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The Walsh–Kaczmarz–Marcinkiewicz means and Hardy spaces

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Abstract

It is known that the maximal operator \({\sigma^{\kappa,*}(f)} := sup_{n \in \mathbf{P}}{|{\sigma}_{n}^{\kappa} (f)|}\) is bounded from the dyadic Hardy space \({H_{p}}\) into the space \({L_{p}}\) for \({p > 2/3}\) [6]. Moreover, Goginava and Nagy showed that \({\sigma^{\kappa,*}}\) is not bounded from the Hardy space \({H_{2/3}}\) to the space \({L_{2/3}}\) [9]. The main aim of this paper is to investigate the case \({0 < p < 2/3}\). We show that the weighted maximal operator \({\tilde{\sigma}^{\kappa,*,p}(f) :=sup_{n\in \mathbf{P}} \frac{|{\sigma}_{n}^\kappa (f)|}{n^{2/p-3}}}\), is bounded from the Hardy space \({H_{p}}\) into the space \({L_{p}}\) for any \({0 < p < 2/3}\). With its aid we provide a necessary and sufficient condition for the convergence of Walsh–Kaczmarz–Marcinkiewicz means in terms of modulus of continuity on the Hardy space \({H_p}\), and prove a strong convergence theorem for this means.

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Authors and Affiliations

  1. Institute of Mathematics and Computer Sciences, University of Nyíregyháza, P.O. Box 166, Nyíregyháza, H-4400, Hungary

    K. Nagy

  2. Institute of Mathematics, Faculty of Exact and Natural Sciences, Tbilisi State University, Chavchavadze str. 1, Tbilisi, 0128, Georgia

    G. Tephnadze

  3. Department of Engineering Sciences and Mathematics, Luleå University of Technology, SE-971 87, Luleå, Sweden

    G. Tephnadze

Authors
  1. K. Nagy
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  2. G. Tephnadze
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Correspondence to G. Tephnadze.

Additional information

Research is supported by project TÁMOP-4.2.2.A-11/1/KONV-2012-0051, Campus Hungary grant B2/4R/15701 and Shota Rustaveli National Science Foundation grants YS15-2.1.1-47 and DI/9/5-100/13.

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Nagy, K., Tephnadze, G. The Walsh–Kaczmarz–Marcinkiewicz means and Hardy spaces. Acta Math. Hungar. 149, 346–374 (2016). https://doi.org/10.1007/s10474-016-0617-y

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  • DOI: https://doi.org/10.1007/s10474-016-0617-y

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