Advertisement

Chinese Annals of Mathematics, Series B

, Volume 40, Issue 4, pp 585–598 | Cite as

Boundedness of Commutators of θ-Type Calderón-Zygmund Operators on Non-homogeneous Metric Measure Spaces

  • Chol Ri
  • Zhenqiu ZhangEmail author
Article

Abstract

Let (X, d, μ) be a metric measure space satisfying both the upper doubling and the geometrically doubling conditions in the sense of Hytönen. In this paper, the authors obtain the boundedness of the commutators of θ-type Calderón-Zygmund operators with RBMO functions from L (μ) into RBMO(μ) and from \(H_{{\rm{at}}}^{1,\;\infty }\left( \mu \right)\) into L1 (μ), respectively. As a consequence of these results, they establish the Lp (μ) boundedness of the commutators on the non-homogeneous metric spaces.

Keywords

Non-homogeneous space θ-Type Calderón-Zygmund operator Commutator RBMO(μ) space \(H_{{\rm{at}}}^{1,\;\infty }\left( \mu \right)\) space 

2000 MR Subject Classification

42B20 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Bui, T. A. and Duong, X. T., Hardy spaces, regularized BMO spaces and the boundedness of Calderón-Zygmund operators on non-homogeneous spaces, J. Geom. Anal., 23, 2013, 895–932.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    Calderón, A. P., Cauchy integrals on Lipschitz curves and related operators, Proc. Natl. Acad. Sci. USA, 53, 1965, 1092–1099.CrossRefzbMATHGoogle Scholar
  3. [3]
    Calderón, A. P., Commutators of singular integral operators, Proc. Natl. Acad. Sci. USA., 74, 1977, 1324–1327.CrossRefzbMATHGoogle Scholar
  4. [4]
    Coifman, R., Rochberg, R. and Weiss, G., Factorization theorems for Hardy spaces in several variables, Ann. Math., 103(3), 1976, 611–635.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    Coifman, R. and Weiss, G., Analyse harmonique non-commutative sur certains espaces homogenes, Lecture Notes in Mathematics, 242, Springer-Verlag, Cham, 1971.CrossRefzbMATHGoogle Scholar
  6. [6]
    Coifman, R. and Weiss, G., Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc., 83, 1977, 569–645.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Fu, X., Hu, G. and Yang, D., A remark on the boundedness of Calderón-Zygmund operators in non-homogeneouspaces, Acta Math. Sinica, (Engl. Ser.), 23, 2007, 449–456.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    Heinenon, J., Lectures on Analysis on Metric Spaces, Springer-Verlag, New York, 2001.CrossRefGoogle Scholar
  9. [9]
    Hytönen, T., A framework for non-homogeneous analysis on metric spaces, and the RBMO space of Tolsa, Publ. Mat., 54, 2010, 485–504.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    Hytönen, T., Liu, S., Yang, D. and Yang, D., Boundedness of Calderón-Zygmund operators on non-homogeneous metric measure spaces, Can. J. Math., 64, 2012, 892–923.CrossRefzbMATHGoogle Scholar
  11. [11]
    Hytönen, T., Yang, D. and Yang, D., The Hardy space H 1 on non-homogeneous metric spaces, Math. Proc. Camb. Phil. Soc., 153, 2012, 9–31.CrossRefzbMATHGoogle Scholar
  12. [12]
    Janson, S., Mean oscillation and commutators of singular integral operators, Ark. Mat., 16(2), 1978, 263–270.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    Liu, S., Yang, D. and Yang, D., Boundedness of Calderón-Zygmund operators on non-homogeneous metric measure spaces: equivalent characterizations, J. Math. Anal. Appl., 386, 2012, 258–272.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    Perez, C., Endpoint estimates for commutators of singular integral operators, J. Funct. Anal., 128(1), 1995, 163–185.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    Tolsa, X., BMO, H1 and Calderón-Zygmund operators for non doubling measures, Math. Ann., 319, 2001, 89–149.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    Xie, R. and Shu, L., θ-type Calderón-Zygmund operators with non-doubling measures, Acta Math. App. Sinica, (Engl. Ser.), 29(2), 2013, 263–280.CrossRefzbMATHGoogle Scholar
  17. [17]
    Yabuta, K., Generalization of Calderón-Zygmund operators, Studia Math., 82, 1985, 17–31.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    Yang, D., Yang, D. and Hu, G., The Hardy space H1 with non-doubling measures and their applications, Lecture Notes in Mathematics, 2084, Springer-Verlag, Cham, 2013.CrossRefGoogle Scholar

Copyright information

© The Editorial Office of CAM and Springer-Verlag Berlin Heidelberg 2019

Authors and Affiliations

  1. 1.Department of MathematicsKim Hyong Jik Normal UniversityPyongYangKorea
  2. 2.School of Mathematical Sciences and the Key Laboratory of Pure Mathematics and CombinatoricsNankai UniversityTianjinChina

Personalised recommendations