Weighted Morrey spaces related to Schrodinger operators with potentials satisfying a reverse Holder inequality and fractional integrals

Let $\mathcal{L}=-\mathrm{\Delta }+V$ be a Schr"odinger operator on ${\mathbb{R}}^d$, $d\ge 3$, where $\mathrm{\Delta }$ is the Laplacian operator on ${\mathbb{R}}^d$ and the nonnegative potential $V$ belongs to the reverse H"older class $R{H}_s$ for $s\ge d\mathrm{/}2$. For given $0<\alpha <d$, the fractional integrals associated to the Schr"odinger operator $\mathcal{L}$ is defined by $\mathcal I_{\alpha}={\mathcal L}^{-{\alpha}/2}$\mathrm{.}Supposethat$b$ is a locally integrable function on ${\mathbb{R}}^d$, the commutator generated by $b$ and $\mathcal I_{\alpha}$isdefinedby$[b,\mathcal I_{\alpha}]f(x)=b(x)\cdot \mathcal I_{\alpha}f(x)-\mathcal I_{\alpha}(bf)(x)$. In this paper, we first introduce some kinds of weighted Morrey spaces related to certain nonnegative potentials belonging to the reverse H"older class $R{H}_s$ for $s\ge d\mathrm{/}2$. Then we will establish the boundedness properties of the fractional integrals $\mathcal I_{\alpha}$ on these new spaces. Furthermore, weighted strong-type estimate for the corresponding commutator $[b,{\mathcal{I}}_{\alpha }]$ in the framework of Morrey spaces is also obtained. The classes of weights, the classes of symbol functions as well as weighted Morrey spaces discussed in this paper are larger than $A_{p,q}$, $\mathrm{B}\mathrm{M}\mathrm{O}({\mathbb{R}}^d)$ and $L^{p,\kappa }(\mu ,\nu )$ corresponding to the classical case (that is $V\equiv 0$).