estimates.**]{} In this paper, we give a precise description of the constant $\underline{C}(\sigma^1)$ which appears in (\[intro2\]) and (\[intro3\]), and generalize these estimates in three directions:

-   We control the symbolic expansion of the commutator in terms of the Poisson brackets. For instance, in the particular case when the symbol $\sigma^1(x,\xi)=\sigma^1(\xi)$ does not depend on $x$, we derive the following estimate (see Th. \[theoIII3\]): if $m_1\in{{\mathbb R}}$ and $n\in{{\mathbb N}}$ are such that $m_1>n$, then for all $s\geq 0$, one has $$\begin{aligned}
        \lefteqn{\left\vert [{\mbox{\textnormal{Op}}}(\sigma^1),\sigma^2]u
        -{\mbox{\textnormal{Op}}}(\{\sigma^1,\sigma^2\}_n)u
        \right\vert_{H^s}}\\
        &\lesssim&C(\sigma^1)\left(
        \vert\nabla^{n+1}\sigma^2\vert_{{\infty}}
        \vert u\vert_{H^{s+m_1-n-1}}+
        \vert \sigma^2\vert_{H^{s+m_1}}
        \vert u\vert_\infty\right),
        \end{aligned}$$ and a precise description of $C(\sigma^1)$ is given; if $\sigma^1(\cdot)$ is regular at the origin, we have a more precise version involving only derivatives of $\sigma^2$, $$\begin{aligned}
        \lefteqn{\left\vert [{\mbox{\textnormal{Op}}}(\sigma^1),\sigma^2]u
        -{\mbox{\textnormal{Op}}}(\{\sigma^1,\sigma^2\}_n)u
        \right\vert_{H^s}}\\
        &\lesssim&C'(\sigma^1)\left(
        \vert\nabla^{n+1}\sigma^2\vert_{{\infty}}
        \vert u\vert_{H^{s+m_1-n-1}}+
        \vert \nabla^{n+1}\sigma^2\vert_{H^{s+m_1-n-1}}
        \vert u\vert_\infty\right).
        \end{aligned}$$ For a similar generalization of (\[intro3\]), see Th. \[theoIII1bis\].

-   We allow $\sigma^2$ to be a pseudo-differential operator and not only a function (Ths. \[theoIII1\], \[theoIII2\], \[theoIII1bis\], \[theoIV1\] and \[theoIV1bis\], and Corollaries \[coroIII3\] and \[coroIV1\]);

-   We give an alternative to (\[intro2\]) allowing the cases $m_1\leq 0$ and $s<0$ (Ths. \[theoIII1\] and \[theoIV1\] and Corollaries \[coroIII3\] and \[coroIV1\]); similarly, we show that negative values of 