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

We establish compactness estimates for $\overline {\partial }_{b}$ 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 in 2010 using 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.