AN ANDREOTTI-GRAUERT THEOREM WITH $L^r$ ESTIMATES.

Eric Amar (1)

1 IMB - Institut de Mathématiques de Bordeaux

By a theorem of Andreotti and Grauert if $\omega $ is a $(p,q)$ -current, $q < n,$ in a Stein manifold, $\bar{\partial }$ closed and with compact support, then there is a solution $u$ to $\bar{\partial }u=\omega $ still with compact support. The aim of this work is to show that if moreover $\omega \in L^{r}(dm),$ where $m$ is a suitable Lebesgue measure on the Stein manifold, then we have a solution $u$ with compact support {\sl and} in $L^{r}(dm). As a consequence of the previous result and again by a duality argument we have $L^{r,loc}(dm)-L^{r,loc}(dm)$ solutions for the $\bar \partial $ equation on suitable domains.

Mots clés: d_bar; compact support

Domaines: Variables complexes [math.CV]

Eric Amar. AN ANDREOTTI-GRAUERT THEOREM WITH $L^r$ ESTIMATES.. 2012. ⟨hal-00676110v4⟩