bp,q

EG

¡ −jn ¢ ¡ ¢ n 2 ≥ fj∗ 2−jn ∼ 2j r ,

j ∈ N.

This yields the desired B-result, s Bp,q

EG

1

(t) ≥ c t− r ,

0 < t < 1.

(8.10)

Moreover, choose for given s and p parameters σ > s and 0 < v < p σ s such that σ − nv = s − np = − nr . By (7.39), Bv,p ,→ Fp,q ; hence (8.10) and Proposition 3.4(iv) complete the lower estimate in the F -setting, Fs

1

EG p,q (t) ≥ c t− r ,

0 < t < 1.

Step 3. Finally we have to verify that in the F -case v = p is the smallest possible exponent satisfying 1/v  ε Z h iv dt ° ° 1 s ° ∗   ≤ c °f |Fp,q t r f (t) t 0

122

Envelopes and sharp embeddings of function spaces

s for all f ∈ Fp,q , whereas in the B-case v = q is optimal. Note that we Fs

Bσ

have EG p,q (t) ∼ EG v,p (t) whenever σ − nv = s − np = − nr so that by Proposition 4.5 together with (7.39) it is sufficient to deal with the B-case only. Clearly, µG (dt) ∼ dt t , and another application of Proposition 4.5 together with Bs

(8.7) and Theorem 4.7(i) immediately provides uG p,q ≤ q. The sharpness can be either taken as a consequence of [Tri99, Cor. 2.5], see also [Tri01, Thm. 15.2], or derived in a way parallel to the argument presented in the proof of Proposition 4.10: we consider a refined construction of the above “extremal” functions ¯fj .¯ Let {bj }j∈N be a sequence of ¢non-negative¡ numbers, ¡ ¢ and let x0 ∈ Rn , ¯x0 ¯ > 4, such that supp ψ 2j · −x0 ∩ supp ψ 2r · −x0 = ∅ for j 6= r, j, r ∈ N0 , and ψ given by (3.36). Then fb (x) :=

∞ X

¡ ¢ n 2j r bj ψ 2j x − x0 ,

x ∈ Rn ,

(8.11)

j=1

can be seen as atomic decomposition of fb ; hence Theorem 7.8 implies ° ° s ° °fb |Bp,q ≤ c kb|`q k . (8.12) ¡ ¢ n Recall fb∗ c 2−jn ≥ c0 bj 2j r , j ∈ N0 . For convenience we may assume b1 = · · · = bJ−1 = 0, where J is suitably chosen such that 2−J ∼ ε given by (4.14). Then by monotonicity arguments, 1    v1  ε  v1 Z h ∞ ∞ iv dt v X X ¡ £ ¢¤ n 1 v  ≥ c1   2−j r fb∗ c2−jn  ≥ c2  bvj  , t r fb∗ (t) t 0

j=J

j=J

i.e., we arrive at kb|`q k ≥ c kb|`v k for arbitrary sequences of non-negative numbers. This obviously requires v ≥ q.

Remark 8.2 Observe that (8.1) implies Proposition 4.10 when 1 < p < ∞ in view of (7.7). Moreover, (8.1) together with Theorem 4.7(i) lead to ³ ´ ³ 1 ´ ³ ´ s , (8.13) EG Lr,p = t− r , p = EG Fp,q where 0 < q ≤ ∞, s > 0, 1 < r < ∞ and 0 < p < ∞ with s − np = − nr ; s that is, we have by (8.6) the embedding Fp,q ,→ Lr,p only, whereas the corresponding envelopes even coincide. This can be interpreted as Lr,p being s indeed the best possible space within the Lorentz scale in which Fp,q can be embedded continuously. On the other hand this is to be understood in s the sense that Lr,p is “as good as” Fp,q – as far as only the growth of the unbounded functions belonging to the spaces under consideration is concerned; (additional) smoothness features (making a big difference between the spaces

Growth envelopes EG of function spaces Asp,q

123

s Lr,p and Fp,q , for instance) are obviously “ignored” by the growth envelope. This is not really astonishing in view of its construction, but worth noticing. The parallel assertion for the B-case, i.e., (8.2) together with Theorem 4.7(i) provide ³ ´ ³ 1 ´ ³ ´ s EG Lr,q = t− r , q = EG Bp,q , (8.14) s the parameters being as above. Again we note by (8.7) that Bp,q can be embedded in Lr,q , whereas their envelopes even coincide. We return to this phenomenon in Section 11.2.

The embedding result (8.7) can (in the Banach space situation) also be found in [Gol87a] and [Kol98]. Moreover, Gol’dman’s result [Gol87b, Thm. 2.1, Cor. 5.1] can be interpreted as the fact that Lr,q is the best possible space within s the Lorentz scale in which Bp,q can be embedded continuously – coinciding with our above interpretation of (8.13), see also [Her68, Thm. 8.5]. Remark 8.3 Forerunners of this result – formulated in a different context – are presented in [Tri99]. This is extended and generalised in [Tri01, Sect. 15]. There one also finds a lot of remarks and references on the long history of related studies; thus we shall only mention some of the most important names and papers briefly: essential contributions were achieved by Peetre [Pee66], Strichartz [Str67], Herz [Her68], as well as in the Russian school by Brudnyi [Bru72], Gol’dman [Gol87c], [Gol87b], Lizorkin [Liz86], Kalyabin, Lizorkin [KL87], Netrusov [Net87], [Net89], see also the book by Ziemer [Zie89]. More recent treatments are, for instance, [CP98] by Cwikel, Pustylnik, [EKP00] by Edmunds, Kerman, Pick and the surveys [Kol98] by Kolyada, [Tar98] by Tartar. There are far more investigations connected in some sense with limiting embeddings; we refer to the survey papers for detailed information.

Remark 8.4 Recently Caetano and Moura obtained parallel results in the (s,Ψ) sub-critical case when studying spaces of generalised smoothness of type Bp,q , (s,Ψ) Fp,q , see Remark 7.4. According to Remark 7.21 we assume 0 < p, q ≤ ∞, σp < s < np . The corresponding sub-critical result in [CM04b, Thm. 4.4] then reads as ³ ´ ³ 1 ´ (s,Ψ) = t− r Ψ(t)−1 , q (8.15) EG Bp,q and

³ ´ ³ 1 ´ (s,Ψ) = t− r Ψ(t)−1 , p . EG Fp,q

(8.16)

Further extensions were obtained by Bricchi and Moura in [BM03] conn cerning spaces Aσ p,q (R ), where the usual (scalar) regularity index σ ∈ R is replaced by a sequence σ = {σj }j∈N0 . Releasing also the subordinate (dyadic) partition of unity one obtains spaces Aσ,N p,q of generalised smoothness; their growth envelopes were studied in [CF06], [CL06]. There is some recent work on anisotropic spaces [MNP06] that leads to the same, i.e., isotropic results.

124

Envelopes and sharp embeddings of function spaces

Finally, growth envelopes of another possible modification of spaces, namely Lorentz-Karamata-Bessel potential spaces H s Lp,q;Ψ (Rn ) with a slowly varying function Ψ are investigated in [GNO04], see also [GO06]. Example 8.5 We return to Example 7.5, s ∈ R, b ∈ R, 0 < p < ∞, 0 < q ≤ ∞. In view of Example 7.22 we assume σp < s < np , referring to the sub-critical case. Then (8.15) implies ´ ¡ s,b ¢ ³ − 1 −b = t r |log t| , q = EG (Lr,q (log L)a ) EG Bp,q (8.17) where 1 < r < ∞ is given by s − np = − nr ; see Theorem 4.7(i). This is the counterpart of (8.14), adapted to this more general setting. Corollary 8.6 Let 1 ≤ p < n, 0 < q ≤ ∞, α > 1q , then (1,−α)

Lip p, q

EG

Proof:

α− q1

1

1

(t) ∼ t− p + n |log t|

,

0
1 . 2

(8.18)

Note that by (7.59) and (7.61) we have 1,−α+ 1

q Bp,min(q,1)

,→

Lip(1,−α) p, q

,→

1,−α+ q1

Bp,∞

.

Now the result follows by Proposition 3.4(iv) and (8.17). We conclude this section with a short digression on weighted spaces of type Asp,q (wα ). Recall our notation (3.48), wα (x) = hxiα ,

α ∈ R,

x ∈ Rn ,

and Remark 3.29. In continuation of the spaces Lp (Rn , w) = Lp (w) intros s duced in (3.47) one can define spaces of type Bp,q (Rn , w) and Fp,q (Rn , w) n completely parallel to Definition 7.1 simply replacing Lp (R ) by Lp (Rn , w), where w is admissible in the sense of Section 3.4. In [HT94a] we have proved that f ∈ Asp,q (Rn , w) if, and only if, wf ∈ Asp,q (Rn ), more precisely, that for such admissible weights w, ° ° ° ° °f |Asp,q (Rn , w)° ∼ °wf |Asp,q (Rn )° (8.19) are equivalent norms. Moreover, we proved in [HT94a, Thm. 2.3] that for A = F, Asp11 ,q1 (w1 ) ,→ Asp22 ,q2 (w2 ) (8.20) if, and only if, s1 −

n n ≥ s2 − p1 p2

and

w2 (x) ≤c<∞ w1 (x)

(8.21)

Growth envelopes EG of function spaces Asp,q

125

for some c > 0 and all x ∈ Rn . Here we assume s1 > s2 , 0 < p1 ≤ p2 < ∞, 0 < q1 , q2 ≤ ∞, and w1 , w2 are admissible weight functions. For A = B we got the same assertion, but in the limiting case s1 − pn1 = s2 − pn2 we have to observe q1 ≤ q2 as is clear from the case w1 = w2 ≡ 1 and (7.32). With w2 ≡ 1 and w1 = wα , α > 0, we thus have under the above assumptions on the parameters, Asp11 ,q1 (wα ) ,→ Asp22 ,q2 . However, for our special weight this can even be extended to values p2 < p1 , as long as p12 < p10 = p11 + α n. Remark 8.7 We proved even a bit more in [HT94a, 2.4], namely that for s ∈ R, 0 < p ≤ ∞ (p < ∞ in the case of F -spaces), 0 < q ≤ ∞, α > 0, and p0 as above, Asp,q (wα ) ,→ weak − Asp0 ,q , (8.22) where in the usual definition of Asp,q , Definition 7.1, the spaces Lp0 are replaced by Lp0 ,∞ . This is essentially based on Lemma 3.33. Due to (8.19) we immediately obtain counterparts for other embeddings, e.g., s Bps00 ,p (w) ,→ Fp,q (w) ,→ Bps11 ,p (w) (8.23) for 0 < p0 < p < p1 ≤ ∞, s ∈ R, s0 − pn0 = s − np = s1 − pn1 , 0 < q ≤ ∞, and w an admissible weight function. Proposition 8.8 Let 0 < q ≤ ∞, 0 < p < ∞, and σp < s < Then ³ ´ ³ 1 s ´ s EG Fp,q (wα ) = t− p + n , p and

³ ´ ³ 1 s ´ s EG Bp,q (wα ) = t− p + n , q .

n p.

Let α > 0. (8.24) (8.25)

P r o o f : Parallel to the proof of Theorem 8.1 and in view of (8.23) it is sufficient to deal with the B-case only as Propositions 3.4(iv) and 4.5 will do s s the rest. Furthermore, by the characterisation of Bp,q - and Fp,q -spaces via local means, see [Tri92, Sect. 2.4.6, 2.5.3], one easily checks that Asp,q (wα ) ,→ Asp,q ,

α > 0.

(8.26)

Proposition 3.4(iv) and Theorem 8.1 imply the upper bound, s Bp,q (wα )

EG

1

s

(t) ≤ c t− p + n ,

t > 0.

Conversely, we found in Step 2 of the proof of Theorem 8.1 that for j ∈ N, ¡ ¢ n fj (x) = 2j r ψ 2j x , x ∈ Rn ,

126

Envelopes and sharp embeddings of function spaces

s are atoms in Bp,q , where ψ is given by (3.36), and the number r, 1 < r < ∞, s denotes again the “foot-point”, i.e., s− np = − nr . In particular, kfj |Bp,q k ≤ 1, ¡ −jn ¢ ∗ jn and fj c 2 ∼ 2 r , j ∈ N. In view of (8.19) and wα (x) ≤ c, for x ∈ supp fj ⊂ K1 (0), j ∈ N, we thus conclude

° ° ° ° ° ° s s ° s ° °fj |Bp,q (wα )° ≤ c °wα fj |Bp,q ≤ c0 °fj |Bp,q ≤ c00 , s again using “local” characterisation of Bp,q such as (sub-) atomic decompositions or local means, cf. Theorem 7.8 or [Tri92, Sect. 2.5.3]. Because of s Bp,q (wα )

EG

¡ −jn ¢ ¡ ¢ n c2 ≥ fj∗ c 2−jn ∼ 2j r ,

j ∈ N,

this completes the B-result for the envelope function. Moreover, together B s (wα ) with (8.26), Theorem 8.1 and Proposition 4.5 this implies uG p,q ≤ q. It remains to check the converse inequality, that is, we claim that 1/v Zε h iv dt ° ° 1 s   ≤ c °f |Bp,q (wα )° t r f ∗ (t) t 

(8.27)

0

s for all f ∈ Bp,q (wα ), if, and only if, v ≥ q. We use a method similar to Step 3 in the proof of Theorem 8.1, recall also our construction in the proof of Proposition 4.10.¯ Let of¢ non-negative ¯ b = {bj }j∈N be a sequence ¡ ¡ numbers, ¢ and let x0 ∈ Rn , ¯x0 ¯ > 4, such that supp ψ 2j · −x0 ∩ supp ψ 2k · −x0 = ∅ for j 6= k, j, k ∈ N0 , and ψ given by (3.36). Then

fb (x) := hxi−α

∞ X

¡ ¢ n 2j r bj ψ 2j x − x0 ,

x ∈ Rn ,

(8.28)

j=1 s s is an atomic decomposition in Bp,q (wα ), as wα f ∈ Bp,q , due to (8.19), ¡ j ¢ 0 jn and 2 r ψ 2 x − x , j ∈ N, are (s, p)K,−1 -atoms, K ≥ 1 can be chosen arbitrarily large (no moment conditions needed). Assuming b = {bj }j∈N ∈ `q , Theorem 7.8 together with (8.19) imply,

° ° ° ° ° ¯ ° s s ° °fb |Bp,q (wα )° ∼ °wα fb |Bp,q ≤ c °b¯`q ° ,

(8.29)

where c is independent of b ∈ `q . On the other hand, ¡ ¢−1 ¡ ¢ n , fb∗ c2−jn ≥ c0 2j r bj wα 2−j x0

j ∈ N.

For convenience we may assume b1 = · · · = bJ−1 = 0, where J ∈ N is

Growth envelopes EG of function spaces Asp,q

127

¡ ¢ suitably chosen such that 2−J ≤ ε, then wα 2−j x0 ∼ 1, hence  

X j=J

1/v bvj 



1/v ∞ X £ −j n ∗ ¡ −jn ¢¤v  ≤ c1  2 r fb c 2 j=J

1/v  ε Z h iv dt 1  ≤ c2  t r fb∗ (t) t °0 s ° ≤ c3 °fb |Bp,q (wα )° ≤ c4 kb|`q k ,

(8.30)

B s (wα )

thus disproving (8.27) for v < q. Now uG p,q ≥ q follows immediately thus – in view of our above remarks – completing the proof.

Remark 8.9 Note that – as in the case of Lp -spaces, see Theorem 4.7(i) and Proposition 4.13 – the locally regular weight wα (x) = hxiα , α > 0 does not change the local singularity behaviour characterised by EG (X), see Theorem 8.1. Again, the result can be transferred without big difficulties to admissible weights w which are bounded from below, w(x) ≥ c, x ∈ Rn . We shall see in Proposition 10.21 that the global behaviour takes care of the weight involved. We conclude this section with the study of spaces Asp,q (wα ), where the weight is given by (3.52). These spaces are defined in a way completely parallel to Definition 7.1 replacing Lp (Rn ) by Lp (Rn , w) = Lp (w), where w is a Muckenhoupt weight in the sense of Section 3.4. Spaces of Besov and Triebel-Lizorkin type with weights w belonging to a Muckenhoupt class Ar have been treated systematically by Bui et al. in [Bui82], [Bui84], and [BPT96], [BPT97]. Later this topic was revived and extended by Rychkov in [Ryc01], including also approaches for locally regular (admissible) weights. Quite recently, the topic was revived in papers of Roudenko [Rou04], [FR04] and Bownik [Bow05], [BH06]. We studied atomic decompositions of spaces Asp,q (Rn , w) in [HPxx]. This will enable us to deal with their growth envelope functions. Note that we do not have the counterpart of (8.19) in this case, whereas assertions (8.22) and (8.23) remain valid. Proposition 8.10 Let 0 < q ≤ ∞, 0 < p < ∞, σp < s < n + s − np . Then ³ ´ ³ 1 s α ´ s (wα ) = t− p + n − n , p EG Fp,q and

³ ´ ³ 1 s α ´ s (wα ) = t− p + n − n , q . EG Bp,q

n p.

Let 0 < α < (8.31) (8.32)

128

Envelopes and sharp embeddings of function spaces

P r o o f : In view of (8.23) we may restrict ourselves to the B-setting only. Let again r denote the number that satisfies 1 < r < ∞, s − np = − nr . Step 1. We first deal with the envelope functions. Note that our assumptions imply α 1 1 := + < 1. (8.33) 0< rα n r s We apply an embedding result for spaces Bp,q (wα ) that can easily be verified using its atomic representation; cf. [HPxx], [Bow05]. Then a sequence space argument as in [KLSS06b, Thm. 1] gives the counterpart of (8.22), and, moreover,

Bps11 ,q (wα ) ,→ Bps22 ,q

if

0 < α ≤ δ = s1 −

n n − s2 + , p1 p2

(8.34)

s1 > s2 , 0 < p1 , p2 ≤ ∞, with p12 < p11 + α n , 0 < q ≤ ∞; see also [HS06]. For convenience we only stated the embedding result (8.34) when q1 = q2 = q; for extensions and further remarks see Remark 11.9 below. Consequently, choosing σ such that 0 < σ < s, and % such that s − np = σ − n% + α, (8.34) leads to s σ Bp,q (wα ) ,→ B%,q (8.35) with σ−

n n n =− −α=− . % r rα

By (8.33) we can apply Theorem 8.1 together with Proposition 3.4(iv) and obtain s Bp,q (wα )

EG

σ B%,q

(t) ≤ c EG

1

1

α

(t) ≤ c0 t− rα = c t− r − n ,

0 < t < 1.

(8.36)

Conversely, observe that ¡ ¢ n fjα (x) = 2j r +jα ψ 2j x ,

x ∈ Rn ,

(8.37)

s are atoms in Bp,q (wα ), where ψ is given by (3.36); we refer to the atomic decomposition result in [HPxx], [Bow05], and also to [HS06]. Hence, ° α s ° ¡ α ¢∗ ¡ −jn ¢ n °fj |Bp,q (wα )° ≤ 1, and fj c2 ∼ 2j r +jα , j ∈ N.

Thus s Bp,q (wα )

EG

¡

c 2−jn

¢

≥

¡ α ¢∗ ¡ −jn ¢ α 1 fj c2 ∼ 2jn( r + n ) ,

j ∈ N,

leading to the inequality converse to (8.36). Step 2. Obviously Step 1, combined with (8.35), Theorem 8.1 and Proposition 4.5 yields B s (wα ) uG p,q ≤ q,

Growth envelopes EG of function spaces Asp,q and it remains to disprove 1/v  ε Z h iv dt ° ° α 1 s   ≤ c °f |Bp,q (wα )° t r + n f ∗ (t) t

129

(8.38)

0

s for some c > 0 and all f ∈ Bp,q (wα ), if v < q. As usual, we construct extremal functions built upon the atoms (8.37), i.e., for a sequence b = {bj }j∈N of non-negative numbers we consider

f α (x) =

∞ X

bj fjα (x) =

j=1

∞ X

¡ ¢ n bj 2j r +jα ψ 2j x ,

x ∈ Rn .

(8.39)

j=1

By the above-mentioned atomic decomposition argument this leads to ° α s ° °f |Bp,q (wα )° ≤ c kb|`q k , (8.40) and

(f α )

∗

¡

¢ n c 2−jn ≥ c0 bj 2j r +jα ,

j ∈ N,

(8.41)

−J

so that – choosing J ∈ N with 2 ≤ ε, and b1 = · · · = bJ−1 = 0, for convenience – (8.38) and (8.40) lead to  1/v  1/v ∞ X X ¡ ¢¤ £ n v ∗  ≤ c1  bvj  2−j r −jα (f α ) c 2−jn  j=J

j=J

1/v  ε Z h iv dt α 1 ∗  ≤ c2  t r + n (f α ) (t) t °0 s ° ≤ c3 °f α |Bp,q (wα )° ≤ c4 kb|`q k .

(8.42)

B s (wα )

Now uG p,q ≥ q follows immediately and – in view of our introductory remark about the F -case – the proof is finished. Remark 8.11 We formulated Proposition 8.10 in the “sub-critical” setting parallel to Proposition 8.8, i.e., assuming s − np < 0. However, inspecting the above proof, we see that this assumption is much stronger than necessary, as – unlike the unweighted sub-critical situation Theorem 8.1 and the wα weighted counterpart in Proposition 8.8, where we need for the foot-point r in Figure 12, given by 1r = p1 − ns , that 1 < r < ∞, – the whole argument in the above weighted setting relies on 1 < rα < ∞ using notation (8.33). Thus we can accordingly weaken our assumptions and realise that Proposition 8.10 remains true for 0 < q ≤ ∞, s > 0, 0 < p < ∞ with ¶ µ n n <α
130

Envelopes and sharp embeddings of function spaces s s σ

11111111 00000000000 11111111111 00000000 00000000000 11111111111 00000000 11111111 00000000000 11111111111 00000000 11111111 00000000000 11111111111 00000000 11111111 00000000000 11111111111 00000000 11111111 00000000000 11111111111 00000000 11111111 00000000000 11111111111 00000000 11111111 00000000000 11111111111 00000000 11111111 00000000000 11111111111 00000000 11111111 s=

s=

n p

n p

+α

s Bp,q (wα )

σ B%,q

s=n

α

1 rα

1− α n

1 p

1

³

1 p

´ −1

1 p

1 %

Figure 13 Then, in particular,

³ ´ ³ s 1 α ´ s EG Bp,q (wα ) = t n − p − n , q .

(8.44)

This extension of admitted parameters, as indicated in Figure 13, obviously does not apply to weights of type wα , see Proposition 8.8.

8.2

Growth envelopes in sub-critical borderline cases

We study the situation s = σp = n

³

1 p

´

−1

+

now. Recall that this refers

to the thick lines in Figure 12. However, in this situation additional care is needed because not all spaces in question are contained in Lloc 1 , recall (7.43) and (7.44). We first consider the “bottom line” of the sub-critical strip in Figure 12; that is, where 1 < p < ∞, and s = 0, restricting q according to (7.43), (7.44). Proposition 8.12 Let 1 < p < ∞. (i) Assume 0 < q ≤ 2. Then ¡ 0 ¢ ³ −1 ´ EG Fp,q = t p, p . (ii) Assume 0 < q ≤ min(p, 2). Then ¶ µ ¡ 0 ¢ 1 B0 with EG Bp,q = t− p , uG p,q

(8.45)

B0

q ≤ uG p,q ≤ p.

(8.46)

Growth envelopes EG of function spaces Asp,q In particular,

Proof:

Step 1.

¡ 0 ¢ ³ −1 ´ EG Bp,p = t p, p ,

131

1 < p ≤ 2.

We use the following embeddings (see (7.32) and (7.7)) s 0 0 ,→ Fp,q ,→ Fp,2 = Lp , Fu,w

(8.47)

where 0 < q ≤ 2, s > 0, s− nu = − np and 0 < w ≤ ∞. Then Theorems 4.7(i), 8.1 and Proposition 3.4(iv) immediately yield F0

1

EG p,q (t) ∼ t− p . Similarly,

s 0 0 Bu,q ,→ Bp,q ,→ Fp,2 = Lp ,

(8.48)

− np ;

n u

for 0 < q ≤ min(p, 2), s > 0, s − = see (7.32) and (7.27). Hence Theorem 4.7(i) and Proposition 3.4(iv) as well as Theorem 8.1 lead to 0 Bp,q

EG

1

(t) ∼ t− p .

