A number of monographs of various aspects of complex analysis in several variables have appeared since the first version of this book was published, but none of them uses the analytic techniques based on the solution of the Neumann Problem as the main tool. The additions made in this third, revised edition place additional stress on results where these methods are particularly important. Thus, a section has been added presenting Ehrenpreis' ``fundamental principle'' in full. The local arguments in this section are closely related to the proof of the coherence of the sheaf of germs of functions vanishing on an analytic set. Also added is a discussion of the theorem of Siu on the Lelong numbers of plurisubharmonic functions. Since the L2 techniques are essential in the proof and plurisubharmonic functions play such an important role in this book, it seems natural to discuss their main singularities.
Author(s): L. Hormander
Series: North-Holland Mathematical Library 7
Edition: 3ed., NH
Publisher: North Holland
Year: 1990
Language: English
Pages: 263
Contents......Page 6
Preface......Page 3
List of Symbols......Page 8
1.1. Preliminaries......Page 10
1.2. Cauchy's integral formula and its applications......Page 11
1.3. The Runge approximation theorem......Page 15
1.4. The Mittag-Leffler theorem......Page 18
1.5. The Weierstrass theorem......Page 23
1.6. Subharmonic functions......Page 25
Notes......Page 30
2.1. Preliminaries......Page 31
2.2. Applications of Cauchy's integral formula in polydiscs......Page 34
2.3. The inhomogeneous Cauchy-Riemann equations in a polydisc......Page 39
2.4. Power series and Reinhardt domains......Page 43
2.5. Domains of holomorphy......Page 45
2.6. Pseudoconvexity and plurisubharmonicity......Page 53
2.7. Runge domains......Page 61
Notes......Page 68
3.1. Preliminaries......Page 70
3.2. Analytic functions of elements in a Banach algebra......Page 77
Notes......Page 84
4.1. Preliminaries......Page 86
4.2. Existence theorems in pseudoconvex domains......Page 91
4.3. Approximation theorems......Page 98
4.4. Existence theorems in L^2 spaces......Page 101
4.5. Analytic functionals......Page 116
Notes......Page 121
5.1. Definitions......Page 123
5.2. L^2 estimates and existence theorems for the \bar{\partial} operator......Page 127
5.3. Embedding of Stein manifolds......Page 138
5.4. Envelopes of holomorphy......Page 146
5.5. The Cousin problems on a Stein manifold......Page 152
5.6. Existence and approximation theorems for sections of an analytic vector bundle......Page 155
5.7. Almost complex manifolds......Page 158
Notes......Page 162
6.1. The Weierstrass preparation theorem......Page 164
6.2. Factorization in the ring A_0 of germs of analytic functions......Page 167
6.3. Finitely generated A_0-modules......Page 170
6.4. The Oka theorem......Page 174
6.5. Analytic sets......Page 176
Notes......Page 185
Summary......Page 186
7.1. Definition of sheaves......Page 187
7/2. Existence of global sections of a coherent analytic sheaf......Page 192
7.3. Cohomology groups with values in a sheaf......Page 201
7.4. The cohomology groups of a Stein manifold with coefficients in a coherent analytic sheaf......Page 207
7.5. The de Rham theorem......Page 214
7.6. Cohomology with bounds and constant coefficient differential equations......Page 215
7.7. Quotients of A^K by submodules, and the Ehrenpreis fundamental principle......Page 236
Notes......Page 257
Bibliography......Page 258
Index......Page 262
