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 $\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.

Mots clés: d_bar; compact support

Domaines: Variables complexes [math.CV]

Eric Amar. AN ANDREOTTI-GRAUERT THEOREM WITH $L^r$ ESTIMATES.. 2012. ⟨hal-00676110v7⟩