Compactness of the Complex Green Operator on CR-Manifolds of Hypersurface Type

The purpose of this article is to study compactness of the complex Green operator on CR manifolds of hypersurface type. We introduce (CR-P_q), a potential theoretic condition on $(0,q)$-forms that generalizes Catlin's property (P_q) to CR manifolds of arbitrary codimension. We prove that if an embedded CR-manifold of hypersurface type satisfies (CR-P_q) and (CR-P_{n-1-q}) and is of real dimension at least five, then the complex Green operator is a compact operator on the Sobolev spaces $H^s_{0,q}(M)$, if $1\leq q \leq n-2$ and $s\geq 0$. We use CR-plurisubharmonic functions to build a microlocal norm that controls the totally real direction of the tangent bundle.