Analytic properties of locally quasisymmetric mappings from Euclidean domains

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We establish basic analytic properties of locally quasisymmetric homeomorphisms f : G → Y, where G is a domain in the Euclidean space ℝn, n ≥ 2, and Y is a metric space with locally finite Hausdorff n-measure. We show that such maps are ACL and satisfy a reverse Hadamard inequality. Furthermore, they are "Sobolev functions" (in a sense recently introduced by Reshetnyak) and satisfy the following form of Lusin's condition (N): if E ⊂ G has measure zero, then f (E) has Hausdorff n-measure zero. These results extend work of Väisälä, who showed that these facts hold when the image embeds in some (high-dimensional) Euclidean space.

Original languageEnglish (US)
Pages (from-to)995-1016
Number of pages22
JournalIndiana University Mathematics Journal
Issue number3
StatePublished - 2000
Externally publishedYes


  • Condition (n)
  • Packing measures
  • Quasisymmetric mappings

ASJC Scopus subject areas

  • Mathematics(all)


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