In this post it was shown that if $\Omega$ and $\Omega'$ are diffeomorphic non-empty open domains in some Euclidean space then the corresponding local Sobolev spaces are diffeomorphic with diffeomorphism given by the corresponding composition operator $$ f \mapsto f\circ \phi , $$ where $\phi:\Omega\rightarrow \Omega'$ is a diffeomorphism; with the diffeomorphism again being the composition operator?
The key point here is not the existence of a diffeomorphism between these Banach spaces, which is obvious, but that it is given by the composition operator.
Let $\mu$ be a measure on a free Carnot group $G$.
I'm wondering, since any free Carnot group $G$ is diffeomorphic to some Euclidean space, then is it true that the Hajlasz-Sobolev space $H_s^p(G;\mu)$ is diffeomorphic to the Hajlasz-Sobolev space $H_s^p(\mathbb{R}^d;\nu)$ where $\nu$ is induced by the (postulated) diffeomorphism?
