Weighted generalization of the Ramadanov theorem and further considerations

TL;DR

This paper investigates the convergence behavior of weighted Bergman kernels on complex domains, providing a weighted generalization of Ramadanov's theorem and characterizing domains where inverse convergence holds.

Contribution

It introduces a weighted version of Ramadanov's theorem, establishing conditions for uniform convergence of weighted Bergman kernels and characterizing domains with inverse convergence properties.

Findings

Weighted Bergman kernels converge uniformly under certain conditions.

A weighted generalization of Ramadanov's theorem is established.

Convergence implies specific domain properties, characterizing inverse theorem applicability.

Abstract

We study the limit behavior of weighted Bergman kernels on a sequence of domains in a complex space $\mathbb{C}^N$, and show that under some conditions on domains and weights, weighed Bergman kernels converge uniformly on compact sets. Then we give a weighted generalization of the theorem given in [S, p. 38], highlighting some special property of the domains, on which the weighted Bergman kernels converge uniformly. Moreover we will show that convergence of weighted Bergman kernels implies this property, which will give a characterization of the domains, for which the inverse of Ramadanov theorem holds.