  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 $(\Omega,{\cal {A}},\mu)$
where $\mu$ is a probability measure and for any $t\in \mathbb{R}$, $T_t$ is
measure-preserving transformation on measure space $(\Omega,{\cal {A}},\mu)$
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 $\displaystyle \lim_{T
\rightarrow +\infty} \frac1{T}\int_{0}^{T} f(T_t\omega) e^{2 i \pi \theta t}
dt$ exists for all $\theta \in \mathbb{R}$. We further present the joining
proof of the amenable group version of Wiener-Wintner theorem due to Weiss and
Ornstein.
