The Kohn-Laplacian and Cauchy-Szeg\"{o} projection on Model Domains

TL;DR

This paper investigates the Kohn-Laplacian and Cauchy-Szeg"o projections on model domains in complex space, providing explicit kernels and establishing their Calderón-Zygmund properties using real analysis techniques.

Contribution

It offers explicit formulas for the Cauchy-Szeg"o kernels on model domains and proves their Calderón-Zygmund nature, advancing understanding of these operators in several complex variables.

Findings

Explicit Cauchy-Szeg"o kernels derived for model domains

Proved kernels are Calderón-Zygmund under suitable quasi-metrics

Enhanced understanding of the Kohn-Laplacian's fundamental solution

Abstract

We study the Kohn-Laplacian and its fundamental solution on some model domains in $\mathbb C^{n+1}$, and further discuss the explicit kernel of the Cauchy-Szeg\"o projections on these model domains using the real analysis method. We further show that these Cauchy-Szeg\"o kernels are Calder\'on-Zygmund kernels under the suitable quasi-metric.