Then $C^r_*({{\mathbb R}}^d)$ is the set of all $u\in {\mathcal S}'({{\mathbb R}}^d)$ such that $$\vert u\vert_{C^r_*}:=
    \sup_{p\geq -1}2^{pr}\vert \varphi_p(D)u\vert_\infty<\infty.$$

We recall the continuous embeddings $H^s({{\mathbb R}}^d)\subset C_*^{s-d/2}({{\mathbb R}}^d)$, for all $s\in{{\mathbb R}}$, and $L^\infty({{\mathbb R}}^d)\subset C^0_*({{\mathbb R}}^d)$.

We now introduce admissible cut-off functions, which play an important role in paradifferential theory ([@Bony; @Meyer] and Appendix B of [@Metivier-Zumbrun]).

\[defiLP0\] A smooth function $\chi(\eta,\xi)$ defined on ${{\mathbb R}}^d\times{{\mathbb R}}^d$ is an admissible cut-off function if and only if:

-   There are $\delta_1$ and $\delta_2$ such that $0<\delta_1<\delta_2<1$ and $$\begin{aligned}
        \forall \vert\xi\vert\geq 1/2,\qquad
        \chi(\eta,\xi)&=&1 \quad\mbox{ for }\quad \vert\eta\vert\leq 
        \delta_1\vert \xi\vert,\\
        \forall \vert\xi\vert\geq 1/2,\qquad
        \chi(\eta,\xi)&=&0 \quad\mbox{ for }\quad \vert\eta\vert\geq 
        \delta_2\vert\xi\vert;
        \end{aligned}$$

-   For all $\alpha,\beta\in{{\mathbb N}}^d$, there is a constant $C_{\alpha,\beta}$ such that $$\label{estadm}
        \forall (\eta,\xi)\in {{\mathbb R}}^{2d},
        \quad\vert\xi\vert\geq 1/2,\qquad
        \left\vert\partial_\xi^\alpha\partial_\eta^\beta 
        \chi(\eta,\xi)\right\vert\leq 
        C_{\alpha,\beta}{\langle\xi\rangle}^{-\vert\alpha\vert-\vert\beta\vert}.$$

A useful admissible cut-off function is given, for all $N\geq 2$, by $$\label{LP7}
    \chi(\eta,\xi)=\sum_{p\geq -1}\psi(2^{-p+N}\eta)\varphi_p(\xi);$$ One can check that it satisfies indeed the conditions of Definition \[defiLP0\] with $\delta_1=2^{-N-2}$ and $\delta_2=2^{1-N}$.

An important property satisfied by admissible cut-off functions is that ${\check}\chi(\cdot,\xi)$ and its derivatives with respect to $\xi$ enjoy good estimates in $L^1$-norm. The next lemma is a simple consequence of the estimates imposed in Definition \[defiLP0\]; we refer 