Approximate differentiability of mappings of Carnot–Carathéodory spaces

Abstract

We study the approximate differentiability of measurable mappings of Carnot–Carathéodory spaces. We show that the approximate differentiability almost everywhere is equivalent to the approximate differentiability along the basic horizontal vector fields almost everywhere. As a geometric tool we prove the generalization of Rashevsky–Chow theorem for \(C^1\)-smooth vector fields. The main result of the paper extends theorems on approximate differentiability proved by Stepanoff (1923, 1925) and Whitney (1951) for Euclidean spaces and by Vodopyanov (2000) for Carnot groups.