Duality of Fourier type with respect to locally compact Abelian groups
type: Item
instance of: scholarly article
title: Duality of Fourier type with respect to locally compact Abelian groups
zbMATH Open document ID: 1130.43001
author: Aicke Hinrichs
author: Hun Hee Lee
published in: Israel Journal of Mathematics
publication date: 2007-04-23T00:00:00Z
review text: The Hausdorff-Young inequality \(\|\widehat{f}\|_{q} \leq \|f\|_{p}, f \in L^{1}\cap L^{p}\), \(\widehat{f} = \mathcal{F}f\) the Fourier transform of \(f\), was initially related to ordinary Fourier series. Let \(X\) be a Banach space with dual denoted \(X'\) and \(G\) a locally compact Abelian group with dual denoted \(G'\). \textit{J. Peetre} [Rend. Sem. Mat. Univ. Padova 42, 15--26 (1969; Zbl 0241.46033)] extended Hausdorff-Young to Banach spaces of `Fourier type \(p\)', i.e., where the Fourier transform can be extended from the algebraic tensor product of \(L^{p}\) and \(X\) to become a bounded linear operator from \(L^{p}_{X}\) to \(L^{q}_{X}\). \textit{M. Milman} [Ann. Mat. Pura Appl., IV. Ser. 136, 317--328 (1984; Zbl 0578.46068)] extended the concept of `Fourier type' to functions on groups, the extension leading to an operator from \(L^{p}_{X}(G)\) to \(L^{q}_{X}(G')\). The Hausdorff-Young inequality can be expressed as \(\| \mathcal{F}_{G}f\| \leq c\|f\|\) for a real \(c= c(p,G)\), \(f \in L^{p}_{X}(G)\), the left hand side having the \(L^{q}_{X}(G')\) norm. This is shown to imply a dual inequality \(\|\mathcal{F}_{G'}{g}\| \leq c'\|g\|\), for all \(g \in L^{p}_{X'}(G')\), where the left hand side has the \(L^{q}_{X'}(G)\) norm; etc. This was known to be true for some groups, such as \(\mathbb{R}\), \(\mathbb{T}\), \(\mathbb{Z}\) and the Cantor group \(\{ 0,1 \} ^{n}\) [see \textit{J. Garcia-Cuerva, K. S. Kazarian, V. I. Kolyada} and \textit{J. L. Torrea}, Russ. Math. Surv. 53, 435--513 (1998; Zbl 0923.46017)]. The object of this paper is to prove all this is true for arbitrary locally compact Abelian \(G\). In their proof the authors first extend Milman's results to products of groups. Then they consider groups which have open subgroups; the proof reduces to that for discrete groups which turn out to be isomorphic to appropriate product groups.
reviewed by: Aubrey Wulfsohn
Mathematics Subject Classification ID: 43A25
Mathematics Subject Classification ID: 46B28
Mathematics Subject Classification ID: 47B48
zbMATH DE Number: 5145476
zbMATH Keywords: Banach space of Fourier type
zbMATH Keywords: locally compact Abelian group
zbMATH Keywords: dual group
zbMATH Keywords: \(L^p\) norms
zbMATH Keywords: Hausdorff-Young inequality
MaRDI profile type: Publication
full work available at URL: s11856-006-0008-0
OpenAlex ID: W2047277612
cites work: On the vector valued Hausdorff-Young inequality
cites work: Vector-valued Hausdorff-Young inequality and applications
cites work: Q3205756
cites work: On the Fourier-Coefficients of Vector-Valued Functions
cites work: Complex interpolation and geometry of Banach spaces
cites work: On the vector-valued Fourier transform and compatibility of operators
cites work: Sur la transformation de Fourier des fonctions à valeurs vectorielles
cites work: Q4212491
DOI: 10.1007/S11856-006-0008-0
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description: scientific article; zbMATH DE number 5145476