Heat kernels in the context of Kato potentials on arbitrary manifolds

TL;DR

This paper introduces Kato control pairs and demonstrates that on any Riemannian manifold, the Kato class contains functions with densities relative to the volume measure, impacting the analysis of Schrödinger operators with singular potentials.

Contribution

It develops the concept of Kato control pairs for heat kernels and establishes the existence of densities in the Kato class on all Riemannian manifolds, extending previous results.

Findings

Kato class functions have densities in L^q(M) for all manifolds.

Local L^1 mean value inequality ensures densities exist.

Results apply to self-adjointness of Schrödinger operators with singular potentials.

Abstract

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 $\mathsf{L}^q_{loc}(M)\subset\mathcal{K}_{loc}(M,g)$. 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.