Acta Mathematica Hungarica

, Volume 157, Issue 2, pp 408–433 | Cite as

Weighted mixed-norm inequality on Doob’s maximal operator and John–Nirenberg inequalities in Banach function spaces

  • W. Chen
  • K.-P. Ho
  • Y. Jiao
  • D. ZhouEmail author


We prove a weighted mixed-norm inequality for the Doob maximal operator on a filtered measure space. We also give some characterizations of martingale BMO spaces in the setting of Banach function spaces. The main method is based on the technique of extrapolation on martingale Banach spaces.

Key words and phrases

mixed-norm inequality weight Doob’s maximal operator extrapolation BMO 

Mathematics Subject Classification

primary 60G46 secondary 60G42 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



The authors thank the referee for careful reading and useful comments.


  1. 1.
    K. Andersen and R. John, Weighted inequalities for vector-valued maximal functions and singular integrals, Studia Math., 69 (1980/81), 19–31Google Scholar
  2. 2.
    Antipa, A.: Doob's inequality for rearrangement-invariant function spaces. Rev. Roumaine Math. Pures Appl. 35, 101–108 (1990)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Aoyama, H.: Lebesgue spaces with variable exponent on a probability space. Hiroshima Math. J. 39, 207–216 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    C. Bennett and R. Sharpley, Interpolation of operators, Pure and Applied Mathematics, vol. 129, Academic Press, Inc. (Boston, MA, 1988)Google Scholar
  5. 5.
    Chen, W., Liu, P.: Weighted inequalities for the generalized maximal operator in martingale spaces. Chin. Ann. Math. Ser. B 32, 781–792 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chen, W., Liu, P.: Weighted norm inequalities for multisublinear maximal operator on martingale spaces. Tohoku Math. J. 66, 539–553 ((2), 2014,)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cruz-Uribe, D., Fiorenza, A.: Variable Lebesgue Spaces. Applied and Numerical Harmonic Analysis, Birkhäuser/Springer (Heidelberg (2013)CrossRefzbMATHGoogle Scholar
  8. 8.
    Cruz-Uribe, D., Fiorenza, A., Martell, J., Pérez, C.: The boundedness of classical operators on variable \(L^p\) spaces. Ann. Acad. Sci. Fenn. Math. 31, 239–264 (2006)MathSciNetzbMATHGoogle Scholar
  9. 9.
    D. Cruz-Uribe, J. Martell, and C. Pérez, Weights, extrapolation and the theory of Rubio de Francia, Operator Theory: Advances and Applications, vol. 215, Birkhäuser/Springer Basel AG (Basel, 2011)Google Scholar
  10. 10.
    Cruz-Uribe, D., Wang, L.: Variable Hardy spaces. Indiana Univ. Math. J. 63, 447–493 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Cruz-Uribe, D., Wang, L.: Extrapolation and weighted norm inequalities in the variable Lebesgue spaces. Trans. Amer. Math. Soc. 369, 1205–1235 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Diening, L.: Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces. Bull. Sci. Math. 129, 657–700 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    L. Diening, R. Harjulehto, P. Hästö, and M. Rǔzička, Lebesgue and Sobolev spaces with variable exponents, Lecture Notes in Mathematics, vol. 2017, Springer (Heidelberg, 2011)Google Scholar
  14. 14.
    Fefferman, C., Stein, E.M.: Some maximal inequalities. Amer. J. Math. 93, 107–115 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    J. García-Cuerva and J. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland Mathematics Studies, vol. 116, North-Holland Publishing Co. (Amsterdam, 1985)Google Scholar
  16. 16.
    L. Grafakos, Modern Fourier Analysis, Second ed., Graduate Texts in Mathematics, 250, Springer (New York, 2009)Google Scholar
  17. 17.
    Ho, K.-P.: Atomic decompositions, dual spaces and interpolations of martingale Hardy-Lorenta-Karamata spaces. Q. J. Math. 65, 985–1009 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Ho, K.-P.: Atomic decomposition of Hardy spaces and characterization of BMO via Banach function spaces. Anal. Math. f38, 173–185 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Ho, K.-P.: John-Nirenberg inequalities on Lebesgue spaces with variable exponents. Taiwanese J. Math. 18, 1107–1118 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Ho, K.-P.: Vector-valued John-Nirenberg inequalities and vector-valued mean osciallations characterization of BMO. Results. Math. 70, 257–270 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Ho, K.-P.: Strong maximal operator on mixed-norm spaces. Ann. Univ. Ferrara Sez. VII Sci. Mat. 62, 275–291 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Hytönen, T., Kemppainen, M.: On the relation of Carleson's embedding and the maximal theorem in the context of Banach space geometry. Math. Scand. 109, 269–284 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    T. Hytönen, J. Neerven, M. Veraar, and L. Weis, Analysis in Banach Spaces, Vol. I, Martingales and Littlewood–Paley theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, vol. 63, Springer (Cham, 2016.)Google Scholar
