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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Compactness estimates for $ \Box_b$ on a CR manifold


Authors: Tran Vu Khanh, Stefano Pinton and Giuseppe Zampieri
Journal: Proc. Amer. Math. Soc. 140 (2012), 3229-3236
MSC (2010): Primary 32W05, 32W10, 32T25
Published electronically: January 25, 2012
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Abstract: 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.


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Additional Information

Tran Vu Khanh
Affiliation: Tan Tao University, Tan Tao University Avenue, Duc Hoa District, Long An Prov- ince, Vietnam
Email: khanh.tran@ttu.edu.vn

Stefano Pinton
Affiliation: Dipartimento di Matematica, Università di Padova, via Trieste 63, 35121 Padova, Italy
Email: pinton@math.unipd.it

Giuseppe Zampieri
Affiliation: Dipartimento di Matematica, Università di Padova, via Trieste 63, 35121 Padova, Italy
Email: zampieri@math.unipd.it

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11190-6
PII: S 0002-9939(2012)11190-6
Received by editor(s): December 30, 2010
Received by editor(s) in revised form: March 29, 2011
Published electronically: January 25, 2012
Communicated by: Franc Forstneric
Article copyright: © Copyright 2012 American Mathematical Society