Cesàro means of Vilenkin-Fourier series (Q2770606)

The author investigates some weighted one- and two-parameter Fejér means with respect to Vilenkin systems. It is known that there is a sharp distance between the so-called bounded and unbounded cases (when the generating sequence of the Vilenkin group is not bounded). For example, in the bounded case the one-parameter Fejér means of the Vilenkin-Fourier series of an integrable function converge a.e. to the function [see \textit{J. Pál} and \textit{P. Simon}, Acta Math. Acad. Sci. Hung. 29, 155-164 (1993; Zbl 0345.42011)]. NEWLINENEWLINENEWLINEOn the other hand, in the unbounded case there is a continuous function on the group such that its Fejér means diverge at a prescribed point [\textit{J. Price}, Can. J. Math. 9, 413-425 (1957; Zbl 0079.09204)]. NEWLINENEWLINENEWLINEHowever, the reviewer proved [\textit{G. Gát}, J. Approx. Theory 101, No. 1, 1-36 (1999; Zbl 0972.42019)] that the Vilenkin-Fejér means of a function belonging to \(L^p\) for some \(p>1\) converge to the function even in the unbounded case. NEWLINENEWLINENEWLINE\textit{W. R. Wade} [Appl. Numer. Harmon. Anal., Boston, MA; Birkhäuser, 41-50 (1999; Zbl 0918.42017)] obtained estimates for the growth of special Vilenkin groups. His main result is a maximal theorem which states that a weighted maximal function of special Fejér means belongs to the Lorentz space \(L^{p,q}\) if \(f\) is from the Hardy-Lorentz space \(H^{p,q}\). Moreover, the maximal operator is bounded from \(H^{p,q}\) to \(L^{p,q}\). NEWLINENEWLINENEWLINEThe author in this work improves these results of Wade. The assumption on the generating sequence of the Vilenkin-group is weakened. Furthermore, he modifies the definition of the maximal function. He applies weights less then in the paper of Wade and proves that the boundedness of the corresponding maximal operator between Lorentz and Hardy-Lorentz spaces remains true.