Bergman-Szeg\H{o} kernel asymptotics in weakly pseudoconvex finite type cases

TL;DR

This paper develops a detailed asymptotic analysis of the Szeg\

Contribution

It extends Fefferman's boundary asymptotics of the Bergman kernel to weakly pseudoconvex domains in c2b2 and generalizes a CR embedding theorem in three dimensions.

Findings

Constructed a pointwise parametrix for the Szeg\

Extended boundary asymptotics to weakly pseudoconvex domains in c2b2.

Generalized a CR embedding theorem of Lempert.

Abstract

We construct a pointwise Boutet de Monvel-Sj\"ostrand parametrix for the Szeg\H{o} kernel of a weakly pseudoconvex three dimensional CR manifold of finite type assuming the range of its tangential CR operator to be closed; thereby extending the earlier analysis of Christ. This particularly extends Fefferman's boundary asymptotics of the Bergman kernel to weakly pseudoconvex domains in $\mathbb{C}^{2}$, in agreement with D'Angelo's example. Finally our results generalize a three dimensional CR embedding theorem of Lempert.