An Integral Kernel for Weakly Pseudoconvex Domains

TL;DR

This paper introduces a new explicit Cauchy-Fantappiè kernel for weakly pseudoconvex domains, capturing boundary geometry and providing tools for complex analysis in these general settings.

Contribution

It presents a novel explicit kernel construction that reflects boundary geometry, expanding analytical tools for weakly pseudoconvex domains.

Findings

Kernel reflects complex boundary geometry

Provides estimates for the associated integral operator

Offers new tools for analysis on weakly pseudoconvex domains

Abstract

A new explicit construction of Cauchy-Fantappi\'e kernels is introduced for an arbitrary weakly pseudoconvex domain with smooth boundary. While not holomorphic in the parameter, the new kernel reflects the complex geometry and the Levi form of the boundary. Some estimates are obtained for the corresponding integral operator, which provide evidence that this kernel and related constructions give useful new tools for complex analysis on this general class of domains.