An andreotti-grauert theorem with $l^r$ estimates

Abstract

By a theorem of Andreotti and Grauert if $ω$ is a $(p,q)$ current, $q < n,$ in a Stein manifold $\displaystyle Ω,\ \bar \partial $ closed and with compact support, then there is a solution $u$ to $\bar \partial u=ω$ still with compact support in $\displaystyle Ω.$ The main result of this work is to show that if moreover $\displaystyle ω\in L^{r}(m),$ 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.