Subellipticity, compactness, H^ estimates and regularity for on weakly q-pseudoconvex/concave domains

For a weakly q -pseudoconvex (resp. q -pseudoconcave) domain in a Stein manifold X of dimension n, we give a sufficient condition for subelliptic estimates for the -Neumann problem. Moreover, we study the compactness of the -Neumann operator N on. Such compactness estimates immediately lead to smoothness of solutions, the closed range property, the L^2 -setting and the Sobolev estimates of N on for any -closed (r, k) -form with k q (resp. k q). Furthermore, we study the -problem with support conditions in for forms of type (r, k), with values in a holomorphic vector bundle. Applications to the ₁ -problem for smooth forms on boundaries of are given.