Weighted Estimates for Marcinkiewicz Integral Operator Associated with Schr？dinger Operator and Its Commutator on Generalized Fractional Morrey Spaces

The aim of this paper is to investigate the boundedness of a Marcinkiewicz integral operator ${\mu }_j^{？}$ associated with Schr？dinger operator and its commutator on weighted generalized fractional Morrey spaces $L^{p,\eta ,\phi }\left({？}^n,\omega \right)$ . Under assumption that the function $\phi$ satisfies doubling conditions, the author proves that the ${\mu }_j^{？}$ is bounded on spaces $L^{p,\eta ,\phi }\left({？}^n,\omega \right)$ , where $0<\eta <\frac{n}{m}$ , $\frac{n}{m\eta }<p<\frac{n}{\eta }$ , and $\omega \in A_p^{\rho ,\theta }\left({？}^n\right)$ . Furthermore, the boundedness of the commutator $\left[b,{\mu }_j^{？}\right]$ formed by $$