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Danka Lučić

Publications and source records attributed to Danka Lučić.

At least 19 recordsLinked to original sources
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

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

Closed BV-extension and $W^{1,1}$-extension sets

This paper studies the relations between extendability of different classes of Sobolev $W^{1,1}$ and $BV$ functions from closed sets in general metric measure spaces. Under the assumption that the metric measure space satisfies a weak $(1,1)$-Poincaré inequality and measure doubling, we prove further properties for the extension sets. In the case of the Euclidean plane, we show that compact finitely connected $BV$-extension sets are always also $W^{1,1}$-extension sets. This is shown via a local quasiconvexity result for the complement of the extension set.

math.MG↗
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↗
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

Approximation by BV-extension sets via perimeter minimization in metric spaces

We show that every bounded domain in a metric measure space can be approximated in measure from inside by closed $BV$-extension sets. The extension sets are obtained by minimizing the sum of the perimeter and the measure of the difference between the domain and the set. By earlier results, in PI-spaces the minimizers have open representatives with locally quasiminimal surface. We give an example in a PI-space showing that the open representative of the minimizer need not be a $BV$-extension domain nor locally John.

math.MG↗
arXiv2023

Mass optimization problem with convex cost

In this paper we consider a mass optimization problem in the case of scalar state function, where instead of imposing a constraint on the total mass of the competitors, we penalize the classical compliance by a convex functional defined on the space of measures. We obtain a characterization of optimal solutions to the problem through a suitable PDE. This generalizes the case considered in the literature of a linear cost and applies to the optimization of a conductor where very low and very high conductivities have both a high cost, and then the study of nonlinear models becomes relevant.

math.OC↗
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↗
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↗
arXiv2021

Representation theorems for normed modules

In this paper we study the structure theory of normed modules, which have been introduced by Gigli. The aim is twofold: to extend von Neumann's theory of liftings to the framework of normed modules, thus providing a notion of precise representative of their elements; to prove that each separable normed module can be represented as the space of sections of a measurable Banach bundle. By combining our representation result with Gigli's differential structure, we eventually show that every metric measure space (whose Sobolev space is separable) is associated with a cotangent bundle in a canonical way.

math.FA↗
arXiv2021

Gamma-convergence of Cheeger energies with respect to increasing distances

We prove a $Γ$-convergence result for Cheeger energies along sequences of metric measure spaces, where the measure space is kept fixed, while distances are monotonically converging from below to the limit one. As a consequence, we show that the infinitesimal Hilbertianity condition is stable under this kind of convergence of metric measure spaces.

math.MG↗
arXiv2020

A short proof of the infinitesimal Hilbertianity of the weighted Euclidean space

We provide a quick proof of the following known result: the Sobolev space associated with the Euclidean space, endowed with the Euclidean distance and an arbitrary Radon measure, is Hilbert. Our new approach relies upon the properties of the Alberti-Marchese decomposability bundle. As a consequence of our arguments, we also prove that if the Sobolev norm is closable on compactly-supported smooth functions, then the reference measure is absolutely continuous with respect to the Lebesgue measure.

math.FA↗