Real Variable Things in Bergman Theory

TL;DR

This paper explores the relationship between real variable methods and Bergman theory, establishing new inequalities, regularity results, and bounds for Bergman kernels in complex analysis.

Contribution

It introduces novel inequalities and bounds linking real variable techniques with Bergman kernel properties, enhancing understanding of their regularity and quantitative behavior.

Findings

Established $L^2$ Hartogs extension theorem using Hardy inequalities

Proved absolute continuity of Bergman kernels on planar domains

Derived lower bounds for the Bergman kernel in terms of capacity and eigenvalues

Abstract

In this article, we investigate the connection between certain real variable things and the Bergman theory. We first use Hardy-type inequalities to give an $L^2$ Hartogs-type extension theorem and an $L^p$ integrability theorem for the Bergman kernel $K_\Omega(\cdot,w)$. We then use the Sobolev-Morrey inequality to show the absolute continuity of Bergman kernels on planar domains with respect to logarithmic capacities. Finally, we give lower bounds of the minimum $\kappa(\Omega)$ of the Bergman kernel $K_\Omega(z)$ in terms of the interior capacity radius for planar domains and the volume density for bounded pseudoconvex domains in $\mathbb C^n$. As a consequence, we show that $\kappa(\Omega)\ge c_0 \lambda_1(\Omega)$ holds on planar domains, where $c_0$ is a numerical constant and $\lambda_1(\Omega)$ is the first Dirichlet eigenvalue of $-\Delta$.