Optimal Extension of Fourier Multiplier Operators in L
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Authors: | G Mockenhaupt S Okada W J Ricker |
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Institution: | 1. Fachbereich 6: Mathematik, Universit?t Siegen, 57068, Siegen, Germany 2. 112 Marconi Crescent, Kambah, ACT, 2902, Australia 3. Math.-Geogr. Fakult?t, Katholische Universit?t Eichst?tt-Ingolstadt, 85072, Eichst?tt, Germany
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Abstract: | Given 1 ≤ p < ∞, a compact abelian group G and a p-multiplier ${\psi : \Gamma \to {\mathbb C}}Given 1 ≤ p < ∞, a compact abelian group G and a p-multiplier
y: G? \mathbb C{\psi : \Gamma \to {\mathbb C}} (with Γ the dual group), we study the optimal domain of the multiplier operator T(p)y : Lp (G) ? Lp (G){T^{(p)}_\psi : L^p (G) \to L^p (G)}. This is the largest Banach function space, denoted by L1(m(p)y){L^1(m^{(p)}_\psi)}, with order continuous norm into which L
p
(G) is embedded and to which T(p)y{ T^{(p)}_\psi} has a continuous L
p
(G)-valued extension. Compactness conditions for the optimal extension are given, as well as criteria for those ψ for which L1(m(p)y) = Lp (G){L^1(m^{(p)}_\psi) = L^p (G)} is as small as possible and also for those ψ for which L1(m(p)y) = L1 (G){L^1(m^{(p)}_\psi) = L^1 (G)} is as large as possible. Several results and examples are presented for cases when
Lp (G) \subsetneqq L1(m(p)y) \subsetneqq L1 (G){L^p (G) \subsetneqq L^1(m^{(p)}_\psi) \subsetneqq L^1 (G)}. |
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