Notes on commutators and Morrey spaces
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)].
scientific article; zbMATH DE number 2053125
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