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The Fefferman-Stein type inequality for the Kakeya maximal operator


Author: Hitoshi Tanaka
Journal: Proc. Amer. Math. Soc. 129 (2001), 2373-2378
MSC (2000): Primary 42B25
DOI: https://doi.org/10.1090/S0002-9939-01-06069-5
Published electronically: January 23, 2001
MathSciNet review: 1823921
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Abstract:

Let $K_\delta$, $0<\delta<<1$, be the Kakeya maximal operator defined as the supremum of averages over tubes of the eccentricity $\delta$. We shall prove the so-called Fefferman-Stein type inequality for $K_\delta$,

\begin{displaymath}\Vert K_\delta f\Vert _{L^p(\mathbf R^d,w)} \le C_{d,p} (\fra... ...ta}))^{\alpha(d)} \Vert f\Vert _{L^p(\mathbf R^d,K_\delta w)}, \end{displaymath}

in the range $(1<p\le(d^2-2)/(2d-3)$, $d\ge3$, with some constants $C_{d,p}$ and $\alpha(d)$independent of $f$ and the weight $w$.


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Additional Information

Hitoshi Tanaka
Affiliation: Department of Mathematics, Gakushuin University, 1-5-1 Mejiro, Toshima-ku, Tokyo 171-8588, Japan
Email: hitoshi.tanaka@gakushuin.ac.jp

DOI: https://doi.org/10.1090/S0002-9939-01-06069-5
Received by editor(s): December 15, 1999
Published electronically: January 23, 2001
Additional Notes: This work was supported by the Japan Society for the Promotion of Sciences and the Fūjyukai Foundation.
Communicated by: Christopher D. Sogge
Article copyright: © Copyright 2001 American Mathematical Society

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