(\[A8\]), then ${\mbox{\textnormal{Op}}}({\sigma_I})$ extends as a continuous mapping on $H^{s+m}({{\mathbb R}}^d)$ with values in $H^s({{\mathbb R}}^d)$, for all $s\in{{\mathbb R}}$. Moreover, $$\forall s\in{{\mathbb R}},\quad\forall u\in H^{s+m}({{\mathbb R}}^d),\qquad
    \left\vert {\mbox{\textnormal{Op}}}({\sigma_I})u\right\vert_{H^s}\lesssim
    M_d^m(\sigma)\vert u\vert_{H^{s+m}},$$ where $M_d^m(\sigma)$ is defined in (\[A2bis\]).

Let us first prove the following lemma, which deals with the action of operators whose symbol satisfies the spectral property (\[A3\]).

\[lemmII1\] Let $m\in{{\mathbb R}}$ and $\sigma(x,\xi)\in  
    C^\infty({{\mathbb R}}^d\times {{\mathbb R}}^d\backslash\{0\})$ be such that $M^m_d(\sigma)<\infty$, where $M^m_d(\cdot)$ is as defined in (\[A2bis\]).\
If moreover $\sigma(x,\xi)$ vanishes for $\vert\xi\vert\leq 1/2$ and satisfies the spectral condition (\[A3\]) then ${\mbox{\textnormal{Op}}}\left(\sigma\right)$ extends as a continuous mapping on $H^{s+m}({{\mathbb R}}^d)$ with values in $H^s({{\mathbb R}}^d)$ for all $s\in{{\mathbb R}}$ and $$\forall u\in H^{s+m}({{\mathbb R}}^d),\qquad
    \left\vert {\mbox{\textnormal{Op}}}\left(\sigma\right)u\right\vert_{H^s}\lesssim
    \sup_{\vert\xi\vert\geq 1/2}\sup_{\vert\beta\vert\leq d}
    \Big({\langle\xi\rangle}^{\vert\beta\vert-m}
    \vert \partial_\xi^\beta\sigma(\cdot,\xi)\vert_\infty\Big) 
    \vert u\vert_{H^{s+m}}.$$

Using (\[LP3\]), we write ${\displaystyle}\sigma(x,\xi)=
\sum_{p\geq -1}\sigma_p(x,\xi)$, with $\sigma_p(x,\xi)=\varphi_p(\xi)\sigma(x,\xi)$. For all $u\in{{\mathcal S}({{\mathbb R}}^d)}$, (\[LP3\]) and (\[LP4\]) yield $$\label{B1}
    {\mbox{\textnormal{Op}}}(\sigma)u=\sum_{p\geq -1}{\mbox{\textnormal{Op}}}(\sigma_p)
    \sum_{\vert p-q\vert\leq 1}\varphi_q(D)u.$$ Let us now define $\widetilde{\sigma_p}(x,\xi):=\sigma_p(2^{-p}x,2^p\xi)$ for all $p\in{{\mathbb N}}$. One obviously has (see e.g. Lemma II.1 of [@Coifman-Meyer]) $\Vert {\mbox{\textnormal{Op}}}(\sigma_p)\Vert_{L^2\to L^2}
=\Vert {\mbox{\textnormal{Op}}}(\widetilde{\sigma_p})\Vert_{L^2\to L^2}$. Moreover, Hwang proved in [@Hwang] that $$\Vert {\mbox{\textnormal{Op}}}(\widetilde{\sigma_p})\Vert_{L^2\to L^2}\leq 
    {\mbox{\textnormal{Cst }}}\sum_{\alpha,\beta\in\{0,1\}^d}
    \left\vert\partial_x^\alpha\partial_\xi^\beta 
    \widetilde{\sigma_p}\right\vert_{L^\infty({{\mathbb R}}^d\times{{\mathbb R}}^d)};$$ it follows therefore from (\[A5\]) that $$\label{B2}
    \Vert {\mbox{\textnormal{Op}}}({\sigma_p})\Vert_{L^2\to L^2}
    \leq 