# Article

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Keywords:
Vilenkin system; Vilenkin group; Nörlund means; martingale Hardy space; maximal operator; Vilenkin-Fourier series; strong convergence; inequality
Summary:
We prove and discuss some new $( H_{p},L_{p})$-type inequalities of weighted maximal operators of Vilenkin-Nörlund means with non-increasing coefficients $\{q_{k}\colon k\geq 0\}$. These results are the best possible in a special sense. As applications, some well-known as well as new results are pointed out in the theory of strong convergence of such Vilenkin-Nörlund means. To fulfil our main aims we also prove some new estimates of independent interest for the kernels of these summability results. \endgraf In the special cases of general Nörlund means $t_{n}$ with non-increasing coefficients analogous results can be obtained for Fejér and Cesàro means by choosing the generating sequence $\{ q_{k}\colon k\geq 0\}$ in an appropriate way.
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