Privileged Coordinates for Carnot–Carathéodory Spaces of Lower Smoothness

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