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Antonín Procházka

Publications and source records attributed to Antonín Procházka.

17 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.

math.FA↗
arXiv2026

Composition operators for holomorphic Lipschitz functions

We study composition operators on spaces of holomorphic Lipschitz functions defined on the open unit ball of a complex Banach space. Our approach is based on the linearization of the symbol through the holomorphic Lipschitz-free spaces, which allow composition operators to be realized as adjoints of linear operators. For spaces with the bounded approximation property, we characterize composition operators between spaces of holomorphic Lipschitz functions vanishing at the origin and describe when composition operators are onto isomorphisms. We further investigate compactness and weak compactness properties of composition operators. In the finite-dimensional setting, compactness and weak compactness are shown to coincide, and a complete characterization is obtained in terms of the symbol. Finally, we analyze the asymptotic behavior of the iterates of composition operators, proving convergence to zero whenever the supremum norm of the symbol is less than one, and we extend several results to the case not vanishing at 0.

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arXiv2025

On curve-flat Lipschitz functions and their linearizations

We show that several operator ideals coincide when intersected with the class of linearizations of Lipschitz maps. In particular, we show that the linearization $\widehat{f}$ of a Lipschitz map $f:M\to N$ is Dunford-Pettis if and only if it is Radon-Nikodým if and only if it does not fix any copy of $L_1$. We also identify and study the corresponding metric property of $f$, which is a natural extension of the curve-flatness.

math.FA↗
arXiv2025

Diversity of Lipschitz-free spaces over countable complete discrete metric spaces

We show that there are uncountably many mutually non-isomorphic Lipschitz-free spaces over countable, complete, discrete metric spaces. Also there is a countable, complete, discrete metric space whose free space does not embed into the free space of any uniformly discrete metric space. This enhanced diversity is a consequence of the fact that the dentability index $D$ presents a highly non-binary behavior when assigned to the free spaces of metric spaces outside of the oppressive confines of compact purely 1-unrectifiable spaces. Indeed, the cardinality of $\{D(\mathcal F(M)): M$ countable, complete, discrete$\}$ is uncountable while $\{D(\mathcal F(M)):M$ infinite, compact, purely 1-unrectifiable$\}=\{ω,ω^2\}$. Similar barrier is observed for uniformly discrete metric spaces as higher values of the dentability index are excluded for their free spaces: $\{D(\mathcal F(M)):M$ infinite, uniformly discrete$\}=\{ω^2,ω^3\}$.

math.FA↗
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.

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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.

math.MG↗
arXiv2021

On Kalton's interlaced graphs and nonlinear embeddings into dual Banach spaces

We study the nonlinear embeddability of Banach spaces and the equi-embeddability of the family of Kalton's interlaced graphs $([\mathbb N]^k,d_{\mathbb K})_k$ into dual spaces. Notably, we define and study a modification of Kalton's property $\mathcal Q$ that we call property $\mathcal{Q}_p$ (with $p \in (1,+\infty]$). We show that if $([\mathbb N]^k,d_{\mathbb K})_k$ equi-coarse Lipschitzly embeds into $X^*$, then the Szlenk index of $X$ is greater than $ω$, and that this is optimal, i.e., there exists a separable dual space $Y^*$ that contains $([\mathbb N]^k,d_{\mathbb K})_k$ equi-Lipschitzly and so that $Y$ has Szlenk index $ω^2$. We prove that $c_0$ does not coarse Lipschitzly embed into a separable dual space by a map with distortion strictly smaller than $\frac{3}{2}$. We also show that neither $c_0$ nor $L_1$ coarsely embeds into a separable dual by a weak-to-weak$^*$ sequentially continuous map.

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arXiv2020

Compact reduction in Lipschitz free spaces

We prove a general principle satisfied by weakly precompact sets of Lipschitz-free spaces. By this principle, certain infinite dimensional phenomena in Lipschitz-free spaces over general metric spaces may be reduced to the same phenomena in free spaces over their compact subsets. As easy consequences we derive several new and some known results. The main new results are: $\mathcal F(X)$ is weakly sequentially complete for every superreflexive Banach space $X$, and $\mathcal F(M)$ has the Schur property and the approximation property for every scattered complete metric space $M$.

