Skip to main content
SpringerLink
Log in

Musielak-Orlicz BMO-type spaces associated with generalized approximations to the identity

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

Abstract

Let X be a space of homogenous type and φ: X × [0,∞) → [0,∞) be a growth function such that φ (·, t) is a Muckenhoupt weight uniformly in t and φ(x, ·) an Orlicz function of uniformly upper type 1 and lower type p ∈ (0, 1]. In this article, the authors introduce a new Musielak-Orlicz BMO-type space BMO φ A (χ) associated with the generalized approximation to the identity, give out its basic properties and establish its two equivalent characterizations, respectively, in terms of the spaces BMO φ A,max (χ) and \(\widetilde{BMO}_A^\phi (\chi )\). Moreover, two variants of the John-Nirenberg inequality on BMO φ A (χ) are obtained. As an application, the authors further prove that the space \(BMO_{\sqrt \Delta }^\phi (\mathbb{R}^n )\), associated with the Poisson semigroup of the Laplace operator Δ on ℝn, coincides with the space \(BMO^\phi (\mathbb{R}^n )\) introduced by Ky.

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. Bonami, A., Feuto, J., Grellier, S.: Endpoint for the DIV-CURL lemma in Hardy spaces. Publ. Mat., 54, 341–358 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bonami, A., Grellier, S.: Hankel operators and weak factorization for Hardy-Orlicz spaces. Colloq. Math., 118, 107–132 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bonami, A., Grellier, S., Ky, L. D.: Paraproducts and products of functions in BMO(ℝn) and H 1(ℝn) through wavelets. J. Math. Pures Appl. (9), 97, 230–241 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bonami, A., Iwaniec, T., Jones, P., et al.: On the product of functions in BMO and H 1. Ann. Inst. Fourier (Grenoble), 57, 1405–1439 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bui, H.: Weighted Hardy spaces. Math. Nachr., 103, 45–62 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bui, T. A., Duong, X. T.: Weighted BMO spaces associated to operators. ArXiv: 1201.5828, 2012, preprint

    Google Scholar 

  7. Christ, M.: A T(b) theorem with remarks on analytic capacity and the Cauchy integral. Colloq. Math., 60/61, 601–628 (1990)

    MathSciNet  Google Scholar 

  8. Cruz-Uribe, D., Neugebauer, C. J.: The structure of the reverse Hölder classes. Trans. Amer. Math. Soc., 347, 2941–2960 (1995)

    MathSciNet  MATH  Google Scholar 

  9. Coifman, R. R., Weiss, G.: Analyse Harmonique Non-commutative sur Certains Espaces Homogènes, Lecture Notes in Math., 242, Springer-Verlag, Berlin-New York, 1971

    MATH  Google Scholar 

  10. Coifman, R. R., Weiss, G.: Extensions of Hardy spaces and their use in analysis. Bull. Amer. Math. Soc., 83, 569–645 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  11. Davies, E. B.: Heat Kernels and Spectral Theory, Cambridge Tracts in Math., Vol. 92, Cambridge University Press, Cambridge, 1990

    Google Scholar 

  12. Deng, D., Duong, X. T., Yan, L.: A characterization of the Morrey-Campanato spaces. Math. Z., 250, 641–655 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. Diening, L.: Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces. Bull. Sci. Math., 129, 657–700 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. Diening, L., Hästö, P., Roudenko, S.: Function spaces of variable smoothness and integrability. J. Funct. Anal., 256, 1731–1768 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Duong, X. T., McIntosh, A.: Singular integral operators with non-smooth kernels on irregular domains. Rev. Mat. Iberoamericana, 15, 233–265 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  16. Duong, X. T., Xiao, J., Yan, L.: Old and new Morrey spaces with heat kernel bounds. J. Fourier Anal. Appl., 13, 87–111 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Duong, X. T., Yan, L.: New function spaces of BMO type, the John-Nirenberg inequality, interpolation, and applications. Comm. Pure Appl. Math., 58, 1375–1420 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  18. Fefferman, C., Stein, E. M.: H p spaces of several variables. Acta Math., 129, 137–193 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  19. Garcá-Cuerva, J., Rubio de Francia, J.: Weighted Norm Inequalities and Related Topics, Amsterdam, North-Holland, 1985

