$$\forall u\in H^{m+t}({{\mathbb R}}^d),\qquad
    \left\vert {\mbox{\textnormal{Op}}}(\sigma_R)u\right\vert_{H^{s+t-d/2}}
    \lesssim N^m_{{{2[\frac{d}{2}]+2}},s}(\sigma)\vert u\vert_{H^{m+t}}$$ and $$\forall u\in C_*^{m+t}({{\mathbb R}}^d),\qquad
    \left\vert {\mbox{\textnormal{Op}}}(\sigma_R)u\right\vert_{H^{s+t}}
    \lesssim N^m_{{{2[\frac{d}{2}]+2}},s}(\sigma)\vert u\vert_{C_*^{m+t}}.$$

\[remagrad1\] For the same reasons as in Remark \[remagrad\], 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, provided that one replaces ${\sigma_R}$ by $\sigma_{R,1}$, where $\sigma_{R,1}$ is defined in (\[decompreste\]).

We only prove the first of the two estimates given in the proposition. The second one is both easier and contained in Th. B of [@Yamazaki1]. The proof we present below is an adaptation of the corresponding result which gives control of the residual term in paraproduct theory (e.g. Th. 2.4.1. of [@Chemin]).\
Using the expression of ${\sigma_R}$ given in (\[A10\]) and a Littlewood-Paley decomposition, one can write $${\mbox{\textnormal{Op}}}({\sigma_R})u=\sum_{r\geq -1}\varphi_r(D){\mbox{\textnormal{Op}}}({\sigma_R})u
    =\sum_{r\geq -1}\varphi_r(D)\sum_{p\geq -1}R_p(\sigma)u,$$ where ${\displaystyle}R_p(\sigma)u:=\sum_{\vert p-q\vert\leq N}
\sigma_{(q)}(x,D)(1-\psi(D))\varphi_p(D)u$, and with $\sigma_{(q)}(\cdot,\xi)=\varphi_q(D)\sigma(\cdot,\xi)$.\
Since ${\mbox{\textnormal{Sp }}}R_p(\sigma)u$ is included in $\vert\xi\vert\leq (1+2^N)2^{p+1}$, there exists $n_0\in{{\mathbb N}}$ such that $\varphi_r(D)R_p(\sigma)u=0$ whenever $r>p+n_0$. Thus, one has in fact $${\mbox{\textnormal{Op}}}({\sigma_R})u=\sum_{r\geq -1}\varphi_r(D)\sum_{p\geq r-n_0}R_p(\sigma)u,$$ and the proposition follows from Lemma \[lemmLP1\] and the estimate $$\label{II2}
    \Big(\sum_{r\geq -1}2^{2r(s+t-d/2)}
    \Big\vert \varphi_r(D)\sum_{p\geq r-n_0}
    R_p(\sigma)u\Big\vert_2^2\Big)^{1/2}
    \lesssim N^m_{{{2[\frac{d}{2}]+2}},s}(\sigma)\vert u\vert_{m+t}.$$ The end of the proof is thus devoted to establishing (\[II2\]).\
Using Prop. \[propA2\] –and with the 