Abstract ....................................................... vi
Chapter 0. Introduction ......................................... 1
Chapter 1. Background and Definitions ........................... 4
§1.1. Notation, terminology and known results ................ 4
§1.2. Hardy spaces and layer potentials ...................... 6
Chapter 2. The Boundary Layer Potentials ........................ 9
§2.1. Compactness of operators К, К* ......................... 9
§2.2. Invertibility of ±½l+K, ±½l+K* ........................ 19
Chapter 3. The Dirichlet problem ............................... 21
§3.1. Lp boundary data ...................................... 21
§3.2. Hardy space boundary data ............................. 23
§3.3. Holder space boundary data ............................ 25
Chapter 4. The Neumann problem ................................. 27
§4.1. Lp boundary data ...................................... 27
§4.2. Hardy space boundary data ............................. 27
§4.3. Holder space boundary data ............................ 29
Chapter 5. Compactness of Layer Potentials, Part II;
The Dirichlet regularity problem ............................ 31
§5.1. Preliminaries ......................................... 31
§5.2. Compactness and invertibility of К on Sobolev space
H1,р ................................................... 33
§5.3. Compactness and invertibility of К on Hardy-Sobolev
space H1,р ............................................. 38
§5.4. Dirichlet regularity problem, Sobolev H1,р(1<p<∞)
data .................................................. 41
§5.5. Dirichlet regularity problem, H1,р((n—1)/n<p≤1) data ... 42
Chapter 6. The equivalence of Hardy space definitions .......... 44
§6.1. Preliminaries ......................................... 44
§6.2. C-suharmonicity ....................................... 45
§6.3. The main step ......................................... 47
§6.4. The equivalence theorem on C1 domains ................. 50
§6.5. The equivalence theorem on Lipschitz domains .......... 51
APPENDIX A. Variable Coefficient Cauchy Integrals .............. 53
APPENDIX B. One Result on the Maximal Operator ................. 65
Bibliography ................................................... 77
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