Sharp $\left( H_{p},L_{p}\right) $ type inequalities of maximal operators of $T$ means with respect to Vilenkin systems with monotone coefficients

TL;DR

This paper establishes new inequalities for maximal operators of T means in Vilenkin systems with monotone coefficients, leading to strong convergence results and demonstrating optimality of these inequalities.

Contribution

It introduces novel (H_{p},L_{p}) inequalities for T means in Vilenkin systems, advancing understanding of their convergence and optimality.

Findings

Proved new (H_{p},L_{p}) inequalities for maximal operators

Established strong convergence theorems for T means

Showed the optimality of the inequalities in a specific sense

Abstract

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