Necessary and Sufficient Conditions for Boundedness of Commutators of the General Fractional Integral Operators on Weighted Morrey Spaces

Abstract

We prove that $b$ is in $\text{L}\text{i}{\text{p}}_{\beta }(\omega )$ if and only if the commutator $[b,L^{-\alpha /2}]$ of the multiplication operator by $b$ and the general fractional integral operator $L^{-\alpha /2}$ is bounded from the weighted Morrey space $L^{p,k}(\omega )$ to $L^{q,kq/p}({\omega }^{1-(1-\alpha /n)q},\omega )$, where , , $1/q=1/p-(\alpha +\beta )/n$, , ${\omega }^{q/p}\in A_{\mathrm{1,}}$ and $r_{\omega }>\mathrm{(1}-\mathrm{k)}/\mathrm{(p}/(q-k))$, and here $r_{\omega }$ denotes the critical index of $\omega$ for the reverse Hölder condition.