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

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.