$L_h^2$-functions in unbounded balanced domains

TL;DR

This paper studies square integrable holomorphic functions on unbounded balanced domains, solving a key problem in dimension two and providing criteria for their existence, with implications for Bergman kernels and metrics.

Contribution

It solves Wiegerinck's problem for balanced domains in dimension two and offers algebraic criteria for square integrability of polynomials in pseudoconvex balanced domains.

Findings

Characterization of $L_h^2$-domains of holomorphy in balanced domains

Algebraic criterion for polynomial square integrability in $ ext{C}^2$

Identification of domains with positive Bergman kernel and metric

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.