I would like to know some non-too-trivial examples of Pansu-differentiable maps between stratified groups (real ones, not $\mathbb{R}^n$, pun intended).
For example, can anyone name a Pansu-differentiable map $f:\mathbb{H}^1\to\mathbb{H}^1$ (from the Heisenberg group of real dimension 3 to itself) which isn't a composition of translations and group homomorphisms?
Definitions
A stratified group is a Lie group $\mathbb{G}$ whose Lie algebra $\mathfrak{g}$ decomposes as a sum $V_1\oplus\ldots\oplus V_m$ of vector spaces, such that $[V_i, V_j]=V_{i+j}$. We can endow the algebra with dilations $\delta_{\lambda}(v_1\oplus\ldots\oplus v_m)=\lambda v_1\oplus \lambda^2v_2\oplus\ldots\oplus \lambda^m v_m$ and by exponentiation we obtain also dilations $\delta_\lambda^{\mathbb{G}}$ on $\mathbb{G}$.
Let us consider two stratified groups $(\mathbb{G},\cdot)$ and $(\mathbb{M}, \odot)$.
A map $f:\mathbb{G}\to\mathbb{M}$ between stratified groups is said to be Pansu differentiable if there exists a group homomorphism $L:\mathbb{G}\to\mathbb{M}$ such that $\delta_{\lambda}^\mathbb{M}\circ L=L\circ\delta_{\lambda}^{\mathbb{G}}$ and such that $\lim_{\lambda\to0}\delta_{1/\lambda}^{\mathbb{M}}(f(v\cdot \delta_\lambda^\mathbb{G}h)\odot f(v)^{-1})=L(h)$ uniformly in $h\in\mathbb{G}$ in some neighbourhood of the origin.
The Heisenberg group is (in short) the group of $3\times 3$ upper triangular matrices of the form $\begin{pmatrix}1 &a&c\\0&1&b\\0&0&1\end{pmatrix}$