Some new characterizations of BLO and Campanato spaces in the Schr\"{o}dinger setting

TL;DR

This paper introduces new characterizations of BLO and Campanato spaces associated with Schr"odinger operators, establishing inequalities and extending previous results to weighted Lebesgue spaces.

Contribution

The authors define new BLO and Campanato spaces in the Schr"odinger setting and provide characterizations and inequalities, extending prior work to weighted Lebesgue spaces.

Findings

Established John--Nirenberg inequality for BLO spaces

Provided new characterizations of BLO and Campanato spaces

Extended results to weighted Lebesgue spaces

Abstract

Let us consider the Schr\"{o}dinger operator $\mathcal{L}=-\Delta+V$ on $\mathbb R^d$ with $d\geq3$, where $\Delta$ is the Laplacian operator on $\mathbb R^d$ and the nonnegative potential $V$ belongs to certain reverse H\"{o}lder class $RH_s$ with $s\geq d/2$. In this paper, the authors first introduce two kinds of function spaces related to the Schr\"{o}dinger operator $\mathcal{L}$. A real-valued function $f\in L^1_{\mathrm{loc}}(\mathbb R^d)$ belongs to the (BLO) space $\mathrm{BLO}_{\rho,\theta}(\mathbb R^d)$ with $0\leq\theta<\infty$ if \begin{equation*} \|f\|_{\mathrm{BLO}_{\rho,\theta}} :=\sup_{\mathcal{Q}}\bigg(1+\frac{r}{\rho(x_0)}\bigg)^{-\theta}\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 $\mathbb R^d$, $\rho(\cdot)$ is the critical radius function in the Schr\"{o}dinger context. For $0<\beta<1$, a real-valued function $f\in L^1_{\mathrm{loc}}(\mathbb R^d)$ belongs to the (Campanato) space $\mathcal{C}^{\beta,\ast}_{\rho,\theta}(\mathbb R^d)$ with $0\leq\theta<\infty$ if \begin{equation*} \|f\|_{\mathcal{C}^{\beta,\ast}_{\rho,\theta}} :=\sup_{\mathcal{B}}\bigg(1+\frac{r}{\rho(x_0)}\bigg)^{-\theta} \bigg(\frac{1}{|B(x_0,r)|^{1+\beta/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 $\mathrm{BLO}_{\rho,\theta}(\mathbb R^d)$ with $0\leq\theta<\infty$ 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.