Carleson Measure Characterization of Solutions to the Heat Equation with a Potential from the Reverse Hölder Class

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.