Bergman kernel and hyperconvexity index

TL;DR

This paper establishes new estimates for the Bergman kernel on hyperconvex domains in complex analysis, linking kernel behavior to the hyperconvexity index and extremal functions, with applications to complex geometry.

Contribution

It provides novel bounds for the Bergman kernel on hyperconvex domains using the hyperconvexity index and extremal functions, advancing understanding of kernel estimates in complex analysis.

Findings

Derived integral bounds for the Bergman kernel involving the hyperconvexity index.

Established pointwise estimates for the normalized Bergman kernel using extremal functions.

Presented applications of these estimates in complex geometric contexts.

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.