Calculus of Variations and Geometric Measure Theory
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M. Romney

Singular quasisymmetric mappings in dimensions two and greater

created by romney on 21 Dec 2018

[BibTeX]

Submitted Paper

Inserted: 21 dec 2018
Last Updated: 21 dec 2018

Year: 2018

ArXiv: 1803.02322 PDF

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

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