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Ramón J. Aliaga

Publications and source records attributed to Ramón J. Aliaga.

At least 19 recordsLinked to original sources
arXiv2026

The Ball-Covering Property in Lipschitz-Free Spaces

We study ball-covering properties in Lipschitz-free spaces. We establish an extension criterion for proving that $\mathcal F(M)$ fails the ball-covering property and apply it to several classes of nonseparable metric spaces. In contrast, we construct a nonseparable uniformly discrete metric space $M$ such that $\mathcal F(M)$ has the uniform ball-covering property and is isomorphic to $\ell_1(2^ω)$. More precisely, $\mathcal F(M)$ has the $α$-ball-covering property for every $α\in[-1,1)$. This example also shows that these quantitative ball-covering properties are not hereditary within the class of Lipschitz-free spaces. Finally, we prove some stability results under sufficiently small bi-Lipschitz perturbations of the metric.

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arXiv2026

Lipschitz-free spaces are strongly unique preduals

We prove that the Lipschitz-free space $\mathcal{F}(M)$ is the strongly unique isometric predual of $\mathrm{Lip}_0(M)$ for every metric space $M$, solving a longstanding open problem of Weaver. The result is proved first for length metric spaces by analyzing the behavior of Lipschitz functions on paths in $M$ and the topological properties of subspaces of functions vanishing on certain barrier-sets of $M$. Then, the result is extended to general metric spaces by embedding them into length spaces in such a way that the restriction operator for Lipschitz functions is weak$^*$ continuous with respect to their selected preduals. Finally, we also give a counterexample to the codimension-one inheritance assertion used in Weaver's original proof for bounded and geodesic metric spaces.

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arXiv2026

On Weak Compactness and Uniform Regularity in Lipschitz free Spaces

We analyze the properties of weakly compact sets in Lipschitz free spaces. Prior research has established that, for a complete metric space $M$, weakly precompact sets in the Lipschitz free space $\mathcal F(M)$ are tight. In this paper, we prove that these sets actually exhibit a stronger property, which we call uniform regularity. However, this condition alone is not sufficient to characterize weakly compact sets, except in the case of scattered metric spaces. On the other hand, if $T$ is an $\mathbb R$-tree, we leverage Godard's isometry between $\mathcal F(T)$ and $L^1(λ_T)$ to obtain an intrinsic characterization of weakly compact sets in $\mathcal F(T)$. This approach allows us to identify conditions that may describe weak compactness across a wider range of spaces. In particular, we provide a characterization of norm-compactness in terms of sums of "large molecules'', while we show that sums of "small molecules'' contain an $\ell_1$-basis.

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arXiv2026

Lipschitz-free spaces and purely 1-unrectifiable metric spaces

The Lipschitz-free space $\mathcal{F}(M)$ is a canonical linearization of a complete metric space $M$ whose topological dual is the space of Lipschitz functions on $M$. We review the properties of $\mathcal{F}(M)$ when the underlying space $M$ is purely 1-unrectifiable, that is, it contains no bi-Lipschitz copy of a subset of $\mathbb{R}$ with positive measure. For compact $M$, this is equivalent to several Banach space properties of $\mathcal{F}(M)$, including the Radon-Nikodým and Schur properties or admitting a predual. We shall see how the study of locally flat Lipschitz functions on $M$ reveals these equivalences, and describe a technique that allows most of them to be transferred to the non-compact setting. This manuscript is an expository text based on results by the author in collaboration with C. Gartland, C. Petitjean and A. Procházka, originally published in a Trans. Amer. Math. Soc. paper, and corresponds to a lecture delivered at the Second Winter School in Geometric Measure Theory at Westlake University, Hangzhou, on February 2026.

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arXiv2026

The unique predual problem for Lipschitz spaces, revisited

In his 2018 paper ``On the unique predual problem for Lipschitz spaces'', N. Weaver published proofs that Banach spaces $\mathrm{Lip}_0(M)$ of Lipschitz functions on a complete metric space $M$ have strongly unique preduals whenever $M$ has finite diameter or is geodesic. A gap was recently noticed in the proof of a crucial lemma that claimed that the property of having a strongly unique predual passes to 1-codimensional weak$^*$-closed subspaces. In this note, we confirm that the lemma is actually false by providing an explicit counterexample. We also expand on some of Weaver's original arguments to provide a new, valid proof of the following particular case: $\mathrm{Lip}_0(M)$ has a strongly unique predual whenever $M$ is a convex subset of a finite-dimensional normed space.

