the proof.

A first important consequence of Propositions \[propII3\] and \[propII4\] is that one can control the action of the operator associated to the ’remainder’ symbol $\sigma-{\sigma_{lf}}-{\sigma_I}$, which is more regular than the full operator if $\sigma(x,\xi)$ is smooth enough in the space variables.

\[propII5\] Let $m\in{{\mathbb R}}$, $s_0>d/2$ and $d/2<t_0\leq s_0$. If for some $r\geq 0$, one has $\sigma\in\Gamma^m_{s_0+r}$ then,\
[**i.**]{} For all $-t_0<s\leq s_0$, the following estimate holds: $$\forall u\in H^{m+t_0-r}({{\mathbb R}}^d),\qquad
    \vert {\mbox{\textnormal{Op}}}(\sigma-{\sigma_I}-{\sigma_{lf}})u\vert_{H^s}\lesssim 
    N_{{{2[\frac{d}{2}]+2}},s+r}^m(\sigma)
    \vert u\vert_{H^{m+t_0-r}}.$$ [**ii.**]{} For all $r'\in{{\mathbb R}}$ (such that $t_0+r'\leq s_0+r$) and $-t_0<s\leq t_0+r'$, one has $$\forall u\in H^{s+m-r'}({{\mathbb R}}^d),\qquad
    \vert {\mbox{\textnormal{Op}}}(\sigma-{\sigma_I}-{\sigma_{lf}})u\vert_{H^s}\lesssim 
    N_{{{2[\frac{d}{2}]+2}},t_0+r'}^m(\sigma)
    \vert u\vert_{H^{s+m-r'}}.$$ [**iii.**]{} For symbols of nonnegative order, i.e. when $m>0$, then for all $s>0$ such that $s+m\leq s_0$, one also has $$\forall u\in C_*^{-r}({{\mathbb R}}^d),\qquad
    \vert {\mbox{\textnormal{Op}}}(\sigma-{\sigma_I}-{\sigma_{lf}})u\vert_{H^s}\lesssim
    N^m_{{{2[\frac{d}{2}]+2}},s+m+r}(\sigma)\vert u\vert_{C^{-r}_*};$$ this estimate still holds for slightly negative values of $r$, namely, if $-m<r$.

One has $\sigma-{\sigma_I}-{\sigma_{lf}}={\sigma_{II}}+{\sigma_R}$, and we are therefore led to control $\vert {\mbox{\textnormal{Op}}}({\sigma_{II}})u\vert_{H^s}$ and $\vert {\mbox{\textnormal{Op}}}({\sigma_R})u\vert_{H^s}$. We first prove point [**i**]{}.\
The estimate on $\vert{\mbox{\textnormal{Op}}}({\sigma_{II}})u\vert_{H^s}$ is given by the first part of Prop. \[propII3\] when $r=0$. When $r>0$, taking $s=s+r$ and $t=r$ in the second part of this proposition gives the result. The estimate on $\vert{\mbox{\textnormal{Op}}}({\sigma_R})u\vert_{H^s}$ is given 