Privileged Coordinates for Carnot–Carathéodory Spaces of Lower Smoothness

Abstract

We describe classes of local coordinates on the Carnot–Caratheodory spaces of lower smoothness which permit the homogeneous approximation of quasimetrics and basis vector fields. We establish the minimal smoothness that is required for these classes to coincide with the class of the already-described privileged coordinates in the infinite smoothness case. Moreover, we apply these results to prove the analogs of the available theorems in the case of the canonical coordinates of the second kind. Also, we prove some convergence theorems in quasimetric spaces.