Geometric and Analytic Properties of Carnot–Carathéodory Spaces under Minimal Smoothness

We describe geometric and analytical results in the theory of non-holonomic spaces under minimal smoothness, which we define following works [1, 2].
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Definition. Fix a connected Riemannian $C^\infty$-manifold $\mathbb{M}$ of topological dimension $N$. The manifold $\mathbb{M}$ is called the Carnot–Carathéodory space if the tangent bundle $T\mathbb{M}$ has a filtration
$$ H \mathbb{M} = H_1 \mathbb{M} \subsetneq H_2 \mathbb{M} \subsetneq \dots \subsetneq H_M \mathbb{M} = T \mathbb{M} $$
by subbundles such that every point $x_0 \in \mathbb{M}$ has a neighborhood $U(x_0) \subset \mathbb{M}$ equipped with a collection of $C^1$-smooth vector fields $X_1, \dots, X_N$ enjoying the following two properties:
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(1) At every point $x \in U(x_0)$ we have a subspace
$$ H_i \mathbb{M} (x) = H_i (x) = \mathrm{span} \{ X_1(x), \dots, X_{\dim H_i}(x) \} \subset T_x \mathbb{M} $$
of the dimension $\dim H_i$ independent of $x$, $i = 1, \dots, M$.
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(2) The inclusion $[H_i, H_j] \subset H_{i+j}$ holds for $i+j \leq M$.
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Moreover, the Carnot–Carathéodory space is called the Carnot manifold if the following third condition holds:
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(3) $H_{j+1} = \mathrm{span} \{ H_j, [H_1, H_j], \dots, [H_k, H_{j+1-k}] \}$, where $k = \lfloor \tfrac{j+1}{2} \rfloor$ for $j = 1, \dots, M-1$.
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Since it is not a priori known whether Carnot manifolds carry Carnot–Carathéodory metric, the Nagel–Stein–Wainger “Box” metric $d_\infty(x,y)$ is used instead in their study. Using results on fine properties of Carnot–Carathéodory spaces [2] we show that Carnot–Carathéodory metric is well-defined proving an analogue of Carathéodory– Rashevskiĭ–Chow theorem.
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Theorem [3]. 1) For every point $g \in \mathbb{M}$ there is a neighborhood $U$ and $C > 0$ such that every point $x \in U$ can be represented as
$$ x = \exp(a_L X_{i_L}) \circ \dots \circ \exp(a_1 X_{i_1})(g) $$
with $i_k \in \{ 1, \dots, \dim H_1 \}$ and $|a_k| \leq C d_\infty(x,g)$ for $k = 1, \dots, L$. Here $L = L(\mathbb{M})$ does not depend on the points $g$, $x$.
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2) In a connected Carnot manifold any two points can be joined by an absolutely continuous curve consisting of finitely many segments of integral lines of vector fields $X_1, \dots, X_{\dim H_1}$.
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This result in turn was utilized in [4] to prove local equivalence of “Box” metric $d_\infty$ and Carnot–Carathéodory metric $d_{cc}$ which immediately implies that:
$\bullet$ locally there is $C>0$ such that $B(x, C^{-1} r) \subset \mathrm{Box}(x,r) \subset B(x, Cr)$;
$\bullet$ the Hausdorff dimension of $\mathbb{M}$ is $\nu = \sum\limits_{k=1}^N \deg X_k$;
$\bullet$ the Hausdorff measure $\mathcal{H}^\nu$ is locally doubling.
As an application of these results to a theory of Sobolev spaces we obtain the Poincaré inequality for John domains in Carnot manifolds.
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Theorem [5]. Let $x_0 \in \mathbb{M}$ and $1 \leq p < \infty$. There are $C_p > 0$ and $r_0 > 0$ such that for every John domain $\Omega \subset B(x_0, r_0)$ of class $J(a,b)$ and every $f \in C^\infty(\overline{\Omega})$ we have
$$ \Vert f - f_\Omega \Vert_{L_p(\Omega)} \leq (\tfrac{b}{a})^\nu \mathrm{diam} (\Omega) \Vert (X_1 f, \dots, X_{\dim H_1} f) \Vert_{L_p(\Omega)} $$
where $f_\Omega = \tfrac{1}{|\Omega|} \int_\Omega f$ and $\nu$ is the Hausdorff dimension of $\mathbb{M}$.