Boundary singularity of Poisson and harmonic Bergman kernels

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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.

Keywords

Harmonic Bergman kernel
Poisson kernel
Pseudodifferential boundary operators

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Research supported by GA AV ČR grant No. IAA100190802, GACR grant No. 201/12/0426 and Czech Ministry of Education, Youth and Sports research plan No. MSM4781305904.