Banach Journal of Mathematical Analysis

Hardy-type space estimates for multilinear commutators of Calderón–Zygmund operators on nonhomogeneous metric measure spaces

Jie Chen and Haibo Lin

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Let (X,d,μ) be a metric measure space satisfying the so-called upper doubling condition and the geometrically doubling condition. Let T be a Calderón–Zygmund operator and let b:=(b1,,bm) be a finite family of \widetilde{RBMO}(μ) functions. In this article, the authors establish the boundedness of the multilinear commutator Tb, generated by T and b from the atomic Hardy-type space H˜fin,b,m,ρ1,q,m+1(μ) into the Lebesgue space L1(μ). The authors also prove that Tb is bounded from the atomic Hardy-type space H˜fin,b,m,ρ1,q,m+2(μ) into the atomic Hardy space H˜1(μ) via the molecular characterization of H˜1(μ).

Article information

Banach J. Math. Anal., Volume 11, Number 3 (2017), 477-496.

Received: 28 April 2016
Accepted: 25 July 2016
First available in Project Euclid: 19 April 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47B47: Commutators, derivations, elementary operators, etc.
Secondary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B35: Function spaces arising in harmonic analysis 30L99: None of the above, but in this section

nonhomogeneous metric measure space multilinear commutator Calderón–Zygmund operator $\widetilde{\mathrm{RBMO}}(\mu)$ space Hardy-type space


Chen, Jie; Lin, Haibo. Hardy-type space estimates for multilinear commutators of Calderón–Zygmund operators on nonhomogeneous metric measure spaces. Banach J. Math. Anal. 11 (2017), no. 3, 477--496. doi:10.1215/17358787-2017-0002.

Export citation


  • [1] T. A. Bui and X. T. Duong, Hardy spaces, regularized BMO spaces and the boundedness of Calderón-Zygmund operators on non-homogeneous spaces, J. Geom. Anal. 23 (2013), no. 2, 895–932.
  • [2] R. R. Coifman, R. Rochberg, and G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math. (2) 103 (1976), no. 3, 611–635.
  • [3] R. R. Coifman and G. Weiss, Analyse harmonique non-commutative sur certains espaces homogènes, Lecture Notes in Math. 242, Springer, Berlin, 1971.
  • [4] R. R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), 569–645.
  • [5] X. Fu, H. Lin, Da. Yang, and Do. Yang, Hardy spaces $H^{p}$ over non-homogeneous metric measure spaces and their applications, Sci. China Math. 58 (2015), no. 2, 309–388.
  • [6] X. Fu, Da. Yang, and Do. Yang, The molecular characterization of the Hardy space $H^{1}$ on non-homogeneous metric measure spaces and its application, J. Math. Anal. Appl. 410 (2014), no. 2, 1028–1042.
  • [7] X. Fu, D. Yang, and W. Yuan, Boundedness of multilinear commutators of Calderón- Zygmund operators on Orlicz spaces over non-homogeneous spaces, Taiwanese J. Math. 16 (2012), no. 6, 2203–2238.
  • [8] J. Heinonen, Lectures on Analysis on Metric Spaces, Universitext, Springer, New York, 2001.
  • [9] T. Hytönen, A framework for non-homogeneous analysis on metric spaces, and the RBMO space of Tolsa, Publ. Mat. 54 (2010), no. 2, 485–504.
  • [10] T. Hytönen and H. Martikainen, Non-homogeneous $Tb$ theorem and random dyadic cubes on metric measure spaces, J. Geom. Anal. 22 (2012), no. 4, 1071–1107.
  • [11] T. Hytönen and H. Martikainen, Non-homogeneous $T1$ theorem for bi-parameter singular integrals, Adv. Math. 261 (2014), 220-273.
  • [12] T. Hytönen, D. Yang, and D. Yang, The Hardy space $H^{1}$ on non-homogeneous metric spaces, Math. Proc. Cambridge Philos. Soc. 153 (2012), no. 1, 9–31.
  • [13] H. Lin, S. Wu, and D. Yang, Boundedness of certain commutators over non-homogeneous metric measure spaces, Anal. Math. Phys. (2016).
  • [14] H. Lin and D. Yang, An interpolation theorem for sublinear operators on non-homogeneous metric measure spaces, Banach J. Math. Anal. 6 (2012), 168–179.
  • [15] H. Lin and D. Yang, Equivalent boundedness of Marcinkiewicz integrals on non-homogeneous metric measure spaces, Sci. China Math. 57 (2014), no. 1, 123–144.
  • [16] Y. Meng and D. Yang, Multilinear commutators of Calderón-Zygmund operators on Hardy-type spaces with non-doubling measures, J. Math. Anal. Appl. 317 (2006), no. 1, 228–244.
  • [17] F. Nazarov, S. Treil, and A. Volberg, Weak type estimates and Cotlar inequalities for Calderón-Zygmund operators on nonhomogeneous spaces, Internat. Math. Res. Notices 1998, no. 9, 463–487.
  • [18] F. Nazarov, S. Treil, and A. Volberg, The $Tb$-theorem on non-homogeneous spaces, Acta Math. 190 (2003), no. 2, 151–239.
  • [19] C. Pérez, Endpoint estimates for commutators of singular integral operators, J. Funct. Anal. 128 (1995), no. 1, 163–185.
  • [20] C. Pérez and R. Trujillo-González, Sharp weighted estimates for multilinear commutators, London Math. Soc. 65 (2002), no. 3, 672–692.
  • [21] Y. Sawano and H. Tanaka, Morrey spaces for non-doubling measures, Acta Math. Sin. (Engl. Ser.) 21 (2005), no. 6, 1535–1544.
  • [22] L. Shu, M. Wang, and M. Qu, Commutators of Hardy type operators on Herz spaces with variable exponents, Acta Math. Sin. (Chin. Ser.) 58 (2015), no. 1, 29–40.
  • [23] C. Tan and J. Li, Littlewood-Paley theory on metric measure spaces with non doubling measures and its applications, Sci. China Math. 58 (2015), no. 5, 983–1004.
  • [24] X. Tolsa, BMO, $H^{1}$, and Calderón-Zygmund operators for non doubling measures, Math. Ann. 319 (2001), no. 1, 89–149.
  • [25] X. Tolsa, Littlewood-Paley theory and the $T(1)$ theorem with non-doubling measures, Adv. Math. 164 (2001), no. 1, 57–116.
  • [26] X. Tolsa, Painlevé’s problem and the semiadditivity of analytic capacity, Acta Math. 190 (2003), no. 1, 105–149.
  • [27] X. Tolsa, The space $H^{1}$ for nondoubling measures in terms of a grand maximal operator, Trans. Amer. Math. Soc. 355 (2003), no. 1, 315–348.
  • [28] X. Tolsa, Analytic Capacity, the Cauchy Transform, and Non-homogeneous Calderón- Zygmund Theory, Progr. Math. 307, Birkhäuser, Cham, 2014.
  • [29] A. Volberg and B. D. Wick, Bergman-type singular operators and the characterization of Carleson measures for Besov-Sobolev spaces on the complex ball, Amer. J. Math. 134 (2012), no. 4, 949–992.
  • [30] D. Yang, D. Yang, and G. Hu, The Hardy Space $H^{1}$ with Non-doubling Measures and Their Applications, Lecture Notes in Math. 2084, Springer, Cham, 2013.