6533b86ffe1ef96bd12cde4b
RESEARCH PRODUCT
Sobolev classes of Banach space-valued functions and quasiconformal mappings
Pekka KoskelaJuha HeinonenNageswari ShanmugalingamJeremy T. Tysonsubject
Discrete mathematicsMathematics::Complex VariablesGeneral MathematicsEberlein–Šmulian theoremMathematics::Analysis of PDEsSobolev inequalitySobolev spaceMathematics::Metric GeometryBesov spaceInterpolation spaceBirnbaum–Orlicz spaceMetric differentialAnalysisMathematicsTrace operatordescription
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 Poincare inequality.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2001-12-01 |