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Weighted endpoint estimates for commutators of Calderón-Zygmund operators


Authors: Yiyu Liang, Luong Dang Ky and Dachun Yang
Journal: Proc. Amer. Math. Soc. 144 (2016), 5171-5181
MSC (2010): Primary 47B47; Secondary 42B20, 42B30, 42B35
DOI: https://doi.org/10.1090/proc/13130
Published electronically: May 3, 2016
MathSciNet review: 3556262
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Abstract: Let $ \delta \in (0,1]$ and $ T$ be a $ \delta $-Calderón-Zygmund operator. Let $ w$ be in the Muckenhoupt class $ A_{1+\delta /n}({\mathbb{R}}^n)$ satisfying $ \int _{{\mathbb{R}}^n}\frac {w(x)}{1+\vert x\vert^n}\,dx<\infty $. When $ b\in {\rm BMO}(\mathbb{R}^n)$, it is well known that the commutator $ [b, T]$ is not bounded from $ H^1(\mathbb{R}^n)$ to $ L^1(\mathbb{R}^n)$ if $ b$ is not a constant function. In this article, the authors find out a proper subspace $ {\mathop \mathcal {BMO}_w({\mathbb{R}}^n)}$ of $ \mathop \mathrm {BMO}(\mathbb{R}^n)$ such that, if $ b\in {\mathop \mathcal {BMO}_w({\mathbb{R}}^n)}$, then $ [b,T]$ is bounded from the weighted Hardy space $ H_w^1(\mathbb{R}^n)$ to the weighted Lebesgue space $ L_w^1(\mathbb{R}^n)$. Conversely, if $ b\in {\rm BMO}({\mathbb{R}}^n)$ and the commutators of the classical Riesz transforms $ \{[b,R_j]\}_{j=1}^n$ are bounded from $ H^1_w({\mathbb{R}}^n)$ to $ L^1_w({\mathbb{R}}^n)$, then $ b\in {\mathop \mathcal {BMO}_w({\mathbb{R}}^n)}$.


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Additional Information

Yiyu Liang
Affiliation: Department of Mathematics, Beijing Jiaotong University, Beijing 100044, People’s Republic of China
Email: yyliang@bjtu.edu.cn

Luong Dang Ky
Affiliation: Department of Mathematics, University of Quy Nhon, 170 An Duong Vuong, Quy Nhon, Binh Dinh, Vietnam
Email: dangky@math.cnrs.fr

Dachun Yang
Affiliation: School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, People’s Republic of China
Email: dcyang@bnu.edu.cn

DOI: https://doi.org/10.1090/proc/13130
Keywords: Calder\'on-Zygmund operator, commutator, Muckenhoupt weight, $\mathrm{BMO}$ space, Hardy space
Received by editor(s): September 13, 2015
Received by editor(s) in revised form: January 30, 2016
Published electronically: May 3, 2016
Additional Notes: The first author was supported by the Fundamental Research Funds for the Central Universities of China (Grant No. 2016JBM065)
The second author was supported by the Vietnam National Foundation for Science and Technology Development (Grant No. 101.02-2014.31)
The third author was the corresponding author, who was supported by the National Natural Science Foundation of China (Grant Nos. 11571039 and 11361020), the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20120003110003) and the Fundamental Research Funds for Central Universities of China (Grant Nos. 2014KJJCA10)
Communicated by: Svitlana Mayboroda
Article copyright: © Copyright 2016 American Mathematical Society