Sobolev and H\"older estimates for the $\overline \partial$ equation on pseudoconvex domains of finite type in $\mathbb C^2$

TL;DR

This paper establishes near-optimal Sobolev and H"older estimates for the $ar{ ext{d}}$ equation on finite type pseudoconvex domains in $ ext{C}^2$, extending previous results through a novel construction of holomorphic support functions.

Contribution

It introduces a new method for constructing holomorphic support functions that enable sharp estimates for the $ar{ ext{d}}$ equation on certain complex domains.

Findings

Almost sharp Sobolev and H"older-Zygmund estimates achieved.

Extension of classical results to finite type pseudoconvex domains.

Novel construction of support functions with precise boundary estimates.

Abstract

We prove a homotopy formula which yields almost sharp estimates in all (positive-indexed) Sobolev and H\"older-Zygmund spaces for the $\overline \partial$ equation on pseudoconvex domains of finite type in $\mathbb C^2$, extending the earlier results of Fefferman-Kohn (1988), Range (1990), and Chang-Nagel-Stein (1992). The main novelty of our proof is the construction of holomorphic support functions that admit precise estimates when the parameter variable lies in a thin shell outside the domain.