Dispersive type estimates for Fourier integrals and applications to hyperbolic systems

TL;DR

This paper develops dispersive estimates for Fourier integrals with parameter-dependent phases using geometric properties of Fresnel surfaces, and applies these results to hyperbolic systems and higher order equations.

Contribution

It introduces new dispersive estimates based on geometric analysis of Fresnel surfaces, extending previous methods to parameter-dependent phases.

Findings

Dispersive estimates are derived for Fourier integrals with phase functions depending on parameters.

Applications to hyperbolic systems show improved decay estimates.

Results enhance understanding of wave propagation in complex media.

Abstract

In this note we provide dispersive estimates for Fourier integrals with parameter-dependent phase functions in terms of geometric quantities of associated families of Fresnel surfaces. The results are based on a multi-dimensional van der Corput lemma due to the first author. Applications to dispersive estimates for hyperbolic systems and scalar higher order hyperbolic equations are also discussed.