Abstract
Let X be a Stein manifold of dimension n and let Omega be a bounded pseudoconvex domain with smooth boundary b Omega in X. If 1 <= q <= n - 2, n >= 3 and if b Omega satisfies both (P-q) and (Pn-q-1), then the Green operator G(q) is a compact operator (and so is G(n-q-1)). Moreover, we show that the compactness in the partial derivative-Neumann problem on locally convexiable domains, yield the corresponding characterization of compactness of the complex Green operator(s) on these domains.