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

DOI:
https://doi.org/10.1090/S0002-9947-1992-1075378-9

MathSciNet review:
1075378

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Abstract: Direct proofs are given for the formal equivalence of the 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 for the double maximal (ergodic) helical transform is extended to actions of and . It is also noted that the spectral measure of a measure-preserving flow has a continuity property at , the Local Ergodic Theorem satisfies a Wiener-Wintner property, and the maximal helical transform is not weak .

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1992-1075378-9

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