Notes on commutators and Morrey spaces
type: Item
instance of: scholarly article
title: Notes on commutators and Morrey spaces
zbMATH Open document ID: 1044.42011
author: Yasuo Komori
author: Takahiro Mizuhara
published in: Hokkaido Mathematical Journal
publication date: 2004-03-08T00:00:00Z
review text: Let \(I_\alpha\), \(0<\alpha<n\), be the fractional integral operator defined by \[ I_\alpha f(x)=\int_{\mathbb R^n}\frac{f(y)}{| x-y| ^{n-\alpha}}\,dy, \] and \([M_b,\,I_\alpha]\) be the commutator defined by \[ [M_b,\,I_\alpha]f(x)=b(x)I_\alpha f(x)-I_\alpha(bf)(x), \] where \(b\in L^1_{loc}(\mathbb R^n)\). In this paper, the authors prove that \([M_b,\,I_\alpha]\) is bounded from the Morrey space \(L^{p,\,\lambda}(\mathbb R^n)\) to the Morrey space \(L^{q,\,\lambda}(\mathbb R^n)\), where \(1<p<\infty\), \(0<\alpha<n\), \(0<\lambda<n-\alpha p\) and \(\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{n-\alpha}\), if and only if \(b\) belongs to \(\text{BMO}(\mathbb R^n)\), which improves the corresponding results of \textit{G. Di Fazio} and \textit{M. A. Ragusa} in [Boll. Unione Mat. Ital., VII. Ser., A 5, No.3, 323-332 (1991; Zbl 0761.42009)].
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 47B47
Mathematics Subject Classification ID: 47B06
Mathematics Subject Classification ID: 47A30
zbMATH DE Number: 2053125
zbMATH Keywords: commutator
zbMATH Keywords: fractional integral
zbMATH Keywords: Morrey space
reviewed by: Dachun Yang
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.14492/hokmj/1350657528
OpenAlex ID: W2060847716
DOI: 10.14492/HOKMJ/1350657528
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description: scientific article; zbMATH DE number 2053125