arXiv subjects

Jan Malý

Publications and source records attributed to Jan Malý.

13 recordsLinked to original sources

arXiv2025

Maximal noncompactness of embeddings into Marcinkiewicz spaces

We develop a new functional-analytic technique for investigating the degree of noncompactness of an operator defined on a quasinormed space and taking values in a Marcinkiewicz space. The main result is a general principle from which it can be derived that such operators are almost always maximally noncompact in the sense that their ball measure of noncompactness coincides with their operator norm. We point out specifications of the universal principle to the case of the identity operator.

math.FA↗

arXiv2023

Weak limit of homeomorphisms in $W^{1,n-1}$ and (INV) condition

Let $Ω,Ω'\subset\mathbb{R}^3$ be Lipschitz domains, let $f_m:Ω\toΩ'$ be a sequence of homeomorphisms with prescribed Dirichlet boundary condition and $\sup_m \int_Ω(|Df_m|^2+1/J^2_{f_m})<\infty$. Let $f$ be a weak limit of $f_m$ in $W^{1,2}$. We show that $f$ is invertible a.e., more precisely it satisfies the (INV) condition of Conti and De Lellis and thus it has all the nice properties of mappings in this class. Generalization to higher dimensions and an example showing sharpness of the condition $1/J^2_f\in L^1$ are also given. Using this example we also show that unlike the planar case the class of weak limits and the class of strong limits of $W^{1,2}$ Sobolev homeomorphisms in $\mathbb{R}^3$ are not the same.

math.FA↗

arXiv2020

Plans on measures and AM-modulus

For measuring families of curves, or, more generally, of measures, $M_p$-modulus is traditionally used. More recent studies use so-called plans on measures. In their fundamental paper \cite{ADS}, Ambrosio, Di Marino and Savaré proved that these two approaches are in some sense equivalent within $1<p<\infty$. We consider the limiting case $p=1$ and show that the $AM$-modulus can be obtained alternatively by the plan approach. On the way, we demonstrate unexpected behavior of the $AM$-modulus in comparison with usual capacities and consider the relations between the $M_1$--modulus and the $AM$--modulus.

math.FA↗

arXiv2020

On BV homeomorphisms

We obtain the rectifiability of the graph of a bounded variation homeomorphism $f$ in the plane and relations between gradients of $f$ and its inverse. Further, we show an example of a bounded variation homeomorphism $f$ in the plane which satisfies the $(N)$ and $(N^{-1})$ properties and strict positivity of Jacobian of both itself and its inverse, but neither $f$ nor $f^{-1}$ is Sobolev.

math.CA↗

arXiv2019

AM-modulus and Hausdorff measure of codimension one in metric measure spaces

Let $Γ(E)$ be the family of all paths which meet a set $E$ in the metric measure space $X$. The set function $E \mapsto AM(Γ(E))$ defines the $AM$--modulus measure in $X$ where $AM$ refers to the approximation modulus. We compare $AM(Γ(E))$ to the Hausdorff measure $co\mathcal H^1(E)$ of codimension one in $X$ and show that $$co\mathcal H^1(E) \approx AM(Γ(E))$$ for Suslin sets $E$ in $X$. This leads to a new characterization of sets of finite perimeter in $X$ in terms of the $AM$--modulus. We also study the level sets of $BV$ functions and show that for a.e. $t$ these sets have finite $co\mathcal H^1$--measure. Most of the results are new also in $\mathbb R^n$.

math.FA↗

arXiv2019

On distributional adjugate and derivative of the inverse

Let $Ω\subset\er^3$ be a domain and let $f\colonΩ\to\er^3$ be a bi-$BV$ homeomorphism. Very recently in \cite{HKL} it was shown that the distributional adjugate of $Df$ (and thus also of $Df^{-1}$) is a matrix-valued measure. In the present paper we show that the components of $\Adj Df$ are equal to components of $Df^{-1}(f(U))$ as measures and that the absolutely continuous part of the distributional adjugate $\Adj Df$ equals to the pointwise adjugate $\adj Df(x)$ a.e. We also show the equivalence of several approaches to the definition of the distributional adjugate.

