$\overline{\partial}$-Estimates on the product of bounded Lipschitz domain

TL;DR

This paper constructs an integral operator to solve the $ar{ ext{d}}$-equation on product domains with Lipschitz boundaries, providing $L^p$ estimates for solutions in complex analysis.

Contribution

It introduces a new integral solution operator for the $ar{ ext{d}}$-problem on product domains with Lipschitz boundaries, extending $L^p$ estimates to these settings.

Findings

Constructed an integral solution operator for $ar{ ext{d}}$-equation.

Established $L^p$ estimates for solutions on product domains.

Extended $ar{ ext{d}}$-problem solutions to Lipschitz boundary domains.

Abstract

Let $D$ be a bounded domain in the complex plane with Lipschitz boundary. In the paper, we construct an integral solution operator $T[f]$ for any $\overline{\partial}$ closed $(0,1)$-form $f\in L^p_{(0,1)}(D^n)$ solving the Cauchy-Riemain equation $\overline{\partial} u=f$ on the product domains $D^n$ and obtain the $L^p$-estimates for all $1<p\le \infty$.