We study the $$\bar{\partial }$$ -equation first in Stein manifold then in complete Kahler 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 Kahler 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 Kahler manifold. This uses and improves, in special cases, our result on Andreotti–Grauert-type theorem.