Fourier multipliers from \(L^p\)-spaces to Morrey spaces on the unit circle
type: Item
instance of: scholarly article
title: Fourier multipliers from \(L^p\)-spaces to Morrey spaces on the unit circle
zbMATH Open document ID: 1303.42003
author: Takashi Izumi
author: Enji Sato
published in: Tokyo Journal of Mathematics
publication date: 2014-09-30T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.tjm/1406552439
review text: The Morrey spaces \(L^{p,\lambda }(\mathbb{T)}\) for \(1\leq p\leq \infty \) and \(0\leq \lambda \leq 1\) are defined by \[ L^{p,\lambda }(\mathbb{T)=}\left\{ f:\sup_{I\text{ interval }}\left( \frac{1 }{|I|^{\lambda }}\int_{I}|f|^{p}dx\right) ^{1/p}<\infty \right\} . \] These are Banach spaces that generalize the classical \(L^{p}\) spaces; indeed, \(L^{p,0}(\mathbb{T)}=L^{p}(\mathbb{T)}\). The authors study Fourier multipliers on these spaces. In analogy with the classical theory, they prove that the spaces of multipliers mapping \(L^{p}\) to \(L^{p,\lambda }\) are distinct for suitable choices of \(p\) and \(\lambda \) . They also show that an absolutely continuous measure belongs to a Lipschitz class \(\mathrm{Lip}(\alpha )\), for some \(0<\alpha <1\), if and only if the measure acts by convolution from \(L^{p}\) to \(L^{p,\lambda }\) for some \( 1<p<\infty \) and \(0<\lambda <1\). The Peetre interpolation theorem for Morrey spaces is an important idea in their proofs.
reviewed by: Kathryn E. Hare
Mathematics Subject Classification ID: 42A45
Mathematics Subject Classification ID: 42A55
Mathematics Subject Classification ID: 43A15
zbMATH DE Number: 6350030
zbMATH Keywords: Fourier multipliers
zbMATH Keywords: Morrey spaces
MaRDI profile type: Publication
OpenAlex ID: W1982915998
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DOI: 10.3836/TJM/1406552439
description: scientific article; zbMATH DE number 6350030