# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1702.08872