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Sobolev estimates for the local extension of $ \bar{\partial}_b$-closed $ (0,1)$-forms on real hypersurfaces in $ \mathbb{C}^n$ with two positive eigenvalues


Author: Sanghyun Cho
Journal: Proc. Amer. Math. Soc. 139 (2011), 4053-4062
MSC (2010): Primary 32V25; Secondary 32W10
DOI: https://doi.org/10.1090/S0002-9939-2011-10828-1
Published electronically: April 11, 2011
MathSciNet review: 2823050
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Abstract: Let $ \mathcal M$ be a smooth real hypersurface in complex space of dimension $ n\ge 3$, and assume that the Levi-form at $ z_0$ on $ \mathcal M$ has at least two positive eigenvalues. We estimate solutions of the local $ \bar{\partial}$-closed extension problem near $ z_0$ for $ (0,1)$-forms in Sobolev spaces. Using this result, we estimate the local solution of tangential Cauchy-Riemann equations near $ z_0$ for $ (0,1)$-forms in Sobolev spaces.


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Additional Information

Sanghyun Cho
Affiliation: Department of Mathematics, Sogang University, Seoul, 121-742, Republic of Korea
Email: shcho@sogang.ac.kr

DOI: https://doi.org/10.1090/S0002-9939-2011-10828-1
Keywords: Tangential Cauchy-Riemann equation, $\bar{\partial}_{b}$-closed extension problem.
Received by editor(s): July 22, 2010
Received by editor(s) in revised form: October 5, 2010
Published electronically: April 11, 2011
Additional Notes: The author was partially supported by KRF-2005-070-C00007 and the Sogang University research fund.
Communicated by: Mei-Chi Shaw
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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