    Google Scholar 

  20. Grafakos, L.: Modern Fourier Analysis, Second edition, Graduate Texts in Mathematics 250, Springer, New York, 2008

    Google Scholar 

  21. Han, Y., Müller, D., Yang, D.: A theory of Besov and Triebel-Lizorkin spaces on metric measure spaces modeled on Carnot-Carathéodory spaces. Abstr. Appl. Anal., Art. ID 893409, 250 pp. (2008)

    Google Scholar 

  22. Hu, G., Meng, Y., Yang, D.: Boundedness of some maximal commutators in Hardy-type spaces with non-doubling measures. Acta Math. Sin., Engl. Series, 23, 1129–1148 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  23. Hu, G., Zhu, Y.: Weighted norm inequalities with general weights for the commutator of Calderón. Acta Math. Sin., Engl. Series, 29, 505–514 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  24. Hou, S., Yang, D., Yang, S.: Lusin area function and molecular characterizations of Musielak-Orlicz Hardy spaces and their applications. Commun. Contemp. Math., 15(6), 1350029, 37pp. (2013)

    Article  MathSciNet  Google Scholar 

  25. Janson, S.: Generalizations of Lipschitz spaces and an application to Hardy spaces and bounded mean oscillation. Duke Math. J., 47, 959–982 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  26. John, F., Nirenberg, L.: On functions of bounded mean oscillation. Comm. Pure Appl. Math., 14, 415–426 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  27. Johnson, R., Neugebauer, C. J.: Homeomorphisms preserving A p. Rev. Mat. Iberoamericana, 3, 249–273 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  28. Journé, J.-L.: Calderón-Zygmund Operators, Pseudodifferential Operators and the Cauchy Integral of Calderón, Lecture Notes in Math., 994, Springer, Berlin, 1983

    Google Scholar 

  29. Ky, L. D.: Bilinear decompositions and commutators of singular integral operators. Trans. Amer. Math. Soc., 365, 2931–2958 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  30. Ky, L. D.: New Hardy spaces of Musielak-Orlicz type and boundedness of sublinear operators. Integral Equations Operator Theory, 78, 115–150 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  31. Ky, L. D.: Endpoint estimates for commutators of singular integrals related to Schrödinger operators. ArXiv: 1203.6335, 2012, preprint

    Google Scholar 

  32. Ky, L. D.: Bilinear decompositions for the product space H 1L ×BMOL. Math. Nachr., Doi: 10.1002/mana.201200101, to appear

  33. Ky, L. D.: On weak*-convergence in H 1L (ℝd). Potential Anal., 39, 355–368 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  34. Lai, Q.: The weighted #-operator and maximal operator on spaces of homogeneous type. Chinese Ann. Math. Ser. A, 8, 600–610 (1987)

    MathSciNet  MATH  Google Scholar 

  35. Lerner, A. K.: Some remarks on the Hardy-Littlewood maximal function on variable L p spaces. Math. Z., 251, 509–521 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  36. Liang, Y., Huang, J., Yang, D.: New real-variable characterizations of Musielak-Orlicz Hardy spaces. J. Math. Anal. Appl., 395, 413–428 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  37. Liang, Y., Liu, L., Yang, D.: An off-diagonal Marcinkiewicz interpolation theorem on Lorentz spaces. Acta Math. Sin., Engl. Series, 27, 1477–1488 (2011)

    Article  MathSciNet  Google Scholar 

  38. Liang, Y., Yang, D.:, Musielak-Orlicz Campanato spaces and applications. J. Math. Anal. Appl., 406, 307–322 (2013)

    Article  MathSciNet  Google Scholar 

  39. Liu, L., Yang, Da., Yang, Do.: Atomic Hardy-type spaces between H 1 and L 1 on metric spaces with non-doubling measures. Acta Math. Sin., Engl. Series, 27, 2445–2468 (2011)

