简介
The book systematically develops the nonlinear potential theory connected with the weighted Sobolev spaces, where the weight usually belongs to Muckenhoupt's class of Ap weights. These spaces occur as solutions spaces for degenerate elliptic partial differential equations. The Sobolev space theory covers results concerning approximation, extension, and interpolation, Sobolev and Poincar脙漏 inequalities, Maz'ya type embedding theorems, and isoperimetric inequalities. In the chapter devoted to potential theory, several weighted capacities are investigated. Moreover, "Kellogg lemmas" are established for various concepts of thinness. Applications of potential theory to weighted Sobolev spaces include quasi continuity of Sobolev functions, Poincar脙漏 inequalities, and spectral synthesis theorems.
目录
Introduction
1 Preliminaries p. 1
1.1 Notation and conventions p. 1
1.2 Basic results concerning weights p. 2
1.2.1 General weights p. 2
1.2.2 A p weights p. 3
1.2.3 Doubling weights p. 6
1.2.4 A ∞ weights p. 7
1.2.5 Proof of Muckenhoupt's maximal theorem p. 10
1.2.6 Boundedness of singular integrals p. 12
1.2.7 Two theorems by Muckenhoupt and Wheeden p. 12
2 Sobolev spaces p. 15
2.1 The Sobolev space p. 16
2.1.1 Approximation results p. 17
2.1.2 Extension theorems p. 19
2.1.3 An interpolation inequality p. 23
2.2 The Sobolev space p. 25
2.3 Hausdorff measures p. 37
2.4 Isoperimetric inequalities p. 40
2.4.1 Preliminary lemmas p. 41
2.4.2 Extensions of some results by David and Semmes p. 44
2.4.3 Isoperimetric inequalities involving lower Minkowski content p. 47
2.4.4 Isoperimetric inequalities with Hausdorff measures p. 49
2.4.5 A boxing inequality p. 53
2.5 Some Sobolev type inequalities p. 54
2.6 Embeddings into p. 58
2.6.1 Introduction p. 59
2.6.2 Embedding theorems p. 62
3 Potential theory p. 69
3.1 Norm inequalities for fractional integrals and maximal functions p. 70
3.1.1 Proof of the main inequality and some corollaries p. 70
3.1.2 An inequality for Bessel potentials p. 75
3.2 Meyers' theory for Incapacities p. 77
3.2.1 Outline of Meyers'theory p. 77
3.2.2 Capacitary measures and capacitary potentials p. 80
3.3 Bessel and Riesz capacities p. 88
3.3.1 Basic properties p. 88
3.3.2 Adams' formula for the capacity of a ball p. 92
3.4 Hausdorff capacities p. 97
3.4.1 Basic properties p. 97
3.4.2 The capacity of a ball p. 99
3.4.3 Non-triviality of p. 104
3.4.4 Local equivalence between and p. 108
3.4.5 Continuity properties p. 110
3.4.6 Prostman's lemma p. 113
3.5 Variational capacities p. 115
3.5.1 The case 1 p. 115
3.5.2 The case p = 1 p. 117
3.5.3 An embedding theorem p. 120
3.6 Thinness: The case 1 p. 121
3.6.1 Preliminary considerations p. 122
3.6.2 A Wolff type inequality p. 124
3.6.3 Proof of the Kellogg property p. 127
3.6.4 A concept of thinness based on a condensor capacity p. 131
3.7 Thinness: The case p = 1 p. 134
4 Applications of potential theory to Sobolev spaces p. 141
4.1 Quasicontinuity p. 141
4.1.1 The case 1 p. 142
4.1.2 The case p = 1 p. 144
4.2 Measures in the dual of p. 148
4.2.1 The case 1 p. 148
4.2.2 The case p = 1 p. 149
4.3 Poincare type inequalities p. 151
4.3.1 The case 1 p. 151
4.3.2 The case p = 1 p. 155
4.4 Spectral synthesis p. 156
4.4.1 The case 1 p. 157
4.4.2 The case p = 1 p. 160
References p. 163
Index p. 171
1 Preliminaries p. 1
1.1 Notation and conventions p. 1
1.2 Basic results concerning weights p. 2
1.2.1 General weights p. 2
1.2.2 A p weights p. 3
1.2.3 Doubling weights p. 6
1.2.4 A ∞ weights p. 7
1.2.5 Proof of Muckenhoupt's maximal theorem p. 10
1.2.6 Boundedness of singular integrals p. 12
1.2.7 Two theorems by Muckenhoupt and Wheeden p. 12
2 Sobolev spaces p. 15
2.1 The Sobolev space p. 16
2.1.1 Approximation results p. 17
2.1.2 Extension theorems p. 19
2.1.3 An interpolation inequality p. 23
2.2 The Sobolev space p. 25
2.3 Hausdorff measures p. 37
2.4 Isoperimetric inequalities p. 40
2.4.1 Preliminary lemmas p. 41
2.4.2 Extensions of some results by David and Semmes p. 44
2.4.3 Isoperimetric inequalities involving lower Minkowski content p. 47
2.4.4 Isoperimetric inequalities with Hausdorff measures p. 49
2.4.5 A boxing inequality p. 53
2.5 Some Sobolev type inequalities p. 54
2.6 Embeddings into p. 58
2.6.1 Introduction p. 59
2.6.2 Embedding theorems p. 62
3 Potential theory p. 69
3.1 Norm inequalities for fractional integrals and maximal functions p. 70
3.1.1 Proof of the main inequality and some corollaries p. 70
3.1.2 An inequality for Bessel potentials p. 75
3.2 Meyers' theory for Incapacities p. 77
3.2.1 Outline of Meyers'theory p. 77
3.2.2 Capacitary measures and capacitary potentials p. 80
3.3 Bessel and Riesz capacities p. 88
3.3.1 Basic properties p. 88
3.3.2 Adams' formula for the capacity of a ball p. 92
3.4 Hausdorff capacities p. 97
3.4.1 Basic properties p. 97
3.4.2 The capacity of a ball p. 99
3.4.3 Non-triviality of p. 104
3.4.4 Local equivalence between and p. 108
3.4.5 Continuity properties p. 110
3.4.6 Prostman's lemma p. 113
3.5 Variational capacities p. 115
3.5.1 The case 1 p. 115
3.5.2 The case p = 1 p. 117
3.5.3 An embedding theorem p. 120
3.6 Thinness: The case 1 p. 121
3.6.1 Preliminary considerations p. 122
3.6.2 A Wolff type inequality p. 124
3.6.3 Proof of the Kellogg property p. 127
3.6.4 A concept of thinness based on a condensor capacity p. 131
3.7 Thinness: The case p = 1 p. 134
4 Applications of potential theory to Sobolev spaces p. 141
4.1 Quasicontinuity p. 141
4.1.1 The case 1 p. 142
4.1.2 The case p = 1 p. 144
4.2 Measures in the dual of p. 148
4.2.1 The case 1 p. 148
4.2.2 The case p = 1 p. 149
4.3 Poincare type inequalities p. 151
4.3.1 The case 1 p. 151
4.3.2 The case p = 1 p. 155
4.4 Spectral synthesis p. 156
4.4.1 The case 1 p. 157
4.4.2 The case p = 1 p. 160
References p. 163
Index p. 171
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