Subellipticity, compactness, $H^{ϵ}$ estimates and regularity for $\bar{\partial }$ on weakly $q$-pseudoconvex/concave domains

Abstract

For a weakly $q$-pseudoconvex (resp. $q$-pseudoconcave) domain $\Omega$ in a Stein manifold $X$ of dimension $n$, we give a sufficient condition for subelliptic estimates for the $\bar{\partial }$-Neumann problem. Moreover, we study the compactness of the $\bar{\partial }$-Neumann operator $N$ on $\Omega$. Such compactness estimates immediately lead to smoothness of solutions, the closed range property, the $L^2$-setting and the Sobolev estimates of $N$ on $\Omega$ for any $\bar{\partial }$-closed $(r,k)$-form with $k⩾q$ (resp. $k⩽q$). Furthermore, we study the $\bar{\partial }$-problem with support conditions in $\Omega$ for forms of type $(r,k)$, with values in a holomorphic vector bundle. Applications to the ${\bar{\partial }}_b$-problem for smooth forms on boundaries of $\Omega$ are given.