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

Abstract

Let $\mathcal L=-\Delta+V$ be a Schr\"odinger operator on $\mathbb R^d$, $d\geq3$, where $\Delta$ is the Laplacian operator on $\mathbb R^d$ and the nonnegative potential $V$ belongs to the reverse H\"older class $RH_s$ for $s\geq d/2$. For given $0<\alpha<d$, the fractional integrals associated to the Schr\"odinger operator $\mathcal L$ is defined by $\mathcal I_{\alpha}={\mathcal L}^{-{\alpha}/2}$.Suppose that $b$ is a locally integrable function on $\mathbb R^d$, the commutator generated by $b$ and $\mathcal I_{\alpha}$ is defined by $[b,\mathcal I_{\alpha}]f(x)=b(x)\cdot \mathcal I_{\alpha}f(x)-\mathcal I_{\alpha}(bf)(x)$. In this paper, we first introduce some kinds of weighted Morrey spaces related to certain nonnegative potentials belonging to the reverse H\"older class $RH_s$ for $s\geq d/2$. 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 $A_{p,q}$, $\mathrm{BMO}(\mathbb R^d)$ and $L^{p,\kappa}(\mu,\nu)$ corresponding to the classical case (that is $V\equiv0$).