math.FA↗

arXiv2018

Mapping Analytic sets onto cubes by little Lipschitz functions

A mapping $f:X\to Y$ between metric spaces is called \emph{little Lipschitz} if the quantity $$ \operatorname{lip}(f(x)=\liminf_{r\to0}\frac{\operatorname{diam} f(B(x,r))}{r} $$ is finite for every $x\in X$. We prove that if a compact (or, more generally, analytic) metric space has packing dimension greater than $n$, then $X$ can be mapped onto an $n$-dimensional cube by a little Lipschitz function. The result requires two facts that are interesing in their own right. First, an analytic metric space $X$ contains, for any $\varepsilon>0$, a compact subset $S$ that embeds into an ultrametric space by a Lipschitz map, and $\dim_P S\geq\dim_P X-\varepsilon$. Second, a little Lipschitz function on a closed subset admits a little Lipschitz extension.

math.CA↗

arXiv2018

Multi-platform Version of StarCraft: Brood War in a Docker Container: Technical Report

We present a dockerized version of a real-time strategy game StarCraft: Brood War, commonly used as a domain for AI research, with a pre-installed collection of AI developement tools supporting all the major types of StarCraft bots. This provides a convenient way to deploy StarCraft AIs on numerous hosts at once and across multiple platforms despite limited OS support of StarCraft. In this technical report, we describe the design of our Docker images and present a few use cases.

cs.AI↗

arXiv2017

Approximation by mappings with singular Hessian minors

Let $Ω\subset\mathbb R^n$ be a Lipschitz domain. Given $1\leq p<k\leq n$ and any $u\in W^{2,p}(Ω)$ belonging to the little Hölder class $c^{1,α}$, we construct a sequence $u_j$ in the same space with $\operatorname{rank}D^2u_j<k$ almost everywhere such that $u_j\to u$ in $C^{1,α}$ and weakly in $W^{2,p}$. This result is in strong contrast with known regularity behavior of functions in $W^{2,p}$, $p\geq k$, satisfying the same rank inequality.

math.AP↗

arXiv2015

Quasiopen and p-path open sets, and characterizations of quasicontinuity

In this paper we give various characterizations of quasiopen sets and quasicontinuous functions on metric spaces. For complete metric spaces equipped with a doubling measure supporting a p-Poincaré inequality we show that quasiopen and p-path open sets coincide. Under the same assumptions we show that all Newton-Sobolev functions on quasiopen sets are quasicontinuous.

math.FA↗

arXiv2013

Luzin's Condition (N) and Modulus of Continuity

In this paper, we establish Luzin's condition (N) for mappings in certain Sobolev-Orlicz spaces with certain moduli of continuity. Further, given a mapping in these Sobolev-Orlicz spaces, we give bounds on the size of the exceptional set where Luzin's condition (N) may fail. If a mapping violates Luzin's condition (N), we show that there is a Cantor set of measure zero that is mapped to a set of positive measure.

math.CA↗

arXiv2013

A low rank property and nonexistence of higher dimensional horizontal Sobolev sets

We establish a "low rank property" for Sobolev mappings that pointwise solve a first order nonlinear system of PDEs, whose smooth solutions have the so-called "contact property". As a consequence, Sobolev mappings from an open set of the plane, taking values in the first Heisenberg group and that have almost everywhere maximal rank must have images with positive 3-dimensional Hausdorff measure with respect to the sub-Riemannian distance of the Heisenberg group. This provides a complete solution to a question raised in a paper by Z. M. Balogh, R. Hoefer-Isenegger and J. T. Tyson. Our approach differs from the previous ones. Its technical aspect consists in performing an "exterior differentiation by blow-up", where the standard distributional exterior differentiation is not possible. This method extends to higher dimensional Sobolev mappings taking values in higher dimensional Heisenberg groups.

math.AP↗

arXiv2010

An elementary way to introduce a Perron-like integral

We give an alternative definition of integral at the generality of the Perron integral and propose an exposition of the foundations of integral theory starting from this new definition. Both definition and proofs needed for the development are unexpectedly simple. We show how to adapt the definition to cover the multidimensional and Stieltjes case and prove that our integral is equivalent to the Henstock-Kurzweil(-Stieltjes) integral.

math.CA↗