The spectral measure and hilbert transform of a measure-preserving transformation
V. F. Gaposhkin gave a condition on the spectral measure of a normal contraction on L2 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. © 1989 American Mathematical Society.
Transactions of the American Mathematical Society
Campbell, J., & Petersen, K. (1989). The spectral measure and hilbert transform of a measure-preserving transformation. Transactions of the American Mathematical Society, 313 (1), 121-129. https://doi.org/10.1090/S0002-9947-1989-0958884-4