Multilinear oscillatory integral operators and geometric stability

TL;DR

This paper establishes sharp decay estimates for multilinear oscillatory integral operators and investigates their stability under smooth perturbations of phases and projections, introducing a novel decomposition method that combines Gabor and wavelet features.

Contribution

It introduces a new decomposition technique that nearly diagonalizes multilinear oscillatory integrals, enhancing understanding of their stability under perturbations.

Findings

Proved sharp decay estimates for multilinear oscillatory integrals.

Demonstrated stability of these estimates under smooth phase and projection perturbations.

Developed a novel decomposition method combining Gabor and wavelet features.

Abstract

In this article we prove a sharp decay estimate for certain multilinear oscillatory integral operators of a form inspired by the general framework of Christ, Li, Tao, and Thiele [6]. A key purpose of this work is to determine when such estimates are stable under smooth perturbations of both the phase and corresponding projections, which are typically only assumed to be linear. The proof is accomplished by a novel decomposition which mixes features of Gabor or windowed Fourier bases with features of wavelet or Littlewood-Paley decompositions. This decomposition very nearly diagonalizes the problem and seems likely to have useful applications to other geometrically-inspired objects in Fourier analysis.