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Measures with Fourier Transforms in L 2 of a Half-space

  • Bassam Shayya (a1)
Abstract

We prove that if the Fourier transform of a compactly supported measure is in L 2 of a half-space, then the measure is absolutely continuous to Lebesgue measure. We then show how this result can be used to translate information about the dimensionality of a measure and the decay of its Fourier transform into geometric information about its support.

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References
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[1] Erdogan, M. B., A bilinear Fourier extension theorem and applications to the distance set problem. Int. Math. Res. Not. 2005, no. 23, 14111425.
[2] Forelli, F., Analytic and quasi-invariant measures. Acta Math. 118(1967), 3359. doi:10.1007/BF02392475
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[6] Mattila, P., Spherical averages of Fourier transforms of measures with finite energy; dimension of intersections and distance sets. Mathematika 34(1987), no. 2, 207228. doi:10.1112/S0025579300013462
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Canadian Mathematical Bulletin
  • ISSN: 0008-4395
  • EISSN: 1496-4287
  • URL: /core/journals/canadian-mathematical-bulletin
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