THE KAKEYA MAXIMAL FUNCTION AND THE SPHERICAL SUMMATION MULTIPLIERS.

obvious cases: n = 1 or p =2; and also Fefferman [4] proved that TX is bounded on L P (Rn) provided that p (X) (n - 1)/4. This result has been sharpened by Tomas [15] to X > (n - 1)/2(n + 1). Finally Carleson and Sjolin [3], Fefferman [6] and Hormander [10] proved that, in R2, TA is bounded on LP whenever X >0 and p(X) 2 we have the natural question: is TX (X >0) bounded on L P(R'), p(X) < p < p'(X)? Our approach to the problem is inspired by the work of Fefferman and it is as follows: The multiplier theorem for TX can be easily reduced to this problem: