${\mathcal}{C}^s$ as the Besov space $B^s_{\infty,\infty}$.

Finally, let us also point out that for any $k\in{\mathbb{N}}$ and $p\in[1,+\infty]$, we have the following chain of continuous embeddings: $$B^k_{p,1}\hookrightarrow W^{k,p}\hookrightarrow B^k_{p,\infty}\,,$$ where $W^{k,p}$ denotes the set of $L^p$ functions with derivatives up to the order $k$ in $L^p$.

Besov spaces have many nice properties which will be recalled throughout the paper whenever they are needed. For the time being, let us just mention that if the conditions $$s\,>\,1\,+\,\frac{N}{p}\qquad\qquad\mbox{ or }\qquad\qquad s\,=\,1\,+\,\frac{N}{p}\quad\mbox{and}\quad r=1$$ holds true, then $B^s_{p,r}$ is an algebra continuously embedded in the set ${\mathcal}{C}^{0,1}$ of bounded Lipschitz functions, and that the gradient operator maps $B^s_{p,r}$ in $B^{s-1}_{p,r}$.

The following result will be also needed.

\[p:CZ\] Let $F:{\mathbb{R}}^N\rightarrow{\mathbb{R}}$ be a smooth homogeneous function of degree $m$ away from a neighborhood of the origin.

Then for all $(p,r)\in[1,+\infty]^2$ and all $s\in{\mathbb{R}}$, the operator $F(D)$ maps $B^s_{p,r}$ in $B^{s-m}_{p,r}$.

The following fundamental lemma describes, by the so-called *Bernstein’s inequalities*, the way derivatives act on spectrally localized functions.

\[l:bern\] Let $0<r<R$. A constant $C$ exists so that, for any nonnegative integer $k$, any couple $(p,q)$ in $[1,+\infty]^2$ with $1\leq p\leq q$ and any function $u\in L^p$, we have, for all $\lambda>0$, $$\displaylines{
{\rm supp}\, \widehat 