The $\overline\partial$-problem with support conditions and pseudoconvexity of general order in K\"ahler manifolds
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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 $\ell<\ell_{0}$.
Abstract
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 $\ell<\ell_{0}$.
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