Skip to main content
SpringerLink
Log in

Littlewood-Paley theory on metric spaces with non doubling measures and its applications

  • Articles
  • Published:
Science China Mathematics Aims and scope Submit manuscript

Abstract

The purpose of this paper is to extend the Littlewood-Paley theory to a geometrically doubling metric space with a non-doubling measure satisfying a weak growth condition. Moreover, we prove that our setting mentioned above, is equivalent to the one introduced and studied by Hytönen (2010) in his remarkable framework, i.e., the geometrically doubling metric space with a non-doubling measure satisfying a so-called upper doubling condition. As an application, we obtain the T1 theorem in this more general setting. Moreover, the Gaussian measure is also discussed.

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

Similar content being viewed by others

References

  1. Bui T A, Duong X T. Hardy spaces, regularized BMO and the boundedness of Calderón-Zygmund operators on non-homogeneous spaces. J Geom Anal, 2013, 23: 895–932

    Article  MATH  MathSciNet  Google Scholar 

  2. Coifman R R, Weiss G. Analyse Harmonique Non-Commutative Sur Certains Espaces Homogènes. Berlin-New York: Springer-Verlag, 1971

    MATH  Google Scholar 

  3. David G, Journé J L, Semmes S. Opérateurs de Calderón-Zygmund, fonctions para-accrétives et interpolation. Rev Mat Iberoamericana, 1985, 1: 1–56

    Article  MATH  Google Scholar 

  4. Deng D G, Han Y S. Harmonic Analysis on Spaces of Homogeneous Type. Berlin: Springer-Verlag, 2009

    MATH  Google Scholar 

  5. Deng D, Han Y, Yang D. Besov spaces with non-doubling measures. Trans Amer Math Soc, 2006, 358: 2965–3001

    Article  MATH  MathSciNet  Google Scholar 

  6. Fu X, Yang D, Yuan W. Boundedness of multilinear commutators of Calderón-Zygmund operators on Orlicz spaces over non-homogeneous spaces. Taiwanese J Math, 2012, 16: 2203–2238

    MATH  MathSciNet  Google Scholar 

  7. García-Cuerva J, Martell J M. Weighted inequalities and vector-valued Calderón-Zygmund operators on nonhomogeneous spaces. Publ Mat, 2000, 44: 613–640

    Article  MATH  MathSciNet  Google Scholar 

  8. García-Cuerva J, Martell J M. Two-weight norm inequalities for maximal operators and fractional integrals on nonhomogeneous spaces. Indiana Univ Math J, 2001, 50: 1241–1280

    Article  MATH  MathSciNet  Google Scholar 

  9. Han Y S, Sawyer E T. Littlewood-Paley Theory on Spaces of Homogeneous Type and the Classical Function Spaces. Providence, RI: Amer Math Soc, 1994

    Google Scholar 

  10. Han Y, Yang D. Triebel-Lizorkin spaces with non-doubling measures. Studia Math, 2004, 162: 105–140

    Article  MATH  MathSciNet  Google Scholar 

  11. Hytönen T. A framework for non-homogeneous analysis on metric spaces, and the RBMO space of Tolsa. Publ Mat, 2010, 54: 485–504

    Article  MATH  MathSciNet  Google Scholar 

  12. Hytönen T, Kairema A. Systems of dyadic cubes in a doubling metric space. Colloq Math, 2012, 126: 1–33

    Article  MATH  MathSciNet  Google Scholar 

  13. Hytönen T, Liu S, Yang D, et al. Boundedness of Calderón-Zygmund operators on non-homogeneous metric measure spaces. Canad J Math, 2012, 64: 892–923

    Article  MATH  MathSciNet  Google Scholar 

  14. Hytönen T, Martikainen H. Non-homogeneous Tb theorem and random dyadic cubes on metric measure spaces. J Geom Anal, 2012, 22: 1071–1107

    Article  MATH  MathSciNet  Google Scholar 

  15. Hytönen T, Martikainen H. On general local Tb theorems. Trans Amer Math Soc, 2012, 364: 4819–4846

    Article  MATH  MathSciNet  Google Scholar 

  16. Hytönen T, Yang D C, Yang D Y. The Hardy Space H 1 on Non-homogeneous Metric Spaces. Math Proc Cambridge Philos Soc, 2012, 153: 9–31

    Article  MATH  MathSciNet  Google Scholar 

  17. Lin H, Yang D. Spaces of type BLO on non-homogeneous metric measure spaces. Front Math China, 2011, 6: 271–292

    Article  MATH  MathSciNet  Google Scholar 

  18. Lin H, Yang D. Equivalent boundedness of Marcinkiewicz integrals on non-homogeneous metric measure spaces. Sci China Math, 2014, 57: 123–144

