Fourier multipliers and spectral measures in Banach function spaces |
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Authors: | Ben de Pagter Werner J. Ricker |
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Affiliation: | 1. Delft Institute of Applied Mathematics, Faculty EEMCS, Delft University of Technology, P.O. Box 5031, GA 2600, Delft, The Netherlands 2. Math.-Geogr.Fakult?t, Katholische Universit?t Eichst?tt-Ingolstadt, D-85072, Eichst?tt, Germany
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Abstract: | It is classical that amongst all spaces Lp (G), 1 ≤ p ≤ ∞, for , or say, only L2 (G) (that is, p = 2) has the property that every bounded Borel function on the dual group Γ determines a bounded Fourier multiplier operator in L2 (G). Stone’s theorem asserts that there exists a regular, projection-valued measure (of operators on L2 (G)), defined on the Borel sets of Γ, with Fourier-Stieltjes transform equal to the group of translation operators on L2 (G); this fails for every p ≠ 2. We show that this special status of L2 (G) amongst the spaces Lp (G), 1 ≤ p ≤ ∞, is actually more widespread; it continues to hold in a much larger class of Banach function spaces defined over G (relative to Haar measure). |
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Keywords: | Fourier multiplier operators Banach function spaces |
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