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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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


Author: Hitoshi Tanaka
Journal: Proc. Amer. Math. Soc. 133 (2005), 763-772
MSC (2000): Primary 42B25
Published electronically: August 20, 2004
MathSciNet review: 2113926
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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:

\begin{displaymath}\Vert K_\delta f\Vert _{L^p(\mathbf{R}^d,w)} \le C (1/\delta)... ...1/\delta))^\alpha \Vert f\Vert _{L^p(\mathbf{R}^d,K_\delta w)} \end{displaymath}

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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Additional Information

Hitoshi Tanaka
Affiliation: Department of Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Tokyo 153-8914, Japan
Email: htanaka@ms.u-tokyo.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9939-04-07623-3
PII: S 0002-9939(04)07623-3
Received by editor(s): October 22, 2003
Published electronically: August 20, 2004
Additional Notes: This work was supported by the Fūjyukai Foundation.
Communicated by: Andreas Seeger
Article copyright: © Copyright 2004 American Mathematical Society