Uniform estimates of the Cauchy-Riemann equation on product domains

TL;DR

This paper proves that the continuity assumption on functions is unnecessary for uniform estimates of the Cauchy-Riemann equation on product domains, extending $L^p$ estimates to all p in [1, ∞].

Contribution

It shows that boundedness suffices for uniform estimates, resolving a question from 1971 and broadening the scope of $L^p$ estimates for the $ar ext{d}$ operator.

Findings

Continuity assumption can be replaced by boundedness.

Uniform estimates hold under weaker conditions.

$L^p$ estimates are valid for all p in [1, ∞].

Abstract

We observe that the continuity assumption on $f$ for the uniform estimates of the canonical solution to $\bar\partial u = f$ on products of $C^2$ bounded planar domains in \cite{DPZ} can be reduced to the boundedness assumption. This completely answers the original question raised by Kerzman in 1971. Moreover, the $L^p$ estimates of $\bar\partial$ is obtained for all $p \in [1, \infty]$.