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

Assani, Idris; Petersen, Karl

doi:10.1090/S0002-9947-1992-1075378-9

The helical transform as a connection between ergodic theory and harmonic analysis
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by Idris Assani and Karl Petersen

Trans. Amer. Math. Soc. 331 (1992), 131-142

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

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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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