Porosity and differentiability of Lipschitz maps from stratified groups to Banach homogeneous groups

Abstract

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 σ-porous set. At all density points of A we establish a pointwise characterization of differentiability in terms of directional derivatives. These results naturally lead us to 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.

Universal differentiability sets and maximal directional derivatives in Carnot groups

“…Note that Theorem 2.9 also holds for Carnot group targets (and even for suitable infinite dimensional targets [26,27]), but we will be concerned mainly with realvalued maps.…”

“…By replacing Euclidean translations and dilations with translations and dilations in the Carnot group, one can define Pansu differentiability of functions between Carnot groups (Definition 2.7). Pansu generalized Rademacher's theorem to Carnot groups, showing that Lipschitz functions between Carnot groups are Pansu differentiable almost everywhere with respect to the Haar measure (Theorem 2.9) [30,27]. This can be applied to show that every Carnot group (other than Euclidean space itself) contains no subset of positive measure which bi-Lipschitz embeds into a Euclidean space [20,24,36].…”

A $C^{m,ω}$ Whitney Extension Theorem for Horizontal Curves in the Heisenberg Group

“…In recent years, it has become clear that a large part of geometric analysis, geometric measure theory, and real analysis in Euclidean spaces may be generalized to the Carnot group setting. See, for example, [2,3,12,13,17,19,20,21,22,23,24].…”

A $C^m$ Lusin Approximation Theorem for Horizontal Curves in the Heisenberg Group

“…In recent years, it has become clear that a large part of geometric analysis, geometric measure theory and real analysis in Euclidean spaces may be generalized to more general settings, see for example [4,6,7,13,14,16,18,22,21,23,25,26,27]. Carnot groups are Lie groups whose Lie algebra admits a stratification.…”