Besicovitch-weighted ergodic theorems with continuous time

TL;DR

This paper proves almost uniform convergence of Besicovitch-weighted ergodic theorems for continuous-time ergodic flows in various function spaces, extending classical results to more general settings.

Contribution

It introduces Besicovitch-weighted ergodic theorems for continuous-time flows in $L^p$ and symmetric spaces, including Orlicz, Lorentz, and Marcinkiewicz spaces, with explicit limit identification.

Findings

Almost uniform convergence in $L^p$-spaces for Besicovitch-weighted ergodic flows.

Extension of ergodic theorems to fully symmetric spaces.

Identification of the ergodic limit in the weighted setting.

Abstract

Given $1\leq p<\infty$, we show that ergodic flows in the $L^p$-space over a $\sigma$-finite measure space generated by strongly continuous semigroups of Dunford-Schwartz operators and modulated by bounded Besicovitch almost periodic functions converge almost uniformly (in Egorov's sense). The corresponding local ergodic theorem is proved with identification of the limit. Then we extend these results to arbitrary fully symmetric spaces, including Orlicz, Lorentz, and Marcinkiewicz spaces.