L

Step 2. Recall that uG p = p by (4.9). Thus Proposition 4.5 together with (8.47) and (8.48) provide A0

uG p,q ≤ p, the parameters being as above. We complete the proof for the F -case and F0

proceed by contradiction. Assume v := uGp,q < p; without restriction of generality we may put v > 1. Choose a number s > 0 such that u1 := s 1 1 p + n < v . Now (8.47) and another application of Proposition 4.5 give Fs

F0

uGu,w ≤ uGp,q = v. Fs

On the other hand we know by Theorem 8.1 that uGu,w = u, thus leading finally to Fs

F0

u = uGu,w ≤ uGp,q = v. This, however, contradicts our setting u > v. In the same way one proves B0

Bs

uG p,q ≥ uG u,q = q. B0

Remark 8.13 To prove uG p,q ≥ q one could also use the following example, 0 n which ¯ 0 ¯ is an adapted version of [Tri99, Sect. 3.2]. Assume x ∈ R with ¯x ¯ > 4. Put f (x) =

∞ X j=1

n

bj 2j p

¡ ¡ j ¢ ¡ ¢¢ ψ 2 x − ψ 2j x − x0 ,

x ∈ Rn ,

(8.49)

132

Envelopes and sharp embeddings of function spaces

¡ ¢ ¡ ¢¢ n ¡ where ψ is the function given by (3.36). Now 2j p ψ 2j x − ψ 2j x − x0 0 is an (0, p)K,0 - atom in Bp,q (first moment conditions are necessary), where K ≥ 1 can be chosen arbitrarily large. Assuming b = {bj }j∈N ∈ `q , a slightly modified version of [Tri99, (3.13)] (or Theorem 7.8), implies ° ° ° ¯ ° 0 ° °f |Bp,q ≤ c °b¯`q ° , (8.50) where c is independent of b ∈ `q . Moreover, assuming that b = {bj }j∈N additionally satisfies n

bj ≥ bj+1 ≥ d 2− p bj > 0,

j ∈ N,

(8.51)

n

for some d ∈ R, 1 < d < 2 p , then ¡ ¢ kn f ∗ 2−kn ∼ bk 2 p ,

k ∈ N.

(8.52)

Thus, for f given by (8.49) (with the additional assumptions on b ∈ `q ), and assuming 1/v  ε Z h iv dt ° ° 1 0 °   ≤ c °f |Bp,q (8.53) t p f ∗ (t) t 0

for some number v, we obtain by (8.50) and (8.52)  

∞ X

1/v bvj 

j=J

1/v 1/v  ε Z h ∞ h iv dt iv X ¡ ¢ n 1  ∼ ∼ 2−j p f ∗ 2−jn  t p f ∗ (t) t 

j=J



0

1/q

∞ X ° ° 0 ° ≤ c °f |Bp,q ≤ c0  bqj 

.

j=1

Now v ≥ q follows immediately for all v satisfying (8.53), but this means B0

uG p,q ≥ q.

³

Finally we study the line s = σp = n

1 p

´ − 1 , where 0 < p ≤ 1, q given

by (7.43), (7.44). ´ ³ Proposition 8.14 Let 0 < p ≤ 1 and s = n p1 − 1 . (i) Assume 0 < q ≤ ∞, and 0 < q ≤ 2 if p = 1. Then s ´ ¡ s ¢ ³ −1 Fp,q Fs EG Fp,q = t , uG with p ≤ uGp,q ≤ 1. In particular,

¡ 0 ¢ ¡ −1 ¢ = t , 1 , EG F1,q

0 < q ≤ 2.

(8.54)

Growth envelopes EG of function spaces Asp,q (ii) Assume 0 < q ≤ 1. Then s ´ ¡ s ¢ ³ −1 Bp,q = t , uG EG Bp,q

with

133

Bs

q ≤ uG p,q ≤ 1.

(8.55)

In particular, µ

¡ s ¢ ¡ −1 ¢ EG Bp,1 = t , 1 , Proof:

0 < p ≤ 1, s = n

¶ 1 −1 . p

Step 1. We have s 0 0 Fp,q ,→ F1,u ,→ F1,2 = h1 ,→ L1 ,

(8.56)

where 0 < u ≤ 2; see (7.10), and [Tri83, Thm. 2.5.8/1, Rem. 2.5.8/4] as well as (7.32) for the embeddings. Thus by Theorem 4.7(i), Fs

F0

EG p,q (t) ≤ c EG 1,u (t) ≤ c0 t−1 . Similarly we have

s 0 0 Bp,q ,→ B1,q ,→ B1,1 ,→ L1 ,

(8.57)

see Proposition 7.11; this implies s Bp,q

EG

0 B1,q

(t) ≤ c EG

(t) ≤ c0 t−1 ,

0 < q ≤ 1.

The corresponding estimates from below can be obtained as in Step 2 of the proof of Theorem 8.1, where r is replaced by 1 now. Moreover, one has to modify the functions fj given by (8.8) slightly, as now first moment conditions s are necessary for atoms in spaces Bp,q with s = σp , see Theorem 7.8. For j ∈ N let the functions fj be given by £ ¡ ¢ ¡ ¢¤ fj (x) := 2jn ψ 2j x − ψ 2j x − x0 , x ∈ Rn , (8.58) ¯ ¯ and x0 ∈ Rn with, say, ¯x0 ¯ > 4; this is a simplified version of Remark 8.13. s The function ψ is given by (3.36). Then fj is an (s, p)°K,0 - atom ° in Bp,q , s ° ° where K ≥ s + 1. Applying Theorem 7.8 we obtain fj |Bp,q ∼ 1 and, ¡ ¢ parallel to Step 2 of the proof of Proposition 8.1, fj∗ 2−jn ∼ 2jn , j ∈ N. Thus again ¢ ¡ ¢ Bs ¡ EG p,q 2−jn ≥ fj∗ 2−jn ∼ 2jn , j ∈ N . The rest is a matter of monotonicity, i.e., 0 B1,q

EG

s Bp,q

(t) ≥ c EG

(t) ≥ c0 t−1 ,

0 < t < 1.

The F -case follows now by elementary embeddings (7.27): choose for a given 0 0 q, 0 < q ≤ 2, a number u, 0 < u ≤ min(q, 1). Then B1,u ,→ F1,q , leading to F0

0 B1,u

EG 1,q (t) ≥ c EG

(t) ≥ c0 t−1 .

134

Envelopes and sharp embeddings of function spaces ´ ³ When 0 < p < 1 and s = n p1 − 1 , we take p0 such that 0 < p0 < p and ´ ³ s put s0 = n p10 − 1 . Consequently (7.39) implies Bps00 ,p ,→ Fp,q , and our just obtained result in the B-case yields Bps00 ,p

Fs

EG p,q (t) ≥ c EG

(t) ≥ c0 t−1 .

Step 2. We look for the smallest possible number v such that 1/v  ε Z ° °  [t f ∗ (t)]v dt  ≤ c °f |Asp,q ° t

(8.59)

0

holds for all f ∈ Asp,q . Clearly (8.56) and (8.57) together with Proposition 4.5 As

and Theorem 4.7(i) immediately yield uG p,q ≤ 1. Furthermore, a slightly modified version of (8.49) serves as an extremal function for the B-setting: with p replaced by 1, as in Remark 8.13, we obtain ° ° ¡ ¢ s ° °f |Bp,q ≤ c kb|`q k , and f ∗ 2−kn ∼ bk 2kn , k ∈ N, where b = {bj }j∈N ∈ `q with the counterpart of (8.51). Hence the B-version of (8.59) implies  

∞ X j=J

1/v bvj 

1/v 1/v  ε Z ∞ X £ −jn ∗ ¡ −jn ¢¤v dt v   ∼ 2 f 2 ∼  [t f ∗ (t)] t 

j=J

 1/q ∞ X ° ° s ° ≤ c °f |Bp,q ≤ c0  bqj 

0

j=1 Bs

for all numbers v satisfying (8.59). Thus uG p,q ≥ q, 0 < q ≤ 1; in particular, ¡ s ¢ ¡ −1 ¢ EG Bp,1 = t ,1 , 0 < p ≤ 1, s = σp . F0

s 0 Moreover, this also leads to uG1,q ≥ 1, 0 < q ≤ 2, because Bp,1 ,→ F1,q by (7.39). This completes the proof of (8.54) in case of p = 1, s = 0, i.e., ¡ 0 ¢ ¡ −1 ¢ EG F1,q = t ,1 , 0 < q ≤ 2. σ s ,→ Fp,q Moreover, let σ > s and σ − nr = s − np = σp . Then by (7.39), Br,p and Proposition 4.5 with our above B-result complete the proof in the F -case, too, Fs

Bσ

uGp,q ≥ uG r,p ≥ p.

Growth envelopes EG of function spaces Asp,q

135

Remark 8.15 Clearly Propositions 8.12 and 8.14 show that the borderline situation, in particular, the determination of the corresponding indices uX G, is rather complicated to handle and not yet solved completely (apart from some special cases). Even worse, a reasonable guess as to what the correct outcome could be is also missing. Concerning the “bottom line” – referring to Proposition 8.12 – one asks whether B-spaces with s = 0 show their “usual” B0

behaviour, i.e., uG p,q = q, independently of the delicate limiting situation, or if B0

they “suffer” from this setting and tend to behave like the F -spaces uG p,q = p – or something ´ in between. The situation is even more obscure on the line ³ s = n p1 − 1 , 0 < p ≤ 1.

8.3

Growth envelopes in the critical case

We deal with spaces Asp,q , where s = np , see Figure 11. Recall that for 0 < p ≤ ∞ (with p < ∞ for F -spaces), and 0 < q ≤ ∞, n/p

if, and only if,

0 < p ≤ 1,

0 < q ≤ ∞,

(8.60)

n/p

if, and only if,

0 < p ≤ ∞,

0 < q ≤ 1,

(8.61)

F p,q ,→ L∞ and B p,q ,→ L∞

where L∞ can be replaced by C; see Proposition 7.13. We shall study the remaining cases now. Theorem 8.16 Let 0 < p < ∞ and 0 < q ≤ ∞. (i)

Let 1 < p < ∞ and

1 p

+

1 p0

= 1, as usual. Then

³ ´ ³ ´ 1 n/p EG F p,q = |log t| p0 , p . (ii) Let 1 < q ≤ ∞ and

1 q

+

1 q0

(8.62)

= 1, as usual. Then

³ ´ ³ ´ 1 n/p EG B p,q = |log t| q0 , q .

(8.63)

P r o o f : This theorem can be found in [Tri01, Thm. 13.2]; we include here – mainly for completeness and later reference – a slightly different proof as presented in [Har01] and based on [ET99b]. Essential ideas, in particular in Step 3, can be found in [Tri01, Thm. 13.2].

136

Envelopes and sharp embeddings of function spaces

Note that due to (7.39) and Propositions 3.4(iv) and 4.5 it is sufficient to deal with the B-case. Step 1.

We study the envelope function first and prove ° ° f ∗ (t) ° n/p ° ≤ c f |B ° ° 0 p,q | log t|1/q

sup 0
(8.64)

n/p

for all f ∈ B p,q , where 0 < p < ∞ and 1 < q ≤ ∞. Assume first q = ∞ so that (8.64) reads as sup 0
° ° f ∗ (t) ° n/p ° ≤ c °f |B p,∞ ° | log t|

(8.65)

n/p

for all f ∈ B p,∞ . Note that by [BS88, (7.22)] one has (at least locally, but this is sufficient for our purpose) bmo ,→ Lexp,1 ,

(8.66)

see (7.11) for the definition of bmo . On the other hand, Proposition 3.18 yields 1 L a EG exp,a (t) ∼ |log t| , 0 < t < , 2 so by Proposition 3.4(iv) and (8.66) it is sufficient to prove n/p

n/p

B p,q ,→ B p,∞ ,→ bmo ,

0 < p < ∞,

0 < q ≤ ∞.

(8.67)

0 , 0 < p < ∞, where hp are This can be seen as follows: by (7.10), hp = Fp,2 the local Hardy spaces. Then by an application of (7.39), −n(1− r1 )

0 h1 = F1,2 ,→ Br,1

,

1 < r < ∞,

0

and because of bmo = (h1 ) , see [Gol79b], we arrive at ¶0 µ ¡ 0 ¢0 −n(1− r1 ) n/r 0 0 Br0 ,∞ = Br,1 ,→ F1,2 = (h1 ) = bmo , 1 < r0 < ∞,

(8.68)

where the first identity is covered by [Tri83, Thm. 2.11.2]. By (7.32) we thus have (8.67) for all p, 0 < p < ∞, and (8.65) is verified. In order to prove (8.64) for 1 < q < ∞ we use a (nonlinear) real interpolation argument as follows. Let T :

f 7−→ f ∗∗

and

µa (dt) =

dt , | log t|a

(8.69)

where f ∗∗ is given by (2.14). Note that one can replace f ∗ in (8.65) by f ∗∗ , see [ET99b, Prop. 2.3] or the extended version of Hardy’s inequality

Growth envelopes EG of function spaces Asp,q

137

in [BR80, Thm. 6.4]. Thus (the so modified version of) (8.65) implies that ¢ ¡¡ ¢ n/p T : B p,∞ −→ L∞ 0, 21 , µ1 . Moreover, 1

1

Z2

Z2 f ∗∗ (t)dt =

0

0

1

Zt

1 t

Z2 f ∗ (s) ds

f ∗ (0)dt

≤

0

0

° ° ° n/p ° ≤ c f (0) = c kf |L∞ k ≤ c0 °f |B p,1 ° , ∗

(8.70)

¢ ¡¡ ¢ n/p by the monotonicity of f ∗ and (7.41). Hence T : B p,1 −→ L1 0, 21 , µ0 , with µ0 (dt) = dt. We use the replacement of f ∗ by f ∗∗ in (8.65) now, because the sub-additivity of f ∗∗ (2.15) immediately gives the Lipschitzcontinuity of T . The nonlinear real interpolation as stated in [Tar72, Thm. 4] leads to ³ ´ ³ ´ n/p n/p −→ L∞ (µ1 ), L∞ (µ0 ) (8.71) T : B p,∞ , B p,1 θ,q

θ,q

with 0 < θ < 1 and 1 < q < ∞. Concerning the right-hand side of (8.71) we may proceed by ³ ´ ³ ´ = L∞ (µ1−θ ), (8.72) ,→ L∞ (µ1 ), L∞ (µ0 ) L∞ (µ1 ), L∞ (µ0 ) θ,∞

θ,q

see [BL76, Thm. 5.4.1]. On the other hand, for 1 < p < ∞ and with 1−θ θ 1 q = ∞ + 1 = θ we conclude by [Tri78a, Thm. 2.4.1(b)], ³

n/p

n/p

B p,∞ , B p,1

´

n/p

θ,q

= B p,q ,

Summarising (8.71) – (8.73) results equivalently, ° ° °T f |L∞ (µ1/q0 )° ≤ c

in

θ=

1 , q

T

: B p,q

1 < p < ∞. n/p

° ° ° n/p ° °f |B p,q ° ,

(8.73)

−→ L∞ (µ1/q0 ), or,

1
In view of (8.69) this yields exactly (8.64) when 1 < p < ∞. However, the remaining case 0 < p ≤ 1 now simply follows by the monotonicity of B-spaces. In the same way as before (8.64) yields n/p Bp,q

EG

1

(t) ≤ c |log t| q0 ,

B n/p

0
1 . 2

1

Step 2. We prove EG p,q (t) ∼ |log t| q0 . We benefit from the construction of extremal functions as presented in [ET99b] by Edmunds and Triebel, recall also Remark 8.13. Let 0 < p < ∞, 1 ≤ q ≤ ∞; put fb (x) =

∞ X j=1

¡ ¢ bj ψ 2j−1 x ,

x ∈ Rn ,

(8.74)

138

Envelopes and sharp embeddings of function spaces ∞

where ψ is the standard function given by (3.36) and b = {bj }j=1 is a sequence of positive, monotonically decreasing numbers with b ∈ `q . Let J ∈ N and put bj ≡ 1, j = 1, . . . , J, and bj ≡ 0 for j > J; then kbj |`q k = J 1/q . Denote by fJ the function given by (8.74) with the abovedescribed sequence; then (understanding (8.74) as an atomic decomposition), we have ° ° 1 ° n/p ° °fJ |B p,q ° ≤ kbj |`q k ∼ J q . On the other hand, ½ fJ∗ (t)

∼

J , t ≤ 2−Jn , |log t|, 2−Jn ≤ t ≤ 1 .

(8.75)

Hence we have for 0 < p < ∞, 1 < q ≤ ∞, n/p Bp,q

EG

1

1

1

(t) ≥ sup J − q fJ∗ (t) ≥ c J − q |log t| ∼ |log t| q0 ,

t ∼ 2−Jn .

(8.76)

J∈N

Step 3.

We come to the indices and have thus to prove that  ε Z ·  0

∗

f (t) | log t|1/q0 +1/v

¸v

 v1

° ° dt  ° n/p ° ≤ c °f |B p,q ° , t

(8.77)

B n/p

is satisfied if, and only if, v ≥ q. First we prove uG p,q ≤ q. Without restriction of generality we may assume 1 < p < ∞ now, the extension to 0 < p ≤ 1 being a matter of (7.32) and Proposition 4.5. Moreover, by (8.65) we may also exclude q = ∞ here and are thus left to verify  q1  ε Z · ∗ ¸q ° ° dt f (t) n/p °  ≤ c °  °f |B p,q ° , | log t| t

1
(8.78)

0

Let f (x) =

∞ X

fj (x)

with fj (x) =

X

λjm ajm (x)

(8.79)

m∈Zn

j=0 n/p

be an atomic decomposition of f ∈ B p,q in the sense of Theorem 7.8. Note that we need no moment conditions in our setting. Here λjm ∈ C, j ∈ N0 , m ∈ Zn , with  Ã ! pq  q1 ∞ X X  < ∞. Λ := kλ|bpq k =  |λjm |p j=0

m∈Zn

(8.80)

Growth envelopes EG of function spaces Asp,q

139

£ ¢ −jn Let χjl be the characteristic function of the interval D 2−jn l n (lo− 1), D 2 on R+ , where D > 0, j ∈ N0 , and l ∈ N. Denote by λ∗jl the nonl∈N

increasing rearrangement of the sequence {λjm }m∈Zn , where j ∈ N0 is fixed. If D > 0 is chosen in an appropriate way, then fj∗ (t) ≤

∞ X

λ∗jl χjl (t),

j ∈ N0 ,

l=1

see, Example 2.4. Let t > 0 be such that D 2−jn (l − 1) ≤ t < D 2−jn l. Then Zt l 1X ∗ 1 ∗∗ fj∗ (τ )dτ ≤ c λjk =: c λ∗∗ fj (t) = (8.81) jl . t l k=1

0

n o∞ is a monotonically decreasing sequence, it As 1 < p < ∞ and λ∗jk k=1 follows by the sequence version of the Hardy-Littlewood maximal inequality (see, for instance, [CS90, Lemma 1.5.3]) that ∞ X ¡

λ∗∗ jl

¢p

l=1

∼

∞ X ¡

λ∗jk

¢p

=

X

|λjm |p .

(8.82)

m∈Zn

k=1

Assume D 2−(k+1)n ≤ t < D 2−kn for some k ∈ N. By the sub-additivity (2.15), and (8.79), (8.81) (with l = 1 and l = 2(j−k)n , respectively) we arrive at ∞ k ∞ X X X f ∗∗ (t) ≤ fj∗∗ (t) ≤ c λ∗∗ λ∗∗ j1 + c j2(j−k)n . j=0

j=0

j=k+1

Our assumption 1 < q < ∞ thus implies 1

Z2 µ 0

f ∗ (t) | log t|

¶q

¶q f ∗∗ (D 2−kn ) k k=1 q q   k ∞ ∞ ∞ X X X X  (8.83) 1 1 λ∗∗  + c2 λ∗∗ ≤ c2 j2(j−k)n k j=0 j1 k j=k+1 k=1 k=1 {z } {z } | | =: Λq1 =: Λq2 ∞

X dt ≤ c1 t

µ

The sequence space version of the Hardy-Littlewood maximal inequality yields for the sum Λq1 that q  k ∞ ∞ X X X ¡ ∗∗ ¢q 1 q ∗∗   λ ≤ c λj1 ≤ c0 Λq , (8.84) Λ1 = k j=0 j1 j=0 k=1

140

Envelopes and sharp embeddings of function spaces

where we used (8.82) and (8.80). Put temporarily Ã

! p1

X

Bj =

p

,

|λjm |

j ∈ N,

m∈Zn

for convenience. Then (8.80) reads as Λq =

∞ X

Bjq . On the other hand,

n j=0 o (8.82) and the monotonicity of the sequence λ∗∗ – based on the monojl l∈N n o tonicity of λ∗jl – imply l∈N

Bjp

∞ ∞ X ¡ ∗∗ ¢p ¡ ¢p X ∼ λjl = λ∗∗ + j1 l=1

≥c

ln

2 X

¡ ∗∗ ¢p λjk

l=1 k=2(l−1)n +1

∞ ³ X

λ∗∗ j2ln

´p

³ ´p 2(l−1)n ≥ c λ∗∗ 2(l−1)n j2ln

l=1

for some c > 0 and any j ∈ N0 , l ∈ N; that is, nl

−p Bj , λ∗∗ j2ln ≤ C 2

j ∈ N0 ,

l ∈ N.

However, this implies q q   ∞ ∞ ∞ X X X (j−k)n −  ≤ c0  ≤ c  p B Blq 2 λ∗∗ (j−k)n j j2 l=0

j=k+1

j=k+1

and, for q > 1, Λq2 =

∞ X k=1

≤ c0

 1 k

∞ X

∞ X

q  ≤ c λ∗∗ j2(j−k)n

j=k+1

∞ ∞ X 1 X q Bl kq

k=1

Blq = c0 Λq .

l=0

(8.85)

l=0

So finally (8.83) together with (8.84) and (8.85) result in 1

Z2 µ 0

f ∗ (t) | log t|

¶q

dt ≤ cΛ. t

(8.86)

Taking the infimum over all representations (8.79) with (8.80) we obtain B n/p

(8.78), i.e., uG p,q ≤ q, 0 < p < ∞, 1 < q ≤ ∞.

Growth envelopes EG of function spaces Asp,q Step 4.

141

B n/p

It remains to show uG p,q ≥ q, i.e., we shall disprove 

Zε ·

 0

f ∗ (t) | log t|1/q0 +1/v

¸v

1/v ° ° dt  ° n/p ° ≤ c °f |B p,q ° , t

(8.87)

n/p

for some c > 0 and all f ∈ B p,q , whenever v < q. We proceed by contradiction; that is, assume v < q. Let J ∈ N and 1

1

bj = j − q (loghji)− v ,

j = 1, . . . , J;

then  J ° ° X ° n/p ° °fb |B p,q ° ≤ c kb|`q k = c  j=1

1/q 1  ≤ c2 < ∞, j (loghji)q/v

since v < q. Note that c2 does not depend on J ∈ N. On the other hand, fb∗ (2−kn ) ≥ c

k X

1

bj ≥ c k bk ∼ k q0 (loghki)

− v1

,

k = 1, . . . , J,

j=1

and so 

Zε ·

 0

fb∗ (t) | log t|1/q0 +1/v

¸v

 v1

dt  ∼ t

Ã

J · ∗ −kn ¸v X f (2 )

k=1

b k 1/q0 +1/v

! v1

à ≥ c

J X

k=1

1 k loghki

! v1 .

Now it is clear that the expression on the right-hand side diverges for J → ∞, n/p such that there are functions fb ∈ B p,q , not satisfying (8.87). This completes the proof.

Remark 8.17 In analogy to (8.13) and (8.14) in Remark 8.2 we see that ³ ´ ³ ³ ´ ´ 1 n/p EG L∞,p (log L)−1 = |log t| p0 , p = EG F p,q , (8.88) where 0 < q ≤ ∞ and 1 < p < ∞; cf. Theorem 4.7(iii) and (8.62). Correspondingly the situation in B-case reads as ³ ´ ³ ³ ´ ´ 1 n/p EG L∞,q (log L)−1 = |log t| q0 , q = EG B p,q , (8.89) where 0 < p < ∞ and 1 < q ≤ ∞. This follows by Theorem 4.7(iii) and (8.63).

