u \subset   B(0,\lambda R)
\;\Longrightarrow\;
\|\nabla^k u\|_{L^q} \leq
 C^{k+1}\lambda^{k+N\left(\frac{1}{p}-\frac{1}{q}\right)}\|u\|_{L^p}\,;\cr
{\rm supp}\, \widehat u \subset \{\xi\in{\mathbb{R}}^N\,/\, r\lambda\leq|\xi|\leq R\lambda\}
\;\Longrightarrow\; C^{-k-1}\lambda^k\|u\|_{L^p}
\leq\|\nabla^k u\|_{L^p}\leq C^{k+1}  \lambda^k\|u\|_{L^p}\,.
}$$

As an immediate consequence of the first Bernstein’s inequality, one gets the following embedding result.

\[c:embed\] The space $B^{s_1}_{p_1,r_1}$ is continuously embedded in the space $B^{s_2}_{p_2,r_2}$ for all indices satisfying $p_1\,\leq\,p_2$ and $$s_2\,<\,s_1-N\left(\frac{1}{p_1}-\frac{1}{p_2}\right)\qquad\hbox{or}\qquad
  s_2\,=\,s_1-N\left(\frac{1}{p_1}-\frac{1}{p_2}\right)\;\;\hbox{and }\;\;r_1\,\leq\,r_2\,.$$

From now on we will focus on the particular case of Hölder spaces.

Paradifferential calculus
-------------------------

Let us now introduce Bony’s decomposition of the product of two tempered distrubutions $u$ and $v$: we will define the paraproduct operator and recall a few nonlinear estimates in Hölder spaces. Constructing the paraproduct operator relies on the observation that, formally, the product $u\,v$, may be decomposed into $$\label{eq:bony}
u\,v\,=\,T_uv\,+\,T_vu\,+\,R(u,v)\,,$$ with $$T_uv\,:=\,\sum_jS_{j-1}u\,\Delta_jv\qquad\hbox{ and }\qquad
R(u,v)\,:=\,\sum_j\sum_{|k-j|\leq1}\Delta_ju\,\Delta_{k}v\,.$$ The above operator $T$ is called “paraproduct”, whereas $R$ is called “remainder”.

The paraproduct and remainder operators have many nice continuity properties. The following ones will be of constant use in this paper.

\[p:op\] For any $s\in{\mathbb{R}}$ and $t>0$, the paraproduct operator $T$ maps $L^\infty\times{\mathcal}{C}^s$ into ${\mathcal}{C}^s$ and ${\mathcal}{C}^{-t}\times{\mathcal}{C}^s$ into ${\mathcal}{C}^{s-t}$, and the following estimates hold: $$\|T_uv\|_{{\mathcal}{C}^s}\,\leq\,C\,\|u\|_{L^\infty}\,\|\nabla v\|_{{\mathcal}{C}^{s-1}}\qquad\hbox{and}\qquad
\|T_uv\|_{{\mathcal}{C}^{s-t}}\,\leq\,C\,\|u\|_{{\mathcal}{C}^{-t}}\,\|\nabla v\|_{{\mathcal}{C}^{s-1}}\,.$$

For any $s_1$ and $s_2$ in ${\mathbb{R}}$ such that $s_1+s_2>0$, 