Algebraic Stability of Oscillatory Integral Estimates: A Calculus for Uniform Estimates

TL;DR

This paper develops an algebraic framework to analyze the stability of oscillatory integral estimates under polynomial phase transformations, providing new insights into their behavior in one and higher dimensions.

Contribution

It introduces a calculus for understanding how oscillatory integral estimates remain stable under algebraic transformations of the phase, especially polynomial compositions.

Findings

Stable estimates under polynomial phase composition in one dimension

Extension of stability results to higher dimensions using van der Corput's lemma

Insights into the algebraic structure underlying oscillatory integral decay estimates

Abstract

Oscillatory integrals arise in many situations where it is important to obtain decay estimates which are stable under certain perturbations of the phase. Examining the structural problems underpinning these estimates leads one to consider sublevel set estimates, which behave nicely under certain algebraic operations such as composition with a polynomial. This motivates us to ask how oscillatory integral estimates behave under such transformations of the phase, and under some natural higher order convexity assumptions we obtain stable estimates under composition with polynomial phases in one dimension, and in higher dimensions in the setting of the higher dimensional van der Corput's lemma of Carbery-Christ-Wright.