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The complex Green operator on CR-submanifolds of $ \mathbb{C}^{n}$ of hypersurface type: Compactness


Author: Emil J. Straube
Journal: Trans. Amer. Math. Soc. 364 (2012), 4107-4125
MSC (2010): Primary 32W10, 32V99
DOI: https://doi.org/10.1090/S0002-9947-2012-05510-3
Published electronically: March 15, 2012
MathSciNet review: 2912447
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Abstract: We establish compactness estimates for $ \overline {\partial }_{b}$ on a compact pseudoconvex CR-submanifold of $ \mathbb{C}^{n}$ of hypersurface type that satisfies property(P). When the submanifold is orientable, these estimates were proved by A. Raich in 2010 using microlocal methods. Our proof deduces the estimates from (a slight extension, when $ q>1$, of) those known on hypersurfaces via the fact that locally, CR-submanifolds of hypersurface type are CR-equivalent to a hypersurface. The relationship between two potential theoretic conditions is also clarified.


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Additional Information

Emil J. Straube
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
Email: straube@math.tamu.edu

DOI: https://doi.org/10.1090/S0002-9947-2012-05510-3
Keywords: Complex Green operator, $\overline{\partial}_b$, compactness, property(P), CR-submanifold of hypersurface type, pseudoconvex CR-submanifold
Received by editor(s): July 5, 2010
Received by editor(s) in revised form: August 9, 2010
Published electronically: March 15, 2012
Additional Notes: This research was supported in part by NSF grant DMS 0758534
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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