Maz'ya Sobolev Spaces in Mathematics I

Sobolev Type Inequalities
1. Auflage 2008
ISBN: 978-0-387-85648-3
Verlag: Springer-Verlag
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 378 Seiten

Autoren/Hrsg.

Maz'ya, Vladimir

Inhaltsverzeichnis

1;Contributors;12
2;Contents;21
3;My Love Affair with the Sobolev Inequality;26
3.1;1 The Trace Inequality;29
3.2;2 A Mixed Norm Inequality;32
3.3;3 A Morrey–Sobolev Inequality;34
3.4;4 A Morrey–Besov Inequality;37
3.5;5 Exponential Integrability;38
3.6;6 Vanishing Exponential Integrability;43
3.7;7 Concluding Remarks;45
3.8;References;46
4;Maximal Functions in Sobolev Spaces;49
4.1;1 Introduction;49
4.2;2 Maximal Function Defined on the Whole Space;51
4.2.1;2.1 Boundedness in Sobolev spaces;51
4.2.2;2.2 A capacitary weak type estimate;56
4.3;3 Maximal Function Defined on a Subdomain;57
4.3.1;3.1 Boundedness in Sobolev spaces;57
4.3.2;3.2 Sobolev boundary values;62
4.4;4 Pointwise Inequalities;65
4.4.1;4.1 Lusin type approximation of Sobolev functions;69
4.5;5 Hardy Inequality;73
4.6;6 Maximal Functions on Metric Measure Spaces;78
4.6.1;6.1 Sobolev spaces on metric measure spaces;79
4.6.2;6.2 Maximal function defined on the whole space;81
4.6.3;6.3 Maximal function defined on a subdomain;86
4.6.4;6.4 Pointwise estimates and Lusin type approximation;88
4.7;References;89
5;Hardy Type Inequalities via Riccati and Sturm– Liouville Equations;92
5.1;1 Introduction;92
5.2;2 Riccati Equations;94
5.3;3 Transition to Sturm–Liouville Equations;98
5.4;4 Hardy Type Inequalities with Weights;100
5.5;5 Poincar´e Type Inequalities;106
5.6;References;108
6;Quantitative Sobolev and Hardy Inequalities, and Related Symmetrization Principles;110
6.1;1 Introduction;110
6.2;2 Symmetrization Inequalities;112
6.2.1;2.1 Rearrangements of functions and function spaces;112
6.2.2;2.2 The Hardy-Littlewood inequality;115
6.2.3;2.3 The Polya-Szegö inequality;119
6.3;3 Sobolev Inequalities ;124
6.3.1;3.1 Functions of Bounded Variation;124
6.3.2;3.2 The case 1 < p < n;126
6.3.3;3.3 The case p > n;129
6.4;4 Hardy Inequalities;131
6.4.1;4.1 The case 1 < p < n;131
6.4.2;4.2 The case p = n;134
6.5;References;136
7;Inequalities of Hardy–Sobolev Type in Carnot– Caratheodory Spaces;140
7.1;1 Introduction;140
7.2;2 Preliminaries;144
7.3;3 Pointwise Hardy Inequalities;150
7.4;4 Hardy Inequalities on Bounded Domains;162
7.5;5 Hardy Inequalities with Sharp Constants;168
7.6;References;172
8;Sobolev Embeddings and Hardy Operators;175
8.1;1 Introduction;175
8.2;2 Hardy Operators on Trees;176
8.3;3 The Poincare Inequality, a(E) and Hardy Type Operators;180
8.4;4 Generalized Ridged Domains;184
8.5;5 Approximation and Other s-Numbers of Hardy Type Operators;192
8.6;6 Approximation Numbers of Embeddings on Generalized Ridged Domains;203
8.7;References;204
9;Sobolev Mappings between Manifolds and Metric Spaces;206
9.1;1 Introduction;206
9.2;2 Sobolev Mappings between Manifolds;208
9.3;3 Sobolev Mappings into Metric Spaces;218
9.3.1;3.1 Density;223
9.4;4 Sobolev Spaces on Metric Measure Spaces;226
9.4.1;4.1 Integration on rectifiable curves;226
9.4.2;4.2 Modulus;228
9.4.3;4.3 Upper gradient;229
9.4.4;4.4 Sobolev spaces N1,p;229
