18 citations · 62 across the 31 of their papers we have counts for
11 papers · 1 filter
A simple proof of reflexivity and separability of 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 supports a -Poincaré inequality, then the Sobolev…
Sobolev embedding for 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, , implies Sobolev embedding theorems for Sobolev spaces defined on metric-measure spa…
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…
A Marcinkiewicz integral type characterization of the Sobolev space
Piotr Hajłasz, Zhuomin Liu
In this paper we present a new characterization of the Sobolev space , which is a higher dimensional version of a result of Waterman. We also provide a new an…
Sobolev spaces, Lebesgue points and maximal functions
Piotr Hajlasz, Zhuomin Liu
In this note we study boundedness of a large class of maximal operators in Sobolev spaces that includes the spherical maximal operator. We also study the size of the set of Lebesgu…
Maximal potentials, maximal singular integrals, and the spherical maximal function
Piotr Hajlasz, Zhuomin Liu
We introduce a notion of maximal potentials and we prove that they form bounded operators from to the homogeneous Sobolev space for all . We appl…
The Lusin theorem and horizontal graphs in the Heisenberg group
Piotr Hajlasz, Jacob Mirra
In this paper we prove that every collection of measurable functions , coincides a.e. with th order derivatives of a function whose derivatives of or…
Sobolev mappings, degree, homotopy classes and rational homology spheres
Pawel Goldstein, Piotr Hajlasz
In the paper we investigate the degree and the homotopy theory of Orlicz-Sobolev mappings between manifolds, where the Young function satisfies a divergence cond…
Sobolev mappings: Lipschitz density is not an isometric invariant of the target
Piotr Hajlasz
If is a compact smooth manifold and is a compact metric space, the Sobolev space is defined through an isometric embedding of into a Banach space. We pro…
On the lack of density of Lipschitz mappings in Sobolev spaces with Heisenberg target
Noel DeJarnette, Piotr Hajlasz, Anton Lukyanenko +1
We study the question: when are Lipschitz mappings dense in the Sobolev space ? Here denotes a compact Riemannian manifold with or without boundary, wh…
Sobolev mappings: Lipschitz density is not a bi-Lipschitz invariant of the target
Piotr Hajlasz
We study a question of density of Lipschitz mappings in the Sobolev class of mappings from a closed manifold into a singular space. The main result of the paper shows that if we ch…