We identify conditions for which a Dirichlet space admitting a sub-Gaussian heat kernel would be in the Da Prato-Debussche regime of the $\Phi^{n+1}$ equation. For this purpose, we use heat kernel based Besov spaces, where regularity of schwartz-type distributions is measured using the small time behavior of the heat kernel. We show some classical PDE tools such as Paraproducts and Schauder's estimate can be re-discovered in this setting, as well as how they are affected by the geometry of these spaces. In particular, the paraproduct theorem has a sharp restriction based on the maximal Holder regularity of the heat kernel $\theta$, which results in $\theta$ appearing in the DPD regime condition.