AN ANDREOTTI-GRAUERT THEOREM WITH $L^r$ ESTIMATES.
Résumé
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 main result 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 $L^{r}(dm).$ We prove it by estimates in $L^{r}$ spaces with weights.\ \par In a second part, we prove directly that there are global $L^{r,loc}(dm)-L^{r,loc}(dm)$ solutions for the $\bar \partial $ equation on Stein manifolds. This gives, again by duality, another proof for the main result.
Domaines
Variables complexes [math.CV]
Origine : Fichiers produits par l'(les) auteur(s)