Sobolev Spaces in Mathematics I (eBook)

Contributors 12
Contents 21
My Love Affair with the Sobolev Inequality 26
1 The Trace Inequality 29
2 A Mixed Norm Inequality 32
3 A Morrey–Sobolev Inequality 34
4 A Morrey–Besov Inequality 37
5 Exponential Integrability 38
6 Vanishing Exponential Integrability 43
7 Concluding Remarks 45
References 46
Maximal Functions in Sobolev Spaces 49
1 Introduction 49
2 Maximal Function Defined on the Whole Space 51
2.1 Boundedness in Sobolev spaces 51
2.2 A capacitary weak type estimate 56
3 Maximal Function Defined on a Subdomain 57
3.1 Boundedness in Sobolev spaces 57
3.2 Sobolev boundary values 62
4 Pointwise Inequalities 65
4.1 Lusin type approximation of Sobolev functions 69
5 Hardy Inequality 73
6 Maximal Functions on Metric Measure Spaces 78
6.1 Sobolev spaces on metric measure spaces 79
6.2 Maximal function defined on the whole space 81
6.3 Maximal function defined on a subdomain 86
6.4 Pointwise estimates and Lusin type approximation 88
References 89
Hardy Type Inequalities via Riccati and Sturm– Liouville Equations 92
1 Introduction 92
2 Riccati Equations 94
3 Transition to Sturm–Liouville Equations 98
4 Hardy Type Inequalities with Weights 100
5 Poincar´e Type Inequalities 106
References 108
Quantitative Sobolev and Hardy Inequalities, and Related Symmetrization Principles 110
1 Introduction 110
2 Symmetrization Inequalities 112
2.1 Rearrangements of functions and function spaces 112
2.2 The Hardy-Littlewood inequality 115
2.3 The Polya-Szegö inequality 119
3 Sobolev Inequalities 124
3.1 Functions of Bounded Variation 124
3.2 The case 1 < p <
3.3 The case p > n
4 Hardy Inequalities 131
4.1 The case 1 < p <
4.2 The case p = n 134
References 136
Inequalities of Hardy–Sobolev Type in Carnot– Caratheodory Spaces 140
1 Introduction 140
2 Preliminaries 144
3 Pointwise Hardy Inequalities 150
4 Hardy Inequalities on Bounded Domains 162
5 Hardy Inequalities with Sharp Constants 168
References 172
Sobolev Embeddings and Hardy Operators 175
1 Introduction 175
2 Hardy Operators on Trees 176
3 The Poincare Inequality, a(E) and Hardy Type Operators 180
4 Generalized Ridged Domains 184
5 Approximation and Other s-Numbers of Hardy Type Operators 192
6 Approximation Numbers of Embeddings on Generalized Ridged Domains 203
References 204
Sobolev Mappings between Manifolds and Metric Spaces 206
1 Introduction 206
2 Sobolev Mappings between Manifolds 208
3 Sobolev Mappings into Metric Spaces 218
3.1 Density 223
4 Sobolev Spaces on Metric Measure Spaces 226
4.1 Integration on rectifiable curves 226
4.2 Modulus 228
4.3 Upper gradient 229
4.4 Sobolev spaces N1,p 229
4.5 Doubling measures 230
4.6 Other spaces of Sobolev type 232
4.7 Spaces supporting the Poincare inequality 235
5 Sobolev Mappings between Metric Spaces 236
5.1 Lipschitz polyhedra 239
References 240
A Collection of Sharp Dilation Invariant Integral Inequalitiesfor Differentiable Functions 244
1 Introduction 244
2 Estimate for a Quadratic Form of the Gradient 247
3 Weighted Garding Inequality for the Biharmonic Operator 251
4 Dilation Invariant Hardy’s Inequalities with Remainder Term 254
5 Generalized Hardy–Sobolev Inequality with Sharp Constant 262
6 Hardy’s Inequality with Sharp Sobolev Remainder Term 265
References 266
Optimality of Function Spaces in Sobolev Embeddings 269
1 Prologue 269
2 Preliminaries 276
3 Reduction Theorems 278
4 Optimal Range and Optimal Domain of Rearrangement- Invariant Spaces 281
5 Formulas for Optimal Spaces Using the Functional f** - f* 284
6 Explicit Formulas for Optimal Spaces in Sobolev Embeddings 287
7 Compactness of Sobolev Embeddings 290
8 Boundary Traces 295
9 Gaussian Sobolev Embeddings 296
References 298
On the Hardy–Sobolev–Maz’ya Inequality and Its Generalizations 301
1 Introduction 301
2 Generalization of the Hardy–Sobolev–Maz’ya Inequality 304
3 The Space D1,2V (O) and Minimizers for the Hardy–Sobolev–Maz’ya Inequality 312
4 Convexity Properties of the Functional Q for p > 2
References 316
Sobolev Inequalities in Familiar and Unfamiliar Settings 318
1 Introduction 318
2 Moser’s Iteration 319
2.1 The basic technique 319
2.2 Harnack inequalities 321
2.3 Poincare, Sobolev, and the doubling property 322
2.4 Examples 329
3 Analysis and Geometry on Dirichlet Spaces 331
3.1 First order calculus 331
3.2 Dirichlet spaces 331
3.3 Local weak solutions of the Laplace and heat equations 333
3.4 Harnack type Dirichlet spaces 335
3.5 Imaginary powers of - A and the wave equation 337
3.6 Rough isometries 339
4 Flat Sobolev Inequalities 341
4.1 How to prove a flat Sobolev inequality? 341
4.2 Flat Sobolev inequalities and semigroups of operators 343
4.3 The RozenblumÒCwikelÒLieb inequality 345
4.4 Flat Sobolev inequalities in the finite volume case 348
4.5 Flat Sobolev inequalities and topology at infinity 349
5 Sobolev Inequalities on Graphs 349
5.1 Graphs of bounded degree 350
5.2 Sobolev inequalities and volume growth 351
5.3 Random walks 352
5.4 Cayley graphs 354
References 358
A Universality Property of Sobolev Spaces in Metric Measure Spaces 363
1 Introduction 363
2 Background 365
3 Dirichlet Forms and N1,2(X) 367
4 Axiomatic Sobolev Spaces and N1,p(X) 374
References 376
Cocompact Imbeddings and Structure of Weakly Convergent Sequences 378
1 Introduction 378
2 Dislocation Space and Weak Convergence Decomposition 380
3 Cocompactness and Minimizers 385
4 Flask Subspaces 389
5 Compact Imbeddings 390
References 392
Index 394