## Weighted endpoint estimates for commutators of Calderón-Zygmund operators

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- by Yiyu Liang, Luong Dang Ky and Dachun Yang PDF
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**144**(2016), 5171-5181 Request permission

## 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 $\int _{{\mathbb R}^n}\frac {w(x)}{1+|x|^n} dx<\infty$. When $b\in \textrm {BMO}(\mathbb R^n)$, 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 ${\mathcal {BMO}_w({\mathbb R}^n)}$ of $\mathrm {BMO}(\mathbb R^n)$ such that, if $b\in {\mathcal {BMO}_w({\mathbb R}^n)}$, 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 $b\in \textrm {BMO}({\mathbb R}^n)$ 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 $b\in {\mathcal {BMO}_w({\mathbb R}^n)}$.## References

- Josefina Álvarez, Richard J. Bagby, Douglas S. Kurtz, and Carlos Pérez,
*Weighted estimates for commutators of linear operators*, Studia Math.**104**(1993), no. 2, 195–209. MR**1211818**, DOI 10.4064/sm-104-2-195-209 - Steven Bloom,
*Pointwise multipliers of weighted BMO spaces*, Proc. Amer. Math. Soc.**105**(1989), no. 4, 950–960. MR**960640**, DOI 10.1090/S0002-9939-1989-0960640-3 - Marcin Bownik, Baode Li, Dachun Yang, and Yuan Zhou,
*Weighted anisotropic Hardy spaces and their applications in boundedness of sublinear operators*, Indiana Univ. Math. J.**57**(2008), no. 7, 3065–3100. MR**2492226**, DOI 10.1512/iumj.2008.57.3414 - Jun Cao, Der-Chen Chang, Dachun Yang, and Sibei Yang,
*Riesz transform characterizations of Musielak-Orlicz-Hardy spaces*, Trans. Amer. Math. Soc.**368**(2016), no. 10, 6979–7018. MR**3471083**, DOI 10.1090/tran/6556 - R. R. Coifman, R. Rochberg, and Guido Weiss,
*Factorization theorems for Hardy spaces in several variables*, Ann. of Math. (2)**103**(1976), no. 3, 611–635. MR**412721**, DOI 10.2307/1970954 - José García-Cuerva,
*Weighted $H^{p}$ spaces*, Dissertationes Math. (Rozprawy Mat.)**162**(1979), 63. MR**549091** - J. García-Cuerva and K. S. Kazarian,
*Calderón-Zygmund operators and unconditional bases of weighted Hardy spaces*, Studia Math.**109**(1994), no. 3, 255–276. MR**1274012**, DOI 10.4064/sm-109-3-255-276 - José García-Cuerva and José L. Rubio de Francia,
*Weighted norm inequalities and related topics*, North-Holland Mathematics Studies, vol. 116, North-Holland Publishing Co., Amsterdam, 1985. Notas de Matemática [Mathematical Notes], 104. MR**807149** - Eleonor Harboure, Carlos Segovia, and José L. Torrea,
*Boundedness of commutators of fractional and singular integrals for the extreme values of $p$*, Illinois J. Math.**41**(1997), no. 4, 676–700. MR**1468874** - F. John and L. Nirenberg,
*On functions of bounded mean oscillation*, Comm. Pure Appl. Math.**14**(1961), 415–426. MR**131498**, DOI 10.1002/cpa.3160140317 - Luong Dang Ky,
*A note on $H_w^p$-boundedness of Riesz transforms and $\theta$-Calderón-Zygmund operators through molecular characterization*, Anal. Theory Appl.**27**(2011), no. 3, 251–264. MR**2844661**, DOI 10.1007/s10496-011-0251-z - Luong Dang Ky,
*Bilinear decompositions and commutators of singular integral operators*, Trans. Amer. Math. Soc.**365**(2013), no. 6, 2931–2958. MR**3034454**, DOI 10.1090/S0002-9947-2012-05727-8 - Carlos Pérez,
*Endpoint estimates for commutators of singular integral operators*, J. Funct. Anal.**128**(1995), no. 1, 163–185. MR**1317714**, DOI 10.1006/jfan.1995.1027 - Liang Song and Lixin Yan,
*Riesz transforms associated to Schrödinger operators on weighted Hardy spaces*, J. Funct. Anal.**259**(2010), no. 6, 1466–1490. MR**2659768**, DOI 10.1016/j.jfa.2010.05.015 - Kôzô Yabuta,
*Pointwise multipliers of weighted BMO spaces*, Proc. Amer. Math. Soc.**117**(1993), no. 3, 737–744. MR**1123671**, DOI 10.1090/S0002-9939-1993-1123671-X

## Additional Information

**Yiyu Liang**- Affiliation: Department of Mathematics, Beijing Jiaotong University, Beijing 100044, People’s Republic of China
- MR Author ID: 946733
- 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
- MR Author ID: 954241
- 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
- MR Author ID: 317762
- Email: dcyang@bnu.edu.cn
- 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
- © Copyright 2016 American Mathematical Society
- 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
- MathSciNet review: 3556262