1 Preliminaries ................................................ 1
1.1 Basics .................................................. 1
1.1.1 Convention ....................................... 1
1.1.2 Notation ......................................... 1
1.1.3 Spaces of Functions and Their Duals .............. 2
1.1.4 Maximal Functions ................................ 3
1.1.5 Integral Inequalities ............................ 4
1.1.6 Distributions .................................... 4
1.1.7 The Fourier Transform ............................ 5
1.1.8 The Riesz Transform and Singular Integrals ....... 5
1.2 Sobolev Spaces and Bessel Potentials .................... 6
1.2.1 Sobolev Spaces ................................... 6
1.2.2 Riesz Potentials ................................. 8
1.2.3 Bessel Potentials ................................ 9
1.2.4 Bessel Kernels .................................. 10
1.2.5 Some Classical Formulas for Bessel Functions .... 11
1.2.6 Bessel Potential Spaces ......................... 13
1.2.7 The Sobolev Imbedding Theorem ................... 14
1.3 Banach Spaces .......................................... 14
1.4 Two Covering Lemmas .................................... 16
2 Lp-Capacities and Nonlinear Potentials ...................... 17
2.1 Introduction ........................................... 17
2.2 A First Version of (α, p)-Capacity ..................... 19
2.3 A General Theory for Lp-Capacities ..................... 24
2.4 The Minimax Theorem .................................... 30
2.5 The Dual Definition of Capacity ........................ 34
2.6 Radially Decreasing Convolution Kernels ................ 38
2.7 An Alternative Definition of Capacity and
Removability of Singularities .......................... 45
2.8 Further Results ........................................ 48
2.9 Notes .................................................. 48
3 Estimates for Bessel and Riesz Potentials ................... 53
3.1 Pointwise and Integral Estimates ....................... 53
3.2 A Sharp Exponential Estimate ........................... 58
3.3 Operations on Potentials ............................... 62
3.4 One-Sided Approximation ................................ 66
3.5 Operations on Potentials with Fractional Index ......... 68
3.6 Potentials and Maximal Functions ....................... 72
3.7 Further Results ........................................ 78
3.8 Notes .................................................. 81
4 Besov Spaces and Lizorkin-Triebel Spaces .................... 85
4.1 Besov Spaces ........................................... 85
4.2 Lizorkin-Triebel Spaces ................................ 91
4.3 Lizorkin-Triebel Spaces, Continued ..................... 97
4.4 More Nonlinear Potentials ............................. 104
4.5 An Inequality of Wolff ................................ 108
4.6 An Atomic Decomposition ............................... 111
4.7 Atomic Nonlinear Potentials ........................... 116
4.8 A Characterization of Lα,p ............................ 122
4.9 Notes ................................................. 125
5 Metric Properties of Capacities ............................ 129
5.1 Comparison Theorems ................................... 129
5.2 Lipschitz Mappings and Capacities ..................... 140
5.3 The Capacity of Cantor Sets ........................... 142
5.4 Sharpness of Comparison Theorems ...................... 146
5.5 Relations Between Different Capacities ................ 148
5.6 Further Results ....................................... 150
5.7 Notes ................................................. 152
6 Continuity Properties ...................................... 155
6.1 Quasicontinuity ....................................... 156
6.2 Lebesgue Points ....................................... 158
6.3 Thin Sets ............................................. 164
6.4 Fine Continuity ....................................... 176
6.5 Further Results ....................................... 180
6.6 Notes ................................................. 185
7 Trace and Imbedding Theorems ............................... 187
7.1 A Capacitary Strong Type Inequality ................... 187
7.2 Imbedding of Potentials ............................... 191
7.3 Compactness of the Imbedding .......................... 195
7.4 A Space of Quasicontinuous Functions .................. 199
7.5 A Capacitary Strong Type Inequality. Another
Approach .............................................. 203
7.6 Further Results ....................................... 208
7.7 Notes ................................................. 213
8 Poincare Type Inequalities ................................. 215
8.1 Some Basic Inequalities ............................... 215
8.2 Inequalities Depending on Capacities .................. 219
8.3 An Abstract Approach .................................. 227
8.4 Notes ................................................. 231
9 An Approximation Theorem ................................... 233
9.1 Statement of Results .................................. 233
9.2 The Case m = 1 ........................................ 239
9.3 The General Case. Outline ............................. 240
9.4 The Uniformly (1, p)-Thick Case ....................... 243
9.5 The General Thick Case ................................ 245
9.6 Proof of Lemma 9.5.2 for m = 1 ........................ 248
9.7 Proof of Lemma 9.5.2 .................................. 251
9.8 Estimates for Nonlinear Potentials .................... 257
9.9 The Case Cm,p(K) = 0 .................................. 263
9.10 The Case Ck,p(K) = 0,l ≤ k < m ........................ 266
9.11 Conclusion of the Proof ............................... 277
9.12 Further Results ....................................... 278
9.13 Notes ................................................. 278
10 Two Theorems of Netrusov ................................... 281
10.1 An Approximation Theorem, Another Approach ............ 281
10.2 A Generalization of a Theorem of Whitney .............. 293
10.3 Further Results ....................................... 301
10.4 Notes ................................................. 302
11 Rational and Harmonic Approximation ........................ 305
11.1 Approximation and Stability ........................... 305
11.2 Approximation by Harmonic Functions in Gradient
Norm .................................................. 312
11.3 Stability of Sets Without Interior .................... 314
11.4 Stability of Sets with Interior ....................... 316
11.5 Approximation by Harmonic Functions and Higher Order
Stability ............................................. 318
11.6 Further Results ....................................... 324
11.7 Notes ................................................. 325
References .................................................... 329
Index ......................................................... 351
List of Symbols ............................................... 363
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