The comparsion principle for viscosity solutions of fully nonlinear subelliptic equations in Carnot groups

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For any Carnot group G and a bounded domain Ω⊂ G, we prove that viscosity solutions in C() of the fully nonlinear subelliptic equation F(u,∇ₕ u, ∇²ₕ u)=0 are unique when F∈ C(R× Rᵐ× S(m)) satisfies (i) F is degenerate subelliptic and decreasing in u or (ii) F is uniformly subelliptic and nonincreasing in u. This extends Jensen's uniqueness theorem from the Euclidean space to the sub-Riemannian setting of the Carnot group.