Necessary and sufficient conditions for boundedness of commutators of the general fractional integral operators on weighted Morrey spaces

We prove that $b$ is in $Lip_{\bz}(\bz)$ 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 weighed Morrey space $L^{p,k}(\omega)$ to $L^{q,kq/p}(\omega^{1-(1-\alpha/n)q},\omega)$, where $0<\beta<1$, $0<\alpha+\beta<n, 1<p<{n}/({\alpha+\beta})$, ${1}/{q}={1}/{p}-{(\alpha+\beta)}/{n},$ $0\leq k<{p}/{q},$ $\omega^{{q}/{p}}\in A_1$ and $ r_\omega> \frac{1-k}{p/q-k},$ and here $r_\omega$ denotes the critical index of $\omega$ for the reverse H\"{o}lder condition.