142

Envelopes and sharp embeddings of function spaces

Remark 8.18 Recall Remark 8.4 on the studies of spaces of generalised smoothness by Caetano and Moura; for the critical case their parallel result in [CM04a, Thm. 4.4] reads as ³ ´ (n/p,Ψ) EG Bp,q = (Φ∞,q0 (t), q) , (8.90) and

³ ´ (n/p,Ψ) EG Fp,q = (Φ∞,p0 (t), p) ,

(8.91)

Ψ is an admissible function, 0¡ <¡ p < ¡where ¡ ¢¢ ¢¢ ∞, 0 < q ≤ ∞, and Ψ 2−j j∈N 6∈ `q0 in the B-case, and Ψ 2−j j∈N 6∈ `p0 in the F -case. The auxiliary function Φr,u : (0, 2−n ] → R, 0 < r, u ≤ ∞, defined as à Z1 y

Φr,u (t) :=

−n ru

Ψ(y)−u

dy y

!1/u ,

t1/n

and introduced in [CM04a], has the additional advantage that both the subcritical and critical result can be summarised using this notation: in our setting we have Ψ ≡ 1 and consequently à Z1 Φr,u (t) :=

y

−n ru

dy y

!1/u ,

(8.92)

t1/n

leading to the following reformulation of Theorems 8.1 and 8.16: Let 0 < q ≤ ∞, s > 0, 1 < r ≤ ∞ and p with 0 < p < ∞ be such that s − np = − nr . ¡ ¢ s (i) Assume 1 < q ≤ ∞ only if s = np . Then EG (Bp,q ) = Φr,q0 (t), q . (ii) Assume 1 < p < ∞ only if s =

n p.

s EG (Fp,q ) =

Then

¡ ¢ Φr,p0 (t), p .

We mention other related work on growth envelopes in the critical case: there is the recent paper on anisotropic results [MNP06], as well as the investigations for spaces of generalised smoothness, [CF06], [CL06]. In the case of spaces with dominating mixed derivatives some envelope results can be found in [KS05], [KS06].

Example 8.19 In view of our Example 7.5 we obtain for 0 < r < ∞, 0 < u ≤ ∞, b ∈ R, 

1/u

Z1 n

Φbr,u (t) = 

−bu

y − r u (1 + | log y|) t1/n

dy  y

1

−b

∼ t− r |log t|

(8.93)

Growth envelopes EG of function spaces Asp,q

143

for 0 < t ≤ 21 , and when r = ∞, b < 1/u, 0 < u ≤ ∞, 

1/u

Z1 −bu

Φb∞,u (t) = 

(1 + | log y|)

dy  y

− b− 1 ∼ |log t| ( u )

(8.94)

1 . 4

(8.95)

t1/n

for 0 < t ≤ 21 ; when b = 1/u > 0, then Φb∞,u (t) ∼ (log |log t|)

1/u

,

0
Let 0 < p < ∞, 0 < q ≤ ∞, s = np , and b < 1/q 0 according to Example 7.22, then by (8.94) and (8.90), ³ ´ ³ ´ 1 −b n/p,b EG Bp,q = |log t| q0 , q , whereas for 1 < q ≤ ∞ and b = 1/q 0 , we obtain ³ ´ ³ ´ 0 n/p,b EG Bp,q = (log |log t|)1/q , q .

Remark 8.20 Studying spaces on a bounded domain Ω ⊂ Rn , say with |Ω| < 1, Theorem 8.16 can be rewritten as n/p

F p,q (Ω) ,→ Lexp, 10 (Ω), p

1 < p < ∞,

0 < q ≤ ∞,

(8.96)

0 < p < ∞,

1 < q ≤ ∞,

(8.97)

and n/p

B p,q (Ω) ,→ Lexp, 10 (Ω), q

recall L∞,∞ (log L)−a = Lexp,a , a ≥ 0. Note that (8.96) and (8.97) are nothing else than the classical results mentioned in Remark 3.28, extended to all reasonable cases in the context of Bor F -spaces. Moreover, the history of papers devoted to critical embeddings in the above sense is very long already; we mentioned in Remark 8.3 some of the relevant papers. Additionally we shall refer to Peetre [Pee66], Bennett, Sharpley [BS88, Ch. 4] and Triebel in [Tri93]. and [ET99b, Rem. 2.6] for an extensive discussion of the history of embeddings of that “critical” type. In view of (7.40) and (7.41) and our preceding comment we may summarise the corresponding results as follows. Recall that we put r10 := 0 when 0 < r < 1, and Lexp,0 = L∞ .

144

Envelopes and sharp embeddings of function spaces

Corollary 8.21 Let Ω ⊂ Rn with |Ω| < 1. Let 0 < p < ∞ and 0 < q ≤ ∞. Then n/p

if, and only if,

a ≥

1 , p0

(8.98)

n/p

if, and only if,

a ≥

1 . q0

(8.99)

F p,q (Ω) ,→ Lexp,a (Ω) and B p,q (Ω) ,→ Lexp,a (Ω)

P r o o f : The cases 0 < p ≤ 1 in (8.98) and 0 < q ≤ 1 in (8.99) are clear, see Proposition 7.13. For the remaining ones, the sufficiency results n/p from (8.96) and (8.97), respectively. Now let F p,q (Ω) ,→ Lexp,a (Ω). We 1 proceed by contradiction; that is, assume a < p0 . Then obviously 1 − a > p1 and we may choose a number r with p1 < 1r < 1 − a. Consequently, 1/r  1 1/r  1 Z2 · ∗ ¸r Z2 ∗ f (t)  dt  f (t) dt   ≤ sup | log t|−(1−a)r    a  1 | log t| | log t| t t 0
0

0

∗

≤ c sup 0
because

1 r

f (t) ≤ c0 kf |Lexp,a (Ω)k , | log t|a n/p

< 1 − a. Our assumption F p,q (Ω) ,→ Lexp,a (Ω) thus implies   

1

Z2 · 0

∗

f (t) | log t|

¸r

1/r dt   t

° ° ° ° n/p ≤ c °f |F p,q (Ω)°

for some r < p. This, however, contradicts the sharpness assertion in (8.62). The proof in the B-case is similar.

Remark 8.22 Note that by (7.40) and (7.41), together with (7.32) and elementary embedding properties of B- and F -spaces it is obvious that Asp,q ,→ As

L∞ in the super-critical case, see Figure 11. Thus we know that EG p,q (t) is bounded in the super-critical case and thus, by our convention, EG (Asp,q ) = ¡ ¢ 1, ∞ when 0 < p ≤ ∞ (p < ∞ in the F -case), s > np , and 0 < q ≤ ∞. Let Ω ⊂ Rn be bounded, say, with |Ω| < 1, and bmo (Ω) be the local space of functions of bounded mean oscillation given by (a suitably adapted modification of) (7.11). Note that by (8.66) and Propositions 3.18 and 3.4(iv)

Growth envelopes EG of function spaces Asp,q

145

we have EGbmo (t) ≤ c |log t| for small t > 0. Conversely, (8.67) and Theorem 8.16(ii) with q = ∞ lead to n/p Bp,∞

EGbmo (t) ≥ c EG

(t) ≥ c0 |log t| .

By the same two embedding arguments, now combined with Proposition 4.5, we obtain ubmo = ∞, i.e., G ³ ´ EG bmo (Ω) = (|log t| , ∞) ; (8.100) see also [Tri01, (13.89)]. We finally turn to the situation p = ∞, excluded so far and start with the F -spaces. We study the connection between spaces of 0 type F∞,q and bmo , appearing (though secretly hidden) in (8.68) already; recall Remark 7.9. In the critical case s = 0, p = ∞, one has for 0 < q ≤ 2, 0 0 F∞,q ,→ F∞,2 = bmo ,→ Lloc 1 .

(8.101) s+n/p

Conversely, Marschall proved in [Mar95, Lemma 16], that Bp,∞ for all s ∈ R, 0 < p < ∞, and 0 < q ≤ ∞; in particular, n/p 0 Bp,∞ ,→ F∞,q ,

s ,→ F∞,q

0 < p < ∞, 0 < q ≤ ∞ ;

(8.102)

the case q ≥ 1 is already covered by [Mar87, Cor. 4]. Combining (8.100), Theorem 8.16(ii), and (8.101), (8.102) we arrive at 0 Corollary 8.23 Let 0 < q ≤ 2, and F∞,q be given by (7.23). Then ¡ 0 ¢ EG F∞,q = (|log t| , ∞) .

0 Recall that in the parallel B-case we have by (7.41) that B∞,q ,→ L∞ 0 if, and only if, when 0 < q ≤ 1. On the other hand, by (7.44) B∞,q ⊂ Lloc 1 0 < q ≤ 2. So we are left to consider the case 1 < q ≤ 2.

Proposition 8.24 Let 1 < q ≤ 2. Then there are positive constants c1 , c2 so that for all small t > 0, 1

0 B∞,q

c1 |log t| q0 ≤ EG

(t) ≤ c2 |log t| .

(8.103)

P r o o f : Note that (8.63), together with Proposition 3.4(iv) and the elemenn/p 0 tary embedding B p,q ,→ B∞,q imply 0 B∞,q

EG

1

(t) ≥ c |log t| q0

146

Envelopes and sharp embeddings of function spaces

for small t > 0 and all admitted q. On the other hand, by [Tri83, pp. 37, 50, 0 0 93] one has B∞,2 ,→ F∞,2 = bmo , thus by (8.66) and (4.13) we immediately arrive at B0 EG ∞,q (t) ≤ c |log t| for any q, 0 < q ≤ 2, and small t > 0.

Chapter 9 Continuity envelopes EC of function spaces Asp,q

We study continuity envelopes, introduced in Chapters 5 and 6, in the context of function spaces of type Asp,q , where 0 < p ≤ ∞ (with p < ∞ in the F -case), 0 < q ≤ ∞, and np ≤ s ≤ np + 1.

9.1

Continuity envelopes in the super-critical case

We deal with the super-critical case of spaces of type Asp,q as introduced in Figure 11, i.e., np < s ≤ np +1. In view of Proposition 7.14 and (7.47) such spaces can be embedded into C. Hence it is reasonable to study their continuity envelope function. On the other hand, when s > np +1, we may conclude that Asp,q are continuously embedded in Lip1 , see (7.45) and (7.46), so that by Proposition 5.3(ii) the corresponding continuity envelope functions are bounded and thus of no further interest. We postpone the borderline case s = np + 1 to the next section.

s

s= ³

1 p, s

n p

+1

´

1 σ

Proposition 9.1 Let 0 < q ≤ ∞ and 0 < s < 1. Then ³ ´ ³ −(1−s) ´ s = t ,q . EC B∞,q

s=

n p

1 p

Figure 14

(9.1)

147

148

Envelopes and sharp embeddings of function spaces

Proof:

Step 1. Recall that by (7.3), 1/q  1 ¸q Z2 · ° ° ω(f, t) dt   s ° °f |B∞,q ∼ kf |Ck +   . ts t

(9.2)

0

(Note that we consider s as fixed in this context, so that dependences of constants appearing on s do not matter; otherwise ° °one has to deal very cares s ° fully with s ↑ 1.) Let f ∈ B∞,q with °f |B∞,q ≤ 1. Then by (9.2) and Proposition 5.3(i) we obtain for any number τ , 0 < τ < 21 ,  c1

ω(f, τ )  τ

Zτ 0

 q1

 q1  τ ¸q Z · ° ° ω(f, t) dt  s ° t(1−s)q−1 dt ≤ c2  ≤ °f |B∞,q ≤ 1, s t t 0

hence

ω(f, τ ) ≤ c τ −(1−s) . τ

This implies

Bs

EC ∞,q (t) ≤ c t−(1−s) for small values of t > 0. Step 2. We show the converse inequality; we consider functions fs given by n o fσ,κ (x) = σ −κ |x|κ+s χ[0,σ) (|x|) + ψ(x)χ[σ,∞) (|x|) ,

x ∈ Rn ,

with 0 < σ < 21 and κ > 0, and the continuous, non-negative, monotonically decreasing function ψ is chosen so that fσ,κ is continuous, kfσ,κ |Ck ∼ σ s < 1. κ+s We obtain ω(fσ,κ , t) ∼ σ −κ [min(t, σ)] and thus ¸q !1/q ω(fσ,κ , t) dt ts t 0   !1/q  à !1/q à Z21    Zσ t−sq−1 dt ≤ c σ −κ tκq−1 dt + σ κ+s    

à Z21 ·

σ

0

≤Cσ

© −κ

σ κ + σ κ+s σ

ª −s

≤ C 0,

s such that kfσ,κ |B∞,q k ≤ c for 0 < σ < 21 . On the other hand, for small t > 0, ω(fσ,κ , t) ω(ft,κ , t) −(1−s) Bs ≥ ∼ t . EC ∞,q (t) ≥ sup 1 t t 0<σ< 2

Continuity envelopes EC of function spaces Asp,q

149

Step 3. It remains to determine the smallest number v satisfying (6.2) where µC (dt) ∼ dt t , that is, 

Zε ·

 0

ω(f, t) t · t−(1−s)

¸v

1/v dt  t

s ≤ c kf |B∞,q k

s for f ∈ B∞,q . But in view of (9.2) and Proposition 4.1, v ≥ q is obvious.

We study spaces Asp,q now belonging to the “super-critical strip” (without the borderlines so far), that is, 0 < s − np < 1, 0 < p < ∞, see Figure 14. Theorem 9.2 Let 0 < p < ∞, 0 < q ≤ ∞,

and

n p

n p

<s<

+ 1. Then

³ ´ ³ −(1+ n −s) ´ s p , q EC Bp,q = t

(9.3)

³ ´ ³ −(1+ n −s) ´ s p , p . EC Fp,q = t

(9.4)

P r o o f : Step 1. We start with the B-setting. Let σ = s − np be given as in σ s and ,→ B∞,q Figure 14, so that 0 < σ < 1. The elementary embedding Bp,q (9.1) imply Bs EC p,q (t) ≤ c t−(1−σ) for small t > 0. Conversely, let ¡ ¢ fj (x) = 2−jσ ϕ 2j x ,

j ∈ N,

(9.5)

where ϕ is of the following type. We start with some (non-smooth) function ½ 0 , |x| ≥ 1, ϕ(x) e = x ∈ Rn , 1 − |x| , |x| ≤ 1, ¡ ¢ such that supp ϕ e 2j · ⊂ {y ∈ Rn : |y| ≤ 2−j }, j ∈ N, and ¡ ¡ j¢ ¢ ω ϕ e 2 · ,t = 2j , t ∼ 2−j , j ∈ N . t Now we mollify the function ϕ e in a standard way and obtain some smooth descendant ϕ with similar properties. We realise that fj given by (9.5) is a s Bp,q - atom we ° (as ° do not need moment conditions again). In particular, this s ° ° means fj |Bp,q ∼ 1. Moreover, ω (fj , t) ∼ 2j(1−σ) , t

t ∼ 2−j ,

j ∈ N,

150

Envelopes and sharp embeddings of function spaces

so that

¡ ¢ ω fj , 2−j ≥ c0 2j(1−σ) , 2 ≥ c 2−j and by standard arguments, Bs EC p,q

¡

−j

¢

Bs

EC p,q (t) ≥ c t−(1−σ) ,

j ∈ N,

0 < t < 1,

as desired. Step 2. We show that 

Zε ·

 0

ω(f, t) tσ

¸v

1/v ° ° dt  s ° ≤ c °f |Bp,q t

(9.6)

Bs

s s σ for all f ∈ Bp,q if, and only if, v ≥ q, i.e., uC p,q = q. By Bp,q ,→ B∞,q Bs

Bσ

and Propositions 9.1 and 6.4 it follows uC p,q ≤ uC ∞,q = q. The converse inequality needs some more effort. For this purpose we construct extremal functions based on a combination of the functions fj given by (9.5). Put f (x) :=

∞ X

¡ ¢ bj 2−jσ ϕ 2j x − y j ,

x ∈ Rn ,

(9.7)

j=1

where the function ϕ behaves as described in Step 1 above, and bj > 0, j∈ Moreover, we¡ can choose y j ∈ Rn , j ∈ N, such that the supports of ¡ N. ¢ ¢ j j k k ϕ 2 · −y and ϕ 2 · −y are disjoint for k 6= j. Note that –¡in view of¢ our above remarks concerning ϕ – the functions gj (x) = 2−jσ ϕ 2j x − y j s are (s, p)K,−1 - atoms (no moment conditions) in Bp,q , where K ≥ s + 1 γ −jσ+j|γ| can be chosen arbitrarily large, |D gj (x)| ≤ 2 = 2−j(s−n/p)+j|γ| , |γ| ≤ K. Let b = {bj }j∈N ∈ `q ; then by Theorem 7.8, ° ° s ° °f |Bp,q ≤ c kb|`q k .

(9.8)

In addition, we shall assume now bj+1 ≤ bj ≤ 21−σ bj+1 ,

j ∈ N.

(9.9)

Then by construction (9.7) (and our preceding remarks concerning the function ϕ ) we obtain ¡ ¢ ω f, 2−j ∼ 2−j sup bk 2k(1−σ) = bj 2−jσ . k≤j

Now it is clear, that (9.9) is needed for this last observation only. Moreover, it is the natural replacement of (8.51) (when dealing with the sub-critical case

Continuity envelopes EC of function spaces Asp,q

151

in terms of growth envelopes). Thus (9.6) implies for those extremal functions f given by (9.7) with (9.8) (in the above-described setting),  

∞ X

1/v bvj 

j=J

1/v " ¡ ¢ #v 1/v Zε · ¸v ∞ X ω f, 2−j dt ω(f, t)   ∼ ∼ 2−jσ tσ t 

j=J

0

 1/q ∞ X ° ° q s 0 ≤ c °f |Bp,q ° ≤ c  bj  j=1

and v ≥ q is obvious. This finishes the proof of (9.3). Step 3. It remains to verify (9.4). Let s0 > s > σ, s0 − then (7.39) implies

n p0

= s−

n p

= σ,

s σ Bps00 ,p ,→ Fp,q ,→ B∞,p

and Propositions 5.3(iii) and 6.4 together with our just obtained B-results (9.3) and Proposition 9.1 complete the proof.

Remark 9.3 Note that with the help of (5.13) and Theorem 8.1 one easily Bs

Fs

derives uC p,q ≤ q and uC p,q ≤ p, respectively, in Theorem 9.2: simply put κ = σ, s = σ + np , 1r = 1−σ n , and make use of the lifting property (7.8) for Asp,q -spaces, and also our argument in Step 2 of the proof of Theorem 9.4 below. Parallel to Remarks 8.2 and 8.17 we mention that Theorem 6.6, Proposition 9.1 and Theorem 9.2 lead to ´ ³ ³ ´ ³ −(1−σ) ´ σ+n/p = t , q = EC Lip(σ,0) EC Bp,q ∞,q for 0 < p ≤ ∞, 0 < q ≤ ∞, 0 < σ < 1, and ´ ³ ³ ´ ³ −(1−σ) ´ σ+n/p = t , p = EC Lip(σ,0) EC Fp,q ∞,p where 0 < p < ∞, 0 < q ≤ ∞, 0 < σ < 1. Comparison with the situation of spaces of generalised smoothness is postponed to Remark 9.7 below.

152

9.2

Envelopes and sharp embeddings of function spaces

Continuity envelopes in the super-critical borderline case

We come to the borderline case s = np + 1. Let 0 < p ≤ ∞ (with p < ∞ for F -spaces), 0 < q ≤ ∞ and α ≥ 0. Then, by Proposition 7.15, 1+n/p Fp,q ,→ Lip(1,−α)

if, and only if,

α≥

1 , p0

(9.10)

1+n/p Bp,q ,→ Lip(1,−α)

if, and only if,

α≥

1 . q0

(9.11)

if, and only if,

0 < p ≤ 1 and

0 < q ≤ ∞,

(9.12)

if, and only if,

0
0 < q ≤ 1,

(9.13)

and

In particular, 1+n/p Fp,q ,→ Lip1

and 1+n/p ,→ Lip1 Bp,q

and

is the (lifted) counterpart of (7.40) and (7.41). Hence, in view of Proposition 5.3(ii) it is clear that spaces given by (9.12) and (9.13), respectively, are of no further interest in our context, because the corresponding envelope functions are bounded. We shall study the remaining cases now. Recall Proposition 5.10; this enables us to give our result in the “borderline” super-critical case when s = np + 1. Theorem 9.4 Let 0 < p < ∞ and 0 < q ≤ ∞. (i)

Let 1 < p < ∞ and

1 p

+

1 p0

= 1, as usual. Then

³

´ ³ ´ 1 1+n/p EC Fp,q = |log t| p0 , p . (ii) Let 1 < q ≤ ∞ and

1 q

+

1 q0

(9.14)

= 1, as usual. Then

³ ´ ³ ´ 1 1+n/p = |log t| q0 , q . EC Bp,q

(9.15)

P r o o f : Step 1. Clearly, by Propositions 5.3(iii) and Theorem 6.6(ii), as well as (9.10), (9.11), we obtain F 1+n/p

EC p,q

1

(t) ≤ c | log t| p0 ,

B 1+n/p

EC p,q

1

(t) ≤ c0 | log t| q0 .

We have to show the converse inequalities. In view of the construction presented for the proof of Theorem 8.16, we look for some extremal functions

Continuity envelopes EC of function spaces Asp,q in the sense of (8.74). We shall use (14.15)-(14.19)] in the following way. ( − 1 e δ2 −y2 h(y) := 0 For L ∈ N0 let hL (y) := h(y) −

153

those constructed by Triebel in [Tri01, Let 0 < δ ≤ 41 and put ) if |y| < δ , y ∈ R. (9.16) if |y| ≥ δ L X

µ` h(`) (y − 1),

`=0

where the numbers µ` ∈ R are chosen so that hL has moment conditions up to order L, Z y k hL (y)dy = 0, k = 0, . . . , L. Setting

hL (y) :=

R Ry

hL (u)du, y ∈ R, we can replace (8.74) by

−∞

fb (x) =

∞ X

n ¡ ¢Y ¡ ¢ bj 2−j+1 hL 2j−1 x1 h 2j−1 xk ,

j=1

x = (x1 , . . . , xn ) .

(9.17)

k=2 ∞

Assume that b = {bj }j=1 is a sequence of non-negative monotonically decreasing numbers, b1 ≥ b2 ≥ . . . ≥ bj ≥ bj+1 ≥ · · · ≥ 0. 1+n/p

Then (9.17) can be interpreted as an atomic decomposition in Ap,q , see [Tri01, Cor. 13.4]. The assertion parallel to the argument presented in the proof of Theorem 8.16 reads as 1/q  1/q  ε ¸q Z · ∞ ° ° X dt ω(f , t) ° b 1+n/p °   ∼ °fb |Bp,q (9.18) ∼ bqj  °, t | log t| t j=1 0

and a similar expression for F -spaces; cf. [Tri01, (14.18)-(14.19)]. There one also finds the useful estimate k X ω(fb , t) ∼ bj t j=1

if t ∼ 2−k

and

k ∈ N.

(9.19)

We proceed as for Theorem 8.16 and put bj ≡ 1, j = 1, . . . , J, and bj ≡ 0 for j > J for some J ∈ N. Denote by fJ the function given° by (9.17) with ° ° 1+n/p ° the above-described sequence; thus – according to (9.18) – °fJ |Bp,q ° ∼ kb|`q k = J 1/q . On the other hand one computes by (9.19), ¡ ¢ ½ ω fJ , 2−k k ,k≤J ∼ . J ,k>J 2−k

154

Envelopes and sharp embeddings of function spaces

Hence we obtain for any J ∈ N, B 1+n/p EC p,q

¡

−J

2

¢

≥ J

− q1

¡ ¢ 1 ω fJ , 2−J ∼ J q0 , −J 2

completing the argument in the B-case. The F -case can be handled in analogy to Theorem 8.16; in particular, (7.39) implies 1+n/r 1+n/p Br,p ,→ Fp,q F 1+n/p

for 0 < r < p leading to EC p,q B 1+n/p

Step 2. We prove uC p,q 

Zε ·

 0

1

(t) ∼ |log t| p0 finally. F 1+n/p

≤ q, uC p,q

ω(f, t) t | log t|

¸p

(9.20)

1/p dt  t

≤ p; that is, we show

° ° ° 1+n/p ° ≤ c °f |Fp,q °

(9.21)

when 1 < p < ∞, 0 < q ≤ ∞, and 1/q  ε ¸q Z · ° ° dt ω(f, t) ° 1+n/p °   ≤ c °f |Bp,q ° t | log t| t

(9.22)

0

when 0 < p < ∞, 1 < q ≤ ∞. We essentially gain from Proposition 5.10 and our preceding results in Section 8.3 now. Recall (7.9), ° n ° ° ° ° ° X ° ∂f ¯¯ n/p ° ° ° n/p ° 1+n/p ° ° ° °f |Ap,q ° ∼ °f |Ap,q ° + ° ∂xk ¯ Ap,q ° .

