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 Y (q) condition, the complex Green operator Gq is exactly (globally) regular if and only if the Szegö projections Sq-1, Sq and a third orthogonal projection S q+1 are exactly (globally) regular. The projection S'q+1 is closely related to the Szegö projection Sq+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 ∂-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. [ABSTRACT FROM AUTHOR]