Compactness of the complex Green operator on CR-manifolds of hypersurface type

Abstract

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 of real dimension at least five satisfies (CR-P q ) and (CR-P n-1-q), then the complex Green operator is a compact operator on the Sobolev spaces $${H^s_{0,q}(M)}$$ and $${H^s_{0,n-1-q}(M)}$$ , if 1 ≤ q ≤ n−2 and s ≥ 0. We use CR-plurisubharmonic functions to build a microlocal norm that controls the totally real direction of the tangent bundle.