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Randomised circular means of Fourier transforms of measures


Authors: Jonathan M. Bennett and Ana Vargas
Journal: Proc. Amer. Math. Soc. 131 (2003), 117-127
MSC (2000): Primary 42B10
DOI: https://doi.org/10.1090/S0002-9939-02-06696-0
Published electronically: August 19, 2002
MathSciNet review: 1929031
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Abstract: We explore decay estimates for $L^1$ circular means of the Fourier transform of a measure on $\mathbb{R}^2$ in terms of its $\alpha$-dimensional energy. We find new upper bounds for the decay exponent. We also prove sharp estimates for a certain family of randomised versions of this problem.


References [Enhancements On Off] (What's this?)

  • 1. J. A. Barceló, Tesis Doctoral, Universidad Autónoma de Madrid, 1988.
  • 2. J. A. Barceló, J. M. Bennett, A. Carbery, A bilinear extension estimate in two dimensions, preprint (2001).
  • 3. J. A. Barceló, A. Ruiz, L. Vega, Weighted estimates for the Helmholtz equation and consequences, JFA Vol. 150 (1997), 2, 356-382. MR 99a:35033
  • 4. J. Bourgain, Hausdorff dimension and distance sets, Israel J. Math. 87 (1994), 193-201. MR 95h:28008
  • 5. A. Carbery, F. Soria, Pointwise Fourier inversion and localisation in $\mathbb{R}^{n}$, Journal of Fourier Analysis and Applications 3, special issue, 847-858 (1997). MR 99c:42018
  • 6. A. Carbery, F. Soria, A. Vargas, preprint.
  • 7. A. Córdoba, The Kakeya Maximal Functions and the Spherical Summation Multipliers, Am. J. Math. 99 (1977), 1-22. MR 56:6259
  • 8. N. H. Katz, T. Tao, Some connections between Falconer's distance set conjecture, and sets of Furstenburg type, New York J. Math. 7 (2001), 149-187.
  • 9. P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, Cambridge Studies in Advanced Mathematics 44. MR 96h:28006
  • 10. P. Mattila, Spherical averages of Fourier transforms of measures with finite energy; dimensions of intersections and distance sets, Mathematika 34 (1987), 207-228. MR 90a:42009
  • 11. P. Sjölin, Estimates of spherical averages of Fourier transforms and dimensions of sets, Mathematika 40 (1993), 322-330. MR 95f:28007
  • 12. G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, 1944. MR 6:64a
  • 13. T. H. Wolff, Decay of circular means of Fourier transforms of measures, Internat. Math. Res. Notices 10 (1999), 547-567. MR 2000k:42016
  • 14. T. H. Wolff, Recent work connected with the Kakeya problem, Prospects in mathematics (Princeton, NJ, 1996), 129-162, Amer. Math. Soc., Providence, RI, 1999. MR 2000d:42010

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Additional Information

Jonathan M. Bennett
Affiliation: Department of Mathematics, University Autonoma de Madrid, 28049 Madrid, Spain
Email: jonathan.bennett@uam.es

Ana Vargas
Affiliation: Department of Mathematics, University Autonoma de Madrid, 28049 Madrid, Spain
Email: ana.vargas@uam.es

DOI: https://doi.org/10.1090/S0002-9939-02-06696-0
Keywords: Fourier transforms, circular means, $\alpha$-energy
Received by editor(s): April 27, 2001
Published electronically: August 19, 2002
Communicated by: Andreas Seeger
Article copyright: © Copyright 2002 American Mathematical Society

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