Weighted Morrey spaces related to certain nonnegative potentials and Riesz transforms

Let $\mathcal{L}=-\mathrm{\Delta }+V$ be a Schr"odinger operator, where $\mathrm{\Delta }$ is the Laplacian on ${\mathbb{R}}^d$ and the nonnegative potential $V$ belongs to the reverse H"older class $R{H}_q$ for $q\ge d$. The Riesz transform associated with the operator $\mathcal{L}=-\mathrm{\Delta }+V$ is denoted by $\mathcal R=\nabla{(-\Delta+V)}^{-1/2}$andthedualRiesztransformisdenotedby$\mathcal R^{\ast}=(-\Delta+V)^{-1/2}\nabla$. In this paper, we first introduce some kinds of weighted Morrey spaces related to certain nonnegative potentials belonging to the reverse H"older class $R{H}_q$ for $q\ge d$. Then we will establish the boundedness properties of the operators $\mathcal{R}$ and its adjoint ${\mathcal{R}}^{\ast }$ on these new spaces. Furthermore, weighted strong-type estimate and weighted endpoint estimate for the corresponding commutators $[b,\mathcal{R}]$ and $[b,{\mathcal{R}}^{\ast }]$ are also obtained. The classes of weights, the classes of symbol functions as well as weighted Morrey spaces discussed in this paper are larger than $A_p$, $$\mathrm{B}\mathrm{M}\mathrm{O}$(\mathbb R^d)$and$L^{p,\kappa}(w)$ corresponding to the classical Riesz transforms ($V\equiv 0$).