Khanh Vu Tran

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

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 $p \in (1, 4)$.