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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

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


Authors: James Campbell and Karl Petersen
Journal: Trans. Amer. Math. Soc. 313 (1989), 121-129
MSC: Primary 28D05; Secondary 47A35, 47A60
DOI: https://doi.org/10.1090/S0002-9947-1989-0958884-4
MathSciNet review: 958884
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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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DOI: https://doi.org/10.1090/S0002-9947-1989-0958884-4
Keywords: Spectral measure, ergodic Hubert transform, maximal inequality
Article copyright: © Copyright 1989 American Mathematical Society