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Construction of Fourier multipliers
Published online by Cambridge University Press: 17 April 2009
Abstract
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The classical Wiener-Pitt phenomenon for measures may be formulated as an existence theorem for Fourier multipliers with irregular, spectral properties and the result has been refined in various ways over the years. The most recent development is due to Zafran, who exhibits abnormal spectral behaviour in multipliers whose transforms vanish at infinity and which operate on Lp where 1 < p < ∞, p ≠ 2. Zafran's methods use Littlewood-Paley theory, interpolation, and the construction of a certain class of measures. We show here how the constructive element of his proof may be considerably simplified and sharpened.
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- Copyright © Australian Mathematical Society 1977
References
[1]Kahane, Jean-Pierre, Séries de Fourier absolument convergentes (Ergebnisse der Mathematik und ihrer Grenzgebiete, 50. Springer-Verlag, Berlin, Heidelberg, New York, 1970).CrossRefGoogle Scholar
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[3]Taylor, Joseph L., Measure algebras (Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, 16. American Mathematical Society, Providence, Rhode Island, 1973).Google Scholar
[4]Zafran, Misha, “Spectra of convolution operators on Lp spaces”, Proc. Nat. Aaad. Sci. U.S.A. 72 (1975), 3285–3286.CrossRefGoogle Scholar
[5]Zafran, Misha, “The spectra of multiplier transformations on the Lp spaces”, Ann. of Math. (2) 103 (1976), 355–374.CrossRefGoogle Scholar
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