On estimates for the $\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.