A noncommutative weak type maximal inequality for modulated ergodic averages with general weights

TL;DR

This paper establishes a weak type maximal inequality for weighted ergodic averages in noncommutative $L_p$-spaces, leading to new ergodic theorems with specific weight sequences and extending to multiparameter cases.

Contribution

It introduces a novel weak type $(p,p)$ maximal inequality for weighted averages of noncommutative operators, enabling new ergodic theorems with general weights.

Findings

Proves weak type $(p,p)$ maximal inequality for noncommutative weighted averages.

Derives modulated ergodic theorems with $q$-Besicovitch and $q$-Hartman weights.

Extends results to multiparameter ergodic averages.

Abstract

In this article, we prove a weak type $(p,p)$ maximal inequality, $1<p<\infty$, for weighted averages of a positive Dunford-Schwarz operator $T$ acting on a noncommutative $L_p$-space associated to a semifinite von Neumann algebra $\mathcal{M}$, with weights in $W_q$, where $\frac{1}{p}+\frac{1}{q}=1$. This result is then utilized to obtain modulated individual ergodic theorems with $q$-Besicovitch and $q$-Hartman sequences as weights. Multiparameter versions of these results are also investigated.