Log-hyperconvexity index and Bergman kernel

TL;DR

This paper establishes quantitative estimates of the Bergman distance and integrability properties of the Bergman kernel for bounded domains in complex space with a certain log-hyperconvexity index, advancing understanding in complex analysis.

Contribution

It provides new bounds on the Bergman distance and integrability conditions of the Bergman kernel based on the domain's log-hyperconvexity index, a novel approach in the field.

Findings

Quantitative estimate of Bergman distance for domains with high log-hyperconvexity index

$A^2( ext{log }A)^q$-integrability of the Bergman kernel established

Conditions on the log-hyperconvexity index for improved estimates

Abstract

We obtain a quantitative estimate of Bergman distance when $\Omega \subset \mathbb{C}^n$ is a bounded domain with log-hyperconvexity index $\alpha_l(\Omega)>\frac{n-1+\sqrt{(n-1)(n+3)}}{2}$, as well as the $A^2(\log A)^q$-integrability of the Bergman kernel $K_{\Omega}(\cdot, w)$ when $\alpha_l(\Omega)>0$.