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

Let $\mathrm{\Omega }\subset {\mathbb{C}}^n$ 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 $\bar{\mathrm{\partial }}$ in $\mathrm{\Omega }$ with supnorm estimates when the datum is in $C^{n-1,\alpha }(\mathrm{\Omega })$.