Variational estimates for the bilinear iterated Fourier integral

TL;DR

This paper establishes pointwise variational Lp bounds for a bilinear Fourier integral operator, extending classical results and suggesting broader applications in multilinear analysis and differential equations.

Contribution

It provides the first variational bounds for bilinear Fourier integral operators, unifying and strengthening several classical harmonic analysis results.

Findings

Proved variational Lp bounds for bilinear Fourier integral operators.

Extended results to multilinear estimates using rough path theory.

Potential applications to ordinary differential equations.

Abstract

We prove pointwise variational Lp bounds for a bilinear Fourier integral operator in a large but not necessarily sharp range of exponents. This result is a joint strengthening of the corresponding bounds for the classical Carleson operator, the bilinear Hilbert transform, the variation norm Carleson operator, and the bi-Carleson operator. Terry Lyon's rough path theory allows for extension of our result to multilinear estimates. We consider our result a proof of concept for a wider array of similar estimates with possible applications to ordinary differential equations.