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

DOI:
https://doi.org/10.1090/S0002-9947-07-04545-X

Published electronically:
November 26, 2007

MathSciNet review:
2366961

Full-text PDF Free Access

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 $( {\mathbf {D}} + {\mathbf {k}} ) A ( {\mathbf {D}}+ {\mathbf {k} })^{T}$ on the $d$-torus, where ${\mathbf {D}} =-i\nabla , {\mathbf {k}}\in {\mathbb {C}} ^{d}$ 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 ${\mathbb R}^{d}$ with singular potential. The absolute continuity of the elliptic operator div$(\omega ( {\mathbf {x}})\nabla )$ on ${\mathbb R}^{d}$ with a positive periodic scalar function $\omega ( {\mathbf {x}} )$ is also studied.

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Additional Information

**Zhongwei Shen**

Affiliation:
Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506

MR Author ID:
227185

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

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.