GRADIENT ESTIMATES FOR THE COMMUTATOR WITH FRACTIONAL DIFFERENTIATION FOR SECOND ORDER ELLIPTIC OPERATORS

doi:10.1007/s10473-019-0505-y

数学物理学报(英文版) ›› 2019, Vol. 39 ›› Issue (5): 1255-1264.doi: 10.1007/s10473-019-0505-y

GRADIENT ESTIMATES FOR THE COMMUTATOR WITH FRACTIONAL DIFFERENTIATION FOR SECOND ORDER ELLIPTIC OPERATORS

陶文宇, 陈艳萍, 李纪黎

Department of Applied Mathematics, School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China

收稿日期:2018-05-06 修回日期:2019-03-04 出版日期:2019-10-25 发布日期:2019-11-11
通讯作者: Yanping CHEN E-mail:yanpingch@126.com
作者简介:Wenyu TAO,E-mail:wytao@xs.ustb.edu.cn;Jili LI,E-mail:wolfljl13972635073@163.com
基金资助:
The second author was supported by NSFC (11471033), NCET of China (NCET-11-0574) and the Fundamental Research Funds for the Central Universities (FRF-BR-16-011A).

GRADIENT ESTIMATES FOR THE COMMUTATOR WITH FRACTIONAL DIFFERENTIATION FOR SECOND ORDER ELLIPTIC OPERATORS

Wenyu TAO, Yanping CHEN, Jili LI

Department of Applied Mathematics, School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China

Received:2018-05-06 Revised:2019-03-04 Online:2019-10-25 Published:2019-11-11
Contact: Yanping CHEN E-mail:yanpingch@126.com
Supported by:
The second author was supported by NSFC (11471033), NCET of China (NCET-11-0574) and the Fundamental Research Funds for the Central Universities (FRF-BR-16-011A).

摘要/Abstract

摘要： Let L=-div(A∇) be a second order divergence form elliptic operator, where A is an accretive, n×n matrix with bounded measurable complex coefficients on Rⁿ. Let L^α/2 (0 < α < 1) denotes the fractional differential operator associated with L and (-∆)^α/2b ∈ L^n/α(Rⁿ). In this article, we prove that the commutator[b, L^α/2] is bounded from the homogenous Sobolev space L_α² (Rⁿ) to L²(Rⁿ).

关键词: commutator, fractional differentiation, elliptic operators, Sobolev space

Abstract: Let L=-div(A∇) be a second order divergence form elliptic operator, where A is an accretive, n×n matrix with bounded measurable complex coefficients on Rⁿ. Let L^α/2 (0 < α < 1) denotes the fractional differential operator associated with L and (-∆)^α/2b ∈ L^n/α(Rⁿ). In this article, we prove that the commutator[b, L^α/2] is bounded from the homogenous Sobolev space L_α² (Rⁿ) to L²(Rⁿ).

Key words: commutator, fractional differentiation, elliptic operators, Sobolev space

中图分类号:

42B20

陶文宇, 陈艳萍, 李纪黎. GRADIENT ESTIMATES FOR THE COMMUTATOR WITH FRACTIONAL DIFFERENTIATION FOR SECOND ORDER ELLIPTIC OPERATORS[J]. 数学物理学报(英文版), 2019, 39(5): 1255-1264.

Wenyu TAO, Yanping CHEN, Jili LI. GRADIENT ESTIMATES FOR THE COMMUTATOR WITH FRACTIONAL DIFFERENTIATION FOR SECOND ORDER ELLIPTIC OPERATORS[J]. Acta mathematica scientia,Series B, 2019, 39(5): 1255-1264.

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GRADIENT ESTIMATES FOR THE COMMUTATOR WITH FRACTIONAL DIFFERENTIATION FOR SECOND ORDER ELLIPTIC OPERATORS

GRADIENT ESTIMATES FOR THE COMMUTATOR WITH FRACTIONAL DIFFERENTIATION FOR SECOND ORDER ELLIPTIC OPERATORS

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