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.
http://resolver.caltech.edu/CaltechAUTHORS:20120110-151151274

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## Abstract

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).

Item Type: | Article | |||||||||
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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. | |||||||||

Classification Code: | 1991 Mathematics Subject Classification: 42B25, 28A78 | |||||||||

Record Number: | CaltechAUTHORS:20120110-151151274 | |||||||||

Persistent URL: | http://resolver.caltech.edu/CaltechAUTHORS:20120110-151151274 | |||||||||

Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | |||||||||

ID Code: | 28736 | |||||||||

Collection: | CaltechAUTHORS | |||||||||

Deposited By: | Jason Perez | |||||||||

Deposited On: | 11 Jan 2012 18:24 | |||||||||

Last Modified: | 11 Jan 2012 18:24 |

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