Estimates for some kakeya-typemaximal operators

Abstract

We use an abstract version of a theorem of Kolmogorov-Seliverstov- Paley to obtain sharp L2 estimates for maximal operators of the form: µβ(f(x) = sup –χєsєβ 1/|S| ∫S |f(x - y)| dy. We consider the cases where β is the class of all rectangles in Rn congru- ent to some dilate of [0, 1]n-l x [0, N-1]; the class congruent to dilates of [0, N-1]n-l x [0, 1]; and, in R2, the class of all rectangles with longest side parallel to a particular countable set of directions that include the lacunary and the uniformly distributed cases.