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 Schrödinger operator on ${\mathbb{R}}^d$, $d\ge 3$, where $\Delta$ is the Laplacian operator on ${\mathbb{R}}^d$ and the nonnegative potential $V$ belongs to the reverse Hölder class $R{H}_s$ for $s\ge d/2$. For given $0<\alpha <d$, the fractional integrals associated to the Schrödinger operator $\mathcal{L}$ is defined by ${\mathcal{I}}_{\alpha }={\mathcal{L}}^{-\alpha /2}$.Suppose that $b$ is a locally integrable function on ${\mathbb{R}}^d$, 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 $R{H}_s$ for $s\ge d/2$. 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 $V\equiv 0$).

30 pages. arXiv admin note: text overlap with arXiv:1801.10217