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

TL;DR

This paper introduces new weighted Morrey spaces related to Schr"odinger operators with potentials in reverse H"older classes, and establishes boundedness of fractional integrals and their commutators on these spaces.

Contribution

It defines novel weighted Morrey spaces associated with Schr"odinger operators and proves boundedness results for fractional integrals and their commutators within these spaces.

Findings

Boundedness of fractional integrals on new Morrey spaces.

Weighted strong-type estimates for commutators.

Extension of classical weight and space classes beyond traditional $A_{p,q}$ and BMO.

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 $V\equiv0$).