Investigations of certain operators with respect to the Vilenkin system (Q1332977)

The author examines Vilenkin analogues of Walsh series results of \textit{P. Simon} for strong convergence [Acta Math. Hung. 49, 425-431 (1987; Zbl 0643.42020)] and for the Sunouchi operator [Acta Math. Hung. 46, 307-310 (1985; Zbl 0591.42019)]. He shows that if \(f\) belongs to the atomic Hardy space \(H^ 1(G_ m)\), for any Vilenkin group \(G_ m\), then \[ \lim_{n\to\infty} \log^{-1} n \sum^ n_{k=0} k^{-1} \| S_ k f\|_ 1= \| f\|_ 1, \] and if \(f\) belongs to the martingale Hardy space \(H(G_ m)\), for a Vilenkin group \(G_ m\) of bounded type, then \[ Tf:= \left(\sum^ \infty_{n=0} | S_{M_ n}f- \sigma_{M_ n} f|^ 2\right)^{{1\over 2}} \] is a bounded operator from \(H(G_ m)\) into \(L^ 1(G_ m)\). He shows this last result never holds if \(G_ m\) is of unbounded type. However, if \(H(G_ m)\) is replaced by \(H^ 1(G_ m)\), then \(T\) can be a bounded operator for some Vilenkin groups of unbounded type. In fact, the author proves that \(T\) is bounded from \(H^ 1(G_ m)\) to \(L^ 2(G_ m)\) if and only if \(m_ k':= M^{-1}_{k+1} \sum^{k-1}_{j= 0} M_{j+1}\log m_ j\) is bounded.