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 $L^{-α/2}$ be the fractional integrals of $L$ for $0<α<n.$ For any locally integrable function $b$, The commutators associated with $L^{-α/2}$ are defined by $[b,L^{-α/2}](f)(x)=b(x)L^{-α/2}(f)(x)-L^{-α/2}(bf)(x)$. When $b\in BMO(ω)$(weighted $BMO$ space) or $b\in BMO$, the author obtain the necessary and sufficient conditions for the boundedness of $[b,L^{-α/2}]$ on weighted Morrey spaces respectively.

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