When $\sigma(x,\xi)=\sigma(x)$ does not depend on $\xi$, one has ${\mbox{\textnormal{Op}}}({\sigma_{lf}})u=\sigma\psi(D)u$, ${\mbox{\textnormal{Op}}}({\sigma_I})u=T_\sigma\widetilde{u}$, ${\mbox{\textnormal{Op}}}({\sigma_{II}})u=T_{\widetilde{u}}\sigma$ and ${\mbox{\textnormal{Op}}}({\sigma_R})u=R(\sigma,\widetilde{u})$, with $\widetilde{u}:=(1-\psi(D))u$ and where $T_f$ denotes the usual paraproduct operator associated to $f$ and $R(f,g)=fg-T_f g-T_gf$ (see [@Bony; @Meyer; @Chemin]).\
[**iii.**]{} In [@Yamazaki1], Yamazaki used a similar decomposition of symbols into three components. We need a fourth one here, namely ${\sigma_{lf}}$, in order to take into account symbols which are not infinitely smooth with respect to $\xi$ at the origin. A fifth component is also introduced in (\[decompreste\]); it is used in the proof of the second parts of Ths. \[theoIII1\]-\[theoIII1bis\].

In the next proposition, we check that ${\sigma_I}$ belongs to the class of paradifferential symbols $\Sigma_s^m$.

\[propA1\] Let $m\in{{\mathbb R}}$, $s_0>d/2$, and let $\sigma\in\Gamma_{s_0}^m$.\
Then, the symbol ${\sigma_I}$ defined in (\[A8\]) belongs to $\Sigma_{s_0}^m$ and, for all $k\in{{\mathbb N}}$ and $s\leq s_0$, $$N^m_{k,s}({\sigma_I})\leq {\mbox{\textnormal{Cst }}}N_{k,s}^m(\sigma)
    \quad\mbox{ and }\quad
    M_k^m(\sigma_I)\leq {\mbox{\textnormal{Cst }}}M_k^m(\sigma).$$

One can write ${\sigma_I}(\cdot,\xi)
=(1-\psi(\xi)){\check}\chi(\cdot,\xi)*\sigma(\cdot,\xi)$, where $\chi(\eta,\xi)$ denotes the admissible cut-off function constructed in (\[LP7\]). The spectral property (\[A3\]) is thus obviously satisfied by ${\sigma_I}$ and the result follows therefore from simple convolution estimates, together with the bounds on the $L^1$-norm on the derivatives $\partial_\xi^\alpha {\check}\chi(\cdot,\xi)$ given in Lemma \[lemmLP0\].

Together with the decomposition 