WEIGHTED NORM INEQUALITIES FOR COMMUTATORS OF THE KATO SQUARE ROOT OF SECOND ORDER ELLIPTIC OPERATORS ON $\mathbb R^n$

doi:10.1007/s10473-022-0404-5

数学物理学报(英文版) ›› 2022, Vol. 42 ›› Issue (4): 1310-1332.doi: 10.1007/s10473-022-0404-5

WEIGHTED NORM INEQUALITIES FOR COMMUTATORS OF THE KATO SQUARE ROOT OF SECOND ORDER ELLIPTIC OPERATORS ON $\mathbb R^n$

陈艳萍¹, 丁勇², 朱凯³

1. Department of Applied Mathematics, School of Mathematics and Physics, University of Science and Technology Beijing, Beijing, 100083, China;
2. Laboratory of Mathematics and Complex Systems, School of Mathematical Sciences, Beijing Normal University, Ministry of Education of China, Beijing, 100875, China;
3. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China

收稿日期:2020-12-11 修回日期:2021-03-31 出版日期:2022-08-25 发布日期:2022-08-23
通讯作者: Kai ZHU,E-mail:zhukai@ucas.ac.cn E-mail:zhukai@ucas.ac.cn
基金资助:
The first author is supported by NSFC (11871096, 11471033). The second author is supported by NSFC (11371057, 11471033, 11571160), SRFDP (20130003110003) and the Fundamental Research Funds for the Central Universities (2014KJJCA10). The third author would like to thank the China Scholarship Council for its support.

WEIGHTED NORM INEQUALITIES FOR COMMUTATORS OF THE KATO SQUARE ROOT OF SECOND ORDER ELLIPTIC OPERATORS ON $\mathbb R^n$

Yanping CHEN¹, Yong DING², Kai ZHU³

1. Department of Applied Mathematics, School of Mathematics and Physics, University of Science and Technology Beijing, Beijing, 100083, China;
2. Laboratory of Mathematics and Complex Systems, School of Mathematical Sciences, Beijing Normal University, Ministry of Education of China, Beijing, 100875, China;
3. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China

Received:2020-12-11 Revised:2021-03-31 Online:2022-08-25 Published:2022-08-23
Contact: Kai ZHU,E-mail:zhukai@ucas.ac.cn E-mail:zhukai@ucas.ac.cn
Supported by:
The first author is supported by NSFC (11871096, 11471033). The second author is supported by NSFC (11371057, 11471033, 11571160), SRFDP (20130003110003) and the Fundamental Research Funds for the Central Universities (2014KJJCA10). The third author would like to thank the China Scholarship Council for its support.

摘要/Abstract

摘要： Let $L=-\mathrm{div}(A\nabla)$ be a second order divergence form elliptic operator with bounded measurable coefficients in ${\Bbb R}^n$. We establish weighted $L^p$ norm inequalities for commutators generated by $\sqrt{L}$ and Lipschitz functions, where the range of $p$ is different from $(1,\infty)$, and we isolate the right class of weights introduced by Auscher and Martell. In this work, we use good-$\lambda$ inequality with two parameters through the weighted boundedness of Riesz transforms $\nabla L^{-1/2}$. Our result recovers, in some sense, a previous result of Hofmann.

关键词: Muckenhoupt weights, commutator, Kato square root, Lipschitz function, elliptic operators

Abstract: Let $L=-\mathrm{div}(A\nabla)$ be a second order divergence form elliptic operator with bounded measurable coefficients in ${\Bbb R}^n$. We establish weighted $L^p$ norm inequalities for commutators generated by $\sqrt{L}$ and Lipschitz functions, where the range of $p$ is different from $(1,\infty)$, and we isolate the right class of weights introduced by Auscher and Martell. In this work, we use good-$\lambda$ inequality with two parameters through the weighted boundedness of Riesz transforms $\nabla L^{-1/2}$. Our result recovers, in some sense, a previous result of Hofmann.

Key words: Muckenhoupt weights, commutator, Kato square root, Lipschitz function, elliptic operators

中图分类号:

42B20

陈艳萍, 丁勇, 朱凯. WEIGHTED NORM INEQUALITIES FOR COMMUTATORS OF THE KATO SQUARE ROOT OF SECOND ORDER ELLIPTIC OPERATORS ON $\mathbb R^n$[J]. 数学物理学报(英文版), 2022, 42(4): 1310-1332.

Yanping CHEN, Yong DING, Kai ZHU. WEIGHTED NORM INEQUALITIES FOR COMMUTATORS OF THE KATO SQUARE ROOT OF SECOND ORDER ELLIPTIC OPERATORS ON $\mathbb R^n$[J]. Acta mathematica scientia,Series B, 2022, 42(4): 1310-1332.

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WEIGHTED NORM INEQUALITIES FOR COMMUTATORS OF THE KATO SQUARE ROOT OF SECOND ORDER ELLIPTIC OPERATORS ON $\mathbb R^n$

WEIGHTED NORM INEQUALITIES FOR COMMUTATORS OF THE KATO SQUARE ROOT OF SECOND ORDER ELLIPTIC OPERATORS ON $\mathbb R^n$

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