$s$ and $m_1$ are possible in (\[intro3\]) (see Ths. \[theoIII1bis\] and \[theoIV1bis\]). For instance, if $\sigma^1$ is a Fourier multiplier of order $m_1\in{{\mathbb R}}$ and $\sigma^2$ is of order $m_2\in{{\mathbb R}}$ with $\sigma^2(x,\xi)=\Sigma^2(v(x),\xi)$ and $v\in H^{\infty}({{\mathbb R}}^d)^p$ then for all $s\in{{\mathbb R}}$ such that $\max\{-t_0,-t_0-m_1\}<s$ (with $t_0>d/2$ arbitrary), $$\begin{aligned}
        \lefteqn{\big\vert \big[\sigma^1(D),\sigma^2(x,D)\big]u
        -{\mbox{\textnormal{Op}}}({\{\sigma^1,\sigma^2\}_n})
        u\big\vert_{H^s}}\\
        & \leq &C(\sigma^1,\vert v\vert_{W^{n+1,\infty}})
        \big(\vert u\vert_{H^{s+m_1+m_2-n-1}}
        +
        \vert v\vert_{H^{(s+m_1\wedge n)_+}}
        \vert u\vert_{H^{m_1+m_2+t_0-m_1\wedge n}}\big).
        \end{aligned}$$

The above results admit generalization to $L^p$-based Sobolev spaces and Besov spaces, but we deliberately chose to work with classical $L^2$-based Sobolev spaces to ease the readability. We refer the reader interested by this kind of generalizations to [@Yamazaki1; @Yamazaki2], [@Marschall] and [@TaylorM0; @TaylorM3] for instance.\
The methods used to prove the above results rely heavily on Bony’s paradifferential calculus [@Bony] as well as on the works of Coifman and Meyer [@Meyer; @Meyer-Coifman3].\
In Section \[sectsymb\], we introduce the class of symbols adapted to our study; they consist in all the symbols $\sigma(x,\xi)$ such that $\sigma(\cdot,\xi)$ belongs to some Sobolev space for all $\xi$. These symbols are decomposed into four components, one of them being the well-known paradifferential symbol associated to $\sigma$, and some basic properties are given.\
In Section \[sectoper\], we study 