Optimal H\"{o}lder regularity for the $\bar\partial$ problem on product domains in $\mathbb C^2$

TL;DR

This paper establishes the optimal Hölder regularity for solutions to the ar problem on product domains in b^2, demonstrating the existence of bounded solution operators that preserve Hölder spaces.

Contribution

It proves the existence of bounded solution operators with optimal regularity for the ar problem on product domains in b^2, extending regularity results.

Findings

Existence of bounded solution operators in Hölder spaces.

Optimal regularity results confirmed by Stein-Kerzman example.

Regularity preservation for the ar problem on product domains.

Abstract

The note concerns the $\bar\partial$ problem on product domains in $\mathbb C^2$. We show that there exists a bounded solution operator from $C^{k, \alpha}$ into itself, $k\in \mathbb Z^+\cup \{0\}, 0<\alpha< 1$. The regularity result is optimal in view of an example of Stein-Kerzman.