Subellipticity, compactness, $H^{\epsilon}$ 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 \geqslant q (resp. k \leqslant 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.