Sobolev classes of Banach space-valued functions and quasiconformal mappings

@article{Heinonen2001SobolevCO,
  title={Sobolev classes of Banach space-valued functions and quasiconformal mappings},
  author={Juha M. Heinonen and Pekka Koskela and Nageswari Shanmugalingam and Jeremy T. Tyson},
  journal={Journal d’Analyse Math{\'e}matique},
  year={2001},
  volume={85},
  pages={87-139}
}
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… 

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