given in (\[A6\]), we shall also need another kind of decomposition, namely, Coifman and Meyer’s decomposition into elementary symbols. The proof of the next proposition is a quite close adaptation of the proof of Prop. II.5 of [@Coifman-Meyer]; it is given in Appendix \[appCM\].

\[propA2\] Let $m\in{{\mathbb R}}$ and $s_0>d/2$, and let $\sigma\in \Gamma_{s_0}^m$. With $\psi\in C^\infty_0({{\mathbb R}}^d)$ as given by (\[LP1\]), one has $$(1-\psi(\xi))\sigma(x,\xi)=
    \sum_{k\in{{\mathbb Z}}^d}\frac{1}{(1+\vert k\vert^2)^{[\frac{d}{2}]+1}}
    p_k(x,\xi){\langle\xi\rangle}^m,$$ with ${\displaystyle}p_k(x,\xi)=\sum_{q\geq -1}c_{k,q}(x)\lambda_k(2^{-q}\xi)$, and where:\
[**i.**]{} The coefficients $c_{k,q}(\cdot)$ are in $H^{s_0}({{\mathbb R}}^d)$ and for all $s\leq s_0$, one has $$\vert c_{k,q}\vert_{H^s}\leq {\mbox{\textnormal{Cst }}}N^m_{{{2[\frac{d}{2}]+2}},s}(\sigma);$$ moreover, for all $p\geq -1$, one has ${\displaystyle}\vert \varphi_p(D)c_{k,q}\vert_{H^s}
    \leq{\mbox{\textnormal{Cst }}}N^m_{{{2[\frac{d}{2}]+2}},s}(\sigma_{(p)})$, where the symbol $\sigma_{(p)}$ is defined as $\sigma_{(p)}(\cdot,\xi)=\varphi_p(D_x)\sigma(\cdot,\xi)$.\
[**ii.** ]{} For all $k\in {{\mathbb Z}}^d$, the functions $\lambda_k(\cdot)$ are smooth and supported in $2/5\leq \vert \xi\vert\leq 12/5$. Moreover, ${\displaystyle}\left\vert{\check}\lambda_k\right\vert_{L^1}$ is bounded from above uniformly in $k\in{{\mathbb Z}}^d$.

Operators {#sectoper}
=========

To any symbol $\sigma(x,\xi)\in C^0({{\mathbb R}}^d\times{{\mathbb R}}^d\backslash\{0\})$, one can associate an operator $\sigma(x,D)={\mbox{\textnormal{Op}}}(\sigma)$ acting on functions whose Fourier transform is smooth and compactly supported in ${{\mathbb R}}^d\backslash\{0\}$: $$\forall x\in {{\mathbb R}}^d,\qquad
    {\mbox{\textnormal{Op}}}(\sigma)u(x)=(2\pi)^{-d}\int_{{{\mathbb R}}^d}
    e^{ix\cdot\xi}\sigma(x,\xi)\widehat{u}(\xi)d\xi.$$ The aim of this section is to study ${\mbox{\textnormal{Op}}}(\sigma)$ when $\sigma\in \Gamma^m_s$. In order to do so,we study successively ${\mbox{\textnormal{Op}}}({\sigma_{lf}})$, ${\mbox{\textnormal{Op}}}({\sigma_I})$, ${\mbox{\textnormal{Op}}}({\sigma_{II}})$ 