ESTIMATES FOR SOME KAKEYA-TYPE MAXIMAL OPERATORS

We use an abstract version of a theorem of Kolmogorov-Seliverstov-Paley to obtain sharp L 2 estimates for maximal operators of the form: M(B) f(x) = sup(x is-an-element-of S is-an-element-of B) 1//S/ integral-\f(x - y)\dy/S. We consider the cases where B is the class of all rectangles in R(n) congruent to some dilate of [0, 1]n-1 x [0, N-1] ; the class congruent to dilates of [0, N-1]n-1 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.