Sobolev classes of Banach space-valued functions and quasiconformal mappings

Juha Heinonen, Pekka Koskela, Nageswari Shanmugalingam, Jeremy T. Tyson

Research output: Contribution to journalArticlepeer-review


We give a definition for the class of Sobolev functions from a metric measure space into a Banach space. We give various characterizations of Sobolev classes and study the absolute continuity in measure of Sobolev mappings in the "borderline case". We show under rather weak assumptions on the source space that quasisymmetric homeomorphisms belong to a Sobolev space of borderline degree; in particular, they are absolutely continuous. This leads to an analytic characterization of quasiconformal mappings between Ahlfors regular Loewner spaces akin to the classical Euclidean situation. As a consequence, we deduce that quasisymmetric maps respect the Cheeger differentials of Lipschitz functions on metric measure spaces with borderline Poincaré inequality.

Original languageEnglish (US)
Pages (from-to)87-138
Number of pages52
JournalJournal d'Analyse Mathematique
StatePublished - Dec 1 2001
Externally publishedYes

ASJC Scopus subject areas

  • Analysis
  • Mathematics(all)


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