Operator-valued Fourier multiplier theorems on \(L_{p}\)(\(X\)) and geometry of Banach spaces.
type: Item
instance of: scholarly article
title: Operator-valued Fourier multiplier theorems on \(L_{p}\)(\(X\)) and geometry of Banach spaces.
zbMATH Open document ID: 1062.42005
DOI: 10.1016/S0022-1236(03)00185-X
author: Maria Girardi
author: Lutz Weis
published in: Journal of Functional Analysis
publication date: 2004-02-03T00:00:00Z
review text: The paper under review considers the extension of scalar-valued multiplier theorems to operator-valued multiplier theorems. Let \(X\), \(Y\) be Banach spaces and \({\mathcal B}(X,Y)\) the space of bounded linear operators from \(X\) to \(Y\). A bounded measurable function \(m\colon\mathbb{R}^N\to {\mathcal B}(X,Y)\) is a Fourier multiplier from \(L_q(\mathbb{R}^N,X)\) to \(L_q(\mathbb{R}^N,Y)\) if the map \(f\in{\mathcal S}(\mathbb{R}^N,X)\mapsto {\mathcal F}^{-1}(m\cdot {\mathcal F} f)\in L_\infty(\mathbb{R}^N,Y)\), where \({\mathcal F}\) is the Fourier transform and \({\mathcal S}\) the Schwartz class, extends to a bounded operator from \(L_q(\mathbb{R}^N,X)\) to \(L_q(\mathbb{R}^N,Y)\). Since multiplier theorems in general do not extend from the scalar case to the Banach space-valued case, some geometrical conditions on the Banach spaces are needed. A Banach space \(X\) has the unconditional martingale difference (UMD) property if all \(X\)-valued martingale difference sequences are unconditional in \(L_p(X)\), \((1<p<\infty)\). Results of \textit{J. Bourgain} [Ark. Mat. 21, 163--168 (1983; Zbl 0533.46008)] and \textit{D. L. Burkholder} [Conference on harmonic analysis in honor of Antoni Zygmund, Vol. 1, 270--286 (1983; Zbl 0532.42001)] show that this property is equivalent to the boundedness on \(L_p(X)\) of vector-valued Hilbert transform. The main result of the paper gives a condition on a bounded measurable function \(m\colon\mathbb{R}^N\setminus\{0\}\to {\mathcal B}(X,Y)\) so that it is a Fourier multiplier from \(L_q(\mathbb{R}^N,X)\) to \(L_q(\mathbb{R}^N,Y)\) \((1<q<\infty)\). The condition depends on the Fourier type of \(X\) and \(Y\) (\(X\), \(Y\) have non-trivial Fourier type since they have the UMD property). The condition is rather technical and uses vector-valued Besov spaces and the \(R\)-boundedness of a certain map induced by \(m\). The concept of \(R\)-boundedness strengthens boundedness in norm: A collection of operators \(\tau\subset B(X,Y)\)…
reviewed by: Guillermo P. Curbera
Mathematics Subject Classification ID: 42B15
Mathematics Subject Classification ID: 46E40
Mathematics Subject Classification ID: 46B09
zbMATH DE Number: 2037117
zbMATH Keywords: operator-valued Fourier multipliers
zbMATH Keywords: UMD Banach spaces
zbMATH Keywords: Fourier type of Banach spaces
MaRDI profile type: Publication
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full work available at URL: s0022-1236(03)00185-x
OpenAlex ID: W1996511671
description: scientific article; zbMATH DE number 2037117