Uniform estimates for cubic oscillatory integrals

TL;DR

This paper derives optimal decay rates for cubic oscillatory integrals in multiple variables under nondegeneracy conditions, with stability under perturbations, using a novel geometric approach.

Contribution

It introduces a new method to obtain stable, optimal decay estimates for cubic oscillatory integrals based on nonisotropic geometric constructions.

Findings

Established optimal decay rates for cubic oscillatory integrals.

Demonstrated stability of estimates under small linear perturbations.

Developed a geometric framework using nonisotropic balls for analysis.

Abstract

This paper establishes the optimal decay rate for scalar oscillatory integrals in $n$ variables which satisfy a nondegeneracy condition on the third derivatives. The estimates proved are stable under small linear perturbations, as encountered when computing the Fourier transform of surface-carried measures. The main idea of the proof is to construct a nonisotropic family of balls which locally capture the scales and directions in which cancellation occurs.