On the Dirichlet problem for the Schrödinger equation with boundary value in BMO space

Abstract

Let $(X,d,\mu )$ be a metric measure space satisfying a $Q$-doubling condition, $Q>1$, and an $L^2$-Poincaré inequality. Let $\mathcal{L}=\mathcal{L}+V$ be a Schrödinger 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 Hölder condition $R{H}_q$ for some $q\ge (Q+1)/2$. We show that a solution to $(\mathcal{L}-{\partial }_t^2)u=0$ on $X\times {\mathbb{R}}_{+}$ satisfies the Carleson condition, $$\underset{B(x_B,r_B)}{\sup }\frac{1}{\mu (B(x_B,r_B))}{\int }_0^{r_B}{\int }_{B(x_B,r_B)}|t\nabla u(x,t){|}^2\frac{\mathrm{d}\mu \mathrm{d}t}{t}<\infty ,$$ if and only if, $u$ can be represented as the Poisson integral of the Schrödinger operator $\mathcal{L}$ with trace in the BMO space associated with $\mathcal{L}$.