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At page 763 of Trace theorems for vector fields - R. Monti, D. Morbidelli the Carnot-Carathéodory metric $d$ induced on $\mathbb{R}^2$ by the vector fields $$X_1 = \partial_x \ \text{ and } \ X_2 = |x|^\alpha \partial_y, \ \alpha >0$$ is computed rather explicitely. In particular they say:

Lemma. Let $\lambda >0$. For all $(x,y), (\xi, \eta) \in \mathbb{R}^2$ with $|x|\ge|\xi|$ $$d((x,y),(\xi,\eta)) \simeq |x-\xi| + \frac{|y-\eta|}{|x|^\alpha} \quad \text{ if } \quad |x|^{\alpha +1} \ge \lambda |y-\eta|$$and $$d((x,y),(\xi,\eta)) \simeq |x-\xi| + |y-\eta|^{1/(\alpha+1)} \quad \text{ if } \quad |x|^{\alpha +1} < \lambda |y-\eta|$$where the equivalence constants depend on $\lambda$.

What I would like to know is if there exists a similar way to estimate the CC metric $\hat{d}$ induced on $\mathbb{R}^3$ by the vector fields (called Martinet field - see e.g. Control of Nonholonomic Systems: from Sub-Riemannian Geometry to Motion Planning - F. Jean, page 18) $$X_1 = \partial_x \ \text{ and } \ X_2 = \partial_y + x^2 \partial_z.$$

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    $\begingroup$ Is by any chance the standard distance estimates for the ball-box theorem useful for your purposes? See, e.g. Proposition 7.25 in "The tangent space in sub-Riemannian geometry" by A. Bellaiche? (There is indeed an analogous statement in F. Jean book in Section 2.2 - Distance estimates). There are also uniform estimates later on in the book, see Theorem 2.3, but these hold in a neighborhood of regular points, and all points with $x=0$ are not regular in the Martinet case. $\endgroup$
    – Raziel
    Commented Sep 25, 2016 at 11:30
  • $\begingroup$ Are you sure that the uniform estimates hold only in a neighborhood of regular points? In "Uniform Estimation of Sub-Riemannian Balls" F. Jean doesn't put that hypothesis in the statement of the theorem (but maybe he may have written something like "$p$ is hereafter assumed to be regular" in some step before, I have to check). $\endgroup$
    – gangrene
    Commented Sep 25, 2016 at 20:38
  • $\begingroup$ Theorem 2.3 assumes that the point is regular. However, what is really needed is the existence of a continuously varying set of privileged coordinates and continuous nilpotent approximation (which always exist in a neighborhood of regular points). Martinet is not equiregular, but you are lucky, since a a continuous varying system of privileged coordinates do exist (Example 2.9). You have to check if the nilpotent approximation is continuous, and it seems to me it is. If so, then 2.3 still holds. Does this uniform estimate (where the weighted norms now depend on the point) help you? $\endgroup$
    – Raziel
    Commented Sep 26, 2016 at 6:14
  • $\begingroup$ Well, I'm afraid it doesn't. I'm still pretty unsure, because the situation is not completely clear in my mind yet (and sub-Riemannian Geometry is definitely not my field). Practically speaking I have to compute integrals, imitating what the authors of the paper I linked in the OP did in the second half of page 766. I'm afraid that Theorem 2.3 is still too weak for that purpose, and that I'd need Theorem 2.4 to be applicable to my case... $\endgroup$
    – gangrene
    Commented Sep 26, 2016 at 13:23

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