Maximal operators of $T$ means with respect to the Vilenkin system

Abstract

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