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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

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.