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

This volume mark's the centenary of the birth of the outstanding mathematician of the 20th century, Sergey Sobolev. It includes new results on the latest topics of the theory of Sobolev spaces, partial differential equations, analysis and mathematical physics.

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