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Locally convex aspects of the Kato and the Dynkin class on manifolds
Differential Geometry · Submitted 22 Jul 2023
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We consider the Kato and the Dynkin class and their local counterparts on a smooth Riemannian manifold as Fr\'{e}chet spaces. | contribution 0.54 | 0.70 | 0.05 | asserted 0.91 |
| s1 | 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. | result 0.98 | 0.87 | 0.60 | asserted 1.00 |
| s2 | Here $L^q_{\phi_g}(X,g)$ is the $L^q$-space which is weighted with the inverse volume function. | background 0.76 | 0.18 | 0.03 | asserted 0.98 |
| s3 | 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. | result 0.86 | 0.94 | 0.29 | asserted 1.00 |
| s4 | 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. | result 0.88 | 0.96 | 0.17 | asserted 1.00 |
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Original 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.
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2 arXiv papers cite this in 2 sentences
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Since then, several results which were well-known in the setting of non-negative Ricci curvature have been generalized to manifolds whose Ricci curvature is in the Kato or in the Dynkin class; see for instance; [ref] .
Asymptotically non-negative Ricci curvature, elliptic Kato constant and isoperimetric inequalities · Isoperimetric inequalities beyond non-negative Ricci curvature · 0.84
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