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Proceedings of the American Mathematical Society

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Compactness estimates for $\Box _b$ on a CR manifold
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by Tran Vu Khanh, Stefano Pinton and Giuseppe Zampieri PDF
Proc. Amer. Math. Soc. 140 (2012), 3229-3236 Request permission


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
  • MR Author ID: 815734
  • Email:
  • Stefano Pinton
  • Affiliation: Dipartimento di Matematica, Università di Padova, via Trieste 63, 35121 Padova, Italy
  • Email:
  • Giuseppe Zampieri
  • Affiliation: Dipartimento di Matematica, Università di Padova, via Trieste 63, 35121 Padova, Italy
  • Email:
  • 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
  • © Copyright 2012 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 140 (2012), 3229-3236
  • MSC (2010): Primary 32W05, 32W10, 32T25
  • DOI:
  • MathSciNet review: 2917095