The CMO-Dirichlet problem for the Schr\"odinger equation in the upper half-space and characterizations of CMO

Let $\mathcal{L}$ be a Schr\"odinger operator of the form $\mathcal{L}=-\Delta+V$ acting on $L^2(\mathbb R^n)$ where the nonnegative potential $V$ belongs to the reverse H\"older class ${RH}_q$ for some $q\geq (n+1)/2$. Let ${CMO}_{\mathcal{L}}(\mathbb{R}^n)$ denote the function space of vanishing mean oscillation associated to $\mathcal{L}$. In this article we will show that a function $f$ of ${ CMO}_{\mathcal{L}}(\mathbb{R}^n)$ is the trace of the solution to $\mathbb{L}u=-u_{tt}+\mathcal{L} u=0$, $u(x,0)=f(x)$, if and only if, $u$ satisfies a Carleson condition $$\sup_{B: \ { balls}}\mathcal{C}_{u,B} :=\sup_{B(x_B,r_B): \ { balls}} r_B^{-n}\int_0^{r_B}\int_{B(x_B, r_B)} \big|t \nabla u(x,t)\big|^2\, \frac{ dx\, dt } {t} <\infty,$$ and $$\lim _{a \rightarrow 0}\sup _{B: r_{B} \leq a} \,\mathcal{C}_{u,B} = \lim _{a \rightarrow \infty}\sup _{B: r_{B} \geq a} \,\mathcal{C}_{u,B} = \lim _{a \rightarrow \infty}\sup _{B: B \subseteq \left(B(0, a)\right)^c} \,\mathcal{C}_{u,B}=0.$$ This continues the lines of the previous characterizations by Duong, Yan and Zhang \cite{DYZ} and Jiang and Li \cite{JL} for the ${ BMO}_{\mathcal{L}}$ spaces, which were founded by Fabes, Johnson and Neri \cite{FJN} for the classical BMO space. For this purpose, we will prove two new characterizations of the ${ CMO}_{\mathcal{L}}(\mathbb{R}^n)$ space, in terms of mean oscillation and the theory of tent spaces, respectively.