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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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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 $w$.
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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