Singular quasisymmetric mappings in dimensions two and greater

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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.

Original languageEnglish
Pages (from-to)479-494
Number of pages16
JournalAdvances in Mathematics
Volume351
DOIs
StatePublished - 31 Jul 2019

Keywords

  • Absolute continuity
  • Metric space
  • Quasiconformal mapping

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