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Uniform Sobolev inequalities and absolute continuity of periodic operators

Authors:	Zhongwei Shen Peihao Zhao

Institution:	Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506 ; Department of Mathematics, Lanzhou University, Lanzhou, Gansu, 730000, People's Republic of China

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 -torus, where ${\bold {D}} =-i\nabla , {\bold {k}}\in {\Bbb {C}} ^{d}$ and 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.

Keywords:	Dirac operator periodic potential absolute continuous spectrum uniform Sobolev inequalities

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