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ON Lp BOUNDS FOR KAKEYA MAXIMAL FUNCTIONS AND THE MINKOWSKI DIMENSION IN ℝ2

Published online by Cambridge University Press:  01 March 1999

U. KEICH
U. KEICH
Affiliation:
Department of Applied Mathematics, Caltech 217-50, Pasadena, CA 91125, USA
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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.

Type
Notes and Papers
Information
Bulletin of the London Mathematical Society , Volume 31 , Issue 2 , March 1999 , pp. 213 - 221
Copyright
© The London Mathematical Society 1999

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