  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.
