Function Theory on Domains of Holomorphy in ℂⁿ

Abstract

In §1 of this chapter we first extend the fundamental vanishing theorem $H_{\overline \partial }^q \left( K \right) = 0\,for\,q \geqslant 1$ for q ≥ 1 on a Stein compactum K proved in Corollary V.2.6 to arbitrary open Stein domains (Theorem 1.4). The proof involves an approximation theorem for holomorphic functions on compact analytic polyhedra which is of independent interest, and which generalizes the classical Runge Approximation Theorem in the complex plane. We also discuss several variations of this approximation theorem. In particular we consider the Runge property for the exhaustion of a pseudoconvex domain D by strictly pseudoconvex domains which arises from the existence of a strictly plurisubharmonic exhaustion function on D. Together with the results of Chapter V this yields the solution of the Levi problem for arbitrary pseudoconvex domains. In §2 we apply these methods to solve the Cauchy-Riemann equations directly on a pseudoconvex domain D, i.e., we show that $H_{\overline \partial }^q \left( D \right) = 0\,for\,q \geqslant 1$ for q ≥ 1, and we prove that this property characterizes Stein domains. §3 deals with some topological properties of Stein domains D, for example, we show that if D ⊂ ℂⁿ, then H ^r(D, ℂ) = 0 for r > n. This section may be skipped without loss of continuity. Finally, in §4 and §5, the vanishing of $H_{\overline \partial }^1$ for Stein domains D is used to generalize to several variables the classical theorems of Mittag-Leffler and Weierstrass on the existence of global meromorphic functions with prescribed poles and zero sets on regions in the complex plane. §5 includes a detailed discussion of the new—strictly higher dimensional—phenomenon of a topological obstruction in the analog of the Weierstrass theorem.