The CMO-Dirichlet problem for the Schrödinger 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}