and the product is the star product (\[star product\]).\
A Heisenberg formal symbol is said *Heisenberg elliptic* if it is invertible in ${\mathcal{S}}_H({\mathbb{R}}^n)$. This is equivalent to say that its *Heisenberg principal symbol* is invertible on ${\mathbb{R}}^n_x \times {\mathbb{R}}^n_p \smallsetminus \{0\}$. The corresponding pseudodifferential operators are in general not elliptic, but only hypoelliptic.

\[subelliptic laplacian\] The sub-elliptic Laplacian is the differential operator $$\Delta_H = \partial_{x_1}^4 +  \ldots + \partial_{x_v}^4 - (\partial_{x_{v+1}}^2 + \ldots + \partial_{x_n}^2)$$ It has Heisenberg principal symbol $\sigma(x,p) = \vert p \vert'^4$, and is therefore Heisenberg elliptic. However, its usual principal symbol, as an ordinary differential operator, is $(x,p) \mapsto \sum_{i=1}^v p_i^4 $, so $\Delta_H$ is clearly not elliptic.\

Heisenberg pseudodifferential operators are compatible with foliated coordinate changes. Therefore, the Heisenberg calculus can be defined globally on foliations by using a partition of unity. Then, for a foliated manifold $M$, we denote by $\Psi_H(M)$ the algebra of properly-supported Heisenberg pseudodifferential operators on $M$, and by $\Psi_{H,c}(M)$ its subalgebra of compactly-supported operators.\
For a (${\mathbb{Z}}_2$-graded) complex vector bundle $E$ over $M$, one defines in the same way the algebra of Heisenberg pseudodifferential operators $\Psi_H(M,E)$ acting on the smooth compactly-supported sections $C^{\infty}_c(M,E)$ of $E$. One always has an exact 