THE <TEX>${\bar{\partial}}$</TEX>-PROBLEM WITH SUPPORT CONDITIONS AND PSEUDOCONVEXITY OF GENERAL ORDER IN KÄHLER MANIFOLDS

Abstract

Let M be an n-dimensional <TEX>$K{\ddot{a}}hler$</TEX> manifold with positive holomorphic bisectional curvature and let <TEX>${\Omega}{\Subset}M$</TEX> be a pseudoconvex domain of order <TEX>$n-q$</TEX>, <TEX>$1{\leq}q{\leq}n$</TEX>, with <TEX>$C^2$</TEX> smooth boundary. Then, we study the (weighted) <TEX>$\bar{\partial}$</TEX>-equation with support conditions in <TEX>${\Omega}$</TEX> and the closed range property of <TEX>${\bar{\partial}}$</TEX> on <TEX>${\Omega}$</TEX>. Applications to the <TEX>${\bar{\partial}}$</TEX>-closed extensions from the boundary are given. In particular, for q = 1, we prove that there exists a number <TEX>${\ell}_0$</TEX> > 0 such that the <TEX>${\bar{\partial}}$</TEX>-Neumann problem and the Bergman projection are regular in the Sobolev space <TEX>$W^{\ell}({\Omega})$</TEX> for <TEX>${\ell}$</TEX> < <TEX>${\ell}_0$</TEX>.