The polynomial sub-Riemannian differentiability of some Hölder mappings of Carnot groups

The Area of Graphs on Arbitrary Carnot Groups with Sub-Lorentzian Structure

“…(5) 0 = (0, 0) and M = max{M, M }; Remark 1.21. Requirement (6) above remains valid whenever we replace 0 with any point; see [36] for more details. Definition 1.22.…”

“…The concept generalizes sub-Riemannian differentiability. Definition 1.28 (see [36] for instance). Given two nilpotent graded Lie groups G and G, consider Ω ⊂ G with ψ : Ω → G andd : ψ(Ω) × G → R + such thatd(x, x) = 0 andd(x, y) =d(y, x) for all points of the domain of definition ofd.…”

“…Remark 1.30. The properties required in Definition 1.28 do not guarantee the unique choice of polynomial hc-differential; for more detail, see [36]. Tacitly we consider the minimal possible approximating polynomial.…”

“…If sharper approximation is necessary, we make the polynomial hc-differential explicit. Theorem 1.32 [36]. Suppose that ϕ ∈ C 1 H (Ω, G).…”

“…Let us introduce the concept of adapted basis and state some results. Definition 1.35 [36]. Given two nilpotent graded groups G and G, consider Ω ⊂ G with a polynomially hc-differentiable mapping ψ : Ω → G. If the coordinates {κ j } N j=1 of the polynomial sub-Riemannian differential D P ψ(x) y regarded with respect to the point ψ(x) in the basis…”