Let $\Omega$ be a bounded open domain in $\mathbb{R}^{n}$ with smooth boundary.$\{X_{1},\cdots,X_{m}\}$ be smooth real vector fields on $\Omega$ Which satisfy the Hormander condition. If $\gamma$ is an absolutely continuous mapping of of an interval $[a,b]$ to $\Omega$, we say that $\gamma$ is a sub-unit mapping, if for almost all $t\in[a,b]$ on has $$ \gamma'(t)=\sum_{j=1}^{m}a_{j}(t)X_{j}(\gamma(t)) $$ with $$\sum_{j=1}^{m}|a_{j}(t)|^{2}\leq 1$$ The Carnot-Caratheodory 'distance' between two points $p,q\in M$ relative to the vector fields $\{X_{1},\cdots,X_{m}\}$ is then defined to be the infimum of those $\delta>0$ such that there is a sub-unit mapping from an interval of length $\delta$ to $\Omega$ joining $p$ and $q$.that is $$ d(x,y)=\inf\{b-a|\exists \gamma(t)~~\text{with}~~ \gamma(a)=x,\gamma(b)=y. \gamma~~\text{is a sub-unit mapping} \} $$ Due to vector fields $\{X_{1},\cdots,X_{m}\}$ satisfy the Hormander condition on $\Omega$. we always have $d(x,y)<0$. we can define the sub-unit ball as $$ B_{d}(x,r)=\{y\in\Omega| d(x,y)<r\} $$ Let $\{Y_{1},\cdots,Y_{q}\}$ be smooth real vector fields choose from $$ \{X_{1},\cdots,X_{m},\cdots[X_{j},X_{k}],\cdots,[X_{j}[X_{k},X_{l}]],\cdots\} $$ and satisfy the following condition:
(1) For every $x\in \Omega$,the vectors $\{Y_{1}(x),\cdots,Y_{q}(x)\}$ span $\mathbb{R}^n$
(2) For every pair of indices $1\leq k,l\leq q$,there are functions $c_{k,l}^{1},\cdots,c_{k,l}^{q}\in C^{\infty}(\Omega)$ such that
$$ [Y_{k},Y_{l}]=\sum_{m=1}^{q}c_{k,l}^{m}(x)Y_{m} $$
For $1\leq m\leq q, c_{k,l}^{m}(x)=0$ if $d_{m}>d_{k}+d_{l}$
here $d_{i}$ means the degree of $Y_{i}$ like N.S.W <Balls and metrics defined by vector fields I: Basic properties> defined . Then for $x,y\in \Omega$ let $AC(x,y;\delta)$ denotes the set of all absolutely continuous mappings $\phi:[0,1]\to \Omega$ such that for almost all $0\leq t\leq 1$.
$$\phi'(x)=\sum_{j=1}^{q}a_{j}(t)Y_{j}(\phi(t)) $$
with
$$ |a_{j}(t)|<\delta^{d_{j}} $$
The metric $\rho$ can be define
$$ \rho(x,y)=\inf\{\delta>0|AC(x,y,\delta)\neq\varnothing\} $$
and for the metric $\rho$, we can define the sub-elliptic ball
$$ B_{\rho}(x,\delta)=\{y\in\Omega|\rho(x,y)<\delta\} $$
In N.S.W <Balls and metrics defined by vector fields I: Basic properties> , we know that the sub-elliptic ball has the ball-box theorem that can estimate the volume of $B_{\rho}(x,\delta)$. but in many articles such as Feffermann and Phong's article <Subelliptic Eigenvalue problems> and Jerison and Sanchez-Calle's article <Estimates for the heat kernel for a sum of squares of vector fields> it relate the volume of $B_{d}(x,\delta)$. Now, What's the relationship between sub-unit ball $B_{d}(x,\delta)$ and sub-elliptic ball $B_{\rho}(x,\delta)$ ? Are they equivalence ? if not, Can we have some property's like ball-box theorem for $B_{d}(x,\delta)$ ? Can we compute the volume of $B_{d}(x,\delta)$ ?
On the other hand, in a old article wrote by David Jerison and Antonio Sanchez-Calle < Subelliptic, Second order differential operators>, the third part the ball $B_{L}(x,r)$ he defined in the article is sub-unit ball $B_{d}(x,r)$, then it wrote "In the Hormander-type case Nagel,Stein, and Wainger have proved that the ball $B_{L}(x,r)$ is comparable to the image under the exponential mapping of an appropriate cube $\cdots$", That confused me so much, because in N.S.W's article , it just deal with the sub-elliptic ball $B_{\rho}(x,r)$ not the sub-unit ball $B_{d}(x,r)$. Can anyone help me? Thank you very much in advance!
