Calculus of Variations and Geometric Measure Theory

M. Romney

Singular quasisymmetric mappings in dimensions two and greater

created by romney on 21 Dec 2018


Submitted Paper

Inserted: 21 dec 2018
Last Updated: 21 dec 2018

Year: 2018

ArXiv: 1803.02322 PDF


For all $n \geq 2$, we construct a metric space $(X,d)$ and a quasisymmetric mapping $f\colon [0,1]^n \rightarrow X$ 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 $E \subset [0,1]^n$ with Lebesgue measure $
>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 $
$ is arbitrarily close to 1, or so that $
=1$. This gives a negative answer to a question of Heinonen and Semmes.

Tags: GeoMeG
Keywords: quasisymmetric mapping, absolute continuity