Damping oscillatory integrals by the Hessian determinant via Schr\"odinger

TL;DR

This paper investigates conditions under which degenerate oscillatory integrals can be made to decay like nondegenerate ones by restricting to regions with a lower Hessian determinant bound, revealing a link to the Schrödinger equation.

Contribution

It establishes when and how degenerate oscillatory integrals can be uniformly estimated using Hessian determinant bounds, connecting oscillatory integral asymptotics with the Schrödinger equation.

Findings

Decay estimates are not always possible in two dimensions.

The results are uniform under phase perturbations and Hessian cutoff.

A geometric approach links oscillatory integrals to the Schrödinger equation.

Abstract

We consider the question of when it is possible to force a degenerate scalar oscillatory integral to decay as fast as a nondegenerate one by restricting the support to the region where the Hessian determinant of the phase is bounded below. We show in two dimensions that the desired outcome is not always possible, but does occur for a broad class of phases which may be described in terms of the Newton polygon. The estimates obtained are uniform with respect to linear perturbation of the phase and uniform in the cutoff value of the Hessian determinant. In the course of the proof, we investigate a geometrically-invariant approach to making uniform estimates of qualitatively nondegenerate oscillatory integrals. The approach illuminates a previously unknown, fundamental relationship between the asymptotics of oscillatory integrals and the Schr\"odinger equation.