P. Hajłasz

activity

20062026

most citedSobolev mappings: Lipschitz density is not a bi-Lipschitz invariant of the target

18 citations · 62 across the 31 of their papers we have counts for

collaborators

43 papers

math.AP2026

Differentiability of Lipschitz mappings into metric spaces: area and co-area formulas

Behnam Esmayli, Paweł Goldstein, Piotr Hajłasz

We give a self-contained exposition of metric differentiability for Lipschitz mappings from sets in Euclidean spaces into arbitrary metric spaces, together with the corresponding a…

math.AP2026

The Trudinger inequality is true for $M^{1, s}$ spaces

Ryan Alvarado, Ahmed Dughayshim, Piotr Hajłasz

We prove the Trudinger inequality with exponent $\frac{s}{s - 1}$ for $M^{1, s}$ Sobolev spaces on metric measure spaces under the sole measure growth assumption $μ (B (x, r)) \geq b r^{s}$ .…

math.DG2025

Hölder continuous mappings, differential forms and the Heisenberg groups

Piotr Hajłasz, Jacob Mirra, Armin Schikorra

We develop analysis of Hölder continuous mappings with applications to geometry and topology of the Heisenberg groups. We cover the theory of distributional Jacobians of Hölder con…

math.GT2024

Gluing diffeomorphisms, bi-Lipschitz mappings and homeomorphisms

Paweł Goldstein, Zofia Grochulska, Piotr Hajłasz

Cerf and Palais independently proved a remarkable result about extending diffeomorphisms defined on smooth balls in a manifold to global diffeomorphisms of the manifold onto itself…

math.CA2023

Constructing diffeomorphisms and homeomorphisms with prescribed derivative

Paweł Goldstein, Zofia Grochulska, Piotr Hajłasz

We prove that for any measurable mapping $T$ into the space of matrices with positive determinant, there is a diffeomorphism whose derivative equals $T$ outside a set of measure le…

math.GT2023★ 2 cited

On the Gromov non-embedding theorem

Piotr Hajłasz, Armin Schikorra

We develop a new method leading to an elementary proof of a generalization of Gromov's theorem about non existence of Hölder embeddings into the Heisenberg group.

math.FA2022

A simple proof of reflexivity and separability of $N^{1, p}$ Sobolev spaces

Ryan Alvarado, Piotr Hajłasz, Lukáš Malý

We present an elementary proof of a well-known theorem of Cheeger which states that if a metric-measure space $X$ supports a $p$ -Poincaré inequality, then the $N^{1, p} (X)$ Sobolev…

math.MG2022

Smooth approximation of mappings with rank of the derivative at most $1$

Paweł Goldstein, Piotr Hajłasz

It was conjectured that if $f \in C^{1} (R^{n}, R^{n})$ satisfies $rank D f \leq m < n$ everywhere in $R^{n}$ , then $f$ can be uniformly approximated by…

math.MG2021

Lipschitz mappings, metric differentiability, and factorization through metric trees

Behnam Esmayli, Piotr Hajłasz

Given a Lipschitz map $f$ from a cube into a metric space, we find several equivalent conditions for $f$ to have a Lipschitz factorization through a metric tree. As an application…

math.CA2020

Lusin-type properties of convex functions and convex bodies

Daniel Azagra, Piotr Hajłasz

We prove that if $f : R^{n} \to R$ is convex and $A \subset R^{n}$ has finite measure, then for any $ε > 0$ there is a convex function $g:\mathbb{R}^n\to\m…

math.CA2020

On an old theorem of Erdös about ambiguous locus

Piotr Hajłasz

Erdös proved in 1946 that if a set $E \subset R^{n}$ is closed and non-empty, then the set, called ambiguous locus or medial axis, of points in $R^{n}$ with the propert…

math.CA2020

The Coarea Inequality

Behnam Esmayli, Piotr Hajłasz

The aim of this paper is to provide a self-contained proof of a general case of the coarea inequality, also known as the Eilenberg inequality. The result is known, but we are not a…

math.CV2019★ 11 cited

Analysis on Metric Spaces

Mario Bonk, Luca Capogna, Piotr Hajłasz +2

This note is a survey of Analysis on Metric spaces, in connection with the upcoming AMS Mathematics Research Communities program in June 2020.

math.GT2019

Linking topological spheres

Piotr Hajłasz

There is a topological embedding $ι : S^{1} \to R^{5}$ such that $π_{3} (R^{5} ∖ ι (S^{1})) = 0$ . Therefore, no $3$ -sphere can be linked with $ι(\mathbb{S}^…

math.FA2019★ 1 cited

Sobolev embedding for $M^{1, p}$ spaces is equivalent to a lower bound of the measure

