BERGMAN KERNEL AND HYPERCONVEXITY INDEX

Let Omega subset of C-n be a bounded domain with the hyperconvexity index alpha(Omega) > 0. Let rho be the relative extremal function of a fixed closed ball in Omega, and set mu := |rho|(1+|log|rho||)(-1) and nu := |rho|(1+|log|rho||)(n). We obtain the following estimates for the Bergman kernel. (1) For every 0 < alpha < alpha(Omega) and 2 <= p < 2 + 2 alpha(Omega)/(2n - alpha(Omega)), there exists a constant C > 0 such that integral(Omega)|K-Omega(., omega)/root K Omega(w)|(p) <= C|mu(w)|(-(p-2)n/alpha) for all w is an element of Omega. (2) For every 0 < r < 1, there exists a constant C > 0 such that |K-Omega(z, w)|(2)/(K-Omega(z)K-Omega(w)) <= C (min{v(z)/mu(w), v(w)/mu(z)})(r) for all z; w is an element of Omega. Various applications of these estimates are given.