文摘
By introducing the concept of 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 𝖫q(M,dϱ), where ϱ has a continuous density with respect to the volume measure μg (where q depends on \(\dim (M)\)). Using a local parabolic 𝖫1-mean value inequality, we prove the existence of such densities for every Riemannian manifold, which in particular implies \(\text {\textsf {L}}^{q}_{\text {loc}}(M)\subset \mathcal {K}_{\text {loc}}(M,g)\). 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.
