Uniform sobolev inequalities and absolute continuity of periodic operators

Zhongwei Shen; Peihao Zhao

doi:10.1090/S0002-9947-07-04545-X

Uniform sobolev inequalities and absolute continuity of periodic operators

Zhongwei Shen, Peihao Zhao

Producción científica: Article › revisión exhaustiva

12 Citas (Scopus)

Resumen

We establish certain uniform L^p- L^q inequalities for a family of second order elliptic operators of the form (D + k)A(D + k) ^T on the d-torus, where D =-i∇, k ∈ ℂ^d and A is a symmetric, positive definite d × 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 ℝ^d with singular potential. The absolute continuity of the elliptic operator div(ω(x)∇) on ℝ^d with a positive periodic scalar function ω(x) is also studied.

Idioma original	English
Páginas (desde-hasta)	1741-1758
Número de páginas	18
Publicación	Transactions of the American Mathematical Society
Volumen	360
N.º	4
DOI	https://doi.org/10.1090/S0002-9947-07-04545-X
Estado	Published - abr 2008

Nota bibliográfica

Funding Information:
We thank the Fermilab staff and the technical staffs of the participating institutions for their vital contributions. This work was supported by the U.S. Department of Energy and National Science Foundation; the Italian Istituto Nazionale di Fisica Nucleare; the Ministry of Education, Science and Culture of Japan; the Natural Sciences and Engineering Research Council of Canada; the National Sci- ence Council of the Republic of China; the Foundation; and the Max Kade Foundation.

ASJC Scopus subject areas

General Mathematics
Applied Mathematics

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10.1090/S0002-9947-07-04545-X

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title = "Uniform sobolev inequalities and absolute continuity of periodic operators",

abstract = "We establish certain uniform Lp- Lq inequalities for a family of second order elliptic operators of the form (D + k)A(D + k) T on the d-torus, where D =-i∇, k ∈ ℂd and A is a symmetric, positive definite d × 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 ℝd with singular potential. The absolute continuity of the elliptic operator div(ω(x)∇) on ℝd with a positive periodic scalar function ω(x) is also studied.",

keywords = "Absolute continuous spectrum, Dirac operator, Periodic potential, Uniform Sobolev inequalities",

author = "Zhongwei Shen and Peihao Zhao",

note = "Funding Information: We thank the Fermilab staff and the technical staffs of the participating institutions for their vital contributions. This work was supported by the U.S. Department of Energy and National Science Foundation; the Italian Istituto Nazionale di Fisica Nucleare; the Ministry of Education, Science and Culture of Japan; the Natural Sciences and Engineering Research Council of Canada; the National Sci- ence Council of the Republic of China; the Foundation; and the Max Kade Foundation.",

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T1 - Uniform sobolev inequalities and absolute continuity of periodic operators

AU - Shen, Zhongwei

AU - Zhao, Peihao

N1 - Funding Information: We thank the Fermilab staff and the technical staffs of the participating institutions for their vital contributions. This work was supported by the U.S. Department of Energy and National Science Foundation; the Italian Istituto Nazionale di Fisica Nucleare; the Ministry of Education, Science and Culture of Japan; the Natural Sciences and Engineering Research Council of Canada; the National Sci- ence Council of the Republic of China; the Foundation; and the Max Kade Foundation.

PY - 2008/4

Y1 - 2008/4

N2 - We establish certain uniform Lp- Lq inequalities for a family of second order elliptic operators of the form (D + k)A(D + k) T on the d-torus, where D =-i∇, k ∈ ℂd and A is a symmetric, positive definite d × 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 ℝd with singular potential. The absolute continuity of the elliptic operator div(ω(x)∇) on ℝd with a positive periodic scalar function ω(x) is also studied.

AB - We establish certain uniform Lp- Lq inequalities for a family of second order elliptic operators of the form (D + k)A(D + k) T on the d-torus, where D =-i∇, k ∈ ℂd and A is a symmetric, positive definite d × 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 ℝd with singular potential. The absolute continuity of the elliptic operator div(ω(x)∇) on ℝd with a positive periodic scalar function ω(x) is also studied.

KW - Absolute continuous spectrum

KW - Dirac operator

KW - Periodic potential

KW - Uniform Sobolev inequalities

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U2 - 10.1090/S0002-9947-07-04545-X

DO - 10.1090/S0002-9947-07-04545-X

M3 - Article

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SN - 0002-9947

VL - 360

SP - 1741

EP - 1758

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

IS - 4

ER -

Uniform sobolev inequalities and absolute continuity of periodic operators

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ASJC Scopus subject areas

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