FOURIER ASYMPTOTICS OF FRACTAL MEASURES

A measure μ on Rn will be called locally uniformly α-dimensional if μ(Br(x)) ≤ crα for all r ≤ 1 and all x, where Br(x) denotes the ball of radius r about x. For f{hook} ε{lunate} L2(dμ), the measure f{hook} dμ is in I′ so (f{hook} dμ) is well-defined. We show it is locally in L2 and sup r≥1rδ-n∫B|(f dμ)^ (ξ)|2 dξ ≤ c ∥f∥2· Under additional hypotheses we show that lim r→∞rδ-n∫B|(f dμ)^ (ξ)|2 (ξ)|2 dξ is comparable in size to ∥f{hook}∥2 2. A number of other related results are established. The special case when α is an integer and μ is the surface measure on a C1 manifold was treated by S. Agmon and L. Hörmander (J. Analyse Math. 30, 1976, 1-38). © 1990.