Let $\Omega \in \mathbb{R}^n$ a bounded open set when $n\geq 2$, and let $X_{1},X_{2},\cdots,X_{m}$ be real smooth vector fields that satisfy Hormander condition on $\Omega$. If we denote $Q(x)$ as the Hormander index of each $x\in \Omega$, then for each $1\leq i\leq Q(x)$ we denote $C_{i}(x)$ as the subspace of the tangent space $T_{x}(\Omega)$ which is spanned by the vector fields $\{X_{J}\}$ with $|J|\leq i$ ($X_{J}$ means $X_{1},X_{2},\cdots X_{m}$ together with all $|J|$ step repeated commutators).
Next we define $$ v(x)=\sum_{i=1}^{Q(x)}i(\dim C_{i}(x)-\dim C_{i-1}(x))\qquad \dim C_{0}(x)=0 $$ From Stein's article "Balls and metrics defined by vector fields I:Basic propoerties" We know that we can define a metrics via the vector fields by $$ d(x,y)=\inf\{\delta>0|\exists\varphi\in C(\delta)\qquad \varphi(0)=x,\varphi(1)=y \}.$$ Here $C(\delta)$ is a class of absolutely continuous mappings $\varphi:[0,1]\to \Omega$ which almost everywhere satisfy the differential equation $$\varphi'(t)=\sum_{j=1}^{q}a_{j}(t)Y_{j}(\varphi(t)) $$ with $|a_{j}(t)|<\delta^{d_{j}}$. Finally, we define the sub-elliptic Ball $ B(x,r)=\{y\in\Omega|d(x,y)<r\}$. I wonder if the volume of $B(x,r)$ has some relation with $r^{v(x)}$? in other words - can we find some positive constant $C_{1},C_{2}$ so that $$ C_{1}r^{v(x)}\leq |B(x,r)|\leq C_{2}r^{v(x)} ?$$
Here $|B(x,r)| $ means the Lebesgue measure of $B(x,r)$ in $\mathbb{R}^n$