Existence theorems for the dbar equation and Sobolev estimates on q-convex domains

In this paper, we study a sufficient condition for subelliptic estimates in the weak Z(k) domain with C3 boundary in an n-dimentionsl Stein manifold X. Consequently, the compactness of the a-Neumann operator N on M is obtained and the closedness ranges of a and a* are presented. The L2-setting and the Sobolev estimates of N on M are proved. We study the a problem with support conditions in M for Xi-valued (p, k) forms, where Xi is the m-times tensor product of holomorphic line bundle Xi circle times m for positive integer m. Moreover, the compactness of the weighted a-Neumann operator is studied on an annular domain in a Stein manifold M = (sic)M1\(sic)M2, between two smooth bounded domains M1 and M2 satisfy (sic)M2 c= M1, M1 is weak Z(k), M2 is weak Z(n - 1 - k), 1 < k < n - 2 with n > 3. In all cases, the closedness of a and a*, global boundary regularity for a and ab are studied.