H\"older estimates for homotopy operators on strictly pseudoconvex domains with $C^2$ boundary

TL;DR

This paper introduces a new homotopy formula for strictly pseudoconvex domains with $C^2$ boundary in complex space, providing Lipschitz space estimates for homotopy operators and boundary regularity results.

Contribution

It develops a novel homotopy formula for such domains and establishes Lipschitz space estimates for the associated operators, advancing regularity theory in complex analysis.

Findings

Derived a new homotopy formula for $C^2$ boundary domains.

Established Lipschitz space estimates for homotopy operators.

Applied estimates to boundary regularity of solutions in Levi-flat spaces.

Abstract

We derive a new homotopy formula for a strictly pseudoconvex domain of $C^2$ boundary in ${\mathbf C}^n$ by using a method of Lieb and Range and obtain estimates in Lipschitz spaces for the homotopy operators. For $r>1$ and $q>0$, we obtain a $\Lambda_{r+{1}/{2}}$ solution $u$ to $\overline\partial u=f$ for $\overline\partial$-closed $(0,q)$ forms $f$ of class $\Lambda_{r}$ on the domain. We apply the estimates to obtain boundary regularities of $\mathcal D$-solutions for a domain in the Levi-flat Euclidean space.