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 $L^2(\mathbf R^n)$ 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 weighted Morrey spaces $L^{p,\kappa}(w)$ when the symbol $b$ belongs to $BMO(\mathbf R^n)$ or the homogeneous Lipschitz space.