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On the Nörlund means of Vilenkin-Fourier series. (English) Zbl 1374.42054

Let \(G\) be a Vilenkin group generated by a bounded sequence \((m_n)\). Hardy spaces of martingales with respect to \(G\) denote by \(H_p(G)\;(0<p<\infty)\). For a sequence of non-negative numbers \(q=(q_n)\) by \(t_{q,n}(f)\) denote the \(n\)-th Nörlund mean of a martingale \(f\). Note that Cesaro means are particular cases of Nörlund ones. For an index \(\alpha>0\) by \(t_{q,\alpha}^{\ast}\) is denoted the weighted maximal operator: \[ t_{q,\alpha}^{\ast}(f)=\sup\limits_{n\in\mathbb{N}}\;\frac{|t_{q,n}(f)|}{\log^{1+\alpha}(n+1)}. \] In the paper there are proved the following two results.
Theorem 1. Let \(0<\alpha\leq 1\) and let \((q_n)\) be a decreasing sequence of positive numbers satisfying conditions: \[ \sup_{n\in\mathbb{N}}\;\frac{n^\alpha}{q_1+\dots+q_{n-1}}<\infty, \]
\[ \sup_{n\in\mathbb{N}}\;\frac{q_n-q_{n+1}}{n^{\alpha-2}}<\infty. \] Then the weighted maximal operator \(t_{q,\alpha}^{\ast}\) is bounded from \(H_{1/(1+\alpha)}(G)\) to \(L_{1/(1+\alpha)}(G)\).
Theorem 2. Let \(0<\alpha < 1\) and let \((q_n)\) be a decreasing sequence of positive numbers satisfying the conditions of Theorem 1. Then there exists a positive constant \(c_\alpha\) such that \[ \frac{1}{\log n}\sum_{k=1}^n \frac{\|t_{q,k}(f)\|_{H_{1/(1+\alpha)}(G)}^{1/(1+\alpha)}}{k}\leq c_\alpha \|f\|_{H_{1/(1+\alpha)}(G)}^{1/(1+\alpha)} \] for every \(f\in H_{1/(1+\alpha)}(G)\).
This theorems generalise the corresponding results obtained earlier in works of I. Blahota, G. Tepnadze and U. Goginava for the case of Cesaro means \(\sigma_\alpha(f)\).

MSC:

42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
42B25 Maximal functions, Littlewood-Paley theory
42B30 \(H^p\)-spaces
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