On the lower bound of Sunouchi's operator with respect to Vilenkin systems (Q1382405)

Let \( m=(m_k,k\in {\mathbb N}\quad ({\mathbb N}:=\{ 0,1,\dots \})\), where \(m_k\in {\mathbb N}\), \(m_k\geq 2\). Then by definition \(G_m=\prod_{k=0}^{\infty} {\mathbb Z}_{m_k}\) and the group of characters for \(G_m\) is an orthonormal Vilenkin system. Let \(S_n f\) and \(\sigma_n f\) be the Vilenkin-Fourier sums and their Fejér means for \(f\in L^1(G_m)\), respectively. If \(M_0:=1\), \(M_{n+1}:=m_n M_n\) \((n\in{\mathbb N})\), then one defines Sunouchi's operator as follows: \[ Tf:=(\sum^{\infty}_{n=0}|S_{M_n}f-\sigma_{M_n}f|^2)^{\frac12} \quad(f\in L^1(G_m)). \] The author considers the Hardy space \( H^1(G_m)\) and the so-called atomic Hardy space \( H(G_m)\) [see \textit{F. Schipp, W.R. Wade, P. Simon} and \textit{J. Pál}, ``Walsh series. An introduction to dyadic harmonic analysis'' (1990; Zbl 0727.42017))]. He proves the following theorems. Theorem 1.1. If \( \sup_{n\in {\mathbb N}}m_n=\infty \), then there exists a function \(f\in L^1(G_m)\), for which \(|Tf|_1<\infty\) and \(|f|_H =\infty\). Theorem 2.1. Let \(f\in L^1(G_m)\), \(S_1 f=0\) and \( \sum_{n=0}^{\infty}m_n^{-2}<\infty \), then \[ |f|_{H^1}\leq c|Tf|_1. \] .