$L^p$ Estimates for the $\bar{\partial }$-Problem on Rational Hartogs Triangles

Abstract

We investigate $L^p$ estimates for the $\bar{\partial }$-problem on rational Hartogs triangles ${\mathbb{H}}_{m/n}=\{(z_1,z_2)\in {\mathbb{C}}^2:|z_1{|}^m<|z_2{|}^n<1\}$. For $p\in (1,\infty )$, we establish the existence of a solution operator that is bounded on $L^p({\mathbb{H}}_{m/n})$. Our approach avoid the need for any {\it a priori} condition on the data. We also show that the canonical solution $K_{{\mathbb{H}}_{m/n}}$ is bounded on $L^p({\mathbb{H}}_{m/n})$ for $p\in (p_0,p_2)$, where $p_0=\frac{2m+2n}{m+n+1+\min \{m,n\}}$ and $p_2=\frac{2m+2n}{m+n-1}$. For classical Hartogs triangle, ${\mathbb{H}}_1$, this establishes boundedness for $p\in (1,4)$.