Weighted Kernel Functions on Planar Domains

TL;DR

This paper investigates how weighted Szegő and Garabedian kernels vary on planar domains, establishing a Ramadanov type theorem, analyzing zeros, boundary behavior, and relations to classical kernels for different weights.

Contribution

It introduces a Ramadanov type theorem for weighted kernels, studies zeros and boundary properties for weights near constant, and explores relations with classical kernel functions.

Findings

Weighted kernels vary continuously with weights.

Zeros of kernels are well-behaved near constant weights.

Connections between weighted kernels and classical kernel functions are established.

Abstract

We study the variation of weighted Szeg\H{o} and Garabedian kernels on planar domains as a function of the weight. A Ramadanov type theorem is shown to hold as the weights vary. As a consequence, we derive properties of the zeros of the weighted Szeg\H{o} and Garabedian kernel for weights close to the constant function $1$ on the boundary. We further study the weighted Ahlfors map and strengthen results concerning its boundary behaviour. Explicit examples of the weighted kernels are presented for certain classes of weights. We highlight an interesting property of the weighted Szeg\H{o} and Garabedian kernels, implicit in Nehari's work, and explore several of its consequences. Finally, we discuss the weighted Carath\'eodory metric, and describe relations of the weighted Szeg\H{o} and Garabedian kernel with certain classical kernel functions.