with pseudo-differential operators with nonregular symbols, the focus is generally on the continuity of such operators on Sobolev or Zygmund spaces (see for instance [@TaylorM0; @TaylorM3]) and not on the derivation of precise (and tame) estimates. In [@Grenier], E. Grenier gave some description of the constants $\underline{C}(\sigma^1)$ in (\[intro1\])-(\[intro2\]) but his results, though sufficient for his purposes, are far from optimal. In this article, we aim at proving more precise versions of (\[intro1\]), (\[intro2\]) and (\[intro3\]), and we also give some extensions of these results. Let us describe roughly some of them:\
[**Action of pseudo-differential operators on Sobolev spaces**]{} (see Corollary \[coroII1\]). Take a symbol $\sigma\in S^m_{1,0}$ of the form $\sigma(x,\xi)=\Sigma(v(x),\xi)$, with $\Sigma$ as described above and $v\in H^\infty$. Then Moser’s tame product estimate can be generalized to pseudo-differential operators of order $m>0$: for all $s>0$, $$\vert {\mbox{\textnormal{Op}}}(\sigma)u\vert_{H^s}
    \lesssim C(\vert v\vert_{\infty})
    (\vert v\vert_{H^{s+m}}\vert u\vert_\infty+\vert u\vert_{H^{s+m}}).$$ Another estimate which does not assume any restriction on the order $m$ and also holds for negative values of $s$ is the following: for all $t_0>d/2$, one has $$\begin{aligned}
    \forall -t_0<s < t_0,&\quad&
    \vert {\mbox{\textnormal{Op}}}(\sigma)u\vert_{H^s}\lesssim C(\vert v\vert_{\infty})
    \vert v\vert_{H^{t_0}}\vert u\vert_{H^{s+m}},\\
    \forall t_0\leq s, &\quad&
    \vert {\mbox{\textnormal{Op}}}(\sigma)u\vert_{H^s}
    \lesssim C(\vert v\vert_{\infty})
    (\vert v\vert_{H^{s}}\vert u\vert_{H^{m+t_0}}
    +\vert u\vert_{H^{s+m}}).\end{aligned}$$ [**Commutator 