ISBN: 3540675884
TITLE: Nonlinear Potential Theory and Weighted Sobolev Spaces
AUTHOR: Turesson, Bengt O.
TOC:

Introduction v
1. Preliminaries 1
1.1. Notation and conventions 1
1.2. Basic results concerning weights 2
1.2.1. General weights 2
1.2.2. A_p weights 3
1.2.3. Doubling weights 6
1.2.4. A_infinite weights
1.2.5. Proof of Muckenhoupt's maximal theorem 10
1.2.6. Boundedness of singular integrals 12
1.2.7. Two theorems by Muckenhoupt and Wheeden 12
2. Sobolev spaces 15
2.1. The Sobolev space W^m, p_omega (Omega) 16
2.1.1. Approximation results 17
2.1.2. Extension theorems 19
2.1.3. An interpolation inequality 23
2.2. The Sobolev space V^m, p_omega ( Omega) 25
2.3. Hausdorff measures 37
2.4. Isoperimetric inequalities 40
2.4.1. Preliminary lemmas 41
2.4.2. Extensions of some results by David and Semmes 44
2.4.3. Isoperimetric inequalities involving lower Minkowski content 47
2.4.4. Isoperimetric inequalities with Hausdorff measures 49
2.4.5. A boxing inequality 53
2.5. Some Sobolev type inequalities 54
2.6. Embeddings into L^q_mi(Omega) 58
2.6.1. Introduction 59
2.6.2. Embedding theorems 62
3. Potential theory 69
3.1. Norm inequalities for fractional integrals and maximal functions 70
3.1.1. Proof of the main inequality and some corollaries 70
3.1.2. An inequality for Bessel potentials 75
3.2. Meyers' theory for U-capacities 77
3.2.1. Outline of Meyers' theory 77
3.2.2. Capacitary measures and capacitary potentials 80
3.3. Bessel and Riesz capacities 88
3.3.1. Basic properties 88
3.3.2. Adams' formula for the capacity of a ball 92
3.4. Hausdorff capacities 97
3.4.1. Basic properties 97
3.4.2. The capacity of a ball 99
3.4.3. Non-triviality of H^N- alpha_omega, infinite 104
3.4.4. Local equivalence between H^N- alpha_omega, r ho and H^N- alpha_omega, infinite 108
3.4.5. Continuity properties 110
3.4.6. Frostman's lemma 113
3.5. Variational capacities 115
3.5.1. The case 1<p<infinite 115
3.5.2. The case p=1 117
3.5.3. An embedding theorem 120
3.6. Thinness: The case 1<p<infinite 121
3.6.1. Preliminary considerations 122
3.6.2. A Wolff type inequality 124
3.6.3. Proof of the Kellogg property 127
3.6.4. A concept of thinness based on a condensor capacity 131
3.7. Thinness: The case p = 1 134
4. Applications of potential theory to Sobolev spaces 141
4.1. Quasicontinuity 141
4.1.1. The case 1<p<infinite 142
4.1.2. The case p=1 144
4.2. Measures in the dual of W^m,p _omega ( Omega) 148
4.2.1. The case 1<p<infinite 148
4.2.2. The case p=1 149
4.3. Poincar type inequalities 151
4.3.1. The case 1<p<infinite 151
4.3.2. The case p=1 155
4.4. Spectral synthesis 156
4.4.1. The case 1<p<infinite 157
4.4.2. The case p=1 160
References 163
Index 171
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