Optimal \(L^p\) Regularity for \(\bar{\partial }\) on the Hartogs Triangle

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