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

TL;DR

This paper characterizes the space of functions with vanishing mean oscillation associated to a Schr"odinger operator, linking boundary traces to solutions satisfying a Carleson measure condition, extending previous BMO characterizations.

Contribution

It provides new characterizations of ${ CMO}_{ ext{L}}( ext{R}^n)$ via Carleson measures, mean oscillation, and tent spaces, for Schr"odinger operators with potentials in reverse H"older classes.

Findings

Characterization of ${ CMO}_{ ext{L}}( ext{R}^n)$ as boundary traces of solutions satisfying Carleson conditions.

Extension of previous BMO characterizations to CMO spaces associated with Schr"odinger operators.

Introduction of new characterizations using mean oscillation and tent space theory.

Abstract

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.