Another proof of the existence of the ergodic Hilbert transform

Petersen, Karl

doi:10.1090/S0002-9939-1983-0691275-2

Another proof of the existence of the ergodic Hilbert transform
HTML articles powered by AMS MathViewer

by Karl Petersen PDF

Proc. Amer. Math. Soc. 88 (1983), 39-43 Request permission

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.

References

A dominated ergodic estimate for

spaces with weights

E. Atencia and F. J. Martín-Reyes, The maximal ergodic Hilbert transform with weights, Pacific J. Math. 108 (1983), no. 2, 257–263. MR 713735

On the comparison of transcendents with certain applications to the theory of definite integrals

147

A.-P. Calderón, Ergodic theory and translation-invariant operators, Proc. Nat. Acad. Sci. U.S.A. 59 (1968), 349–353. MR 227354, DOI 10.1073/pnas.59.2.349
Mischa Cotlar, A unified theory of Hilbert transforms and ergodic theorems, Rev. Mat. Cuyana 1 (1955), 105–167 (1956) (English, with Spanish summary). MR 84632

Ergodic

spaces

22

Richard Hunt, Benjamin Muckenhoupt, and Richard Wheeden, Weighted norm inequalities for the conjugate function and Hilbert transform, Trans. Amer. Math. Soc. 176 (1973), 227–251. MR 312139, DOI 10.1090/S0002-9947-1973-0312139-8
Roger L. Jones, An analog of the Marcinkiewicz integral in ergodic theory, Studia Math. 68 (1980), no. 3, 281–289. MR 599150, DOI 10.4064/sm-68-3-281-289
Lynn H. Loomis, A note on the Hilbert transform, Bull. Amer. Math. Soc. 52 (1946), 1082–1086. MR 19155, DOI 10.1090/S0002-9904-1946-08713-3
Norbert Wiener, The ergodic theorem, Duke Math. J. 5 (1939), no. 1, 1–18. MR 1546100, DOI 10.1215/S0012-7094-39-00501-6