The Fefferman-Stein type inequality for the Kakeya maximal operator
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- by Hitoshi Tanaka PDF
- Proc. Amer. Math. Soc. 129 (2001), 2373-2378 Request permission
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 $K_\delta$, \[ \|K_\delta f\|_{L^p(\mathbf R^d,w)} \le C_{d,p} (\frac {1}{\delta })^{d/p-1} (\log (\frac {1}{\delta }))^{\alpha (d)} \|f\|_{L^p(\mathbf R^d,K_\delta w)}, \] in the range $(1<p\le (d^2-2)/(2d-3)$, $d\ge 3$, with some constants $C_{d,p}$ and $\alpha (d)$ independent of $f$ and the weight $w$.References
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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
- 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
- © Copyright 2001 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 129 (2001), 2373-2378
- MSC (2000): Primary 42B25
- DOI: https://doi.org/10.1090/S0002-9939-01-06069-5
- MathSciNet review: 1823921