Let $��\subset {\mathbb C}^n$ be a bounded domain with the hyperconvexity index $��(��)>0$. Let $\varrho$ be the relative extremal function of a fixed closed ball in $��$ and set $��:=|\varrho|(1+|\log|\varrho||)^{-1}$, $��:=|\varrho|(1+|\log|\varrho||)^n$. We obtain the following estimates for the Bergman kernel: (1) For every $00$ such that $\int_��|\frac{K_��(\cdot,w)}{\sqrt{K_��(w)}}|^{p}\le C |��(w)|^{-\frac{(p-2) n}��}$ for all $w\in ��$. (2) For every $00$ such that $ \frac{|K_��(z,w)|^2}{K_��(z)K_��(w)}\le C (\min\{\frac{��(z)}{��(w)},\frac{��(w)}{��(z)}\})^r $ for all $z,w\in ��$. Various application of these estimates are given.

Minor changes. To appear in Analysis & PDE