math.FA↗
arXiv2020

Supports in Lipschitz-free spaces and applications to extremal structure

We show that the class of Lipschitz-free spaces over closed subsets of any complete metric space $M$ is closed under arbitrary intersections, improving upon the previously known finite-diameter case. This allows us to formulate a general and natural definition of supports for elements in a Lipschitz-free space $\mathcal F(M)$. We then use this concept to study the extremal structure of $\mathcal F(M)$. We prove in particular that $(δ(x) - δ(y))/d(x,y)$ is an exposed point of the unit ball of $\mathcal F(M)$ whenever the metric segment $[x,y]$ is trivial, and that any extreme point which can be expressed as a finitely supported perturbation of a positive element must be finitely supported itself. We also characterise the extreme points of the positive unit ball: they are precisely the normalized evaluation functionals on points of $M$.

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arXiv2019

Embeddings of Lipschitz-free spaces into $\ell_1$

We show that, for a separable and complete metric space $M$, the Lipschitz-free space $\mathcal F(M)$ embeds linearly and almost-isometrically into $\ell_1$ if and only if $M$ is a subset of an $\mathbb R$-tree with length measure 0. Moreover, it embeds isometrically if and only if the length measure of the closure of the set of branching points of $M$ (taken in any minimal $\mathbb R$-tree that contains $M$) is negligible. We also prove that, for any subset $M$ of an $\mathbb R$-tree, every extreme point of the unit ball of $\mathcal F(M)$ is an element of the form $(δ(x)-δ(y))/d(x,y)$ for $x\neq y\in M$.

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arXiv2018

On the coarse geometry of James spaces

In this note we prove that the Kalton interlaced graphs do not equi-coarsely embed into the James space $\mathcal J$ nor into its dual $\mathcal J^*$. It is a particular case of a more general result on the non equi-coarse embeddability of the Kalton graphs into quasi-reflexive spaces with a special asymptotic stucture. This allows us to exhibit a coarse invariant for Banach spaces, namely the non equi-coarse embeddability of this family of graphs, which is very close to but different from the celebrated property $\mathcal Q$ of Kalton. We conclude with a remark on the coarse geometry of the James tree space $\mathcal J \mathcal T$ and of its predual.

math.FA↗
arXiv2018

On exposed points of Lipschitz free spaces

In this note we prove that a molecule $d(x,y)^{-1}(δ(x)-δ(y))$ is an exposed point of the unit ball of a Lispchitz free space $\mathcal F(M)$ if and only if the metric segment $[x,y]=\{z \in M \; : \; d(x,y)=d(z,x)+d(z,y) \}$ is reduced to $\{x,y\}$. This is based on a recent result due to Aliaga and Pernecká which states that the class of Lipschitz free spaces over closed subsets of M is closed under arbitrary intersections when M has finite diameter.

math.FA↗
arXiv2017

A characterisation of the Daugavet property in spaces of Lipschitz functions

We study the Daugavet property in the space of Lipschitz functions $\operatorname{Lip}_0(M)$ for a complete metric space $M$. Namely we show that $\operatorname{Lip}_0(M)$ has the Daugavet property if and only if $M$ is a length space. This condition also characterises the Daugavet property in the Lipschitz free space $\mathcal{F}(M)$. Moreover, when $M$ is compact, we show that either $\mathcal{F}(M)$ has the Daugavet property or its unit ball has a strongly exposed point. If $M$ is an infinite compact subset of a strictly convex Banach space then the Daugavet property of $\operatorname{Lip}_0(M)$ is equivalent to the convexity of $M$.

math.FA↗
arXiv2016

A characterisation of octahedrality in Lipschitz-free spaces

We characterise the octahedrality of Lipschitz-free space norm in terms of a new geometric property of the underlying metric space. We study the metric spaces with and without this property. Quite surprisingly, metric spaces without this property cannot embed isometrically into $\ell_1$ and similar Banach spaces.

math.FA↗
arXiv2014

Low distortion embeddings into Asplund Banach spaces

We give a simple example of a countable metric space $M$ that does not embed bi-Lipschitz with distortion strictly less than 2 into any Asplund space. Actually, if $M$ embeds with distortion strictly less than 2 to a Banach space $X$, then $X$ contains an isomorphic copy of $\ell_1$. We also show that the space $M$ does not embed with distortion strictly less than $2$ into $\ell_1$ itself but it does embed isometrically into a space that is isomorphic to $\ell_1$.

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