Inserted: 6 jun 2017
Last Updated: 26 oct 2019
Journal: Ann. Mat. Pura Appl.
Let $f$ be a Lipschitz map from a subset $A$ of a stratified group to a Banach homogeneous group. We show that directional derivatives of $f$ act as homogeneous homomorphisms at density points of $A$ outside a $\sigma$-porous set. At density points of $A$ we establish a pointwise characterization of differentiability in terms of directional derivatives. We use these new results to obtain an alternate proof of almost everywhere differentiability of Lipschitz maps from subsets of stratified groups to Banach homogeneous groups satisfying a suitably weakened Radon-Nikodym property. As a consequence we also get an alternative proof of Pansu's Theorem.
Keywords: Carnot-Carathéodory distance, Carnot group, Rademacher's theorem, differentiability, Porous set, stratified group, Banach homogeneous group, Lipschitz map