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Autumn 2017 $L^p$ Fourier transformation on non-unimodular locally compact groups
Marianne Terp
Adv. Oper. Theory 2(4): 547-583 (Autumn 2017). DOI: 10.22034/AOT.1709-1231

Abstract

Let $G$ be a locally compact group with modular function $\Delta$ and left regular representation $\lambda$. We define the $L^p$ Fourier transform of a function $f \in L^p(G)$, $1 \le p \le 2$, to be essentially the operator $\lambda(f)\Delta^{\frac{1}{q}}$ on $L^2(G)$ (where $\frac{1}{p}+\frac{1}{q}=1$) and show that a generalized Hausdorff-Young theorem holds. To do this, we first treat in detail the spatial $L^p$ spaces $L^p(\psi_0)$, $1 \le p \le \infty$, associated with the von Neumann algebra $M=\lambda(G)^{\prime\prime}$ on $L^2(G)$ and the canonical weight $\psi_0$ on its commutant. In particular, we discuss isometric isomorphisms of $L^2(\psi_0)$ onto $L^2(G)$ and of $L^1(\psi_0)$ onto the Fourier algebra $A(G)$. Also, we give a characterization of positive definite functions belonging to $A(G)$ among all continuous positive definite functions.

Citation

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Marianne Terp. "$L^p$ Fourier transformation on non-unimodular locally compact groups." Adv. Oper. Theory 2 (4) 547 - 583, Autumn 2017. https://doi.org/10.22034/AOT.1709-1231

Information

Received: 19 August 2017; Accepted: 19 September 2017; Published: Autumn 2017
First available in Project Euclid: 4 December 2017

zbMATH: 1374.39040
MathSciNet: MR3730047
Digital Object Identifier: 10.22034/AOT.1709-1231

Subjects:
Primary: 39B82
Secondary: 44B20‎, 46C05

Rights: Copyright © 2017 Tusi Mathematical Research Group

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Vol.2 • No. 4 • Autumn 2017
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