{\mbox{\textnormal{Cst }}}M^m_d(\sigma) 2^{pm}.$$ The result follows therefore from (\[B1\]), (\[B2\]) and Lemma \[lemmLP1\] if we can prove that for all $p\in{{\mathbb N}}$, ${\displaystyle}{\mbox{\textnormal{Op}}}(\sigma_p)\sum_{\vert p-q\vert\leq 1}\varphi_q(D)u$ has its spectrum supported in an annulus $A 2^{p-1}\leq \vert\eta\vert\leq B 2^{p+1}$ for some $A,B>0$. Since this is an easy consequence of the spectral property (\[A3\]), the proof of the lemma is complete.

The proof of the proposition is now very simple. One just has to apply Lemma \[lemmII1\] to ${\sigma_I}$, and to use Prop. \[propA1\].

The next proposition gives details on the action of ${\mbox{\textnormal{Op}}}({\sigma_{II}})$.

\[propII3\] Let $m\in{{\mathbb R}}$ and $s_0>d/2$, and let $\sigma\in \Gamma_{s_0}^m$.\
If ${\sigma_{II}}$ is defined as in (\[A9\]) then ${\mbox{\textnormal{Op}}}({\sigma_{II}})$ extends as an operator on any Sobolev space and one has, for all $s\leq s_0$ and $t>0$, $$\forall u\in C_*^{t+m},\qquad 
    \left\vert{\mbox{\textnormal{Op}}}({\sigma_{II}})u\right\vert_{H^s}\lesssim 
    N_{{{2[\frac{d}{2}]+2}},s}^m(\sigma)\vert u\vert_{C_*^{t+m}},$$ and $$\forall u\in C_*^{-t+m},\qquad 
    \left\vert{\mbox{\textnormal{Op}}}({\sigma_{II}})u\right\vert_{H^{s-t}}\lesssim 
    N_{{{2[\frac{d}{2}]+2}},s}^m(\sigma)\vert u\vert_{C_*^{-t+m}},$$ where $N^m_{{{2[\frac{d}{2}]+2}},s}(\sigma)$ is defined in (\[A1\]).

\[remagrad\] [**i.**]{} One can replace the quantity $N^m_{{{2[\frac{d}{2}]+2}},s}(\sigma)$ by $N^m_{{{2[\frac{d}{2}]+2}},s-k}(\nabla_x^k\sigma)$, $k\in{{\mathbb N}}$, in the estimates of the proposition. This follows from the fact that $\vert f\vert_{H^s}\leq {\mbox{\textnormal{Cst }}}\vert \nabla^k f\vert_{H^{s-k}}$, $k\in{{\mathbb N}}$, whenever $\widehat{f}$ vanishes in a neighborhood of the origin, and from the observation that one can replace $\sigma$ by $(1-\psi(D_x))\sigma$ in the 