On compactness of the $\overline {\partial }$-Neumann problem and Hankel operators
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- by Mehmet Çeli̇k and Sönmez Şahutoğlu PDF
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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.References
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Additional Information
- Mehmet Çeli̇k
- Affiliation: Department of Mathematics and Information Sciences, University of North Texas at Dallas, 7300 Houston School Road, Dallas, Texas 75241
- MR Author ID: 869210
- Email: Mehmet.Celik@unt.edu
- Sönmez Şahutoğlu
- Affiliation: Department of Mathematics & Statistics, University of Toledo, 2801 West Bancroft Street, Toledo, Ohio 43606
- ORCID: 0000-0003-0490-0113
- Email: sonmez.sahutoglu@utoledo.edu
- 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
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 140 (2012), 153-159
- MSC (2010): Primary 32W05; Secondary 47B35
- DOI: https://doi.org/10.1090/S0002-9939-2011-11350-9
- MathSciNet review: 2833527