# The Cauchy-Leray integral: counter-examples to the $L^p$-theory

## TL;DR

This paper demonstrates the necessity of certain geometric and regularity conditions for the $L^p$-boundedness of the Cauchy-Leray integral in complex spaces, through explicit counter-examples.

## Contribution

It establishes the optimality of previously known hypotheses by providing elementary counter-examples that show the necessity of strong $ ext{C}$-linear convexity and regularity conditions.

## Key findings

- Strong $ ext{C}$-linear convexity is necessary for $L^p$-boundedness.
- Regularity of order 2 is required for the $L^p$-theory.
- Counter-examples demonstrate the failure of boundedness without these conditions.

## Abstract

We prove the optimality of the hypotheses guaranteeing the $L^p$-boundedness for the Cauchy-Leray integral in $\mathbb C^n$, $n\geq 2$, obtained in [LS-4]. Two domains, both elementary in nature, show that the geometric requirement of strong $\mathbb C$-linear convexity, together with regularity of order 2, are both necessary.