arXiv subjects

Enrico Pasqualetto

Publications and source records attributed to Enrico Pasqualetto.

At least 19 recordsLinked to original sources

arXiv2026

Convergence of metric measure spaces via embeddings in the Urysohn universal space

We study different notions of convergence of metric measure spaces by means of isometric embeddings into the Urysohn universal metric space $\mathbb U$. Due to the universality of $\mathbb U$, the collection $\mathbb X_1$ of isomorphism classes of normalised metric measure spaces can be canonically identified with the quotient (set) $\mathscr P_\sim(\mathbb U)=\mathscr P(\mathbb U)/\sim$ of the space $\mathscr P(\mathbb U)$ of Borel probability measures on $\mathbb U$, where $μ\simν$ if $ν$ is the pushforward of $μ$ under an isometry between their respective supports. By making crucial use of the ultrahomogeneity of $\mathbb U$, we show that, under the above identification, Gromov's box topology on $\mathbb X_1$ coincides with the quotient topology induced by the weak topology of $\mathscr P(\mathbb U)$. More quantitatively, the truncated $1$-Wasserstein distance on $\mathscr P(\mathbb U)$ induces a complete and separable distance ${\sf d}_{\rm mG}$ on $\mathbb X_1\cong\mathscr P_\sim(\mathbb U)$, which metrises the quotient topology of $\mathscr P_\sim(\mathbb U)$ and is Hölder equivalent to the box distance $\square$.

math.MG↗

arXiv2026

On the equivalence of BV notions in metric measure spaces

The aim of the paper is to compare in detail several notions of $BV$ space of functions of bounded variation in metric measure spaces $({\rm X},\mathsf{d},\mathfrak{m})$. Informally, they can be grouped in two classes, either by a relaxation procedure starting from a class of nicer functions (and with different notions of pseudo-gradient in the relaxation procedure) or by requiring good behaviour along a rich class of absolutely continuous curves. In the second approach, richness can be understood according to the notion of approximation modulus of [O. Martio, Adv. Calc. Var., 9 (2016)] or according to the notion of test plan introduced in [L. Ambrosio, N. Gigli, and G. Savaré, Invent. Math., 195 (2014)]. Extending [L. Ambrosio and S. Di Marino, J. Funct. Anal., 266 (2014)], we prove that all these approaches are isometrically equivalent in any locally complete metric measure space.

math.FA↗

arXiv2026

A Bochner-type integration theory for random normed modules

We develop a measure and integration theory for random normed modules. Given a probability space $({\rm X},Σ,\mathfrak m)$, we introduce and study measures taking values into the space $L^0(\mathfrak m)$ of $\mathfrak m$-measurable functions quotiented up to $\mathfrak m$-a.e. equality. Moreover, we develop a Bochner-type integration theory with respect to an $L^0(\mathfrak m)$-valued measure $μ$, for maps whose target ${\rm M}$ is a complete random normed module with base $({\rm X},Σ,\mathfrak m)$, or equivalently an $L^0(\mathfrak m)$-Banach $L^0(\mathfrak m)$-module. Inter alia, we prove versions of the Radon-Nikodým theorem and of the Riesz-Markov-Kakutani representation theorem for $L^0(\mathfrak m)$-valued measures. We also outline several applications of our integration theory: we introduce a notion of martingale with values in a complete random normed module, we propose a definition of random Radon-Nikodým property and we discuss random sets of finite perimeter.

