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, $k\ge 1$. We construct a $\overline{\partial }$ solution operator (depending on $k$) that gains $\frac{1}{2}$ 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 $\frac{1}{2}$ 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.

arXiv comment25 pages. To appear in Trans. Amer. Math. Soc

References in corpus (1)

• Sobolev $\frac{1}{2}$ estimates for $\overline{\partial }$ equations on strictly pseudoconvex domains with $C^2$ boundary