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

Abstract

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