An Estimate for the Kakeya Maximal Operator on Functions of Square Radial Type

Abstract

The small Kakeya maximal operator, $M_{a,N}$, in $\mathbf{R}^d$ is defined by averages on cylinders with the width $a$ and the height $Na$. We show that the inequality $\lVert M_{a,N}f\rVert_d \leq C\log N\lVert f\rVert_d$ holds for the functions of square radialy type, where $C$ is a constant depending only on $d$.