  In this paper we prove that on a CR manifold of hypersurface type that
satisfies the weak $Y(q)$ condition, the complex Green operator $G_q$ is
exactly (globally) regular if and only if the Szeg\"o projections $S_{q-1},
S_q$ and a third orthogonal projection $S'_{q+1}$ are exactly (globally)
regular. The projection $S'_{q+1}$ is closely related to the Szeg\"o projection
$S_{q+1}$ and actually coincides with it if the space of harmonic
$(0,q+1)$-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 $\bar\partial$-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 $Z(q)$ condition on a Stein manifold.
