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Another proof of the existence of the ergodic Hilbert transform


Author: Karl Petersen
Journal: Proc. Amer. Math. Soc. 88 (1983), 39-43
MSC: Primary 28D05; Secondary 42A50
DOI: https://doi.org/10.1090/S0002-9939-1983-0691275-2
MathSciNet review: 691275
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Abstract: We give a direct proof of the existence of the ergodic Hilbert transform $ \sum _{k = - \infty }^{\infty '}f({T^k}x)/k$, where $ T:X \to X$ is a measure-preserving transformation and $ f$ is an integrable function.


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  • [1] E. Atencia and A. de la Torre, A dominated ergodic estimate for $ {L^p}$ spaces with weights (preprint).
  • [2] E. Atencia and F. J. Martin-Reyes, The maximal ergodic Hilbert transform with weights (preprint). MR 713735 (85e:28025)
  • [3] G. Boole, On the comparison of transcendents with certain applications to the theory of definite integrals, Philos. Trans. Roy. Soc. London 147 (1857), 745-803.
  • [4] A. P. Calderón, Ergodic theory and translation-invariant operators, Proc. Nat. Acad. Sci. U. S. A. 59 (1968), 349-353. MR 0227354 (37:2939)
  • [5] M. Cotlar, A unified theory of Hilbert transforms and ergodic theorems, Rev. Mat. Cuyana 1 (1955), 105-167. MR 0084632 (18:893d)
  • [6] Alberto de la Torre, Ergodic $ {H^1}$ spaces, Bol. Soc. Mat. Mexicana 22 (1977), 10-22.
  • [7] R. Hunt, B. Muckenhoupt and R. Wheeden, Weighted norm inequalities for the conjugate function and Hilbert transform, Trans. Amer. Math. Soc. 176 (1973), 227-251. MR 0312139 (47:701)
  • [8] Roger L. Jones, An analog of the Marcinkiewicz integral in ergodic theory, Studia Math. 68 (1980), 281-289. MR 599150 (82m:28034)
  • [9] Lynn H. Loomis, A note on the Hilbert transform, Bull. Amer. Math. Soc. 52 (1946), 1082-1086. MR 0019155 (8:377d)
  • [10] Norbert Wiener, The ergodic theorem, Duke Math. J. 5 (1939), 1-18. MR 1546100

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1983-0691275-2
Article copyright: © Copyright 1983 American Mathematical Society

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