Function spaces and potential theory

Cover......Page 1
Title Page......Page 4
Copyright......Page 5
Preface......Page 6
Table of Contents......Page 10
1.1.2 Notation......Page 14
1.1.3 Spaces of Functions and Their Duals......Page 15
1.1.4 Maximal Functions......Page 16
1.1.6 Distributions......Page 17
1.1.8 The Riesz Transform and Singular Integrals......Page 18
1.2.1 Sobolev Spaces......Page 19
1.2.2 Riesz Potentials......Page 21
1.2.3 Bessel Potentials......Page 22
1.2.4 Bessel Kernels......Page 23
1.2.5 Some Classical Formulas for Bessel Functions......Page 24
1.2.6 Bessel Potential Spaces......Page 26
1.3 Banach Spaces......Page 27
1.4 Two Covering Lemmas......Page 29
2.1 Introduction......Page 30
2.2 A First Version of (a, p)-Capacity......Page 32
2.3 A General Theory for La-Capacities......Page 37
2.4 The Minimax Theorem......Page 43
2.5 The Dual Definition of Capacity......Page 47
2.6 Radially Decreasing Convolution Kernels......Page 51
2.7 An Alternative Definition of Capacity and Removability of Singularities......Page 58
2.9 Notes......Page 61
3.1 Pointwise and Integral Estimates......Page 66
3.2 A Sharp Exponential Estimate......Page 71
3.3 Operations on Potentials......Page 75
3.4 One-Sided Approximation......Page 79
3.5 Operations on Potentials with Fractional Index......Page 81
3.6 Potentials and Maximal Functions......Page 85
3.7 Further Results......Page 91
3.8 Notes......Page 94
4.1 Besov Spaces......Page 98
4.2 Lizorkin-Triebel Spaces......Page 104
4.3 Lizorkin-Triebel Spaces, Continued......Page 110
4.4 More Nonlinear Potentials......Page 117
4.5 An Inequality of Wolff......Page 121
4.6 An Atomic Decomposition......Page 124
4.7 Atomic Nonlinear Potentials......Page 129
4.8 A Characterization of L......Page 135
4.9 Notes......Page 138
5.1 Comparison Theorems......Page 142
5.2 Lipschitz Mappings and Capacities......Page 153
5.3 The Capacity of Cantor Sets......Page 155
5.4 Sharpness of Comparison Theorems......Page 159
5.5 Relations Between Different Capacities......Page 161
5.6 Further Results......Page 163
5.7 Notes......Page 165
6. Continuity Properties......Page 168
6.1 Quasicontinuity......Page 169
6.2 Lebesgue Points......Page 171
6.3 Thin Sets......Page 177
6.4 Fine Continuity......Page 189
6.5 Further Results......Page 193
6.6 Notes......Page 198
7.1 A Capacitary Strong Type Inequality......Page 200
7.2 Imbedding of Potentials......Page 204
7.3 Compactness of the Imbedding......Page 208
7.4 A Space of Quasicontinuous Functions......Page 212
7.5 A Capacitary Strong Type Inequality. Another Approach......Page 216
7.6 Further Results......Page 221
7.7 Notes......Page 226
8.1 Some Basic Inequalities......Page 228
8.2 Inequalities Depending on Capacities......Page 232
8.3 An Abstract Approach......Page 240
8.4 Notes......Page 244
9.1 Statement of Results......Page 246
9.2 The Case m = I......Page 252
9.3 The General Case. Outline......Page 253
9.4 The Uniformly (1, p)-Thick Case......Page 256
9.5 The General Thick Case......Page 258
9.6 Proof of Lemma 9.5.2 for m = 1......Page 261
9.7 Proof of Lemma 9.5.2......Page 264
9.8 Estimates for Nonlinear Potentials......Page 270
9.9 The Case C^m.p(K) = 0......Page 276
9.10 The Case C^k,p(K) = 0, 1 < k < m......Page 279
9.11 Conclusion of the Proof......Page 290
9.13 Notes......Page 291
10.1 An Approximation Theorem, Another Approach......Page 294
10.2 A Generalization of a Theorem of Whitney......Page 306
10.3 Further Results......Page 314
10.4 Notes......Page 315
11.1 Approximation and Stability......Page 318
11.2 Approximation by Harmonic Functions in Gradient Norm......Page 325
11.3 Stability of Sets Without Interior......Page 327
11.4 Stability of Sets with Interior......Page 329
11.5 Approximation by Harmonic Functions and Higher Order Stability......Page 331
11.6 Further Results......Page 337
11.7 Notes......Page 338
References......Page 342
Index......Page 364
List of Symbols......Page 376
Back Cover......Page 382