Supnorm estimates for $\bar\partial$ on product domains in $\mathbb{C}^n$

Abstract

Let $\Omega\subset\mathbb{C}^n$ be a product of one-dimensional open bounded domains with $C^{1,\alpha}$ boundary, where $0<\alpha<1$. Using methods from complex analysis in one variable, we construct an integral operator that solves $\bar\partial$ in $\Omega$ with supnorm estimates when the datum is in $C^{n-1,\alpha}(\Omega)$.