Real Analysis Exchange

Norm Estimates for the Kakeya Maximal Function with Respect to General Measures

Themis Mitsis

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Abstract

We generalize Bourgain's theorem on the two dimensional Kakeya maximal function by proving norm estimates with respect to measures satisfying certain conditions. We use this to extend the classical result of Davies on the Hausdorff dimension of Kakeya sets in the plane.

Article information

Source
Real Anal. Exchange, Volume 27, Number 2 (2001), 563-572.

Dates
First available in Project Euclid: 2 June 2008

Permanent link to this document
https://projecteuclid.org/euclid.rae/1212412855

Mathematical Reviews number (MathSciNet)
MR1922668

Zentralblatt MATH identifier
1048.42021

Subjects
Primary: 42B25: Maximal functions, Littlewood-Paley theory 28A78: Hausdorff and packing measures

Keywords
Kakeya maximal function Hausdorff measures

Citation

Mitsis, Themis. Norm Estimates for the Kakeya Maximal Function with Respect to General Measures. Real Anal. Exchange 27 (2001), no. 2, 563--572. https://projecteuclid.org/euclid.rae/1212412855


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References

  • J. Bourgain, Besicovitch type maximal operators and applications to Fourier Analysis, Geom. Funct. Anal. 1 (1991), 147–187.
  • A. Cordoba, The Kakeya maximal function and spherical summation multipliers, Amer. J. Math. 99 (1977), 1–22.
  • R. O. Davies, Some remarks on the Kakeya problem, Proc. Cambbridge Philos. Soc. 69 (1971), 417–421.
  • I. J. Schoenberg, On the Besicovitch-Perron solution of the Kakeya problem Studies in Mathematical Analysis-Essays in Honour of G. Pólya, 359–363, Stanford University Press, 1962.