Vector calculus for tamed Dirichlet spaces

Braun, Mathias

doi:10.48550/arxiv.2108.12374

Cited by 3 publications

(6 citation statements)

References 65 publications

(229 reference statements)

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“…Gauss-Green integration by parts formulae for sets of finite perimeter and vector fields with such low regularity in the Euclidean setting have been studied in [CTZ09,CP20]. Later on, in [BCM19] the theory has been partially extended to locally compact RCD(K, ∞) metric measure spaces (see also the recent [Br21]). Here, fully exploiting the finite dimensionality assumption N < ∞ and the regularity theory for sets of finite perimeter, we achieve a quite complete extension of the Euclidean results, sharpening those in [BCM19].…”

Section: Pointwise Behaviour Of the Unit Normal And Operations With S...mentioning

confidence: 99%

Constancy of the dimension in codimension one and locality of the unit normal on $\mathrm{RCD}(K,N)$ spaces

Bruè,

Pasqualetto,

Semola

2021

Preprint

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The aim of this paper is threefold. We first prove that, on RCD(K, N ) spaces, the boundary measure of any set with finite perimeter is concentrated on the n-regular set Rn, where n ≤ N is the essential dimension of the space. After, we discuss localization properties of the unit normal providing representation formulae for the perimeter measure of intersections and unions of sets with finite perimeter. Finally, we study Gauss-Green formulae for essentially bounded divergence measure vector fields, sharpening the analysis in [BCM19]. These tools are fundamental for the development of a regularity theory for local perimeter minimizers on RCD(K, N ) spaces in [MS21].

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Section: Pointwise Behaviour Of the Unit Normal And Operations With S...mentioning

confidence: 99%

Constancy of the dimension in codimension one and locality of the unit normal on $\mathrm{RCD}(K,N)$ spaces

Bruè,

Pasqualetto,

Semola

2021

Preprint

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“…A short proof of Corollary 4.14 is included for convenience. The integrated Bochner inequality from Corollary 4.15 -according to the notion of Hessian established in [5] -follows from [5,Cor. 8.3].…”

Section: Taming For Almost Smooth Lr Manifoldsmentioning

confidence: 86%

“…In view of the above mentioned diversity of examples as well as recent breakthroughs in metric geometry for nonuniform curvature bounds [5,8,18,26,56], we predict the class of LR manifolds -regarded as prototypes of Riemannian spaces with singularities -to have great potential for near future research, which this article aims to initiate.…”

Section: Introductionmentioning

confidence: 99%

“…The prominent role of the Kato class in mathematical physics has been well established in the past decades in a series of works [2,32,49]. Recently, its interest in the context of singular Ricci bounds has grown as well [5,6,7,9,18,24,26,43] (also with applications to Ricci flow and general relativity). The study of the class of LR manifolds in this latter setting constitutes the last part of our work.…”

Section: Introductionmentioning

confidence: 99%

“…v. The crucial step to prove Theorem 4.9 is contained in Lemma 4.10. A similar strategy as in its proof has been pursued for different regularity results within a general second order calculus for tamed spaces, see e.g [5, . Cor.…”

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confidence: 99%

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Heat kernel bounds and Ricci curvature for Lipschitz manifolds

Braun¹,

Rigoni²

2021

Preprint

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Given any d-dimensional Lipschitz Riemannian manifold (M, g) with heat kernel p, we establish uniform upper bounds on p which can always be decoupled in space and time. More precisely, we prove the existence of a constant C > 0 and a bounded Lipschitz function R : M → (0, ∞) such that for every x ∈ M and every t > 0,This allows us to identify suitable weighted Lebesgue spaces w.r.t. the given volume measure as subsets of the Kato class induced by (M, g). In the case ∂M = ∅, we also provide an analogous inclusion for Lebesgue spaces w.r.t. the surface measure on ∂M .We use these insights to give sufficient conditions for a possibly noncomplete Lipschitz Riemannian manifold to be tamed, i.e. to admit a measure-valued lower bound on the Ricci curvature, formulated in a synthetic sense.

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Constancy of the dimension in codimension one and locality of the unit normal on $\RCD(K,N)$ spaces

Bruè¹,

Pasqualetto²,

Semola³

2023

Annali Scuola Normale Superiore - Classe Di Scienze

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The aim of this paper is threefold. We first prove that, on RCD(K, N ) spaces, the boundary measure of any set with finite perimeter is concentrated on the n-regular set Rn, where n ≤ N is the essential dimension of the space. After, we discuss localization properties of the unit normal providing representation formulae for the perimeter measure of intersections and unions of sets with finite perimeter. Finally, we study Gauss-Green formulae for essentially bounded divergence measure vector fields, sharpening the analysis in [21]. These tools are fundamental for the development of a regularity theory for local perimeter minimizers on RCD(K, N ) spaces in [50].

show abstract

Vector calculus for tamed Dirichlet spaces

Cited by 3 publications

References 65 publications

Constancy of the dimension in codimension one and locality of the unit normal on $\mathrm{RCD}(K,N)$ spaces

Constancy of the dimension in codimension one and locality of the unit normal on $\mathrm{RCD}(K,N)$ spaces

Heat kernel bounds and Ricci curvature for Lipschitz manifolds

Constancy of the dimension in codimension one and locality of the unit normal on $\RCD(K,N)$ spaces

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