On a square function with respect to Vilenkin system (Q2714193)

The so-called Sunouchi operator \(S\) is well-known in the Walsh-Fourier analysis. It is proved by Sunouchi that \(S\) characterizes the \(L^p\) spaces for \(p>1\). This is not the case for \(p=1\), namely \(S\) does not form a bounded map from \(L^1\) into \(L^1\). The purpose of this paper is to investigate the following Sunouchi-like operator: NEWLINE\[NEWLINE Tf := \left( \sum_{n=0}^{\infty}\sum_{j=1}^{m_n-1}|S_{jM_n}f-\sigma_{jM_n}f|^2 \right)^{\frac{1}{2}} \quad (f\in L^1). NEWLINE\]NEWLINE The author proves NEWLINENEWLINENEWLINE\noindent \textbf{Theorem.} Let \(m\) (the generating sequence of the Vilenkin system) be bounded and \(0<p\leq 1\). Then \(T : H^p\to L^p\) is bounded. Moreover, there exists positive constants \(c_p, C_p\) depending only on \(p\) such that for all \(f\in H^p\) with \(\widehat f(0)=0\) we have NEWLINE\[NEWLINE c_p\|f\|_{H^p} \leq \|Tf\|_p \leq C_p\|f\|_{H^p}.NEWLINE\]NEWLINE NEWLINENEWLINE\noindent \textbf{Corollary.} Assume the boundedness of \(m\). Then \(T\) is of weak type \((1,1)\). Moreover, for all \(1<p<\infty\) there exists a constant \(C_p\) depending only on \(p\) such that \( \|Tf\|_p \leq C_p\|f\|_p (f\in L^p).\)