Bergman kernel and hyperconvexity index

Abstract

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.

Minor changes. To appear in Analysis & PDE

References in corpus (4)

• The Bergman projection on fat Hartogs triangles: L^p boundedness
• A global estimate for the Diederich--Fornaess index of weakly pseudoconvex domains
• The Kobayashi balls of C-convex domains
• Off-diagonal decay of toric Bergman kernels

Cited by in corpus (2)

• On the $p-$Bergman theory
• Diederich-Fornaess and Steinness indices for abstract CR manifolds