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

Overview & Synopsis

We extend the classical van der Corput inequality to the real line. As a consequence, we obtain a simple proof of he classical Wiener-Wintner theorem for the R-action which assert that for any family of maps (T_t)_{ t∈R} acting on the Lebesgue measure space (Ω, A, µ) where µ is a probability measure and for any t ∈ R, T_t is measure-preserving transformation on measure space (Ω, A, µ) with T_t • T_s = T_{t+s }, for any t, s ∈ R. Then, for any f ∈ L^1 (µ), there is a a single null set off which lim_{T →+∞} 1/T \int_{0}^{T} f (T_t ω)e 2iπθt dt exists for all θ ∈ R. We further present the joining proof of the amenable group version of Wiener-Wintner theorem due to Weiss and Ornstein.