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

Yanping CHEN; Yong DING; Kai ZHU

doi:10.1007/s10473-022-0404-5

Acta mathematica scientia, Series B >

2022 , Vol. 42 >Issue 4: 1310 - 1332

DOI: https://doi.org/10.1007/s10473-022-0404-5

Articles

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

Yanping CHEN ,
Yong DING ,
Kai ZHU

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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 date: 2020-12-11

Revised date: 2021-03-31

Online published: 2022-08-23

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.

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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

Cite this article

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 . DOI: 10.1007/s10473-022-0404-5

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