Bergman kernel and hyperconvexity index

Abstract

Let $Ω\subset {\mathbb C}^n$ be a bounded domain with the hyperconvexity index $α(Ω)>0$. Let $\varrho$ be the relative extremal function of a fixed closed ball in $Ω$ and set $μ:=|\varrho|(1+|\log|\varrho||)^{-1}$, $ν:=|\varrho|(1+|\log|\varrho||)^n$. We obtain the following estimates for the Bergman kernel: (1) For every $0<α<α(Ω)$ and $2\le p<2+\frac{2α(Ω)}{2n-α(Ω)}$, there exists a constant $C>0$ such that $\int_Ω|\frac{K_Ω(\cdot,w)}{\sqrt{K_Ω(w)}}|^{p}\le C |μ(w)|^{-\frac{(p-2) n}α}$ for all $w\in Ω$. (2) For every $0 0$ such that $ \frac{|K_Ω(z,w)|^2}{K_Ω(z)K_Ω(w)}\le C (\min\{\frac{ν(z)}{μ(w)},\frac{ν(w)}{μ(z)}\})^r $ for all $z,w\in Ω$. Various application of these estimates are given.