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 nonnegative potential $V$ belongs to the reverse Hölder class ${RH}_q$ for some $q\ge (n+1)/2$. 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 $$\underset{B:balls}{\sup }{\mathcal{C}}_{u,B}:=\underset{B(x_B,r_B):balls}{\sup }{r}_B^{-n}{\int }_0^{r_B}{\int }_{B(x_B,r_B)}|t\nabla u(x,t){|}^2\frac{dxdt}{t}<\infty ,$$ and $$\underset{a\to 0}{\lim }\underset{B:r_B\le a}{\sup }{\mathcal{C}}_{u,B}=\underset{a\to \infty }{\lim }\underset{B:r_B\ge a}{\sup }{\mathcal{C}}_{u,B}=\underset{a\to \infty }{\lim }\underset{B:B\subseteq {\left(B(0,a)\right)}^c}{\sup }{\mathcal{C}}_{u,B}=0.$$ 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.

31 pages