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Sobolev classes of Banach space-valued functions and quasiconformal mappings

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Abstract

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.

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Author notes

  1. Nageswari Shanmugalingam

    Present address: Department of Mathematics, University of Texas, 78712-1082, Austin, TX, USA

Authors and Affiliations

  1. Department of Mathematics, University of Michigan, 48109-1109, Ann Arbor, MI, USA

    Juha Heinonen

  2. Department of Mathematics, University of Jyväskylä, P.O. Box 35 (MAD), FIN-40351, Jyväskylä, Finland

    Pekka Koskela

  3. Department of Mathematics, N. U. I. Maynooth Co., Kildare, Ireland

    Nageswari Shanmugalingam

  4. Department of Mathematics, Suny at Stony Brook, 11794-3651, Stony Brook, NY, USA

    Jeremy T. Tyson

Authors
  1. Juha Heinonen
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  2. Pekka Koskela
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  3. Nageswari Shanmugalingam
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  4. Jeremy T. Tyson
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Additional information

J. H. supported by NSF grant DMS9970427. P. K. supported by the Academy of Finland, project 39788. N. S. supported in part by Enterprise Ireland. J. T. T. supported by an NSF Postdoctoral Research Fellowship.

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Heinonen, J., Koskela, P., Shanmugalingam, N. et al. Sobolev classes of Banach space-valued functions and quasiconformal mappings. J. Anal. Math. 85, 87–139 (2001). https://doi.org/10.1007/BF02788076

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