Optimal $L^p$ regularity for $\bar\partial$ on the Hartogs triangle

TL;DR

This paper establishes sharp weighted $L^p$ estimates for the $ar ext{d}$ equation on product domains, especially the Hartogs triangle, demonstrating optimal regularity results and providing explicit counterexamples.

Contribution

It proves weighted $L^p$ estimates for the canonical solutions on product domains and verifies the sharpness of $L^p$ regularity on the Hartogs triangle.

Findings

$L^p$ solutions exist for $p ext{ in } [4, \infty)$ on the Hartogs triangle.

Constructs examples showing no $L^{p+ ext{epsilon}}$ solutions exist, confirming sharpness.

Provides weighted $L^p$ estimates for solutions on product domains.

Abstract

In this paper, we prove weighted $L^p$ estimates for the canonical solutions on product domains. As an application, we show that if $p\in [4, \infty)$, the $\bar\partial$ equation on the Hartogs triangle with $L^p$ data admits $L^p$ solutions with the desired estimates. For any $\epsilon>0$, by constructing an example with $L^p$ data but having no $L^{p+\epsilon}$ solutions, we verify the sharpness of the $L^p$ regularity on the Hartogs triangle.