  24. 24.
    Izuki, M., Nakai, E., Sawano, Y.: Wavelet characterization and modular inequalities for weighted Lebesgue spaces with variable exponent. Ann. Acad. Sci. Fenn. Math. 40, 551–571 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Izumisawa, M., Kazamaki, N.: Weighted norm inequalities for martingales. Tôhoku Math. J. 29, 115–124 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Jawerth, B.: Weighted inequalities for maximal operators: linearization, localization and factorization. Amer. J. Math. 108, 361–414 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Jiao, Y., Peng, L., Liu, P.: Atomic decompositions of Lorentz martingale spaces and applications. J. Funct. Space Appl. 7, 153–166 (2009)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Jiao, Y., Wu, L., Popa, M.: Operator-valued martingale transforms in rearrangement invariant spaces and applications. Sci. China Math. 56, 831–844 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Jiao, Y., Wu, L., Yang, A., Yi, R.: The predual and John-Nirenberg inequalities on generalized BMO martingale spaces. Trans. Amer. Math. Soc. 369, 537–553 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Jiao, J., Xie, G., Zhou, D.: Dual spaces and John-Nirenberg inequalities on martingale Hardy-Lorenta-Karamata spaces. Q. J. Math. 66, 605–623 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Jiao, J., Zhou, D., Hao, Z., Chen, W.: Martingale Hardy spaces with variable exponents. Banach J. Math. Anal. 10, 750–770 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Kikuchi, M.: Characterization of Banach function space that preserve the Burkholder-square-function inequality. Illinois J. Math. 47, 867–882 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Kikuchi, M.: On the Davis inequality in Banach function space. Math. Nachr. 281, 697–708 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Kikuchi, M.: On some inequalities for Doob decompositions in Banach function spaces. Math. Z. 265, 865–887 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Kurtz, D.: Classical operators on mixed-normed spaces with product weights. Rocky Mountain J. Math. 37, 269–283 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    A. Lerner, On a dual property of the maximal operator on weighted variable \(L^ p\) spaces (preprint), arXiv:1509.07664 (2015)
  37. 37.
    J. Lindenstrauss and L. Tzafriri, Classical Banach spaces. II, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 97, Springer-Verlag (Berlin–New York, 1979)Google Scholar
  38. 38.
    R. Long, Martingale Spaces and Inequalities, Peking University Press (Beijing); Friedr. Vieweg & Sohn (Braunschweig, 1993)Google Scholar
  39. 39.
    Miyamoto, T., Nakai, E., Sadasue, G.: Martingale Orlicz-Hardy spaces. Math. Nachr. 285, 670–686 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Muckenhoupt, B.: Weighted norm inequalities for the Hardy maximal function. Trans. Amer. Math. Soc. 165, 207–226 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    E. Nakai and G. Sadasue, Martingale Morrey–Campanato spaces and fractional integrals, J. Funct. Spaces Appl. (2012), Article ID 673929, 29 ppGoogle Scholar
  42. 42.
    Nakai, E., Sadasue, G.: Maximal function on generalized martingale Lebesgue space with variable exponent. Stat. Probab. Lett. 83, 2168–2171 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    E. Nakai, G. Sadasue, and Y. Sawano, Martingale Morrey–Hardy and Campanato–Hardy spaces, J. Funct. Spaces Appl., (2013), Article ID 690258, 14 ppGoogle Scholar
  44. 44.
    Nakai, E., Sawano, Y.: Hardy spaces with variable exponents and generalized Campanato spaces. J. Funct. Anal. 262, 3665–3748 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Sawyer, E.: A characterization of a two-weight norm inequality for maximal operators. Studia Math. 75, 1–11 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Tanaka, H., Terasawa, Y.: Positive operators and maximal operators in a filtered measure space. J. Funct. Anal. 264, 920–946 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Weisz, F.: Dual spaces of multi-parameter martingale Hardy spaces. Q. J. Math. 67, 137–145 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    F. Weisz, Martingale Hardy Spaces and their Applications in Fourier Analysis, Lecture Notes in Mathematics, vol. 1568, Springer-Verlag (Berlin, 1994)Google Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2018

Authors and Affiliations

  1. 1.School of Mathematical SciencesYangzhou UniversityYangzhouChina
  2. 2.Department of Mathematics and Information TechnologyThe Hong Kong Institute of EducationTai Po, Hong KongChina
  3. 3.School of Mathematical SciencesCentral South UniversityChangshaChina

Personalised recommendations