Estimates for the Kakeya Maximal Operator on Radial Functions in Rⁿ

Abstract

For a real number N > 1, the Kakeya maximal operator K _N is defined on locally integrable functions fof R ⁿas

$$ {K_N}f\left( x \right) = \mathop {\sup }\limits_{x \in R \in {B_N}} \frac{1}{{\left| R \right|}}\int {_R} \left| {f\left( y \right)} \right|dy $$

where B _N denotes the class of all rectangles in R ⁿof eccentricity N, that is, congruent with any dilate of the rectangle [0,1]^n-1x [0, N], and where x007C;Ax007C; represents the Lebesgue measure of the set A.

Partially supported by NSF grant DMS-8610730 and by DGITYC