Function Theory in the Unit Ball of Cn (Grundlehren der mathematischen Wissenschaften)

Function Theory in the Unit Ball of C^n.^ From the reviews: "...The book is easy on the reader. The prerequisites are minimal—just the standard graduate introduction to real analysis, complex analysis (one variable), and functional analysis. This presentation is unhurried and the author does most of the work. ...certainly a valuable reference book, and (even though there are no exercises) could be used as a text in advanced courses." __R. Rochberg__ in __Bulletin of the London Mathematical Society__. "...an excellent introduction to one of the most active research fields of complex analysis. ...As the author emphasizes, the principal ideas can be presented clearly and explicitly in the ball, specific theorems can be quickly proved. ...Mathematics lives in the book: main ideas of theorems and proofs, essential features of the subjects, lines of further developments, problems and conjectures are continually underlined. ...Numerous examples throw light on the results as well as on the difficulties." __C. Andreian Cazacu__ in __Zentralblatt für Mathematik__ Around 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. Around 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 Function Theory in the Unit Ball of C n. From the reviews: "...The book is easy on the reader. The prerequisites are minimal—just the standard graduate introduction to real analysis, complex analysis (one variable), and functional analysis. This presentation is unhurried and the author does most of the work. ...certainly a valuable reference book, and (even though there are no exercises) could be used as a text in advanced courses." R. Rochberg in Bulletin of the London Mathematical Society . "...an excellent introduction to one of the most active research fields of complex analysis. ...As the author emphasizes, the principal ideas can be presented clearly and explicitly in the ball, specific theorems can be quickly proved. ...Mathematics lives in the book: main ideas of theorems and proofs, essential features of the subjects, lines of further developments, problems and conjectures are continually underlined. ...Numerous examples throw light on the results as well as on the difficulties." C. Andreian Cazacu in Zentralblatt für Mathematik Front Matter....Pages i-xiii Preliminaries....Pages 1-22 The Automorphisms of B ....Pages 23-35 Integral Representations....Pages 36-46 The Invariant Laplacian....Pages 47-64 Boundary Behavior of Poisson Integrals....Pages 65-90 Boundary Behavior of Cauchy Integrals....Pages 91-119 Some L p -Topics....Pages 120-160 Consequences of the Schwarz Lemma....Pages 161-184 Measures Related to the Ball Algebra....Pages 185-203 Interpolation Sets for the Ball Algebra....Pages 204-233 Boundary Behavior of H ∞ -Functions....Pages 234-252 Unitarily Invariant Function Spaces....Pages 253-277 Moebius-Invariant Function Spaces....Pages 278-287 Analytic Varieties....Pages 288-299 Proper Holomorphic Maps....Pages 300-329 The $$ \bar \partial $$ -Problem....Pages 330-363 The Zeros of Nevanlinna Functions....Pages 364-386 Tangential Cauchy-Riemann Operators....Pages 387-402 Open Problems....Pages 403-417 Back Matter....Pages 419-436