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

Abstract

We prove the optimality of the hypotheses guaranteeing the $L^p$-boundedness for the Cauchy-Leray integral in ${\mathbb{C}}^n$, $n\ge 2$, 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.