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arXiv · 2108.12374

Vector calculus for tamed Dirichlet spaces

Mathias Braun
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Abstract

In the language of $L^\infty$-modules proposed by Gigli, we introduce a first order calculus on a topological Lusin measure space $(M,\mathfrak{m})$ carrying a quasi-regular, strongly local Dirichlet form $\mathscr{E}$. Furthermore, we develop a second order calculus if $(M,\mathscr{E},\mathfrak{m})$ is tamed by a signed measure in the extended Kato class in the sense of Erbar, Rigoni, Sturm and Tamanini. This allows us to define e.g. Hessians, covariant and exterior derivatives, Ricci curvature, and second fundamental form.

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Mathias Braun. 2021-08-27. Vector calculus for tamed Dirichlet spaces. https://arxiv.org/abs/2108.12374

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