shows that $$\label{intro1}
    \forall u\in H^{s+m_1}({{\mathbb R}}^d),\qquad
    \vert {\mbox{\textnormal{Op}}}(\sigma^1)u\vert_{H^s}\leq \underline{C}(\sigma^1)
    \vert u\vert_{H^{s+m_1}}.$$

Concerning the study of commutators, Taylor (following works of Moser [@Moser] and Kato-Ponce [@Kato-Ponce]) proved in [@TaylorM0] that for all $\sigma^1\in S^{m_1}_{1,0}$, with $m_1>0$, and all $\sigma^2\in H^\infty({{\mathbb R}}^d)$, one has for all $s\geq 0$, $$\label{intro2}
    \big\vert [{\mbox{\textnormal{Op}}}(\sigma^1),\sigma^2]u\big\vert_{H^s}
    \leq \underline{C}(\sigma^1)\big(\vert\sigma^2\vert_{W^{1,\infty}} 
    \vert u\vert_{H^{s+m_1-1}}
    +\vert \sigma^2\vert_{H^{s+m_1}}\vert u\vert_\infty\big);$$ another well-known commutator estimate is the so-called Calderon-Coifman-Meyer estimate: if $m_1\geq 0$ then for all $s\geq 0$ and $t_0>d/2$ such that $s+m_1\leq t_0+1$, one has (see Prop. 4.2 of [@TaylorM] for instance): $$\label{intro3}
    \big\vert [{\mbox{\textnormal{Op}}}(\sigma^1),\sigma^2]u\big\vert_{H^s}
    \leq \underline{C}(\sigma^1)\vert \sigma^2\vert_{H^{t_0+1}}
    \vert u\vert_{H^{s+m_1-1}}.$$ A drawback of (\[intro1\]), (\[intro2\]) and (\[intro3\]) is that the dependence of the constant $\underline{C}(\sigma^1)$ on $\sigma^1$ is not specified. This may cause these estimates to be inoperative in the study of some nonlinear PDE; indeed, when solving such an equation by an iterative scheme, one is led to study the pseudo-differential operator corresponding to the linearized equations around some reference state. Generally, the symbol of this operator can be written $\sigma(x,\xi)=\Sigma(v(x),\xi)$, where $\Sigma(v,\xi)$ is smooth with respect to $v$ and of order $m$ with respect to $\xi$, while $v(\cdot)$ belongs to some Sobolev space $H^s({{\mathbb R}}^d)$. For instance, in the study of 