Besicovitch-weighted ergodic theorems with continuous time

Abstract

Given $1\le p<\infty$, we show that ergodic flows in the $L^p$-space over a $\ddot{I}ƒ$-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.