Submitted Paper
Inserted: 21 dec 2018
Last Updated: 21 dec 2018
Year: 2018
Abstract:
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 $
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.
Tags:
GeoMeG
Keywords:
quasisymmetric mapping, absolute continuity