Solutions of the $\bar \partial $-equation on Stein and on K\"ahler manifold with compact support

We study the $\bar \partial$-equation first in Stein manifold then in complete K\"ahler manifolds. The aim is to get $L^{r}$ and Sobolev estimates on solutions with compact support. In the Stein case we get that for any $(p,q)$-form $\omega$ in $L^{r}$ with compact support and $\bar \partial$-closed there is a $(p,q-1)$-form $u$ in $W^{1,r}$ with compact support and such that $\bar \partial u=\omega .$ In the case of K\"ahler manifold, we prove and use estimates on solutions on Poisson equation with compact support and the link with $\bar \partial$ equation is done by a classical theorem stating that the Hodge laplacian is twice the $\bar \partial$ (or Kohn) Laplacian in a K\"ahler manifold. This uses and improves, in special cases, our result on Andreotti-Grauert type theorem.