Sharp $(H_p,L_p)$ and $(H_p,\text{weak}-L_p)$ type inequalities of weighted maximal operators of $T$ means with respect to Vilenkin systems

TL;DR

This paper establishes optimal weighted maximal inequalities for $T$ means with respect to Vilenkin systems, extending known results and providing new insights into harmonic analysis in this context.

Contribution

It proves the sharpness of $(H_p,L_p)$ and $(H_p,\text{weak}-L_p)$ inequalities for weighted maximal operators of $T$ means in Vilenkin systems, including new applications.

Findings

Optimality of weighted inequalities established

Extension of results to Vilenkin systems with monotone coefficients

New applications demonstrated in harmonic analysis

Abstract

We discuss $(H_p,L_p)$ and $(H_p,\text{weak}-L_p)$ type inequalities of weighted maximal operators of $T$ means with respect to the Vilenkin systems with monotone coefficients, considered in \cite{tut4} and prove that these results are the best possible in a special sense. As applications, both some well-known and new results are pointed out.