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1
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Change of variables formula under minimal assumptions

100%
Hajłasz P.
Colloquium Mathematicum
|
1993
|
tom 64
|
nr 1
93-101
EN
In the previous papers concerning the change of variables formula (in the form involving the Banach indicatrix) various assumptions were made about the corresponding transformation (see e.g. [BI], [GR], [F], [RR]). The full treatment of the case of continuous transformation is given in [RR]. In [BI] the transformation was assumed to be continuous, a.e. differentiable and with locally integrable Jacobian. In this paper we show that none of these assumptions is necessary (Theorem 2). We only need the a.e. existence of approximate partial derivatives. In Section 3 we consider the general form of the change of variables formula for Sobolev mappings. The author wishes to thank Professor Bogdan Bojarski for many stimulating conversations and suggestions.
2
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A Sard type theorem for Borel mappings

100%
Hajłasz P.
Colloquium Mathematicum
|
1994
|
tom 67
|
nr 2
217-221
EN
We find a condition for a Borel mapping $f:ℝ^m → ℝ^n$ which implies that the Hausdorff dimension of $f^{-1}(y)$ is less than or equal to m-n for almost all $y ∈ ℝ^n$.
3
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A counterexample to the $L^{p}$-Hodge decomposition

100%
Hajłasz P.
Banach Center Publications
|
1996
|
tom 33
|
nr 1
79-83
EN
We construct a bounded domain $Ω ⊂ ℝ^2$ with the cone property and a harmonic function on Ω which belongs to $W_0^{1,p}(Ω)$ for all 1 ≤ p < 4/3. As a corollary we deduce that there is no $L^p$-Hodge decomposition in $L^{p}(Ω,ℝ^2)$ for all p > 4 and that the Dirichlet problem for the Laplace equation cannot be in general solved with the boundary data in $W^{1,p}(Ω)$ for all p > 4.
4
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A new characterization of the Sobolev space

100%
Hajłasz P.
Studia Mathematica
|
2003
|
tom 159
|
nr 2
263-275
EN
The purpose of this paper is to provide a new characterization of the Sobolev space $W^{1,1}(ℝⁿ)$. We also show a new proof of the characterization of the Sobolev space $W^{1,p}(ℝⁿ)$, 1 ≤ p < ∞, in terms of Poincaré inequalities.
5
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On Conditions for Unrectifiability of a Metric Space

64%
Hajłasz P., Malekzadeh S.
Analysis and Geometry in Metric Spaces
|
2015
|
tom 3
|
nr 1
EN
We find necessary and sufficient conditions for a Lipschitz map f : E ⊂ ℝk → X into a metric space to satisfy ℋk(f(E)) = 0. An interesting feature of our approach is that despite the fact that we are dealing with arbitrary metric spaces, we employ a variant of the classical implicit function theorem. Applications include pure unrectifiability of the Heisenberg groups.
6
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Geodesics in the Heisenberg Group

64%
Hajłasz P., Zimmerman S.
Analysis and Geometry in Metric Spaces
|
2015
|
tom 3
|
nr 1
EN
We provide a new and elementary proof for the structure of geodesics in the Heisenberg group Hn. The proof is based on a new isoperimetric inequality for closed curves in R2n.We also prove that the Carnot- Carathéodory metric is real analytic away from the center of the group.
7
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The Lusin Theorem and Horizontal Graphs in the Heisenberg Group

64%
Hajłasz P., Mirra J.
Analysis and Geometry in Metric Spaces
|
2013
|
tom 1
295-301
EN
In this paper we prove that every collection of measurable functions fα , |α| = m, coincides a.e. withmth order derivatives of a function g ∈ Cm−1 whose derivatives of order m − 1 may have any modulus of continuity weaker than that of a Lipschitz function. This is a stronger version of earlier results of Lusin, Moonens-Pfeffer and Francos. As an application we construct surfaces in the Heisenberg group with tangent spaces being horizontal a.e.
8
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How to get rid of one of the weights in a two-weight Poincaré inequality?

64%
Franchi B., Hajłasz P.
Annales Polonici Mathematici
|
2000
|
tom 74
|
nr 1
97-103
EN
We prove that if a Poincaré inequality with two different weights holds on every ball, then a Poincaré inequality with the same weight on both sides holds as well.
9
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On the differentiability of solutions of quasilinear elliptic equations

64%
Hajłasz P., Strzelecki P.
Colloquium Mathematicum
|
1993
|
tom 64
|
nr 2
287-291
10
Content available remote

Pointwise inequalities for Sobolev functions and some applications

64%
Bojarski B., Hajłasz P.
Studia Mathematica
|
1993
|
tom 106
|
nr 1
77-92
EN
We get a class of pointwise inequalities for Sobolev functions. As a corollary we obtain a short proof of Michael-Ziemer's theorem which states that Sobolev functions can be approximated by $C^m$ functions both in norm and capacity.
11
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Polynomial asymptotics and approximation of Sobolev functions

64%
Hajłasz P., Kałamajska A.
Studia Mathematica
|
1995
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tom 113
|
nr 1
55-64
EN
We prove several results concerning density of $C_{0}^{∞}$, behaviour at infinity and integral representations for elements of the space $L^{m,p} = {⨍ | ∇^{m}⨍ ∈ L^p}$.
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