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A Kakeya maximal estimate for regulus strips

Shukun Wu

Abstract

We prove Kakeya-type estimates for regulus strips. As a result, we obtain another epsilon improvement over the Kakeya conjecture in $\mathbb{R}^3$, by showing that the regulus strips in the ${\rm SL}_2$ example are essentially disjoint. We also establish an $L^p$ inequality regarding Nikodym-type maximal function in the first Heisenberg group.

A Kakeya maximal estimate for regulus strips

Abstract

We prove Kakeya-type estimates for regulus strips. As a result, we obtain another epsilon improvement over the Kakeya conjecture in , by showing that the regulus strips in the example are essentially disjoint. We also establish an inequality regarding Nikodym-type maximal function in the first Heisenberg group.

Paper Structure

This paper contains 3 sections, 11 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['r-strip-thm']}
  3. An explanation for Lemma \ref{['rs-wp']}

Key Result

Theorem 1.5

Suppose $\mathcal{S}\subset \bar{\mathcal{S}}$ is a collection of regulus strips satisfying the two-dimensional ball condition two-dim-ball. Let $Y:\mathcal{S}\to[0,1]^3$ be a shading such that $Y(S)\subset S\cap[0,1]^3$ for any $S\in\mathcal{S}$. Let $\lambda\in(0,1]$. Suppose $|Y(S)|\geq \lambda |

Theorems & Definitions (32)

  • Conjecture 1.1
  • Definition 1.2
  • Definition 1.3: Regulus strip
  • Definition 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Definition 1.7
  • Corollary 1.8
  • Lemma 2.1: $\rm{SL}_2$ is linear
  • proof
  • ...and 22 more