A remark on the dimension of the Bergman space of some Hartogs domains

TL;DR

This paper investigates the Bergman space of certain Hartogs domains, proving it is either trivial or infinite dimensional, depending on the properties of the defining subharmonic function.

Contribution

It establishes a dichotomy for the dimension of the Bergman space on a class of Hartogs domains based on the subharmonic function u.

Findings

Bergman space is either trivial or infinite dimensional.

The result depends on the properties of the subharmonic function u.

Provides a clear criterion for the dimension of the Bergman space.

Abstract

Let D be a Hartogs domain of the form D={(z,w) \in CxC^N : |w| < e^{-u(z)}} where u is a subharmonic function on C. We prove that the Bergman space of holomorphic and square integrable functions on D is either trivial or infinite dimensional.