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 a_{m}(T), 1 < p ≤ q < ∞ | p. 39 |
Estimates for a_{n}(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 L_{p} | 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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