Hardy Operators, Function Spaces and Embeddings

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  • Format: Hardcover
  • Copyright: 2004-09-15
  • Publisher: Springer Nature
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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

Preliminariesp. 1
Hausdorff and Minkowski dimensionsp. 1
The area and coarea formulaep. 3
Approximation numbersp. 6
Inequalitiesp. 9
Hardy-type Operatorsp. 11
Introductionp. 11
Boundedness of Tp. 12
Compactness of Tp. 17
Approximation numbers of Tp. 23
The Hardy operator on a finite intervalp. 24
The general case: Preliminariesp. 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 classesp. 51
Hardy-type operators on treesp. 55
Analysis on treesp. 55
Boundedness of Tp. 57
Compactness of T and its approximation numbersp. 58
Notesp. 59
Banach function spacesp. 63
Introductionp. 63
Definitionsp. 64
Rearrangementsp. 69
Rearrangement-invariantspacesp. 84
Examplesp. 90
Lorentz, Lorentz-Zygmund and generalised Lorentz-Zygmund spacesp. 90
Orliczspacesp. 96
Lorentz-Karamataspacesp. 108
Decompositionsp. 121
Operatorsofjointweaktypep. 125
Definitionsp. 125
Operatorsofstrongandweaktypep. 128
Bessel-Lorentz-Karamata-potential spacesp. 133
Abstract Sobolev spacesp. 133
Bessel-Lorentz-Karamata-potential spacesp. 134
Sub-limiting embeddingsp. 139
Limiting embeddingsp. 140
Super-limiting embeddingsp. 144
Examplesp. 152
Other spacesp. 155
Notesp. 158
Poincaré and Hardy inequalitiesp. 161
Introductionp. 161
Poincaré inequalities in BFSsp. 164
Poincaré and Friedrichs inequalitiesp. 164
Examplesp. 174
Higher-order casesp. 183
Concrete spacesp. 185
Classes of domainsp. 185
Sobolev and Poincaré inequalitiesp. 193
Hardyinequalitiesp. 207
Notesp. 217
Generalised ridged domainsp. 219
Introductionp. 219
Ridges and skeletonsp. 220
Simple ridges in <$>{\op R}^2<$>p. 224
Generalised ridged domainsp. 228
Measure of non-compactnessp. 234
Analysis on GRDp. 244
The map T and its approximate inverse Mp. 245
Equivalentembeddingsp. 249
Equivalent Poincaré inequalitiesp. 251
Compactness of Ep. 252
Local compactnessp. 252
Measureofnon-compactnessp. 254
EmbeddingTheoremsp. 261
The Poincaré inequality and ¿(E)p. 266
Notesp. 273
Approximation numbers of Sobolev embeddingsp. 275
Introductionp. 275
Some quotient space normsp. 277
Dirichlet-Neumann bracketing in Lpp. 282
Further asymptotic estimates for a GRD ¿p. 294
Notesp. 305
Referencesp. 307
Author Indexp. 319
Subject Indexp. 323
Notation Indexp. 325
Table of Contents provided by Publisher. All Rights Reserved.

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