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Equivalence of linear and trilinear Kakeya conjectures in three dimensions

Cristian Rios, Eric T. Sawyer

TL;DR

This work proves the equivalence of the dual Kakeya maximal operator conjecture and its disjoint trilinear dual form in three dimensions by translating the problem into a single-scale Fourier square function framework on the paraboloid and exploiting parabolic rescaling. The authors develop a modulated, single-scale square-function extension theory $\mathcal{E}^{\operatorname{square}}(\otimes_{1}L^{\infty}\rightarrow L^{q};\varepsilon)$ for $q>3$ and show it is equivalent to the linear dual Kakeya conjecture; they also extend the equivalence to the disjoint trilinear setting via a parallel modulated square-function framework. A detailed pigeonholing argument, following Bourgain–Guth and RiSa, is adapted to handle single-scale wavelet projections and three interaction regimes (separated, clustered, dipole) using parabolic rescaling, enabling an absorption-based conclusion that the key single-scale bounds imply the Kakeya conjectures. The results unify linear and trilinear Kakeya phenomena through Fourier-extension techniques on the paraboloid, with potential implications for transverse multilinear Kakeya and related Fourier-analytic problems. The paper thus advances our understanding of Kakeya-type phenomena by linking maximal-operator estimates to square-function and paraboloid-extension frameworks, offering a robust approach for future sharp bounds.

Abstract

We prove the equivalence of two Kakeya conjectures: 1.The Kakeya maximal operator conjecture 2.The disjoint trilinear dual form of the Kakeya maximal operator conjecture

Equivalence of linear and trilinear Kakeya conjectures in three dimensions

TL;DR

This work proves the equivalence of the dual Kakeya maximal operator conjecture and its disjoint trilinear dual form in three dimensions by translating the problem into a single-scale Fourier square function framework on the paraboloid and exploiting parabolic rescaling. The authors develop a modulated, single-scale square-function extension theory for and show it is equivalent to the linear dual Kakeya conjecture; they also extend the equivalence to the disjoint trilinear setting via a parallel modulated square-function framework. A detailed pigeonholing argument, following Bourgain–Guth and RiSa, is adapted to handle single-scale wavelet projections and three interaction regimes (separated, clustered, dipole) using parabolic rescaling, enabling an absorption-based conclusion that the key single-scale bounds imply the Kakeya conjectures. The results unify linear and trilinear Kakeya phenomena through Fourier-extension techniques on the paraboloid, with potential implications for transverse multilinear Kakeya and related Fourier-analytic problems. The paper thus advances our understanding of Kakeya-type phenomena by linking maximal-operator estimates to square-function and paraboloid-extension frameworks, offering a robust approach for future sharp bounds.

Abstract

We prove the equivalence of two Kakeya conjectures: 1.The Kakeya maximal operator conjecture 2.The disjoint trilinear dual form of the Kakeya maximal operator conjecture

Paper Structure

This paper contains 12 sections, 6 theorems, 131 equations.

Table of Contents

  1. Introduction
  2. Organization of the proof
  3. Preliminaries
  4. The Fourier square function
  5. Linear and trilinear equivalence of Kakeya maximal operator conjectures
  6. A detailed proof of the square function theorem
  7. The pigeonholing argument of Bourgain and Guth
  8. Case 1: Separated interaction
  9. Case 2: Clustered interaction
  10. Case 3: Dipole interaction
  11. Completing the proof of the theorem
  12. Concluding remarks

Key Result

Theorem 8

The linear Kakeya Conjecture linear Kakeya is equivalent to the disjoint trilinear Kakeya Conjecture trilinear Kakeya. Moreover, these Kakeya conjectures are equivalent to their Fourier counterparts, Conjectures ssFsfec and ssFsftec below.

Theorems & Definitions (35)

  • Definition 1
  • Definition 2
  • Definition 3
  • Conjecture 4: Dual form of the Kakeya maximal operator conjecture in $\mathbb{R}^{3}$
  • Definition 5
  • Definition 6
  • Conjecture 7: The disjoint trilinear dual form of the Kakeya maximal operator conjecture in $\mathbb{R}^{3}$
  • Theorem 8
  • Definition 9
  • Definition 11
  • ...and 25 more