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http://dx.doi.org/10.4134/JKMS.j140768

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

Saber, Sayed (Mathematics Department Faculty of Science Beni-Suef University)
Publication Information
Journal of the Korean Mathematical Society / v.53, no.6, 2016 , pp. 1211-1223 More about this Journal
Abstract
Let M be an n-dimensional $K{\ddot{a}}hler$ 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) $\bar{\partial}$-equation with support conditions in ${\Omega}$ and the closed range property of ${\bar{\partial}}$ on ${\Omega}$. Applications to the ${\bar{\partial}}$-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 ${\bar{\partial}}$-Neumann problem and the Bergman projection are regular in the Sobolev space $W^{\ell}({\Omega})$ for ${\ell}$ < ${\ell}_0$.
Keywords
${\bar{\partial}}$; ${\bar{\partial}}_b$ and ${\bar{\partial}}$-Neumann operators; pseudoconvex domains; CR manifolds;
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