[**iii.**]{} If $f\in F$ and $g\in G$, $F$ and $G$ being two Banach spaces, the notation $\vert f\vert_{F}\lesssim \vert g\vert_{G}$ means that $\vert f\vert_{F}\leq C\vert g\vert_{G}$ for some constant $C$ which does not depend on $f$ nor $g$.\
[**iv.**]{} Here, ${{\mathcal S}({{\mathbb R}}^d)}$ denotes the Schwartz space of rapidly decaying functions, and for any distribution $f\in {\mathcal S}'({{\mathbb R}}^d)$, we write respectively $\widehat{f}$ and $\check{f}$ its Fourier and inverse Fourier transform.\
[**v.**]{} We use the classical notation $f(D)$ to denote the Fourier multiplier, namely, $\widehat{f(D)u}(\cdot)=f(\cdot)\widehat{u}(\cdot)$.

Brief reminder of Littlewood-Paley theory
-----------------------------------------

We recall in this section basic facts in Littlewood-Paley theory.

Throughout this article, $\psi\in C_0^\infty({{\mathbb R}}^d)$ denotes a smooth bump function such that $$\label{LP1}
    \psi(\xi)=1\quad\mbox{ if } \quad \vert\xi\vert\leq1/2 
    \quad\mbox{ and } \quad \psi(\xi)=0
    \quad\mbox{ if } \quad \vert\xi\vert\geq 1,$$ and we define $\varphi\in C_0^\infty({{\mathbb R}}^d)$ as $$\label{LP2}
    \varphi(\xi)=\psi(\xi/2)-\psi(\xi),\qquad\forall\xi\in{{\mathbb R}}^d,$$ so that $\varphi$ is supported in the annulus $1/2\leq \vert\xi\vert\leq 2$, and one has $$\label{LP3} 
    1=\psi(\xi)+\sum_{p\geq 0}\varphi(2^{-p}\xi),\qquad\forall \xi\in{{\mathbb R}}^d.$$ For all $p\in{{\mathbb Z}}$, we introduce the functions $\varphi_p$, supported in $2^{p-1}\leq\vert\xi\vert\leq 2^{p+1}$, and defined as $$\label{LP4}
    \varphi_p=0\quad\mbox{if} \quad p<-1,\qquad
    \varphi_{-1}=\psi,\qquad
    \varphi_p(\cdot)=\varphi(2^{-p}\cdot)
    \quad\mbox{if} \quad p\geq 0.$$ This allows us to give the classical definition of Zygmund spaces:

Let $r\in{{\mathbb R}}$. 