Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Bilinear Kakeya Inequality in the Heisenberg Group

Yannis Galanos

Abstract

We prove a bilinear Kakeya inequality in the first Heisenberg group and a sharp bilinear Kakeya estimate for Euclidean curved tubes in $\R^2$. By adapting an argument of Fässler, Pinamonti and Wald involving Heisenberg projections, we show that the latter implies the former. We prove the estimate for curved tubes using a combination of techniques developed by Pramanik, Yang and Zahl, Wolff and Schlag. We introduce a novel broadness hypothesis inspired by works of Zahl, which rules out bush-type configurations that break transversal structure. We argue that such a hypothesis is needed for proving the bilinear estimates we present. We also introduce necessary additional linear terms to the estimate to counteract Szemerédi--Trotter-type clustering phenomena.

A Bilinear Kakeya Inequality in the Heisenberg Group

Abstract

We prove a bilinear Kakeya inequality in the first Heisenberg group and a sharp bilinear Kakeya estimate for Euclidean curved tubes in . By adapting an argument of Fässler, Pinamonti and Wald involving Heisenberg projections, we show that the latter implies the former. We prove the estimate for curved tubes using a combination of techniques developed by Pramanik, Yang and Zahl, Wolff and Schlag. We introduce a novel broadness hypothesis inspired by works of Zahl, which rules out bush-type configurations that break transversal structure. We argue that such a hypothesis is needed for proving the bilinear estimates we present. We also introduce necessary additional linear terms to the estimate to counteract Szemerédi--Trotter-type clustering phenomena.

Paper Structure

This paper contains 15 sections, 21 theorems, 155 equations.

Table of Contents

  1. Introduction and Main Results
  2. An Overview of the Problem
  3. Kakeya in the Heisenberg Group
  4. The Bilinear Curved Kakeya Estimate in $\mathbb{R}^2$
  5. Proof of the Bilinear Heisenberg Kakeya Estimate
  6. Preliminary Reductions and Heisenberg Projections
  7. Reducing Theorem \ref{['thm:main']} to Theorem \ref{['thm:kakeya']}
  8. A Bilinear Kakeya Estimate for Parabolas
  9. Tangencies of Planar Curves and Incidence Counting for Curvilinear Rectangles
  10. Coarse-Scale Rectangles
  11. The Broadness Argument
  12. On the Sharpness of the Main Estimates
  13. Sharpness of Planar Curved Kakeya Estimate
  14. Sharpness of the Bilinear Tube Estimate
  15. Concluding Remarks

Key Result

Theorem 1.2

Let $\varepsilon>0$. For every $\delta\in(0,1)$ and every collection $\mathcal{T}$ of Heisenberg $\delta$-tubes with $\delta^2$-separated directions in $\mathbb{S}^1$ we have $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (52)

  • Definition 1.1
  • Theorem 1.2: Fässler--Pinamonti--Wald FPW
  • Theorem 1.3: Pramanik--Yang--Zahl PYZ
  • Definition 1.4
  • Conjecture 1
  • Example 1
  • Definition 1.5: Broadness Condition for Horizontal Directions
  • Theorem 1.6
  • Definition 1.7: Bipartite pair
  • Definition 1.8: Curvilinear Rectangles
  • ...and 42 more