Compactness of $\Box_b$ in a CR manifold

This note is aimed at simplifying current literature about compactness estimates for the Kohn-Laplacian on CR manifolds. The approach consists in a tangential basic estimate in the formulation given by the first author in \cite{Kh10} which refines former work by Nicoara \cite{N06}. It has been proved by Raich \cite{R10} that on a CR manifold of dimension $2n-1$ which is compact pseudoconvex of hypersurface type embedded in $\C^n$ 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 in \cite{C84}, implies compactness estimates for the Kohn-Laplacian $\Box_b$ in degree $k$ for any $k$ satisfying $q\leq k\leq n-1-q$. The same result is stated by Straube in \cite{S10} without the assumption of orientability. We regain these results by a simplified method and extend the conclusions in two directions. First, the CR manifold is no longer required to be embedded. Second, when $(CR-P_q)$ holds for $q=1$ (and, in case $n=1$, under the additional hypothesis that $\dib_b$ has closed range on functions) we prove compactness also in the critical degrees $k=0$ and $k=n-1$.

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