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 $$W^{k,p}_{\beta }(D)$$ for a homotopy operator on a bounded strictly pseudoconvex domain D of $$C^2$$ boundary in $${\mathbb {C}}^n$$ . As a result, we show that given any $$2n< p < \infty $$ , $$k > 1$$ , $$q \ge 1$$ , and a $$\overline{\partial }$$ -closed (0, q) form $$\varphi $$ of class $$W^{k,p}(D)$$ , there exists a solution u to $$\overline{\partial } u = \varphi $$ , such that $$u \in W^{k,p}_{\frac{1}{2} - \epsilon }(D)$$ for any $$\epsilon > 0$$ . If $$k=1$$ , then we can take p to be any value between 1 and $$\infty $$ . In other words, the solution gains almost $$\frac{1}{2}$$ -derivative in a suitable sense.