the remainder operator $R$ maps ${\mathcal}{C}^{s_1}\times{\mathcal}{C}^{s_2}$ into ${\mathcal}{C}^{s_1+s_2}$ continuously.

Combining the above proposition with Bony’s decomposition , we easily get the following “tame estimate”:

\[c:op\] Let $u$ be a bounded function such that $\nabla u\in {\mathcal}{C}^{s-1}$ for some $s>0$.

Then for any $v\in{\mathcal}{C}^s$ we have $u\,v\,\in\,{\mathcal}{C}^s$ and there exists a constant $C$, depending only on $N$ and $s$, such that $$\|u\,v\|_{{\mathcal}{C}^s}\,\leq\, C\Bigl(\|u\|_{L^\infty}\,\|v\|_{{\mathcal}{C}^s}\,+\,\|v\|_{L^\infty}\,\|\nabla u\|_{{\mathcal}{C}^{s-1}}\Bigr)\,.$$

In our computations we will often have to handle compositions between a paraproduct operator and a Fourier multiplier. The following lemma (see the proof e.g. in [@D1997]) provides us with estimates for the commutator operator.

\[l:comm\] Let $m\in{\mathbb{R}}$, $R>0$ and $f\in{\mathcal}{C}^\infty({\mathbb{R}}^N)$ be a homogeneous smooth function of degree $m$ out of the ball $B(0,R)$.

Then, there exists a constant $C$, depending only on $R$, such that, for all $s\in{\mathbb{R}}$ and all $\sigma<1$, one has: $$\label{est:comm}
 \left\|\left[T_u,f(D)\right]v\right\|_{{\mathcal}{C}^{s-m+\sigma}}\,\leq\,
\frac{C}{1-\sigma}\,\|\nabla u\|_{{\mathcal}{C}^{\sigma-1}}\,\|v\|_{{\mathcal}{C}^s}\,.$$

Let us now quote another result (see [@D2010] for the proof of the former part, [@D1997] for the proof of the latter), pertaining to the composition of functions in Besov spaces, which will be of great importance in the sequel.

\[p:comp\]

-   Let $I$ be an open interval of ${\mathbb{R}}$ and $F:I\rightarrow{\mathbb{R}}$ a smooth function.

    Then for all compact 