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arXiv2026

Lipschitz-free spaces over non-porous subsets of $\mathbb{R}^n$

We prove that the Lipschitz-free space $\mathcal{F}(M)$ contains a complemented copy of $\mathcal{F}(\mathbb{Z}^n)$ whenever $M\subset\mathbb{R}^n$ is not porous. Consequently, if $M\subset\mathbb{R}^n$ is uniformly discrete and not porous then $\mathcal{F}(M)$ is isomorphic to $\mathcal{F}(\mathbb{Z}^n)$.

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arXiv2026

Almost preserved extreme points

In this paper we introduce the notion of an almost preserved extreme point (APEP) of a set as a weakening of the concept of preserved extreme points, and we systematically study such points. As a main result, we prove that a Banach space $X$ has the Radon-Nikodým property (RNP) if and only if every closed, convex, and bounded subset of the space has an APEP. Similarly, we prove that $X$ has the RNP if and only if the unit ball of every equivalent renorming has an APEP. We further investigate APEPs of the unit ball of classical Banach spaces, absolute sums, Lipschitz-free spaces, and projective tensor products. In the latter setting, our work also describes the preserved extreme points in the unit ball under the assumption that every bounded operator is compact, thereby partially solving an open problem.

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arXiv2026

Lipschitz extension and Lipschitz-free spaces over nets in normed spaces

We consider subsets $S$ of a metric space $M$ such that Lipschitz mappings defined on $S$ can be extended to Lipschitz mappings on $M$, and we show that the union of such subsets has the same property under appropriate geometric conditions. We then derive several consequences to the isomorphic structure and classification of Lipschitz and Lipschitz-free spaces. Our main result is that the Lipschitz-free space $\mathcal{F}(M)$ is isomorphic to its countable $\ell_1$-sum when $M$ is either a net $N_X$ in any Banach space $X$ or the integer grid $\mathbb{Z}_{\ell_1}$ in $\ell_1$. We also prove that the Lipschitz space $\mathrm{Lip}_0(\mathbb{Z}_{\ell_1})$ is isomorphic to $\mathrm{Lip}_0(\ell_1)$ and that $\mathrm{Lip}_0(N_X)$ contains a complemented copy of $\mathrm{Lip}_0(X)$, among other results. This answers questions raised by Albiac, Ansorena, Cúth and Doucha and Candido, Cúth and Doucha, respectively, and extends previous results by the same authors as well as Hájek and Novotný.

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arXiv2026

A solution to the extreme point problem and other applications of Choquet theory to Lipschitz-free spaces

We prove that every element of a Lipschitz-free space admits an expression as a convex series of elements with compact support. As a consequence, we conclude that all extreme points of the unit ball of Lipschitz-free spaces are elementary molecules, solving a long-standing problem. We also deduce that all elements of a Lipschitz-free space with the Radon-Nikodým property can be expressed as convex integrals of molecules. Our results are based on a recent theory of integral representation for functionals on Lipschitz spaces which draws on classical Choquet theory, due to the third named author.

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arXiv2026

Integral representations of projective norm-attaining tensors

We introduce a Bochner integral approach to projective norm attainment in tensor products of Banach spaces by defining the class of integral projective norm-attaining tensors. This framework provides a broader, measure-theoretic approach to the study of projective norm attainment in tensor products of Banach spaces. We show that every integral norm-attaining tensor can be approximated in norm by norm-attaining tensors with finite representations. As a consequence, the Bishop-Phelps type density problem for classical norm-attaining tensors is equivalent to the corresponding density problem for integral norm-attaining tensors. Moreover, we prove that if an integral projective norm-attaining tensor represented by a Radon measure is an extreme point, then it must be an elementary tensor. We further investigate weaker topological versions of integral norm-attainment, including weak and weak$^*$ integral representations, providing sufficient conditions for the existence of Bochner representations. Finally, we extend known constructions of projective tensor products containing non-norm-attaining tensors to the integral setting. We show, for instance, that $L_1\widehat{\otimes}_πL_p$ and the real $c_0\widehat{\otimes}_πL_p$ contain non-norm-attaining tensors for $1<p<\infty$.

