A stationary phase type estimate

TL;DR

This paper develops a stationary phase estimate tailored for parameter-dependent phases, avoiding assumptions on critical point positions or phase gradient bounds, with explicit dependence of constants on derivatives.

Contribution

It introduces a stationary phase estimate that handles parameter dependence without requiring critical point position or gradient bounds, with explicit constant dependencies.

Findings

Provides a new stationary phase estimate for parameter-dependent phases.

Constants depend explicitly on derivatives of the phase and symbols.

Applicable to phases without specified critical point behavior.

Abstract

The purpose of this note is to prove a stationary phase estimate well adapted to parameter dependent phases. In particular, no discussion is made on the positions (and behaviour) of critical points, no lower or upper bound on the gradient of the phase is assumed, and the dependence of the constants with respect to derivatives of the phase and symbols is explicit.