On estimates for the ∂ˉ\displaystyle \bar \partial equation in Stein manifolds

We generalize to intersection of strictly $c$ -convex domains in Stein manifold, $L^{r}-L^{s}$ and Lipschitz estimates for the solutions of the $\bar \partial$ equation done by Ma and Vassiliadou for domains in ${\mathbb{C}}^{n}.$ For this we use a Docquier-Grauert holomorphic retraction plus the raising steps method I introduce earlier. This gives results in the case of domains with low regularity, ${\mathcal{C}}^{3},$ for their boundary.Comment: I follow the nice suggestions done by the referee which substantially simplify the proof of theorem 4.1. Also typos are corrected and the presenrtation is slightly modifie

An andreotti-grauert theorem with lrl^r estimates

By a theorem of Andreotti and Grauert if $\omega$ is a $(p,q)$ current, $q < n,$ in a Stein manifold $\displaystyle \Omega ,\ \bar \partial$ closed and with compact support, then there is a solution $u$ to $\bar \partial u=\omega$ still with compact support in $\displaystyle \Omega .$ The main result of this work is to show that if moreover $\displaystyle \omega \in L^{r}(m),$ where $m$ is a suitable Lebesgue measure on the Stein manifold, then we have a solution $u$ with compact support {\sl and} in $L^{s}(m),\ \frac{1}{s}=\frac{1}{r}-\frac{1}{2(n+1)}.$We prove it by estimates in$L^{r}$ spaces with weights.Comment: Thanks to the referee, the presentation is highly enhanced and some typos are fixed. This will appear in:Annali de la Scuola Norm. Sup. di Pis