A1 Journal article (refereed)
Singular quasisymmetric mappings in dimensions two and greater (2019)
Romney, M. (2019). Singular quasisymmetric mappings in dimensions two and greater. Advances in Mathematics, 31, 479-494. https://doi.org/10.1016/j.aim.2019.05.022
JYU authors or editors
Publication details
All authors or editors: Romney, Matthew
Journal or series: Advances in Mathematics
ISSN: 0001-8708
eISSN: 1090-2082
Publication year: 2019
Volume: 31
Issue number: 0
Pages range: 479-494
Publisher: Academic Press
Publication country: United Kingdom
Publication language: English
DOI: https://doi.org/10.1016/j.aim.2019.05.022
Publication open access: Other way freely accessible online
Publication channel open access:
Publication is parallel published (JYX): https://jyx.jyu.fi/handle/123456789/64478
Web address where publication is available: https://arxiv.org/abs/1803.02322
Abstract
For all n ≥2, we construct a metric space (X, d)and a quasisymmetric mapping f:[0, 1]n→X with the property that f−1 is not absolutely continuous with respect to the Hausdorff n-measure on X. That is, there exists a Borel set E⊂[0, 1]n with Lebesgue measure |E| >0 such that f(E) has Hausdorff n-measure zero. The construction may be carried out so that X has finite Hausdorff n-measure and |E| is arbitrarily close to 1, or so that |E| =1. This gives a negative answer to a question of Heinonen and Semmes.
Keywords: complex analysis; metric spaces
Free keywords: quasiconformal mapping; metric space; absolute continuity
Contributing organizations
Related projects
- Geometry of subRiemannian groups: regularity of finite-perimeter sets, geodesics, spheres, and isometries with applications and generalizations to biLipschitz homogenous spaces
- Le Donne, Enrico
- Research Council of Finland
- GeoMeG Geometry of Metric groups
- Le Donne, Enrico
- European Commission
Ministry reporting: Yes
Reporting Year: 2019
JUFO rating: 3