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

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 (Tt)t∈R(Tt)t∈R acting on the Lebesgue measure space (Ω,A,μ)(Ω,A,μ), where μμ is a probability measure and for any t∈Rt∈R, TtTt is measure-preserving transformation on measure space (Ω,A,μ)(Ω,A,μ) with Tt∘Ts=Tt+sTt∘Ts=Tt+s, for any t,s∈Rt,s∈R. Then, for any f∈L1(μ)f∈L1(μ), there is a single null set off which $\displaystyle \lim_{T \rightarrow +\infty} \frac{1}{T}\int_{0}^{T} f(T_t\omega) e^{2 i \pi \theta t} dt$limT→+∞1T∫0Tf(Ttω)e2iπθtdt exists for all θ∈θ∈\RRR. We further present the joining proof of the amenable group version of Wiener-Wintner theorem due to Ornstein and Weiss.