Skip to main content
SpringerLink
Log in

Boundedness of Some Maximal Commutators in Hardy-type Spaces with Non-doubling Measures

  • ORIGINAL ARTICLES
  • Published:
Acta Mathematica Sinica, English Series Aims and scope Submit manuscript

Abstract

Let μ be a non-negative Radon measure on ℝd which satisfies only some growth conditions. Under this assumption, the boundedness in some Hardy-type spaces is established for a class of maximal Calderón–Zygmund operators and maximal commutators which are variants of the usual maximal commutators generated by Calderón–Zygmund operators and RBMO(μ) functions, where the Hardytype spaces are some appropriate subspaces, associated with the considered RBMO(μ) functions, of the Hardy space H 1(μ) of Tolsa.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Fu, X., Hu, G., Yang, D.: A Remark on boundedness of Calderón–Zygmund operators in non-homogeneous spaces. Acta Mathematica Sinica, English Series, 23(3), 449–456 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  2. García-Cuerva, J., Martell, J. M.: On the existence of principal values for the Cauchy integral on weighted Lebesgue spaces of non-doubling measures. J. Fourier Anal. Appl., 7, 469–487 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  3. Nazarov, F., Treil, S., Volberg, A.: Cauchy integral and Calderón–Zygmund operators on nonhomogeneous spaces. Internat. Math. Res. Notices, 15, 703–726 (1997)

    Article  Google Scholar 

  4. Nazarov, F., Treil, S., Volberg, A.: Accretive system Tb-theorems on nonhomogeneous spaces. Duke Math. J., 113, 259–312 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  5. Nazarov, F., Treil, S., Volberg, A.: The Tb-theorem on non-homogeneous spaces. Acta Math., 190, 151–239 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  6. Orobitg, J., Pérez, C.: Ap weights for nondoubling measures in Rn and applications. Trans. Amer. Math. Soc., 354, 2013–2033 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  7. Sawano, Y., Tanaka, H.: Morrey spaces for non-doubling measures. Acta Mathematica Sinica, English Series, 21(6), 1535–1544 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  8. Tolsa, X.: Littlewood-Paley theory and the T(1) theorem with non-doubling measures. Adv. Math., 164, 57–116 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  9. Tolsa, X.: The space H 1 for nondoubling measures in terms of a grand maximal operator. Trans. Amer. Math. Soc., 355, 315–348 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  10. Tolsa, X.: Painlevé's problem and the semiadditivity of analytic capacity. Acta Math., 190, 105–149 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  11. Verdera, J.: The fall of the doubling condition in Calderón–Zygmund theory. Publ. Mat., Vol. Extra, 275–292 (2002)

  12. Hu, G., Meng, Y., Yang, D.: Estimates for maximal singular integral operators in non-homogeneous spaces. Proc. Roy. Soc. Edinb., 136, 351–364 (2006)

    MATH  Google Scholar 

  13. Uchiyama, A.: The factorization of Hp on the space of homogeneous type. Pacific. J. Math., 92, 453–468 (1981)

    MATH  MathSciNet  Google Scholar 

  14. Tolsa, X.: BMO, H1 and Calderón–Zygmund operators for non doubling measures. Math. Ann., 319, 89–149 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  15. Chen, W., Miao, C.: Vector valued commutators on non-homogeneous spaces. Taiwanese J. Math., to appear

  16. Meng, Y., Yang, D.: Multilinear commutators of Calderón–Zygmund operators on Hardy-type spaces with non-doubling measures. J. Math. Anal. Appl., 317, 228–244 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  17. Hu, G., Meng, Y., Yang, D.: New atomic characterization of H 1 space with non-doubling measures and its applications. Math. Proc. Camb. Phil. Soc., 138, 151–171 (2005)

    Article  MATH  Google Scholar 

  18. Tolsa, X.: Characterization of the atomic space H 1 for non doubling measures in terms of a grand maximal operator. http://www.math.chalmers.se/Math/Research/Preprints/, Chalmers Univ. of Technology, Mathematics, Preprint 2000:2

  19. Bownik, M.: Boundedness of operators on Hardy spaces via atomic decompositions. Proc. Amer. Math. Soc., 133, 3535–3542 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  20. Liu, H., Zhang, X.: The compactness of commutators on the Heisenberg group. Acta Mathematica Sinica, English Series, 22(2), 425–430 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  21. Softova, L.: Singular integrals and commutators in generalized Morrey spaces. Acta Mathematica Sinica, English Series, 22(3), 757–766 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  22. Sun, Y., Su, W.: An endpoint estimate for the commutator of singular integrals. Acta Mathematica Sinica, English Series, 21, 1249–1258 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  23. Pérez, C., Pradolini, D.: Sharp weighted endpoint estimates for commutators of singular integral. Michigan Math. J., 49, 23–37 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  24. Pérez, C., Trujillo-González, R.: Sharp weighted estimates for multilinear commutators. J. London Math. Soc., 65, 672–692 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  25. Grafakos, L.: Estimates for maximal singular integrals. Colloq. Math., 96, 167–177 (2003)

    MATH  MathSciNet  Google Scholar 

  26. Nazarov, F., Treil, S., Volberg, A.: Weak type estimates and Cotlar’s inequalities for Calderón–Zygmund operators on non-homogeneous spaces. Internat. Math. Res. Notices, 9, 463–487 (1998)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, University of Information Engineering, 1001–747, Zhengzhou, 450002, P. R. China

    Guo En Hu

  2. School of Information, Renmin University of China, Beijing, 100872, P. R. China

    Yan Meng

  3. School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, P. R. China

    Da Chun Yang

Authors
  1. Guo En Hu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yan Meng
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Da Chun Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Guo En Hu.

Additional information

The third (corresponding) author is supported by Program for New Century Excellent Talents in University (NCET-04-0142) of China

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hu, G.E., Meng, Y. & Yang, D.C. Boundedness of Some Maximal Commutators in Hardy-type Spaces with Non-doubling Measures. Acta Math Sinica 23, 1129–1148 (2007). https://doi.org/10.1007/s10114-007-0938-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-007-0938-4

Keywords

MR (2000) Subject Classification

Search

Navigation