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.
Categories: math.CV