Some new $\left(H_{p},L_{p}\right)$ type inequalities of maximal operators of Vilenkin-N\"orlund means with non-decreasing coefficients

TL;DR

This paper establishes new $ ext{(H}_p, ext{L}_p)$ inequalities for maximal operators of Vilenkin-Nörlund means with non-decreasing coefficients, leading to strong convergence theorems and extending known results in harmonic analysis.

Contribution

It introduces the best possible $ ext{(H}_p, ext{L}_p)$ inequalities for these operators, advancing the understanding of their convergence properties.

Findings

Proved new $ ext{(H}_p, ext{L}_p)$ inequalities for maximal operators.

Established strong convergence theorems for Vilenkin-Nörlund means.

Extended and improved upon existing results in the field.

Abstract

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