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On the Ergodic Averages and the Ergodic Hilbert Transform

Published online by Cambridge University Press:  20 November 2018

L. M. Fernández-Cabrera ,
F. J. Martín-Reyes  and
J. L. Torrea
L. M. Fernández-Cabrera
Affiliation:
Escuela Universitaria de Estadística Universidad Complutense Madrid Spain e–mail: [email protected]
F. J. Martín-Reyes
Affiliation:
Análisis Maternático Facultad de Ciencias Universidad de Málaga 29071–Málaga Spain e–mail: [email protected]
J. L. Torrea
Affiliation:
Departamento de Matemáticas Facultad de Ciencias Universidad Autónoma de Madrid 28049 Madrid Spain e–mail: [email protected]
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Abstract

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Let T be an invertible measure-preserving transformation on a σ-finite measure space (X, μ) and let 1 < p < ∞. This paper uses an abstract method developed by José Luis Rubio de Francia which allows us to give a unified approach to the problems of characterizing the positive measurable functions v such that the limit of the ergodic averages or the ergodic Hilbert transform exist for all f ∈ Lp(νdμ). As a corollary, we obtain that both problems are equivalent, extending to this setting some results of R. Jajte, I. Berkson, J. Bourgain and A. Gillespie. We do not assume the boundedness of the operator Tf(x) = f(Tx) on Lp(νdμ). However, the method of Rubio de Francia shows that the problems of convergence are equivalent to the existence of some measurable positive function u such that the ergodic maximal operator and the ergodic Hilbert transform are bounded from LP(νdμ) into LP(udμ). We also study and solve the dual problem.

Keywords

28D0542B25Ergodic Hilbert transformergodic maximal operatormeasure-preserving transformationsvector valued inequalitiesweights
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 47 , Issue 2 , 01 April 1995 , pp. 330 - 343
Copyright
Copyright © Canadian Mathematical Society 1995

References

[AT] Atencia, E. and de, A. la Torre, A dominated ergodic estimate for If-spaces with weights, Studia Math. (1)74(1982), 35–47.Google Scholar
[AM] Atencia, E. and Martín-Reyes, F.J., The maximal ergodic Hilbert transform with weights, Pacific J. Math. (2) 108(1983), 257–263.Google Scholar
[AP] Assani, I. and Petersen, K., The helical transform as a connection between Ergodic Theory and Harmonic Analysis, Trans. Amer. Math. Soc. 331(1992), 131–142.Google Scholar
[BBG] Berkson, I., Bourgain, J. and Gillespie, A., On the almost everywhere convergence of ergodic averages for power-bounded operators on LP-subspaces, Integral Equations Operator Theory 14(1991), 678–715.Google Scholar
[FT] Fernández-Cabrera, L.M. and Torrea, J.L., Vector valued inequalities with weights, Publ. Mat. 37(1993), 177–208.Google Scholar
[GM] Gallardo, D. and Martín-Reyes, F.J., On the almost everywhere existence of the ergodic Hilbert transform, Proc. Amer. Math. Soc. 105(1989), 636–643.Google Scholar
[GR] García-Cuerva, J. and Rubio, J.L. de Francia, Weighted Norm Inequalities and Related Topics, North- Holland, 1985.Google Scholar
[HMW] Hunt, R.A., Muckenhoupt, B. and Wheeden, R., Weighted norm inequalities for the conjugate function and Hilbert transform, Trans. Amer. Math. Soc. 176(1973), 227–251.Google Scholar
[J] Jajte, R., On the existence of the ergodic Hilbert transform, Ann. Probab. 15(1987), 831–835.Google Scholar
[Ml] Martin-Reyes, F.J., Inequalities for the ergodic maximal function and convergence of the averages in weighted LP-spaces, Trans. Amer. Math. Soc. 296(1986), 61–82.Google Scholar
[M2] Martin-Reyes, F.J., On a pointwise ergodic theorem for multiparameter groups, Proceedings of the International Conference On Almost Everywhere Convergence in Probability and Ergodic Theory, Academic Press, 1989. 267–279.Google Scholar
[MT] Martín-Reyes, F.J. and de, A. la Torre, On the almost everywhere convergence of the ergodic averages, Ergodic Theory Dynamical Systems 10(1990), 141–149.Google Scholar
[PI] Petersen, K., Another Proof of The Existence of the Ergodic Hilbert Transform, Proc. Amer. Math. Soc. 88(1983), 39–44.Google Scholar
[P2] Petersen, K., Ergodic Theory, Cambridge Stud. Adv. Math. 2, Cambridge Univ. Press, 1983.Google Scholar
[RT] Ruiz Blasco, F.J. and Torrea, J.L., Vector-valued Calderón-Zygmund Theory and Carleson Measures on Spaces of Homogeneous Nature, Studia Math. 88(1990), 221–243.Google Scholar
[S] Sato, R., On a pointwise ergodic theorem for positive operators, Studia Math. 97(1990), 71–84.Google Scholar