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Enrico Pasqualetto

Publications and source records attributed to Enrico Pasqualetto.

At least 37 records · Page 2Linked to original sources
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↗
arXiv2023

Asymptotic isoperimetry on non collapsed spaces with lower Ricci bounds

This paper studies sharp and rigid isoperimetric comparison theorems and asymptotic isoperimetric properties for small and large volumes on $N$-dimensional ${\rm RCD}(K,N)$ spaces $(X,\mathsf{d},\mathscr{H}^N)$. Moreover, we obtain almost regularity theorems formulated in terms of the isoperimetric profile and enhanced consequences at the level of several functional inequalities. Most of our statements are new even in the classical setting of smooth, non compact manifolds with lower Ricci curvature bounds. The synthetic theory plays a key role via compactness and stability arguments.

math.DG↗
arXiv2023

Sharp isoperimetric comparison on non-collapsed spaces with lower Ricci bounds

This paper studies sharp isoperimetric comparison theorems and sharp dimensional concavity properties of the isoperimetric profile for non smooth spaces with lower Ricci curvature bounds, the so-called $N$-dimensional ${\rm RCD}(K,N)$ spaces $(X,\mathsf{d},\mathscr{H}^N)$. The absence of most of the classical tools of Geometric Measure Theory and the possible non existence of isoperimetric regions on non compact spaces are handled via an original argument to estimate first and second variation of the area for isoperimetric sets, avoiding any regularity theory, in combination with an asymptotic mass decomposition result of perimeter-minimizing sequences. Most of our statements are new even for smooth, non compact manifolds with lower Ricci curvature bounds and for Alexandrov spaces with lower sectional curvature bounds. They generalize several results known for compact manifolds, non compact manifolds with uniformly bounded geometry at infinity, and Euclidean convex bodies.

math.DG↗
arXiv2023

Topological regularity of isoperimetric sets in PI spaces having a deformation property

We prove topological regularity results for isoperimetric sets in PI spaces having a suitable deformation property, which prescribes a control on the increment of the perimeter of sets under perturbations with balls. More precisely, we prove that isoperimetric sets are open, satisfy boundary density estimates and, under a uniform lower bound on the volumes of unit balls, are bounded. Our results apply, in particular, to the class of possibly collapsed $\mathrm{RCD}(K,N)$ spaces. As a consequence, the rigidity in the isoperimetric inequality on possibly collapsed $\mathrm{RCD}(0,N)$ spaces with Euclidean volume growth holds without the additional assumption on the boundedness of isoperimetric sets. Our strategy is of interest even in the Euclidean setting, as it simplifies some classical arguments.

math.MG↗
arXiv2023

On the integration of Banach modules and its applications to vector calculus on ${\sf RCD}$ spaces

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.

math.FA↗
arXiv2023

Projective and injective tensor products of Banach $L^0$-modules

We study projective and injective tensor products of Banach $L^0$-modules over a $σ$-finite measure space. En route, we extend to Banach $L^0$-modules several technical tools of independent interest, such as quotient operators, summable families, and Schauder bases.

math.FA↗
arXiv2023

An axiomatic theory of normed modules via Riesz spaces

We introduce and study an axiomatic theory of $V$-normed $U$-modules, where $V$ is a Riesz space and $U$ is an $f$-algebra; the spaces $U$ and $V$ also have some additional structure and are required to satisfy a compatibility condition. Roughly speaking, a $V$-normed $U$-module is a module over $U$ that is endowed with a pointwise norm operator taking values in $V$. The aim of our approach is to develop a unified framework, which is tailored to the differential calculus on metric measure spaces, where as $U$ and $V$ one can take many different spaces of functions.

math.FA↗
arXiv2023

Variational problems concerning length distances in metric spaces

Given a locally compact, complete metric space $({\rm X},{\sf D})$ and an open set $Ω\subseteq{\rm X}$, we study the class of length distances $\sf d$ on $Ω$ that are bounded from above and below by fixed multiples of the ambient distance $\sf D$. More precisely, we prove that the uniform convergence on compact sets of distances in this class is equivalent to the $Γ$-convergence of several associated variational problems. Along the way, we fix some oversights appearing in the previous literature.

