The Fourier transforms of Herz spaces on certain groups
type: Item
instance of: scholarly article
title: The Fourier transforms of Herz spaces on certain groups
zbMATH Open document ID: 0537.43007
DOI: 10.1007/BF02349628
author: C. W. Onneweer
published in: Monatshefte für Mathematik
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/178182
review text: Let G be a locally compact Abelian group with a suitable family of compact open subgroups in the sense of \textit{R. E. Edwards} and \textit{G. I. Gaudry} [''Littlewood-Paley and Multiplier Theory'' (1977; Zbl 0464.42013)] and let X be its dual group. Let \(K(\alpha\),p,q;G) and \(K(\alpha\),p,q;X) denote the Beurling-Herz spaces on G and X, as defined by the author in [Lect. Notes Math. 939, 106-121 (1981; Zbl 0507.46026)]. Theorem: For \(1<p\leq 2\) and \(0\leq \alpha<1-1/p\) the Fourier transform maps \(K(\alpha\),p,p;G) continuously into K(-\(\alpha\),p',2;X). The proof requires two results that are of independent interest: an extension of the Hausdorff-Young inequality to certain weighted \(L_ p\)- spaces on G and a generalization of the Littlewood-Paley theorem to the same weighted \(L_ p\)-spaces.
Mathematics Subject Classification ID: 43A15
Mathematics Subject Classification ID: 46E35
zbMATH DE Number: 3853713
zbMATH Keywords: totally disconnected group
zbMATH Keywords: Herz space
zbMATH Keywords: Hausdorff-Young inequality
zbMATH Keywords: Littlewood-Paley theorem
zbMATH Keywords: weighted \(L_ p\)-spaces
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3853713