Almost uniform convergence for noncommutative Vilenkin-Fourier series

TL;DR

This paper proves almost uniform convergence of noncommutative Vilenkin-Fourier series by establishing maximal inequalities and employing advanced noncommutative harmonic analysis techniques.

Contribution

It introduces noncommutative maximal inequalities for Cesàro means of Vilenkin-Fourier series, extending classical convergence results to a noncommutative setting.

Findings

Established noncommutative maximal inequalities.

Proved almost uniform convergence of Cesàro means.

Utilized noncommutative square function and Calderón-Zygmund decomposition.

Abstract

In the present paper, we study almost uniform convergence for noncommutative Vilenkin-Fourier series. Precisely, we establish several noncommutative (asymmetric) maximal inequalities for the Ces\`{a}ro means of the noncommutative Vilenkin-Fourier series, which in turn give the corresponding almost uniform convergence. The primary strategy in our proof is to explore a noncommutative generalization of Sunouchi square function operator, and the very recent advance of the noncommutative Calder\'{o}n-Zygmund decomposition.