    Article  MATH  Google Scholar 

  40. Long, R., Yang, L.: BMO functions in spaces of homogeneous type. Sci. Sinica Ser. A, 27, 695–708 (1984)

    MathSciNet  MATH  Google Scholar 

  41. Muckenhoupt, B., Wheeden, R.: On the dual of weighted H 1 of the half-space. Studia Math., 63, 57–79 (1978)

    MathSciNet  MATH  Google Scholar 

  42. Muckenhoupt, B., Wheeden, R.: Weighted bounded mean oscillation and the Hilbert transform. Studia Math., 54, 221–237 (1975/76)

    MathSciNet  Google Scholar 

  43. Musielak, J.: Orlicz Spaces and Modular Spaces, Lecture Notes in Math., 1034, Springer-Verlag, Berlin, 1983

    MATH  Google Scholar 

  44. Nakai, E., Yabuta, K.: Pointwise multipliers for functions of bounded mean oscillation. J. Math. Soc. Japan, 37, 207–218 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  45. Rao, M., Ren, Z.: Applications of Orlicz Spaces, Dekker, New York, 2000

    Google Scholar 

  46. Rao, M., Ren, Z.: Theory of Orlicz Spaces, Dekker, New York, 1991

    MATH  Google Scholar 

  47. Si, Z., Xue, Q.: λ-central BMO estimates for commutators of multilinear maximal operators. Acta Math. Sin., Engl. Series, 29, 729–742 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  48. Strömberg, J.-O.: Bounded mean oscillation with Orlicz norms and duality of Hardy spaces. Indiana Univ. Math. J., 28, 511–544 (1979)

    Article  MathSciNet  Google Scholar 

  49. Strömberg, J.-O., Torchinsky, A.: Weighted Hardy Spaces, Lecture Notes in Math., 1381, Springer-Verlag, Berlin, 1989

    MATH  Google Scholar 

  50. Tang, L.: New function spaces of Morrey-Campanato type on spaces of homogeneous type. Illinois J. Math., 51, 625–644 (2007)

    MathSciNet  MATH  Google Scholar 

  51. Viviani, B. E.: An atomic decomposition of the predual of BMO(ρ). Rev. Mat. Iberoamericana, 3, 401–425 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  52. Yang, D., Yang, S.: Local Hardy spaces of Musielak-Orlicz type and their applications. Sci. China Math., 55, 1677–1720 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  53. Yang, D., Yang, S.: Musielak-Orlicz Hardy spaces associated with operators and their applications. J. Geom. Anal., 24, 495–570 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  54. Yang, D., Zhou, Y.: Some new characterizations on spaces of functions with bounded mean oscillation. Math. Nachr., 283, 588–614 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  55. Yang, D., Zhou, Y.: New properties of Besov and Triebel-Lizorkin spaces on RD-spaces. Manuscripta Math., 134, 59–90 (2011)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing, 100875, P. R. China

    Shao Xiong Hou & Da Chun Yang

  2. School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, P. R. China

    Si Bei Yang

Authors
  1. Shao Xiong Hou
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Da Chun Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Si Bei Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Da Chun Yang.

Additional information

The second author is supported by National Natural Science Foundation of China (Grant Nos. 11171027 and 11361020), the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20120003110003) and the Fundamental Research Funds for Central Universities of China (Grant Nos. 2012LYB26, 2012CXQT09 and lzujbky-2014-18)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hou, S.X., Yang, D.C. & Yang, S.B. Musielak-Orlicz BMO-type spaces associated with generalized approximations to the identity. Acta. Math. Sin.-English Ser. 30, 1917–1962 (2014). https://doi.org/10.1007/s10114-014-3181-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-014-3181-9

Keywords

MR(2010) Subject Classification

Search

Navigation