Journal of the
Korean Mathematical Society JKMS

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

Keywords: $\overline\partial$, $\overline\partial_b$ and $\overline\partial$-Neumann operators, pseudoconvex domains, CR manifolds

MSC numbers: 32F10, 32W05, 32W10, 35J20, 35J60