How Rough Local Geometry Makes Treating Singular Equations Even Harder

We identify conditions for which a Dirichlet space(a metric measure space with diffusion) 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. In the process, we show how many nontrivial parts of the solution theory such as construction of paraproducts and energy estimates are made more difficult by the roughness of the underlying geometry. These difficulties in fact produce a more restrictive regime than one may first expect by typical scaling heuristics.