is supported in $A\leq\vert\xi\vert\leq B$, there exists $n_0\in {{\mathbb N}}$ such that for all $\xi\in{{\mathbb R}}^d$ and $q\geq -1$, one has ${\displaystyle}\lambda(2^{-q}\xi)=\lambda(2^{-q}\xi)\sum_{\vert r-q\vert\leq n_0}
\varphi_r(\xi)$. Therefore, one can write $$\begin{aligned}
    \vert\lambda(2^{-q}D)u\vert_\infty&=&
    \Big\vert \lambda(2^{-q}D)\sum_{\vert r-q\vert\leq n_0}\varphi_r(D)
    u\Big\vert_\infty\\
    &\leq&\vert{\check}\lambda\vert_{L^1}\sum_{\vert r-q\vert\leq n_0}
    \vert\varphi_r(D)u\vert_\infty,\end{aligned}$$ so that the lemma follows from the very definition of Zygmund spaces.

In order to prove the first part of Prop. \[propII3\], take any $t>0$ and use the lemma to remark that (\[II1\]) yields $$\begin{aligned}
    \nonumber
    I&\leq& {\mbox{\textnormal{Cst }}}N^m_{{{2[\frac{d}{2}]+2}},0}(\sigma_{(p)})\sum_{q\geq -1}
    2^{-qt}\left\vert
    \psi(2^{-p+N}D)u\right\vert_{C^t_*}\\
    \label{bis}
    &\leq& {\mbox{\textnormal{Cst }}}N^m_{{{2[\frac{d}{2}]+2}},0}(\sigma_{(p)})
    \left\vert u\right\vert_{C_*^{t}},\end{aligned}$$ since $\sum_{q\geq -1}2^{-qt}<\infty$. From Lemma \[lemmLP1\] and the definition of $N^m_{{{2[\frac{d}{2}]+2}},0}(\cdot)$, it is obvious that $$\label{truc}
    \sum_{p\geq -1}2^{2ps}N^m_{{{2[\frac{d}{2}]+2}},0}(\sigma_{(p)})^2\leq {\mbox{\textnormal{Cst }}}N^m_{{{2[\frac{d}{2}]+2}},s}(\sigma)^2,$$ so that (\[bis\]) and Lemma \[lemmLP1\] give the result.\
To prove the second part of the proposition, proceed as above to obtain $$\begin{aligned}
    I&\leq& {\mbox{\textnormal{Cst }}}N^m_{{{2[\frac{d}{2}]+2}},0}(\sigma_{(p)})\left(\sum_{q=-1}^{p-N+1}2^{qt}
    \right)\left\vert u\right\vert_{C_*^{-t}},\\
    &\leq&  {\mbox{\textnormal{Cst }}}N^m_{{{2[\frac{d}{2}]+2}},0}(\sigma_{(p)})2^{pt}
    \left\vert u\right\vert_{C_*^{-t}};\end{aligned}$$ the end of the proof is done as for the first part of the proposition.

We finally turn to study ${\mbox{\textnormal{Op}}}({\sigma_R})$:

\[propII4\] Let $m\in{{\mathbb R}}$ and $s_0>d/2$, and let $\sigma\in\Gamma^m_{s_0}$.\
If ${\sigma_R}$ is as given in (\[A10\]) and if $s+t>0$ and $s\leq s_0$ then ${\mbox{\textnormal{Op}}}({\sigma_R})$ extends as a continuous operator on $H^{m+t}({{\mathbb R}}^d)$ with values in $H^{s+t-\frac{d}{2}}({{\mathbb R}}^d)$. Moreover, 