Regularity equivalence of the Szegö projection and the complex Green operator

In this paper we prove that on a CR manifold of hypersurface type that satisfies the weak condition, the complex Green operator is exactly (globally) regular if and only if the Szegö projections and a third orthogonal projection are exactly (globally) regular. The projection is closely related to the Szegö projection and actually coincides with it if the space of harmonic -forms is trivial. This result extends the important and by now classical result by H. Boas and E. Straube on the equivalence of the regularity of the -Neumann operator and the Bergman projections on a smoothly bounded pseudoconvex domain. We also prove an extension of this result to the case of bounded smooth domains satisfying the weak condition on a Stein manifold.