COMPACTNESS ESTIMATES FOR square(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 <= q <= n-1/2, a generalization of the one introduced by Catlin, implies compactness estimates for the Kohn-Laplacian square(b) in any degree k satisfying q <= k <= 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 partial derivative(b). For n >= 3, this refines former work by Raich and Straube and separately by Straube.