Ryan Alvarado, Przemysław Górka, Piotr Hajłasz

It has been known since 1996 that a lower bound for the measure, $μ (B (x, r)) \geq b r^{s}$ , implies Sobolev embedding theorems for Sobolev spaces $M^{1, p}$ defined on metric-measure spa…

math.FA2019★ 2 cited

A note on metric-measure spaces supporting Poincaré inequalities

Ryan Alvarado, Piotr Hajłasz

Using a method of Korobenko, Maldonado and Rios we show a new characterization of doubling metric-measure spaces supporting Poincaré inequalities without assuming a priori that the…

math.CA2018

Jacobians of $W^{1, p}$ homeomorphisms, case $p = [n /2]$

Paweł Goldstein, Piotr Hajłasz

We investigate a known problem whether a Sobolev homeomorphism between domains in $R^{n}$ can change sign of the Jacobian. The only case that remains open is when $f\in W^{1…

math.GT2018★ 4 cited

An implicit function theorem for Lipschitz mappings into metric spaces

Piotr Hajłasz, Scott Zimmerman

We prove a version of the implicit function theorem for Lipschitz mappings $f : R^{n + m} \supset A \to X$ into arbitrary metric spaces. As long as the pull-back of the Hausdor…

math.CA2018

$C^{1}$ mappings in $R^{5}$ with derivative of rank at most $3$ cannot be uniformly approximated by $C^{2}$ mappings with derivative of rank at most 3

Paweł Goldstein, Piotr Hajłasz

We find a counterexample to a conjecture of Gałęski by constructing for some positive integers $m < n$ a mapping $f \in C^{1} (R^{n}, R^{n})$ satisfying $\mathrm{rank}\, Df\…

math.CA2018

Topologically nontrivial counterexamples to Sard's theorem

Paweł Goldstein, Piotr Hajłasz, Pekka Pankka

We prove the following dichotomy: if $n = 2, 3$ and $f \in C^{1} (S^{n + 1}, S^{n})$ is not homotopic to a constant map, then there is an open set $Ω \subset S^{n + 1}$ …

math.CA2018

Topological obstructions to continuity of Orlicz-Sobolev mappings of finite distortion

Paweł Goldstein, Piotr Hajłasz

In the paper we investigate continuity of Orlicz-Sobolev mappings $W^{1, P} (M, N)$ of finite distortion between smooth Riemannian $n$ -manifolds, $n \geq 2$ , under the assumption that…

math.CA2017

Finite distortion Sobolev mappings between manifolds are continuous

Paweł Goldstein, Piotr Hajłasz, Mohammad Reza Pakzad

We prove that if $M$ and $N$ are Riemannian, oriented $n$ -dimensional manifolds without boundary and additionally $N$ is compact, then Sobolev mappings $W^{1, n} (M, N)$ of finite dis…

math.MG2017

Weak BLD mappings and Hausdorff measure

Piotr Hajłasz, Soheil Malekzadeh, Scott Zimmerman

We prove that if $Φ : X \to Y$ a mapping of weak bounded length distortion from a quasiconvex and complete metric space $X$ to any metric space $Y$ , then for any Lipschitz mapping $f:…

math.MG2017

The ( $n + 1$ )-Lipschitz homotopy group of the Heisenberg group $H^{n}$

Piotr Hajłasz

We prove that for $n \geq 2$ , the Lipschitz homotopy group $π_{n + 1}^{Lip} (H^{n}) \neq = 0$ of the Heisenberg group $H^{n}$ is nontrivial.

Differentiability of Lipschitz mappings into metric spaces: area and co-area formulas

The Trudinger inequality is true for M1,s spaces

Hölder continuous mappings, differential forms and the Heisenberg groups

Gluing diffeomorphisms, bi-Lipschitz mappings and homeomorphisms

Constructing diffeomorphisms and homeomorphisms with prescribed derivative

On the Gromov non-embedding theorem

A simple proof of reflexivity and separability of N1,p Sobolev spaces

Smooth approximation of mappings with rank of the derivative at most 1

Lipschitz mappings, metric differentiability, and factorization through metric trees

Lusin-type properties of convex functions and convex bodies

On an old theorem of Erdös about ambiguous locus

The Coarea Inequality

Analysis on Metric Spaces

Linking topological spheres

Sobolev embedding for M1,p spaces is equivalent to a lower bound of the measure

A note on metric-measure spaces supporting Poincaré inequalities

Jacobians of W1,p homeomorphisms, case p=[n/2]

An implicit function theorem for Lipschitz mappings into metric spaces

C1 mappings in R5 with derivative of rank at most 3 cannot be uniformly approximated by C2 mappings with derivative of rank at most 3

Topologically nontrivial counterexamples to Sard's theorem

Topological obstructions to continuity of Orlicz-Sobolev mappings of finite distortion

Finite distortion Sobolev mappings between manifolds are continuous

Weak BLD mappings and Hausdorff measure

The (n+1)-Lipschitz homotopy group of the Heisenberg group Hn

The Trudinger inequality is true for $M^{1, s}$ spaces

A simple proof of reflexivity and separability of $N^{1, p}$ Sobolev spaces

Smooth approximation of mappings with rank of the derivative at most $1$

Sobolev embedding for $M^{1, p}$ spaces is equivalent to a lower bound of the measure

Jacobians of $W^{1, p}$ homeomorphisms, case $p = [n /2]$

$C^{1}$ mappings in $R^{5}$ with derivative of rank at most $3$ cannot be uniformly approximated by $C^{2}$ mappings with derivative of rank at most 3

The ( $n + 1$ )-Lipschitz homotopy group of the Heisenberg group $H^{n}$