(9.23)

k=1

We start with the F -case, that is, 1 < p < ∞. Let first q < ∞. Apply (5.12) with u = 1, r = p; then Theorem 8.16, i.e.,   

1

Z2 · 0

∗

g (t) | log t|

¸p

1/p dt   t

° ° ° n/p ° ≤ c °g|F p,q ° ,

(9.24)

and the F -part of (9.23) yield 1/p  ε 1/p  ε ¸p ¸p Z · Z · ∗ ° ° dt |∇f | (t) dt ω(f, t) ° 1+n/p °    ≤ c ≤ c0 °f |Fp,q ° t | log t| t | log t| t 0

0

1+n/p

for all f ∈ C 1 ∩ Fp,q . The rest is done by completion. The same method applies in the B-case when q < ∞, now using (5.12) with r = q, u = 1

Continuity envelopes EC of function spaces Asp,q (recall our assumption q > 1, that is u = 1 > of (9.24), 

Zε ·

 0

ω(f, t) t| log t|

¸q

1 q

155

= 1r ) and the B-counterpart

1/q  ε 1/q ¸q Z · ∗ ° ° dt  |∇f | (t) dt ° 1+n/p °  ≤ c ≤ c0 °f |Bp,q °. t | log t| t 0

Finally, when q = ∞ we shall prove sup 0
° ° ω(f, t) ° 1+n/p ° ≤ c °f |Bp,∞ ° t | log t|

(9.25)

1+n/p

for all f ∈ Bp,∞ . (Note that the completion argument does not work here.) Let ϕ ∈ S be some standard smooth function compactly supported near the origin, ¡ see ¢ (7.1). We apply the counterpart of (5.12) (with r = ∞) to fj = F −1 ϕ 2−j · Ff , j ∈ N, and obtain (9.25) for any fj . The right-hand sides can be estimated uniformly with respect to j, thus sup 0
° ° ω(fj , t) ° 1+n/p ° ≤ c °f |Bp,∞ ° t | log t|

for any j ∈ N. Now fj converges pointwise to f and (9.25) follows, as well as its counterpart in the F -case. Step 3. It remains to show the sharpness of v = p (in the F -case) and v = q (in the B-case). This works exactly as for Theorem 8.16, now with the extremal functions given by (9.17), (9.18) (and the parallel F -assertion), where we benefit from (9.19) again.

Remark 9.5 Combining Theorem 6.6(ii) and (9.14), (9.15), we arrive at ³ ´ ³ ´ ´ ³ 1 1+n/p p0 , p EC Lip(1,−1) = E F , (9.26) = |log t| p,q ∞, p C with 1 < p < ∞, 0 < q ≤ ∞, and ´ ³ ³ ³ ´ ´ 1 1+n/p q0 , q = |log t| B , EC Lip(1,−1) = E ∞, q p,q C

(9.27)

with 0 < p ≤ ∞, 1 < q ≤ ∞, respectively; see also Proposition 9.6 below. This situation is similar to Remarks 8.2 and 8.17 when dealing with growth envelopes; the corresponding envelopes coincide whereas the underlying spaces do not; cf. [Har00b, Cor. 13, 20] and its extension by Neves [Nev01a]. In addition to the more or less historic references we gave in Remarks 8.3 and 8.20 already, which are partly connected with the super-critical case, too, we shall mention the results by Br´ezis, Wainger [BW80], by Bourdaud and Lanza de Cristoforis [BL02], approaches based on extrapolation by Edmunds, Krbec

156

Envelopes and sharp embeddings of function spaces

[EK95], Krbec, Schmeisser [KS01], and recently by Neves [Nev01a]. The borderline case was already studied by Zygmund [Zyg45], [Zyg77]. Concerning limiting embeddings we also contributed with some papers [EH99], [EH00], [Har00b]. Proposition 9.6 Let 1 < q ≤ ∞. Then ´ ³ ³ ´ 1 1 EC B∞,q = |log t| q0 , q .

P r o o f : Step 1. As before, we gain from Proposition 5.3(iii) and Theorem 6.6(ii) together with (9.11) and obtain B1

1

EC ∞,q (t) ≤ c |log t| q0 . As for the lower bound, note that 1+n/p 1 Bp,q ,→ B∞,q ;

(9.28)

hence another application of Proposition 5.3(iii) together with our preceding result from Theorem 9.4, in particular Step 1, gives the desired estimate, B1

1

EC ∞,q (t) ∼ |log t| q0 . B1

Step 2. We determine the critical index uC ∞,q . We conclude by (9.28), B1

(9.15) and Proposition 6.4 that uC ∞,q ≥ q; hence it remains to show 1/q  ε ¸q Z · ° ° dt ω(f, t) 1 °   ≤ c °f |B∞,q , (9.29) t | log t| t 0

1 or, in other words, B∞,q ,→ Lip(1,−1) ∞, q . This is covered by Proposition 7.15(i) with p = ∞, v = q. Note that another simple and elegant proof of (9.29) for 1 < q < ∞ was obtained by Bourdaud, Lanza de Cristoforis in [BL02, Prop. 1], combining Marchaud’s inequality (2.22) and Hardy’s inequality (see [BS88, Ch. 3, Lemma 3.9]).

Remark 9.7 We already mentioned in Remark 8.18 parallel results for spaces of generalised smoothness and introduced the function Φr,u : (0, 2−n ] → R, 0 < r, u ≤ ∞, !1/u à Z1 dy u −u −n . Φr,u (t) = y r Ψ(y) y t1/n

In [HM04] and [CH05] we proved that for 0 < p, q ≤ ∞ (with p < ∞ in the F -case), 0 < σ ≤ 1, s = np + σ and Ψ a continuous admissible function, then

Continuity envelopes EC of function spaces Asp,q ³ ¡ ¢−1 ´ (i) with the additional assumption Ψ 2−j 6∈ `q0 when s = j∈N

³ ´ ³ (s,Ψ) n EC Bp,q = Φ 1−σ , ³ ¡ ¢−1 ´ (ii) and, assuming that Ψ 2−j

j∈N

q 0 (t

n

´ ), q ,

6∈ `p0 for s =

³ ´ ³ (s,Ψ) n EC Fp,q = Φ 1−σ ,

p0 (t

n

n p

157 n p

+ 1, (9.30)

+ 1,

´ ), p .

(9.31)

Obviously our above results for Ψ ≡ 1 can be rewritten using the function Φ as presented in Remark 8.18. Releasing also the subordinate (dyadic) partition of unity one obtains spaces Aσ,N p,q of generalised smoothness; their continuity envelopes are studied in [HM06]. Further results on continuity envelopes related to Lorentz-Karamata-Bessel potential spaces H s Lp,q;Ψ (Rn ) with a slowly varying function Ψ were obtained in [GNO05]. Example 9.8 We return to Example 7.5, s ∈ R, b ∈ R, 0 < p < ∞, 0 < q ≤ ∞. In view of Example 7.22 we assume np < s ≤ np + 1, referring to the super-critical case. Let σ = s − np . Then (9.30) and (8.93) imply for 0 < σ < 1, ³ ´ ³ ´ ³ ´ −b σ+n/p,b EC Bp,q = t−(1−σ) |log t| , q = EC Lip(σ,b) ; (9.32) ∞, q see Theorem 6.6(iii). When 0 < p, q ≤ ∞, σ = s − np = 1, and b < 1/q 0 according to Example 7.22, then by (8.94) and (9.30), ³ ´ ³ ´ 1 −b 1+n/p,b EC Bp,q = |log t| q0 , q , whereas for 1 < q ≤ ∞ and b = 1/q 0 , we obtain ³ ´ ³ ´ 0 1+n/p,b EC Bp,q = (log |log t|)1/q , q .

Corollary 9.9 Let n < p ≤ ∞, 0 < q ≤ ∞, α > 1q , then (1,−α)

Lip p, q

EC

n

(t) ∼ t− p |log t|

α− q1

,

0
1 , 2

(9.33)

(appropriately modified for p = ∞). P r o o f : The case p = ∞ is covered by Theorem 6.6(ii), where we even obtained a complete envelope result, ´ ³ ³ ´ α− 1 = |log t| q , q . EC Lip(1,−α) ∞, q

158

Envelopes and sharp embeddings of function spaces

Assume now n < p < ∞, such that 0 < σ = 1 − (7.61), 1,−α+ 1

q Bp,min(q,1)

,→

Lip(1,−α) p, q

n p

,→

< 1. We use (7.59) and 1,−α+ q1

(9.34)

Bp,∞

and Proposition 5.3(iii) and (9.32).

9.3

Continuity envelopes in the critical case

We return to the critical case, already studied in Section 8.3; that is, we n/p consider spaces Ap,q , see Figure 11. In view of (7.40) and (7.41) (where L∞ can be replaced by C) we deal with the remaining cases now, not covered by Theorem 8.16 (in terms of growth envelopes EG ). Theorem 9.10 Let 0 < p ≤ ∞ and 0 < q ≤ ∞. (i)

Assume 0 < p ≤ 1. Then F n/p

EC p,q (t) ∼ t−1 , and

F n/p

p ≤ uC p,q

0 < t < 1,

≤ ∞.

(9.35)

(ii) Assume 0 < q ≤ 1. Then B n/p

EC p,q (t) ∼ t−1 , and

B n/p

q ≤ uC p,q Proof:

0 < t < 1,

≤ ∞.

(9.36) n/p

Step 1. We have by Proposition 7.13 that Ap,q ,→ C for the adAn/p

mitted parameters. Thus Theorem 6.6(iv) immediately provides EC p,q (t) ≤ c t−1 , 0 < t < 1. Conversely, note that our construction of the functions fj in the proof of Proposition 9.2, that is, in (9.5), works for σ = 0, too. This yields the lower estimate in the B-case (no moment conditions). Moreover, by (7.39) we have n/p n/r Br,p ,→ F p,q , (9.37) where 0 < r < p ≤ 1 and 0 < q ≤ ∞. Thus by the preceding observation, F n/p

B n/r

EC p,q (t) ≥ c EC r,p (t) ≥ c0 t−1 .

Continuity envelopes EC of function spaces Asp,q B n/p

159

F n/p

Step 2. We show that uC p,q ≥ q, uC p,q ≥ p (the upper estimates in (9.35) and (9.36) are clear). Note that the extremal functions (9.7) serve for the proof of the necessity here, too; that is, the construction presented in the proof of Proposition 9.2, in particular Step 2, works also for σ = 0. Thus we B n/p

obtain uC p,q ≥ q. Turning to the F -case, let r be such that 0 < r < p ≤ 1, then (9.37) and Proposition 6.4 complete the argument for (9.35).

Remark 9.11 We briefly discuss the obvious gaps in (9.35) and (9.36). At B n/p

F n/p

first glance one might be tempted to prove that uC p,q = q, uC p,q = p; however, our methods presented so far fail necessarily in this limiting case: assume we would like to show that 1/q  ε Z ° ° dt ° n/p ° q   [ω(f, t)] ≤ c °f |B p,q ° (9.38) t 0

n/p

holds for all f ∈ B p,q , 0 < p ≤ ∞, 0 < q ≤ 1. The “lifting argument”, however, as applied in Step 2 of the proof of Theorem 9.4 quite effectively, cannot be used as our setting now refers to Proposition 5.10(iii), but with κ = 0. This is probably not true in general, but at least not covered by Proposition 5.10. Moreover, for several reasons it is not so clear whether the B n/p

F n/p

conjecture uC p,q = q, uC p,q = p is the right one for that case.

Chapter 10 Envelope functions EG and EC revisited

In this section we return to a more general point of view concerning both growth and continuity envelope functions. First we study assertions like Corollaries 3.14, 3.16 in a more general context; then we investigate in what cases the envelope function EGX can be “realised” by some f ∈ X in the sense that X EGX (t) ∼ f ∗ (t), 0 < t < ε; similarly for EC . Finally, as mentioned before, we turn to global settings for growth envelope functions, i.e., we consider EGX (t) for t → ∞.

10.1

Spaces on R+

In this section we insert a short digression on (envelopes of) spaces on R+ = [0, ∞) (equipped with | · |) first. We pose the question as to whether, say, EGX ∈ X, and this makes sense only in such spaces. Secondly, we shall study the question as to whether there exists a function in X that realises the growth envelope function EGX , i.e., EGX (t) ∼ f ∗ (t),

0 < t < 1,

for some f ∈ X. Such a function will be called an “enveloping function”; this is postponed to Section 10.2 below. Obviously, these topics are closely related. We first summarise what is already known. In ¤ order to simplify the setting £ further we regard only spaces X on Ω = 0, 21 in the sequel. Corollary 10.1 (i)

Let 0 < p < ∞, 0 < q ≤ ∞, and a ∈ R. Then L

EG p,q

(log L)a

∈ Lp,q (log L)a

if, and only if,

q = ∞.

161

162

Envelopes and sharp embeddings of function spaces

(ii) Let a ≥ 0. Then L

EG exp,a ∈ Lexp,a . (iii) We have EGbmo ∈ bmo . P r o o f : Part (i) follows from Corollaries 3.14, 3.16, (ii) is a consequence of Corollary 3.19, and (8.100) implies (iii), i.e., EGbmo (t) ∼ |log t|, t > 0 small, and [BS88, Ch. 5, Sect. 7]. Also taking the index uX G as defined by Definition 4.2 into account, one observes the following peculiarity: whenever EGX ∈ X, then uX G = ∞; see Theorem 4.7 and (8.100). Thus the following assertion seems natural. X be a³function Proposition 10.2 Let X ,→ Lloc 1 ´ space °with ¯ EG° ∈ X and ° X¯ ° X EGX 6≡ 0. Then uX X ° ≥ 1. G = ∞, i.e., EG (X) = EG , ∞ , and °EG

P r o o f : Step 1. We need not exclude the case X ,→ L∞ as Proposition 3.4(iii) together with Remark 4.4 cover that case. So assume now X 6,→ L∞ ; it is clear by Remark 4.3 that uX G ≤ ∞. It remains to check whether (4.5) is also satisfied for some v < ∞. On the other hand, put f = EGX in (4.5); then (4.5) with v < ∞ means  ε 1/v Z ° ¯ ° ° °  µG (dt) ≤ c °EGX ¯ X °

(10.1)

0

for some c > 0. Recall that µG is the associated Borel measure with respect to the distribution function log EGX (or some differentiable function equivalent to EGX , respectively), and EGX % ∞ when t ↓ 0. Consequently the left-hand side of (10.1) does not converge and thus our assumption v < ∞ leads to a contradiction. ° ° ° ° Step 2. We assume EGX ∈ X and EGX 6= 0 in X, i.e., °EGX |X ° > 0. Then ° °−1 ° ° g := EGX °EGX |X ° is well-defined, g ∈ X and kg|Xk = 1. Hence ³ EGX (t) ≥ g ∗ (t) =

´∗ EGX (t) E X (t) ° ° =° G ° ° X ° ° X ° °EG |X ° °EG |X °

by Proposition 3.4(i); this gives the result.

Envelope functions EG and EC revisited

163

Remark 10.3 Note that in our above examples in Corollary 10.1 we always have ° ° ° X ¯¯ ° (10.2) °EG X ° ∼ 1. This is due to the fact that all these spaces are rearrangement-invariant spaces which can be equivalently re-normed to rearrangement-invariant spaces of type M (X), kf |M (X)k = sup f ∗∗ (t)ϕX (t), for the definition of the maximal t>0

function f ∗∗ (t) and the fundamental function ϕX (t) we refer to (2.14) and (3.29), respectively; for Lorentz spaces of type M (X) see [BS88, Ch. 2, Sect. 5]. In view of Propositions 3.4(i) and 3.21, together with (EGX )∗∗ (t) ∼ (MEGX )(t) ∼ EGX (t) in all above-mentioned examples, we immediately obtain (10.2). X

X

Concerning EC it obviously makes no sense to ask whether EC ∈ X with X being a function space on Ω = [0, 21 ], for – apart from the not X very interesting case when EC is bounded, i.e., X ,→ Lip1 – we know that X X EC (t) % ∞ when t ↓ 0, such that EC 6∈ X for all X ,→ C. However, one may replace this by the question X

t EC (t) ∈ X X

It is clear by Corollary 5.7(i) that t EC X

0 ≤ t EC (t) ≤ 2

sup

?

is uniformly bounded, for

kf |Ck ≤ 2 kid : X → Ck ,

(10.3)

kf |Xk≤1

recall X ,→ C. For convenience, put X

eX (t) := t EC (t),

t ≥ 0.

(10.4)

Corollary 5.7 yields that lim eX (t) = 0, eX is monotonically increasing in t↓0

t > 0 and concave, and eX is uniformly bounded, see (10.3). We shall now study the following question: for which spaces X is it true that eX ∈ X. We look for a counterpart of Corollary 10.1 and collect some examples. Corollary 10.4 (i)

We have

eC ∈ C.

(ii) Let 0 < a ≤ 1. Then (iii) Let 0 < q ≤ ∞, α > (1,−α)

1 q

a

eLip ∈ Lipa . (with α ≥ 0 when q = ∞). Then

eLip q, ∞ ∈ Lip(1,−α) q, ∞

if, and only if,

q = ∞.

164

Envelopes and sharp embeddings of function spaces

(iv) Let 0 < a < 1, 0 < q ≤ ∞, α ∈ R. Then (a,−α)

eLip∞, q ∈ Lip(a,−α) ∞, q

if, and only if,

q = ∞.

P r o o f : Proposition 5.15 covers (i), whereas Proposition 5.12 yields for (ii) a Lipa −(1−a) that EC (t) ∼ t for 0 < t < 1, hence eLip (t)° ∼ max (ta°, 1) for ¡ Lipa ¢ a ¯ t ≥ 0. Then ω e , t ∼ ta and by (2.25) we obtain °eLip ¯ Lipa ° ∼ 1. Lip(1,−α) ∞, q

α− 1

(t) ∼ (1 ³− log t) q ´ for small t > 0, cf. (1,−α) α− 1 Lip(1,−α) (5.23). Thus sup e ∞, q (t) ∼ cα,q and ω eLip ∞, q , t ∼ t (1 − log t) q Concerning (iii) we use EC t>0

for small t > 0. The result follows by the definition of Lip(1,−α) now imme∞, q diately, see (2.35). Lip(a,−α) ∞, q

Finally, to prove (iv) we use EC

(t) ∼ t−(1−a) (1 − log t)

α

for small

Lip(a,−α) ∞, q

t > 0, cf. Proposition 5.14. Hence sup e (t) ∼ ca,α,q and t>0 ³ (a,−α) ´ α ω eLip∞, q , t ∼ ta |log t| for small t > 0. The definition of Lip(a,−α) ∞, q completes the proof. Again, Theorem 6.6 now suggests the counterpart of Proposition 10.2 as we observe that whenever eX ∈ X, we also have uX = ∞. C Proposition 10.5 Let X ,→ C be a non-trivial function space with eX ∈ X. Then eX is a constant) this implies uX = ∞, i.e., EC (X) = C ³ ´ (unless ° X¯ ° X ° ¯ ° E , ∞ , and e X ≥ 1. C

P r o o f : Step 1. Parallel to the proof of Proposition 10.2 we conclude that the case X ,→ Lip1 is covered by Proposition 5.3(ii) together ¡with Re¢ mark 6.3; we turn to the remaining cases now. We shall prove that ω eX , t ∼ eX (t) for small t > 0. Afterwards we proceed as in the proof of Proposition 10.2, i.e., put f = e¡X in ¢(6.2) and argue as in case of EG (X). So it remains to show ω eX , t ∼ eX (t). Corollary 5.7 implies that ψC = eX is increasing, concave near 0, and lim eX (t) = 0. But this yields the assertion. t↓0

Let eX ∈ X. Note that eX ≡ 0 would imply that X contains ° °−1 constants only, but this is excluded. Then g := eX °eX |X ° is well-defined, g ∈ X and kg|Xk = 1. Hence ¡ ¢ ω eX , t X . e (t) ≥ keX |Xk ¡ ¢ In view of our above observation ω eX , t ∼ eX (t) the result is obtained. Step 2.

Envelope functions EG and EC revisited

165

° ¯ ° Remark 10.6 Observe that °eX ¯ X ° ∼ 1 for all our examples in Corollary 10.4. We review our results in Sections 8.1, 8.3, 9.1. ¤ £ Corollary 10.7 Let all spaces be defined on Ω = 0, 21 . (i) Let 0 < q ≤ ∞, s > 0, 1 < r < ∞ and 0 < p < ∞ be such that s − p1 = − 1r . Then s Bp,q

EG

s ∈ Bp,q

if, and only if,

q = ∞.

(ii) Let 1 < q ≤ ∞, and 0 < p < ∞. Then 1/p Bp,q

EG

1/p ∈ Bp,q

if, and only if,

q = ∞.

(iii) Let 0 < p ≤ ∞, 0 < q ≤ ∞, 0 < σ < 1, and s = σ + p1 . Then s

s eBp,q ∈ Bp,q

if, and only if,

q = ∞.

(iv) Let 0 < p ≤ ∞, and 1 < q ≤ ∞. Then 1+1/p

eBp,q

1+1/p ∈ Bp,q

if, and only if,

q = ∞.

P r o o f : By Theorems 8.1 and 8.16 together with Proposition 10.2 it is immediately clear that only B-spaces with q = ∞ can satisfy EGX ∈ X as othAs

erwise uG p,q < ∞ which contradicts EGX ∈ X. So it remains to verify that in 1/p s the sub-critical case t−1/r ∈ Bp,∞ , s − p1 = − 1r (locally), and | log t| ∈ Bp,∞ , 0 < p < ∞, referring to the critical case. For p ≥ 1 a straightforward calculation based on (7.3) was sufficient, but otherwise the atomic characterisation seems to be better adapted: we start with the sub-critical case, i.e., s − p1 = − 1r . Let ϕ be a smooth cut-off function supported near t = 0; take, for instance, the standard one from (7.1). Let ψj (t) = ϕ(2j t) − ϕ(2j+1 t), j ∈ N0 , 0 < t < 1, build a partition of unity; then 1

1

t− r = ϕ(t)t− r ∼

∞ X

1 1 1 2−j (s− p ) ψj (t)ϕ(t)t− r 2j (s− p ) , 0 < t < 1, (10.5) {z } | j=0 := aj (t)

© ª where the aj (t), j ∈ N0 , are supported near s ∈ [0, 1] : s ∼ 2−j , such that 1 j 1 t− r ∼ 2 r ∼ 2−j (s− p ) , t ∈ supp aj . Hence (10.5) can be understood as an 1 atomic decomposition of t− r (near 0, no moment conditions) with coefficients 1 s . Concerning λj ≡ 1, i.e., kλ|`∞ k = 1. Theorem 7.8 then implies t− r ∈ Bp,∞

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Envelopes and sharp embeddings of function spaces

the critical case we return to our construction (8.74); in particular, with ϕ(t) as above, and ψ(t) = h(t) given by (9.16), i.e., a one-dimensional version of ψ from (3.36), we consider ∞ X

¡ ¢ ψ 2j−1 t ϕ(2t),

(10.6)

j=1

supported near t = 0. Then for small t > 0, ∞ X

¡ ¢ ψ 2j−1 t ϕ(2t) ∼

j=1

[| log t|]

X

1 ∼ | log t|,

j=1

i.e., (10.6) can be interpreted as an atomic decomposition for | log t| near 0 (no moment conditions) with λj ∼ 1 and thus kλ|`∞ k ∼ 1. Consequently 1/p | log t| ∈ Bp,∞ , 0 < p < ∞ (locally). Concerning (iii), (iv), Theorems 9.2, 9.4 imply that only B-spaces with q = ∞ can satisfy eX ∈ X, see Proposition 10.5. So we have to show σ+1/p for 0 < σ < 1, 0 < p ≤ ∞ (at least locally), and that tσ ∈ Bp,∞ 1+1/p t| log t| ∈ Bp,∞ , 0 < p ≤ ∞. For the super-critical case (iii) we proceed parallel to the sub-critical one in (i), where (10.5) is now replaced by t

σ

σ

= ϕ(t) t

∼

∞ X

2−j (σ+ p − p ) ψj (t) ϕ(t) tσ 2jσ , 1

1

0 < t < 1,

(10.7)

j=0

the rest is similar. Concerning (iv) we return to the extremal functions fb as constructed by Triebel in [Tri01, (14.15)-(14.19)]; see also (9.18). Put bj ≡ 1; then this is essentially the integrated version of (10.6), ∞ X

−j+1

2

¡ ¢ Ψ 2j−1 t ϕ(2t),

Zz Ψ(z) =

j=1

ψ(u)du,

(10.8)

−∞

where ψ(t), ϕ(t) are as above; note that we need no moment conditions. One checks that ∞ X j=1

¡ ¢ 2−j+1 Ψ 2j−1 t ϕ(2t) ∼ t |log t| ,

0
1 , 2

and (10.8) can be understood as the atomic decomposition of t |log t| (near 0). Now (9.18) and the particular choice of the sequence b ∈ `∞ imply 1+1/p t |log t| ∈ Bp,∞ , 0 < p ≤ ∞. Remark 10.8 Triebel studied a related question in [Tri01, Sect. 17.1], asking under what conditions there are functions f ∈ Asp,q such that f ∗ (t)

Envelope functions EG and EC revisited

167

are equivalent to the corresponding growth or continuity envelope or ω(f,t) t functions. By the same arguments as above only B-spaces with q = ∞ are left to consider; Triebel applies these outcomes showing that certain Green’s n functions (of (id − ∆)− 2 for the n-dimensional critical case, for instance) generate the corresponding envelope functions.

