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Transactions of the American Mathematical Society

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



The helical transform as a connection between ergodic theory and harmonic analysis

Authors: Idris Assani and Karl Petersen
Journal: Trans. Amer. Math. Soc. 331 (1992), 131-142
MSC: Primary 28D05; Secondary 42A20, 42A50
MathSciNet review: 1075378
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Abstract: Direct proofs are given for the formal equivalence of the $ {L^2}$ boundedness of the maximal operators corresponding to the partial sums of Fourier series, the range of a discrete helical walk, partial Fourier coefficients, and the discrete helical transform. Strong $ (2, 2)$ for the double maximal (ergodic) helical transform is extended to actions of $ {\mathbb{R}^d}$ and $ {\mathbb{Z}^d}$. It is also noted that the spectral measure of a measure-preserving flow has a continuity property at $ \infty $, the Local Ergodic Theorem satisfies a Wiener-Wintner property, and the maximal helical transform is not weak $ (1, 1)$.

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Keywords: Carleson-Hunt Theorem, Hilbert transform, Local Ergodic Theorem, Wiener-Wintner property, maximal inequality, large sieve, spectral measure
Article copyright: © Copyright 1992 American Mathematical Society