Weighted Morrey spaces related to Schrödinger operators with nonnegative potentials and fractional integrals (Q2173551)

Summary: Let \(\mathcal{L}=-\Delta+V\) be a Schrödinger operator on \(\mathbb{R}^d\), \(d\geq 3\), where \(\Delta\) is the Laplacian operator on \(\mathbb{R}^d\), and the nonnegative potential \(V\) belongs to the reverse Hölder class \(\text{RH}_s\) with \(s\geq d/2\). For given \(0<\alpha<d\), the fractional integrals associated with the Schrödinger operator \(\mathcal{L}\) is defined by \(\mathcal{I}_\alpha= \mathcal{L}^{-\alpha / 2}\). Suppose that \(b\) is a locally integrable function on \(\mathbb{R}^d\) and 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 Hölder class \(\text{RH}_s\) with \(s\geq d/2\). 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 space 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 \(A_{p,q}\), \(\text{BMO} (\mathbb{R}^d)\), and \(L^{p,\kappa} (\mu,\nu)\) corresponding to the classical case (that is \(V\equiv 0)\).