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

Tanaka, Hitoshi

doi:10.1090/S0002-9939-04-07623-3

The Fefferman-Stein type inequality for the Kakeya maximal operator in Wolff’s range
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Proc. Amer. Math. Soc. 133 (2005), 763-772 Request permission

Abstract:

Let $K_\delta$, $0<\delta \ll 1$, be the Kakeya (Nikodým) maximal operator defined as the supremum of averages over tubes of eccentricity $\delta$. The (so-called) Fefferman-Stein type inequality: \[ \|K_\delta f\|_{L^p(\mathbf {R}^d,w)} \le C (1/\delta )^{d/p-1}(\log (1/\delta ))^\alpha \|f\|_{L^p(\mathbf {R}^d,K_\delta w)} \] is shown in the range $1<p\le (d+2)/2$, where $C$ and $\alpha$ are some constants depending only on $p$ and the dimension $d$ and $w$ is a weight. The result is a sharp bound up to $\log (1/\delta )$-factors.

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