for every multi-index $\alpha, \beta$, one has the following estimate $$\vert \partial^\beta_x \partial^\alpha_p \sigma(x,p) \vert \leq C_{K,\alpha,\beta}(1+\vert p \vert')^{m - \langle \alpha \rangle}$$

We shall focus on the smaller class of *classical Heisenberg symbols*. For this, we first define the *Heisenberg dilations* $$\lambda \cdot (p_1, \ldots, p_v, p_{v+1}, \ldots, p_n) = (\lambda p_1, \ldots, \lambda p_v, \lambda^2 p_{v+1}, \ldots, \lambda^2 p_n)$$ for any non-zero $\lambda \in {\mathbb{R}}$ and non-zero $p \in {\mathbb{R}}^n$.

Then, a Heisenberg pseudodifferential symbol $\sigma$ of order $m$ is called *classical* if it has an asymptotic expansion when $\vert p \vert' \to \infty$ $$\label{classical hpdo}
\sigma(x,p) \sim \sum_{j \geq 0} \sigma_{m-j}(x,p)$$ where $\sigma_{m-j}(x,p)$ are *Heisenberg homogeneous* functions, that is, for any non zero $\lambda \in {\mathbb{R}}$, $$\sigma_{m-j}(x,\lambda \cdot p) = \lambda^{m-j} \sigma_{m-j}(x, p)$$ The *Heisenberg principal symbol* is the symbol of higher degree in the expansion (\[classical hpdo\]).\
To such a symbol $\sigma$ of order $m$, one associates its left-quantization as the linear map: $$P : C^\infty_c({\mathbb{R}}^n) \to C^\infty({\mathbb{R}}^n), \qquad Pf(x) = \frac{1}{(2\pi)^n} \int_{{\mathbb{R}}^n} e^{{\textup{\textbf{i}}}x \cdot p} \sigma(x,p) \hat{f}(p) dp$$ where $\hat{f}$ denotes the Fourier transform of the function $f$. We shall say that $P$ is a *classical Heisenberg pseudodifferential operator of order $m$*. If $P$ is 