Hostname: page-component-7b7567c8c-lpg2g Total loading time: 0 Render date: 2026-11-15T12:33:14.937Z Has data issue: false hasContentIssue false
  • English
  • Français

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)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

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.

Information

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

References

[1]Akcoglu, M. A.. A pointwise ergodic theorem in Lp-spaces. Canad. J. Math. 27(5) (1975), 1075–1082.CrossRefGoogle Scholar
[2]Assani, I. and Lin, M.. On the one-sided ergodic Hilbert transform, ergodic theory and related fields. Contemp. Math. 430 (2007), 21–39.CrossRefGoogle Scholar
[3]Campbell, J. T.. Spectral analysis of the ergodic Hilbert transform. Indiana Univ. Math. J. 35(2) (1986), 379–390.CrossRefGoogle Scholar
[4]Cohen, G. and Lin, M.. The one-sided ergodic Hilbert transform of normal contractions. Preprint.Google Scholar
[5]Çömez, D. and Lin, M.. Mean ergodicity of L1 contractions and pointwise ergodic theorems. Almost Everywhere Convergence, II (Evanston, IL, 1989). Academic Press, Boston, MA, 1991, pp. 113–126.CrossRefGoogle Scholar
[6]Cotlar, M.. A unified theory of Hilbert transforms and ergodic theorems. Rev. Mat. Cuyana 1 (1956), 105–167.Google Scholar
[7]Cuny, C.. Pointwise ergodic theorems with rate and application to limit theorems for stationary processes. Preprint.Google Scholar
[8]Derriennic, Y. and Lin, M.. Fractionnal Poisson equations and ergodic theorems for fractionnal coboundaries. Israel J. Math. 123 (2001), 93–130.CrossRefGoogle Scholar
[9]Dowker, Y. N. and Erdös, P.. Some examples in ergodic theory. Proc. London Math. Soc. 9(3) (1959), 227–241.CrossRefGoogle Scholar
[10]Dunford, N. and Schwartz, J. T.. Linear Operators. Part I. General Theory. Wiley, New York, 1988, Reprint of the 1958 original. Wiley Classics LibraryGoogle Scholar
[11]Gaposhkin, V. F.. Spectral criteria for existence of generalized ergodic transforms. Theory Probab. Appl. 41(2) (1996), 247–264.CrossRefGoogle Scholar
[12]Halmos, P.. A nonhomogeneous ergodic theorem. Trans. Amer. Math. Soc. 66 (1949), 284–288.Google Scholar
[13]Izumi, S.. A non-homogeneous ergodic theorem. Proc. Imp. Acad., Tokyo 15 (1939), 189–192.Google Scholar
[14]Del Junco, A. and Rosenblatt, J.. Counterexamples in ergodic theory and number theory. Math. Ann. 245(3) (1979), 185–197.CrossRefGoogle Scholar
[15]Krengel, U.. Ergodic Theorems. de Gruyter, Berlin, 1985.CrossRefGoogle Scholar
[16]Roth, J.-P.. Reformulation et extension de certains théorèmes ergodiques. Ann. Inst. H. Poincaré Probab. Statist. 26(3) (1990), 437–450.Google Scholar
[17]Zygmund, A.. Trigonometric Series, corrected 2nd edn. Cambridge University Press, Cambridge, 1969.Google Scholar