Metric Definition of µ-homeomorphisms S. Kallunki ∗ and P. Koskela ∗ Dedicated to Fred and Lois Gehring Abstract We give sufficient metric conditions for a homeomorphism to be- long to a Sobolev class. We give an improvement on the result of Gehring’s on the metric definition of quasiconformality where “lim- sup” is replaced with “liminf”. 1 Introduction The analytic definition of quasiconformality declares that a homeomorphism f between domains Ω and Ω ′ in R n is quasiconformal if f ∈ W 1,n loc (Ω, Ω ′ ) and there exists a constant K so that |Df (x)| n ≤ KJ f (x) a.e. in Ω. Because the Jacobian of any homeomorphism f ∈ W 1,1 loc (Ω, Ω ′ ) is locally in- tegrable, the regularity assumption on f in this definition can naturally be relaxed to f ∈ W 1,1 loc (Ω, Ω ′ ). There has been considerable interest recently in so-called µ-homeomorphisms that form a natural generalization of the con- cept of a quasiconformal mapping in dimension two. To be more precise, we consider homeomorphisms f ∈ W 1,1 loc (Ω, Ω ′ ) so that (1) |Df (x)| n ≤ K (x)J f (x) a.e. in Ω 0 2000 Mathematics Subject Classification: 30C62, 30C65 * Both authors partially supported by the Academy of Finland, project 41933, and S.K. by the foundation Vilho, Yrj¨ o ja Kalle V¨ ais¨al¨ an rahasto. This research was completed when P.K. was visiting at the University of Michigan as the Fred and Lois Gehring professor. 1