Solutions of the $\bar{\partial }$-equation on Stein and on Kähler manifold with compact support

Abstract

We study the $\bar{\partial }$-equation first in Stein manifold then in complete Kähler 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 Kähler 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 Kähler manifold. This uses and improves, in special cases, our result on Andreotti-Grauert type theorem.

arXiv commentA Section on Stein manifold was added to clarify the links with a previous work done by C. Laurent-Tiebaut. So the title was also changed