论文标题
关于BMO空间中边界值的Schrödinger方程的Dirichlet问题
On the Dirichlet problem for the Schrödinger equation with boundary value in BMO space
论文作者
论文摘要
令$（x，d，μ）$为满足$ Q $ $ Q $ - Q> 1 $的度量空间，以及$ l^2 $-poincaré不平等。 Let $\mathscr{L}=\mathcal{L}+V$ 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 $RH_q$ for some $q\ge （Q+1）/2 $。我们证明了$（\ Mathscr {l} - \ partial_t^2）u = 0 $ on $ x \ times \ times \ mathbb {r} _+$满足carleson条件，$ \ sup_ {b（x_b，r_b，r_b）} \ frac {1} {b（b（b） \ int_ {0}^{r_b} \ int_ {b（x_b，r_b）} | t \ nabla u（x，x，x，t）|^2 \ frac {\ mathrm {d}μ\ mathrm {d} Schrödinger运算符$ \ Mathscr {l} $带有与$ \ Mathscr {l} $相关的BMO空间中的跟踪。
Let $(X,d,μ)$ be a metric measure space satisfying a $Q$-doubling condition, $Q>1$, and an $L^2$-Poincaré inequality. Let $\mathscr{L}=\mathcal{L}+V$ 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 $RH_q$ 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, $$\sup_{B(x_B,r_B)}\frac{1}{μ(B(x_B,r_B))} \int_{0}^{r_B} \int_{B(x_B,r_B)} |t\nabla u(x,t)|^2 \frac{\mathrm{d}μ\mathrm{d} t}{t}<\infty,$$ if and only if, $u$ can be represented as the Poisson integral of the Schrödinger operator $\mathscr{L}$ with trace in the BMO space associated with $\mathscr{L}$.
