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 (Tt) t∈R acting on the Lebesgue measure space (Ω, A, µ), where µ is a probability measure and for any t ∈ R, Tt is measure-preserving transformation on measure space (Ω, A, µ) with Tt • Ts = T t+s , for any t, s ∈ R. Then, for any f ∈ L 1 (µ), there is a single null set off which lim T →+∞ 1 T T 0 f

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... (Ttω)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 Ornstein and Weiss .