# Boundary Behavior of Holomorphic Functions of Several Complex Variables. (MN-11)

ELIAS M. STEIN
Pages: 84
https://www.jstor.org/stable/j.ctt130hkc1

1. Front Matter
(pp. i-iv)
2. Preface
(pp. v-vi)
3. Introduction
(pp. vii-ix)

In classical function theory of one complex variable there is a very close connection between the boundary behavior of holomorphic functions in a domain, and the corresponding problem for harmonic functions in that domain. As a consequence there is one "potential theory" (that of the Laplacian,$\frac{{{\partial }^{2}}}{\partial {{\text{x}}^{2}}}+\frac{{{\partial }^{2}}}{\partial {{\text{y}}^{2}}}$) which is a fundamental tool for all domains. In the case of more than one complex variable this is no longer so. In the general context the appropriate potential theory (insofar as there is one) should depend on the particular domain considered, and ought, more precisely, to reflect the interplay of the...

(pp. x-x)
5. Chapter I, first part: Review of potential theory in ℝN
(pp. 1-14)

This chapter contains a brief review of known facts from potential theory in ℝNand several complex variables.

Let${\cal {D}}$be a bounded smooth domain in ℝN. Smooth will mean that the boundary is of class C2. (See also the discussion in §2 below.) In what follows in this chapter the class C1+εwould suffice - with slight modifications of the argument. The results would become essentially more difficult if the boundary were only C1(or more generally satisfy a Lipschitz condition). However, since many of the applications to complex analysis require a "parabolic" approach, and the definition of pseudo-convexity...

6. Chapter I, second part: Review of some topics in several complex variables
(pp. 15-31)

We now consider the standard complex n-dimensional Euclidean space ℂn. If we were to disregard the complex structure of ℂn, keeping only the real structure, we would be led to the usual identification of ℂnwith ℝN, where N = 2n. In terms of this identification holomorphic functions in ℂnare harmonic in ℝN, and this explains the relevance of the previous material. Nevertheless for the study of holomorphic functions we need those objects which are intrinsically related to the domain in question and which reflect more intimately the complex structure of ℂn.

We begin with the Bergman kernel. For...

7. Chapter II: Fatou's theorem
(pp. 32-53)

In this chapter we prove an analogue of Fatou's theorem for holomorphic functions in a bounded domain${\cal {D}}$with smooth boundary. It is to be emphasized that here assumptions of pseudo-convexity play no role. The main idea is to try to make estimates analogous to (8.14) for holomorphic functions while by-passing explicit information about the Poisson-Szegö kernel for${\cal {D}}$- knowledge about which we would, in any case, have little hope of getting.

Let${\cal {D}}$be a bounded domain in ℂnwith smooth (C2) boundary. For each$\zeta \in \partial \cal{D}$, let νζdenote the unit outward normal at ζ. For each...

8. Chapter III. Potential theory for strictly pseudo-convex domains
(pp. 54-69)

In this chapter we shall assume that${\cal {D}}$is a smooth (C2) subdomain of ℂnwhich is strictly pseudo-convex (see the definition below). The assumption of strict pseudo-convexity will allow us to introduce an appropriate metric in${\cal {D}}$, and it is the potential theory of the Laplace-Beltrami operator of that metric which is the basic tool in the proof of theorem 12 below.

Recall (see §7) that for each$\zeta \in \partial \cal{D}$we have considered the splitting of ℂngiven by ℂn= Nζ⊕ Cζ, and the induced splitting of the tangent plane${{\text{T}}_{\zeta}}=\text{N}_{\zeta}^{{\mathrm O}}\oplus {{\text{C}}_{\zeta}}$. Let now λ(z) be any...

9. Bibliography
(pp. 70-72)