Ergodic transforms associated to general averages
Volume 199 / 2010
Studia Mathematica 199 (2010), 107-143
MSC: 47A35, 37A40, 42B25.
DOI: 10.4064/sm199-2-1
Abstract
Jones and Rosenblatt started the study of an ergodic transform which is analogous to the martingale transform. In this paper we present a unified treatment of the ergodic transforms associated to positive groups induced by nonsingular flows and to general means which include the usual averages, Cesàro-$\alpha$ averages and Abel means. We prove the boundedness in $L^p$, $1< p< \infty$, of the maximal ergodic transforms assuming that the semigroup is Cesàro bounded in $L^p$. For $p=1$ we find that the maximal ergodic transforms are of weak type $(1,1)$. Convergence results are also proved. We give some general examples of Cesàro bounded semigroups.
