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

TL;DR

This paper resolves Kerzman's long-standing open problem by establishing sup-norm estimates for the Cauchy-Riemann equation on polydiscs and certain bounded product domains in complex spaces.

Contribution

It provides the first solution to Kerzman's problem on sup-norm estimates for ar on polydiscs and extends the results to broader product domains with specific boundary conditions.

Findings

Established sup-norm estimates for ar on polydiscs in imensional complex space.

Extended the estimates to bounded product domains with smooth or piecewise smooth boundaries.

Solved a problem open since 1971, advancing complex analysis and PDE theory.

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 $\Omega$ either is simply connected with $C^{1,\alpha}$ boundary or satisfies a uniform exterior ball condition with piecewise $C^1$ boundary.