Boundary singularity of Poisson and harmonic Bergman kernels

TL;DR

This paper characterizes the boundary behavior of Poisson and harmonic Bergman kernels on smooth bounded domains, extending classical results and employing advanced pseudodifferential operator calculus.

Contribution

It provides a comprehensive description of boundary singularities for these kernels using Boutet de Monvel calculus, generalizing known results for Bergman kernels.

Findings

Complete boundary behavior description for Poisson and harmonic Bergman kernels

Application of Boutet de Monvel calculus to boundary singularities

Extension of classical boundary behavior results to more general operators

Abstract

We give a complete description of the boundary behaviour of the Poisson kernel and the harmonic Bergman kernel of a bounded domain with smooth boundary, which in some sense is an analogue of the similar description for the usual Bergman kernel on a strictly pseudoconvex domain due to Fefferman. Our main tool is the Boutet de Monvel calculus of pseudodifferential boundary operators, and in fact we describe the boundary singularity of a general potential, trace or singular Green operator from that calculus.