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arXiv2026

Lipschitz spaces over non-porous sets

Let $M$ be a subset of $\mathbb{R}^n$. If $M$ is not porous, in particular if it has positive $n$-dimensional Lebesgue measure, we prove that the Lipschitz spaces $\mathrm{Lip}_0(M)$ and $\mathrm{Lip}_0(\mathbb{R}^n)$ are linearly isomorphic. The result also holds more generally if $\mathbb{R}^n$ is replaced with a Carnot group equipped with its Carnot-Carathéodory metric.

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arXiv2025

Pełczyński's property (V$^*$) in Lipschitz-free spaces

We prove that Pelczyński's property (V$^*$) is locally determined for Lipschitz-free spaces, and obtain several sufficient conditions for it to hold. We deduce that $\mathcal{F}(M)$ has property (V$^*$) when the complete metric space $M$ is locally compact and purely 1-unrectifiable, a Hilbert space, or belongs to a class of Carnot-Carathéodory spaces satisfying a bi-Hölder condition, including Carnot groups.

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arXiv2025

De Leeuw representations of functionals on Lipschitz spaces

Let $\mathrm{Lip}_0(M)$ be the space of Lipschitz functions on a complete metric space $(M,d)$ that vanish at a point $0\in M$. We investigate its dual $\mathrm{Lip}_0(M)^*$ using the de Leeuw transform, which allows representing each functional on $\mathrm{Lip}_0(M)$ as a (non-unique) measure on $β\widetilde{M}$, where $\widetilde{M}$ is the space of pairs $(x,y)\in M\times M$, $x\neq y$. We distinguish a set of points of $β\widetilde{M}$ that are "away from infinity", which can be assigned coordinates belonging to the Lipschitz realcompactification $M^{\mathcal{R}}$ of $M$. We define a natural metric $\bar{d}$ on $M^{\mathcal{R}}$ extending $d$ and we show that optimal (i.e. positive and norm-minimal) de Leeuw representations of well-behaved functionals are characterised by $\bar{d}$-cyclical monotonicity of their support, extending known results for functionals in $\mathcal{F}(M)$, the predual of $\mathrm{Lip}_0(M)$. We also extend the Kantorovich-Rubinstein theorem to normal Hausdorff spaces, in particular to $M^{\mathcal{R}}$, and use this to characterise measure-induced and majorisable functionals in $\mathrm{Lip}_0(M)^*$ as those admitting optimal representations with additional finiteness properties. Finally, we use de Leeuw representations to define a natural L-projection of $\mathrm{Lip}_0(M)^*$ onto $\mathcal{F}(M)$ under some conditions on $M$.

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arXiv2024

Convex integrals of molecules in Lipschitz-free spaces

We introduce convex integrals of molecules in Lipschitz-free spaces $\mathcal{F}(M)$ as a continuous counterpart of convex series considered elsewhere, based on the de Leeuw representation. Using optimal transport theory, we show that these elements are determined by cyclical monotonicity of their supports, and that under certain finiteness conditions they agree with elements of $\mathcal{F}(M)$ that are induced by Radon measures on $M$, or that can be decomposed into positive and negative parts. We also show that convex integrals differ in general from convex series of molecules. Finally, we present some standalone results regarding extensions of Lipschitz functions which, combined with the above, yield applications to the extremal structure of $\mathcal{F}(M)$. In particular, we show that all elements of $\mathcal{F}(M)$ are convex series of molecules when $M$ is uniformly discrete and identify all extreme points of the unit ball of $\mathcal{F}(M)$ in that case.

