Log-hyperconvexity index and Bergman kernel

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 ${\alpha }_l(\Omega )>0$.