\((L^ 1,H)\)-type estimations for some operators with respect to the Walsh-Paley system (Q1074803)

The following operators are examined: \[ U_ pf=(\sum^{\infty}_{n=1}n^{-1}| S_ nf-\sigma_ nf|^ p)^{1/p},\quad Tf=(\sum^{\infty}_{n=0}| S_{2^ n}f- \sigma_{2^ n}f|^ 2)^{1/2}, \] where \(p\geq 1\), \(f\in L^ 1[0,1]\), \(S_ nf\) and \(\sigma_ nf\) denote the n-th partial sum resp. (C,1) mean of the Walsh-Fourier series of f. It is known [see \textit{G. I. Sunouchi}, Proc. Am. Math. Soc. 2, 5-11 (1951; Zbl 0044.071) and Tôhoku Mat. J. 16, 228-237 (1964; Zbl 0146.089)] that \(U_ p\) \((p\geq 2)\) and T are \(L^ r\) bounded if \(1<r<\infty\) and the analogous statements for \(r=1\) are not true. In this note it is proved that T as mapping from the dyadic Hardy space into \(L^ 1[0,1]\) is bounded, but the same statement fails to hold for \(U_ p\) \((p\geq 1)\).