arXiv subjects

Jakub Takáč

Publications and source records attributed to Jakub Takáč.

7 recordsLinked to original sources

arXiv2026

Perturbations of measures and sets having curves in d directions

We show that whenever a separable subset $S$ of a complete metric space $X$ admits a $d$-dimensional weak tangent field, the set $S$ is close to being $d$-dimensional in the following sense. Whenever $μ$ is a Borel finite measure on $X$ supported on $S$, then a typical $1$-Lispchitz map (in the sense of Baire category) into a Euclidean space maps $μ$-almost all of $S$ into a set of Hausdorff dimension at most $d$. When taking $d=0$, this implies that any $1$-purely unrectifiable set is typically carried into a Hausdorff $0$-dimensional set up to a $μ$-null set. We show that the result is sharp in Euclidean spaces and, more generally, in strictly convex Banach spaces of finite dimension.

math.MG↗

arXiv2025

Failure of Lang's Flat Chain Conjecture and non-regularity of the prescribed Jacobian equation

We show that Lang's Flat Chain Conjecture (that is, without requiring finite mass of the underlying currents) fails for metric $k$-currents in $\mathbb{R}^d$ whenever $d\geq 2$ and $k\in\{1, \dots, d\}$. In all other cases, it holds. The original conjecture due to Ambrosio and Kirchheim remains open. We first connect Lang's conjecture to a regularity statement concerning the prescribed Jacobian equation near $L^\infty$. We then show that the equation does not have the required regularity. For a Lipschitz vector field $π$, its derivative $\mathrm{D}π$ exists a.e. and is identified with a matrix. Our non-regularity results for the prescribed Jacobian equation quantify how "small" the set \begin{equation*} \operatorname{conv}(\{\operatorname{det}\mathrm{D} π: \operatorname{Lip}(π)\leq L\})\subset L^\infty \end{equation*} is for every $L>0$. The symbol "$\operatorname{conv}$" stands for the convex hull. The "smallness" is quantified in topological terms and is used to show that Lang's Flat Chain Conjecture fails.

math.FA↗

arXiv2025

Structure of Metric $1$-currents: approximation by normal currents and representation results

We prove the $1$-dimensional flat chain conjecture in any complete and quasiconvex metric space, namely that metric $1$-currents can be approximated in mass by normal $1$-currents. The proof relies on a new Banach space isomorphism theorem, relating metric $1$-currents and their boundaries to the Arens-Eells space. As a by-product, any metric $1$-current in a complete and separable metric space can be represented as the integral superposition of oriented $1$-rectifiable sets, thus dropping a finite dimensionality condition from previous results of Schioppa [Schioppa Adv. Math. 2016, Schioppa J. Funct. Anal. 2016]. The connection between the flat chain conjecture and the representation result is provided by a structure theorem for metric $1$-currents in Banach spaces, showing that any such current can be realised as the restriction to a Borel set of a boundaryless normal $1$-current. This generalizes, to any Banach space, the $1$-dimensional case of a recent result of Alberti-Marchese in Euclidean spaces [Alberti-Marchese 2023]. The argument of Alberti-Marchese requires the strict polyhedral approximation theorem of Federer for normal $1$-currents, which we obtain in Banach spaces.

math.MG↗

arXiv2024

Typical Lipschitz images of rectifiable metric spaces

This article studies typical 1-Lipschitz images of $n$-rectifiable metric spaces $E$ into $\mathbb{R}^m$ for $m\geq n$. For example, if $E\subset \mathbb{R}^k$, we show that the Jacobian of such a typical 1-Lipschitz map equals 1 $\mathcal{H}^n$-almost everywhere and, if $m>n$, preserves the Hausdorff measure of $E$. In general, we provide sufficient conditions, in terms of the tangent norms of $E$, for when a typical 1-Lipschitz map preserves the Hausdorff measure of $E$, up to some constant multiple. Almost optimal results for strongly $n$-rectifiable metric spaces are obtained. On the other hand, for any norm $|\cdot|$ on $\mathbb{R}^m$, we show that, in the space of 1-Lipschitz functions from $([-1,1]^n,|\cdot|_\infty)$ to $(\mathbb{R}^m,|\cdot|)$, the $\mathcal{H}^n$-measure of a typical image is not bounded below by any $Δ>0$.

math.MG↗

arXiv2024

Almost compact embeddings between Orlicz and Lorentz spaces

We characterize when an Orlicz space $L^A$ is almost compactly (uniformly absolutely continuously) embedded into a Lorentz space $L^{p,q}$ in terms of a balance condition involving parameters $p,q\in[1,\infty]$, and a Young function $A$. In the course of the proof, we develop a new method based on an inequality of Young type involving the measure of level sets of a given function.

math.FA↗

arXiv2022

Optimality problems in Orlicz spaces

In mathematical modelling, the data and solutions are represented as measurable functions and their quality is oftentimes captured by the membership to a certain function space. One of the core questions for an analysis of a model is the mutual relationship between the data and solution quality. The optimality of the obtained results deserves a special focus. It requires a careful choice of families of function spaces balancing between their expressivity, i.e. the ability to capture fine properties of the model, and their accessibility, i.e. its technical difficulty for practical use. This paper presents a unified and general approach to optimality problems in Orlicz spaces. Orlicz spaces are parametrized by a single convex function and neatly balance the expressivity and accessibility. We prove a general principle that yields an easily verifiable necessary and sufficient condition for the existence or the non-existence of an optimal Orlicz space in various tasks. We demonstrate its use in specific problems, including the continuity of Sobolev embeddings and boundedness of integral operators such as the Hardy--Littlewood maximal operator and the Laplace transform.

math.FA↗

arXiv2022

Optimality of function spaces for kernel integral operators

We explore boundedness properties of kernel integral operators acting on rearrangement-invariant (r.i.) spaces. In particular, for a given r.i. space $X$ we characterize its optimal range partner, that is, the smallest r.i. space $Y$ such that the operator is bounded from $X$ to $Y$. We apply the general results to Lorentz spaces to illustrate their strength.

math.FA↗