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

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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

Submission of proposals for consideration Suggestions for publication, in the form of outlines and representative samples, are invited by the Editorial Board for assessment. Intending authors should approach one of the main editors or another member of the Editorial Board, citing the relevant AMS subject classifications. Alternatively, outlines may be sent directly to the publisher’s offices. Refereeing is by members of the board and other mathematical authorities in the topic concerned, throughout the world. Preparation of accepted manuscripts On acceptance of a proposal, the publisher will supply full instructions for the preparation of manuscripts in a form suitable for direct photo-lithographic reproduction. Specially printed grid sheets can be provided. Word processor output, subject to the publisher’s approval, is also acceptable. Illustrations should be prepared by the authors, ready for direct reproduction without further improvement. The use of hand-drawn symbols should be avoided wherever possible, in order to obtain maximum clarity of the text. The publisher will be pleased to give guidance necessary during the preparation of a typescript and will be happy to answer any queries. Important note In order to avoid later retyping, intending authors are strongly urged not to begin final preparation of a typescript before receiving the publisher’s guidelines. In this way we hope to preserve the uniform appearance of the series. CRC Press, Taylor and Francis Group 24-25 Blades Court Deodar Road London SW15 2NU UK

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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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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 Brézis-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 Brézis 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 António 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 Brézis, 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 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


− 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
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 Brézis 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 s}) ,

s ≥ 0.

(2.2)

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

11

12


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


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


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 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 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.


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 Hölder-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 Hölder-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 HölderZygmund 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 0 such that for all λ, 0 < λ < 1, ω(f, t)p ≤ c λ−α . 1−λ 0 −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)


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


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


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
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


where we used Hölder’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 Hölder’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 Hölder’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

1, then by H¨


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


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 Hölder’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 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


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 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 0.

(3.16)

(iii) Let [R, µ] = [Rn , | · |], and As = Kcs1/n (0) with µ(As ) = |Kcs1/n (0)| = s for appropriately ³chosenć > 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.


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 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


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).


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


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)


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 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


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 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


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


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
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)


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 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 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


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 Hölder’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


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 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 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

Ã Z21 ·

min(s, t) t | log t|α

0

α− q1

Ã Zs ·

≤ c1 |log s|

0

+ c2 s |log s|

α− q1

α− q1

0 α− q1

Ã 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

Ã Zs

¸q

.

¸q

dt t

!1/q

!1/q + c4

+ c4 ≤ C,

(5.24)

86


(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 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 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


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 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


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 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


(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,∞

166


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 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.

168


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
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
0. We shall see also that global


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

172


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 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


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 σp = n p − 1 , 0 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 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 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 °

174


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 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 → ∞,


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 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


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 0, c, c0 > 0 such that for all f ∈ X ⊂ Lloc 1 sup κ(t) 0 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 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


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 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


such that Ã Z∞ kfγ |Lp0 ,u k ≥ 2

iu dt h 1 t p0 fγ∗ (t) t

!1/u

Ã 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]

190


mentioned in Remark 5.11, ¢ 1 ω(f, t) ∗∗ ¡ ≤ c |∇f | t2n−1 + 3 sup τ − 2 ω(f, τ ) 2 t 0 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


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 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


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 0, sup 0 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], [Kön86], [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


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


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 Hölder’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


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 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 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 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 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


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 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)

) ,n 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 Ã 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, Zemánek [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


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 [Kön86, 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. König 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], [Kön86] 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 Hölder 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

Hölder-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

Figure Figure Figure Figure Figure Figure Figure Figure Figure

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13 14 14 15 15 18 24 26 36

Figure 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

48

Figure 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

111

Figure 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

119 130

Figure 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

147

Figure 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

188

227

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

Envelopes And Sharp Embeddings Of Function Spaces

Homogeneous Spaces and Equivariant Embeddings

CRC Research Notes in Mathematics Series)

Differential Geometry (Research Notes In mathematics Series)

CRC Research Notes in Mathematics Series)

Homogeneous Spaces and Equivariant Embeddings

Topological Function Spaces (Mathematics and its Applications)

Distributions and Analytic Functions (Pitman Research Notes in Mathematics Series)

Distributions and Analytic Functions (Pitman Research Notes in Mathematics Series)

Singular Points of Smooth Mappings (Research Notes in Mathematics Series)

Singular Points of Smooth Mappings (Research Notes in Mathematics Series)

Local Bifurcation and Symmetry (Research Notes in Mathematics Series)

Theory of function spaces

Distributions and Analytic Functions (Pitman Research Notes in Mathematics Series)

Approximation of Hilbert Space Operators (Research Notes in Mathematics Series)

Function Spaces

Function Spaces

Function Spaces and Applications

Theory of function spaces

Almost-periodic Functions in Abstract Spaces (Research Notes Inmathematics Series)

Closed Graph Theorems and Webbed Spaces (Research Notes in Mathematics Series)

Closed Graph Theorems and Webbed Spaces (Research Notes in Mathematics Series)

Operator Theory in Function Spaces (Pure and Applied Mathematics)

Theory of Function Spaces III (Monographs in Mathematics) (v. 3)

Topics in Galois Theory (Research Notes in Mathematics, Volume 1) (Research Notes in Mathematics)

Sobolev Spaces (Springer Series in Soviet Mathematics)

Multiplicative Functionals on Topological Algebras (Research Notes in Mathematics Series)

Quantum Integrable Systems (Research Notes in Mathematics Series) ( CRC Press )

Hardy-type Inequalities (Pitman Research Notes in Mathematics Series)

Measure, Integration and Function Spaces

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