Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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 $k=n-1$.

21
The categories of this paper are: adap-org
3
The categories of this paper are: physics
36
The categories of this paper are: cmp-lg
0
The categories of this paper are: hep-ph
11
The categories of this paper are: hep-lat
5
The categories of this paper are: gr-qc
6
The categories of this paper are: astro-ph
28
The categories of this paper are: alg-geom
33
The categories of this paper are: plasm-ph
22
The categories of this paper are: chao-dyn
29
The categories of this paper are: funct-an
25
The categories of this paper are: patt-sol
15
The categories of this paper are: nucl-ex
32
The categories of this paper are: atom-ph
16
The categories of this paper are: stat
19
The categories of this paper are: eess