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ON L p BOUNDS FOR THE KAKEYA MAXIMAL FUNCTION

We prove that the bound on the L norms of the Kakeya type maximal functions studied by Cordoba [2], and by Bourgain [1] are sharp for p > 2. The proof is based on a construction originally due to Schoenberg [5], for which we provide an alternative derivation. We also show that r log(1/r) is the exact Minkowski dimension of the class of Kakeya sets in R, and prove that the exact Hausdorff dimension of these sets is between r log(1/r) and r log(1/r) [log log(1/r)].