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

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 $\mathbb{R}$-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 a single null set off which $\underset{T\to +\infty }{\lim }\frac{1}{T}{\int }_0^{T}f(T_t\omega )e^{2i\pi \theta t}dt$ exists for all $\theta \in \mathbb{R}$. We further present the joining proof of the amenable group version of Wiener-Wintner theorem due to Weiss and Ornstein.

12 pages. Scientific Comments and questions are welcome