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An $L^{4/3}$ $SL_2$ Kakeya maximal inequality

Terence L. J. Harris

Abstract

It is shown that $SL_2$ Besicovitch sets of measure zero exist in $\mathbb{R}^3$. The proof is constructive and uses point-line duality analogously to Kahane's construction of measure zero Besicovitch sets in the plane. A corollary is that the $SL_2$ Kakeya maximal inequality cannot hold with uniform constant. A counterexample is given to show that the $SL_2$ Kakeya maximal inequality cannot hold for $p> 3/2$; even in the model case where the $δ$-tubes have $δ$-separated directions and the cardinality of the tube family is $\sim δ^{-2}$. It is then shown that, with $C_ε δ^{-ε}$ loss, the $SL_2$ Kakeya maximal inequality does hold if $p \leq 4/3$, whenever the tubes satisfy a 2-dimensional ball condition (equivalent to the Wolff axioms in the $SL_2$ case). The proof is via an $L^{4/3}$ inequality for restricted families of projections onto planes. For both inequalities, the range $4/3 < p \leq 3/2$ remains an open problem. A related $L^{6/5}$ inequality is derived for restricted projections onto lines. Finally, an application is given to generic intersections of sets in $\mathbb{R}^3$ with "light rays" and "light planes".

An $L^{4/3}$ $SL_2$ Kakeya maximal inequality

Abstract

It is shown that Besicovitch sets of measure zero exist in . The proof is constructive and uses point-line duality analogously to Kahane's construction of measure zero Besicovitch sets in the plane. A corollary is that the Kakeya maximal inequality cannot hold with uniform constant. A counterexample is given to show that the Kakeya maximal inequality cannot hold for ; even in the model case where the -tubes have -separated directions and the cardinality of the tube family is . It is then shown that, with loss, the Kakeya maximal inequality does hold if , whenever the tubes satisfy a 2-dimensional ball condition (equivalent to the Wolff axioms in the case). The proof is via an inequality for restricted families of projections onto planes. For both inequalities, the range remains an open problem. A related inequality is derived for restricted projections onto lines. Finally, an application is given to generic intersections of sets in with "light rays" and "light planes".
Paper Structure (6 sections, 15 theorems, 150 equations)

This paper contains 6 sections, 15 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. SL2 Besicovitch sets of measure zero
  3. Train tracks of planks
  4. An L4/3 inequality for projections onto planes
  5. An L6/5 inequality for projections onto lines
  6. Some open problems

Key Result

Theorem 1.1

Let $\mathbb{T}$ be a collection of $SL_2$ tubes in $\mathbb{R}^3$ of length 1 and radius $\delta$, satisfying the condition for any tube $S$ of length 2 and radius $r$, and any $r \in [\delta, 1]$. Then for any $\epsilon >0$,

Theorems & Definitions (30)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Corollary 2.3
  • proof
  • ...and 20 more