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1980 | Buch

Preliminaries Kapitel lesen Erstes Kapitel lesen

Function Theory in the Unit Ball of ℂ n

verfasst von: Walter Rudin

Verlag: Springer New York

Buchreihe : Grundlehren der mathematischen Wissenschaften

Enthalten in: Professional Book Archive

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Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Preliminaries
Abstract
Throughout this book, ℂ will denote the complex field, and ℂ n will be the cartesian product of n copies of ℂ; here n is any positive integer. The points of ℂ n are thus ordered n-tuples z = (z1, …,z n ), where each zi ∈ ℂ. Algebraically, ℂ n is an n-dimensional vector space over ℂ. Topologically, ℂ n is the euclidean space R2n of real dimension 2n.
Walter Rudin
Chapter 2. The Automorphisms of B
Abstract
The main purpose of this chapter is the description of the biholomorphic maps of B onto B. These will simply be called the automorphisms of B. They form a group, Aut(B), which may also be called the Moebius group of B, since the automorphisms turn out to be linear fractional transformations.
Walter Rudin
Chapter 3. Integral Representations
Abstract
(L p ⋂ H)(B). As usual, n is a fixed positive integer, n ≥ 1, B = B n is the open unit ball of ℂ n , and v is the normalized Lebesgue measure on ℂ n defined in §1.4.1.
Walter Rudin
Chapter 4. The Invariant Laplacian
Abstract
Suppose Ω is an open subset of B, f ∈ C 2 (Ω), and a ∈ Ω. We define
$$\left( {\tilde \Delta f} \right)\left( a \right) = \Delta \left( {f \circ {\varphi _a}} \right)\left( 0 \right),$$
(1)
where ϕa is the involution defined in §2.2.1, and Δ is the ordinary Laplacian given by
$$\Delta g = 4\sum\limits_{i = 1}^n {{D_i}{{\bar D}_i}g} $$
(2)
as in §1.3.4.
Walter Rudin
Chapter 5. Boundary Behavior of Poisson Integrals
Abstract
The principal result of the present chapter is Korányi’s Theorem 5.4.5—the fact that certain maximal functions associated to invariant Poisson integrals are of weak type (1, 1). The existence of what we call “K-limits” at almost all points of S follows easily from this.
Walter Rudin
Chapter 6. Boundary Behavior of Cauchy Integrals
Abstract
One major difference between the Poisson kernel and the Cauchy kernel is that the former is positive and the latter is not. The positivity of P(z, ζ) made it possible to be rather unsubtle in the proof of the basic maximal theorem 5.4.5: μ was replaced ∣μ∣ and it was then just a matter of making size estimates without giving too much away. No cancellation effects came into play.
Walter Rudin
Chapter 7. Some L p -Topics
Abstract
Fix n ≥ 1, write B n for B, temporarily, let s be a nonnegative integer, and let P be the orthogonal projection of ℂ n+s onto ℂ n (regarding ℂ n as the subspace of ℂ n+s defined by z n+1 = ··· = z n+s = 0).
Walter Rudin
Chapter 8. Consequences of the Schwarz Lemma
Abstract
The familiar classical Schwarz lemma deals with functions defined in the open unit disc U ⊂ ℂ, and asserts the following:
(a)
If f: → U U is holomorphic, then ∣f′(0)∣ < 1, except when f(λ) = cλ for some c∈ℂ with ∣c∣ = 1.
 
