On invariant domains of holomorphy

Abstract

Let D be a domain in C and F = F(D) be a subclass in the class O = O(D) of functions holomorphic in D. Recall that D is called an F-domain of holomorphy iff there exists a function f ∈ F(D) which cannot be holomorphically extended across the boundary of D. If, for example, we take for F(D) the class Otemp(D) of temperate holomorphic functions in D (i.e. holomorphic functions in D growing less than a power of distance to the boundary of D) then the notion of Otemp-domains of holomorphy will coincide with the usual notion of (O)domains of holomorphy according to Pflug [10]. On the other hand, considering as F(D) the class H∞(D) of bounded holomorphic functions we obtain the notion of H∞-domains of holomorphy which is quite different from the notion of (O)domains of holomorphy (cf. Sibony [13]). In this paper we are interested in the case when D is invariant under the action of a compact Lie group K and F(D) coincides with the class O(D) of K-invariant holomorphic functions in D. From first examples of O-domains of holomorphy it becomes clear that this notion differs much from the usual notion of domains of holomorphy. Consider, e.g., the ring D = {1 < |z| < 2} in C with the action of the circle group S given by rotations. Then the only S-invariant holomorphic functions in D are constants so they extend holomorphically across the boundary of D to all of C (note that D is a domain of holomorphy in this example). Later on we shall give several (less trivial) examples of that sort. This article based on recent results by Peter Heinzner, Xiangyu Zhou and the author (cf. [4], [5], [12], [16]) contains some general assertions about O-domains of holomorphy and their holomorphic hulls