Weighted Sobolev $L^p$ estimates for homotopy operators on strictly pseudoconvex domains with $C^2$ boundary

Abstract

We derive estimates in a weighted Sobolev space $W_{\mu }^{k,p}(D)$ for a homotopy operator on a bounded strictly pseudoconvex domain $D$ of $C^2$ boundary in ${\C}^n$. As a result, we show that given any $2n<p<\infty$, $k>1$, $q\ge 1$, and a $\dbar$-closed $(0,q)$ form $\var$ of class $W^{k,p}(D)$, there exist a solution $u$ to $\dbar u = \var$ such that $u \in W^{k,p}_{\yh-\ve}(D)$ for any $\ve > 0$. If $k=1$, then we can take $p$ to be any value between $1$ and $\infty$. In other words, the solution gains almost $\yh$-derivative in a suitable sense.

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