The spectral measure and Hilbert transform of a measure-preserving transformation

Campbell, James; Petersen, Karl

doi:10.1090/S0002-9947-1989-0958884-4

The spectral measure and Hilbert transform of a measure-preserving transformation
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by James Campbell and Karl Petersen PDF

Trans. Amer. Math. Soc. 313 (1989), 121-129 Request permission

Abstract:

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. We prove that unitary operators which come from measure-preserving transformations satisfy a stronger version of this condition. This follows from the fact that the rotated ergodic Hubert transform is a continuous function of its parameter. The maximal inequality on which the proof depends follows from an analytic inequality related to the Carleson-Hunt Theorem on the a.e. convergence of Fourier series.

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