Maximal averages and packing of one dimensional sets
type: Item
instance of: scholarly article
title: Maximal averages and packing of one dimensional sets
zbMATH Open document ID: 0903.42006
author: Thomas H. Wolff
published in: Documenta Mathematica
publication date: 1998-08-06T00:00:00Z
full work available at URL: https://eudml.org/doc/232273
full work available at URL: Wolff.MAN.html
review text: The author summarizes some recent work in real analysis related to Kakeya type maximal functions, considered in the classical situations of lines and circles. One defines a Kakeya set to be a compact set \(E\subset \mathbb R^n\) which contains a unit line segment in each direction, \(\forall e\in \mathbb P^{n-1} \exists x\in \mathbb R^n: x+te\in E\) for all \(t\in [-1/2, 1/2]\) where one regards \(\mathbb P^{n-1}\) as being the unit sphere with antipodal points identified. If \(\delta\) is a small positive number and \(f:\mathbb R^n\to \mathbb R\) then one defines the Kakeya maximal function of \(f\), \(f_\delta^\ast:\mathbb P^{n-1}\to \mathbb R\), via \(f_\delta^\ast(e)= \sup_a | T_e^\delta(a)| ^{-1}\int_{f_\delta^\ast(a)}| f(x)| dx\) where \(f_\delta^\ast(a)\) is the cylinder centered at \(a\) with length 1, cross section radius \(\delta\) and axis in the \(e\) direction. Also one defines the \(\delta\)-entropy \(\mathcal N_\delta(E)\) to be the maximum possible cardinality for a \(\delta\)-separated subset. Then the following three questions are open if \(n\geq 3\). 1. Is it true that if \(E\) is a Kakeya set in \(\mathbb R^n\) then \(\limsup_{\delta\to 0} \log \mathcal N_\delta(E)/\log \delta^{-1} = n\)? 2. Is it true that a Kakeya set in \(\mathbb R^n\) must have Hausdorff dimension \(n\)? 3. Is the following estimate true \(\forall \varepsilon >0\;\exists C_\varepsilon: \| f_\delta^\ast \| _{L^n(\mathbb P^{n-1})}\leq C_\varepsilon\delta^{-\varepsilon} \| f\| _{L^n(\mathbb P^{n-1})}\)? The author discusses the partial results that have been proved on this problem and some work on a class of related problems involving circles in the plane, quoting some recent works of several authors (Bourgain, Kolasa, Mistis, Schlag, Wolff, etc).
Mathematics Subject Classification ID: 42B25
zbMATH DE Number: 1184309
zbMATH Keywords: Kakeya set
zbMATH Keywords: Kakeya type maximal functions
reviewed by: Kôzô Yabuta
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1184309