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Localisation and weighted inequalities for spherical Fourier means. (English) Zbl 1161.42005

Authors’ abstract: We establish certain equivalences between the localisation properties with respect to spherical Fourier means of the support of a given Borel measure and the \(L^2\)-rate of decay of the Fourier extension operator associated to it. This, in turn, is intimately connected with the property that the \(X\)-ray transform of the measure be uniformly bounded. Geometric properties of sets supporting such a measure are studied.

MSC:

42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
42B15 Multipliers for harmonic analysis in several variables
42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
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