van der Corput inequality for real line and Wiener-Wintner theorem for amenable groups

Year 2021, Volume: 4 Issue: 4, 420 - 427, 13.12.2021

El Abdalaoui El Houcein

https://doi.org/10.33205/cma.1029202

Abstract

We extend the classical van der Corput inequality to the real line. As a consequence, we obtain a simple proof of the Wiener-Wintner theorem for the $R$-action which assert that for any family of maps $(T_t{)}_{t\in R}$ acting on
the Lebesgue measure space $(\Omega ,A,\mu )$, where $\mu$ is a probability measure and for any $t\in R$, $T_t$ is measure-preserving transformation on measure space $(\Omega ,A,\mu )$ with
$T_t\circ T_s=T_{t+s}$, for any $t,s\in R$. Then, for any
$f\in L^1\left(\mu \right)$, there is a single null set off which  $\displaystyle \lim_{T \rightarrow +\infty} \frac{1}{T}\int_{0}^{T} f(T_t\omega) e^{2 i \pi \theta t} dt$
exists for all $\theta \in$$R$. We further present the joining proof of the amenable group version of Wiener-Wintner theorem due to Ornstein and Weiss.

Keywords

van der Corput inequality, Wiener-Wintner theorem, joinings, amenable group

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