Function Theory in the Unit Ball of CnAround 1970, an abrupt change occurred in the study of holomorphic functions of several complex variables. Sheaves vanished into the back ground, and attention was focused on integral formulas and on the "hard analysis" problems that could be attacked with them: boundary behavior, complex-tangential phenomena, solutions of the J-problem with control over growth and smoothness, quantitative theorems about zero-varieties, and so on. The present book describes some of these developments in the simple setting of the unit ball of en. There are several reasons for choosing the ball for our principal stage. The ball is the prototype of two important classes of regions that have been studied in depth, namely the strictly pseudoconvex domains and the bounded symmetric ones. The presence of the second structure (i.e., the existence of a transitive group of automorphisms) makes it possible to develop the basic machinery with a minimum of fuss and bother. The principal ideas can be presented quite concretely and explicitly in the ball, and one can quickly arrive at specific theorems of obvious interest. Once one has seen these in this simple context, it should be much easier to learn the more complicated machinery (developed largely by Henkin and his co-workers) that extends them to arbitrary strictly pseudoconvex domains. In some parts of the book (for instance, in Chapters 14-16) it would, however, have been unnatural to confine our attention exclusively to the ball, and no significant simplifications would have resulted from such a restriction. |
Contents
| 1 | |
201 | 19 |
Chapter 3 | 36 |
Chapter 5 | 65 |
Chapter 15 | 66 |
Chapter 6 | 91 |
Chapter 7 | 120 |
Chapter 8 | 161 |
Chapter 12 | 253 |
Chapter 13 | 278 |
The Invariant Laplacian | 280 |
Proper Holomorphic Maps | 300 |
Chapter 16 | 330 |
Chapter 17 | 364 |
Tangential CauchyRiemann Operators | 387 |
Chapter 19 | 403 |
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algebra pattern Assume Aut(B Banach space biholomorphic Borel Borel set boundary bounded Cauchy integral Cauchy-Riemann equations compact sets compact subset complex complex-tangential constant converges convex Corollary curve D₁ define definition denote disc domain e₁ example exists extends f₁ finite follows formula Fubini's theorem ƒ e A(B h₁ H¹(B harmonic Hence Henkin measure Hº(B holds holomorphic functions holomorphic map HP(B hypothesis implies inequality inner function isometry K₁ K₂ Lebesgue Lebesgue measure Lemma Lindelöf's theorem linear M-harmonic M-invariant Math neighborhood Note open set orthogonal peak set PI)-set pluriharmonic Poisson integrals polydisc polynomials proof of Theorem Proposition prove radial region replaced restricted K-limit Rudin satisfies Schwarz lemma shows space subharmonic subspace Suppose uniformly variables vector z₁ zero-variety zeros