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.

12 pages