Maximal operators of Vilenkin-N\"orlund means

TL;DR

This paper establishes new inequalities for maximal operators of Vilenkin-Nörlund means with monotone coefficients, demonstrating their almost everywhere convergence and optimality, with applications to known and novel results.

Contribution

It introduces new $H_{p}$ and weak-$L_{p}$ inequalities for these maximal operators and proves their optimality, extending the understanding of Vilenkin-Nörlund means.

Findings

Proved new inequalities for maximal operators with monotone coefficients.

Established almost everywhere convergence of Vilenkin-Nörlund means.

Showed the optimality of these results in a specific sense.

Abstract

In this paper we prove and discuss some new $\left(H_{p},weak-L_{p}\right) $ type inequalities of maximal operators of Vilenkin-N\"orlund means with monotone coefficients. We also apply these results to prove a.e. convergence of such Vilenkin-N\"orlund means. It is also proved that these results are the best possible in a special sense. As applications, both some well-known and new results are pointed out.