Exact sequences and estimates for the $\overline{\partial}$-problem

TL;DR

This paper investigates Sobolev estimates for solutions to the inhomogeneous Cauchy-Riemann equations on annuli in complex space, establishing exact sequences relating cohomology of the annulus, envelope, and hole, and providing solutions with prescribed support and Sobolev estimates.

Contribution

It introduces a novel method to relate Dolbeault cohomology of annuli with that of their parts, enabling precise Sobolev estimates and solutions with prescribed support.

Findings

Established exact sequences relating cohomology of annuli, envelope, and hole.

Derived Sobolev estimates for solutions of the $ar{ ext{d}}$-problem.

Constructed solutions with prescribed support and Sobolev bounds.

Abstract

We study Sobolev estimates for solutions of the inhomogenous Cauchy-Riemann equations on annuli in $\cx^n$, by constructing exact sequences relating the Dolbeault cohomology of the annulus with respect to Sobolev spaces of forms with those of the envelope and the hole. We also obtain solutions with prescibed support and estimates in Sobolev spaces using our method.