English

Hausdorff measure of critical set for Luzin $N$ condition

Functional Analysis 2025-10-14 v1

Abstract

It is well-known that there is a Sobolev homeomorphism fW1,p([1,1]n,[1,1]n)f\in W^{1,p}([-1,1]^n,[-1,1]^n) for any p<np<n which maps a set CC of zero Lebesgue nn-dimensional measure onto the set of positive measure. We study the size of this critical set CC and characterize its lower and upper bounds from the perspective of Hausdorff measures defined by a general gauge function.

Keywords

Cite

@article{arxiv.2005.06559,
  title  = {Hausdorff measure of critical set for Luzin $N$ condition},
  author = {Anna Doležalová and Marika Hrubešová and Tomáš Roskovec},
  journal= {arXiv preprint arXiv:2005.06559},
  year   = {2025}
}