  In this paper, we study time-asymptotic propagation phenomena for a class of
dispersive equations on the line by exploiting precise estimates of oscillatory
integrals. We propose first an extension of the van der Corput Lemma to the
case of phases which may have a stationary point of real order and amplitudes
allowed to have an integrable singular point. The resulting estimates provide
optimal decay rates which show explicitly the influence of these two particular
points. Then we apply these abstract results to solution formulas of a class of
dispersive equations on the line defined by Fourier multipliers. Under the
hypothesis that the Fourier transform of the initial data has a compact support
or an integrable singular point, we derive uniform estimates of the solutions
in space-time cones, describing their motions when the time tends to infinity.
The method permits also to show that symbols having a restricted growth at
infinity may influence the dispersion of the solutions: we prove the existence
of a cone, depending only on the symbol, in which the solution is
time-asymptotically localized. This corresponds to an asymptotic version of the
notion of causality for initial data without compact support.
