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Spherical averages of Fourier transforms of measures with finite energy; dimensions of intersections and distance sets

  • Pertti Mattila (a1)

Let μ, be a positive Radon measure with compact support in the euclidean n-space ℝn. Introducing the Fourier transform

and the averages over the spheres

we can write the α-energy, 0 < α < n, of μ as

where the positive constants c1 and c2 depend only on n and α. The second equality is based on the Plancherel formula and the fact that where .

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K2. J.-P. Kahane . Sur la dimensions des intersections, Aspects of Mathematics and its Applications, J. A. Barroso editor (Elsevier Science Publishers B.V., 1986, 419430).

L. N. S. Landkof . Foundations of Modern Potential Theory (Springer, 1972).

M1. P. Mattila . Hausdorff dimension and capacities of intersections of sets in n-space. Acta Math., 152 (1984), 77105.

M. B. Muckenhoupt . Weighted norm inequalities for the Hardy maximal function. Trans. Amer. Math. Soc., 165 (1972), 207226.

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  • ISSN: 0025-5793
  • EISSN: 2041-7942
  • URL: /core/journals/mathematika
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