Generalized Lipschitz numbers, fine differentiability, and quasiconformal mappings

Abstract

We introduce a generalized version of the local Lipschitz number lip u, and show that it can be used to characterize Sobolev functions u ∈ W 1,p loc (R n ), 1 ≤ p ≤ ∞. This concept turns out to be fruitful for studying, and for establishing new connections between, a wide range of topics including fine differentiability, Rademacher's theorem, regularity of maximal functions, quasiconformal mappings, Alberti's rank one theorem, as well as generalizations to metric measure spaces.