Function Theory in the Unit Ball of Cn: (241 Grundlehren der Mathematischen Wissenschaften)

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.

Table of Contents:
1 Preliminaries.- 1.1 Some Terminology.- 1.2 The Cauchy Formula in Polydiscs.- 1.3 Differentiation.- 1.4 Integrals over Spheres.- 1.5 Homogeneous Expansions.- 2 The Automorphisms of B.- 2.1 Cartan's Uniqueness Theorem.- 2.2 The Automorphisms.- 2.3 The Cayley Transform.- 2.4 Fixed Points and Affine Sets.- 3 Integral Representations.- 3.1 The Bergman Integral in B.- 3.2 The Cauchy Integral in B.- 3.3 The Invariant Poisson Integral in B.- 4 The Invariant Laplacian.- 4.1 The Operator $$ \tilde \Delta $$.- 4.2 Eigenfunctions of $$ \tilde \Delta $$.- 4.3 ?-Harmonic Functions.- 4.4 Pluriharmonic Functions.- 5 Boundary Behavior of Poisson Integrals.- 5.1 A Nonisotropic Metric on S.- 5.2 The Maximal Function of a Measure on S.- 5.3 Differentiation of Measures on S.- 5.4 K-Limits of Poisson Integrals.- 5.5 Theorems of Calderon, Privalov, Plessner.- 5.6 The Spaces N(B) and Hp(B).- 5.7 Appendix: Marcinkiewicz Interpolation.- 6 Boundary Behavior of Cauchy Integrals.- 6.1 An Inequality.- 6.2 Cauchy Integrals of Measures.- 6.3 Cauchy Integrals of Lp-Functions.- 6.4 Cauchy Integrals of Lipschitz Functions.- 6.5 Toeplitz Operators.- 6.6 Gleason's Problem.- 7 Some Lp-Topics.- 7.1 Projections of Bergman Type.- 7.2 Relations between Hp and Lp ? H.- 7.3 Zero-Varieties.- 7.4 Pluriharmonic Majorants.- 7.5 The Isometries of Hp(B).- 8 Consequences of the Schwarz Lemma.- 8.1 The Schwarz Lemma in B.- 8.2 Fixed-Point Sets in B.- 8.3 An Extension Problem.- 8.4 The Lindelof-?irka Theorem.- 8.5 The Julia-Caratheodory Theorem.- 9 Measures Related to the Ball Algebra.- 9.1 Introduction.- 9.2 Valskii's Decomposition.- 9.3 Henkin's Theorem.- 9.4 A General Lebesgue Decomposition.- 9.5 A General F. and M. Riesz Theorem.- 9.6 The Cole-Range Theorem.- 9.7 Pluriharmonic Majorants.- 9.8 The Dual Space of A(B).- 10 Interpolation Sets for the Ball Algebra.- 10.1 Some Equivalences.- 10.2 A Theorem of Varopoulos.- 10.3 A Theorem of Bishop.- 10.4 The Davie-Oksendal Theorem.- 10.5 Smooth Interpolation Sets.- 10.6 Determining Sets.- 10.7 Peak Sets for Smooth Functions.- 11 Boundary Behavior of H?-Functions.- 11.1 A Fatou Theorem in One Variable.- 11.2 Boundary Values on Curves in S.- 11.3 Weak*-Convergence.- 11.4 A Problem on Extreme Values.- 12 Unitarily Invariant Function Spaces.- 12.1 Spherical Harmonics.- 12.2 The Spaces H(p, q).- 12.3 U-Invariant Spaces on S.- 12.4 U-Invariant Subalgebras of C(S).- 12.5 The Case n = 2.- 13 Moebius-Invariant Function Spaces.- 13.1.?-Invariant Spaces on S.- 13.2.?-Invariant Subalgebras of C0(B).- 13.3.?-Invariant Subspaces of C(B).- 13.4 Some Applications.- 14 Analytic Varieties.- 14.1 The Weierstrass Preparation Theorem.- 14.2 Projections of Varieties.- 14.3 Compact Varieties in ?n.- 14.4 Hausdorff Measures.- 15 Proper Holomorphic Maps.- 15.1 The Structure of Proper Maps.- 15.2 Balls vs. Polydiscs.- 15.3 Local Theorems.- 15.4 Proper Maps from B to B.- 15.5 A Characterization of B.- 16 The $$ {\bar \partial } $$ -Problem.- 16.1 Differential Forms.- 16.2 Differential Forms in ?n.- 16.3 The $$ {\bar \partial } $$ -Problem with Compact Support.- 16.4 Some Computations.- 16.5 Koppelman's Cauchy Formula.- 16.6 The $$ {\bar \partial } $$ -Problem in Convex Regions.- 16.7 An Explicit Solution in B.- 17 The Zeros of Nevanlinna Functions.- 17.1 The Henkin-Skoda Theorem.- 17.2 Plurisubharmonic Functions.- 17.3 Areas of Zero-Varieties.- 18 Tangential Cauchy-Riemann Operators.- 18.1 Extensions from the Boundary.- 18.2 Unsolvable Differential Equations.- 18.3 Boundary Values of Pluriharmonic Functions.- 19 Open Problems.- 19.1 The Inner Function Conjecture.- 19.2 RP-Measures.- 19.3 Miscellaneous Problems.