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

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.