Riemannian approximation in Carnot groups

Abstract

We present self-contained proofs of the stability of the constants in the volume doubling property and the Poincaré and Sobolev inequalities for Riemannian approximations in Carnot groups. We use an explicit Riemannian approximation based on the Lie algebra structure that is suited for studying nonlinear subelliptic partial differential equations. Our approach is independent of the results obtained in [11].