A fractal Plancherel theorem
A measure μ on Rn is called locally and uniformly h-dimensional if μ(Br(x)) ≤ h(r) for all x ∈ Rn and for all 0 < r < 1, where h is a real valued function. If f ∈ L2(μ) and Fμf denotes its Fourier transform with respect to μ, it is not true (in general) that Fμf ∈ L2 (e.g. [10]). Howev...
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2009
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| Acceso en línea: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_01471937_v34_n1_p69_Molter http://hdl.handle.net/20.500.12110/paper_01471937_v34_n1_p69_Molter |
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| Sumario: | A measure μ on Rn is called locally and uniformly h-dimensional if μ(Br(x)) ≤ h(r) for all x ∈ Rn and for all 0 < r < 1, where h is a real valued function. If f ∈ L2(μ) and Fμf denotes its Fourier transform with respect to μ, it is not true (in general) that Fμf ∈ L2 (e.g. [10]). However in this paper we prove that, under certain hypothesis on h, for any f ∈ L2(μ) the L2-norm of its Fourier transform restricted to a ball of radius r has the same order of growth as rnh(r-1) when r → ∞. Moreover we prove that the ratio between these quantities is bounded by the L2(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) = xα. |
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