On the almost everywhere existence of the ergodic Hilbert transform

Authors:
Diego Gallardo and F. J. Martín-Reyes

Journal:
Proc. Amer. Math. Soc. **105** (1989), 636-643

MSC:
Primary 28D05; Secondary 47A35

DOI:
https://doi.org/10.1090/S0002-9939-1989-0939964-1

MathSciNet review:
939964

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a finite measure space, an invertible measure-preserving transformation and a positive measurable function. For , we prove that the ergodic Hubert transform exists a.e. for every in if and only if a.e. We also solve the problem for . In this case the condition is a.e. If the transformation is ergodic, the characterizing conditions become that and , respectively. These characterizations, together with some recent results, give, for , that exists a.e. for every in if and only if the sequence of the Césàro-averages converge a.e. for every in . This equivalence has recently been obtained by Jajte for a unitary operator, not necessarily positive, acting on .

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1989-0939964-1

Keywords:
Almost everywhere convergence,
Cesàro-averages,
ergodic Hilbert transform,
ergodic maximal Hilbert transform,
ergodic maximal operator,
measure preserving transformations,
weights

Article copyright:
© Copyright 1989
American Mathematical Society