10.2

Enveloping functions

As already announced we deal with functions f ∈ X now that (locally) X realise the growth or continuity envelope function, EGX and EC , respectively. This idea originates from discussions with H. Triebel. All spaces are defined on Rn , equipped with the Lebesgue measure, unless otherwise stated. Definition 10.9 Let ε > 0. be a quasi-normed function space with growth envelope (i) Let X ,→ Lloc 1 function EGX . Then fG ∈ X, kfG |Xk ≤ 1, is called (growth) enveloping function in X, if EGX (t) ∼ fG∗ (t), 0 < t < ε. (ii) Let X ,→ C be a quasi-normed function space with continuity envelope X function EC . Then fC ∈ X, kfC |Xk ≤ 1, is called a (continuity) enveloping function in X, if X

EC (t) ∼

ω(fC , t) , t

0 < t < ε.

Of course, the particular growth/continuity enveloping function will depend on the function space X; for lucidity we shall but omit the additional index if possible. The number ε > 0 is arbitrary, but meant to be small. For convenience, we may think of ε < 1 in general, and ε < 21 when logterms are involved, or ε < ε0 (a, α) in Corollary 10.13 below, but this is not important. At the moment we are only interested in local enveloping functions – and will thus also omit this special notation. Remark 10.10 By (obvious counterparts of) Propositions 10.2 and 10.5 there X cannot exist enveloping functions in spaces X with uX G < ∞ or uC < ∞, res spectively. This excludes, in particular, spaces of type Fp,q . We shall see X below, that uX G = ∞ or uC = ∞ is necessary, but not sufficient for the existence of associate enveloping functions in X. In continuation of Proposition 10.2 and Remark 10.3 let °us mention ° that in all cases when X possesses ° X ° an enveloping function, then °EG |X ° ∼ 1.

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Envelopes and sharp embeddings of function spaces

Corollary 10.11 (i)

Let ε > 0 be small and all spaces be defined on Ω ⊆ Rn .

Let 0 < p < ∞, and a ∈ R. Then n

−a

fG (x) = |x|− p |log |x||

χK

ε (0)

(x)

is a growth enveloping function in Lp,∞ (log L)a . (ii) Let a ≥ 0, and Ω = K1 (0). Then a

fG (x) = |log |x|| χK

ε (0)

(x)

is a growth enveloping function in Lexp,a . (iii) Assume Ω = K1 (0), and let ϕ be given by (7.1). Then fG (x) = |log |x|| ϕ(x) is a growth enveloping function of bmo . (iv) Let 0 < p < ∞, and σp < s <

n p.

Let ϕ be given by (7.1). Then n

fG (x) = |x|s− p ϕ(x) s is a growth enveloping function in Bp,∞ .

(v) Let 0 < p < ∞, and let the function ϕ be given by (7.1). Then fG (x) = |log |x|| ϕ(2x) n/p

is a growth enveloping function in Bp,∞ . P r o o f : In view of Lemma 3.10(ii), in particular, (3.14) and (3.15) with s = 1, κ = p1 , and r = p, part (i) follows from Proposition 3.15. In the same way Lemma 3.10(iv) with s = 1 and Proposition 3.18 imply (ii). The local version of (7.11) together with (8.100) lead to (iii). Concerning (iv) and (v) we stress similar arguments as for the proof of Corollary 10.7(iii), (iv): let ϕ be given by (7.1), and ¡ ¢ ¡ ¢ ψj (x) = ϕ 2j x − ϕ 2j+1 x , x ∈ Rn , j ∈ N0 , (10.9) build a partition of unity in K1 (0). Then for x ∈ K1 (0), x 6= 0, n

n

|x|s− p = |x|s− p ϕ(x) ∼

∞ X

2−j (s− p ) ψj (x)ϕ(x), | {z } j=0 := aj0 (x) n

(10.10)

Envelope functions EG and EC revisited

169

where the aj0 (x), j ∈ N0 , are supported in cQj0 . Hence (10.10) can be n p ϕ(x), (no moment conditions) seen as an atomic decomposition° of |x|s− ° s ° ° with coefficients λj0 ≡ 1, i.e., fG |Bp,∞ ≤ c kλ|`∞ k ≤ c0 . Moreover, s

1

fG∗ (t) ∼ t n − p , 0 < t < 1, so Theorem 8.1 completes the argument for (iv). In the critical case, let ψ be given by (3.36); then we claim that fG (x) ∼

∞ X

¡ ¢ ψ 2j−1 x ϕ(2x),

(10.11)

j=1

near 0, because for small x, ∞ X

[| log |x||]

X

¡ ¢ ψ 2j−1 x ϕ(2x) ∼

j=1

1 ∼ | log |x||.

j=1

Interpreting ° this as °an atomic decomposition again (no moment conditions), ° n/p ° we obtain °fG |Bp,∞ ° ≤ c kλ|`∞ k ≤ c0 . The rest is obvious, n/p Bp,∞

fG∗ (t) ∼ | log t| ∼ EG

(t),

0 < t < ε,

recall Theorem 8.16(ii). We come to continuity enveloping functions. Remark 10.12 Obviously the first candidate to deal with is X = C, the space of bounded uniformly continuous functions. In view of Proposition 5.15 the question can thus be reformulated to find some continuous function fC such that ω (fC , t) ∼ 1 for all small t > 0. This, however, is impossible for a continuous function, see also the (pointwise) construction in (5.26) for n → ∞. Thus C possesses no continuity enveloping function, though uC = ∞, i.e., C this also yields the insufficiency of this condition for the existence of fC . Corollary 10.13 Let ε > 0, and all spaces be defined on Ω ⊆ Rn . (i) Let 0 < a ≤ 1, b ∈ R (with b ≥ 0 if a = 1), and ϕ be given by (7.1). Then b fC (x) = |x|a |log |x|| ϕ(2x) is a continuity enveloping function in Lip(a,−b) ∞,∞ , in particular, fC (x) = |x|a ϕ(x) is associated to Lipa . (ii) Let 0 < p ≤ ∞,

n p

<s<

n p

+ 1. Let ϕ be given by (7.1). Then n

fC (x) = |x|s− p ϕ(x) s is a continuity enveloping function in Bp,∞ .

170

Envelopes and sharp embeddings of function spaces

(iii) Let 0 < p ≤ ∞, and ϕ be given by (7.1). Then fC (x) = |x| |log |x|| ϕ(2x) 1+n/p

is a continuity enveloping function in Bp,∞ . P r o o f : For convenience we shall assume that ε < min Then clearly b ω (fC , t) = ta |log t| , 0 < t < ε, ° ° ° ° leads to °fC |Lip(a,−b) ∞,∞ ° ≤ c, and

¡1

2, e

−b/a

¢

in (i).

ω (fC , t) , 0 < t < ε, t recall Propositions 5.12, 5.13 and 5.14. For (ii) we argue similar to °Corol° s °≤ c lary 10.11 by atomic decomposition. We already know that °fC |Bp,∞ s− n by (10.10) and the construction gives ω (fC , t) = t p , 0 < t < 1. In view of Proposition 9.1 and Theorem 9.2 this ° ° concludes the proof of (ii). Finally, in ° 1+n/p ° (iii) it remains to prove °fC |Bp,∞ ° ≤ c, as the rest is obvious by construction, Theorem 9.4 and Proposition 9.6, Lip(a,−b) ∞,∞

EC

(t) ∼

ω (fC , t) , 0 < t < ε. t We “integrate” the atomic decomposition used in Corollary 10.11(v) and proceed just as for Theorem 9.4, in particular (9.17): let h be given by (9.16), and Zy1 n Y Ψ(y) = h(yj ) h(u)du, y = (y1 , . . . , yn ) ∈ Rn . (10.12) B 1+n/p

EC p,∞ (t) ∼ |log t| ∼

j=2

−∞

Then ∞ X

¡ ¢ 2−j+1 Ψ 2j−1 x ϕ(2x) ∼ |x| |log |x|| ,

x ∈ Kε (0),

(10.13)

j=1

can be understood as the atomic decomposition of° fC near °0 (no moment ° 1+n/p ° conditions needed), see [Tri01, Cor. 13.4]. Hence °fC |Bp,∞ ° ≤ c as desired.

10.3

Global versus local assertions

As already mentioned several times, we have studied local assertions so far, meaning the behaviour of EGX (t) for small t > 0. We shall see also that global

Envelope functions EG and EC revisited

171

assertions are of some interest, in particular in limiting or weighted situations. Some results are contained in [Harxx]. First we recall what is already known from Section 3.2 in this context. For convenience, we shall restrict ourselves to the situation of Rn equipped with the Lebesgue measure always, [Rn , | · |] as underlying space. Corollary 10.14 Let 0 < p, q ≤ ∞ (with q = ∞ when p = ∞). (i) Then 1 L EG p,q (t) ∼ t− p , t → ∞.

(10.14)

(ii) Assume a ∈ R, then L

EG p,q Proof:

(log L)a

1

−a

(t) ∼ t− p (1 + | log t|)

,

t → ∞.

(10.15)

This is only a reformulation of (3.22) and (3.24).

Proposition 10.15 Let 1 ≤ p < ∞ and k ∈ N0 . Then Wpk

EG

1

(t) ∼ t− p ,

t → ∞.

P r o o f : By definition (2.37), Wpk (Rn ) ,→ Lp (Rn ), thus (10.14) and Proposition 3.4(iv) yield Wpk

EG

1

(t) ≤ c t− p ,

t → ∞.

Conversely, we return to our construction in Section 3.4, and modify the functions fR from (3.35) slightly by ¡ ¢ n f R (x) = R− p ψ R−1 x , x ∈ Rn , where ψ(x) is given by (3.36). Similar to the proof of Proposition 3.25 we are led to ³ X ´1/p ° R k° ° ° p °f |Wp ° ≤ R−|α|p kDα ψ|Lp k ≤ °ψ|Wpk ° , |α|≤k

now assuming R > 1. Let t be large and choose R0 = R0 (t) = dt1/n such 1/n that R0 > (t/|ωn |) > 1, i.e., R0−n t < |ωn | for appropriate d > 0. Hence (2.9) and (3.39) imply Wpk

EG

(t) ≥ c1 sup

R>1

the proof is finished.

¡ R ¢∗ ¡ ¢∗ 1 −n (t) ≥ c1 f R0 (t) ≥ c2 R0 p ≥ c3 t− p , f

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Envelopes and sharp embeddings of function spaces

Remark 10.16 If we compare Propositions 3.25 and 10.15 we see that the additional smoothness assumption of Wpk unlike Lp , that is, k ∈ N0 , is wellWpk

reflected in the local singularity behaviour, EG

(t), 0 < t < 1, whereas globWpk

ally the spaces all share the same integrability, EG Proposition 10.17 Let 1 < p < ∞, 0 < α < L (wα )

EG p

L (wα )

(t) ∼ EG p

n p0 . α

L

(t) ∼ EG p (t), t → ∞.

Then 1

(t) ∼ t− n − p ,

t → ∞.

(10.16)

P r o o f : In case of wα the result is already covered by Proposition 3.35. 1 α L (w ) Consider now wα , then Lemma 3.33 and (10.14) imply EG p α (t) ≤ t−( n + p ) for t → ∞. Conversely, inspecting the proof of Proposition 3.30 we observe that with extremal functions fs given by (3.11) with r = p0 , As = Kcs1/n (0), with c > 0 such that µ(As ) = |Kcs1/n (0)| = s, we get for large s, 1

kfs |Lp (wα )k = s− p0

³Z

´ p1 ³Z ³ ´ p1 ´ αp 1 2 wα (x)p dx 1 + c0 s2/n = s− p 0 dx

As 00

≤c s Here p0 is given by resulting in L (wα )

EG p

− p1

1 p0

0

=

As α n

s |As | 1 p

+

1 p

α n.

≤

C. 1

On the other hand, fs∗ (t) = s− p0 χ[0,s) (t), 1

α

1

(t) ≥ sup fs∗ (t) ≥ c sup s− p0 ∼ t− n − p , s>t

s>0

t → ∞.

Remark 10.18 Obviously, by Propositions 3.30, 3.35, the weight functions wα (x) = hxiα , and wα (x) = |x|α ∈ Ap show the same influence on the underlying Lp -space as regards global assertions (integrability) in contrast to local assertions. It was certainly interesting to study this subject in a more general context, whereas one should expect rather qualitative characterisations then.

Theorem 10.19 Let 0 < q ≤ ∞, 0 < p ≤ ∞ (with p < ∞ for A = F ), and s > σp . Then Asp,q

EG

1

(t) ∼ t− p ,

t → ∞.

(10.17)

P r o o f : We shall present the argument for p < ∞ only; when A = B, everything can be extended to p = ∞ without difficulties. We start with the

Envelope functions EG and EC revisited

173

estimate from above and assume first 1 ≤ p < ∞. Then by (7.7), (7.26), (7.30), (8.56), (8.57) we have Asp,q ,→ Lp ,

(10.18)

so that Proposition 3.4(iv) and (10.14) imply the estimate from above, Asp,q

EG

1

(t) ≤ c t− p ,

t → ∞.

(10.19)

Dealing with 0 < p < 1, the difficulty in extending (10.18) results from questions of convergence, i.e., that Asp,q consists of tempered distributions f ∈ S 0 , whereas this is not the case for L³p with ´ 0 < p < 1; cf. [Tri92, 1 Rem. 2.3.2/3]. However, for s > σp = n p − 1 , 0 < p < 1, we always have at least Asp,q ,→ L1 . Furthermore, we can even estimate Lp -norms for f ∈ Asp,q due to the following characterisation: Let s > σp , and % = ϕ1 = ϕ(2−1 ·) − ϕ, where ϕ is given by (7.1). Then 1/q ∞ Z ° ° ° ° dt q s ° °f |Bp,q  , ∼ kf |Lp k +  t−sq °F −1 (% (t·) Ff ) |Lp ° t

(10.20)

0

and ° ° 1/q ° ° Z∞ ¯ ° ° ° ° ¯ ¯q dt ¯ ° ° s ° −sq ¯ −1 °f |Fp,q ¯  ∼ kf |Lp k + ° t F (% (t·) Ff ) (·) ¯Lp ° , (10.21) ° ° t ° ° 0 cf. [Tri92, Rem. 2.3.3]. This immediately finishes the proof of the upper estimate for° 0 < p < 1: let t > 0 be arbitrarily large, then for each f ∈ Asp,q with ° °f |Asp,q ° ≤ 1, (10.20) and (10.21), respectively, imply kf |Lp k ≤ c, such that 1

(10.14) leads to f ∗ (t) ≤ c0 t− p , i.e., we obtain (10.19) for all p, 0 < p < ∞. For the converse, we make use of the equivalent norms (10.20), (10.21) again and a homogeneity argument. Let R > 0; then straightforward calculation shows that kf (R·)|Lp k = R−n/p kf |Lp k and ¡ ¡ ¤¢ F −1 (% (t·) F [f (R·)]) (x) = F −1 % (t·) [R−n Ff R−1 ·) (x) = F −1 (% (Rt·) Ff ) (Rx),

x ∈ Rn ,

hence ° −1 ° ° ° °F (% (t·) F [f (R·)]) |Lp ° = R−n/p °F −1 (% (Rt·) Ff ) |Lp °

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Envelopes and sharp embeddings of function spaces

We apply (10.20) to fR = f (R·) and thus obtain   ¶q1 µ Z∞  ° ° ° ° n q dt s ° °f (R·)|Bp,q t−sq °F −1 (% (Rt·) Ff ) |Lp ° ≤ c1 R− p kf |Lp k +  t  0   ¶q1 µZ∞  ° ° n q dτ τ −sq °F −1 (% (τ ·) Ff ) |Lp ° = c1 R− p kf |Lp k + Rs   τ 0 ´ ³ ° ° n n s ° ≤ c2 max R− p , Rs− p °f |Bp,q With obvious modifications one can show that (10.21) implies ´° ³ ° ° ° n n s ° s ° °f (R·)|Fp,q , ≤ c max R− p , Rs− p °f |Fp,q so that we obtain for 0 < R ≤ 1, ° ° ° ° °f (R·)|Asp,q ° ≤ c R− np °f |Asp,q ° .

(10.22)

Assume now 0 < R ≤ 1 and consider functions n

ψR (x) = R p ψ (Rx) ,

x ∈ Rn ,

(10.23)

where ψ is given by (3.36). Then by the above estimates, ° ° ° ° °ψR |Asp,q ° ≤ c °ψ|Asp,q ° ≤ c0 , 0 < R ≤ 1, n

∗ and ψR (t) ∼ R p for t ∼ R−n , 0 < R ≤ 1. Let t > 1 be large, then −1/n R0 ∼ t ∈ (0, 1), and up to possible normalisations this yields Asp,q

EG

(t) ≥

sup 0
n/p

∗ ∗ ψR (t) ≥ ψR (t) ∼ R0 0

∼ t−1/p ,

and the proof is finished.

Remark 10.20 Obviously the assumption s > σp is essentially used in the above argument. It is not yet clear what will happen in the borderline case s = σp in the global setting. (s,Ψ)

Dealing with spaces of generalised smoothness Ap,q , 0 < p < ∞, 0 < q ≤ ∞, s > σp , and Ψ slowly varying, we obtain as a direct consequence of Proposition 10.19, A(s,Ψ) p,q

EG

1

(t) ∼ t− p ,

in view of Remark 7.21, in particular, (7.62).

t → ∞,

Envelope functions EG and EC revisited

175

We finally turn to weighted Besov spaces again; recall our local results Propositions 8.8 and 8.10. For convenience we restrict ourselves to the case p > 1. Proposition 10.21 Let 1 < p < ∞, 0 < q ≤ ∞, s > 0, and 0 ≤ α < Then Asp,q (wα )

EG

Asp,q (wα )

(t) ∼ EG

α

1

(t) ∼ t− n − p ,

t → ∞.

n p0 .

(10.24)

P r o o f : We deal first with wα and start with the estimate from above and combine for that reason the (weighted) embedding results (8.20) and Proposition 10.17. In particular, due to (8.20), which is based upon (8.19), we have the counterpart of (10.18), Asp,q (wα ) ,→ Lp (wα ),

(10.25)

such that Proposition 3.4(iv) and (10.16) imply for t → ∞, Asp,q (wα )

EG

L (wα )

(t) ≤ c EG p

1

α

(t) ≤ c0 t− p − n .

The counterpart of (10.25) for wα is covered by [Bui82, Thms. 2.6, 2.8], Asp,q (wα ) ,→ Lp (wα ),

(10.26)

and Propositions 3.4(iv) and 10.17 complete the upper estimate in (10.24). We prove the converse estimate and adapt the argument from the proof of Proposition 10.19 suitably. Instead of (10.23) we consider now functions n

%R (x) = R p +α % (Rx) , with

x ∈ Rn ,

% = ϕ1 = ϕ(2−1 ·) − ϕ,

0 < R < 1,

(10.27) (10.28)

and ϕ is given by (7.1). Then, obviously, supp % ⊂ {x ∈ Rn : 1 < |x| < 4}, and hence, for small 0 < R < 1, ¡ ¢ wα R−1 x ∼ R−α , x ∈ supp %. (10.29) Thus by the above argument (10.22) for 0 < R < 1, and (8.19), ° ° ° ° °%R |Asp,q (wα )° ≤ c1 R np +α °wα % (R·) |Asp,q ° ° ¡ ° ° ° ¢ ≤ c2 Rα °wα R−1 · %|Asp,q ° ≤ c3 °%|Asp,q ° ≤ c4 where for the penultimate inequality we applied (10.29) and some characterisation of spaces Asp,q via local means, see [Tri92, Sect. 2.4.6, 2.5.3]. For

176

Envelopes and sharp embeddings of function spaces

convenience, assume that there is a number η ∈ (1, 2) such that ϕ(x) < for |x| > η, and ϕ(x) > 21 for |x| < η. Consequently,

1 2

¯½ ¾¯ ¯ ¯ ¯ x ∈ Rn : %(x) > 1 ¯ ≥ |{x ∈ Rn : η < |x| < 2η}| ≥ |ωn |η n ≥ |ωn |, ¯ 2 ¯ i.e., %∗ (|ωn |) ≥ n

1 . 2

(10.30)

n

Then %∗R (t) ∼ R p +α %∗ (R−n t) ≥ c R p +α for t ∼ R−n . For t > 1 large, choose R0 ∼ t−1/n ∈ (0, 1), and up to possible normalisation this yields Asp,q (wα )

EG

n

(t) ≥

sup 0
%∗R (t) ≥ %∗R0 (t) ∼ R0p

+α

1

α

∼ t− p − n .

It remains to deal with the weight wα (x) = |x|α . We consider functions ¡ ¢ n gj (x) = 2−j ( p +α) % 2−j x ,

j ∈ N, n

with % given by (10.28). As calculated above this implies gj∗ (t) ≥ 2−j p −jα for t ∼ 2jn , j ∈ N, leading to Asp,q (wα )

EG

1

α

(t) ≥ sup gj∗ (t) ≥ c t− p − n ,

t → ∞.

j∈N

All that is left to prove is ° ° °gj |Asp,q (wα )° ≤ c,

j ∈ N.

Let k be a compactly supported C ∞ function on Rn with X k(x − m) = 1, x ∈ Rn . m∈Zn

Then we have for all x ∈ Rn , gj (x) = 2−j ( p +α) n

X

¡ ¢ k(x − m) % 2−j x

m∈Zn

∼ 2−j ( p +α) n

X

¡ ¢ k(x − m) % 2−j x ,

(10.31) j ∈ N.

(10.32)

|m|∼2j

¡ ¢ On the other hand, a0m (x) := k(x − m) % 2−j x can be regarded as 1K -atoms located near Q0m , m ∈ Zn , in the sense of Definition 7.7(i), and thus (10.32) represents a special atomic representation of gj . By the already mentioned (Muckenhoupt) weighted version of Theorem 7.8 in [HPxx], [Bow05],

Envelope functions EG and EC revisited

177

this yields ° X °p ° ° ° °gj |Asp,q (wα )°p ≤ c1 2−j ( np +α)p ° χ0m |Lp (wα ) ° ° |m|∼2j

≤ c2 2−j (

n p +α

)p

Z ³ X

Rn

≤ c3 2−j ( p +α)p n

´p χ0m (x) |x|αp dx

|m|∼2j

Z

X

|x|αp dx

|m|∼2j Q 0m

≤ c4 2−j ( p +α)p n

X

|m|αp

|m|∼2j

X

≤ c5 2−j ( p +α)p+jαp n

1

|m|∼2j

≤ c6 2−jn+jn ≤ c6 , where χ0m stands for the characteristic function of Q0m , m ∈ Zn . This completes the argument.

Remark 10.22 Parallel to Remarks 10.16 and 10.18 we review our above global characterisations in comparison with the local (sub-critical) ones obtained in Section 8.1 and observe that in unweighted as well as weighted cases the additional smoothness s > σp – in contrast to the underlying Lp -space – influences the local singularity behaviour unlike the global one. Moreover, the different weights wα = hxiα and wα = |x|α cause globally the same asymptotic behaviour for the growth envelope function, whereas this is different locally. Avoiding limiting or borderline situations it also turns out that globally there is no difference between B- and F -spaces; in particular, there is no dependence on the fine index q.

Chapter 11 Applications

We present a few applications of our envelope results. Though our original intention to introduce envelopes of function spaces was certainly of a different flavour, concentrating essentially on some better understanding of rather complicated function spaces, our studies admit quite a number of interesting results: rather direct there is a link to Hardy-type inequalities and limiting embeddings, presented first. With the help of some rather astonishing “lift arguments” for envelopes, as described in Section 11.2, we can finally tackle problems of compactness, that is, asymptotic estimates for entropy and approximation numbers of compact embeddings, with surprisingly sharp results. In this section we present some general results, but otherwise exemplify the possible applications with a few cases only.

