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.