The CMO-Dirichlet Problem for the Schrödinger Equation in the Upper Half-Space and Characterizations of CMO

Abstract

Let $\mathcal {L}$ be a Schrödinger operator of the form $\mathcal {L}=-\Delta +V$ acting on $L^2({\mathbb {R}}^n)$ where the non-negative potential V belongs to the reverse Hölder class $\mathrm{RH}_q$ for some $q\ge (n+1)/2$. Let $\mathrm{CMO}_{\mathcal {L}}(\mathbb {R}^n)$ denote the function space of vanishing mean oscillation associated to $\mathcal {L}$. In this article, we will show that a function f of $\mathrm{CMO}_{\mathcal {L}}(\mathbb {R}^n)$ 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

$$\begin{aligned} \sup _{B: \ \mathrm{balls}}\mathcal {C}_{u,B} :=\sup _{B(x_B,r_B): \ \mathrm{balls}} r_B^{-n}\int _0^{r_B}\int _{B(x_B, r_B)} \big |t \nabla u(x,t)\big |^2\, \frac{ \mathrm{dx}\, \mathrm{dt} }{t} <\infty , \end{aligned}$$

and

$$\begin{aligned} \lim _{a \rightarrow 0}\sup _{B: r_{B} \le a} \,\mathcal {C}_{u,B} = \lim _{a \rightarrow \infty }\sup _{B: r_{B} \ge a} \,\mathcal {C}_{u,B} = \lim _{a \rightarrow \infty }\sup _{B: B \subseteq \left( B(0, a)\right) ^c} \,\mathcal {C}_{u,B}=0. \end{aligned}$$

This continues the lines of the previous characterizations by Duong et al. (J Funct Anal 266(4):2053–2085, 2014) and Jiang and Li (ArXiv:2006.05248v1) for the $\mathrm{BMO}_{\mathcal {L}}$ spaces, which were founded by Fabes et al. (Indiana Univ Math J 25:159–170, 1976) for the classical BMO space. For this purpose, we will prove two new characterizations of the $\mathrm{CMO}_{\mathcal {L}}(\mathbb {R}^n)$ space, in terms of mean oscillation and the theory of tent spaces, respectively.