Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Maximal Operators and Characterization of Hardy Spaces


It is known that the maximal operator σ* ƒ is of type (Hp,Lp) if the Vilenkin group G is bounded and \(p > \tfrac{1}{2}\). We prove a maximal converse inequality which characterizes the space Hp by means of the operator σƒ:= supn |σMnƒ|, for bounded groups.

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


  1. [1]

    M. Avdispahić, N. Memić and F. Weisz, Maximal functions, Hardy spaces and Fourier multiplier theorems on unbounded Vilenkin groups, J. Math. Anal. Appl., 390 (2012), 68–73.

  2. [2]

    I. Blahota, G. Gát and U. Goginava, Maximal operators on Fejér means of Vilenkin-Fourier series, J. Ineq. Pure Appl. Math., 7 (2006), Article ID 149.

  3. [3]

    N. Fujii, A maximal inequality for H1 functions on the generalized Walsh–Paley group, Proc. Amer. Math. Soc, 77 (1979), 111–116.

  4. [4]

    G. Gát, Cesaro means of integrable functions with respect to unbounded Vilenkin systems, J. Approx. Theory, 124 (2003), 25–43.

  5. [5]

    U. Goginava, The maximal operator of Marcinkiewicz–Fejér means of the d-dimensional Walsh-Fourier series, East J. Approx., 12 (2006), 295–302.

  6. [6]

    J. Pál and P. Simon, On a generalization of the concept of derivative, Acta Math. Acad. Sci. Hungar., 29) (1977), 155–164.

  7. [7]

    F. Schipp, W. R. Wade, P. Simon and J. Pál, Walsh series. An Introduction to Dyadic Harmonic Aanalysis, Akadémiai Kiadó (Budapest), Adam Hilger (Bristol, New York, 1990).

  8. [8]

    P. Simon, Investigations with respect to the Vilenkin system, Ann. Univ. Sci. Budapest. Sect. Math., 27 (1982), 87–101.

  9. [9]

    N. Ja. Vilenkin, On a class of complete orthonormal systems, Izv. Akad. Nauk SSSR, Ser. Math., 11 (1947), 363–400 (in Russian); translation in Amer. Math. Soc. Transl., 28 (1963), 1–35.

  10. [10]

    F. Weisz, Bounded operators on weak Hardy spaces and applications, Acta Math. Hungar., 80 (1998), 249–264.

  11. [11]

    F. Weisz, Cesàro summability of one- and two-dimensional Walsh—Fourier series, Analysis Math., 22 (1996), 229–242.

  12. [12]

    F. Weisz, Cesàro summability of two-dimensional Walsh-Fourier series, Trans. Amer. Math. Soc, 348 (1996), 2196–2181.

  13. [13]

    F. Weisz, Martingale Hardy Spaces and their Applications in Fourier Analysis, Lect. Notes in Math., vol. 1568, Springer ((Berlin, Heidelberg, New York, 1994).

  14. [14]

    F. Weisz, θ-summability of Fourier series, Acta Math. Hungar., 103 (2004), 139–176.

Download references


The authors would like to thank the referees for their valuable remarks and suggestions.

Author information

Correspondence to N. Memić.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Memić, N., Sadiković, S. Maximal Operators and Characterization of Hardy Spaces. Anal Math 46, 119–131 (2020).

Download citation

Key words and phrases

  • Vilenkin group
  • Hardy space
  • maximal operator

Mathematics Subject Classification

  • 42C10