## 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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### References:

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