pseudodifferential calculus on foliations
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Let $M$ be a foliated manifold of dimension $n$, and let ${\mathcal{F}}$ be the integrable sub-bundle of the tangent bundle $TM$ of $M$ which defines the foliation. Denote $v$ the dimension of the leaves and $h = n-v$ their codimension.\
The fundamental idea of the Heisenberg calculus is that *longitudinal* vector fields (with respect to to the foliation) have *order 1*, whereas *transverse* vector fields have *order $\leq$ 2*. We shall now describe the symbolic calculus allowing to do so, following Connes and Moscovici [@CM1995].\
Let $(x_1, \ldots , x_n)$ be a foliated local coordinate system of $M$, i.e, the vector fields $\frac{\partial}{\partial x_1}, \ldots , \frac{\partial}{\partial x_v}$ locally span ${\mathcal{F}}$, so that $\frac{\partial}{\partial x_{v+1}}, \ldots , \frac{\partial}{\partial x_n}$ are transverse to the leaves of the foliation. Then, we set $$\begin{aligned}
& \vert p \vert' = (p_1^4 + \ldots + p_v^4 + p_{v+1}^2 + \ldots + p_{n}^2)^{1/4} {\vspace{2mm}}& \langle \alpha \rangle = \alpha_1 + \ldots + \alpha_v + 2\alpha_{v+1} + \ldots 2\alpha_n \end{aligned}$$ for every $p \in {\mathbb{R}}^n$, $\alpha \in {\mathbb{N}}^n$.

A smooth function $\sigma(x,p) \in C^\infty({\mathbb{R}}^n_x \times {\mathbb{R}}^n_p)$ is a *Heisenberg symbol of order $m \in {\mathbb{R}}$* if over any compact subset $K\subset{\mathbb{R}}^n_x$ and 