Theory of Functions on Complex Manifolds

Cover......Page 1
Title......Page 3
Preface......Page 5
Contents......Page 7
1.1. Holomorphic functions......Page 9
1.2. Application of the Cauchy-Green formula to the 8-equation......Page 17
1.3. Domains of holomorphy......Page 19
1.4. Plurieubharmonic functions......Page 23
1.5. Pseudoconvex sets ......Page 32
1.6. Preliminaries concerning differential forms......Page 43
1.7. The differential forms w(v) and w(u)......Page 46
1.8. Leray maps and the operators Bd, Bd, Ld and Bd......Page 48
1.9. The Martinelli-Boohner formula......Page 54
1.10. The Leray formula......Page 55
1.11. The Koppelman formula......Page 57
1.12. The Koppelman-Leray formula......Page 59
Exercises, remarks and problems (one complex variable)......Page 61
Exercises, remarks and problems (several complex variables)......Page 64
2.1. Formula for solving the -equation in a strictly convex open sets......Page 67
2.2. Holder estimates......Page 69
2.3. Solution of the 8-equation with Holder estimates in G* strictly peeudoconvex open sets in Cn......Page 73
2.4. The support function Φ()......Page 77
2.6. The Oka-Hefer lemma and solution of......Page 79
2.6. Formula for solving the 8-equation with Holder estimates in G* strictly peeudoconvex open sets in C......Page 82
2.7. Oka-Weil approximation......Page 83
2.9. Uniform approximation......Page 86
2.10. The 8-equation and Cousin problems in holomorphic vector bundles over complex manifolds......Page 87
2.11. Fredholm solvability of the 8-equation on complex manifolds with strictly peeudoconvex O-boundary......Page 89
2.12. Solvability of the 8-equation on complex manifolds with strictly plurieubharmonic exhausting O-function......Page 92
2.13. Solution of the Levi problem for complex manifolds......Page 97
Notes......Page 101
Exeroises, remarks and problems......Page 102
3.1. The Koppelman-Leray formula......Page 107
3.2. Uniform estimates for the 8-equation......Page 116
3.3. Improvement of the estimates on the boundary......Page 125
3.4. Decomposition of singularities......Page 131
3.5. Uniform approximation......Page 139
3.6. Bounded extension of holomorphic functions from complex planes......Page 144
Notes......Page 154
Exercises, remarks and problems......Page 155
Summary......Page 158
4.1. Coherent analytic sheaves......Page 159
4.2. The section a(z,) and the function ()......Page 161
4.3. The Martinelli-Bochner formula and the Leray formula......Page 163
4.4. The Leray-Norguet formula......Page 169
4.6. The Koppelman formula and the Koppelman-Leray formula......Page 173
4.6. The Koppelman-Leray-Norguet formula......Page 180
4.7. The Weil formula,......Page 182
4.8. The support functions F and F......Page 185
4.9. Formulas for real non-degenerate strictly pseudoconvex polyhedra......Page 188
4.10. Formulas for strictly pseudoconvex open sets with non-emooth boundary......Page 192
4.11. Bounded extension of holomorphic functions from complex eubmanifolds......Page 195
4.12. Formulas in holomorphic vector bundles......Page 197
Notes......Page 198
Exercises, remarks and problems......Page 199
Appendix 1. Estimation of some integrals......Page 202
Appendix 2. On Banach\'s open mapping theorem......Page 209
Bibliography......Page 212
List of symbols......Page 223
Subject index......Page 225