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Published March 1999 | Published
Journal Article Open

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

Keich, U.

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).

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.

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