On the growth of the Bergman kernel near an infinite-type point

TL;DR

This paper provides new estimates for the Bergman kernel near infinite-type boundary points in certain complex domains, including non-convex pseudoconvex domains, based on a mild structural condition.

Contribution

It introduces a mild structural condition that enables optimal bounds for the Bergman kernel near infinite-type points, extending previous results to non-convex domains.

Findings

Established upper and lower bounds for the Bergman kernel near infinite-type points.

Extended estimates to non-convex pseudoconvex domains.

Quantified the flatness of boundary points affecting kernel growth.

Abstract

We study diagonal estimates for the Bergman kernels of certain model domains in $\mathbb{C}^2$ near boundary points that are of infinite type. To do so, we need a mild structural condition on the defining functions of interest that facilitates optimal upper and lower bounds. This is a mild condition; unlike earlier studies of this sort, we are able to make estimates for non-convex pseudoconvex domains as well. This condition quantifies, in some sense, how flat a domain is at an infinite-type boundary point. In this scheme of quantification, the model domains considered below range -- roughly speaking -- from being ``mildly infinite-type'' to very flat at the infinite-type points.