Subellipticity, compactness, Hε estimates and regularity for ∂̄ on weakly q-pseudoconvex/concave domains

For a weakly q -pseudoconvex (resp. q -pseudoconcave) domain Ω 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 Ω . Such compactness estimates immediately lead to smoothness of solutions, the closed range property, the L² -setting and the Sobolev estimates of N on Ω for any ∂̄ -closed (r, k) -form with k q (resp. k q ). Furthermore, we study the ∂̄ -problem with support conditions in Ω for forms of type (r, k) , with values in a holomorphic vector bundle. Applications to the ∂̄b -problem for smooth forms on boundaries of Ω are given.