Optimal extension of Fourier multiplier operators in \(L^p(G)\)
type: Item
instance of: scholarly article
title: Optimal extension of Fourier multiplier operators in \(L^p(G)\)
zbMATH Open document ID: 1206.43003
author: Werner J. Ricker
author: Gerd Mockenhaupt
author: S. Okada
published in: Integral Equations and Operator Theory
publication date: 2011-01-12T00:00:00Z
review text: Let \(G\) be a compact abelian group. For \(1 \leq p < \infty\), in this paper the authors study the optimal domain (a Banach function space containing \(L^p(G)\)) extension question for the Fourier multiplier operator \(T_{\psi}^{(p)}\), with values in \(L^p(G)\). The recent monograph by \textit{S. Okada, W. J. Ricker} and \textit{E. A. Sánchez Pérez} [Optimal domain and integral extension of operators, acting in function spaces. Basel: Birkhäuser (2008; Zbl 1145.47027)] is a good source for background material on this topic. We now quote some results without introducing the notation. The authors identify \(L^1(m_\psi^{(p)})\) as those \(f \in L^1(G)\) for which \((\psi . (\chi_A))^\wedge \in (L^p(G))^\wedge\) for all \(A \in {\mathcal B}(G)\). They also show that \(T^{(p)}_\psi\) is upper semi-Fredholm (lower semi-Fredholm) if and only if it is Fredholm. When this happens, the operator is a finite rank perturbation of an invertible \(p\)-multiplier operator and \(L^1(m_\psi^{(p)}) = L^p(G)\).
reviewed by: T.S.S.R.K. Rao
Mathematics Subject Classification ID: 43A22
Mathematics Subject Classification ID: 46G10
Mathematics Subject Classification ID: 47B07
Mathematics Subject Classification ID: 28B05
Mathematics Subject Classification ID: 43A15
zbMATH DE Number: 5835394
zbMATH Keywords: compact abelian groups
zbMATH Keywords: multipliers
zbMATH Keywords: Banach function spaces
MaRDI profile type: Publication
full work available at URL: s00020-010-1829-0
OpenAlex ID: W2015819363
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DOI: 10.1007/S00020-010-1829-0
description: scientific article; zbMATH DE number 5835394