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