# The role of an integration identity in the analysis of the Cauchy-Leray transform

## TL;DR

This paper investigates the Cauchy-Leray transform's properties on specific domains, showing dense definability and analyzing the impact of boundary regularity and convexity assumptions on its boundedness.

## Contribution

It extends previous results by demonstrating dense definability of the Cauchy-Leray transform on domains where boundedness fails under weaker boundary regularity or convexity.

## Key findings

- Dense definability of the Cauchy-Leray transform on certain domains
- Failure of $L^p$-boundedness without strong boundary regularity
- Impact of convexity assumptions on the transform's properties

## Abstract

The purpose of this paper is to complement the results in [LS-1] by showing the dense definability of the Cauchy-Leray transform for the domains that give the counterexamples of [LS-1], where $L^p$-boundedness is shown to fail when either the "near" $C^2$ boundary regularity, or the strong $\mathbb C$-linear convexity assumption is dropped.