Skip to main content
Log in

On the maximal operator of (C, α)-means of Walsh–Kaczmarz–Fourier series

  • Published:
Ukrainian Mathematical Journal Aims and scope

Simon [J. Approxim. Theory, 127, 39–60 (2004)] proved that the maximal operator σα,κ,* of the (C, α)-means of the Walsh–Kaczmarz–Fourier series is bounded from the martingale Hardy space H p to the space L p for p > 1 / (1 + α), 0 < α ≤ 1. Recently, Gát and Goginava have proved that this boundedness result does not hold if p ≤ 1 / (1 + α). However, in the endpoint case p = 1 / (1 + α ), the maximal operator σα,κ,* is bounded from the martingale Hardy space H 1/(1+α) to the space weak- L 1/(1+α). The main aim of this paper is to prove a stronger result, namely, that, for any 0 < p ≤ 1 / (1 + α), there exists a martingale fH p such that the maximal operator σα,κ,* f does not belong to the space L p .

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

Access this article

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. A. A. Šneider, “On series with respect to the Walsh functions with monotone coefficients,” Izv. Akad. Nauk SSSR, Ser. Mat., 12, 179–192 (1948).

    Google Scholar 

  2. F. Schipp, “Pointwise convergence of expansions with respect to certain product systems,” Anal. Math., 2, 63–75 (1976).

    Google Scholar 

  3. W. S. Young, “On the a.e. convergence of Walsh–Kaczmarz–Fourier series,” Proc. Amer. Math. Soc., 44, 353–358 (1974).

    MATH  MathSciNet  Google Scholar 

  4. V. A. Skvortsov, “On Fourier series with respect to the Walsh–Kaczmarz system,” Anal. Math., 7, 141–150 (1981).

    Article  MATH  MathSciNet  Google Scholar 

  5. G. Gát, “On (C, 1 ) summability of integrable functions with respect to the Walsh–Kaczmarz system,” Stud. Math., 130, 135–148 (1998).

    MATH  Google Scholar 

  6. P. Simon, “On the Cesàro summability with respect to the Walsh–Kaczmarz system,” J. Approxim. Theory, 106, 249–261 (2000).

    Article  MATH  Google Scholar 

  7. F. Weisz, “ϑ-summability of Fourier series,” Acta Math. Hungar., 103, No. 1–2, 139–176 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  8. P. Simon, “(C, α) summability of Walsh–Kaczmarz–Fourier series,” J. Approxim. Theory, 127, 39–60 (2004).

    Article  MATH  Google Scholar 

  9. G. Gát and U. Goginava, “The weak type inequality for the maximal operator of the (C, α) -means of the Fourier series with respect to the Walsh–Kaczmarz system,” Acta Math. Hungar., 125, No. 1–2, 65–83 (2009).

    Article  MATH  MathSciNet  Google Scholar 

  10. F. Schipp, W. R. Wade, P. Simon, and J. Pál, Walsh Series. An Introduction to Dyadic Harmonic Analysis, Adam Hilger, Bristol–New York (1990).

    MATH  Google Scholar 

  11. F. Weisz, Summability of Multi-Dimensional Fourier Series and Hardy Space, Kluwer, Dordrecht (2002).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 62, No. 2, pp. 158–166, February, 2010.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Goginava, U., Nagy, K. On the maximal operator of (C, α)-means of Walsh–Kaczmarz–Fourier series. Ukr Math J 62, 175–185 (2010). https://doi.org/10.1007/s11253-010-0342-6

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-010-0342-6

Keywords

Navigation