The Cauchy Transform and Henkin Operator on Convex Domains of Maximal Type F$F$ in ℂ2$\mathbb {C}^{2}$

Abstract

Let ${\Omega }$ be a smoothly bounded, convex domain in $\mathbb {C}^{2}$, satisfying the maximal type $F$. In this paper, we consider the boundary Lipschitz regularity and Gevrey regularity of the Cauchy transform $\mathcal {C}[u]$ on ${\Omega }$, with an application of the Henkin operator for $\bar {\partial }$-equation. Here, the notion of maximal type $F$ contains all domains of strictly finite type and many cases of infinite type in the sense of Range.