Asymptotic estimates of oscillatory integrals with general phase and singular amplitude: Applications to dispersive equations

TL;DR

This paper develops van der Corput type estimates for oscillatory integrals with stationary points and singular amplitudes, and applies these to analyze decay rates of solutions to dispersive equations with various initial conditions.

Contribution

It introduces new asymptotic estimates for oscillatory integrals with singularities and stationary points, and applies them to dispersive PDEs to understand decay behavior.

Findings

Explicit decay estimates depending on stationary points and singularities.

Analysis of the impact of frequency band limitations on dispersion.

Proof of optimal decay rates under certain conditions.

Abstract

In this paper, we furnish van der Corput types estimates for oscillatory integrals with respect to a large parameter, where the phase is allowed to have a stationary point of real order and the amplitude to have an integrable singularity. The resulting estimates show explicitly the influence of these two particular points on the decay. These results are then applied to the solutions of a family of dispersive equations whose generators are Fourier multipliers. We explore the effect of a limitation to compact frequency bands and of singular frequencies of the initial condition on the decay. Uniform estimates in space-time cones as well as L-infinity-norm estimates are furnished and the optimality of the decay rates is proved under certain hypotheses. Moreover the influence of a growth limitation at infinity of the symbols on the dispersion is exhibited.