Commutator Theorems for Fractional Integral Operators on Weighted Morrey Spaces

Abstract

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