11.1

Hardy inequalities and limiting embeddings

As a first application we can derive some Hardy-type inequalities. This follows immediately from our envelope results above together with the monotonicity (4.2), see Propositions 4.1, 3.4(v) and 5.3(iv): there are constants and g ∈ Y ,→ C, ε > 0, c, c0 > 0 such that for all f ∈ X ⊂ Lloc 1 sup κ(t) 0
and sup κ(t) 0
f ∗ (t) ≤ c kf |Xk , EGX (t) ω(g, t) Y

t EC (t)

≤ c kg|Y k ,

respectively, if, and only if, κ is bounded. Corollary 11.1 Let ε > 0 be small, κ(t) be a positive monotonically decreasing function on (0, ε], and 0 < v ≤ ∞. (i) Assume that X 6,→ L∞ is a (quasi-) normed function space on Rn , satisfying the assumptions of Lemma 3.8; let EGX be the corresponding 179

180

Envelopes and sharp embeddings of function spaces growth envelope function. Then 

Zε "

 0

f ∗ (t) κ(t) X EG (t)

1/v

#v

µG (dt)

≤ c kf |Xk

for some c > 0 and all f ∈ X if, and only if, κ is bounded and uX G ≤v ≤ ∞, with the modification sup

κ(t)

t∈(0,ε)

f ∗ (t) ≤ c kf |Xk EGX (t)

(11.1)

if v = ∞. In particular, if κ is an arbitrary non-negative function on (0, ε], then (11.1) holds if, and only if, κ is bounded. (ii) Let X ,→ C be a function space on Rn with (5.8), X 6,→ Lip1 and X continuity envelope function EC . Then à Zε " κ(t) 0

ω(f, t) X

t EC (t)

#v

!1/v µC (dt)

≤ c kf |Xk

for some c > 0 and all f ∈ X if, and only if, κ is bounded and uX ≤v≤ C ∞, with the modification sup t∈(0,ε)

κ(t) ω(f, t) ≤ c kf |Xk t ECX (t)

(11.2)

if v = ∞. In particular, if κ is an arbitrary non-negative function on (0, ε], then (11.2) holds if, and only if, κ is bounded. Obviously we can establish a lot of assertions similar to Corollary 11.2 by a combination of Corollary 11.1 and our envelope results presented in the last sections. We restrict ourselves to an example only, related to spaces of type Asp,q (wα ), recall Proposition 8.10. Corollary 11.2 Let 0 < q ≤ ∞, s > 0, 1 < r < ∞ and p with 0 < p < ∞ be such that s − np = − nr . Let 0 < α < rn0 . Let κ(t) be a positive monotonically decreasing function on (0, ε] and let 0 < v ≤ ∞. (i) Then

1/v  ε Z h iv dt ° ° α 1 s + ∗   ≤ c °f |Bp,q (wα )° κ(t) t r n f (t) t 0

Applications

181

s for some c > 0 and all f ∈ Bp,q (wα ) if, and only if, κ is bounded and q ≤ v ≤ ∞, with the modification ° ° α 1 s (wα )° (11.3) sup κ(t) t r + n f ∗ (t) ≤ c °f |Bp,q t∈(0,ε)

if v = ∞. In particular, if κ is an arbitrary non-negative function on (0, ε], then (11.3) holds if, and only if, κ is bounded. (ii) Then

1/v Zε h iv dt ° ° α 1 s + ∗   ≤ c °f |Fp,q (wα )° κ(t) t r n f (t) t 

0

s for some c > 0 and all f ∈ Fp,q (wα ) if, and only if, κ is bounded and p ≤ v ≤ ∞, with the modification ° ° α 1 s (wα )° (11.4) sup κ(t) t r + n f ∗ (t) ≤ c °f |Fp,q t∈(0,ε)

if v = ∞. In particular, if κ is an arbitrary non-negative function on (0, ε], then (11.4) holds if, and only if, κ is bounded. Remark 11.3 In [HM04, Cor. 3.4] and [CH05, Cor. 4.1] we obtained similar results when dealing with (unweighted) spaces of generalised smoothness (s,Ψ) Ap,q . Another application concerns necessary conditions for (limiting) embedding assertions. Propositions 3.4(iv) and 5.3(iii) as well as Propositions 4.5 and 6.4, respectively, lead to necessary conditions for embeddings X1 ,→ X2 .

(11.5)

We introduce some notation: For Xi ⊂ Lloc or Xi ,→ C, i = 1, 2, respectively, 1 let (X ,X ) qG 1 2 (t)

:=

EGX1 (t) EGX2 (t)

X1

(X ,X ) qC 1 2 (t)

,

:=

EC (t) X2

EC (t)

,

0 < t < ε.

(11.6)

We may assume that ε > 0 is chosen sufficiently small, say, ε ≤ τ0G (X2 ), given by (3.3), and ε ≤ τ0C (X2 ), according to (5.2), such that (11.6) makes sense. Thus (11.5) and Propositions 3.4(iv) and 5.3(iii) imply that (X1 ,X2 )

sup qG

(t) ≤ c < ∞,

0 < ε ≤ τ0G (X2 ),

(11.7)

(t) ≤ c0 < ∞,

0 < ε ≤ τ0C (X2 ),

(11.8)

0
or

(X1 ,X2 )

sup qC

0
182

Envelopes and sharp embeddings of function spaces

respectively. Moreover, Propositions 4.5 and 6.4 lead to X2 1 uX G ≤ uG

(X1 ,X2 )

if X1 ,→ X2

and

qG

if X1 ,→ X2

and

qC

(t) ∼ 1, 0 < t < ε,

(11.9)

and 1 2 uX ≤ uX C C

(X1 ,X2 )

(t) ∼ 1, 0 < t < ε. (11.10)

We exemplify this for some well-known situations first. Example 11.4 We consider the embedding Bps11 ,q1 (Rn ) ,→ Bps22 ,q2 (Rn ),

(11.11)

where 0 < pi , qi ≤ ∞, si ∈ R, i = 1, 2. Let s1 ≥ s2 ; for convenience we assume pi < ∞. Then we can easily derive the counterpart of (7.32), namely that (11.11) implies p1 ≤ p2 , δ ≥ 0, and q1 ≤ q2 when δ = 0, where we used notation (7.34). This can be seen as follows: Note first that in view of the lift result (7.8) we have (11.11) if, and only if, −σ −σ Bps11 ,q (Rn ) ,→ Bps22 ,q (Rn ), 1 2

σ ∈ R.

(11.12)

Choose first σ < min (s1 − σp1 , s2 − σp2 ), then we can apply Theorem 10.19 −σ −σ ) = ∞, and together with (11.11), to Bpsii ,q , i = 1, 2, which yields τ0G (Bps22 ,q 2 i (11.12), (B1σ ,B2σ )

qG

s2 −σ (Bps1 −σ ,q ,Bp ,q )

(t) := qG

1

1

2

2

1

1

(t) ∼ t p2 − p1 ,

t → ∞.

(11.13)

Thus (11.7) leads to 1 1 − ≤ 0, p2 p1

i.e., p1 ≤ p2 .

(11.14)

n p2

(11.15)

Next we assume that s1 − σp1 > s2 −

and choose σ ∈ R in (11.12) such that n < σ < min (s1 − σp1 , s2 − σp2 ) , s2 − p2 hence

n . (11.16) p2 Now we are prepared to complete the argument in this case, as by assumption (11.16) we can apply Theorems 8.1, 8.16(ii), and (7.41), (7.26) and Proposition 3.4(iii) to obtain for all small t, 0 < t < ε,  s1 −σ − 1   t n p1 , s1 − σ < pn1 , 0 < q1 ≤ ∞     s −σ 1 (B1σ ,B2σ ) − 2n + p1 0 2 n qG (t) ∼ t . (11.17) q1 , 1 < , |log t| s − σ = q ≤ ∞ 1 1   p1     c , otherwise s1 − σ > σp1 ,

σp2 < s2 − σ <

Applications

183

Thus (11.7) and (11.17) imply for all cases covered by (11.15), n n ≥ s2 − σ − , s1 − σ − p1 p2 i.e., δ ≥ 0.

(11.18)

In case of δ = 0 we would like to conclude q1 ≤ q2 . For that reason note that (B σ ,B σ ) our choice of σ as in (11.16) leads to qG 1 2 (t) ∼ 1 in view of (11.17), such that (11.9) and Theorem 8.1 provide the additional condition Bσ

Bσ

q1 = uG 1 ≤ uG 2 = q2

when δ = 0,

(11.19)

i.e., the well-known (sufficient) conditions for the embedding (11.11). It remains to discuss the case not covered by (11.15), that is, n (11.20) s1 − σp1 ≤ s2 − . p2 We disprove (11.11) in that case. Choose σ ∈ R in (11.12) such that n < σ < s1 − σp1 , s1 − (11.21) p1 then (11.20) immediately implies s2 − σ > Proposition 3.4(iii), and Theorem 10.19 give Bps2 −σ ,q

EG

2

2

(t) ∼ 1,

n p2

> σp2 and (7.41), (7.26),

0 < t < ε.

(11.22)

On the other hand, by (11.21) and Theorem 8.1 we conclude Bps1 −σ ,q

EG

1

1

(t) ∼ t

s1 −σ 1 n − p1

,

0 < t < ε,

(11.23)

which led to a contradiction with (11.22) and Proposition 3.4(iv) if (11.12) and hence (11.11) was satisfied. Remark 11.5 The cases pi = ∞ could be incorporated using (11.8) for the local argument instead (with an appropriate lift, i.e., suitably chosen number σ), whereas the global one already covers this case. Moreover, instead of (11.9) one could argue in (11.19) by means of (11.7) and Theorem 8.16(ii), modifying the lift (11.16) properly (e.g., s2 = pn2 ), at least for qi > 1. (s,Ψ)

In [CH05, Prop. 4.3] we used envelope results for spaces Ap,q , see Remarks 8.18 and 9.7, to prove necessary conditions for the embedding 1 ,Ψ1 ) 2 ,Ψ2 ) A(s ,→ A(s p1 ,q1 p2 ,q2

with s1 ≥ s2 , 0 < p1 ≤ p2 ≤ ∞, 0 < q1 , q2 ≤ ∞, and Ψ1 , Ψ2 admissible functions. In particular, we studied the limiting case s1 − pn1 = s2 − pn2 only and obtained precisely the counterparts of the sufficiency results by Moura in [Mou02, Prop. 1.1.13(iv)-(vi)].

184

Envelopes and sharp embeddings of function spaces

s,b Example 11.6 We return to our Example 7.5 concerning spaces Bp,q . Let s1 ≥ s2 , b1 , b2 ∈ R, 0 < p1 ≤ p2 ≤ ∞, and 0 < q1 , q2 ≤ ∞. We assume that

s1 −

n n = s2 − . p1 p2

Our result [CH05, Prop. 4.3] reads in this case as ( b1 − b2 ≥ 0 s1 ,b1 s2 ,b2 Bp1 ,q1 ,→ Bp2 ,q2 if, and only if, b1 − b2 > q12 −

, q 1 ≤ q2 1 q1

) . (11.24)

, q1 > q2

This situation was already known, see [Leo98, Thm. 1]. Moreover, in the special setting s1 = s2 , p1 = p2 , let us denote q := q1 , r := q2 , α := −b1 , β := −b2 , then (11.24) can be reformulated into ( ) β ≥ α , q ≤ r s,−α s,−β Bp,q ,→ Bp,r if, and only if, , β − 1r > α − 1q , q > r see the parallel result Proposition 2.23 for Lipschitz spaces. We return to weighted situations now and first review Lemma 3.33. Corollary 11.7 Let 1 < p < ∞, 0 ≤ α < Let 0 < r, u ≤ ∞.

n p0 ,

and p0 be given by

1 p0

=

1 p

+α n.

(i) Let wα (x) = hxiα be given by (3.48); then Lp (wα ) ,→ Lr,u

(11.25)

if, and only if, ( p0 ≤ r ≤ p,

and

p ≤ u ≤ ∞, r = p

or

r = p0

0 < u ≤ ∞, p0 < r < p

) .

(11.26)

(ii) Let wα (x) = |x|α be given by (3.52); then Lp (wα ) ,→ Lr,u

(11.27)

if, and only if, r = p0 ,

and

u ≥ p.

(11.28)

P r o o f : The sufficiency in (ii) is given by Lemma 3.33, in particular by (3.54), together with the monotonicity of Lorentz spaces Lr,u in u. Concerning (i), we have Lp (wα ) ,→ Lp

and

Lp (wα ) ,→ Lp0 ,p ,

(11.29)

Applications

185

by (3.50) and (3.54), respectively, which implies the sufficiency in (11.26) for r = p and r = p0 , using again monotonicity of Lorentz spaces. Otherwise, in case of p0 < r < p we strengthen real interpolation arguments (3.55) for the target spaces in (11.29) to obtain Lp (wα ) ,→ (Lp , Lp0 ,p )θ,u = Lr,u , where θ ∈ (0, 1) is chosen such that 1−θ θ 1 = + , r p p0 and u is arbitrary. It remains to verify the necessity of (11.26) and (11.28); we use our envelope results. Assume (11.25); then Propositions 3.30 and 10.17, in particular, (3.49) and (10.16) give ( 1 ) ,0
1

EG r,u (t) ∼ t− r ,

t > 0.

(11.31)

Then (11.7), (11.25), (11.30) and (11.31) imply 1 1 ≥ , r p

and

1 1 α 1 ≤ + = , r p n p0

i.e., p0 ≤ r ≤ p. Moreover, when p = r, then (11.9), (11.30), (11.31) lead to L (wα )

uG p

L

= p ≤ u = uG p,u ,

see Theorem 4.7(i) and Proposition 4.13. To complete the proof of (i) we have to show that Lp (wα ) ,→ Lp0 ,u (11.32) implies u ≥ p. We assume u < p and disprove (11.32). For that purpose, consider the function ¾ ½ −n |x| p , |x| < 1 −γ n , x ∈ Rn ; fγ (x) = (1 + |log |x||) |x|− p0 , |x| > 1 then easy calculations show that fγ ∈ Lp (wα ) if, and only if, γ > the other hand, ) ( 1 t− p , t < 1 −γ ∗ , fγ (t) ∼ (1 + |log t|) 1 t− p0 , t > 1

1 p.

On

186

Envelopes and sharp embeddings of function spaces

such that à Z∞ kfγ |Lp0 ,u k ≥ 2

iu dt h 1 t p0 fγ∗ (t) t

!1/u

à Z∞ −γu

∼

(log t) 2

dt t

!1/u

which does not converge for γ ≤ u1 . Thus fγ ∈ Lp (wα ) \ Lp0 ,u for all γ with u1 ≤ γ < p1 in contrast to (11.32). In case of (ii), we have to replace (11.30) by L (wα )

EG p

1

α

(t) ∼ t− n − p ,

t > 0,

see Proposition 3.35, in particular (3.58). Following the above scheme, this immediately leads to p0 ≤ r ≤ p0 and u ≥ p when r = p0 , i.e., (11.28). We end this section with another consequence in the sense of the above Example 11.4, dealing now with weighted Besov spaces instead. We complete the characterisation (8.34), at least for pi > 1. Corollary 11.8 Let 1 < p1 < ∞, 1 < p2 ≤ ∞, 0 < q ≤ ∞, i = 1, 2, s1 > s2 , 0 < α < pn0 . Then 1 Bps11 ,q (wα ) ,→ Bps22 ,q (11.33) implies α ≤ δ,

and

α 1 1 ≤ + . p2 n p1

(11.34)

P r o o f : We proceed just as in the Example 11.4 above. Instead of the operator Iσ given by (7.6) we can use the operator Jσ introduced in [Bui82, Thm. 2.8], ¡ ¢σ/2 Jσ = F −1 1 + 4π 2 |ξ|2 , σ ∈ R, that maps

n Jσ : Asp,q (Rn , w) −→ As−σ p,q (R , w)

(11.35)

isomorphically for all weights w ∈ A∞ , 0 < p ≤ ∞, 0 < q ≤ ∞, s ∈ R. This is the needed counterpart of (7.8). So we have (11.12) again; with the same choice of σ as above (now with σp1 = σp2 = 0), i.e., σ < s2 , we can apply −σ −σ Theorem 10.19 to Bps22 ,q , and Proposition 10.21 to Bps11 ,q (wα ), leading to 2 1 −σ τ0G (Bps22 ,q ) = ∞ again and substitute (11.13) by 2 α s2 −σ (Bps11 −σ ,q1 (w ),Bp2 ,q2 )

qG thus

1

(t) ∼ t p2

− p1 − α n

1 α 1 ≤ + p2 p1 n

1

,

t → ∞,

(11.36) (11.37)

Applications

187 (B σ (wα ),B σ )

2 in view of (11.7). We study the local behaviour of qG 1 (t), using Theorem 8.1 and Proposition 8.10, more precisely, its extension discussed in Remark 8.11. We distinguish the following cases: n n n < s2 < s2 + α ≥ s2 + α (i) s1 − (ii) s2 ≤ s1 − (iii) s1 − p1 p1 p1

Starting with (i), choose σ in (11.35) such that ¶ µ n n < σ < s2 , max s1 − , s2 − p1 p2 hence Proposition 8.10 and Theorem 8.1 give for small t, 0 < t < ε, α Bps1 −σ ,q (w )

EG

1

1

(t) ∼ t

s1 −σ 1 n − p1

−α n

,

Bps2 −σ ,q

EG

2

2

(t) ∼ t

s2 −σ 1 n − p2

,

(11.38)

respectively, that is, (B1σ (wα ),B2σ )

qG

(t) ∼ t−

δ−α n

,

0 < t < ε.

Thus (11.7) implies δ ≥ α. In case (ii) we can choose σ such that ¶ ¶ µ µ n n n − α, s2 − < σ < min s2 , s1 − −α+n . max s1 − p1 p2 p1 Using the extended version of Proposition 8.10, i.e., (8.44) instead, we can argue in the same way as above and obtain (11.34). Finally, we come to (iii); this leads directly to δ ≥ s2 + α − s2 +

n ≥ α. p2

Remark 11.9 Comparison of (8.34) and (11.34) shows that by the envelope methods used above we only obtain p12 ≤ p11 + α n , whereas the necessary , and sufficient condition reads as p12 < p11 + α see for instance [HS06] and n [KLSS06b]. Moreover, there are extensions to different q-parameters, too, in the sense that (8.34) holds if, and only if, α ≤ δ when q1 ≤ q2 , and α < δ for q1 > q2 .

11.2

Envelopes and lifts

Recall that EGX (t) is bounded when X ,→ L∞ , see Proposition 3.4(iii), X whereas EC (t) is only defined for X ,→ C. Thus it might not appear very

188

Envelopes and sharp embeddings of function spaces X2

interesting at first glance to study the interplay of EGX1 and EC in general – at least not when the spaces X1 and X2 coincide, X1 = X2 . We may, however, observe some phenomena granted that X1 and X2 are connected in a suitable way; we shall try to interpret and generalise this afterwards.

s s=

n p

s2 Bp,q

³

1

s=n

s1 Bp,q

1 p

´ −1

σ 1 r

1 p

1

Figure 15

We consider the following situation. Let 0 < p < ∞ and 0 < q ≤ ∞. Assume (as indicated in Figure 15) that s1 = np − nr for some r, 1 < r < ∞, and s2 = σ + np for some σ with 0 < σ < 1. We consider the case that s2 = s1 + 1; that is, where σ = 1 − nr . (Note that the assumptions on σ Asp,q

thus imply r > n.) Furthermore, by Theorem 8.1 we know EG As+1 p,q

whereas Theorem 9.2 yields EC that case

1

(t) ∼ t− r ,

(t) ∼ t−(1−σ) . Consequently we obtain in

As+1

− r1

EC p,q (t) ∼ t−(1−σ) = (tn )

Asp,q

∼ EG

(tn ) .

Likewise, for 0 < p < n and 0 < q ≤ 1 Theorems 9.10 and 8.1 (with r = n) lead to An/p−1 An/p −1 EC p,q (t) ∼ t−1 = (tn ) r ∼ EG p,q (tn ) . A similar behaviour can be observed when dealing with the borderline cases, n/p 1+n/p B p,q and Bp,q , respectively, B 1+n/p

EC p,q

1

n/p Bp,q

(t) ∼ |log t| q0 ∼ EG

(tn ) ,

Applications

189

and a parallel result for the F -case. However, the log-function spoils the interplay of t and tn in that case. Turning to the envelopes EG or EC , it thus appears reasonable to define ³ ´ n EG (X) := EGX (tn ) , uX G , where uX G is given as in Definition 4.2. Then Theorems 8.1 and 9.2, as well as Theorems 8.16 and 9.4 lead to   n n  0 < p < ∞, s = p − r , n < r < ∞  ¡ ¢ n s Asp,q = Fp,q , (11.39) EG (Asp,q ) = EC As+1 if 1 < p < ∞, s = np , p,q  s  1 < q ≤ ∞, s = np , Asp,q = Bp,q where we assume in general 0 < p, q ≤ ∞. When r = n, i.e., s = have at least the corresponding result for the envelope functions, Asp,q

EG

As+1

(tn ) ∼ EC p,q (t),

n p

− 1, we

(11.40)

see Theorems 8.1 (with r = n) and 9.10. Does this reflect a more general behaviour, that is, in what sense can this particular result be extended? So far we only collected results “associated” in the above sense, but achieved (almost) independently of each other. The more desirable was a direct link and |∇f |∗ (tn ) or |∇f |∗∗ (tn ) ) for, say, f ∈ X ,→ C 1 . We between ω(f,t) t return to Proposition 5.10, in particular to estimate (5.11), Zt

n 1

∗

s n −1 |∇f | (s)ds

ω(f, t) ≤ c

(11.41)

0 1

n

for t > 0 and all f ∈ C (R ). Plainly, this estimate plays an essential role X1 in our subsequent study of EC and EGX2 , where X1 ,→ C and X2 ⊂ Lloc 1 are such that |∇f | ∈ X2 for f ∈ X1 (this setting is motivated by our above observations). We first discuss the “optimality” of (11.41). Recall that we have by (11.41) for n = 1, ω(f, t) ∗∗ ≤ c |f 0 | (t), t

0 < t < ε,

f ∈ C 1 (R).

(11.42)

So one can ask whether a replacement of (5.11) in the sense of (11.42), i.e., ω(f, t) ∗∗ ≤ c |∇f | (tn ) , t

0 < t < ε,

(11.43)

was true for all f ∈ C 1 (Rn ) and dimension n > 1. Obviously, (11.43) was sharper than (11.41), and also implied Triebel’s result [Tri01, Prop. 12.16]

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Envelopes and sharp embeddings of function spaces

mentioned in Remark 5.11, ¢ 1 ω(f, t) ∗∗ ¡ ≤ c |∇f | t2n−1 + 3 sup τ − 2 ω(f, τ ) 2 t 0<τ ≤t

(11.44)

for some small ε > 0 and all 0 < t < ε and all f ∈ C 1 (Rn ); we refer to [Har01, Sect. 6.3]. However, (11.43) cannot hold in general when n > 1; we give some argument disproving (11.43). 1 Assume (11.43) was true for n > 1. Let f ∈ Wn1 (Rn ) = Fn,2 (Rn ); by 1 n density arguments we may furthermore suppose that f ∈ Fn,2 (R )∩C0∞ (Rn ). 1 ∗∗ 0 Then by (7.9) |∇f | ∈ Fn,2 = Ln , leading to |∇f | (τ ) ≤ Cn τ − n , τ > 0, and (11.43) then implies ω(f, t) ≤ c t |∇f |

∗∗

1 −n

(tn ) ≤ c0 t (tn )

= c0

1 for small t > 0. In other words, all f ∈ Fn,2 (Rn ) ∩ C0∞ (Rn ) (and by the n 1 usual density arguments then all f ∈ Fn,2 (R ), too) are (locally) bounded. This, however, is wrong: recall (7.40) with p = n > 1; cf. Proposition 7.13. On the other hand, one can also rely on a result of Stein in [Ste81] stating that if a function f on Rn satisfies ∇f ∈ Ln,1 locally, then f is equimeasurable with a continuous function, see also [DS84]. Moreover, there is a remark that the result is sharp in the following sense: taking g 6∈ Ln,1 with f = |x|−(n−1) ∗ g, then there is a positive ge, equi-measurable with |g|, such that the resulting f is unbounded near every point; see also [Ste70, Ch. 8] and [Kol89, §5] for further details. So (11.41) – stating exactly that |∇f | belongs to Ln,1 locally – is the best possible result (in that sense) and (11.43) – referring to |∇f | ∈ Ln – cannot hold. The essential difference to the one-dimensional case is obvious in this setting as L1,1 = L1 , but Lp,1 (Rn ) is properly contained in Lp (Rn ) for any p > 1. Hence for n > 1 we are left with the two estimates (11.41) and (11.44) (instead of (11.43)) and try to compare them. At first glance it seems that our estimate (11.41) might be slightly sharper: though both estimates in question gave rise to the estimate (5.12), only (11.41) implies (5.13). The case n = 1 is clear: the second term in (11.44) disappears and we have (11.42) again.

