×

The spectral measure and Hilbert transform of a measure-preserving transformation. (English) Zbl 0675.28010

V. F. Gaposhkin gave a condition on the spectral measure of a normal contraction on \(L^ 2\) sufficient to imply that the operator satisfies the pointwise ergodic theorem. The authors prove that unitary operators T given by \(Tf=f\circ \tau\) for an invertible measure preserving \(\tau\) satisfy a stronger version of this condition. If \(T=\int^{\pi}_{- \pi}e^{i\lambda t}E(dt)\) is the spectral representation of T, then \(\lim_{k\to \infty}[E((-\epsilon_ k,0))f]=0\quad a.e.\) for all \(f\in L_ 2\) and all nonnegative null sequences \((\epsilon_ k)\). This is deduced from a theorem asserting that the rotated Hilbert transform is a continuous function of its parameter. The maximal inequality used in the proof follows from an analytic inequality related to the Carleson-Hunt theorem on the a.e. convergence of Fourier series.
Reviewer: U.Krengel

MSC:

28D05 Measure-preserving transformations
47A35 Ergodic theory of linear operators
47A60 Functional calculus for linear operators
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] James T. Campbell, Spectral analysis of the ergodic Hilbert transform, Indiana Univ. Math. J. 35 (1986), no. 2, 379 – 390. · Zbl 0575.47023 · doi:10.1512/iumj.1986.35.35023
[2] Mischa Cotlar, A unified theory of Hilbert transforms and ergodic theorems, Rev. Mat. Cuyana 1 (1955), 105 – 167 (1956) (English, with Spanish summary). · Zbl 0071.33402
[3] Richard Duncan, Some pointwise convergence results in \?^{\?}(\?), 1<\?<\infty , Canad. Math. Bull. 20 (1977), no. 3, 277 – 284. · Zbl 0384.47009 · doi:10.4153/CMB-1977-043-7
[4] V. F. Gaposhkin, An individual ergodic theorem for normal operators in \?\(_{2}\), Funktsional. Anal. i Prilozhen. 15 (1981), no. 1, 18 – 22, 96 (Russian). · Zbl 0457.47012
[5] Hörmander, [1973] An introduction to complex analysis in several variables, North-Holland, Amsterdam · Zbl 0271.32001
[6] Carlos E. Kenig and Peter A. Tomas, Maximal operators defined by Fourier multipliers, Studia Math. 68 (1980), no. 1, 79 – 83. · Zbl 0442.42013
[7] Ulrich Krengel, Ergodic theorems, De Gruyter Studies in Mathematics, vol. 6, Walter de Gruyter & Co., Berlin, 1985. With a supplement by Antoine Brunel. · Zbl 0575.28009
[8] Karl Petersen, Another proof of the existence of the ergodic Hilbert transform, Proc. Amer. Math. Soc. 88 (1983), no. 1, 39 – 43. · Zbl 0521.28014
[9] Norbert Wiener and Aurel Wintner, Harmonic analysis and ergodic theory, Amer. J. Math. 63 (1941), 415 – 426. · Zbl 0025.06504 · doi:10.2307/2371534
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.