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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?


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 $$\begin{pmatrix}1 &a&c\\0&1&b\\0&0&1\end{pmatrix}$$

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I think I solved my own problem. The map $$\Phi(x,y,x)=(x^2/2+y, x^2/2-x+y, z+x^3/12)$$ is P-differentiable with P-differential $$\begin{pmatrix}x&1&0\\x-1&1&0\\0&0&1\end{pmatrix}\;.$$ In general, given a map $g=(g^1,g^2):\mathbb{R}^2\to\mathbb{R}^2$ such that $$\det Jg\equiv k\in\mathbb{R}$$ we can solve the differential equations $$\left\{\begin{array}{rcl} 2f_x&=&g^1g^2_x-g^2g^1_x+ky\\ 2f_y&=&g^1g^2_y-g^2g^1_y-kx\end{array}\right.$$ and write the map $$\Psi(x,y,z)=(g^1(x,y), g^2(x,y), kz+f(x,y))$$ which turns out to be P-differentiable at every point.

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