A note on fractional type integrals in the Schrödinger setting

Assume $\mathcal{L}=-\mathrm{\Delta }+V$ is a Schrödinger operator on ${\mathbb{R}}^d$, where $V$ belongs to certain reverse Hölder class $R{H}_{\sigma }$ with $\sigma \ge d\mathrm{/}2$. We consider the class of $A_{p,q}$ weights associated to $\mathcal{L}$, denoted by $A_{p,q}^{\mathcal{L}}({\mathbb{R}}^d)$, which include the classical Muckenhoupt $A_{p,q}({\mathbb{R}}^d)$ weights. We obtain the quantitative $A_{p,q}^{\mathcal{L}}({\mathbb{R}}^d)$ estimates for fractional integrals associated to the Schrödinger operator. Particularly, the quantitative weighted endpoint bound for fractional integrals associated to the Schrödinger operator is first established, which was missing in the literature of Li et al. \cite{LRW}. Moreover, we generalize weighted endpoint inequalities to weighted mixed weak type inequalities for fractional type integrals in the Schrödinger setting.