Di Fazio, Giuseppe; Ragusa, M. A. Commutators and Morrey spaces. (English) Zbl 0761.42009 Boll. Unione Mat. Ital., VII. Ser., A 5, No. 3, 323-332 (1991). A locally \(L^ p\) function \(f\) is said to belong to the Morrey space \(L^{p,\lambda}(\mathbb{R}^ n)\) if \[ \| f\|_{p,\lambda}^ p=\sup_{x,\rho}\rho^{ -\lambda}\int_{| x-y|\leq \rho}| f(y)|^ p dy<\infty. \] As is known, the commutators between the Calderón-Zygmund singular integral operators and the multiplication operator by a function \(a(x)\) are bounded on \(L^ p (\mathbb{R}^ n)\), \(1<p<\infty\), if and only if \(a(x)\) belongs to the John-Nirenberg space \(\roman{BMO}\). The authors show that the same result holds for the Morrey space, in place of \(L^ p\). Commutators between a fractional integral operator and \(a(x)\) are also dealt with in the Morrey space as in the \(L^ p\) space. Reviewer: K.Yabuta (Nara) Cited in 2 ReviewsCited in 79 Documents MSC: 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B25 Maximal functions, Littlewood-Paley theory 42B30 \(H^p\)-spaces 46E99 Linear function spaces and their duals Keywords:commutators; Calderón-Zygmund singular integral operators; multiplication operator; John-Nirenberg space; BMO; Morrey space; fractional integral operator PDFBibTeX XMLCite \textit{G. Di Fazio} and \textit{M. A. Ragusa}, Boll. Unione Mat. Ital., VII. Ser., A 5, No. 3, 323--332 (1991; Zbl 0761.42009)