math.FA↗

arXiv2026

Functions of bounded variation and Lipschitz algebras in metric measure spaces

Given a unital algebra $\mathscr A$ of locally Lipschitz functions defined over a metric measure space $({\mathrm X},{\mathsf d},\mathfrak m)$, we study two associated notions of function of bounded variation and their relations: the space ${\mathrm BV}_{\mathrm H}({\mathrm X};\mathscr A)$, obtained by approximating in energy with elements of $\mathscr A$, and the space ${\mathrm BV}_{\mathrm W}({\mathrm X};\mathscr A)$, defined through an integration-by-parts formula that involves derivations acting in duality with $\mathscr A$. Our main result provides a sufficient condition on the algebra $\mathscr A$ under which ${\mathrm BV}_{\mathrm H}({\mathrm X};\mathscr A)$ coincides with the standard metric BV space ${\mathrm BV}_{\mathrm H}({\mathrm X})$, which corresponds to taking as $\mathscr A$ the collection of all locally Lipschitz functions. Our result applies to several cases of interest, for example to Euclidean spaces and Riemannian manifolds equipped with the algebra of smooth functions, or to Banach and Wasserstein spaces equipped with the algebra of cylinder functions. Analogous results for metric Sobolev spaces ${\mathrm H}^{1,p}$ of exponent $p\in(1,\infty)$ were previously obtained by several different authors.

math.FA↗

arXiv2026

A categorical perspective on extended metric-topological spaces

Motivated by the analysis and geometry of metric-measure structures in infinite dimensions, we study the category of extended metric-topological spaces, along with many of its distinguished subcategories (such as the one of compact spaces). One of the main achievements is the proof of the bicompleteness (i.e. of the existence of all small limits and colimits) of the aforementioned categories.

math.CT↗

arXiv2025

A topological characterization of indecomposable sets of finite perimeter

We prove that a set of finite perimeter is indecomposable if and only if it is, up to a choice of suitable representative, connected in the 1-fine topology. This gives a topological characterization of indecomposability which is new even in Euclidean spaces. Our approach relies crucially on the metric space theory of functions of bounded variation, and we are able to prove our main result in a complete, doubling metric measure space supporting a $1$-Poincaré inequality and having the two-sidedness property (this class includes all Riemannian manifolds, Carnot groups, and ${\sf RCD}(K,N)$ spaces with $K\in\mathbb R$ and $N<\infty$). As an immediate corollary, we obtain an alternative proof of the decomposition theorem for sets of finite perimeter into maximal indecomposable components.

math.MG↗

arXiv2025

Preduals of metric BV spaces

We study the predual of the space of functions of bounded variation defined over a metric measure space $({\rm X},{\sf d},\mathfrak m)$ with $\mathfrak m$ finite. More specifically, for any exponent $p\in(1,\infty)$ we construct an isometric predual of the space ${\rm BV}_p({\rm X})$ of $p$-integrable functions of bounded variation, which we equip with the norm $\|f\|_{{\rm BV}_p({\rm X})}:=\|f\|_{L^p({\rm X})}+|Df|({\rm X})$. Moreover, we prove that the standard BV space ${\rm BV}({\rm X}):={\rm BV}_1({\rm X})$, which fails to have a predual for some choices of the metric measure space, does have a predual in the case where $({\rm X},{\sf d},\mathfrak m)$ is a PI space (i.e. a doubling metric measure space supporting a weak $(1,1)$-Poincaré inequality) of finite diameter. Along the way, we also develop a basic theory of BV functions in the setting of extended metric-topological measure spaces, which is of independent interest.

math.FA↗

arXiv2025

Metric Sobolev spaces II: dual energies and divergence measures

This is the second of two works concerning the Sobolev calculus on metric measure spaces and its applications. In this work, we focus on several approaches to vector calculus in the non-smooth setting of complete and separable metric spaces equipped with a boundedly-finite Borel measure. More precisely, we study different notions of (co)vector fields and derivations appearing in the literature, as well as their mutual relation. We also carry forward a thorough investigation of gradients, divergence measures, and Laplacian measures, together with their applications in potential analysis (for example, regarding the condenser capacity) and in the study of duality properties of Sobolev spaces. Most of the results are obtained for the full range of exponents $p\in[1,\infty)$ and without finiteness assumption on the measure.

math.FA↗

arXiv2025

Spaces with Riemannian curvature bounds are universally infinitesimally Hilbertian

We show that a metric space $X$ that, at every point, has a Gromov-Hausdorff tangent with the splitting property (i.e. every geodesic line splits off a factor $\mathbb{R}$), is universally infinitesimally Hilbertian (i.e. $W^{1,2}(X,μ)$ is a Hilbert space for every measure $μ$). This connects the infinitesimal geometry of $X$ to its analytic properties and is, to our knowledge, the first general criterion guaranteeing universal infinitesimal Hilbertianity. Using it we establish universal infinitesimal Hilbertianity of finite dimensional RCD-spaces. We moreover show that (possibly infinite dimensional) Alexandrov spaces are universally infinitesimally Hilbertian and construct an isometric embedding of tangent modules.

