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On the a.s. convergence of the one-sided ergodic Hilbert transform

Published online by Cambridge University Press:  03 February 2009

CHRISTOPHE CUNY
CHRISTOPHE CUNY*
Affiliation:
Equipe ERIM, University of New Caledonia, BPR4 - 98851 Nouméa Cedex, New Caledonia (email: cuny@univ-nc.nc)
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Abstract

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We show that for T a Dunford–Schwartz operator on a σ-finite measure space (X,Σ,μ) and f∈L1(X,μ), whenever the one-sided ergodic Hilbert transform ∑ n≥1(Tnf/n) converges in norm, it converges μ-a.s. A similar result is obtained for any positive contraction of some fixed Lp(X,Σ,μ), p>1. Applying our result to the case where T is the (unitary) operator induced by a measure-preserving (invertible) transformation, we obtain a positive answer to a question of Gaposhkin.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 6 , December 2009 , pp. 1781 - 1788
Copyright
Copyright © Cambridge University Press 2009

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