Sobolev regularity of the canonical solutions to $\bar\partial$ on product domains

TL;DR

This paper proves that the canonical solution operator for the ar equation on product domains in complex space is bounded in Sobolev spaces, establishing sharp regularity results with implications for complex analysis.

Contribution

It demonstrates Sobolev regularity of the ar solution operator on product domains, extending understanding of regularity in several complex variables.

Findings

Boundedness of the ar solution operator in Sobolev spaces on product domains

Sharpness of Sobolev regularity results via Kerzman-type examples

Extension of regularity results to higher-order Sobolev spaces

Abstract

Let $\Omega$ be a product domain in $\mathbb C^n, n\ge 2$, where each slice has smooth boundary. We observe that the canonical solution operator for the $\bar\partial$ equation on $\Omega$ is bounded in $W^{k,p}(\Omega)$, $k\in \mathbb Z^+, 1<p<\infty$. This Sobolev regularity is sharp in view of Kerzman-type examples.