$C^{-1}\|u\|_{B^{s}_{p,r}}\leq\|{\nabla}u\|_{B^{s-1}_{p,r}}\leq
    C\|u\|_{B^{s}_{p,r}}.$

2.  If $p_{1}<p_{2}$ and $r_{1}\leq r_{2}$ then $B^{s}_{p_{1},r_{1}}\hookrightarrow
    B^{s-N(1/p_{1}-1/p_{2})}_{p_{2},r_{2}}$.

3.  $B^{s^{'}}_{p,r_{1}}\hookrightarrow B^{s}_{p,r}$ if $s^{'}> s$ or if $s=s^{'}$ and $r_{1}\leq r$.

\[interpolation\]

Let now recall a few product laws in Besov spaces coming directly from the paradifferential calculus of J-M. Bony (see [@BJM]) and rewrite on a generalized form in [@AP] by H. Abidi and M. Paicu (in this article the results are written in the case of homogeneous sapces but it can easily generalize for the nonhomogeneous Besov spaces).

\[produit1\] We have the following laws of product:

-   For all $s\in{\mathbb{R}}$, $(p,r)\in[1,+\infty]^{2}$ we have: $$\|uv\|_{B^{s}_{p,r}}\leq
    C(\|u\|_{L^{\infty}}\|v\|_{B^{s}_{p,r}}+\|v\|_{L^{\infty}}\|u\|_{B^{s}_{p,r}})\,.
    \label{2.2}$$

-   Let $(p,p_{1},p_{2},r,\lambda_{1},\lambda_{2})\in[1,+\infty]^{2}$ such that:$\frac{1}{p}\leq\frac{1}{p_{1}}+\frac{1}{p_{2}}$, $p_{1}\leq\lambda_{2}$, $p_{2}\leq\lambda_{1}$, $\frac{1}{p}\leq\frac{1}{p_{1}}+\frac{1}{\lambda_{1}}$ and $\frac{1}{p}\leq\frac{1}{p_{2}}+\frac{1}{\lambda_{2}}$. We have then the following inequalities:\
    if $s_{1}+s_{2}+N\inf(0,1-\frac{1}{p_{1}}-\frac{1}{p_{2}})>0$, $s_{1}+\frac{N}{\lambda_{2}}<\frac{N}{p_{1}}$ and $s_{2}+\frac{N}{\lambda_{1}}<\frac{N}{p_{2}}$ then: $$\|uv\|_{B^{s_{1}+s_{2}-N(\frac{1}{p_{1}}+\frac{1}{p_{2}}-\frac{1}{p})}_{p,r}}\lesssim\|u\|_{B^{s_{1}}_{p_{1},r}}
    \|v\|_{B^{s_{2}}_{p_{2},\infty}},
    \label{2.3}$$ when $s_{1}+\frac{N}{\lambda_{2}}=\frac{N}{p_{1}}$ (resp $s_{2}+\frac{N}{\lambda_{1}}=\frac{N}{p_{2}}$) we replace $\|u\|_{B^{s_{1}}_{p_{1},r}}\|v\|_{B^{s_{2}}_{p_{2},\infty}}$ (resp $\|v\|_{B^{s_{2}}_{p_{2},\infty}}$) by $\|u\|_{B^{s_{1}}_{p_{1},1}}\|v\|_{B^{s_{2}}_{p_{2},r}}$ (resp $\|v\|_{B^{s_{2}}_{p_{2},\infty}\cap L^{\infty}}$), if $s_{1}+\frac{N}{\lambda_{2}}=\frac{N}{p_{1}}$ and $s_{2}+\frac{N}{\lambda_{1}}=\frac{N}{p_{2}}$ we take $r=1$.\
    If $s_{1}+s_{2}=0$, $s_{1}\in(\frac{N}{\lambda_{1}}-\frac{N}{p_{2}},\frac{N}{p_{1}}-\frac{N}{\lambda_{2}}]$ and $\frac{1}{p_{1}}+\frac{1}{p_{2}}\leq 1$ then: $$\|uv\|_{B^{-N(\frac{1}{p_{1}}+\frac{1}{p_{2}}-\frac{1}{p})}_{p,\infty}}\lesssim\|u\|_{B^{s_{1}}_{p_{1},1}}
    \|v\|_{B^{s_{2}}_{p_{2},\infty}}.
    \label{2.4}$$ If $|s|<{\frac{N}{p}}$ for $p\geq2$ and $-\frac{N}{p^{'}}<s<{\frac{N}{p}}$ else, we have: $$\|uv\|_{B^{s}_{p,r}}\leq C\|u\|_{B^{s}_{p,r}}\|v\|_{B^{{\frac{N}{p}}}_{p,\infty}\cap L^{\infty}}.
    \label{2.5}$$

In the sequel $p$ will be either $p_{1}$ or $p_{2}$ and in this case $\frac{1}{\lambda}=\frac{1}{p_{1}}-\frac{1}{p_{2}}$ if $p_{1}\leq p_{2}$, resp $\frac{1}{\lambda}=\frac{1}{p_{2}}-\frac{1}{p_{1}}$ if $p_{2}\leq p_{1}$.

\[produit2\] Let $r\in [1,+\infty]$, $1\leq p\leq p_{1}\leq +\infty$ 