for instance to Appendix B of [@Metivier-Zumbrun] for a proof.

\[lemmLP0\] Let $\chi(\eta,\xi)$ be an admissible cut-off function. Then for all $\beta\in{{\mathbb N}}^d$, there exists a constant $C_\beta$ such that $$\forall \xi\in {{\mathbb R}}^d,
    \qquad  
    \big\vert \partial_\xi^\beta 
    {\check}\chi(\cdot,\xi)\big\vert_{L^1({{\mathbb R}}^d)}\leq 
    C_\beta{\langle\xi\rangle}^{-\vert\beta\vert}.$$

Finally, we end this section with the classical characterization of Sobolev spaces (see for instance Th. 2.2.1 of [@Chemin] or Lemma 9.4 of [@TaylorM3]).

\[lemmLP1\] Let $(u_p)_{p\geq -1}$ be a sequence of ${\mathcal S}'({{\mathbb R}}^d)$ such that for all $p\geq 0$, $\widehat{u_p}$ is supported in $A2^{p-1}\leq\vert\xi\vert\leq B2^{p+1}$, for some $A,B>0$, and such that $\widehat{u_{-1}}$ is compactly supported.\
If, for some $s\in{{\mathbb R}}$, ${\displaystyle}\sum_{p\geq -1}2^{2ps}\vert u_p\vert_2^2<\infty$, then $$\sum_{p\geq -1}u_p=:u\in H^s({{\mathbb R}}^d)\quad\mbox{ and }\quad
    \vert u\vert_{H^s}^2\leq {\mbox{\textnormal{Cst }}}\sum_{p\geq -1}2^{2ps}\vert u_p\vert_2^2.$$ Conversely, if $u\in H^s({{\mathbb R}}^d)$ then $$\sum_{p\geq -1}2^{2ps}\vert\varphi_p(D)u\vert_2^2\leq{\mbox{\textnormal{Cst }}}\vert u\vert_{H^s}^2.$$

Symbols {#sectsymb}
=======

As said in the introduction, we are led to consider nonregular symbols $\sigma(x,\xi)$ such that $$\label{A0}
    \sigma(x,\xi)={\Sigma}(v(x),\xi),$$ where $v\in C^0({{\mathbb R}}^d)^p$ for some $p\in {{\mathbb N}}$, while ${\Sigma}$ is a smooth function belonging to the class $C^\infty({{\mathbb R}}^p,{\mathcal M}^m)$ defined below.

\[defiA0\] Let $p\in{{\mathbb N}}$, $m\in {{\mathbb R}}$ and let ${\Sigma}$ be a function defined over ${{\mathbb R}}^p_v\times{{\mathbb R}}^d_\xi$. We say that ${\Sigma}\in C^\infty({{\mathbb R}}^p,{\mathcal M}^m)$ if

-   $\Sigma_{\vert_{{{\mathbb 