Heat kernels in the context of Kato potentials on arbitrary manifolds

By introducing the concept of \emph{Kato control pairs} for a given Riemannian minimal heat kernel, we prove that on every Riemannian manifold $(M,g)$ the Kato class $\mathcal{K}(M,g)$ has a subspace of the form ${\mathsf{L}}^q(M,d\varrho )$, where $\varrho$ has a continuous density with respect to the volume measure ${\mu }_g$ (where $q$ depends on $\dim (M)$). Using a local parabolic ${\mathsf{L}}^1$-mean value inequality, we prove the existence of such densities for every Riemannian manifold, which in particular implies ${\mathsf{L}}_{loc}^q(M)\subset {\mathcal{K}}_{loc}(M,g)$. Based on previously established results, the latter local fact can be applied to the question of essential self-adjointness of Schr"odinger operators with singular magnetic and electric potentials. Finally, we also provide a Kato criterion in terms of minimal Riemannian submersions.