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

Abstract

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

Thanks to the referee, the presentation is highly enhanced and some typos are fixed. This will appear in:Annali de la Scuola Norm. Sup. di Pisa