No abstract

In this paper, we investigate the compactness theory of the complex Green operator on smooth, embedded, orientable CR manifolds of hypersurface type that satisfy the weak Y (q) condition. The sufficient condition that we define is an adaption of the CR-Pq property for weak Y (q) manifolds and does not require that the CR manifold is the boundary of a domain.We also provide several non-pseudoconvex examples (and a level q) for which the complex Green operator is compact.

The purpose of this paper is to establish sufficient conditions for closed range estimates on (0, q)-forms, for some fixed q, 1 ≤ q ≤ n − 1, for ∂b in both L 2 and L 2 -Sobolev spaces in embedded, not necessarily pseudoconvex CR manifolds of hypersurface type. The condition, named weak Y (q), is both more general than previously established sufficient conditions and easier to check. Applications of our estimates include estimates for the Szegö projection as well as an argument that the harmonic forms have the same regularity as the complex Green operator. We use a microlocal argument and carefully construct a norm that is well-suited for a microlocal decomposition of form. We do not require that the CR manifold is the boundary of a domain. Finally, we provide an example that demonstrates that weak Y (q) is an easier condition to verify than earlier, less general conditions.