John–Nirenberg Type Inequalities for Musielak–Orlicz Campanato Spaces on Spaces of Homogeneous Type

  • Duong Quoc Huy
  • Luong Dang KyEmail author


Let X be a space of homogeneous type in the sense of Coifman and Weiss. Let φ : X × [0, ) → [0, ) be such that φ(x,⋅) is an Orlicz function and φ(⋅, t) is a Muckenhoupt A(X) weight uniformly in t. In this paper, we propose John–Nirenberg type inequalities for Musielak–Orlicz Campanato spaces on spaces of homogeneous type. As an application, we show the coincidence between the space BMO(X) and the weighted space BMOw(X) whenever wA(X).


Musielak–Orlicz function BMO space Campanato space John–Nirenberg inequality Spaces of homogeneous type 

Mathematics Subject Classification (2010)

43A85 46E30 



The authors would like to thank the referees for their careful reading and helpful suggestions.

Funding Information

This work is supported by the Vietnam National Foundation for Science and Technology Development (Grant No. 101.02-2016.22)


  1. 1.
    Anderson, R.C., Cruz-Uribe, D., Moen, K.: Logarithmic bump conditions for calderón–zygmund operators on spaces of homogeneous type. Publ. Mat. 59, 17–43 (2015)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bonami, A., Cao, J., Ky, L.D., Liu, L., Yang, D., Yuan, W.: Multiplication between Hardy spaces and their dual spaces. (submitted)Google Scholar
  3. 3.
    Calderón, A.P.: Inequalities for the maximal function relative to a metric. Stud. Math. 57, 297–306 (1976)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Campanato, S.: Proprietà di una famiglia di spazi funzionali. Ann. Scuola Norm. Sup. Pisa 18, 137–160 (1964)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Coifman, R.R., Weiss, G.: Extensions of Hardy spaces and their use in analysis. Bull. Am. Math. Soc. 83, 569–645 (1977)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Fefferman, C., Stein, E.M.: h p spaces of several variables. Acta Math. 129, 137–193 (1972)MathSciNetCrossRefGoogle Scholar
  7. 7.
    John, F., Nirenberg, L.: On functions of bounded mean oscillation. Commun. Pure Appl. Math. 14, 415–426 (1961)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Ky, L.D.: New Hardy spaces of Musielak–Orlicz type and boundedness of sublinear operators. Integr. Equ. Oper. Theory 78, 115–150 (2014)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Li, W.: John–nirenberg type inequalities for the Morrey–Campanato spaces. J. Inequal. Appl. 2008, 239414 (2008)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Liang, Y., Yang, D.: Musielak–orlicz Campanato spaces and applications. J. Math. Anal. Appl. 406, 307–322 (2013)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Mateu, J., Mattila, P., Nicolau, A., Orobitg, J.: BMO For nondoubling measures. Duke Math. J. 102, 533–565 (2000)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Morrey, C.B.: On the solutions of quasi-linear elliptic partial differential equations. Trans. Am. Math. Soc. 43, 126–166 (1938)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Muckenhoupt, B., Wheeden, R.L.: Weighted bounded mean oscillation and the Hilbert transform. Stud. Math. 54, 221–237 (1976)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Rafeiro, H., Samko, N., Samko, S.: Morrey–Campanato spaces: an overview. In: Karlovich, Y.I., et al. (eds.) Operator Theory, Pseudo-Differential Equations, and Mathematical Physics. Operator Theory Advances and Applications, vol. 228, pp 293–323. Birkhäuser/Springer, Basel (2013)Google Scholar
  15. 15.
    Strömberg, J., Torchinsky, A.: Weighted Hardy Spaces. Lecture Notes in Mathematics, vol. 1381. Springer, Berlin (1989)Google Scholar
  16. 16.
    Trong, N.N., Tung, N.T.: Weighted BMO type spaces associated to admissible functions and applications. Acta. Math. Vietnam. 41, 209–241 (2016)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Yang, D., Liang, Y., Ky, L.D.: Real-Variable Theory of Musielak–Orlicz Hardy Spaces. Lecture Notes in Mathematics, vol. 2182. Springer, Cham (2017)Google Scholar

Copyright information

© Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  1. 1.Department of Natural Science and TechnologyTay Nguyen UniversityDak LakVietnam
  2. 2.Applied Analysis Research Group, Faculty of Mathematics and StatisticsTon Duc Thang UniversityHo Chi Minh CityVietnam

Personalised recommendations