Taiwanese Journal of Mathematics

Maximal Multilinear Commutators on Non-homogeneous Metric Measure Spaces

Jie Chen and Haibo Lin

Full-text: Open access

Abstract

Let $(\mathcal{X},d,\mu)$ be a metric measure space satisfying the so-called upper doubling condition and the geometrically doubling condition. Let $T_*$ be the maximal Calderón-Zygmund operator and $\vec{b} := (b_1,\ldots,b_m)$ be a finite family of $\widetilde{\operatorname{RBMO}}(\mu)$ functions. In this paper, the authors establish the boundedness of the maximal multilinear commutator $T_{*,\vec{b}}$ generated by $T_*$ and $\vec{b}$ on the Lebesgue space $L^p(\mu)$ with $p \in (1, \infty)$. For $\vec{b} = (b_1,\ldots,b_m)$ being a finite family of Orlicz type functions, the weak type endpoint estimate for the maximal multilinear commutator $T_{*,\vec{b}}$ generated by $T_*$ and $\vec{b}$ is also presented. The main tool to deal with these estimates is the smoothing technique.

Article information

Source
Taiwanese J. Math., Volume 21, Number 5 (2017), 1133-1160.

Dates
Received: 27 June 2016
Revised: 21 January 2017
Accepted: 24 January 2017
First available in Project Euclid: 1 August 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1501599186

Digital Object Identifier
doi:10.11650/tjm/7976

Mathematical Reviews number (MathSciNet)
MR3707887

Zentralblatt MATH identifier
06871362

Subjects
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

Keywords
non-homogeneous metric measure space maximal Calderón-Zygmund operator $\widetilde{\operatorname{RBMO}}$ function Orlicz type space commutator

Citation

Chen, Jie; Lin, Haibo. Maximal Multilinear Commutators on Non-homogeneous Metric Measure Spaces. Taiwanese J. Math. 21 (2017), no. 5, 1133--1160. doi:10.11650/tjm/7976. https://projecteuclid.org/euclid.twjm/1501599186


