  In this paper, we consider the uniform estimate for the oscillatory integral
with stationary phase, which was previously studied by Alazard-Burq-Zuily. We
significantly reduce the order of required regularity condition on the phase
and amplitude functions for the uniform estimate. We also study estimates for
the oscillatory integrals of which phase and amplitude functions depend on the
oscillation parameter. The novelty of this article lies in the use of the wave
packet decomposition, which transforms the decay estimate for the oscillatory
integral to the disjointness property of the supports of wave packets. The
latter is geometric in its nature and less sensitive to the smoothness of the
phase and amplitude functions.
