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

Cristian Rios, Eric Sawyer

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.

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

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.

Paper Structure

This paper contains 16 sections, 5 theorems, 121 equations.

Table of Contents

  1. Introduction
  2. Statements of main theorems
  3. Easy directions of the proof
  4. Proof that disjoint trilinear extension implies Fourier extension
  5. The pigeonholing argument of Bourgain and Guth
  6. Case 1: Separated interaction
  7. Case 2: Clustered interaction
  8. Case 3: Dipole interaction
  9. Completing the proof
  10. A smooth Alpert characterization
  11. Initial setup and statement of the main Alpert characterization
  12. Convolution of $\nu$-disjoint singular measures on the paraboloid
  13. Case 1: $s_{3}$ is large
  14. Case 2: $s_{2}$ is small
  15. Wrapup of the proof
  16. ...and 1 more sections

Key Result

Theorem 3

The Fourier extension conjecture holds for the paraboloid $\mathbb{P}^{2}$ in $\mathbb{R}^{3}$if and only if for every $q>3$ there is $\nu>0$ such that the disjoint trilinear inequality $\mathcal{E}_{\mathop{\rm disj}\nu}\left( \otimes_{3}L^{q}\rightarrow L^{\frac{q}{3}};\varepsilon\right)$ holds f

Theorems & Definitions (23)

  • Conjecture 1: Fourier extension
  • Definition 2
  • Theorem 3
  • Remark 4
  • Definition 5
  • Theorem 6
  • proof : Proof of the implication $\left( 3\right) \Longrightarrow\left( 1\right)$ in Theorem \ref{['main']}
  • proof : Proof of Theorem \ref{['Loc lin']}
  • proof : Proof continued
  • proof : Proof continued
  • ...and 13 more