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

Cristian Rios, Eric T. Sawyer

TL;DR

The paper proves that the Fourier extension conjecture for the paraboloid in $\mathbb{R}^3$ is equivalent to a local single-scale, smooth Alpert trilinear inequality with a mild scale factor $2^{\varepsilon s}$. It achieves this by replacing the linear analysis with a single-scale disjoint trilinear framework built from smooth Alpert frames and by adapting Bourgain–Guth pigeonholing together with parabolic rescaling to the Alpert setting. The key technical advance is a dilation-invariant, single-scale trilinear estimate that follows from a local multilinear bound, enabling a complete reduction from linear to trilinear control at fixed scales. This localization sharpens the previous multiscale approach (RiSa) and provides a new pathway to analyze the paraboloid restriction problem, with potential implications for further refinements and related restriction phenomena. The results offer a precise, scale-local mechanism to translate multilinear geometric control into linear extension estimates, facilitating more targeted SEP (scale-enhanced projection) techniques in harmonic analysis of curved manifolds.

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.

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

TL;DR

The paper proves that the Fourier extension conjecture for the paraboloid in is equivalent to a local single-scale, smooth Alpert trilinear inequality with a mild scale factor . It achieves this by replacing the linear analysis with a single-scale disjoint trilinear framework built from smooth Alpert frames and by adapting Bourgain–Guth pigeonholing together with parabolic rescaling to the Alpert setting. The key technical advance is a dilation-invariant, single-scale trilinear estimate that follows from a local multilinear bound, enabling a complete reduction from linear to trilinear control at fixed scales. This localization sharpens the previous multiscale approach (RiSa) and provides a new pathway to analyze the paraboloid restriction problem, with potential implications for further refinements and related restriction phenomena. The results offer a precise, scale-local mechanism to translate multilinear geometric control into linear extension estimates, facilitating more targeted SEP (scale-enhanced projection) techniques in harmonic analysis of curved manifolds.

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.

Paper Structure

This paper contains 12 sections, 8 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Smooth Alpert frames
  4. Equivalence of linear and trilinear single scale inequalities
  5. Reduction to the characterization in RiSa
  6. Proof of Theorem \ref{['single main Alpert']}
  7. Proof of Theorem \ref{['without']} on local single scale inequalities
  8. Dilation invariance
  9. The pigeonholing argument of Bourgain and Guth
  10. Case 1: Separated interaction
  11. Case 2: Clustered interaction
  12. Completing the proof of Theorem \ref{['Loc lin']}

Key Result

Theorem 1

Suppose $0<\delta<1$ and $\kappa\in\mathbb{N}$ with $\kappa>\frac{20}{\delta}$. Then the linear Fourier extension conjecture (FEC) for the paraboloid $\mathbb{P}^{2}$ in $\mathbb{R}^{3}$holds if and only if for every $q>3$ there is $\nu>0$ depending on $q$, such that the single scale disjointtriline holds for all $s\in\mathbb{N}$, $\varepsilon>0$, all single scale smooth Alpert pseudoprojections $

Theorems & Definitions (19)

  • Theorem 1
  • Theorem 2: Saw7
  • Definition 4
  • Definition 5
  • Theorem 6: single scale Fourier and disjoint trilinear Fourier
  • Lemma 7
  • proof
  • Definition 8
  • Theorem 9: RiSa
  • Definition 10
  • ...and 9 more