The Aronsson equation for absolute minimizers of L ∞ - functionals associated with vector fields satisfying Hörmander ’ s condition

Given a Carnot-Carathéodory metric space (R, dcc) generated by vector fields {Xi} m i=1 satisfying Hörmander’s condition, we prove in theorem A that any absolute minimizer u ∈ W 1,∞ cc (Ω) to F (v,Ω) = supx∈Ω 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 ∞-Laplacian equation (1.7). If the Carnot-Carathédory space is a Carnot group 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 ∞-Laplacian equation is established in G. §