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Let K δ , 0 < δ << 1, be the Kakeya (Nikodym) maximal operator defined as the supremum of averages over tubes of eccentricity δ. The (so-called) Fefferman-Stein type inequality: ∥K δ f∥ Lp(Rd(Rd,w) ≤ C(1/δ) d/p-1 (log(1/δ)) α ∥f∥ Lp(Rd,Kδw) is shown in the range 1 < p < (d + 2)/2, where C and a are some constants depending only on p and the dimension d and w is a weight. The result is a sharp bound up to log(1/δ)-factors.

The Fefferman-Stein type inequality for the Kakeya maximal operator in Wolff’s range