AMAR, Eric

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AMAR, Eric
Institut de Mathématiques de Bordeaux [IMB]

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We study the $\bar \partial $-equation first in Stein manifold then in complete K\"ahler manifolds. The aim is to get $L^{r}$ and Sobolev estimates on solutions with compact support. In the Stein case we get that for any $(p,q)$-form $\omega $ in $L^{r}$ with compact support and $\bar \partial $-closed there is a $(p,q-1)$-form $u$ in $W^{1,r}$ with compact support and such that $\bar \partial u=\omega .$ In the case of K\"ahler manifold, we prove and use estimates on solutions on Poisson equation with compact support and the link with $\bar \partial $ equation is done by a classical theorem stating that the Hodge laplacian is twice the $\bar \partial $ (or Kohn) Laplacian in a K\"ahler manifold. This uses and improves, in special cases, our result on Andreotti-Grauert type theorem.< Réduire

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Solutions of the $\bar \partial $-equation on Stein and on K\"ahler manifold with compact support

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Résumé en anglais

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