Fourier multipliers and spectral measures in Banach function spaces
type: Item
instance of: scholarly article
title: Fourier multipliers and spectral measures in Banach function spaces
zbMATH Open document ID: 1163.43003
author: Werner J. Ricker
author: B. de Pagter
published in: Positivity
publication date: 2009-03-27T00:00:00Z
review text: Let \(G\) be a locally compact abelian group and denote its dual group by \(\Gamma\). In the paper under review \(G\) will always be infinite. Haar measure on the Borel \(\sigma\)-algebra \(\mathcal{B}(G)\) is denoted by \(\lambda\). Let \(L^0(G)\) denote the space of all \(\mathbb{C}\)-valued, \(\mathcal{B}(G)\)-measurable functions. The space \(L^0(G)\) is a complex Dedekind complete vector lattice for the \(\lambda\)-a.e. pointwise order on the real valued functions. A Banach function space on \(G\) is an ideal \(E \subseteq L^0(G)\) equipped with a norm \(\| \cdot \|_E\) with respect to which \(E\) is a Banach lattice. The authors only consider Banach function spaces whose carrier is equal to \(G\), that is, for every \(A \in \mathcal{B}(G)\) with \(\lambda(A) > 0\) there exists \(B \in \mathcal{B}(G)\) with \(B \subseteq A\) and \(\lambda(B) > 0\) such that \(\chi_B \in E\). For \(f \in L^0(G)\) and \(g \in G\), the translate \(\tau_g f\) of \(f\) by \(g\) is defined by \((\tau_g f)(u) = f(u-g), u \in G\). A Banach function space \(E \subseteq L^0(G)\) is called translation invariant whenever \(\tau_g f \in E\) and \(\| \tau_g f \|_E = \| f \|_E\) for all \(f \in E\) and \(g \in G\). For a Banach function space \(E\) on a locally compact abelian group \(G\), the space of all bounded linear operators on \(E\) will be denoted by \(\mathcal{L}(E)\); when equipped with the strong operator topology it will be denoted by \(\mathcal{L}_s(E)\). For non-compact \(G\) and \(E\) a Banach function space over \((G, \mathcal{B}(G), \lambda)\) we assume that \(E \cap L^2(G)\) is dense in \(E\). Then a function \(\varphi \in L^{\infty}(\Gamma)\) is called an \(E\)-multiplier if there exists an operator \(R_{\varphi} \in \mathcal{L}(E)\) mapping \(E \cap L^2(G)\) into itself and satisfying \[ (R_{\varphi} f)^{\wedge} = \varphi \cdot \widehat{f}, f \in E \cap L^2(G), \] where the above equality is as elements of \(L^2(\Gamma)\). If \(E\) is a Banach function space over \((\mathb…
reviewed by: Michael J. Puls
Mathematics Subject Classification ID: 43A15
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 46G10
Mathematics Subject Classification ID: 47D03
Mathematics Subject Classification ID: 43A25
zbMATH DE Number: 5534431
zbMATH Keywords: Fourier multiplier operators
zbMATH Keywords: Banach function spaces
MaRDI profile type: Publication
full work available at URL: s11117-008-2210-2
OpenAlex ID: W2065287212
cites work: Q4039801
cites work: The Hilbert Transform on Rearrangement-Invariant Spaces
cites work: Q5654646
cites work: Q3598957
cites work: Spectral properties of translation operators in certain function spaces
cites work: A Spectral Theorem for Lp Translations
cites work: Q4150490
cites work: Q4188284
cites work: \(R\)-bounded representations of \(L^1(G)\)
cites work: Operator algebras generated by commuting projections: a vector measure approach
cites work: Q5890171
cites work: Q3711080
DOI: 10.1007/S11117-008-2210-2
description: scientific article; zbMATH DE number 5534431