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 |
|---|---|
| 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 |
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| 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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