Commutator Theorems for Fractional Integral Operators on Weighted Morrey Spaces

Let $L$ be the infinitesimal generator of an analytic semigroup on $L^2(R^n)$ with Gaussican kernel bounds, and let $L^{-\alpha/2}$ be the fractional integrals of $L$ for $0<\alpha<n.$ For any locally integrable function $b$, The commutators associated with $L^{-\alpha/2}$ are defined by $[b,L^{-\alpha/2}](f)(x)=b(x)L^{-\alpha/2}(f)(x)-L^{-\alpha/2}(bf)(x)$. When $b\in BMO(\omega)$(weighted $BMO$ space) or $b\in BMO$, the author obtain the necessary and sufficient conditions for the boundedness of $[b,L^{-\alpha/2}]$ on weighted Morrey spaces respectively.