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

Let (X, d, μ) be a metric measure space satisfying Q-doubling condition (Q > 1) and an L2-Poincaré inequality. $${\cal L} = {\cal + V$$ Schrödinger operator on X, where L}$$ is non-negative generalized by Dirichlet form, V Muckenhoupt weight that satisfies reverse Hölder RHq for some q ⩾ 1)/2. We show solution to $$({\cal - \partial _t^2)u 0$$ X × ℝ+ the Carleson $$\mathop {{\rm{sub}}}\limits_{B({x_B},{r_B})} {1 \over {\mu (B({x_B},{r_B}))}}\int_0^{{r_B}} {\int_{B({x_B},{r_B})} {{{\left| {t\nabla u(x,t)} \right|}^2}{{d\mu dt} t} < \infty } $$ if only u can represented as Poisson integral of ℒ with trace in BMO (bounded mean oscillation) associated ℒ.