Fourier asymptotics of fractal measures

Abstract

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 ƒ ϵ L2(dμ), the measure ƒ dμ is in I′ so (ƒ dμ) is well-defined. We show it is locally in L2 and supr⩾1rδ−n∫Br(y)|(f dμ)^ (ξ)|2 dξ ⩽ c ∥f∥2· Under additional hypotheses we show that limr→∞rδ−n∫Br(y)|(f dμ)^ (ξ)|2 (ξ)|2 dξ is comparable in size to ∥ƒ∥22. 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).