The Cauchy Integral in $\mathbb C^n$ for domains with minimal smoothness

TL;DR

This paper establishes $L^p$ regularity of the Cauchy-Leray integral on bounded domains in complex space with minimal boundary smoothness of class $C^{1,1}$, under a convexity condition.

Contribution

It proves $L^p$ regularity of the Cauchy-Leray integral for domains with minimal boundary smoothness and a natural convexity condition, extending previous results.

Findings

$L^p$ regularity proven for $C^{1,1}$ boundaries

Applicable to convex domains in $ ext{C}^n$

Extends regularity results to less smooth domains

Abstract

We prove $L^p(b\mathcal{D})$-regularity of the Cauchy-Leray integral for bounded domains $\mathcal{D}\subset\mathbb C^n$ whose boundary satisfies the minimal regularity condition of class $C^{1,1}$, together with a naturally occurring notion of convexity.