I. Function Spaces 5
II. Partial Differential Equations 6
III. Analysis and Applications 7
Contributors 8
Authors 9
Function Spaces 14
Contents 16
Hardy Inequalities for Nonconvex Domains 21
1 Introduction 21
2 Main Results 22
3 Proof of the Main Results 25
4 Remarks 29
References 31
Distributions with Slow Tails and Ergodicity of Markov Semigroups in Infinite Dimensions 33
1 Weak Forms of Poincare Type Inequalities 33
2 Lp-Embeddings under Weak Poincare 37
3 Growth of Moments and Large Deviations 40
4 Relations for Lp-Like Pseudonorms 42
5 Isoperimetric and Capacitary Conditions 44
6 Convex Measures 53
7 Examples. Perturbation 56
8 Weak Poincare with Oscillation Terms 59
9 Convergence of Markov Semigroups 65
10 Markov Semigroups and Weak Poincare 71
11 L虏 Decay to Equilibrium in Infnite Dimensions 78
11.1 Basic inequalities and decay to equilibrium in the product case 78
11.2 Semigroup for an infnite system with interaction. 80
11.3 L虏 decay 82
12 Weak Poincare Inequalities for Gibbs Measures 86
References 98
On Some Aspects of the Theory of Orlicz\u2013Sobolev Spaces 100
1 Introduction 100
2 Background 101
3 Orlicz\u2013Sobolev Embeddings 104
3.1 Embeddings into Orlicz spaces 105
6 Trace Inequalities 117
References 121
4 Modulus of Continuity 111
5 Differentiability Properties 114
Mellin Analysis of Weighted Sobolev Spaces with Nonhomogeneous Norms on Cones 124
1 Introduction 124
2 Notation: Weighted Sobolev Spaces on Cones 126
2.1 Weighted spaces with homogeneous norms 127
2.2 Weighted spaces with nonhomogeneous norms 128
3 Characterizations by Mellin Transformation Techniques 130
3.1 Mellin characterization of spaces with homogeneous norms 130
3.2 Mellin characterization of seminorms 133
3.3 Spaces defined by Mellin norms 138
3.4 Spaces defined by weighted seminorms 141
3.5 Mellin characterization of spaces with nonhomogeneous norms 144
4 Structure of Spaces with Nonhomogeneous Norms in the Critical Case 148
4.1 Weighted Sobolev spaces with analytic regularity 148
4.2 Mellin regularizing operator in one dimension 149
4.3 Generalized Taylor expansions 151
5 Conclusion 154
References 155
Optimal Hardy\u2013Sobolev\u2013Maz'ya Inequalities with Multiple Interior Singularities 156
1 Introduction 156
2 Improved Hardy Inequalities with Multiple Singularities 161
3 Hardy\u2013Sobolev\u2013Maz'ya Inequalities 169
References 177
Sharp Fractional Hardy Inequalities in Half-Spaces 180
1 Introduction and Main Results 180
2 Proofs 182
2.1 General Hardy inequalities 182
References 186
Collapsing Riemannian Metrics to Sub-Riemannian and the Geometry of Hypersurfaces in Carnot Groups 187
1 Introduction 187
2 Hypersurfaces in Carnot Groups of Step 2 190
3 The Limit as e鈫? of the Rescaled e\u2013Volume Forms on M 194
4 Orthonormal Basis 197
5 Geometric Guantities with respect to the Collapsing Metrics 202
6 First and Second Variation Formulas for H-Perimeter 215
References 223
Sobolev Homeomorphisms and Composition Operators 225
1 Introduction 225
2 Composition Operators in Sobolev Space 227
3 Proof of the Main Result 234
References 237
Extended Lp Dirichlet Spaces 239
1 Introduction 239
2 The Case for an Lp Theory for Dirichlet Forms 240
3 Bessel Potential Spaces and (r; p)-Capacities 242
4 Extended Lp-Dirichlet Spaces 245
References 255
Characterizations for the Hardy Inequality 257
1 Introduction 257
2 Maz'ya Type Characterization 259
3 The Capacity Density Condition 261
4 Characterizations in the Borderline Case 265
5 Eigenvalue Problem 267
References 270
Geometric Properties of Planar BV -Extension Domains 273
1 Introduction 273
2 Preliminaries 275
3 Proofs of the Results 284
4 Examples 287
References 289
On a New Characterization of Besov Spaces with Negative Exponents 291
1 Introduction 291
2 The Left-Hand Side Inequality (1.4) 292
3 The Right-Hand Side Inequality (1.4) 296
3.1 The case 0 < s < 2 296
3.2 The general case 298
4 A Regularity Result for the Green Operator 301
References 302
Isoperimetric Hardy Type and Poincare Inequalities on Metric Spaces 303
1 Introduction 303
2 Background 305
3 Hardy Isoperimetric Type 307
4 Model Riemannian Manifolds 309
5 E. Milman's Equivalence Theorems 311
6 Some Spaces That Are not of Isoperimetric Hardy Type 312
References 315
Gauge Functions and Sobolev Inequalities on Fluctuating Domains 317
1 Introduction 317
2 Gauge Functions 319
3 Gauged Poincare Inequalities 325
4 Gauged Capacitary Inequalities 332
5 Fluctuating Domains 335
References 337
A Converse to the Maz'ya Inequality for Capacities under Curvature Lower Bound 339
1 Introduction 339
2 Definitions and Preliminaries 342
2.1 Isoperimetric inequalities 342
2.2 Functional inequalities 343
2.3 Known connections 345
3 Capacities 346
3.1 1-Capacity and isoperimetric profiles 346
3.2 q-Capacitary and weak Orlicz\u2013Sobolev inequalities 347
3.3 q-Capacitary and strong Orlicz\u2013Sobolev inequalities 349
3.4 Passing between q-capacitary inequalities 351
3.5 Combining everything 353
4 The Converse Statement 354
4.1 Case q 鈮?2 357
4.2 Case 1 < q 4.2 Case 1 < q 鈮?2 360
4.2.1 Semigroup Approach 360
4.2.2 Capacity Approach 363
References 365
Pseudo-Poincare Inequalities and Applications to Sobolev Inequalities 367
1 Introduction 367
2 Sobolev Inequality and Volume Growth 369
3 The Pseudo-Poincare Approach to Sobolev Inequalities 370
4 Pseudo-Poincare Inequalities 372
5 Pseudo-Poincare Inequalities and the Liouville Measure 375
6 Homogeneous Spaces 379
7 Ricci Curvature Bounded Below 382
8 Domains with the Interior Cone Property 385
References 389
The p-Faber-Krahn Inequality Noted 391
1 The p-Faber-Krahn Inequality Introduced 391
2 The p-Faber-Krahn Inequality Improved 394
3 The p-Faber-Krahn Inequality Characterized 400
References 407
References to Maz'ya's Publications Made in Volume I 411
Index 409