Hardy Operators, Function Spaces and Embeddings

by Edmunds, D. E.; Evans, W. Desmond

• ISBN13: 9783540219729
• ISBN10: 3540219722
• Format: Hardcover
• Copyright: 2004-09-15
• Publisher: Springer Nature

Summary

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.

Table of Contents

Preliminaries	p. 1
Hausdorff and Minkowski dimensions	p. 1
The area and coarea formulae	p. 3
Approximation numbers	p. 6
Inequalities	p. 9
Hardy-type Operators	p. 11
Introduction	p. 11
Boundedness of T	p. 12
Compactness of T	p. 17
Approximation numbers of T	p. 23
The Hardy operator on a finite interval	p. 24
The general case: Preliminaries	p. 31
Estimates for am(T), 1 < p ≤ q < ∞	p. 39
Estimates for an(T) when p = 1 or q = ∞	p. 42
Approximation numbers of T when 1 ≤ q < p ≤ ∞	p. 43
Asymptotic results for p = q ∈ (1, ∞)	p. 43
The cases p = 1, ∞	p. 50
l¿ and l¿, w classes	p. 51
Hardy-type operators on trees	p. 55
Analysis on trees	p. 55
Boundedness of T	p. 57
Compactness of T and its approximation numbers	p. 58
Notes	p. 59
Banach function spaces	p. 63
Introduction	p. 63
Definitions	p. 64
Rearrangements	p. 69
Rearrangement-invariantspaces	p. 84
Examples	p. 90
Lorentz, Lorentz-Zygmund and generalised Lorentz-Zygmund spaces	p. 90
Orliczspaces	p. 96
Lorentz-Karamataspaces	p. 108
Decompositions	p. 121
Operatorsofjointweaktype	p. 125
Definitions	p. 125
Operatorsofstrongandweaktype	p. 128
Bessel-Lorentz-Karamata-potential spaces	p. 133
Abstract Sobolev spaces	p. 133
Bessel-Lorentz-Karamata-potential spaces	p. 134
Sub-limiting embeddings	p. 139
Limiting embeddings	p. 140
Super-limiting embeddings	p. 144
Examples	p. 152
Other spaces	p. 155
Notes	p. 158
Poincaré and Hardy inequalities	p. 161
Introduction	p. 161
Poincaré inequalities in BFSs	p. 164
Poincaré and Friedrichs inequalities	p. 164
Examples	p. 174
Higher-order cases	p. 183
Concrete spaces	p. 185
Classes of domains	p. 185
Sobolev and Poincaré inequalities	p. 193
Hardyinequalities	p. 207
Notes	p. 217
Generalised ridged domains	p. 219
Introduction	p. 219
Ridges and skeletons	p. 220
Simple ridges in <$>{\op R}^2<$>	p. 224
Generalised ridged domains	p. 228
Measure of non-compactness	p. 234
Analysis on GRD	p. 244
The map T and its approximate inverse M	p. 245
Equivalentembeddings	p. 249
Equivalent Poincaré inequalities	p. 251
Compactness of E	p. 252
Local compactness	p. 252
Measureofnon-compactness	p. 254
EmbeddingTheorems	p. 261
The Poincaré inequality and ¿(E)	p. 266
Notes	p. 273
Approximation numbers of Sobolev embeddings	p. 275
Introduction	p. 275
Some quotient space norms	p. 277
Dirichlet-Neumann bracketing in Lp	p. 282
Further asymptotic estimates for a GRD ¿	p. 294
Notes	p. 305
References	p. 307
Author Index	p. 319
Subject Index	p. 323
Notation Index	p. 325
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