On Lp Bounds for Kakeya Maximal Functions and the Minkowski Dimension in R2
Identifieur interne : 000558 ( Main/Exploration ); précédent : 000557; suivant : 000559On Lp Bounds for Kakeya Maximal Functions and the Minkowski Dimension in R2
Auteurs : U. Keich [États-Unis]Source :
- Bulletin of the London Mathematical Society [ 0024-6093 ] ; 1999-03.
Abstract
We prove that the bound on the Lp 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 r2 log (1/r) is the exact Minkowski dimension of the class of Kakeya sets in R2, and prove that the exact Hausdorff dimension of these sets is between r2 log (1/r) and r2 log (1/r) [log log (1/r)]2+ε. 1991 Mathematics Subject Classification 42B25, 28A78.
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DOI: 10.1112/S0024609398005372
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