(b)
If also f(0) = 0, then ∣f(λ)∣ < ∣λ∣ for every λ∈U\{0}, except when f(λ) = cλ, as in (a)
 
Walter Rudin
Chapter 9. Measures Related to the Ball Algebra
Abstract
This chapter deals with two types of topics. The material of Sections 9.2 and 9.3 is function-theoretic. The measures that are discussed there are intimately related to the holomorphic functions in B. On the other hand, Sections 9.4 and 9.5 describe some measure-theoretic aspects of the theory of function algebras in general. These would not become any simpler by specializing to the ball algebra. Both aspects are used in the proof of the Cole-Range theorem (Section 9.6), which is one of several modern generalizations of the classical theorem of F. and M. Riesz.
Walter Rudin
Chapter 10. Interpolation Sets for the Ball Algebra
Abstract
We shall be concerned with compact sets K ⊂ S. For convenience, we introduce six labels, (Z), (P), (I), PI), (N), and (TN) to denote certain properties that K may or may not have in relation to the ball algebra A(B).
Walter Rudin
Chapter 11. Boundary Behavior of H∞-Functions
Abstract
The objective of this preliminary section is Theorem 11.1.2, a one-variable Fatou-type theorem for nonholomorphic functions that will be needed in the proof of Theorem 11.2.4.
Walter Rudin
Chapter 12. Unitarily Invariant Function Spaces
Abstract
This chapter deals with a subject that is basically a topic in harmonic analysis and which, at first glance, may seem to have little to do with our principal concern, namely with holomorphic functions. Nevertheless, one of its main results (Theorem 12.3.6) will be essential later in the classification of Moebius-invariant spaces, an obviously function-theoretic topic.
Walter Rudin
Chapter 13. Moebius-Invariant Function Spaces
Abstract
A space Y of functions with domain S, or B, or \(\bar B\), is said to be Moebius-invariant, or simply ℳ-invariant, if f º ψ ∈ Y for all f ∈ Y and all ψ ∈ Aut(B).
Walter Rudin
Chapter 14. Analytic Varieties
Abstract
This chapter contains a brief introduction to analytic varieties. It is quite elementary, but will be sufficient for the material that follows.
Walter Rudin
Chapter 15. Proper Holomorphic Maps
Abstract
Let X and Y be topological spaces. A continuous map f: X → Y is said to be proper if f−1(K) is compact in X for every compact K ⊂ Y.
Walter Rudin
Chapter 16. The -Problem
Abstract
Differential forms are often introduced purely algebraically (Spivak [1], Gunning-Rossi [1]), as members of a graded ring, or simply as “formal sums” that are to be manipulated according to the rules of “exterior algebra,” but they can also be defined as complex-valued functions whose domain is the collection of all suitably differentiable surfaces of the appropriate dimension. We shall sketch this second approach, omitting all proofs; they are elementary, but long-winded and repetitious. Details may be found in Rudin [16]. The main purpose of this introductory section is to recall the basic facts and to establish notation.
Walter Rudin
Chapter 17. The Zeros of Nevanlinna Functions
Abstract
In Theorem 7.3.3 we saw that the zero-variety Z(f) of a function f ∈ N(B) satisfies the Blaschke condition. The Henkin-Skoda theorem asserts the converse: if a zero-variety V in B satisfies the Blaschke condition, then V = Z(f) for some f ∈ N(B). (Actually, both Henkin and Skoda proved this in strictly pseudoconvex domains.)
Walter Rudin
Chapter 18. Tangential Cauchy-Riemann Operators
Abstract
The theme of this section is that holomorphic functions in a region Ω ⊂ ℂ n that are smooth on \(\bar \Omega \) satisfy “tangential Cauchy-Riemann equations” on ∂Ω when n > 1, and that, conversely, all functions defined on a portion M of the boundary that satisfy these equations extend to be holomorphic on one or even on both sides of M, provided certain geometric conditions hold.
Walter Rudin
Chapter 19. Open Problems
Abstract
We define an inner function in B to be a nonconstant f ∈ H∞(B) whose radial limits f* satisfy ∣f*(ζ)∣ = 1 for almost all ζ ∈ S.
Walter Rudin
Backmatter
Metadaten
Titel
Function Theory in the Unit Ball of ℂ n
verfasst von
Walter Rudin
Copyright-Jahr
1980
Verlag
Springer New York
Electronic ISBN
978-1-4613-8098-6
Print ISBN
978-1-4613-8100-6
DOI
https://doi.org/10.1007/978-1-4613-8098-6