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

  • Pertti Mattila (a1)
  • DOI:
  • Published online: 01 February 2010

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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This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

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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