The complex Green operator on CR-submanifolds of ${\mathbb{C}}^n$ of hypersurface type: compactness

We establish compactness estimates for ${\bar{\mathrm{\partial }}}_b$ on a compact pseudoconvex CR-submanifold of ${\mathbb{C}}^n$ of hypersurface type that satisfies property(P). When the submanifold is orientable, these estimates were proved by A.~Raich via microlocal methods. Our proof deduces the estimates from (a slight extension, when $q>1$, of) those known on hypersurfaces via the fact that locally, CR-submanifolds of hypersurface type are CR-equivalent to a hypersurface. The relationship between two potential theoretic conditions is also clarified.