A Fractal Plancherel Theorem

Abstract

A measure µ on R n is called locally and uniformly h-dimensional if µ(Br(x)) ≤ h(r) for all x ∈ R n and for all 0 < r < 1, where h is a real valued function. If f ∈ L 2 (µ) and Fµf denotes its Fourier transform with respect to µ, it is not true (in general) that Fµf ∈ L 2 (e.g. [10]). However in this paper we prove that, under certain hypothesis on h, for any f ∈ L 2 (µ) the L 2 -norm of its Fourier transform restricted to a ball of radius r has the same order of growth as r nh(r −1 ) when r → ∞. Moreover we prove that the ratio between these quantities is bounded by the L 2 (µ)-norm of f (Theorem 3.2). By imposing certain restrictions on the measure µ, we can also obtain a lower bound for this ratio (Theorem 4.3). These results generalize the ones obtained by Strichartz in [10] where he considered the particular case in which h(x) = x α .