A Solution Operator for the $\overline\partial$ Equation in Sobolev Spaces of Negative Index

TL;DR

This paper constructs a $ar{ ext{d}}$ solution operator in Sobolev spaces with negative index for strictly pseudoconvex domains, gaining half a derivative, using homotopy formulas and novel anti-derivative techniques.

Contribution

It introduces a new $ar{ ext{d}}$ solution operator that gains half a derivative in Sobolev spaces of negative index, applicable to domains with limited boundary regularity.

Findings

Constructed solution operators gain 0.5 derivative in Sobolev spaces.

Operators work for domains with $C^{k+2}$ boundary, $k \\geq 1$.

Extended results to $C^{\\infty}$ domains, gaining 0.5 derivative for all $s$.

Abstract

Let $\Omega$ be a strictly pseudoconvex domain in $\mathbb{C}^n$ with $C^{k+2}$ boundary, $k \geq 1$. We construct a $\overline\partial$ solution operator (depending on $k$) that gains $\frac12$ derivative in the Sobolev space $H^{s,p} (\Omega)$ for any $1<p<\infty$ and $s>\frac{1}{p} -k$. If the domain is $C^{\infty}$, then there exists a $\overline\partial$ solution operator that gains $\frac12$ derivative in $H^{s,p}(\Omega)$ for all $s \in \mathbb{R}$. We obtain our solution operators via the method of homotopy formula. A novel technique is the construction of ``anti-derivative operators'' for distributions defined on bounded Lipschitz domains.