Some estimates for commutators of fractional integrals associated to operators with Gaussian kernel bounds on weighted Morrey spaces

Let $L$ be the infinitesimal generator of an analytic semigroup on $L^2({\mathbb{R}}^n)$ with Gaussian kernel bound, and let $L^{-\alpha \mathrm{/}2}$ be the fractional integrals of $L$ for $0<\alpha <n$. In this paper, we will obtain some boundedness properties of commutators $[b,L^{-\alpha \mathrm{/}2}]$ on the weighted Morrey spaces $L^{p,\kappa }(w)$ when the symbol $b$ belongs to $BMO({\mathbb{R}}^n)$ or homogeneous Lipschitz space.