This paper aims to study isometries of the $1$-Wasserstein space $\mathcal{ W}_1(\mathbf{ G})$ over Carnot groups endowed with horizontally strictly convex norms. Well-known examples of horizontally strictly convex norms on Carnot groups are the Heisenberg group $\mathbb{ H}^n$ endowed with the Heisenberg-Kor\'anyi norm, or with the Naor-Lee norm; and $H$-type Iwasawa groups endowed with a Kor\'anyi-type norm. We prove that on a general Carnot group there always exists a horizontally strictly convex norm. The main result of the paper says that if $(\mathbf{ G},N_{ \mathbf{ G}})$ is a Carnot group where $N_{ \mathbf{ G}}$ is a horizontally strictly convex norm on $\mathbf{ G}$, then the Wasserstein space $\mathcal{ W}_1(\mathbf{ G})$ is isometrically rigid. That is, for every isometry $\Phi:\mathcal{ W}_1(\mathbf{ G})\to\mathcal{ W}_1(\mathbf{ G})$ there exists an isometry $\psi:\mathbf{ G}\to \mathbf{ G}$ such that $\Phi=\psi_{ \#}$.
Categories: math.MG math-ph math.FA math.MP