with ${\displaystyle}{\lim_{J \to +\infty} \limsup_{n \to +\infty} \sup_{t' < t} | \gamma_{n}^{J}(t')|  = 0}$. ]{}*

For the convenience of the reader, we recall the usual definition of Besov spaces. We refer the reader to [@BCD], from page $63$, for a detailed presentation of the theory and analysis of homogeneous Besov spaces.

*[ Let $s$ be in $\R$, $(p,r)$ in $[1,+\infty]^2$ and $u$ in $\mathcal{S'}$. A tempered distribution $u$ is an element of the Besov space $\dot{B}^{s}_{p,r}$ if $u$ satifies ${\displaystyle}{\lim_{j \to \infty}\, || \dot{S}_{j} u ||_{L^\infty} = 0 }$ and $$\| u\|_{\dot{B}^{s}_{p,r}} \eqdefa \Bigl(\sum_{j \in \Z} 2^{jrs}\,\,|| \dot{\Delta}_{j} u ||^{r}_{L^p}\Bigr)^{\frac{1}{r}}  < \infty,$$ where $\dot{\Delta}_{j}$ is a frequencies localization operator (called Littlewood-Paley operator), defined by $$\dot{\Delta}_{j}u(\xi) \eqdefa \mathcal{F}^{-1}\bigl(\varphi(2^{-j}|\xi|)\widehat{u}(\xi)\bigr),$$ with $\varphi \in \mathcal{D}([\frac{1}{2},2])$, such that ${\displaystyle}{\sum_{j \in \Z} \varphi(2^{-j}t) = 1}$, for any $t>0$. ]{}*

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\[equivalence norm besov\] Notice that the characterization of Besov spaces with positive indices in terms of finite differences is equivalent to the above definition (cf [@BCD]). In the case where the regularity index is between $0$ and $1$, one has the following property. Let $s$ be in $]0,1[$ and $(p,r)$ in $[1,\infty]^2$. A constant $C$ exists such that, for any $u \in \mathcal{S{'}}$, $$C^{-1}\, \| u\|_{\dot{B}^{s}_{p,r}} 