Compactness estimates for $\Box_b$ on a CR manifold

This paper aims to state compactness estimates for the Kohn-Laplacian on an abstract CR manifold in full generality. The approach consists of a tangential basic estimate in the formulation given by the first author in his thesis, which refines former work by Nicoara. It has been proved by Raich that on a CR manifold of dimension $2n-1$ which is compact pseudoconvex of hypersurface type embedded in the complex Euclidean space and orientable, the property named ``$(CR-P_q)$'' for $1\leq q\leq \frac {n-1}2$, a generalization of the one introduced by Catlin, implies compactness estimates for the Kohn-Laplacian $\Box _b$ in any degree $k$ satisfying $q\leq k\leq n-1-q$. The same result is stated by Straube without the assumption of orientability. We regain these results by a simplified method and extend the conclusions to CR manifolds which are not necessarily embedded nor orientable. In this general setting, we also prove compactness estimates in degree $k=0$ and $k=n-1$ under the assumption of $(CR-P_1)$ and, when $n=2$, of closed range for ${\bar \partial }_b$. For $n\geq 3$, this refines former work by Raich and Straube and separately by Straube.