Norm estimates for the Kakeya maximal function with respect to general measures.
type: Item
instance of: scholarly article
title: Norm estimates for the Kakeya maximal function with respect to general measures.
zbMATH Open document ID: 1048.42021
author: Themis Mitsis
published in: Real Analysis Exchange
publication date: 2004-08-06T00:00:00Z
review text: Let \(0 < \delta \ll 1\) and let \(f\) be a locally integrable real-valued function on \(R^2\). For each direction \(e \in S^1\), let \(K_\delta f(e)\) be the quantity \[ K_\delta f(e) := \sup_T \frac{1}{| T| } \int_T | f| \] where \(T\) ranges over all \(1 \times \delta\) tubes oriented in the direction \(e\). The author addresses the question of establishing sharp weighted \(L^p\) bounds of the form \[ \| K_\delta f \| _{L^p(d\mu)} \leq A \| f \| _{L^p(R^2)} \] for various measures \(\mu\) on the unit circle and \(p \geq 2\). In particular, he establishes the bound \[ A \leq C_p \left(1 + \int_1^{1/\delta} \phi(1/r)\;dr\right)^{1/p} \] where \(C_p\) depends only on \(p\) and \(\phi(r) = \sup_{x \in S^1} \mu(B(x,r))\). This bound is then shown to be sharp in the case of Ahlfors-regular measures \(\mu\). For \(\mu\) equal to arclength measure, these results are due to Bourgain. The author also extends these results from regular measures to measures with finite \(s\)-energy for some \(0 < s < 1\), but with an additional logarithmic loss in the constants, though the author conjectures that in some cases this logarithmic loss can be removed. As an application he introduces the notion of an \(A\)-Kakeya set \(E\) for subsets \(A\) of the unit circle (sets which contain a unit line segment in the directions of \(A\)), and obtains the Hausdorff dimension bound \(\dim(E) \geq \dim(A) + 1\).
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 28A78
Mathematics Subject Classification ID: 28A80
zbMATH DE Number: 2083992
zbMATH Keywords: Ahlfors-regular measures
zbMATH Keywords: Kakeya maximal function
reviewed by: Terence Tao
MaRDI profile type: Publication
OpenAlex ID: W2122974868
DOI: 10.14321/REALANALEXCH.27.2.0563
description: scientific article; zbMATH DE number 2083992