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Ergodic transforms associated to general averages

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Aimar H., Bernardis A., Martín-Reyes F.

Studia Mathematica

2010

tom 199

nr 2

107-143

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-α averages and Abel means. We prove the boundedness in $L^{p}$, 1 < p < ∞, 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.

Ograniczanie wyników

1 Studia Mathematica

1 Aimar H.

1 Bernardis A.

1 Martín-Reyes F.

1 2010

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Ergodic transforms associated to general averages