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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Uniform Sobolev inequalities and absolute continuity of periodic operators


Authors: Zhongwei Shen and Peihao Zhao
Journal: Trans. Amer. Math. Soc. 360 (2008), 1741-1758
MSC (2000): Primary 35J10, 42B15
Published electronically: November 26, 2007
MathSciNet review: 2366961
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Abstract | References | Similar Articles | Additional Information

Abstract: We establish certain uniform $ L^{p}-L^{q}$ inequalities for a family of second order elliptic operators of the form $ ( {\bold {D}} + {\bold {k}} ) A ( {\bold {D}}+ {\bold {k} })^{T}$ on the $ d$-torus, where $ {\bold {D}} =-i\nabla , {\bold {k}}\in {\Bbb {C}} ^{d}$ and $ A$ is a symmetric, positive definite $ d\times d$ matrix with real constant entries. Using these Sobolev type inequalities, we obtain the absolute continuity of the spectrum of the periodic Dirac operator on $ {\Bbb R}^{d}$ with singular potential. The absolute continuity of the elliptic operator div $ (\omega ( {\bold {x}})\nabla )$ on $ {\Bbb R}^{d}$ with a positive periodic scalar function $ \omega ( {\bold {x}} )$ is also studied.


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Additional Information

Zhongwei Shen
Affiliation: Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506
Email: shenz@ms.uky.edu

Peihao Zhao
Affiliation: Department of Mathematics, Lanzhou University, Lanzhou, Gansu, 730000, People’s Republic of China
Email: zhaoph@lzu.edu.cn

DOI: http://dx.doi.org/10.1090/S0002-9947-07-04545-X
PII: S 0002-9947(07)04545-X
Keywords: Dirac operator, periodic potential, absolute continuous spectrum, uniform Sobolev inequalities
Received by editor(s): July 13, 2005
Published electronically: November 26, 2007
Additional Notes: The first author was supported in part by the NSF (DMS-0500257). The second author was supported in part by the NSF of Gansu Province, China (ZS021-A25-002-Z) and the NSFC (10371052).
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.