Some characterizations of BMO and Lipschitz spaces in the Schrödinger setting

We consider the Schrödinger operator $\mathcal{L}=-\mathrm{\Delta }+V$ on ${\mathbb{R}}^d$, $d\ge 3$, where the nonnegative potential $V$ belongs to the reverse Hölder class $R{H}_s$ for some $s\ge d\mathrm{/}2$. A real-valued function $f\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1({\mathbb{R}}^d)$ belongs to the (BMO) space ${\mathrm{B}\mathrm{M}\mathrm{O}}_{\rho ,\theta }({\mathbb{R}}^d)$ with $0<\theta <\mathrm{\infty }$ if \begin{equation*} |f|{$\mathrm{B}\mathrm{M}\mathrm{O}${ρ,θ}} :=\sup_{B(x_0,r)}\bigg(1+\frac{r}{ρ(x_0)}\bigg)^{-θ}\bigg(\frac{1}{|B(x_0,r)|}\int_{B(x_0,r)}\big|f(x)-f_{B}\big|,dx\bigg), \end{equation*} where the supremum is taken over all balls $B(x_0,r)\subset {\mathbb{R}}^d$, $\rho (\cdot )$ is the critical radius function in the Schrödinger context and \begin{equation*} f_{B}:=\frac{1}{|B(x_0,r)|}\int_{B(x_0,r)}f(y),dy. \end{equation*} A real-valued function $f\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1({\mathbb{R}}^d)$ belongs to the (Lipschitz) space ${\mathrm{L}\mathrm{i}\mathrm{p}}_{\beta }^{\rho ,\theta }({\mathbb{R}}^d)$ with $0<\beta <1$ and $0<\theta <\mathrm{\infty }$ if \begin{equation*} |f|{$\mathrm{L}\mathrm{i}\mathrm{p}$β^{ρ,θ}} :=\sup{B(x_0,r)}\bigg(1+\frac{r}{ρ(x_0)}\bigg)^{-θ} \bigg(\frac{1}{|B(x_0,r)|^{1+β/d}}\int{B(x_0,r)}\big|f(x)-f_{B}\big|,dx\bigg). \end{equation*} It can be easily seen that ${\mathrm{B}\mathrm{M}\mathrm{O}}_{\rho ,\theta }({\mathbb{R}}^d)$ (or ${\mathrm{L}\mathrm{i}\mathrm{p}}_{\beta }^{\rho ,\theta }({\mathbb{R}}^d)$) is a function space which is larger than the classical BMO (or Lipschitz) space. In this paper, we give some new characterizations of BMO and Lipschitz spaces associated with the Schrödinger operator $\mathcal{L}$. We extend some previous works of Bongioanni–Harboure–Salinas and Liu–Sheng to the weighted case. The classes of weights considered here are larger than the classical Muckenhoupt classes.