On the convergence of partial sums with respect to Vilenkin system on the martingale Hardy spaces

TL;DR

This paper investigates the convergence behavior of subsequences of partial sums in Vilenkin systems within martingale Hardy spaces for 0<p<1, providing characterizations, conditions, and optimality results.

Contribution

It offers new necessary and sufficient conditions for convergence of partial sums in martingale Hardy spaces, extending and refining previous results in harmonic analysis.

Findings

Characterizations of boundedness of subsequences of partial sums

Necessary and sufficient conditions for modulus of continuity

Optimality of the convergence results

Abstract

In this paper we derive characterizations of boundedness of the subsequences of partial sums with respect to Vilenkin system on the martingale Hardy spaces when $ 0<p<1 $. Moreover, we find necessary and sufficient conditions for the modulus of continuity of $f\in H_{p}$ martingales, which provide convergence of subsequences of partial sums on the martingale Hardy spaces. 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.