Locally convex aspects of the Kato and the Dynkin class on manifolds

TL;DR

This paper investigates the local properties of the Kato and Dynkin classes on Riemannian manifolds, establishing conditions under which the Ricci curvature's negative part belongs to these classes and exploring their independence from the metric.

Contribution

It proves the invariance of local Kato and Dynkin classes under metric changes and demonstrates the density of smooth functions, advancing the understanding of Schrödinger semigroups on manifolds.

Findings

Ricci curvature's negative part in Kato class under Gaussian heat kernel bounds

Local Kato and Dynkin classes are metric-independent Fréchet spaces

Density of smooth compactly supported functions in the local Kato class

Abstract

We consider the Kato and the Dynkin class and their local counterparts on a smooth Riemannian manifold as Fr\'{e}chet spaces. Based on recent results by Carron, Mondello and Tewodrose we show that for a Riemannian manifold $(X,g)$ of dimension $m\geq 2$ with spectral negative part $\sigma^-_g$ of the Ricci curvature in $L^q_{\phi_g}(X,g)+L^\infty(X,g)$ for some $q>m/2$, the function $\sigma^-_g$ is in the Kato class of $(X,g)$ if and only if $(X,g)$ satisfies a Gaussian upper heat kernel bound for small times and is locally volume doubling. Here $L^q_{\phi_g}(X,g)$ is the $L^q$-space which is weighted with the inverse volume function. By establishing a localization result for the Dynkin norm, we prove that the local Kato class and the local Dynkin class do not depend on the chosen Riemannian metric and thus can be defined as Fr\'{e}chet spaces on arbitrary smooth manifolds. Moreover, we prove that smooth compactly supported functions are dense in the local Kato class and we use this result to prove that Schr\"odinger semigroups with Kato decomposable potentials are space-time continuous.