Estimates $ L^{r}-L^{s}$ for solutions of the $\bar \partial $ equation in strictly pseudo convex domains in ${\mathbb{C}}^{n}.$

Abstract

We prove estimates for solutions of the $\bar \partial u=\omega $ equation in a strictly pseudo convex domain $ \Omega $ in ${\mathbb{C}}^{n}.$ For instance if the $ (p,q)$ current $\omega $ has its coefficients in $L^{r}(\Omega )$ with $1\leq r<2(n+1)$ then there is a solution $u$ in $L^{s}(\Omega )$ with $\ \frac{1}{s}=\frac{1}{r}-\frac{1}{2(n+1)}.$ We also have $BMO$ and Lipschitz estimates for $r\geq 2(n+1).$ These results were already done by S. Krantz in the case of $(0,1)$ forms and just for the $L^{r}-L^{s}$ part by L. Ma and S. Vassiliadou for general $(p,q)$ forms. To get the complete result we propose another approach, based on Carleson measures of order $\alpha $ and on the subordination lemma.