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Potential Analysis

, Volume 46, Issue 1, pp 119–134 | Cite as

Heat Kernels in the Context of Kato Potentials on Arbitrary Manifolds

  • Batu GüneysuEmail author
Article
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Abstract

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 𝖫 q (M,dϱ), where ϱ has a continuous density with respect to the volume measure μ g (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.

Keywords

Heat kernel estimates Kato potentials Parabolic mean value inequality 

Mathematics Subject Classification (2010)

31C12 58J35 58J65 

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© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Humboldt-Universität zu BerlinBerlinGermany

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