A weighted inequality for the Kakeya maximal operator

Let X,5 be the Kakeya Maximal Operator defined as the supremum of averages over parallelepipeds of eccentricity a . We show that X,5 satisfies 11%f!LP((0) ? If the conjecture were true,we would obtain, by interpolation, the following LP estimates: (2) IIA~fI1v(Rn) _ < (o 1)1f IIILP(Rn) for p ? n, lCn,p (a.)nlPl (log)an IIfIILP(Rn) for 1

Mathematics Subject Classification. Primary 42B25.