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

Part of: Classical measure theory

Published online by Cambridge University Press:  26 February 2010

Pertti Mattila
Pertti Mattila
Affiliation:
Department of Mathematics, University of Helsinki, Helsinki, Finland.
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Extract

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 .

MSC classification

Secondary: 28A75: Length, area, volume, other geometric measure theory
Type
Research Article
Information
Mathematika , Volume 34 , Issue 2 , December 1987 , pp. 207 - 228
Copyright
Copyright © University College London 1987

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