A remark on irregularity of the $\overline {\partial }$-Neumann problem on non-smooth domains
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- by Sönmez Şahutoğlu
- Proc. Amer. Math. Soc. 136 (2008), 2529-2533
- DOI: https://doi.org/10.1090/S0002-9939-08-09206-X
- Published electronically: March 4, 2008
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Abstract:
It is an observation due to J. J. Kohn that for a smooth bounded pseudoconvex domain $\Omega$ in $\mathbb {C}^n$ there exists $s>0$ such that the $\overline {\partial }$-Neumann operator on $\Omega$ maps $W^s_{(0,1)}(\Omega )$ (the space of $(0,1)$-forms with coefficient functions in $L^2$-Sobolev space of order $s$) into itself continuously. We show that this conclusion does not hold without the smoothness assumption by constructing a bounded pseudoconvex domain $\Omega$ in $\mathbb {C}^{2}$, smooth except at one point, whose $\overline {\partial }$-Neumann operator is not bounded on $W^s_{(0,1)}(\Omega )$ for any $s>0$.References
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Bibliographic Information
- Sönmez Şahutoğlu
- Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1043
- ORCID: 0000-0003-0490-0113
- Email: sonmez@umich.edu
- Received by editor(s): August 21, 2006
- Received by editor(s) in revised form: April 23, 2007
- Published electronically: March 4, 2008
- Communicated by: Mei-Chi Shaw
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 136 (2008), 2529-2533
- MSC (2000): Primary 32W05
- DOI: https://doi.org/10.1090/S0002-9939-08-09206-X
- MathSciNet review: 2390523