The Fefferman-Stein type inequality for the Kakeya maximal operator in Wolff’s range
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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: \[ \|K_\delta f\|_{L^p(\mathbf {R}^d,w)} \le C (1/\delta )^{d/p-1}(\log (1/\delta ))^\alpha \|f\|_{L^p(\mathbf {R}^d,K_\delta w)} \] 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.References
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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
- 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
- © Copyright 2004 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 133 (2005), 763-772
- MSC (2000): Primary 42B25
- DOI: https://doi.org/10.1090/S0002-9939-04-07623-3
- MathSciNet review: 2113926