On the maximal operators of $T$ means with respect to Walsh-Kaczmarz system

TL;DR

This paper establishes new inequalities for the maximal operators of $T$ means with monotone coefficients in the Walsh-Kaczmarz system, demonstrating their optimality and applying them to prove almost everywhere convergence.

Contribution

It introduces new $(H_p, L_{p, au})$ inequalities for maximal operators of $T$ means with monotone coefficients in the Walsh-Kaczmarz system, showing their optimality.

Findings

New $(H_p, L_{p, au})$ inequalities for maximal operators.

Optimality of the inequalities in a specific sense.

Almost everywhere convergence of $T$ means.

Abstract

In this paper we prove and discuss some new $\left( H_p,L_{p,\infty}\right)$ type inequalities of the maximal operators of $T$ means with monotone coefficients with respect to Walsh-Kaczmarz system. 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. In particular, we apply these results to prove a.e. convergence of such $T$ means.