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Van der Corput inequality for real line and Wiener-Wintner theorem for amenable groups. (English) Zbl 1499.11244

Summary: 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 RR-action which assert that for any family of maps \((T_t)_{t\in \mathbb{R}}\) acting on the Lebesgue measure space \(( \Omega, \mathcal{A},\mu)\), where \(\mu\) is a probability measure and for any \(t\in\mathbb{R}\), \(T_t\) is measure-preserving transformation on measure space \(( \Omega, \mathcal{A},\mu)\) with \(T_t\circ T_s=T_{t+s}\), for any \(t,s \in\mathbb{R}\). Then, for any \(f \in L^1(\mu)\), 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 \(\theta \in \mathbb{R}\). We further present the joining proof of the amenable group version of Wiener-Wintner theorem due to Ornstein and Weiss.

MSC:

11K06 General theory of distribution modulo \(1\)
28D05 Measure-preserving transformations
47A35 Ergodic theory of linear operators
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