  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.
