Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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)}$.


References [Enhancements On Off] (What's this?)

  • [1] 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
  • [2] Steven Bloom, Pointwise multipliers of weighted BMO spaces, Proc. Amer. Math. Soc. 105 (1989), no. 4, 950-960. MR 960640, https://doi.org/10.2307/2047058
  • [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, https://doi.org/10.1512/iumj.2008.57.3414
  • [4] 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, https://doi.org/10.1090/tran/6556
  • [5] 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 0412721
  • [6] José García-Cuerva, Weighted $ H^{p}$ spaces, Dissertationes Math. (Rozprawy Mat.) 162 (1979), 63. MR 549091
  • [7] 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
  • [8] 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
  • [9] 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
  • [10] F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961), 415-426. MR 0131498
  • [11] 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, https://doi.org/10.1007/s10496-011-0251-z
  • [12] Luong Dang Ky, Bilinear decompositions and commutators of singular integral operators, Trans. Amer. Math. Soc. 365 (2013), no. 6, 2931-2958. MR 3034454, https://doi.org/10.1090/S0002-9947-2012-05727-8
  • [13] Carlos Pérez, Endpoint estimates for commutators of singular integral operators, J. Funct. Anal. 128 (1995), no. 1, 163-185. MR 1317714, https://doi.org/10.1006/jfan.1995.1027
  • [14] 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, https://doi.org/10.1016/j.jfa.2010.05.015
  • [15] Kôzô Yabuta, Pointwise multipliers of weighted BMO spaces, Proc. Amer. Math. Soc. 117 (1993), no. 3, 737-744. MR 1123671, https://doi.org/10.2307/2159136

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 47B47, 42B20, 42B30, 42B35

Retrieve articles in all journals with MSC (2010): 47B47, 42B20, 42B30, 42B35


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

American Mathematical Society