The role of an integration identity in the analysis of the Cauchy-Leray transform (Q724406)

The authors previously proved [Adv. Math. 264, 776-830 (2014; Zbl 1317.32007)] that if \(D\) is a bounded strongly \(\mathbb{C}\)-linearly convex domain in \(\mathbb{C}^{n}\), \(n\geq 2\), with boundary of class \(C^{1,1}\), then the Cauchy-Leray transform defines a bounded mapping on \(L^{p}(bD)\) when \(1< p< \infty\). Subsequently, they constructed two counterexample domains to demonstrate that neither of the main hypotheses on \(D\) can be essentially weakened [``The Cauchy-Leray integral: counterexamples to the \(L^{p}\)-theory'', Preprint, \url{arxiv:1701.03812}]. Now they show that the Cauchy integral of each counterexample domain does transform every function of class \(C^{1}(bD)\) into a function continuous on the closure of \(D\), even though the transformation does not extend to be a bounded operator on \(L^{p}(bD)\).