Heat kernel bounds and Ricci curvature for Lipschitz manifolds

Given any $d$-dimensional Lipschitz Riemannian manifold $(M,g)$ with heat kernel $\mathsf{p}$, we establish uniform upper bounds on $\mathsf{p}$ which can always be decoupled in space and time. More precisely, we prove the existence of a constant $C>0$ and a bounded Lipschitz function $R\colon M \to (0,\infty)$suchthatforevery$x\in M$andevery$t>0$, \begin{align*} \sup_{y\in M} $\mathsf{p}$(t,x,y) \leq C\min{t, R^2(x)}^{-d/2}. \end{align*} This allows us to identify suitable weighted Lebesgue spaces w.r.t. the given volume measure as subsets of the Kato class induced by $(M,g)$. In the case $\mathrm{\partial }M\mathrm{\ne }\mathrm{\varnothing }$, we also provide an analogous inclusion for Lebesgue spaces w.r.t. the surface measure on $\mathrm{\partial }M$. We use these insights to give sufficient conditions for a possibly noncomplete Lipschitz Riemannian manifold to be tamed, i.e. to admit a measure-valued lower bound on the Ricci curvature, formulated in a synthetic sense.