On some variants of the Kakeya problem
type: Item
instance of: scholarly article
title: On some variants of the Kakeya problem
zbMATH Open document ID: 1019.42013
published in: Pacific Journal of Mathematics
publication date: 2000-06-22T00:00:00Z
review text: The authors give the following two results: A Borel set in \(\mathbb R^d\), \(d\geq 3\), which contains a sphere of every radius has Hausdorff dimension \(d\). A Borel set in \(\mathbb R^2\) which contains a circle of every radius has Hausdorff dimension at least \(11/6\). They derive these by showing the following estimates for a related maximal function. Let \[ C_\delta(x,r)=\{z\in\mathbb R^d;\;r-\delta<|z-x|<r+\delta\} \] and \[ M_\delta f(x)=\sup_{x}|C_\delta(x,r)|^{-1}\int_{C_\delta(x,r)}|f(y)|dy. \] Then \[ \|M_\delta f\|_{L^2(1/2,2)}\leq C(\log(1/\delta)^{1/2}\|f\|_2 \] when \(d\geq 3\), and \[ \|M_\delta f\|_{L^q(1/2,2)}\leq C_{p,q}\delta^{-1/2(3/p-1)}\|f\|_p \] for \(1<p<8/3\), \(q\leq 2p'\) when \(d=2\). As a related result, they show that any cylindrically symmetric Borel set in \(\mathbb R^3\) containing a line segment in every direction has Hausdorff dimension at least \(17/6\). To prove this , they give an estimate for a maximal function similar to the above.
reviewed by: Kôzô Yabuta
Mathematics Subject Classification ID: 42B25
zbMATH DE Number: 1463699
zbMATH Keywords: maximal functions
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: Besicovitch set
zbMATH Keywords: Kakeya set
author: L. A. Kolasa
author: Thomas H. Wolff
MaRDI profile type: Publication
full work available at URL: pjm.1999.190.111
OpenAlex ID: W2049920021
Wikidata QID: Q56341377
DOI: 10.2140/PJM.1999.190.111
description: scientific article; zbMATH DE number 1463699