Capacities, Green function and Bergman functions

TL;DR

This paper provides quantitative estimates of the Green function, lower bounds for the Bergman kernel, and the Bergman distance in bounded pseudoconvex domains using logarithmic capacity, with applications to holomorphic motions.

Contribution

It introduces new bounds relating capacity, Green function, and Bergman kernel for pseudoconvex domains, advancing understanding of their geometric and analytic properties.

Findings

Bergman kernel satisfies $K_ Omega(z) ightrightarrows ext{constant} imes ext{distance}^{-2}$

Quantitative estimates of Green function using capacity

Lower bounds for Bergman kernel and distance in bounded domains

Abstract

Using the logarithmic capacity, we give quantitative estimates of the Green function, as well as lower bounds of the Bergman kernel for bounded pseudoconvex domains in $\mathbb C^n$ and the Bergman distance for bounded planar domains. In particular, it is shown that the Bergman kernel satisfies $K_\Omega(z)\gtrsim \delta_\Omega(z)^{-2}$ for any bounded pseudoconvex domain with $C^0-$boundary. An application to holomorphic motions is given.