9.4.5;4.5 Doubling measures;230
9.4.6;4.6 Other spaces of Sobolev type;232
9.4.7;4.7 Spaces supporting the Poincare inequality;235
9.5;5 Sobolev Mappings between Metric Spaces;236
9.5.1;5.1 Lipschitz polyhedra;239
9.6;References;240
10;A Collection of Sharp Dilation Invariant Integral Inequalitiesfor Differentiable Functions;244
10.1;1 Introduction;244
10.2;2 Estimate for a Quadratic Form of the Gradient;247
10.3;3 Weighted Garding Inequality for the Biharmonic Operator;251
10.4;4 Dilation Invariant Hardy’s Inequalities with Remainder Term;254
10.5;5 Generalized Hardy–Sobolev Inequality with Sharp Constant;262
10.6;6 Hardy’s Inequality with Sharp Sobolev Remainder Term;265
10.7;References;266
11;Optimality of Function Spaces in Sobolev Embeddings;269
11.1;1 Prologue;269
11.2;2 Preliminaries;276
11.3;3 Reduction Theorems;278
11.4;4 Optimal Range and Optimal Domain of Rearrangement- Invariant Spaces;281
11.5;5 Formulas for Optimal Spaces Using the Functional f** - f*;284
11.6;6 Explicit Formulas for Optimal Spaces in Sobolev Embeddings;287
11.7;7 Compactness of Sobolev Embeddings;290
11.8;8 Boundary Traces;295
11.9;9 Gaussian Sobolev Embeddings;296
11.10;References;298
12;On the Hardy–Sobolev–Maz’ya Inequality and Its Generalizations;301
12.1;1 Introduction;301
12.2;2 Generalization of the Hardy–Sobolev–Maz’ya Inequality;304
12.3;3 The Space D1,2V (O) and Minimizers for the Hardy–Sobolev–Maz’ya Inequality;312
12.4;4 Convexity Properties of the Functional Q for p > 2;313
12.5;References;316
13;Sobolev Inequalities in Familiar and Unfamiliar Settings;318
13.1;1 Introduction;318
13.2;2 Moser’s Iteration ;319
13.2.1;2.1 The basic technique;319
13.2.2;2.2 Harnack inequalities;321
13.2.3;2.3 Poincare, Sobolev, and the doubling property;322
13.2.4;2.4 Examples;329
13.3;3 Analysis and Geometry on Dirichlet Spaces ;331
13.3.1;3.1 First order calculus;331
13.3.2;3.2 Dirichlet spaces;331
13.3.3;3.3 Local weak solutions of the Laplace and heat equations;333
13.3.4;3.4 Harnack type Dirichlet spaces;335
13.3.5;3.5 Imaginary powers of - A and the wave equation;337
13.3.6;3.6 Rough isometries;339
13.4;4 Flat Sobolev Inequalities;341
13.4.1;4.1 How to prove a flat Sobolev inequality?;341
13.4.2;4.2 Flat Sobolev inequalities and semigroups of operators;343
13.4.3;4.3 The RozenblumÒCwikelÒLieb inequality;345
13.4.4;4.4 Flat Sobolev inequalities in the finite volume case;348
13.4.5;4.5 Flat Sobolev inequalities and topology at infinity;349
13.5;5 Sobolev Inequalities on Graphs;349
13.5.1;5.1 Graphs of bounded degree;350
13.5.2;5.2 Sobolev inequalities and volume growth;351
13.5.3;5.3 Random walks;352
13.5.4;5.4 Cayley graphs;354
13.6;References;358
14;A Universality Property of Sobolev Spaces in Metric Measure Spaces;363
14.1;1 Introduction;363
14.2;2 Background;365
14.3;3 Dirichlet Forms and N1,2(X);367
14.4;4 Axiomatic Sobolev Spaces and N1,p(X);374
14.5;References;376
15;Cocompact Imbeddings and Structure of Weakly Convergent Sequences;378
15.1;1 Introduction;378
15.2;2 Dislocation Space and Weak Convergence Decomposition;380
15.3;3 Cocompactness and Minimizers;385
15.4;4 Flask Subspaces;389
15.5;5 Compact Imbeddings;390
15.6;References;392
16;Index;394