L_h^2-Functions in Unbounded Balanced Domains

Peter Pflug,Włodzimierz Zwonek

doi:10.1007/s12220-016-9754-3

Peter Pflug, Włodzimierz Zwonek

Open Access

https://doi.org/10.1007/s12220-016-9754-3

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Abstract

We investigate problems related with the existence of square integrable holomorphic functions on (unbounded) balanced domains. In particular, we solve the problem of Wiegerinck for balanced domains in dimension two. We also give a description of L_h^2-domains of holomorphy in the class of balanced domains and present a purely algebraic criterion for homogeneous polynomials to be square integrable in a pseudoconvex balanced domain in mathbb {C}^2. This allows easily to decide which pseudoconvex balanced domain in mathbb {C}^2 has a positive Bergman kernel and which admits the Bergman metric.

Highlights

As a starting point for the considerations presented in the paper, the following problem of Wiegerinck from 1984 may serve
Allow its dimension to be only either or for an arbitrary pseudoconvex domain D ⊂ Cn? Wiegerinck confirmed the dichotomy in dimension one and gave an example of an unbounded non-pseudoconvex domain
Employing the potential theoretic methods applied by Jucha, we present an effective and simple algebraic criterion for a homogeneous polynomial to be square integrable on pseudoconvex balanced domains in dimension two

Summary

Introduction

As a starting point for the considerations presented in the paper, the following problem of Wiegerinck from 1984 (see [24]) may serve. The problems of positive definiteness, being an holomorphy or Bergman complete in the class of bounded balanced pseudoconvex domains, are either trivial or fully understood, the situation in the unbounded case is still unclear. Recall that for an arbitrary bounded pseudoconvex domain D the fact that it is an holomorphy is equivalently described by the boundary behavior of its Bergman kernel (see [16]). Let us once more emphasize that the above results except for their description of notions related with balanced domains may serve as results on approximation of homogeneous, logarithmically plurisubharmonic functions with the help of limits of functions involving homogeneous polynomials (see for instance Proposition 2.1, Theorems 3.1 and 3.2 in [22]) which constitute an example of classical type of results in the theory of plurisubharmonic functions

Bergman Spaces in Two-Dimensional Balanced Domains

Description of Balanced Domains in C2 with Trivial Bergman Spaces

Square Integrable Homogeneous Polynomials