Strong convergence theorem for Vilenkin-Fejér means (Q2875462)

Let \(S_k\) denote the \(k\)th partial sum of the Vilenkin-Fourier series of \(f\) defined on the Vilenkin group \(G_m=\prod_{k=0}^\infty \mathbb{Z}_{m_k}\), \(m=(m_0,m_1,\dots)\in\mathbb{Z}\) where \(m_k\in \{0,1,2,\dots\}\) and \(\sup_k m_k<\infty\). One defines a basis \(\{\psi_n\}\) for \(L^2(G_m)\) in terms of the generalized Rademacher products \(r_k(x)=e^{2\pi i x_k/m_k} \), \(\psi_n=\prod_{k=0}^\infty r_k^{n_k}(x)\) where \(n=\sum_{j=0}^\infty n_j M_j\) and \(M_j\) is defined iteratively by \(M_0=1\) and \(M_{k+1}=m_k M_k\). The Vilenkin-Fourier series is the expansion \(f=\sum_{k=0}^{\infty} \widehat{f}(k) \, \psi_k\) in terms of the orthonormal family \(\psi_n\) for \(L^2(G_m)\) when \(G_m\) is equipped with the standard Haar measure. The partial sum operators are \(S_n (f)=\sum_{k=0}^{n-1} \widehat{f}(k) \, \psi_k\) and the Fejér means are \(\sigma_n(f)=\frac{1}{n}\sum_{k=1}^n S_k(f)\). For a martingale \(f=\{f^{(n)}, n=0,1,2,\dots\}\) with respect to the \(\sigma\)-algebra generated by the neighborhoods \(I_n(x)\), \(x\in G_m\) defined by \(x_k=y_k\), \(k=1,\dots, n\) (and taking the \(k\)th coordinate to be \(\mathbb{Z}_{m_k}\), \(k>n\)) one defines the martingale maximal function \(f^\ast=\sup_n f^{(n)}\) and defines membership in the Hardy martingale spaces \(H^p(G_m)\) (\(0<p<\infty\)) by the condition \(f^\ast\in L^p(G_m)\). The main result of this work is that if \(0<p\leq 1/2\) then there is an absolute constant \(c_p>0\) such that for all \(f\in H^p(G_m)\) and all \(n=2,3,\ldots\), \([\frac{1}{\log^{[1/2+p]}n}\sum_{k=1}^n \frac{\| \sigma_k f\|_p^p}{k^{2(1-p)}}\leq c_p\| f\|_{H_p^p}\, .]\) The result is also proved to be sharp in the sense that the inequality fails when \(k^{2(1-p)}\) is replaced by any sequence that grows more slowly than \(k^{2(1-p)}\). The authors view the results as a natural extension of those of \textit{F. Weisz} [Anal. Math. 22, No. 3, 229--242 (1996; Zbl 0866.42020)] establishing that \(| \sigma_k|_{L^p}\leq c_p | f|_{H^p}\) for \(p>1/2\) whereas there exists a martingale \(f\in H^{1/2}\) such that \(\sup_n\| \sigma_n f\|_{1/2}=+\infty\). The main theorem was proved separately by Tephnadze for the Walsh system (\(m_k=2\) for all \(k\)) (\textit{G. Tephnadze} [Acta Math. Hung. 142, No. 1, 244--259 (2014; Zbl 1313.42086)]).