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On the integration of Banach modules and its applications to vector calculus on ${\sf RCD}$ spaces
Functional Analysis · Submitted 23 Aug 2023
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim The aim of this paper is to connect the vector calculus on the lower dimensional leaves with the one on the base space.
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A finite-dimensional ${\sf RCD}$ space can be foliated into sufficiently regular leaves, where a differential calculus can be performed. Two important examples are given by the measure-theoretic boundary of the superlevel set of a function of bounded variation and the needle decomposition associated to a Lipschitz function. The aim of this paper is to connect the vector calculus on the lower dimensional leaves with the one on the base space. In order to achieve this goal, we develop a general theory of integration of $L^0$-Banach $L^0$-modules of independent interest. Roughly speaking, we study how to `patch together' vector fields defined on the leaves that are measurable with respect to the foliation parameter.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | A finite-dimensional ${\sf RCD}$ space can be foliated into sufficiently regular leaves, where a differential calculus can be performed. | background 0.79 | 0.26 | 0.16 | asserted 1.00 |
| s1 | Two important examples are given by the measure-theoretic boundary of the superlevel set of a function of bounded variation and the needle decomposition associated to a Lipschitz function. | background 0.48 | 0.56 | 0.04 | asserted 1.00 |
| s2 | The aim of this paper is to connect the vector calculus on the lower dimensional leaves with the one on the base space. | contribution 0.83 | 0.23 | 0.08 | asserted 0.90 |
| s3 | In order to achieve this goal, we develop a general theory of integration of $L^0$-Banach $L^0$-modules of independent interest. | contribution 0.71 | 0.90 | 0.05 | asserted 0.99 |
| s4 | Roughly speaking, we study how to `patch together' vector fields defined on the leaves that are measurable with respect to the foliation parameter. | method 0.67 | 0.86 | 0.06 | asserted 0.87 |
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Original abstract
A finite-dimensional ${\sf RCD}$ space can be foliated into sufficiently regular leaves, where a differential calculus can be performed. Two important examples are given by the measure-theoretic boundary of the superlevel set of a function of bounded variation and the needle decomposition associated to a Lipschitz function. The aim of this paper is to connect the vector calculus on the lower dimensional leaves with the one on the base space. In order to achieve this goal, we develop a general theory of integration of $L^0$-Banach $L^0$-modules of independent interest. Roughly speaking, we study how to `patch together' vector fields defined on the leaves that are measurable with respect to the foliation parameter.
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Builds on 1 paper
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In this section, we introduce a notion of Banach module bundle consistent with a disjoint measure-valued map, which generalises the notion of strong Banach bundle studied in [ref] .
Duals and pullbacks of normed modules · Integration of Banach modules · 0.96
Uses the method or tool of 1 paper
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Defining \(𝒱ℳ\) and the map \(ψ𝒱→L^0(μ)^+\) as \(ψ(v)|v|\), we are in a position to apply [ref] to \((𝒱,ψ)\).
An axiomatic theory of normed modules via Riesz spaces · The theory of Banach modules · 0.96
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Similar results, but without integrals of \(L^0\)-Banach \(L^0\)-modules, previously appeared in [ref] (for the squared distance functions from a point) and in [ref] (for arbitrary BV functions). \(b)\) Klartag's localisation technique [ref] has been generalised from the Riemannian setting to metric measure spaces verifying synthetic lower Ricci bounds, see the survey [ref] and the references therein.
Calculus and fine properties of functions of bounded variation on RCD spaces · Introduction · 0.91
Cites as support 1 paper
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Moreover, it was proved in [ref] after [ref] that [equation] thus in particular \(P(E,·)≪Cap\) for every set of finite perimeter \(E⊆\).
Calculus and fine properties of functions of bounded variation on RCD spaces · Functions of bounded variation on PI spaces · 0.96
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