math.MG↗

arXiv2025

Tensor products of measurable Banach bundles

We study injective and projective tensor products of measurable Banach bundles. More precisely, given two separable measurable Banach bundles ${\bf E}$, ${\bf F}$ defined over a probability space $({\rm X},Σ,\mathfrak m)$, we construct two measurable Banach bundles ${\bf E}\hat\otimes_\varepsilon{\bf F}$ and ${\bf E}\hat\otimes_π{\bf F}$ over $({\rm X},Σ,\mathfrak m)$ such that $Γ({\bf E}\hat\otimes_\varepsilon{\bf F})\congΓ({\bf E})\hat\otimes_\varepsilonΓ({\bf F})$ and $Γ({\bf E}\hat\otimes_π{\bf F})\congΓ({\bf E})\hat\otimes_πΓ({\bf F})$, where ${\bf G}\mapstoΓ({\bf G})$ is the map assigning to a measurable Banach bundle ${\bf G}$ its space of $L^\infty(\mathfrak m)$-sections, while $Γ({\bf E})\hat\otimes_\varepsilonΓ({\bf F})$ and $Γ({\bf E})\hat\otimes_πΓ({\bf F})$ denote the injective and projective tensor products, respectively, of $Γ({\bf E})$ and $Γ({\bf F})$ in the sense of $L^\infty(\mathfrak m)$-Banach $L^\infty(\mathfrak m)$-modules. In combination with previous results, this provides a fiberwise representation of the injective tensor product $\mathscr M\hat\otimes_\varepsilon\mathscr N$ and the projective tensor product $\mathscr M\hat\otimes_π\mathscr N$ of two countably-generated $L^\infty(\mathfrak m)$-Banach $L^\infty(\mathfrak m)$-modules $\mathscr M$, $\mathscr N$.

math.FA↗

arXiv2025

A variational approach to the volume-preserving anisotropic mean curvature flow in 2D

In this article, we introduce a variational algorithm, in the spirit of the minimizing movements scheme, to model the volume-preserving anisotropic mean curvature flow in 2D. We show that this algorithm can be used to prove the existence of classical solutions. Moreover, we prove that this algorithm converges to the global solution of the equation.

math.AP↗

arXiv2025

Maps of bounded variation from PI spaces to metric spaces

We study maps of bounded variation defined on a metric measure space and valued into a metric space. Assuming the source space to satisfy a doubling and Poincaré property, we produce a well-behaved relaxation theory via approximation by simple maps. Moreover, several equivalent characterizations are given, including a notion in weak duality with test plans.

math.FA↗

arXiv2025

A note on Laplacian bounds, deformation properties and isoperimetric sets in metric measure spaces

In the setting of length PI spaces satisfying a suitable deformation property, it is known that each isoperimetric set has an open representative. In this paper, we construct an example of a length PI space (without the deformation property) where an isoperimetric set does not have any representative whose topological interior is non-empty. Moreover, we provide a sufficient condition for the validity of the deformation property, consisting in an upper Laplacian bound for the squared distance functions from a point. Our result applies to essentially non-branching ${\sf MCP}(K,N)$ spaces, thus in particular to essentially non-branching ${\sf CD}(K,N)$ spaces and to many Carnot groups and sub-Riemannian manifolds. As a consequence, every isoperimetric set in an essentially non-branching ${\sf MCP}(K,N)$ space has an open representative, which is also bounded whenever a uniform lower bound on the volumes of unit balls is assumed.

