Regularity equivalence of the Szeg\"o projection and the complex Green operator

TL;DR

This paper establishes a precise equivalence between the regularity of the complex Green operator and certain Szeg"o projections on CR manifolds satisfying the weak Y(q) condition, extending classical results to new geometric contexts.

Contribution

It proves a new equivalence of regularity conditions for the complex Green operator and Szeg"o projections on CR manifolds, extending classical results to weak Z(q) domains on Stein manifolds.

Findings

Regularity of the Green operator is equivalent to regularity of Szeg"o projections.

The third projection S'_{q+1} coincides with the Szeg"o projection S_{q+1} if harmonic forms are trivial.

Extension of classical regularity equivalence results to weak Z(q) domains on Stein manifolds.

Abstract

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.