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

, (1.1)was developed by Crandall-Lions [CL] and Evans [E1,2] in 1980’s. This idea, togetherwith Jensen’s celebrated uniqueness theorem [J1], provides a very satisfactory theory onexistence, uniqueness, and compactness theorem of weak solutions of (1.1). The theory ofviscosity solutions has been very powerful in many applications, and we refer to the user’sguide [CIL] by Crandall-Ishii-Lions for many such applications.In recent years there has been an explosion of interest in the study of analysis onsub-Riemannian, or Carnot-Carath´edory spaces. The corresponding developments in thetheory of partial differential equations of subelliptic type have prompted people to considerfully nonliear equations in Carnot groups. For examples, motivated by the very importantwork of Jensen [J2] on absolute minimizing Lipschitz extensions (or ALMEs, a notion firstintroduced by Aronsson [A]) and viscosity solutions to the ∞-laplacian equation in theEuclidean space, Bieske [B], Bieske-Capogna [BC], and Wang [W1] have studied absoluteminimizing horizontal Lipschitz extensions and viscosity solutions to the ∞-sublaplacianequation on Carnot groups. In particular, the notion of viscosity solutions has been ex-tended to fully nonlinear subelliptic equation (see [B]) and the uniqueness of viscositysolution of ∞-sublaplacian eqaution on any Carnot group was established by Wang [W1].It is well-known (cf. the monographs [CC] by Caffarelli-Cabr´e and [G] by Gutierrez) thatboth convexity and the Monge-Amp´ere equation:det(∇

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