Compactness estimates for 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

MathSciNet review:
2917095

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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 which is compact pseudoconvex of hypersurface type embedded in the complex Euclidean space and orientable, the property named ``'' for , a generalization of the one introduced by Catlin, implies compactness estimates for the Kohn-Laplacian in any degree satisfying . 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 and under the assumption of and, when , of closed range for . For , 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:
https://doi.org/10.1090/S0002-9939-2012-11190-6

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