Uniform Sobolev inequalities and absolute continuity of periodic operators期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

Uniform Sobolev inequalities and absolute continuity of periodic operators

Authors:	Zhongwei Shen Peihao Zhao

Affiliation:	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 ${bold {D}} =-inabla , {bold {k}}in {Bbb {C}} ^{d}$ and is a symmetric, positive definite 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 on ${Bbb R}^{d}$ with a positive periodic scalar function is also studied.

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

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