On $\overline\partial$ homotopy formulae for product domains: Nijenhuis-Woolf's formulae and optimal Sobolev estimates

TL;DR

This paper develops homotopy formulae for the $ar{ ext{d}}$ operator on product domains with various boundary conditions, providing solutions with optimal Sobolev regularity across all derivatives and integrability levels.

Contribution

It introduces new homotopy operators for product domains that achieve optimal Sobolev estimates for the $ar{ ext{d}}$ equation, extending previous results to more general boundary types.

Findings

Constructed homotopy formulae for product domains.

Achieved solutions with optimal Sobolev regularity.

Applicable to domains with Lipschitz, pseudoconvex, and convex boundaries.

Abstract

We construct homotopy formulae $f=\overline\partial\mathcal H_qf+\mathcal H_{q+1}\overline\partial f$ for $(0,q)$ forms on the product domain $\Omega_1\times\dots\times\Omega_m$, where each $\Omega_j$ is either a bounded Lipschitz domain in $\mathbb C^1$, a bounded strongly pseudoconvex domain with $C^2$ boundary, or a smooth convex domain of finite type. Such homotopy operators $\mathcal H_q$ yield solutions to the $\overline\partial$ equation with optimal Sobolev regularity $W^{k,p}\to W^{k,p}$ simultaneously for all $k\in\mathbb Z$ and $1<p<\infty$.