Tohoku Mathematical Journal

A note on the Kakeya maximal operator and radial weights on the plane

Hiroki Saito and Yoshihiro Sawano

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Abstract

We obtain an estimate of the operator norm of the weighted Kakeya (Nikodým) maximal operator without dilation on $L^2(w)$. Here we assume that a radial weight $w$ satisfies the doubling and supremum condition. Recall that, in the definition of the Kakeya maximal operator, the rectangle in the supremum ranges over all rectangles in the plane pointed in all possible directions and having side lengths $a$ and $aN$ with $N$ fixed. We are interested in its eccentricity $N$ with $a$ fixed. We give an example of a non-constant weight showing that $\sqrt{\log N}$ cannot be removed.

Article information

Source
Tohoku Math. J. (2), Volume 68, Number 4 (2016), 639-649.

Dates
First available in Project Euclid: 4 February 2017

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1486177220

Digital Object Identifier
doi:10.2748/tmj/1486177220

Mathematical Reviews number (MathSciNet)
MR3605452

Zentralblatt MATH identifier
1364.42023

Subjects
Primary: 42B25: Maximal functions, Littlewood-Paley theory

Keywords
Kakeya maximal operator radial weight

Citation

Saito, Hiroki; Sawano, Yoshihiro. A note on the Kakeya maximal operator and radial weights on the plane. Tohoku Math. J. (2) 68 (2016), no. 4, 639--649. doi:10.2748/tmj/1486177220. https://projecteuclid.org/euclid.tmj/1486177220


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