Bergman kernel and hyperconvexity index

Abstract

Let $\Omega \subset {ℂ}^n$ be a bounded domain with the hyperconvexity index $\alpha \left(\Omega \right)>0$. Let $\varrho$ be the relative extremal function of a fixed closed ball in $\Omega$, and set $\mu :=\left|\varrho \right|{\left(1+|log|\varrho \left|\right|\right)}^{-1}$ and $\nu :=\left|\varrho \right|{\left(1+|log|\varrho \left|\right|\right)}^n$. We obtain the following estimates for the Bergman kernel. (1) For every $0<\alpha <\alpha \left(\Omega \right)$ and $2\le p<2+2\alpha \left(\Omega \right)∕\left(2n-\alpha \left(\Omega \right)\right)$, there exists a constant $C>0$ such that ${\int }_{\Omega }|K_{\Omega }\left(\cdot ,w\right)∕\sqrt{K_{\Omega }\left(w\right)}{|}^p\le C|\mu \left(w\right){|}^{-\left(p-2\right)n∕\alpha }$ for all $w\in \Omega$. (2) For every $0<r<1$, there exists a constant $C>0$ such that $|K_{\Omega }\left(z,w\right){|}^2∕\left(K_{\Omega }\left(z\right)K_{\Omega }\left(w\right)\right)\le C{\left(min\left\{\nu \left(z\right)∕\mu \left(w\right),\nu \left(w\right)∕\mu \left(z\right)\right\}\right)}^r$ for all $z,w\in \Omega$. Various applications of these estimates are given.