Transformation Formula for the Reduced Bergman Kernel and its Application

Abstract

In this article, we prove the transformation formula for the reduced Bergman kernels under proper holomorphic correspondences between bounded domains in the complex plane. As a corollary, we obtain the transformation formula for the reduced Bergman kernels under proper holomorphic maps. We also establish the transformation formula for the weighted reduced Bergman kernels under proper holomorphic maps. Finally, we provide an application of this transformation formula.

The reduced Bergman kernel and its properties

“…In a previous work (see [3]), we proved a transformation formula for the weighted reduced Bergman kernels under proper holomorphic maps between bounded planar domains. In continuation of our previous effort, this note is a step forward in creating a dictionary between the Bergman kernel and the reduced Bergman kernel.…”