Solving the Kerzman's problem on the sup-norm estimate for $\bar\partial$ on product domains

TL;DR

This paper resolves a long-standing open problem by establishing sup-norm estimates for the Cauchy-Riemann equation on polydiscs and bounded product domains in complex space, advancing understanding in several complex variables.

Contribution

It provides the first solution to Kerzman's problem on sup-norm estimates for $ar{ ext{d}}$ on polydiscs and extends this to bounded product domains with smooth boundaries.

Findings

Solved Kerzman's problem on polydiscs in complex space.

Extended the solution to bounded product domains with $C^{1,eta}$ boundaries.

Established new sup-norm estimates for the $ar{ ext{d}}$ operator.

Abstract

In this paper, the author solves the long term open problem of Kerzman on sup-norm estimate for Cauchy-Riemann equation on polydisc in $n$-dimensional complex space. The problem has been open since 1971. He also extends and solves the problem on a bounded product domain $\Omega^n$, where $\Omega$ is any bounded domain in $\mathbb{C}$ with $C^{1,\alpha}$ boundary for some $\alpha>0$.