Function Theory in the Unit Ball of ℂn

1. Front Matter

   Pages i-xiii

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2. Preliminaries

   • Walter Rudin

   Pages 1-22

3. The Automorphisms of B

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   Pages 23-35

4. Integral Representations

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   Pages 36-46

5. The Invariant Laplacian

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   Pages 47-64

6. Boundary Behavior of Poisson Integrals

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   Pages 65-90

7. Boundary Behavior of Cauchy Integrals

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   Pages 91-119

8. Some L^p-Topics

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   Pages 120-160

9. Consequences of the Schwarz Lemma

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   Pages 161-184

10. Measures Related to the Ball Algebra

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    Pages 185-203

11. Interpolation Sets for the Ball Algebra

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    Pages 204-233

12. Boundary Behavior of H^∞-Functions

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    Pages 234-252

13. Unitarily Invariant Function Spaces

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    Pages 253-277

14. Moebius-Invariant Function Spaces

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    Pages 278-287

15. Analytic Varieties

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    Pages 288-299

16. Proper Holomorphic Maps

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    Pages 300-329

17. The \(\bar \partial \)-Problem

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    Pages 330-363

18. The Zeros of Nevanlinna Functions

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    Pages 364-386

19. Tangential Cauchy-Riemann Operators

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    Pages 387-402

20. Open Problems

    • Walter Rudin

    Pages 403-417

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
• Funktionentheorie
• Smooth function
• complex analysis
• convergence
• differential equation
• holomorphic function
• integral
• interpolation
• maximum
• minimum
• operator

• Department of Mathematics, University of Wisconsin, Madison, USA

  Walter Rudin

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