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By introducing the concept of  for a given Riemannian minimal heat kernel, we prove that on every Riemannian manifold (,) the Kato class  has a subspace of the form (,d), where  has a continuous density with respect to the volume measure   (where  depends on ). Using a local parabolic -mean value inequality, we prove the existence of such densities for every Riemannian manifold, which in particular implies . Based on previously established results, the latter local fact can be applied to the question of essential self-adjointness of Schrödinger operators with singular magnetic and electric potentials. Finally, we also provide a Kato criterion in terms of minimal Riemannian submersions.

Heat Kernels in the Context of Kato Potentials on Arbitrary Manifolds