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.
Categories: math.CA math.AP