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

Abstract

Let μ, be a positive Radon measure with compact support in the euclidean n-space ℝn. Introducing the Fourier transformand the averages over the spheres we can write the α-energy, 0 < α < n, of μ aswhere 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 .