math.MG↗

arXiv2025

Derivations and Sobolev functions on extended metric-measure spaces

We investigate the first-order differential calculus over extended metric-topological measure spaces. The latter are quartets $\mathbb X=(X,τ,{\sf d},\mathfrak m)$, given by an extended metric space $(X,{\sf d})$ together with a weaker topology $τ$ (satisfying suitable compatibility conditions) and a finite Radon measure $\mathfrak m$ on $(X,τ)$. The class of extended metric-topological measure spaces encompasses all metric measure spaces and many infinite-dimensional metric-measure structures, such as abstract Wiener spaces. In this framework, we study the following classes of objects: - The Banach algebra ${\rm Lip}_b(X,τ,{\sf d})$ of bounded $τ$-continuous ${\sf d}$-Lipschitz functions on $X$. - Several notions of Lipschitz derivations on $X$, defined in duality with ${\rm Lip}_b(X,τ,{\sf d})$. - The metric Sobolev space $W^{1,p}(\mathbb X)$, defined in duality with Lipschitz derivations on $X$. Inter alia, we generalise both Weaver's and Di Marino's theories of Lipschitz derivations to the extended setting, and we discuss their connections. We also introduce a Sobolev space $W^{1,p}(\mathbb X)$ via an integration-by-parts formula, along the lines of Di Marino's notion of Sobolev space, and we prove its equivalence with other approaches, studied in the extended setting by Ambrosio, Erbar and Savaré. En route, we obtain some results of independent interest, among which are: - A Lipschitz-constant-preserving extension result for $τ$-continuous ${\sf d}$-Lipschitz functions. - A novel and rather robust strategy for proving the equivalence of Sobolev-type spaces defined via an integration-by-parts formula and those obtained with a relaxation procedure. - A new description of an isometric predual of the metric Sobolev space $W^{1,p}(\mathbb X)$.

math.FA↗

arXiv2024

Vector calculus on weighted reflexive Banach spaces

We study first-order Sobolev spaces on reflexive Banach spaces via relaxation, test plans, and divergence. We show the equivalence of the different approaches to the Sobolev spaces and to the related tangent bundles.

math.FA↗

arXiv2024

Smooth approximations preserving asymptotic Lipschitz bounds

The goal of this note is to prove that every real-valued Lipschitz function on a Banach space can be pointwise approximated on a given $σ$-compact set by smooth cylindrical functions whose asymptotic Lipschitz constants are controlled. This result has applications in the study of metric Sobolev and BV spaces: it implies that smooth cylindrical functions are dense in energy in these kinds of functional spaces defined over any weighted Banach space.

math.FA↗

arXiv2024

Metric Sobolev spaces I: equivalence of definitions

This is the first of two works concerning the Sobolev calculus on metric measure spaces and its applications. In this work, we focus on several notions of metric Sobolev space and on their equivalence. More precisely, we give a systematic presentation of first-order $p$-Sobolev spaces, with $p\in[1,\infty)$, defined over a complete and separable metric space equipped with a boundedly-finite Borel measure. We focus on three different approaches: via approximation with Lipschitz functions; by studying the behaviour along curves, in terms either of the curve modulus or of test plans; via integration-by-parts, using Lipschitz derivations with divergence. Eventually, we show that all these approaches are fully equivalent. We emphasise that no doubling or Poincaré assumption is made, and that we allow also for the exponent $p=1$. A substantial part of this work consists of a self-contained and partially-revisited exposition of known results, which are scattered across the existing literature, but it contains also several new results, mostly concerning the equivalence of metric Sobolev spaces for $p=1$.

math.FA↗

arXiv2024

Yet another proof of the density in energy of Lipschitz functions

We provide a new, short proof of the density in energy of Lipschitz functions into the metric Sobolev space defined by using plans with barycenter (and thus, a fortiori, into the Newtonian-Sobolev space). Our result covers first-order Sobolev spaces of exponent $p\in(1,\infty)$, defined over a complete and separable metric space endowed with a boundedly-finite Borel measure. Our proof is based on a completely smooth analysis: first we reduce the problem to the Banach space setting, where we consider smooth functions instead of Lipschitz ones, then we rely on classical tools in convex analysis and on the superposition principle for normal $1$-currents. Along the way, we obtain a new proof of the density in energy of smooth cylindrical functions in Sobolev spaces defined over a separable Banach space endowed with a finite Borel measure.

math.FA↗