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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We prove that the bound on the L^p norms of the Kakeya type maximal functions studied by Cordoba  and Bourgain  are sharp for p > 2. The proof is based on a construction originally due to Schoenberg , 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).
|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|
|Usage Policy:||No commercial reproduction, distribution, display or performance rights in this work are provided.|
|Deposited By:||Jason Perez|
|Deposited On:||11 Jan 2012 18:24|
|Last Modified:||11 Jan 2012 18:24|
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