Lemma 11.10 Let n > 1. There is a c > 0 such that for all 0 < t < 1 and all f ∈ C 1 (Rn ), Zt

n 1

∗

s n −1 |∇f | (s)ds ≤ c1 t |∇f |

∗∗

° ¡ 2n−1 ¢ 1 ° t + c2 t2− n °f |C 1 ° .

(11.45)

0

Proof: Zt

2n−1 tZ

n

s 0

We split the integral on the left-hand side of (11.45) as follows,

1 n −1

∗

|∇f | (s)ds =

s 0

Zt 1 n −1

n

∗

|∇f | (s)ds + t2n−1

1

∗

s n −1 |∇f | (s)ds. (11.46)

Applications

191

We deal with the second term first and use the monotonicity of the maximal function g ∗∗ as well as g ∗ (s) ≤ g ∗∗ (s), Zt

n

s

1 n −1

∗

|∇f | (s)ds ≤ |∇f |

∗∗

¡ 2n−1 ¢ t

t2n−1

Zt

n 1

∗∗

s n −1 ds ≤ c t |∇f |

¡

¢ t2n−1 .

t2n−1

It remains to consider the first term on the right-hand side of (11.46); we verify that 2n−1 tZ ° 1 1 ° ∗ s n −1 |∇f | (s)ds ≤ c t2− n °f |C 1 ° . 0

This is an immediate consequence of our assumption f ∈ C 1 , ° ° ∗ |∇f | (0) = k |∇f | |L∞ k ≤ c °f |C 1 ° . ∗

Hence the monotonicity of |∇f | (s) implies 2n−1 tZ

s

1 n −1

° ° |∇f | (s)ds ≤ c °f |C 1 °

2n−1 tZ

° 1 1 ° s n −1 ds = c0 t2− n °f |C 1 ° .

∗

0

0

The lemma is proved. Obviously the estimate for the second term on the right-hand side in (11.45) is very rough and can probably be improved. On the other hand, a second term is surely necessary in general; for assume we (only) had 2n−1 tZ 1

∗

∗∗

s n −1 |∇f | (s)ds ≤ c t |∇f |

¡

¢ t2n−1 ,

0

for all small t > 0 and f ∈ C 1 , i.e., (by the definition of the maximal function g ∗∗ ) 2n−1 tZ

2n−1 tZ

s 0

1 n −1

∗

|∇f | (s)ds ≤ c t

∗

−(2n−2)

|∇f | (s)ds.

(11.47)

0 ∗

Now a simple example of a function f with |∇f | (s) ∼ s−κ disproves (11.47) if we choose n1 < κ < 1: then the left-hand side of (11.47) diverges whereas the right-hand side does not. So a second term is needed for “compensation” in general. Of course, splitting the integral in (11.46) not with t2n−1 < tn as an intermediate point, but with, say, tn+ε < tn , one can improve the first term on the right-hand side of (11.45) at the expense of the

192

Envelopes and sharp embeddings of function spaces

latter one, Zt

n 1

∗

s n −1 |∇f | (s)ds ≤ cε

©

∗∗

t |∇f |

¡

° °ª ¢ tn+ε + t1+ε °f |C 1 ° ;

0

this argument resembles [Tri01, Rem. 12.17]. 1 Note, that (11.44) leads to a right-hand side like (11.45), but with t2− n 2 in the latter term replaced by t , which is smaller for 0 < t < 1. So a combination of (11.41) and (11.45) results in an estimate less sharp than (11.44), but due to the partly rather rough arguments is it not clear at the moment, whether (11.44) or (11.41) are better in general. Nevertheless, for our purpose estimate (11.41) was completely sufficient; recall Proposition 5.10. We come back to our “lifting” problem for the envelopes. Let X ⊂ Lloc 1 be some function space on Rn of regular distributions with, say, X 6,→ L∞ . Denote by X ∇ ⊂ X the following subspace © ª X ∇ = g ∈ Lloc : Dα g ∈ X, |α| ≤ 1 1 with

(11.48)

X ° ° °g|X ∇ ° ∼ kg|Xk + kDα g |Xk . |α|=1

We assume that X ∇ ,→ C; this setting is obviously motivated by X = Asp,q , see (9.23). In view of (11.39) and (11.40) we study the problem under which assumptions one has ¡ ¢ n EG (X) = EC X ∇ or, at least, X∇

EGX (tn ) ∼ EC We have ° no complete ° ° ° that ° |∇f | |X ° ≤

(t),

0 < t < ε.

answer, but a partial one. Let f ∈ X ∇ ∩ C 1 be such ° ° °f |X ∇ ° ∼ 1 and t > 0 small. Then by (11.41),

1 ω(f, t) ≤ c t t

Zt

n

s

1 n −1

0

1 ∗ |∇f | (s)ds ≤ c t

Zt

n 1

s n −1 EGX (s)ds 0

° ° for all f ∈ X ∇ ∩ C 1 , °f |X ∇ ° = 1, and small t > 0. Assuming further that, for instance, C0∞ is dense in X ∇ , then this implies X∇ EC (t)

1 ≤ c t

Zt

n

s 0

1 n −1

EGX (s)ds

1 ∼ t

Zt EGX (σ n ) dσ. 0

(11.49)

Applications

193

In view of our above-mentioned examples (11.39) we would like to estimate (11.49) further by c EGX (tn ); this refers to the question whether 1 t

Zt EGX (σ n ) dσ ≤ c EGX (tn )

(11.50)

0

is true for some c > 0, and all small t > 0. Reformulating (11.50) we thus achieved the convergence of ¡ −(k+J)n ¢ ∞ X X −k EG 2 2 ≤ C (11.51) EGX (2−Jn ) k=0 uniformly with respect to J ∈ N, for, say, J ≥ J0 , implies X∇

EC

(t) ≤ c EGX (tn )

(11.52)

for all small t, 0 < t < ε. Clearly (11.51) is satisfied for   0 < κ < n1 , µ ∈ R, µ X −κ κ=0 , µ > 0, EG (τ ) ∼ τ |log τ | with  , µ < −1 ; κ = n1

(11.53) n/p−1

this covers all cases in (11.39) apart from the limiting case when X = Bp,q , n/p X ∇ = B p,q , 0 < p < n, 0 < q ≤ 1, – reflecting that (11.51) is only sufficient for (11.52). X∇ Turning to the indices uX G and uC , respectively, one concludes uX C

∇

≤ uX G

(11.54)

provided that there is some c > 0 such that for all k ∈ N, " ¡ −kn ¢ #r k−J X X E G 2 ≤ c, 2−ν% X ∇ ¡ −(k−ν) ¢ EC 2 ν=0

(11.55)

X where r = uX G and % < r (in case of r = uG ≤ 1 we may admit % = r). The proof of this fact copies that one of Proposition 5.10, Step 3. Note that X∇

(5.12) and (5.13) are certain examples: the first one with EC (t) ∼ EGX (t) ∼ u |log t| is performed directly in Step 3 of the proof of Proposition 5.10, whereas X∇

1

(5.13) is related to the setting EC (t) ∼ t−(1−κ) , EGX (t) ∼ t− n (1−κ) , 0 < κ < 1. Obviously condition (11.55) is satisfied in that case, too, " ¡ ¢ #r k−J · k(1−κ) ¸r k−J X X EGX 2−kn 2 −ν% −ν% ∼ 2 2 (k−ν)(1−κ) X ∇ ¡ −(k−ν) ¢ 2 EC 2 ν=0 ν=0 ∼

k−J X ν=0

2−ν(%−r(1−κ)) ≤ c

194

Envelopes and sharp embeddings of function spaces

if we choose % such that r(1 − κ) < % < r. X∇

Assume we knew already that EC (t) ∼ EGX (tn ) for small t > 0. Then condition (11.55) can be reformulated as 

k−J X

2−ν% 

ν=0

X∇

EC

X∇

EC

¡

2−k

¢ r ¢ ≤ c

¡ 2−(k−ν)

(11.56)

for some c > 0 independent of k ∈ N. Recall our difficulties with the situation of continuity envelopes on the critical line in Section 9.3; this refers s s+1 to the situation X = Bp,q , X ∇ = Bp,q , with n > 1, 0 < p < n, s = np − 1, 0 < q ≤ 1, and s Bp,q

EG

B s+1

(tn ) ∼ EC p,q (t) ∼ t−1 ,

and a similar result for F -spaces; see Theorems 8.1 (with r = n) and 9.10. Consequently (11.56) reads as the question whether k X ν=0

¢ r · k ¸r k k X X 2 −ν%   ∇¡ ∼ 2 = 2−ν(%−r) ¢ (k−ν) X 2 E 2−(k−ν) ν=0 ν=0 

2−ν%

X∇

EC

¡

2−k

C

converges independently of k ∈ N. This, however, fails because of % ≤ r. So condition (11.56) reflects the additional problems appearing on the critical line exactly. Nevertheless we shall finally collect the above considerations for further use, though the answer is not yet complete – missing links “converse” to (11.49) and (11.54) in general case so far. Corollary 11.11 Let the spaces X, X ∇ be given as above. (i)

There exists c > 0 such that 1 ≤ c t

X∇ EC (t)

Zt

n 1

s n −1 EGX (s)ds.

(11.57)

0

for all small t, 0 < t < ε. Moreover, if there is a number C > 0 such that for all large J ∈ N, J ≥ J0 , ¡ −(k+J)n ¢ ∞ X X −k EG 2 ≤ C, (11.58) 2 EGX (2−Jn ) k=0 then (11.57) can be replaced by X∇

EC

(t) ≤ c EGX (tn ) .

(11.59)

Applications

195

(ii) Assume there is a number c > 0 such that for all k ∈ N, " ¡ −kn ¢ #r k X X E G 2 ≤ c, 2−ν% X ∇ ¡ −(k−ν) ¢ EC 2 ν=0

(11.60)

where r = uX and % < r (in case of r = uX G G ≤ 1 we may admit % = r). Then ∇ uX ≤ uX (11.61) G. C X∇

(t) ∼ EGX (tn ), (11.60) can be replaced by r  X ∇ ¡ −k ¢ k X E 2 C 2−ν%  X ∇ ¡ ¢  ≤ c. −(k−ν) EC 2 ν=0

In particular, when EC

Inequalities converse to (11.52) and (11.54) are missing so far; further studies in the sense of [JMP91] are necessary, and – in view of our results (11.39), (11.40) – also promising. Remark 11.12 Note that (11.58) and (11.60) are only sufficient to get (11.59) n/p and (11.61), respectively, but not necessary; recall the situation for X = Bp,q , n/p−1 0 < p < ∞, 1 < q ≤ ∞, or X = W1n , X = Bp,q , 0 < p < n, 0 < q ≤ 1, for instance.

11.3

Compact embeddings

We briefly discuss some questions related to compactness. We already mentioned that – turning to spaces on bounded domains defined by restriction – most of our results for (growth or continuity) envelopes can be transferred immediately. For convenience, we shall only regard the unit ball U ⊂ Rn as an underlying domain in this section. We study an embedding between two function spaces defined on U , and possible links between its compactness and the envelopes of the involved spaces. Recall our notation (11.6). Clearly, by Propositions 3.4(iv) and 5.3(iii) there cannot be a continuous embedding X1 ,→ X2 at all whenever (X1 ,X2 )

sup qG

0
(t) = ∞,

or

(X1 ,X2 )

sup qC

(t) = ∞.

(11.62)

0
So for a continuous embedding (not to speak of compactness so far) it is at (X ,X ) (X ,X ) least necessary that qG 1 2 (t) or qC 1 2 (t) are bounded, see also the discussion on “limiting” embeddings in Section 11.1. Moreover, granted the embed(X ,X ) (X ,X ) ding X1 ,→ X2 was continuous, the boundedness of qG 1 2 (t), qC 1 2 (t)

196

Envelopes and sharp embeddings of function spaces

is not sufficient for its compactness: Triebel claimed in [Tri01, 14.6] that, roughly speaking, some embedding cannot be compact when the envelopes of (X ,X ) (X ,X ) source and target spaces coincide, i.e., qG 1 2 (t) ∼ 1 or qC 1 2 (t) ∼ 1. He gave a proof for a special example; we shall generalise this now. Proposition 11.13 Let X1 (U ), X2 (U ) be function spaces of regular distributions with X1 (U ) ,→ X2 (U ). (i) Assume that C ∞ (U ) is dense in X2 (U ), X2 (U ) ,→ C(U ), X2 (U ) 6,→ (X ,X ) Lip1 (U ), and qC 1 2 (t) ∼ 1, 0 < t < ε. Then id : X1 (U ) −→ X2 (U ) cannot be compact. (ii) Assume that there is some c > 0 such that for small t > 0 we have EGX2 (t) ≤ c EGX2 (2t), let the simple functions be dense in X2 (U ), (X ,X ) X2 (U ) 6,→ L∞ (U ), and qG 1 2 (t) ∼ 1, 0 < t < ε. Then id : X1 (U ) −→ X2 (U ) cannot be compact. X1

(X ,X )

X2

P r o o f : Let qC 1 2 (t) ∼ 1, i.e., EC (t) ∼ EC (t), 0 < t < ε, and assume id : X1 (U ) −→ X2 (U ) was compact; then for all γ > 0 we have a finite γ-net {f1 , . . . , fN (γ) } ⊂ X2 , kfj |X2 k ≤ 1, such that for all f ∈ X1 , kf |X1 k ≤ 1, it holds min kf − fj |X2 k < γ, j=1,...,N (γ)

where we may assume additionally that fj ∈ C ∞ (U ), j = 1, . . . , N (γ). Then by Proposition 5.3(iv), or (6.2) and the monotonicity of the left-hand sides of this equation in v, respectively, we obtain for small δ > 0, sup 0
ω(f − fj , t) X2

t EC (t)

≤ c kf − fj |X2 k < γ,

(11.63)

i.e., uniformly for all t, 0 < t < δ, ω(f, t) ω(fj , t) X2 X2 < γ EC (t) + ≤ γ EC (t) + cδ , t t

(11.64)

because of the smoothness of fj . Take the supremum over f ∈ X1 , kf |X1 k ≤ 1, and restrict γ to small values, then we obtain X1

X2

EC (t) ≤ γ EC (t) + cδ , X1

0 < t < δ,

X2

and thus by our assumption EC (t) ∼ EC (t), X2

EC (t) ≤ cδ,γ ,

0 < t < δ.

This, however, contradicts X2 (U ) 6,→ Lip1 (U ) by Proposition 5.3(ii).

(11.65)

Applications

197

Dealing with the situation in (ii), (11.63) has to be replaced by sup 0
(f − fj )∗ (t) ≤ c kf − fj |X2 k < γ, EGX2 (t)

based on Proposition 3.4(v), or (4.5), respectively. Hence by the special subadditivity of the non-increasing rearrangement and our additional assumption we arrive at µ ¶ µ ¶ µ ¶ t t t + fj∗ ≤ γ c EGX2 (t) + fj∗ f ∗ (t) < γ EGX2 2 2 2 uniformly for all t, 0 < t < δ. Assuming that the simple functions are dense in X2 , we thus have the counterpart of (11.65), leading to the same kind of contradiction as above. Consequently the corresponding embedding id : X1 (U ) −→ X2 (U ) can only be compact when (X1 ,X2 )

lim inf qG t↓0

(t) = 0,

or

(X1 ,X2 )

lim inf qC t↓0

(t) = 0.

(11.66)

Another more direct approach between envelope results and compactness studies may be offered by estimates of entropy numbers and approximation numbers, respectively, in terms of moduli of continuity. We briefly recall these concepts. Let A1 linear and any given the image

and A2 be two complex (quasi-) Banach spaces and let T be a continuous operator from A1 into A2 . If T is compact then for ε > 0 there are finitely many balls in A2 of radius ε which cover T U1 of the unit ball U1 = {a ∈ A1 : ka|A1 k ≤ 1}.

Definition 11.14 Let A1 and A2 be two complex (quasi-) Banach spaces, k ∈ N and let T : A1 → A2 be a linear and continuous operator from A1 into A2 . (i) The k th entropy number ek of T is the infimum of all numbers ε > 0 such that there exist 2k−1 balls in A2 of radius ε which cover T U1 . (ii) The k th approximation number ak of T is the infimum of all numbers kT − Sk where S runs through the collection of all continuous linear maps from A1 to A2 with rank S < k, ak (T ) = inf{kT − Sk : S ∈ L(A1 , A2 ), rank S < k}.

198

Envelopes and sharp embeddings of function spaces

For details and properties of approximation numbers we refer to [CS90], [EE87], [K¨on86] and [Pie87] (restricted to the case of Banach spaces), and [ET96] for some extensions to quasi-Banach spaces. Among other features we only want to mention the multiplicativity of entropy numbers and approximation numbers: let A1 , A2 and A3 be complex (quasi-) Banach spaces and T1 : A1 −→ A2 , T2 : A2 −→ A3 two operators in the sense of Definition 11.14. Then ek1 +k2 −1 (T2 ◦ T1 ) ≤ ek1 (T1 ) ek2 (T2 ), k1 , k2 ∈ N, (11.67) and ak1 +k2 −1 (T2 ◦ T1 ) ≤ ak1 (T1 ) ak2 (T2 ),

k1 , k2 ∈ N.

(11.68)

is compact .

(11.69)

Note that one has in general lim ek (T ) = 0

k→∞

if, and only if,

T

The last equivalence justifies the saying that entropy numbers measure “how compact” an operator acts. This is one reason to study the asymptotic behaviour of entropy numbers (that is, their decay) for compact operators in detail. Dealing with approximation numbers there is no complete counterpart of (11.69) in general; in particular, one only has lim an (T ) = 0

n→∞

implies

T is compact.

(11.70)

It is known that it may happen that limn→∞ an (T ) = α(T ) > 0 for some compact T ∈ L(A, B) when B fails to have the approximation property, see [EE87] and [Pie80, Prop. 10.1.3, 10.1.4]. It is known that many spaces, like Hilbert spaces, Lp , 1 ≤ p ≤ ∞, C(K), and `p , 1 ≤ p < ∞, possess this property, but there exist spaces without it, cf. [Enf73], [Pie80, Thm. 10.4.7]. Moreover, approximation numbers – unlike entropy numbers – can be regarded as special s-numbers, a concept introduced by Pietsch [Pie80, Sect. 11]. Example 11.15 Let m ∈ N, and X a real Banach space with dim X = m < ∞, idX ∈ L(X, X) the natural embedding map. Then 2−

k−1 m

≤ ek (idX ) ≤ 4 2−

and

½ ak (idX ) =

k−1 m

,

k ∈ N,

1 , k = 1, . . . , m, 0 , k > m.

In general, one can show for real Banach spaces X and Y , and T ∈ L(X, Y ), that rank T = m if, and only if, ∃c>0

∀ n ∈ N : c 2−

n−1 m

≤ en (T ) ≤ 4kT k2−

n−1 m

,

(11.71)

Applications

199

see [EE87, Prop. II.1.3], [CS90, Prop. 1.3.1]. The counterpart for approximation numbers reads as ak (T ) = 0

if, and only if,

rank T < k.

(11.72)

Example 11.16 We briefly mention a second famous example, the diagonal operator D : `p −→ `p . Let (σk )k be a monotonically decreasing sequence, σ1 ≥ σ2 ≥ · · · ≥ 0, and consider Dσ : `p −→ `p ,

Dσ : x = (ξk )k 7−→ (σk ξk )k ,

1 ≤ p ≤ ∞.

For convenience, let `p be real; then ak (Dσ ) = σk ,

k ∈ N,

and sup 2− m∈N

k−1 m

1

(σ1 · · · σm ) m ≤ ek (Dσ ) ≤ 6 sup 2− m∈N

k−1 m

1

(σ1 · · · σm ) m ,

see [Pie80, Thms. 11.3.2, 11.11.4], [K¨on86], [GKS87], [CS90, Prop. 1.3.2]. Remark 11.17 In view of the above simple examples one might be tempted to find a general relation between entropy numbers and approximation numbers. Obviously, Example 11.15 implies that an estimate of the type ek (T ) ≤ c ak (T ), k ∈ N, cannot hold in general, whereas it is, for instance, always true that limk→∞ ek (T ) ≤ am (T ), m ∈ N; cf. [CS90, Lemma 2.5.2]. Clearly, the converse inequality, am (T ) ≤ c em (T ) – though being true in the context of (real) Hilbert spaces, [CS90, Thm. 3.4.2] – cannot hold in general either, since there are compact operators T with limk→∞ ak (T ) > 0 (if the target space fails to have the approximation property), but limk→∞ ek (T ) = 0 by (11.69). However, replacing the term-wise estimates by particularly weighted averages, one obtains final answers of the following type: Let 0 < r < ∞, X, Y Banach spaces, and T ∈ L(X, Y ), then there exists a constant c = c(r) > 0, such that for m ∈ N, sup k=1,...,m

1

k r ek (T ) ≤ c

sup k=1,...,m

1

k r ak (T ),

see [Car81], [CS90, Thm. 3.1.1]. There are parallel results in [CS90, Thm. 3.1.2] refining the above `∞ -setting. Another extension was found by Triebel in [Tri94]: There exists c > 0 such that for all k ∈ N, ek (T ) ≤ c ak (T ), assuming that there is some c0 > 0 with a2j−1 (T ) ≤ c0 a2j (T ) for all j ∈ N, and T ∈ L(A, B) is compact. More generally, if there is a positive increasing

200

Envelopes and sharp embeddings of function spaces

function f on N with f (2j ) ≤ c f (2j−1 ) for some c > 0 and all j ∈ N, this implies the existence of some C > 0 such that for all k ∈ N, sup f (j) ej (T ) ≤ C sup f (j) aj (T ).

1≤j≤k

(11.73)

1≤j≤k

The interplay between continuity envelopes and approximation numbers relies on the following outcome. Corollary 11.18 Let X(U ) be a Banach space with X(U ) ,→ C(U ). There exists c > 0 such that for all k ∈ N, ³ 1´ 1 (11.74) ak+1 (id : X(U ) −→ C(U )) ≤ c k − n ECX k − n .

P r o o f : We apply the following estimate obtained by Carl and Stephani in [CS90, Thm. 5.6.1]: Let (Y, d) be a compact metric space, X an arbitrary Banach space, and T : X → C(Y ) compact, then ak+1 (T ) ≤

sup

ω (T f, εk (Y )) ,

k ∈ N,

(11.75)

kf |Xk≤1

where εk (Y ) are the usual (non-dyadic) entropy numbers, i.e., the infimum of all numbers ε > 0 such that there exist m ≤ k balls of radius ε which cover Y . Adapted to our setting, Y = U ⊂ Rn , T = id, the result follows X immediately from Definition 5.1 of EC , taking εk (U ) ∼ k −1/n into account, k ∈ N. We combine Corollaries 11.18 and 11.11(i) and adapt the notation (11.48) to our setting: Let X(U ) be some function space of regular distributions with X(U ) 6,→ L∞ (U ). Let X ∇ (U ) = {g ∈ D0 (U ) : Dα g ∈ X(U ), |α| ≤ 1} be the subspace of X(U ) normed by X ° ° °g|X ∇ (U )° = kDα g|X(U )k . |α|≤1

Let C(U ) stand for the space of all complex-valued bounded uniformly continuous functions on U , equipped with the sup-norm as usual. −1

Corollary 11.19 Let X(U ), X ∇ (U ) be given with X(U ) ,→ B ∞,∞ (U ) and X ∇ (U ) ,→ C(U ). Let EGX satisfy (11.58), and assume that there is a bounded (linear) lift operator L mapping X(U ) into X ∇ (U ) such that its inverse −1 L−1 exists and maps C(U ) into B∞,∞ (U ). Then there is some c > 0 such that ¡ ¢ ¡ ¢ 1 −1 ak idX : X(U ) −→ B∞,∞ (U ) ≤ c k − n EGX k −1 , k ∈ N. (11.76)

Applications

201

P r o o f : We combine Corollary 11.18 for compact embeddings in C(U ) (as target space) with the properties of the operator L and its inverse L−1 and Corollary 11.11(i) (which requires (11.58)); in particular, using the bounded 0 −1 lift L : X(U ) −→ X ∇ (U ) with L−1 : B∞,∞ (U ) −→ B∞,∞ (U ), the decomposition ¡ ¢ ¡ ¢ 0 idX = L−1 ◦ C(U ) ,→ B∞,∞ (U ) ◦ idX ∇ : X ∇ (U ) −→ C(U ) ◦ L, an application of Corollary 11.11(i) for idX ∇ , together with the multiplicativity of approximation numbers conclude the argument. Dealing with entropy numbers, we conclude from (11.73) and Corollaries 11.18, 11.19 the following result. Corollary 11.20 Let f : N → R be a positive and increasing function satisfying ¡ ¢ ¡ ¢ f 2k ≤ c f 2k−1 (11.77) for some c > 0 and all k ∈ N. (i)

Let X(U ) be a Banach space with X(U ) ,→ C(U ). There exists C > 0 such that for all m ∈ N, sup 1≤k≤m

f (k) ek (id : X(U ) −→ C(U )) ≤ C sup 1≤k≤m

1

X

f (k) k − n EC

³

´ 1 k− n .

