$\tau(x,\xi)=\sigma(x,\xi)-\Sigma(0,\xi)$, one has $\tau(\cdot,\xi)\in H^{s_0}({{\mathbb R}}^d)$ for all $\xi\in{{\mathbb R}}^d$; moreover:

-   One has $\tau_{\vert_{{{\mathbb R}}^d\times\{\vert\xi\vert\leq 1\}}}
        \in L^\infty(\{\vert\xi\vert\leq 1\}; H^{s_0}({{\mathbb R}}^d))$;

-   For all $\beta\in {{\mathbb N}}^d$ and $0\leq s\leq s_0$, $$\sup_{\xi\in{{\mathbb R}}^d,\vert\xi\vert\geq 1/4}
        \langle \xi\rangle^{\vert\beta\vert-m}
        \left\vert\partial_\xi^\beta 
        \tau(\cdot,\xi)\right\vert_{H^s}
        \leq C'_{s,\beta}(\vert v\vert_{L^\infty} \vert)
        \vert v\vert_{H^s},$$ where $C'_{s,\beta}(\cdot)$ is some nondecreasing function depending only on the $C_{\alpha,\beta}(\cdot)$, $\vert\alpha\vert\leq [s]+2$, introduced in Definition \[defiA0\].

This is a simple consequence of Moser’s inequality (see e.g. Prop. 3.9 of [@TaylorM3]) and the properties of $\Sigma$ set forth in Definition \[defiA0\].

The previous lemma motivates the introduction of the following class of symbols (see also [@Yamazaki1; @Marschall] for similar symbol classes).

\[defiA1\] Let $m\in{{\mathbb R}}$ and $s_0>d/2$. A symbol $\sigma(x,\xi)$ belongs to the class $\Gamma_{s_0}^m$ if and only if

-   One has $\sigma_{\vert_{{{\mathbb R}}^d\times\{\vert\xi\vert\leq 1\}}}
        \in L^\infty(\{\vert\xi\vert\leq 1\}; H^{s_0}({{\mathbb R}}^d))$;

-   For all $\beta\in {{\mathbb N}}^d$, one has $$\sup_{\xi\in{{\mathbb R}}^d,\vert\xi\vert\geq 1/4}
        \langle \xi\rangle^{\vert\beta\vert-m}
        \left\vert\partial_\xi^\beta 
        \sigma(\cdot,\xi)\right\vert_{H^{s_0}}<\infty .$$

We now set some terminology concerning the regularity of the symbols at the origin.

\[defireg\] We say that $\Sigma\in C^\infty({{\mathbb R}}^p,{\mathcal M}^m)$ is *$k$-regular* at the origin if $\Sigma_{\vert_{{{\mathbb R}}^p\times\{\vert\xi\vert\leq 1\}}}\in 
    C^\infty({{\mathbb R}}^p;W^{k,\infty}(\{\vert\xi\vert\leq 1\}))$.\
Similarly, we say that $\sigma\in \Gamma^m_{s_0}$ is *$k$-regular* at the origin if $\sigma_{\vert_{{{\mathbb R}}^d\times\{\vert\xi\vert\leq 1\}}}
    