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Function Theory in the Unit Ball of Cn by Walter Rudin

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Function Theory in the Unit Ball of Cn written by Walter Rudin , Professor of Mathematics, University of Wisconsin. Published by Springer New York , Year: 1980, ISBN: 978-1-4613-8100-6,978-1-4613-8098-6. Around 1970, an abrupt change occurred in the study of holomorphic functions of several complex variables. Sheaves vanished into the background, 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. Since the Contents lists the topics that are covered, this may be the place to mention some that might have been included but were not:

Function Theory in the Unit Ball of Cn written by Walter Rudin cover the following topics.

  • 1. Preliminaries
    2. The Automorphisms of B
    3. Integral Representations
    4. The Invariant Laplacian
    5. Boundary Behavior of Poisson Integrals
    6. Boundary Behavior of Cauchy Integrals
    7. Some U -Topics
    8. Consequences of the Schwarz Lemma
    9. Measures Related to the Ball Algebra
    10. Interpolation Sets for the Ball Algebra
    11. Boundary Behavior of HCO-Functions
    12. Unitarily Invariant Function Spaces
    13. Moebius-Invariant Function Spaces
    14. Analytic Varietie
    15. Proper Holomorphic Maps
    16. The a-Problem
    17. The Zeros of Nevanlinna Functions
    18. Tangential Cauchy-Riemann Operators
    19. Open Problems


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