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

TL;DR

This paper develops supnorm estimates for the $ar ext{d}$ operator on product domains in complex space, using one-variable complex analysis techniques to construct solutions with controlled norms.

Contribution

It introduces a new integral operator approach that provides supnorm estimates for $ar ext{d}$ on product domains with $C^{1,eta}$ boundaries, extending previous results.

Findings

Constructed an integral operator solving $ar ext{d}$ with supnorm bounds.

Achieved solutions for data in $C^{n-1,eta}( ext{domain})$.

Extended complex analysis techniques to higher-dimensional product domains.

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 $C^{n-1,\alpha}(\Omega)$.