Trilinear characterizations of the Fourier extension conjecture on the paraboloid in three dimensions

TL;DR

This paper establishes an equivalence between a local trilinear extension inequality on the paraboloid in three dimensions and the Fourier restriction conjecture, introducing a wavelet-based variant that characterizes the conjecture.

Contribution

The authors prove the equivalence of a local trilinear extension inequality with the Fourier restriction conjecture and introduce a wavelet-based variant that captures the weakest form of this inequality.

Findings

Proved the equivalence between the local trilinear extension inequality and the Fourier restriction conjecture.

Introduced a wavelet-based variant that characterizes the conjecture.

Identified the weakest inequality form that still characterizes the Fourier extension conjecture.

Abstract

We prove that a local trilinear extension inequality on the paraboloid in three dimensions is equivalent to the Fourier restriction conjecture, and then we prove a variant involving smooth Alpert wavelets that represents the weakest such inequality the authors could find that characterizes the Fourier extension conjecture.