Uniform Stationary Phase Estimate with Limited Smoothness

In this paper, we consider uniform estimates for the oscillatory integrals with stationary phase, which were previously studied by Farah–Rousset–Tzvetkov and Alazard–Burq–Zuily. We significantly reduce the order of required regularity 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 has been widely utilized in the study of the Fourier restriction problems. This allows one to obtain the decay estimate for the oscillatory integral from disjointness of the spatial supports of wave packets. The latter is geometric in its nature and less sensitive to the smoothness of the phase and amplitude functions.