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We prove estimates for solutions of the $\bar \partial u=\omega $ equation ina strictly pseudo convex domain $ \Omega $ in ${\mathbb{C}}^{n}.$ For instanceif 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 Lipschitzestimates for $r\geq 2(n+1).$ These results were already done by S. Krantz inthe 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 proposeanother approach, based on Carleson measures of order $\alpha $ and on thesubordination lemma.

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