Some new characterizations of BLO and Campanato spaces in the Schrödinger setting

Let us consider the Schrödinger operator $\mathcal{L}=-\mathrm{\Delta }+V$ on ${\mathbb{R}}^d$ with $d\ge 3$, where $\mathrm{\Delta }$ is the Laplacian operator on ${\mathbb{R}}^d$ and the nonnegative potential $V$ belongs to certain reverse Hölder class $R{H}_s$ with $s\ge d\mathrm{/}2$. In this paper, the authors first introduce two kinds of function spaces related to the Schrödinger operator $\mathcal{L}$. A real-valued function $f\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1({\mathbb{R}}^d)$ belongs to the (BLO) space ${\mathrm{B}\mathrm{L}\mathrm{O}}_{\rho ,\theta }({\mathbb{R}}^d)$ with $0\le \theta <\mathrm{\infty }$ if \begin{equation*} |f|{$\mathrm{B}\mathrm{L}\mathrm{O}${ρ,θ}} :=\sup_{$\mathcal{Q}$}\bigg(1+\frac{r}{ρ(x_0)}\bigg)^{-θ}\bigg(\frac{1}{|Q(x_0,r)|} \int_{Q(x_0,r)}\Big[f(x)-\underset{y\in$\mathcal{Q}$}{$\mathrm{e}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{f}$},f(y)\Big],dx\bigg), \end{equation*} where the supremum is taken over all cubes $\mathcal{Q}=Q(x_0,r)$ in ${\mathbb{R}}^d$, $\rho (\cdot )$ is the critical radius function in the Schrödinger context. For $0<\beta <1$, a real-valued function $f\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1({\mathbb{R}}^d)$ belongs to the (Campanato) space ${\mathcal{C}}_{\rho ,\theta }^{\beta ,\ast }({\mathbb{R}}^d)$ with $0\le \theta <\mathrm{\infty }$ if \begin{equation*} |f|{$\mathcal{C}$^{β,\ast}{ρ,θ}} :=\sup_{$\mathcal{B}$}\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)-\underset{y\in$\mathcal{B}$}{$\mathrm{e}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{f}$},f(y)\Big],dx\bigg), \end{equation*} where the supremum is taken over all balls $\mathcal{B}=B(x_0,r)$ in ${\mathbb{R}}^d$. Then we establish the corresponding John–Nirenberg inequality suitable for the space ${\mathrm{B}\mathrm{L}\mathrm{O}}_{\rho ,\theta }({\mathbb{R}}^d)$ with $0\le \theta <\mathrm{\infty }$ and $d\ge 3$. Moreover, we give some new characterizations of the BLO and Campanato spaces related to $\mathcal{L}$ on weighted Lebesgue spaces, which is the extension of some earlier results.