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-$ P_q $ 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.
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