Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the Dirichlet problem for the Schr\"odinger equation with boundary
  value in BMO space   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 $\mathscr{L}$.

0
The categories of this paper are: hep-ph
37
The categories of this paper are: supr-con
31
The categories of this paper are: ao-sci
17
The categories of this paper are: q-fin
35
The categories of this paper are: mtrl-th
25
The categories of this paper are: patt-sol
8
The categories of this paper are: hep-ex
12
The categories of this paper are: nucl-th
28
The categories of this paper are: alg-geom
30
The categories of this paper are: q-alg
27
The categories of this paper are: comp-gas
15
The categories of this paper are: nucl-ex
21
The categories of this paper are: adap-org
29
The categories of this paper are: funct-an
7
The categories of this paper are: hep-th
3
The categories of this paper are: physics