Bergman Kernels, Hartogs Domains, and the Wiegerinck Problem

Abstract

Let $L^2_h(\Omega,\mu)$ denote the space of holomorphic functions on $\Omega$ which are square-integrable with respect to the weight $\mu:\Omega\to [0,\infty)$, where $\Omega$ is a domain in $\mathbb{C}^n$. When $\mu$ is sufficiently well-behaved, the space $L^2_h(\Omega,\mu)$ possesses a unique sesqui-holomorphic function $K_{\Omega,\mu}:\Omega\times\Omega\to\mathbb{C}$ such that $$ f(z)=\int_{\Omega}f(\zeta)K_{\Omega,\mu}(z,\zeta)\mu(\zeta)\text{d}\text{Volume}(\zeta) $$ known as the Bergman kernel.

This dissertation contains a variety of results concerning Bergman spaces. The Bergman kernel and Wiegerinck problem (whether a nontrivial Bergman space must have infinite dimension) have particular focus.

In Chapter 2, we show that, by changing the weight, one may create zeroes in the Bergman kernel without changing the associated space of holomorphic functions. We also provide a construction of a weight on $\mathbb{C}$ whose Bergman kernel has an infinite number of zeroes.

In Chapter 3, we expand some of the results of Jucha to show that a complete $N$-circled Hartogs domain has infinite-dimensional Bergman space whenever its associated plurisubharmonic function has a neighborhood on which it is strictly plurisubharmonic. This agrees with a work of Gallagher et al. We follow this up with some sufficient conditions for the infinite-dimensionality of a complete $N$-circled Hartogs domain based on various forms of the Ohsawa-Takegoshi extension theorem. We also address a question of Pflug and Zwonek.

In Chapter 4, we directly compute a coefficient which relates the $L^2$-norm of a holomorphic function on a complete $N$-circled Hartogs domain to the weighted $L^2$-norm of an associated function over the base domain. We then use this relationship to compute explicit formulae of the Bergman kernel for generalized Hartogs triangles with rational index, in an alternative manner to Edholm and McNeal. We also provide an alternative proof of the well-known ``inflation'' identity of Boas, Fu, and Straube. Other relationships of this type are presented.