math.MG↗
arXiv2022

On the reflexivity properties of Banach bundles and Banach modules

In this paper we investigate some reflexivity-type properties of separable measurable Banach bundles over a $σ$-finite measure space. Our two main results are the following: - The fibers of a bundle are uniformly convex (with a common modulus of convexity) if and only if the space of its $L^p$-sections is uniformly convex for every $p\in(1,\infty)$. - The fibers of a bundle are reflexive if and only if the space of its $L^p$-sections is reflexive. These results generalise the well-known corresponding ones for Lebesgue-Bochner spaces.

math.FA↗
arXiv2022

Subgraphs of $\rm BV$ functions on $\rm RCD$ spaces

In this work we extend classical results for subgraphs of functions of bounded variation in $\mathbb{R}^n\times\mathbb{R}$ to the setting of $\mathsf{X}\times\mathbb{R}$, where $\mathsf{X}$ is an ${\rm RCD}(K,N)$ metric measure space. In particular, we give the precise expression of the push-forward onto $\mathsf{X}$ of the perimeter measure of the subgraph in $\mathsf{X}\times\mathbb{R}$ of a $\rm BV$ function on $\mathsf{X}$. Moreover, in properly chosen good coordinates, we write the precise expression of the normal to the boundary of the subgraph of a $\rm BV$ function $f$ with respect to the polar vector of $f$, and we prove change-of-variable formulas.

math.MG↗
arXiv2022

The metric-valued Lebesgue differentiation theorem in measure spaces and its applications

We prove a version of the Lebesgue Differentiation Theorem for mappings that are defined on a measure space and take values into a metric space, with respect to the differentiation basis induced by a von Neumann lifting. As a consequence, we obtain a lifting theorem for the space of sections of a measurable Banach bundle and a disintegration theorem for vector measures whose target is a Banach space with the Radon-Nikodým property.

math.FA↗
arXiv2022

Duals and pullbacks of normed modules

We give a general description of the dual of the pullback of a normed module. Ours is the natural generalization to the context of modules of the well-known fact that the dual of the Lebesgue-Bochner space $L^p([0,1],B)$ consists - quite roughly said - of $L^q$ maps from $[0,1]$ to the dual $B'$ of $B$ equipped with the weak$^*$ topology. In order to state our result, we study various fiberwise descriptions of a normed module that are of independent interest.

math.FA↗
arXiv2022

A short proof of the existence of master test plans

The aim of this brief note is to provide a quick and elementary proof of the following known fact: on a metric measure space whose Sobolev space is separable, there exists a test plan that is sufficient to identify the minimal weak upper gradient of every Sobolev function.

math.FA↗
arXiv2022

Variational problems concerning sub-Finsler metrics in Carnot groups

This paper is devoted to the study of geodesic distances defined on a subdomain of a given Carnot group, which are bounded both from above and from below by fixed multiples of the Carnot-Carathéodory distance. We show that the uniform convergence (on compact sets) of these distances can be equivalently characterized in terms of $Γ$-convergence of several kinds of variational problems. Moreover, we investigate the relation between the class of intrinsic distances, their metric derivatives and the sub-Finsler convex metrics defined on the horizontal bundle.

math.AP↗
arXiv2021

Pullback of a quasiconformal map between arbitrary metric measure spaces

We prove that every (geometrically) quasiconformal homeomorphism between metric measure spaces induces an isomorphism between the cotangent modules constructed by Gigli. We obtain this by first showing that every continuous mapping $φ$ with bounded outer dilatation induces a pullback map $φ^*$ between the cotangent modules of Gigli, and then proving the functorial nature of the resulting pullback operator. Such pullback is consistent with the differential for metric-valued locally Sobolev maps introduced by Gigli-Pasqualetto-Soultanis. Using the consistency between Gigli's and Cheeger's cotangent modules for PI spaces, we prove that quasiconformal homeomorphisms between PI spaces preserve the dimension of Cheeger charts, thereby generalizing earlier work by Heinonen-Koskela-Shanmugalingam-Tyson. Finally, we show that if $φ$ is a given homeomorphism with bounded outer dilatation, then $φ^{-1}$ has bounded outer dilatation if and only if $φ^{*}$ is invertible and $φ^{-1}$ is Sobolev. In contrast to the setting of Euclidean spaces, Carnot groups, or more generally, Ahlfors regular PI spaces, the Sobolev regularity of $φ^{-1}$ needs to be assumed separately.

math.MG↗