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A FRACTAL PLANCHEREL THEOREM

机译：分形普朗切定理

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摘要

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 F_u f denotes its Fourier transform with respect to μ, it is not true (in general) that Fμ f∈ 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 T →∞. 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.
机译：如果所有x∈R〜n且所有0

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