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References

  • T. A. Bui, Boundedness of maximal operators and maximal commutators on non-homogeneous spaces, in AMSI International Conference on Harmonic Analysis and Applications, 22–36, Proc. Centre Math. Appl. Austral. Nat. Univ. 45, Austral. Nat. Univ., Canberra, 2013.
  • 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.
  • J. Chen and H. Lin, Hardy-type space estimates for multilinear commutators of Calderón-Zygmund operators on nonhomogeneous metric measure spaces, Banach J. Math. Anal. (to appear).
  • W. Chen, Y. Meng and D. Yang, Calderón-Zygmund operators on Hardy spaces without the doubling condition, Proc. Amer. Math. Soc. 133 (2005), no. 9, 2671–2680.
  • W. Chen and C. Miao, Vector valued commutators on non-homogeneous spaces, Taiwanese J. Math. 11 (2007), no. 4, 1127–1141.
  • W. Chen and E. Sawyer, A note on commutators of fractional integrals with $\orn{RBMO}(\mu)$ functions, Illinois J. Math. 46 (2002), no. 4, 1287–1298.
  • R. R. Coifman and G. Weiss, Analyse Harmonique Non-commutative sur Certains Espaces Homogènes, Étude de certaines intégrales singulières, Lecture Notes in Mathematics 242, Springer-Verlag, Berlin-New York, 1971.
  • ––––, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), no. 4, 569–645.
  • X. Fu, H. Lin, D. Yang and D. Yang, Hardy spaces $H^p$ over non-homogeneous metric measure spaces and their applications, Sci. China Math. 58 (2015), no. 2, 309–388.
  • X. Fu, D. Yang and D. 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.
  • 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.
  • J. García-Cuerva and J. M. Martell, On the existence of principal values for the Cauchy integral on weighted Lebesgue spaces for non-doubling measures, J. Fourier Anal. Appl. 7 (2001), no. 5, 469–487.
  • J. Heinonen, Lectures on Analysis on Metric Spaces, Universitext, Springer-Verlag, New York, 2001.
  • G. Hu, Y. Meng and D. Yang, New atomic characterization of $H^1$ space with non-doubling measures and its applications, Math. Proc. Cambridge Philos. Soc. 138 (2005), no. 1, 151–171.
  • ––––, Multilinear commutators of singular integrals with non doubling measures, Integral Equations Operator Theory 51 (2005), no. 2, 235–255.
  • ––––, Endpoint estimate for maximal commutators with non-doubling measures, Acta Math. Sci. Ser. B Engl. Ed. 26 (2006), no. 2, 271–280.
  • 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.
  • T. Hytönen, S. Liu, D. Yang and D. Yang, Boundedness of Calderón-Zygmund operators on non-homogeneous metric measure spaces, Canad. J. Math. 64 (2012), no. 4, 892–923.
  • 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.
  • ––––, Non-homogeneous $T1$ theorem for bi-parameter singular integrals, Adv. Math. 261 (2014), 220–273.
  • 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.
  • L. Li and Y.-S. Jiang, Estimates for maximal multilinear commutators on non-homogeneous spaces, J. Math. Anal. Appl. 355 (2009), no. 1, 243–257.
  • H. Lin, S. Wu and D. Yang, Boundedness of certain commutators over non-homogeneous metric measure spaces, Anal. Math. Phys. 7 (2017), no. 2, 187–218.
  • 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.
  • S. Liu, Y. Meng and D. Yang, Boundedness of maximal Calderón-Zygmund operators on non-homogeneous metric measure spaces, Proc. Roy. Soc. Edinburgh Sect. A 144 (2014), no. 3, 567–589.
  • S. Liu, D. Yang and D. Yang, Boundedness of Calderón-Zygmund operators on non-homogeneous metric measure spaces: equivalent characterizations, J. Math. Anal. Appl. 386 (2012), no. 1, 258–272.
  • 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 (1998), no. 9, 463–487.
  • ––––, The $Tb$-theorem on non-homogeneous spaces, Acta Math. 190 (2003), no. 2, 151–239.
  • Y. Sawano and H. Tanaka, Morrey spaces for non-doubling measures, Acta Math. Sin. (Engl. Ser.) 21 (2005), no. 6, 1535–1544.
  • ––––, Sharp maximal inequalities and commutators on Morrey spaces with non-doubling measures, Taiwanese J. Math. 11 (2007), no. 4, 1091–1112.
  • C. Segovia and J. L. Torrea, Vector-valued commutators and applications, Indiana Univ. Math. J. 38 (1989), no. 4, 959–971.
  • 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.
  • ––––, Some characterizations of upper doubling conditions on metric measure spaces, Math. Nachr. 290 (2017), no. 1, 142–158.
  • X. Tolsa, BMO, $H^1$, and Calderón-Zygmund operators for non doubling measures, Math. Ann. 319 (2001), no. 1, 89–149.
  • ––––, Littlewood-Paley theory and the $T(1)$ theorem with non-doubling measures, Adv. Math. 164 (2001), no. 1, 57–116.
  • ––––, Painlevé's problem and the semiadditivity of analytic capacity, Acta Math. 190 (2003), no. 1, 105–149.
  • ––––, The space $H^1$ for nondoubling measures in terms of a grand maximal operator, Trans. Amer. Math. Soc. 355 (2003), no. 1, 315–348.
  • ––––, Analytic Capacity, the Cauchy Transform, and Non-homogeneous Calderón-Zygmund Theory, Progress in Mathematics 307, Birkhäuser/Springer, Cham, 2014.
  • A. Volberg and B. D. Wick, Bergman-type singular integral operators and the characterization of Carleson measures for Besov-Sobolev spaces on the complex ball, Amer. J. Math. 134 (2012), no. 4, 949–992.
  • R. Xie, H. Gong and X. Zhou, Commutators of multilinear singular integral operators on non-homogeneous metric measure spaces, Taiwanese J. Math. 19 (2015), no. 3, 703–723.
  • D. Yang, D. Yang and X. Fu, The Hardy space $H^1$ on non-homogeneous spaces and its applications–-a survey, Eurasian Math. J. 4 (2013), no. 2, 104–139.
  • D. Yang, D. Yang and G. Hu, The Hardy Space $H^1$ with Non-doubling Measures and Their Applications, Lecture Notes in Mathematics 2084, Springer, Cham, 2013.