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On L p Bounds for Kakeya Maximal Functions and the Minkowski Dimension in R 2
Author(s) -
Keich U.
Publication year - 1999
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609398005372
Subject(s) - mathematics , dimension (graph theory) , minkowski–bouligand dimension , minkowski space , hausdorff dimension , combinatorics , upper and lower bounds , class (philosophy) , discrete mathematics , mathematical analysis , fractal dimension , fractal , geometry , artificial intelligence , computer science
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 . 1991 Mathematics Subject Classification 42B25, 28A78.