#### Vol. 10, No. 6, 2017

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Bergman kernel and hyperconvexity index

### Bo-Yong Chen

Vol. 10 (2017), No. 6, 1429–1454
##### 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 $\mu :=|\varrho |{\left(1+|log|\varrho ||\right)}^{-1}$ and $\nu :=|\varrho |{\left(1+|log|\varrho ||\right)}^{n}$. 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 ${\int }_{\Omega }|{K}_{\Omega }\left(\phantom{\rule{0.3em}{0ex}}\cdot \phantom{\rule{0.3em}{0ex}},w\right)∕\sqrt{{K}_{\Omega }\left(w\right)}{|}^{p}\le C|\mu \left(w\right){|}^{-\left(p-2\right)n∕\alpha }$ for all $w\in \Omega$. (2) For every $0, 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.

 Dedicated to Professor John Erik Fornaess on the occasion of his 70th birthday

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