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On L^p Bounds for Kakeya Maximal Functions and the Minkowski Dimension in R^2

Keich, U. (1999) On L^p Bounds for Kakeya Maximal Functions and the Minkowski Dimension in R^2. Bulletin of the London Mathematical Society, 31 (2). pp. 213-221. ISSN 0024-6093. doi:10.1112/S0024609398005372.

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We prove that the bound on the L^p norms of the Kakeya type maximal functions studied by Cordoba [2] and Bourgain [1] are sharp for p > 2. The proof is based on a construction originally due to Schoenberg [5], for which we provide an alternative derivation. We also show that r^2 log (1/r) is the exact Minkowski dimension of the class of Kakeya sets in R^2, and prove that the exact Hausdorff dimension of these sets is between r^2 log (1/r) and r^2 log (1/r) [log log (1/r)]^(2+ε).

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Additional Information:© 1999 London Mathematical Society. Received November 25, 1997; Revision received June 11, 1998. I should like to express my gratitude to Tom Wolff for his invaluable advice.
Issue or Number:2
Classification Code:1991 Mathematics Subject Classification: 42B25, 28A78
Record Number:CaltechAUTHORS:20120110-151151274
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:28736
Deposited On:11 Jan 2012 18:24
Last Modified:09 Nov 2021 17:00

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