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

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.