A single scale smooth Alpert trilinear characterization of the Fourier extension conjecture on the paraboloid in three dimensions

Abstract

We show that the Fourier extension conjecture on the paraboloid in three dimensions is equivalent to a local single scale smooth Alpert trilinear inequality, which is an improvement of an analogous multiscale trilinear inequality in arXiv:2506.03992.

16 pages. The multiscale inequality that we use from arXiv:2506.03992 was recently weakened, and in this version we use an additional observation (2.8), along with a new argument for the proof of Theorem 1 on page 7, to complete our proof using the weaker inequality from arXiv:2506.03992. Main results unchanged