subset $J\subset I$ and all $s>0$, there exists a constant $C$ such that, for all function $u$ valued in $J$ and with gradient in ${\mathcal}{C}^{s-1}$, we have $\nabla(F\circ u)\in {\mathcal}{C}^{s-1}$ and $$\|\nabla(F\circ u)\|_{{\mathcal}{C}^{s-1}}\,\leq\,C\,\|\nabla u\|_{{\mathcal}{C}^{s-1}}\,.$$

-   Let $s>0$ and $m\in{\mathbb{N}}$ be such that $m>s$. Let $u\in {\mathcal}{C}^s$ and $\psi\in{\mathcal}{C}^m_b$ such that the Jacobian of $\psi^{-1}$ is bounded.

    Then $u\circ\psi\,\in\,{\mathcal}{C}^s$. Moreover, if $s\in\,]0,1[\,$ the following estimate holds: $$\left\|u\circ\psi\right\|_{{\mathcal}{C}^s}\,\leq\,C\,\left(1+\left\|\nabla\psi\right\|_{L^\infty}\right)\,\|u\|_{{\mathcal}{C}^s}\,.$$

Finally, let us introduce the notion of paravector-field.

\[d:pvec-f\] Let $X$ be a vector-field with coefficients in ${\mathcal}{S}'$. We can formally define the paravector-field operator $T_X$ in the following way: for all $u\in{\mathcal}{S}'$, $$T_Xu\,:=\,\sum_{i=1}^N\,T_{X^i}\d_iu\,.$$

The following result (see [@D1999], section 2 for the proof) says that the paravector-field operator is, in a certain sense, the principal part of the derivation $\d_X$.

\[l:T\_X\] For all vector field $X\in{\mathcal}{C}^s$ and all $u\in{\mathcal}{C}^t$, we have:

-   if $\,t<1$ and $s+t>1$, then $$\left\|\d_Xu\,-\,T_Xu\right\|_{{\mathcal}{C}^{s+t-1}}\,\leq\,\frac{C}{(1-t)\,(s+t-1)}\,\|X\|_{{\mathcal}{C}^s}\,\|\nabla u\|_{{\mathcal}{C}^{t-1}}\,;$$

-   if $\,t<0$, $s<1$ and $s+t>0$, then $$\left\|T_Xu\,-\,{{\rm div}\,}\left(u\,X\right)\right\|_{{\mathcal}{C}^{s+t-1}}\,\leq\,\frac{C}{t\,(s+t)\,(s-1)}\,\|X\|_{{\mathcal}{C}^s}\,\|u\|_{{\mathcal}{C}^t}\,;$$

-   if $\,t<1$ and $s+t>0$, then $$\left\|\d_Xu\,-\,T_Xu\right\|_{{\mathcal}{C}^{s+t-1}}\,\leq\,\frac{C}{(s+t)\,(1-t)}\,{\widetilde}{\|}X\|_{{\mathcal}{C}^s}\,\|\nabla u\|_{{\mathcal}{C}^{t-1}}\,.$$

Moreover, first and last inequalities are still true even in the case $\,t=1$, provided that one replaces $\|\nabla u\|_{{\mathcal}{C}^0_*}$ with $\|\nabla u\|_{L^\infty}$, while the second is still true even if $\,t=0$, with $\|u\|_{L^\infty}$ instead of $\|u\|_{{\mathcal}{C}^0_*}$.

We 