Ergodic transforms associated to general averages

H. Aimar¹ ; A. L. Bernardis¹ ; F. J. Martín-Reyes²

¹ IMAL-CONICET Güemes 3450 3000 Santa Fe, Argentina
² Departamento de Análisis Matemático Facultad de Ciencias Universidad de Málaga 29071 Málaga, Spain

Studia Mathematica, Tome 199 (2010) no. 2, pp. 107-143

Résumé

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.

Détail

DOI : 10.4064/sm199-2-1

Keywords: jones rosenblatt started study ergodic transform which analogous martingale transform paper present unified treatment ergodic transforms associated positive groups induced nonsingular flows general means which include usual averages ces ro alpha averages abel means prove boundedness infty maximal ergodic transforms assuming semigroup ces bounded maximal ergodic transforms weak type convergence results proved general examples ces bounded semigroups