    Article  MATH  MathSciNet  Google Scholar 

  19. Liu L, Yang D. BLO spaces associated with the Ornstein-Uhlenbeck operator. Bull Sci Math, 2008, 132: 633–649

    Article  MATH  MathSciNet  Google Scholar 

  20. Liu L, Yang D. Characterizations of BMO associated with Gauss measures via commutators of local fractional integrals. Israel J Math, 2010, 180: 285–315

    Article  MATH  MathSciNet  Google Scholar 

  21. Liu L, Yang D. Boundedness of Littlewood-Paley operators associated with Gauss measures. J Inequal Appl, 2010, Art ID 643948, 41pp

    Google Scholar 

  22. Liu S, Yang D, Yang D. Boundedness of Calderón-Zygmund operators on non-homogeneous metric measure spaces: Equivalent characterizations. J Math Anal Appl, 2012, 386: 258–272

    Article  MATH  MathSciNet  Google Scholar 

  23. Maas J, Van Neerven J, Portal P. Whitney coverings and the tent space T 1,q(γ) for the Gaussian measure. Ark Mat, 2012, 50: 379–395

    Article  MATH  MathSciNet  Google Scholar 

  24. Maas J, Van Neerven J, Portal P. Non-tangential maximal functions and conical square functions with respect to the Gaussian measure. Publ Mat, 2011, 55: 313–341

    Article  MATH  MathSciNet  Google Scholar 

  25. Mateu J, Mattila P, Nicolau A, et al. BMO for non doubling measures. Duke Math J, 2000, 102: 533–565

    Article  MATH  MathSciNet  Google Scholar 

  26. Mauceri G, Meda S. BMO and H 1 for the Ornstein-Uhlenbeck operator. J Funct Anal, 2007, 252: 278–313

    Article  MATH  MathSciNet  Google Scholar 

  27. Mauceri G, Meda S, Sjögren P. Endpoint estimates for first-order Riesz transforms associated to the Ornstein-Uhlenbeck operator. Rev Mat Iberoamericana, 2012, 28: 77–91

    Article  MATH  Google Scholar 

  28. Nazarov F, Treil S, Volberg A. Cauchy integral and Calderón-Zygmund operators on nonhomogeneous spaces. Int Math Res Not, 1997, 15: 703–726

    Article  MathSciNet  Google Scholar 

  29. Nazarov F, Treil S, Volberg A. Weak type estimates and Cotlar inequalities for Calderón-Zygmund operators on nonhomogeneous spaces. Int Math Res Not, 1998, 9: 463–487

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  31. Saitoh S. Integral Transforms, Reproducing Kernels and Their Applications. New York: Addison Wesley Longman, 1997

    MATH  Google Scholar 

  32. Sawano Y, Tanaka H. Morrey spaces for non-doubling measures. Acta Math Sin Engl Ser, 2005, 21: 1535–1544

    Article  MATH  MathSciNet  Google Scholar 

  33. Sawano Y, Tanaka H. Besov-Morrey spaces and Triebel-Lizorkin-Morrey spaces for non-doubling measures. Math Nachr, 2009, 282: 1788–1810

    Article  MATH  MathSciNet  Google Scholar 

  34. Tchoundja E. Carleson measures for the generalized Bergman spaces via a T(1)-type theorem. Ark Mat, 2008, 46: 377–406

    Article  MATH  MathSciNet  Google Scholar 

  35. Tolsa X. A T(1) theorem for non-doubling measures with atoms. Proc London Math Soc (3), 2001, 82: 195–228

    Article  MATH  MathSciNet  Google Scholar 

  36. Tolsa X. Analytic capacity and Calderón-Zygmund theory with non doubling measures. In: Seminar of Mathematical Analysis. Colecc Abierta, vol. 71. Seville: University Sevilla Secr Publ, 2004, 239-271

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

    Article  MATH  MathSciNet  Google Scholar 

  38. Tolsa X. BMO, H 1, and Calderón-Zygmund operators for non doubling measures. Math Ann, 2001, 319: 89–149

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  40. Volberg A, Wick B D. Bergman-type singular integral operators and the characterization of Carleson measures for Besov-Sobolev spaces and the complex ball. Amer J Math, 2012, 134: 949–992

    Article  MATH  MathSciNet  Google Scholar 

  41. Yang D, Yang D, Hu G. The Hardy Space H 1 with Non-Doubling Measures and Their Applications. Berlin: Springer-Verlag, 2013

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Shantou University, Shantou, 515063, China

    ChaoQiang Tan

  2. Department of Mathematics, Macquarie University, Sydney, NSW, 2109, Australia

    Ji Li

Authors
  1. ChaoQiang Tan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ji Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ji Li.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tan, C., Li, J. Littlewood-Paley theory on metric spaces with non doubling measures and its applications. Sci. China Math. 58, 983–1004 (2015). https://doi.org/10.1007/s11425-014-4950-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-014-4950-8

Keywords

MSC(2010)

Search

Navigation