R}}^p\times\{\vert\xi\vert\leq 1\}}}\in 
        C^\infty({{\mathbb R}}^p;L^\infty(\{\vert\xi\vert\leq 1\}))$;

-   For all $\alpha\in {{\mathbb N}}^p$, $\beta\in {{\mathbb N}}^d$, there exists a nondecreasing function $C_{\alpha,\beta}(\cdot)$ such that $$\sup_{\xi\in{{\mathbb R}}^d,\vert\xi\vert\geq 1/4}
        \langle \xi\rangle^{\vert\beta\vert-m}
        \left\vert\partial_v^\alpha\partial_\xi^\beta \Sigma(v,\xi)\right\vert
        \leq C_{\alpha,\beta}(\vert v \vert).$$

One can write the symbol $\sigma(x,\xi)$ given in (\[symbDN\]) under the form $\sigma(x,\xi)=\Sigma(\nabla a,\xi)$, with $\Sigma(v,\xi)=
    \sqrt{(1+\vert v\vert^2)\vert \xi\vert^2-(v\cdot \xi)^2}$. One can check that $\Sigma\in C^\infty({{\mathbb R}}^d,{\mathcal M}^1)$.

[**i.** ]{}We do not assume in this article that the symbols are smooth at the origin with respect to $\xi$ (for instance, (\[symbDN\]) has singular derivatives at the origin). This is the reason why the estimate in Definition \[defiA0\] is taken over frequencies away from the origin, namely $\vert\xi\vert\geq 1/4$.\
[**ii.**]{} When $p=0$, then $C^\infty({{\mathbb R}}^p,{\mathcal M}^m)$ coincides with the class ${\mathcal M}^m$ of symbols of Fourier multipliers of order $m$.

Let us remark now that if $\sigma(x,\xi)$ is as in (\[A0\]), then one can write $$\sigma(x,\xi)=\left[ \sigma(x,\xi)-\Sigma(0,\xi)\right] 
    +\Sigma(0,\xi);$$ the interest of such a decomposition is that the second term is a simple Fourier multiplier while the first one is in $H^s({{\mathbb R}}^d)$ if $v\in H^s({{\mathbb R}}^d)^p$, $s>d/2$:

\[lemmA0\] Let $p\in{{\mathbb N}}$, $m\in{{\mathbb R}}$, $s_0>d/2$ and take $v\in H^{s_0}({{\mathbb R}}^d)^p$ and $\Sigma\in C^\infty({{\mathbb R}}^p,{\mathcal M}^m)$; set also $\sigma(x,\xi):=\Sigma(v(x),\xi)$.\
Defining 