Necessary and sufficient conditions for boundedness of commutators of fractional integral operators on classical Morrey spaces

Abstract

We prove that $b$ is in $BMO(\R^n)$ if and only if the commutator $[b,I_{\alpha}]$ of the multiplication operator by $b$ and the fractional integral operator $I_{\alpha}$ is bounded from the classical Morrey space $L^{p,\lambda}(\R^n)$ to $L^{q,\mu}(\R^n)$, where $1<p< \infty$, $0<\alpha <n$, $0<\lambda <n- \alpha p$, $1/q=1/p-\alpha /n$ and $\lambda /p = \mu /q$. Also we will show that $b$ is in $\dot{\Lambda}_{\beta}(\R^n)$ if and only if the commutator $[b,I_{\alpha}]$ is bounded from the classical Morrey space $L^{p,\lambda}(\R^n)$ to $L^{q,\mu}(\R^n)$ or from $L^{p,\lambda}(\R^n)$ to $L^{q,\lambda}(\R^n)$, where $\alpha$ and $\beta$ satisfy some conditions.