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reads u(p) = u(0) + hD0 u(0), p1 i + hD1 u(0), p2 i 1 + h(D02 u(0))∗ ¡ p1 , p1 i 2 + o|p|2G , where we have written p1 = (pi,1 )1≤i≤m1 and p2 = (pi,2 )1≤i≤m2 . At another point p0 , we get the horizontal Taylor formula by left-translation. Lemma 8. If u : G 7→ R is a smooth function near p0 we have −1 u(p) = u(p0 )+hD0 u(p0 ), (p−1 0 ¡ p)1 i + hD1 u(p0 ), (p0 ¡ p)2 i 1 −1 −1 2 + h(D02 u(p0 ))∗ (p−1 0 ¡ p)1 , (p0 ¡ p)1 i + o(|p0 ¡ p|G ) 2 as p → p0 .
We continue the development of the theory as we did in the Heisenberg group case. 4.3. Subelliptic Jets. Let u be an upper-semicontinuous real function in G. The second order superjet of u at p0 is defined as 2,+ J (u, p0 ) = (Ρ, Ξ, X ) ∈ Rm1 Ă— Rm2 Ă— S m1 (R) such that −1 u(p) 5 u(p0 ) + hΡ, (p−1 0 ¡ p)1 i + hΞ, (p0 ¡ p)2 i
1 −1 2 −1 + hX (p−1 p| ) ¡ p) i + o(|p ¡ p) , (p 1 1 G 0 0 0 2 Similarly, for lower-semincontinuous v, we define the second order subjet J 2,− (v, p0 ) = (Ρ, Ξ, Y) ∈ Rm1 Ă— Rm2 Ă— S m1 (R) such that −1 v(p) ≼ v(p0 ) + hΡ, (p−1 0 ¡ p)1 i + hΞ, (p0 ¡ p)2 i
1 −1 −1 −1 2 + hY(p0 ¡ p)1 , (p0 ¡ p)1 i + o(|p0 p|G ) 2 As before, one way to get jets is by using smooth functions that touch u from above or below. Let Γ2 denote the class of function φ such that D0 φ, D1 φ and D02 φ are continuous. We define 2,+ K (u, p0 ) = (D0 Ď•(p0 ), D1 Ď•(p0 ),(D2 Ď•(p0 ))∗ ) : Ď• ∈ Γ2 Ď•(p0 ) = u(p0 ) Ď•(p) ≼ u(p), p 6= p0 in a neighborhood of p0 . The set K 2,+ (u, p0 ) is defined analogously.