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On compactness of the $ \overline{\partial}$-Neumann problem and Hankel operators

Authors: Mehmet Çelik and Sönmez Şahutoğlu
Journal: Proc. Amer. Math. Soc. 140 (2012), 153-159
MSC (2010): Primary 32W05; Secondary 47B35
Published electronically: August 29, 2011
MathSciNet review: 2833527
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Abstract: Let $ \Omega=\Omega_1\setminus \overline{\Omega}_2$, where $ \Omega_1$ and $ \Omega_2$ are two smooth bounded pseudoconvex domains in $ \mathbb{C}^n, n\geq 3,$ such that $ \overline{\Omega}_2\subset \Omega_1.$ Assume that the $ \overline{\partial}$-Neumann operator of $ \Omega_1$ is compact and the interior of the Levi-flat points in the boundary of $ \Omega_2$ is not empty (in the relative topology). Then we show that the Hankel operator on $ \Omega$ with symbol $ \phi, H^{\Omega}_{\phi},$ is compact for every $ \phi\in C(\overline{\Omega})$ but the $ \overline{\partial}$-Neumann operator on $ \Omega$ is not compact.

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Additional Information

Mehmet Çelik
Affiliation: Department of Mathematics and Information Sciences, University of North Texas at Dallas, 7300 Houston School Road, Dallas, Texas 75241

Sönmez Şahutoğlu
Affiliation: Department of Mathematics & Statistics, University of Toledo, 2801 West Bancroft Street, Toledo, Ohio 43606

Keywords: $\overline{\partial}$-Neumann problem, Hankel operators, non-pseudoconvex domains
Received by editor(s): August 24, 2010
Published electronically: August 29, 2011
Additional Notes: The second author is supported in part by the University of Toledo’s Summer Research Awards and Fellowships Program
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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