The Aronsson equation for absolute minimizers of $L^{\mathrm{\infty }}$-functionals associated with vector fields satisfying H\"ormander's condition

Given a Carnot-Carath'eodory metric space $(R^n,d_{\text{cc}})$ generated by vector fields $\{X_i{\}}_{i=1}^m$ satisfying H"ormander’s condition, we prove in theorem A that any absolute minimizer $u\in W_{\text{cc}}^{1,\mathrm{\infty }}(\text{Om})$ to $F(v,\text{Om})={\sup }_{x\in \text{Om}}f(x,Xv(x))$ is a viscosity solution to the Aronsson equation (1.6), under suitable conditions on $f$. In particular, any AMLE is a viscosity solution to the subelliptic $\mathrm{\infty }$-Laplacian equation (1.7). If the Carnot-Carath'edory space is a Carnot group $\mathbf{G}$ and $f$ is independent of $x$-variable, we establish in theorem C the uniquness of viscosity solutions to the Aronsson equation (1.13) under suitable conditions on $f$. As a consequence, the uniqueness of both AMLE and viscosity solutions to the subelliptic $\mathrm{\infty }$-Laplacian equation is established in $\mathbf{G}$