Some new characterizations of BLO and Campanato spaces in the Schrödinger setting
Cong Chen, Hua Wang
Abstract
Let us consider the Schrödinger operator $\mathcal{L}=-Δ+V$ on $\mathbb R^d$ with $d\geq3$, where $Δ$ is the Laplacian operator on $\mathbb R^d$ and the nonnegative potential $V$ belongs to certain reverse Hölder class $RH_s$ with $s\geq d/2$. In this paper, the authors first introduce two kinds of function spaces related to the Schrödinger operator $\mathcal{L}$. A real-valued function $f\in L^1_{\mathrm{loc}}(\mathbb R^d)$ belongs to the (BLO) space $\mathrm{BLO}_{ρ,θ}(\mathbb R^d)$ with $0\leqθ<\infty$ if \begin{equation*} \|f\|_{\mathrm{BLO}_{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{ess\,inf}}\,f(y)\Big]\,dx\bigg), \end{equation*} where the supremum is taken over all cubes $\mathcal{Q}=Q(x_0,r)$ in $\mathbb R^d$, $ρ(\cdot)$ is the critical radius function in the Schrödinger context. For $0<β<1$, a real-valued function $f\in L^1_{\mathrm{loc}}(\mathbb R^d)$ belongs to the (Campanato) space $\mathcal{C}^{β,\ast}_{ρ,θ}(\mathbb R^d)$ with $0\leqθ<\infty$ 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{ess\,inf}}\,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 $\mathrm{BLO}_{ρ,θ}(\mathbb R^d)$ with $0\leqθ<\infty$ and $d\geq3$. 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.