On the Dirichlet problem for the Schr\"odinger equation with boundary value in BMO space

TL;DR

This paper establishes a characterization of solutions to the Schr"odinger equation on metric measure spaces with boundary values in BMO space, linking Carleson measure conditions to Poisson integral representations.

Contribution

It proves an equivalence between Carleson measure conditions and BMO boundary trace representations for Schr"odinger equation solutions in a general metric measure space setting.

Findings

Solutions satisfy Carleson condition iff they are Poisson integrals with BMO boundary trace.

Characterizes boundary behavior of Schr"odinger equation solutions in metric measure spaces.

Extends classical PDE boundary value theory to non-Euclidean settings.

Abstract

Let $(X,d,\mu)$ be a metric measure space satisfying a $Q$-doubling condition, $Q>1$, and an $L^2$-Poincar\'{e} inequality. Let $\mathscr{L}=\mathcal{L}+V$ be a Schr\"odinger operator on $X$, where $\mathcal{L}$ is a non-negative operator generalized by a Dirichlet form, and $V$ is a non-negative Muckenhoupt weight that satisfies a reverse H\"older condition $RH_q$ for some $q\ge (Q+1)/2$. We show that a solution to $(\mathscr{L}-\partial_t^2)u=0$ on $X\times \mathbb{R}_+$ satisfies the Carleson condition, $$\sup_{B(x_B,r_B)}\frac{1}{\mu(B(x_B,r_B))} \int_{0}^{r_B} \int_{B(x_B,r_B)} |t\nabla u(x,t)|^2 \frac{\mathrm{d}\mu\mathrm{d} t}{t}<\infty,$$ if and only if, $u$ can be represented as the Poisson integral of the Schr\"odinger operator $\mathscr{L}$ with trace in the BMO space associated with $\mathscr{L}$.