Hardy operators, function spaces and embeddings

Classical Sobolev spaces, based on Lebesgue spaces on an underlying domain with smooth boundary, are not only of considerable intrinsic interest but have for many years proved to be indispensible in the study of partial differential equations and variational problems. Many developments of the basic theory since its inception arise in response to concrete problems, for example, with the (ubiquitous) sets with fractal boundaries.

The theory will probably enjoy substantial further growth, but even now a connected account of the mature parts of it makes a useful addition to the literature. Accordingly, the main themes of this book are Banach spaces and spaces of Sobolev type based on them; integral operators of Hardy type on intervals and on trees; and the distribution of the approximation numbers (singular numbers in the Hilbert space case) of embeddings of Sobolev spaces based on generalised ridged domains.

This timely book will be of interest to all those concerned with the partial differential equations and their ramifications. A prerequisite for reading it is a good graduate course in real analysis.

1 Preliminaries	p. 1
1.1 Hausdorff and Minkowski dimensions	p. 1
1.2 The area and coarea formulae	p. 3
1.3 Approximation numbers	p. 6
1.4 Inequalities	p. 9
2 Hardy-type Operators	p. 11
2.1 Introduction	p. 11
2.2 Boundedness of T	p. 12
2.3 Compactness of T	p. 17
2.4 Approximation numbers of T	p. 23
2.4.1 The Hardy operator on a finite interval	p. 24
2.4.2 The general case: Preliminaries	p. 31
2.4.3 Estimates for a m (T), 1	p. 39
2.4.4 Estimates for a n (T) when p = 1 or q = ∞	p. 42
2.4.5 Approximation numbers of T when 1 ≤ q	p. 43
2.4.6 Asymptotic results for p = q ∈ (1, ∞)	p. 43
2.4.7 The cases p = 1, ∞	p. 50
2.5 l ¿ and l ¿, w classes	p. 51
2.6 Hardy-type operators on trees	p. 55
2.6.1 Analysis on trees	p. 55
2.6.2 Boundedness of T	p. 57
2.7 Compactness of T and its approximation numbers	p. 58
2.8 Notes	p. 59
3 Banach function spaces	p. 63
3.1 Introduction	p. 63
3.1.1 Definitions	p. 64
3.2 Rearrangements	p. 69
3.3 Rearrangement-invariantspaces	p. 84
3.4 Examples	p. 90
3.4.1 Lorentz, Lorentz-Zygmund and generalised Lorentz-Zygmund spaces	p. 90
3.4.2 Orliczspaces	p. 96
3.4.3 Lorentz-Karamataspaces	p. 108
3.4.4 Decompositions	p. 121
3.5 Operatorsofjointweaktype	p. 125
3.5.1 Definitions	p. 125
3.5.2 Operatorsofstrongandweaktype	p. 128
3.6 Bessel-Lorentz-Karamata-potential spaces	p. 133
3.6.1 Abstract Sobolev spaces	p. 133
3.6.2 Bessel-Lorentz-Karamata-potential spaces	p. 134
3.6.3 Sub-limiting embeddings	p. 139
3.6.4 Limiting embeddings	p. 140
3.6.5 Super-limiting embeddings	p. 144
3.7 Examples	p. 152
3.8 Other spaces	p. 155
3.9 Notes	p. 158
4 Poincaré and Hardy inequalities	p. 161
4.1 Introduction	p. 161
4.2 Poincaré inequalities in BFSs	p. 164
4.2.1 Poincaré and Friedrichs inequalities	p. 164
4.2.2 Examples	p. 174
4.2.3 Higher-order cases	p. 183
4.3 Concrete spaces	p. 185
4.3.1 Classes of domains	p. 185
4.3.2 Sobolev and Poincaré inequalities	p. 193
4.4 Hardyinequalities	p. 207
4.5 Notes	p. 217
5 Generalised ridged domains	p. 219
5.1 Introduction	p. 219
5.1.1 Ridges and skeletons	p. 220
5.1.2 Simple ridges in {{\op R}}^2	p. 224
5.2 Generalised ridged domains	p. 228
5.3 Measure of non-compactness	p. 234
5.4 Analysis on GRD	p. 244
5.4.1 The map T and its approximate inverse M	p. 245
5.4.2 Equivalentembeddings	p. 249
5.4.3 Equivalent Poincaré inequalities	p. 251
5.5 Compactness of E	p. 252
5.5.1 Local compactness	p. 252
5.5.2 Measureofnon-compactness	p. 254
5.6 EmbeddingTheorems	p. 261
5.7 The Poincaré inequality and ¿(E)	p. 266
5.8 Notes	p. 273
6 Approximation numbers of Sobolev embeddings	p. 275
6.1 Introduction	p. 275
6.2 Some quotient space norms	p. 277
6.3 Dirichlet-Neumann bracketing in L p	p. 282
6.4 Further asymptotic estimates for a GRD ¿	p. 294
6.5 Notes	p. 305
References	p. 307
Author Index	p. 319
Subject Index	p. 323
Notation Index	p. 325

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