Mathematika

Research Article

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

Pertti Mattilaa1

a1 Department of Mathematics, University of Helsinki, Helsinki, Finland.

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

S0025579300013462_eqnU1

and the averages over the spheres

S0025579300013462_eqnU2

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

S0025579300013462_eqnU3

S0025579300013462_eqnU4

S0025579300013462_eqnU5

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 .

(Received September 08 1987)

Key Words:

  • 28A75: MEASURE AND INTEGRATION; Classical measure theory; Other geometric measure theory.