Weighted Szeg\H{o} Kernels on Planar Domains

TL;DR

This paper investigates the properties of weighted Szeg ext{o} and Garabedian kernels on planar domains, extending known results from the unweighted case and expressing certain kernels as rational combinations of unweighted Szeg ext{o} kernels.

Contribution

It introduces a weighted Kerzman-Stein formula, analyzes the boundary smoothness of weighted kernels, and expresses reduced Bergman kernels as rational combinations of unweighted Szeg ext{o} kernels.

Findings

Weighted Szeg ext{o} kernels depend smoothly on the weight.

Properties of unweighted kernels extend to weights close to constant.

Reduced Bergman kernels can be expressed as rational combinations of Szeg ext{o} kernels.

Abstract

We study properties of weighted Szeg\H{o} and Garabedian kernels on planar domains. Motivated by the unweighted case as explained in Bell's work, the starting point is a weighted Kerzman-Stein formula that yields boundary smoothness of the weighted Szeg\H{o} kernel. This provides information on the dependence of the weighted Szeg\H{o} kernel as a function of the weight. When the weights are close to the constant function $1$ (which corresponds to the unweighted case), it is shown that some properties of the unweighted Szeg\H{o} kernel propagate to the weighted Szeg\H{o} kernel as well. Finally, it is shown that the reduced Bergman kernel and higher order reduced Bergman kernels can be written as a rational combination of three unweighted Szeg\H{o} kernels and their conjugates, thereby extending Bell's list of kernel functions that are made up of simpler building blocks that involve the Szeg\H{o} kernel.