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A comparison of trilinear testing conditions for the paraboloid Fourier extension and Kakeya conjectures in three dimensions

Eric T. Sawyer

TL;DR

This work investigates the relationship between the trilinear characterizations of the paraboloid Fourier extension conjecture in $\mathbb{R}^3$ and the Kakeya conjecture by developing a smooth Alpert testing framework and a Conversion Theorem that equates modulated and unmodulated testing. The authors build a technical foundation of doubly smooth Alpert frames, dual expansions, and well-localized operators, then prove that modulated trilinear inequalities can be converted to unmodulated ones, enabling a unified analysis. A key component is the decomposition of modulated wavelets into Kakeya-type smooth Alpert polynomials and the use of Kakeya square-function techniques with parabolic invariance and pigeonholing to transfer multiscale control to a single scale. The results illuminate how Fourier-extension and Kakeya-type problems in three dimensions can be compared on common testing grounds and suggest strategies toward resolving their long-standing conjectures.

Abstract

We compare the smooth Alpert testing condition for the paraboloid Fourier extension conjecture in <cite>RiSa3</cite> to the modulated testing condition for the Kakeya conjecture in <cite>RiSa2</cite>. To this end, the modulated testing condition is converted to a certain restricted smooth Alpert testing condition for the paraboloid Fourier extension conjecture.

A comparison of trilinear testing conditions for the paraboloid Fourier extension and Kakeya conjectures in three dimensions

TL;DR

This work investigates the relationship between the trilinear characterizations of the paraboloid Fourier extension conjecture in and the Kakeya conjecture by developing a smooth Alpert testing framework and a Conversion Theorem that equates modulated and unmodulated testing. The authors build a technical foundation of doubly smooth Alpert frames, dual expansions, and well-localized operators, then prove that modulated trilinear inequalities can be converted to unmodulated ones, enabling a unified analysis. A key component is the decomposition of modulated wavelets into Kakeya-type smooth Alpert polynomials and the use of Kakeya square-function techniques with parabolic invariance and pigeonholing to transfer multiscale control to a single scale. The results illuminate how Fourier-extension and Kakeya-type problems in three dimensions can be compared on common testing grounds and suggest strategies toward resolving their long-standing conjectures.

Abstract

We compare the smooth Alpert testing condition for the paraboloid Fourier extension conjecture in <cite>RiSa3</cite> to the modulated testing condition for the Kakeya conjecture in <cite>RiSa2</cite>. To this end, the modulated testing condition is converted to a certain restricted smooth Alpert testing condition for the paraboloid Fourier extension conjecture.

Paper Structure

This paper contains 12 sections, 10 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Paraboloid Fourier extension conjecture in $\mathbb{R}^{3}$
  3. Kakeya conjecture in $\mathbb{R}^{3}$
  4. Conversion to an unmodulated trilinear Fourier conjecture
  5. A comparison of trilinear characterizations
  6. Supporting theorems
  7. Doubly smooth Alpert frames
  8. A dual expansion and the $SS^{\ast}$ operator
  9. Well localized estimates for $SS^{\ast}$
  10. Proof of the Conversion Theorem
  11. Smooth Alpert decomposition of modulated wavelets
  12. Passage from multiscale to singe scale and completion of the proof

Key Result

Theorem 3

Fix $0<\delta<1$ 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 the $\left( \delta ,\kappa\right)$-Alpert trilinear Fourier extension Conjecture tFec holds.

Theorems & Definitions (25)

  • Definition 1
  • Conjecture 2
  • Theorem 3: C. Rios and E. Sawyer RiSa2
  • Conjecture 4: strong Kakeya maximal operator conjecture
  • Definition 5
  • Definition 6
  • Definition 7
  • Definition 8
  • Conjecture 9: modulated single scale Fourier square function disjoint trilinear extension conjecture
  • Theorem 10: C. Rios and E. Sawyer RiSa2
  • ...and 15 more