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CHAPTER Ⅰ Elementary Local Properties of Holomorphic Functions 1

1．Holomorphic Functions 1

2．Holomorphic Maps 18

3．Zero Sets of Holomorphic Functions 31

Notes for ChapterⅠ 40

CHAPTER Ⅱ Domains of Holomorphy and Pseudoconvexity 42

1．Elementary Extension Phenomena 43

2．Natural Boundaries and Pseudoconvexity 48

3．The Convexity Theory of Cartan and Thullen 67

4．Plurisubharmonic Functions 82

5．Characterizations of Pseudoconvexity 92

Notes for Chapter Ⅱ 101

CHAPTER Ⅲ Differential Forms and Hermitian Geometry 104

1．Calculus on Real Differentiable Manifolds 105

2．Complex Structures 122

3．Hermitian Geometry in Cn 131

Notes for Chapter Ⅲ 143

CHAPTER Ⅳ Integral Representations in Cn 144

1．The Bochner-Martinelli-Koppelman Formula 145

2．Some Applications 159

3．The General Homotopy Formula 168

4．The Bergman Kernel 179

Notes for Chapter Ⅳ 187

CHAPTER Ⅴ The Levi Problem and the Solution of ？ on Strictly Pseudoconvex Domains 193

1．A Parametrix for ？ on Strictly Pseudoconvex Domains 194

2．A Solution Operator for ？ 199

3．The Lipschitz 1/2-Estimate 206

Notes for Chapter Ⅴ 214

CHAPTER Ⅵ Function Theory on Domains of Holomorphy in Cn 216

1．Approximation and Exhaustions 217

2．？-Cohomological Characterization of Stein Domains 227

3．Topological Properties of Stein Domains 229

4．Meromorphic Functions and the Additive Cousin Problem 232

5．Holomorphic Functions with Prescribed Zeroes 239

6．Preview:Cohomology of Coherent Analytic Sheaves 252

Notes for Chapter Ⅵ 272

CHAPTER Ⅶ Topics in Function Theory on Strictly Pseudoconvex Domains 275

1．A Cauchy Kernel for Strictly Pseudoconvex Domains 276

2．Uniform Approximation on D 282

3．The Kernel of Henkin and Ramirez 285

4．Gleason's Problem and Decomposition in A(D) 292

5．Lp Estimates for Sol utions of ？ 296

6．Approximation of Holomorphic Functions in Lp Norm 305

7．Regularity Properties of the Bergman Projection 309

8．Boundary Regularity of Biholomorphic Maps 326

9．The Reflection Principle 342

Notes for Chapter Ⅶ 351

Appendix A 358

Appendix B 360

Appendix C 362

Bibliography 365

Glossary of Symbols and Notations 376

Index 381