(11.78)

−1

(ii) Let X(U ), X ∇ (U ) be given with X(U ) ,→ B ∞,∞ (U ) and X ∇ (U ) ,→ C(U ). Let EGX satisfy (11.58), and assume that there is a bounded (linear) lift operator L mapping X(U ) into X ∇ (U ) such that its −1 inverse L−1 exists and maps C(U ) into B∞,∞ (U ). Then there is some C > 0 such that for all m ∈ N, ¡ ¢ −1 sup f (k) ek id : X(U ) −→ B∞,∞ (U ) 1≤k≤m

≤ C

sup 1≤k≤m

¡ ¢ 1 f (k) k − n EGX k −1 .

(11.79)

Obviously, (11.74) and (11.76) only provide upper estimates for the corresponding approximation numbers; we shall discuss the sharpness of these bounds in different settings, and start with a short account on what is known for spaces of type Asp,q . Let −∞ < s2 ≤ s1 < ∞, 0 < p1 , p2 ≤ ∞ ( p1 , p2 < ∞ in the F -case), 0 < q1 , q2 ≤ ∞, and idA = id : Asp11 ,q1 (U ) −→ Asp22 ,q2 (U ),

202

Envelopes and sharp embeddings of function spaces

where the spaces Asp,q (U ) are given by (7.24). Then idA is continuous when ¶ µ 1 1 − ≥0 δ+ := s1 − s2 − n (11.80) p1 p2 + and q1 ≤ q2 if δ+ = 0 in the B-case. Furthermore, idA becomes compact when δ+ > 0; cf. [ET96, (2.5.1/10)]. The extension to values p2 < p1 – compared with the Rn - setting – is due to H¨older’s inequality. In this situation Edmunds and Triebel proved in [ET89], [ET92] (see also [ET96, Thm. 3.3.3/2]) that ek (idA ) ∼ k −

s1 −s2 n

,

k ∈ N,

(11.81)

where s1 ≥ s2 , 0 < p1 , p2 ≤ ∞ (p1 , p2 < ∞ in the F -case), 0 < q1 , q2 ≤ ∞, and δ+ > 0. In the case of approximation numbers the situation is more complicated; the result of Edmunds and Triebel in [ET96, Thm. 3.3.4], partly improved by Caetano [Cae98] and Skrzypczak [Skr05] reads as ak (idA ) with

µ κ =

∼

k−

δ+ n

¶

min(p01 , p2 ) −1 2

µ · min

+

−κ

,

k ∈ N,

(11.82)

δ 1 , n min(p01 , p2 )

¶ ,

(11.83)

and δ is given by (7.34). The additional exponent κ only appears when p1 < 2 < p2 . The above asymptotic result is almost complete now, apart from the restriction that (p1 , p2 ) 6= (1, ∞) when 0 < p1 < 2 < p2 ≤ ∞. In particular, when p2 = ∞, s1 ≥ s2 , 0 < p1 ≤ ∞, 0 < q1 , q2 ≤ ∞, and n > 0, δ+ = δ = s1 − s2 − p1 then ¡ ¢ s1 −s2 s2 ek id : Bps11 ,q1 (U ) −→ B∞,q (U ) ∼ k − n , 2

k ∈ N,

(11.84)

and ¡ ¢ s2 ak id : Bps11 ,q1 (U ) −→ B∞,q (U ) 2   s1 −s2 1 − n +p   1 , 2 ≤ p ≤ ∞ k   1     s −s 1 1 2 − n +2 , 1 < p1 < 2, s1 − s2 > n , ∼ k   i p0 h     1 2− 1   − s1 −s n p1 2 , 1 < p1 < 2, s1 − s2 ≤ n k

(11.85)

(and two-sided estimates for 0 < p1 ≤ 1); see [ET96, Thm. 3.3.4], [Cae98], [Skr05].

Applications

203

Remark 11.21 In [HM04] and [CH05] we studied entropy numbers and approximation numbers, respectively, for compact embeddings of spaces of generalised smoothness. Restricted to our Example 7.5, s ∈ R, b ∈ R, 0 < p < ∞, 0 < q ≤ ∞, with np < s ≤ np + 1, the compactness is an immediate consequence of the above argument and (7.62). In [HM06, Prop. 4.4, Example 4.8] we proved that for k ∈ N, s1 −s2 ¡ ¢ 1 −b s1 ,b s2 ak id : Bp,q (U ) −→ B∞,∞ (U ) ∼ k − n + p (1 + log k) ,

(11.86)

assuming 2 ≤ p ≤ ∞ for convenience, and 0 < q ≤ ∞, np < s1 − s2 < np + 1, b ∈ R. For s2 = 0 the target space can be replaced by C(U ). In the case of s = np + 1, 2 ≤ p ≤ ∞, we obtained in [CH05, Prop. 4.10], 1

−b

c1 k − n (1 + log k)

³ ´ 1+n/p,b s2 ≤ ak id : Bp,q (U ) −→ B∞,∞ (U ) ) ( 1 −b (1 + log k) q0 , b < q10 1 −n , ≤ c2 k 1 (log (1 + log k)) q0 , b = q10

for k ∈ N, see also Example 9.8. The counterpart for entropy numbers, can be found in [Leo00, Thm. 3], ¡ ¢ s1 −s2 −b s1 ,b s2 ek id : Bp,q (U ) −→ B∞,∞ (U ) ∼ k − n (1 + log k) , where 0 < p, q ≤ ∞, s1 − s2 >

n p,

(11.87)

b ∈ R, and k ∈ N.

We consider the natural embedding operators

and

id1X : X(U ) −→ C(U ),

(11.88)

−1 id2X : X(U ) −→ B∞,∞ (U ),

(11.89)

where the spaces X(U ) are defined by restriction. Remark 11.22 One can also introduce spaces of type Lip(1,−α) p, q (U ), or C(U ), by the usual adaption of the corresponding definitions, e.g., Lip(1,−α) (U ) as the set of those f ∈ C(U ) such that ° ° ° ° (1,−α) (U )° = kf |L∞ (U )k + °f |Lip

sup

x ∈ U, h ∈ Rn 0 < |h| < 1/2

|(∆h f )(x)| α |h| |log |h||

(11.90)

is finite. Standard procedures show that there is a bounded extension map from X(U ) to X(Rn ) in these cases; see, for example, [EE87, pp. 250-251].

204

Envelopes and sharp embeddings of function spaces

We discuss a few cases for X and begin with special settings for (11.88). Example 11.23 Let X = Asp,q with n n < s < + 1, p p or s=

n + 1, p

0 < p ≤ ∞,

0 < p ≤ ∞,

0 < q ≤ ∞,

1 < q ≤ ∞.

For convenience, we may restrict ourselves to B-spaces only, though the F case can be handled completely parallel. Note that – in view of (7.31) – the above assumption δ+ > 0 for compactness implies s > np , 0 < q ≤ ∞; see also our remarks in Section 7.2 or Proposition 11.13(ii) (together with the corresponding results in the previous sections). Obviously, s id1Bp,q (U ) −→ C(U ) : Bp,q s

remains compact for s > np + 1, but our envelope concept is not adapted appropriately for this higher smoothness; so this loss of information causes very weak estimates only (and will not be discussed further). We conclude from our results in Section 9.1 and (11.74), (11.78), ³ ´ ³ ´ ek id1Bp,q ≤ c ak id1Bp,q s s ) ( s 1 , np < s < np + 1 , 0 < q ≤ ∞ k− n + p 0 . ≤c 1 1 k − n (loghki) q0 , s = np + 1 , 1 < q ≤ ∞ We compare this result with (11.84), (11.85) (with s2 = 0) and realise that for 0 < s − np < 1 (i.e., in the “super-critical strip”) we are led to the ¡ ¢ correct upper estimates for ak id1Bp,q apart from the case 1 < p < 2, s whereas otherwise – as well as for entropy numbers – our method provides a less sharp upper bound only. This, however, is not very surprising, as, firstly, the direct link is given between approximation numbers and envelopes (hence the entropy numbers being only some by-product in that sense), and, secondly, our continuity envelope functions are “made” for 0 ≤ s − np ≤ 1 only; otherwise they lack some interesting information. The more astonishing observation in our opinion is rather the sharpness of the results otherwise. Example 11.24 As a second case for X from (11.88) we regard Lipschitz (1,−α) (U ), α > 0. The spaces, X = Lip(1,−α) p, q . We begin with X(U ) = Lip compactness of id1Lip : Lip(1,−α) (U ) −→ C(U )

Applications

205

is a consequence of [EH00, Cor. 3.19]. Now Theorem 6.6(ii) and (11.74) yield ¡ ¢ ¡ ¢ 1 α ek id1Lip ≤ c ak id1Lip ≤ c0 k − n (loghki) , which by [EH00, Cor. 3.19(i)] gives the exact asymptotic behaviour both for entropy numbers and approximation numbers. Assume 0 < q ≤ ∞, n < p ≤ ∞, α > 1q , then Corollary 9.9 and (11.74) imply for id1 (1,−α) : Lip(1,−α) p, q (U ) −→ C(U ) Lip p, q

and k ∈ N, µ ek id1

¶

µ ≤ c ak id1

(1,−α)

Lip p, q

¶ 1

(1,−α)

Lip p, q

α− q1

1

≤ c0 k − n + p (loghki)

.

In view of (9.34) and (11.86) this is the precise asymptotic description, µ ¶ 1 1 α− 1 1 ak id (1,−α) ∼ k − n + p (loghki) q , k ∈ N, (11.91) Lip p, q

with 0 < q ≤ ∞, n < p ≤ ∞, α > 1q , whereas (11.87) and (9.34) lead to better estimates for entropy numbers in that case, µ ¶ 1 α− 1 ek id1 (1,−α) ∼ k − n (loghki) q , k ∈ N. (11.92) Lip p, q

We consider some cases for X in (11.89). s Example 11.25 Let again X = Bp,q , now with

n  p −1<s<     σp < s < n , p n  s = p,     s = 0,

n p,

0 < p ≤ n, 0 < q ≤ ∞ n < p ≤ ∞, 0 < q ≤ ∞ 0 < p < ∞, 1 < q ≤ ∞ n < p < ∞, 0 < q ≤ min(p, 2)

          

.

Similar to our above remarks, the compactness assumption δ+ > 0 excludes s ≤ np − 1, whereas s > np is omitted because of (necessarily) weaker estimates using (inappropriately adapted) growth envelope techniques. Dealing with the (sub-) critical case we are led to ´ ³ s+1 1 ≤ c k − n + p , k ∈ N, ak id2Bp,q s

206

Envelopes and sharp embeddings of function spaces

if σp < s < np , 0 < q ≤ ∞, or s = 0, 1 < p < ∞, 0 < q ≤ min(p, 2) by our results in Section 8.1, 8.3 and (11.76); recall notation (7.2). Comparison with (11.85) confirms the sharpness in case of σp < s < np , n ≥ 2; otherwise we can repeat our above discussion. This argument applies to entropy numbers, too. Note that the existence of the lift operator L can be verified applying usual restriction-extension procedures and the lift operator Iσ in Rn given s s−σ by (7.6), which maps Bp,q isomorphically onto Bp,q for all admitted parameters. Alternatively one can also use regular elliptic differential operators adapted to U ; see [Tri78a, Thm. 4.9.2] for the case 1 < p < ∞, 1 ≤ q ≤ ∞, and [Tri83, Thm. 4.3.4] for the extensions to 0 < p, q ≤ ∞, which are based on more recent techniques of Fourier multipliers. Example 11.26 Finally, we consider X = Lp (log L)a , with   n < p < ∞, a ∈ R p = n, a > 0  p = ∞, a ≤ 0 The compactness of −1 id2Lp (log L)a : Lp (log L)a (U ) −→ B∞,∞ (U )

is confirmed by [ET96, (2.5.1/10), Props. 2.6.1/1,2, Thm. 3.4.3/1] together with (11.80) and a duality argument. The existence of a bounded linear lift is covered by [ET96, Thm. 2.6.3], at least for n ≤ p < ∞. Theorem 4.7(ii) combined with (11.53) for µ = p1 , κ = −a, and (11.76) provides ( ak (id2Lp (log L)a ) ≤ c

1

1

−a

k − n + p (loghki) −a (loghki)

) ,n1

.

(11.93)

This observation (11.93) led us in [CH03] to the sharp asymptotic estimate: ¡ ¢ 1 1 −a −1 ak id : Lp (log L)a (U ) −→ B∞,∞ (U ) ∼ k − n + p (loghki) , for k ∈ N, where a > 0, n < p < ∞ for n ≥ 2, and 2 ≤ p < ∞ when n = 1. The additional restriction p ≥ 2 is caused by the lower estimate, but was somehow to be expected in the case of approximation numbers, see (11.85). Instead of Corollary 11.20(ii) and Theorem 4.7(ii) we apply (11.84) and Lp+ε (U ) ,→ Lp (log L)a (U ) ,→ Lp (U ), ε > 0, a > 0, and conclude ¡ ¢ 1 −1 ek id : Lp (log L)a (U ) −→ B∞,∞ (U ) ∼ k − n ,

k ∈ N,

for a > 0, n < p < ∞. As for the limiting case p = n, there are entropy number results in [ET96, Sect. 3.4], [EN98], [Cae00], but – as far as we know – no complete results for approximation numbers yet.

Applications

207

Remark 11.27 It is clear that the above method can be applied to a lot of situations whenever envelope results are at hand already, e.g., certain weighted spaces, spaces of generalised smoothness, etc. Usually, by the arguments explicated above, one would expect sharp asymptotic results for approximation numbers (at least in suitably adapted situations), that can be transferred to “dual” embedding operators without difficulties. Concerning entropy numbers the present approach seems too weak in general. Remark 11.28 The study of entropy numbers and approximation numbers of embeddings between function spaces is closely related to the distribution of eigenvalues of (degenerate) elliptic operators, as the books [ET96] and [Tri97] show. We conclude this section with a brief description of the background and the context for some further possible applications of our results. The motivation comes from Carl’s inequality giving an excellent link to possible applications, in particular, between entropy numbers and eigenvalues of some compact operators. The setting is the following. Let A be a complex (quasi-) Banach space and T ∈ L(A) compact. Then the spectrum of T (apart from the point 0) consists only of eigenvalues of finite algebraic multiplicity. Let {µk (T )}k∈N be the sequence of all non-zero eigenvalues of T , repeated according to algebraic multiplicity and ordered such that |µ1 (T )| ≥ |µ2 (T )| ≥ · · · ≥ 0. Then Carl’s inequality states that à k !1/k Y n ≤ inf 2 2k en (T ), |µm (T )| n∈N

m=1

In particular, we have |µk (T )| ≤

k ∈ N.

√ 2 ek (T ).

(11.94)

This result was originally proved by Carl in [Car81] and Carl and Triebel in [CT80] when A is a Banach space. An extension to quasi-Banach spaces is given in [ET96, Thm. 1.3.4]. When A is a Banach space, Zem´anek [Zem80] could prove 1 lim (ek (T m )) m = r(T ), k ∈ N, m→∞

where r(T ) is the spectral radius of T , see also [EE87, Cor. II.1.7]. Concerning estimates from below, i.e., converse to (11.94), and the connection between approximation numbers and eigenvalues, it is reasonable to concentrate on the Hilbert space setting first. Let H be a complex Hilbert space and T ∈ L(H) compact, the non-zero eigenvalues of which are denoted by {µk (T )}k∈N again; then T ∗ T has a non-negative, self-adjoint, compact square root |T |, and for all k ∈ N, ak (T ) = µk (|T |),

(11.95)

208

Envelopes and sharp embeddings of function spaces

see [EE87, Thm. II.5.10]. Hence, if in addition T is non-negative and selfadjoint, then the approximation numbers of T coincide with its eigenvalues. Moreover, a famous inequality of Weyl, see [K¨on86, Thm. 1.b.5], states that for all n ∈ N, n n Y Y |µj (T )| ≤ aj (T ), j=1

j=1

from which it follows that for all n ∈ N and all p ∈ (0, ∞), n X

|µj (T )|p ≤

j=1

n X

apj (T ).

j=1

Outside Hilbert spaces the results are less good but still very interesting. Let A be a complex Banach space, T ∈ L(A) compact, and {µk (T )}k∈N its eigenvalue sequence. K¨onig proved that for all m ∈ N and all p ∈ (0, ∞), µX m

|µj (T )|p

¶ p1 ≤ Kp

µX m

apj (T )

¶ p1 ,

j=1

j=1

√ √ where Kp = 2e/ p if 0 < p < 1, and Kp = 21/p 2e, if 1 ≤ p < ∞. For details and further remarks we refer to [CS90], [EE87], [K¨on86] and [Pie87]. In view of Carl’s inequality (11.94) and (11.95) one may thus obtain upper and lower estimates for eigenvalues from the study of entropy and approximation numbers, respectively, at least in Hilbert space settings when T ∈ L(H) is compact, non-negative and self-adjoint, see (11.95). Sometimes one can furthermore prove that the root spaces coincide and then – by some tricky bootstrapping techniques – “shift” estimates (originally proved in Hilbert spaces) to (quasi-) Banach spaces. The problem to determine ek (T ) or ak (T ), respectively, can often be reduced further to the study of entropy numbers or approximation numbers of suitable embeddings assuming that one has corresponding H¨older inequalities available. Another possible application is connected with the so-called “negative spectrum” and the Birman-Schwinger principle as described in [Sch86, Ch. 8, Sect. 5]. Let A be a self-adjoint operator acting in a Hilbert space H and let A be positive. Let V be a closable operator acting in H and suppose that K : H → H is a compact linear operator such that Ku = V A−1 V ∗ u

for all

u ∈ dom(V A−1 V ∗ )

where V ∗ is the adjoint of V . Assume that dom(A) ∩ dom(V ∗ V ) is dense in H. Then the above-mentioned result provides: A − V ∗ V has a self-adjoint extension G with pure point spectrum in (−∞, 0] such that # {σ(G) ∩ (−∞, 0] } ≤ # {k ∈ N : |λk | ≥ 1}

Applications

209

where {λk } is the sequence of eigenvalues of K, counted according to their multiplicity and ordered by decreasing modulus. In particular, we consider the behaviour of the “negative spectrum” σ(Gν ) ∩ (−∞, 0] of the self-adjoint unbounded operator

where

Gν = a(x, D) − νb2 (x)

as ν → ∞

(11.96)

a(x, D) ∈ Ψκ 1,γ ,

0 ≤ γ < 1,

(11.97)

κ > 0,

is assumed to be a positive-definite and self-adjoint pseudodifferential operator in L2 and b(x) is a real-valued function. We know from former considerations, cf. [HT94b, 2.4, 5.2], that o n √ (11.98) #{σ(Gν ) ∩ (−∞, 0]} ≤ # k ∈ N : 2 ek ≥ ν −1 ¡ ¢ with ek = ek b(x) b(x, D) b(x) and b(x, D) = a−1 (x, D) ∈ Ψ−κ 1,γ . These are essentially the applications we have in mind for using our results on entropy numbers and approximation numbers of compact embeddings. This programme was carried out in [HT94b], [ET96], first, and [Tri97], [Har98], [Har00a], [EH00] in different settings afterwards; we refer to these papers and books for details.

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Symbols

[a]

11

Bp,q

(s,Ψ)

104

EG (X)

n

189

bac

21

s Bp,q (w)

124

eX

163

{a}

21

C

20

f∗

12

a+

11

Cm

20

f ∗∗

17

|α|

20

C (1,−α)

23

fC

167

#M

31

Cs

21

fG

167

r0

11

C0∞

20

ϕX

52

∼

11

χA

19

Φr,u

142

,→

11

Dα

20

F, F −1

101

hxi

11

Di

28

s Fp,q

102

ak (T )

198

δ

109

s F∞,q

106

Ap

61

δ+

202

Fp,q

(s,Ψ)

104

Asp,q

103

∆m h

20

s Fp,q (w)

124

Asp,q (w)

124

EC

X

75

hp

103

bmo

103

EGX

39

Hps

102

bpq

104

ek (T )

198

Iσ

102

s,b Bp,q

104

EC (X)

91

Kr (x0 )

45

s Bp,q

102

EG (X)

65

Lexp,a

19

223

224

Symbols

Lip(1,−α) p, q

22

|ωn |

15

uX C

91

Lip(1,−α)

22

ωr (f, t)p

20

uX G

65

Lip(a,−α) ∞, q

25

p∗

28

wα

59

Lip1 , Lipa

21

qC

(X1 ,X2 )

181

wα

61

Lp

16

qG

(X1 ,X2 )

181

weak − Asp,q

125

Lp,q (log L)a

17

Qνm

105

Wpk

26

Lp,q

17

S

101

X∇

192

Lp (w)

59

S0

101

µf

11

σp

102

Index admissible function, 104 approximation numbers, 198 atomic decomposition, 106 atoms, 105

entropy numbers, 198 enveloping function continuity, 167 growth, 167 equi-measurable, 13

Banach function space, 19 Besov spaces s of type Bp,q , 102 s of type Bp,q (w), 124

fundamental function, 52

(s,Ψ) Bp,q , 104 s,b Bp,q , 104

of type of type bounded mean oscillation function space bmo, 103 continuity envelope, 91 continuity envelope function, 75 for Lip(1,−α) p, q , 157 continuity envelopes s s for Bp,q , Fp,q in the borderline case, 152, 156 in the critical case, 158 in the super-critical case, 147, 149 for Lipa , 93 for Lip(a,−α) ∞, q , 93 (1,−α) for Lip ∞, q , 93 s,b for Bp,q in the super-critical case, 157 for C, 93 for Wpk , 95 for W1n , 98 1+n/p for Wp , 96 difference operator ∆m h , 20 distribution function, 11

225

growth envelope, 65 growth envelope function, 39 for Lip(1,−α) p, q , 124 global assertions s s for Bp,q (wα ), Fp,q (wα ), 175 s s for Bp,q , Fp,q , 172 for Lp (wα ), 172 for Lp (wα ), 172 for Lpq , 171 for Lpq (log L)a , 171 for Wpk , 171 growth envelopes s s for Bp,q , Fp,q in borderline cases, 131, 133 in the critical case, 135, 145 in the sub-critical case, 120 in weighted cases, 125, 128 for Lexp,a , 68 for Lp,q (log L)a , 67 for bmo, 145 s,b for Bp,q in the critical case, 143 in the sub-critical case, 124 0 for F∞,q , 145 for Lp (wα ), 73 for Lp (wα ), 73 for Wpk , 69

226

Index n/p

for Wp

, 71

H¨older-Zygmund spaces of type C (1,−α) , 23 of type C s , 21 Hardy spaces hp , 103 inequality Carl’s, 207 Hardy-Littlewood, 13 Marchaud’s, 21 Weyl’s, 208 Lebesgue spaces of type Lp , 16 of type Lp (w), 59 lift operator, 102 Lipschitz spaces of type Lipa , 21 of type Lip(a,−α) ∞, q , 25 of type Lip(1,−α) , 22 of type Lip(1,−α) p, q , 22 Lorentz spaces Lp,q , 16 Lorentz-Zygmund spaces Lp,q (log L)a , 17 maximal function, 17 modulus of continuity ωr (f, t)p , 20 rearrangement-invariant space, 52 sequence spaces of type bpq , 104 slowly varying function, 104 smooth dyadic resolution of unity, 101 Sobolev spaces of type Hps , 102 of type Wpk , 26 Triebel-Lizorkin spaces s of type Fp,q , 102 s of type Fp,q (w), 124

(s,Ψ)

of type Fp,q

, 104

weights admissible, 59 Muckenhoupt class Ap , 61 Zygmund spaces Lexp,a , 19

List of Figures

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Figure 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Figure 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Figure 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Figure 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Figure 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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