Let \((X,d,\mu )\) be a space of homogeneous type in the sense of Coifman–Weiss, which satisfies a n-doubling property with \(n>1\) , and supports an \(L^2\) -Poincaré inequality. Consider the time independent Schrödinger operator \(\mathscr {L}=\mathcal {L}+V\) on X, where \(\mathcal {L}\) is a non-negative operator generalized by a Dirichlet form, and V is a non-negative Muckenhoupt weight which admits a reverse Hölder inequality of order q for some \(q>\max \{1,n/2\}\) . Without the \(C^1\) -regularity hypothesis, we derive that a solution u to the parabolic Schrödinger equation \(\partial _tu+\mathscr {L}u=0\) on \(X\times \mathbb {R}_+\) satisfies the Carleson measure condition \(\begin{aligned} \sup _{B(x_B,r_B)}\frac{1}{\mu (B(x_B,r_B))}\int _{0}^{r^2_B}\int _{B(x_B,r_B)}(|t\partial _tu|^2+|\sqrt{t}\nabla _x u|^2)\textrm{d}\mu \frac{\textrm{d}t}{t}<\infty \end{aligned}\) if and only if u can be represented as the Gaussian integral of a \(\textrm{BMO}_\mathscr {L}\) -function f. As an application, some limiting behaviors of the Carleson measure/BMO function are also considered.