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arXiv2024

A note on nonseparable Lipschitz-free spaces

We prove that several classical Banach space properties are equivalent to separability for the class of Lipschitz-free spaces, including Corson's property ($\mathcal{C}$), Talponen's Countable Separation Property, or being a Gâteaux differentiability space. On the other hand, we single out more general properties where this equivalence fails. In particular, the question whether the duals of non-separable Lipschitz-free spaces have a weak$^*$ sequentially compact ball is undecidable in ZFC. Finally, we provide an example of a nonseparable dual Lipschitz-free space that fails the Radon-Nikodým property.

math.FA↗
arXiv2024

Delta-points and their implications for the geometry of Banach spaces

We show that the Lipschitz-free space with the Radon--Nikodým property and a Daugavet point recently constructed by Veeorg is in fact a dual space isomorphic to $\ell_1$. Furthermore, we answer an open problem from the literature by showing that there exists a superreflexive space, in the form of a renorming of $\ell_2$, with a $Δ$-point. Building on these two results, we are able to renorm every infinite-dimensional Banach space with a $Δ$-point. Next, we establish powerful relations between existence of $Δ$-points in Banach spaces and their duals. As an application, we obtain sharp results about the influence of $Δ$-points for the asymptotic geometry of Banach spaces. In addition, we prove that if $X$ is a Banach space with a shrinking $k$-unconditional basis with $k < 2$, or if $X$ is a Hahn--Banach smooth space with a dual satisfying the Kadets--Klee property, then $X$ and its dual $X^*$ fail to contain $Δ$-points. In particular, we get that no Lipschitz-free space with a Hahn--Banach smooth predual contains $Δ$-points. Finally we present a purely metric characterization of the molecules in Lipschitz-free spaces that are $Δ$-points, and we solve an open problem about representation of finitely supported $Δ$-points in Lipschitz-free spaces.

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arXiv2023

A relative version of Daugavet-points and the Daugavet property

We introduce relative versions of Daugavet-points and the Daugavet property, where the Daugavet-behavior is localized inside of some supporting slice. These points present striking similarities with Daugavet-points, but lie strictly between the notions of Daugavet- and $Δ$-points. We provide a geometric condition that a space with the Radon--Nikodým property must satisfy in order to be able to contain a relative Daugavet-point. We study relative Daugavet-points in absolute sums of Banach spaces, and obtain positive stability results under local polyhedrality of the underlying absolute norm. We also get extreme differences between the relative Daugavet property, the Daugavet property, and the diametral local diameter 2 property. Finally, we study Daugavet- and $Δ$-points in subspaces of $L_1(μ)$-spaces. We show that the two notions coincide in the class of all Lipschitz-free spaces over subsets of $\mathbb{R}$-trees. We prove that the diametral local diameter 2 property and the Daugavet property coincide for arbitrary subspaces of $L_1(μ)$, and that reflexive subspaces of $L_1(μ)$ do not contain $Δ$-points. A subspace of $L_1[0,1]$ with a large subset of $Δ$-points, but with no relative Daugavet-point, is constructed.

math.FA↗
arXiv2021

Purely 1-unrectifiable metric spaces and locally flat Lipschitz functions

We characterize compact metric spaces whose locally flat Lipschitz functions separate points uniformly as exactly those that are purely 1-unrectifiable, resolving a problem of Weaver. We subsequently use this geometric characterization to answer several questions in Lipschitz analysis. Notably, it follows that the Lipschitz-free space $\mathcal{F}(M)$ over a compact metric space $M$ is a dual space if and only if $M$ is purely 1-unrectifiable. Furthermore, we establish a compact determinacy principle for the Radon-Nikodým property (RNP) and deduce that, for any complete metric space $M$, pure 1-unrectifiability is actually equivalent to some well-known Banach space properties of $\mathcal{F}(M)$ such as the RNP and the Schur property. A direct consequence is that any complete, purely 1-unrectifiable metric space isometrically embeds into a Banach space with the RNP. Finally, we provide a possible solution to a problem of Whitney by finding a rectifiability-based description of 1-critical compact metric spaces, and we use this description to prove the following: a bounded turning tree fails to be 1-critical if and only if each of its subarcs has $σ$-finite Hausdorff 1-measure.

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