are defined by[^2] $$\Delta_j:=0\ \hbox{ if }\ j\leq-2,\quad\Delta_{-1}:=\chi(D)\quad\hbox{and}\quad
\Delta_j:=\varphi(2^{-j}D)\ \text{ if }\  j\geq0.$$ We also introduce the following low frequency cut-off operator: $$S_ju:=\chi(2^{-j}D)=\sum_{k\leq j-1}\Delta_{k}\quad\text{for}\quad j\geq0.$$ The following classical properties will be used freely throughout the paper:

-   for any $u\in{\mathcal}{S}',$ the equality $u=\sum_{j}\Delta_ju$ holds true in ${\mathcal}{S}'$;

-   for all $u$ and $v$ in ${\mathcal}{S}'$, the sequence $\left(S_{j-1}u\,\,\Delta_jv\right)_{j\in{\mathbb{N}}}$ is spectrally supported in dyadic annuli.

One can now define what a (non-homogeneous) Besov space $B^s_{p,r}$ is.

\[d:besov\] Let $u$ be a tempered distribution, $s$ a real number, and $1\leq p,r\leq+\infty.$ We define the space $B^s_{p,r}$ as the set of distributions $u\in{\mathcal}{S}'$ such that $$\|u\|_{B^s_{p,r}}:=\bigg\|\bigl(2^{js}\,
\|\Delta_j  u\|_{L^p}\bigr)_{j\geq-1}\bigg\|_{\ell^r}\;<\;+\infty\,.$$

From the above definition, it is easy to show that for all $s\in{\mathbb{R}}$, the Besov space $B^s_{2,2}$ coincides with the non-homogeneous Sobolev space $H^s$, while for all $s\in\,{\mathbb{R}}_+\!\!\setminus\!{\mathbb{N}}$, the space $B^s_{\infty,\infty}$ is actually the Hölder space ${\mathcal}{C}^s$.

If $s\in{\mathbb{N}}$, instead, we set ${\mathcal}{C}^s_*:=B^s_{\infty,\infty}$, to distinguish it from the space ${\mathcal}{C}^s$ of the differentiable functions with continuous partial derivatives up to the order $s$. Moreover, the strict inclusion ${\mathcal}{C}^s_b\,{\hookrightarrow}\,{\mathcal}{C}^s_*$ holds, where ${\mathcal}{C}^s_b$ denotes the subset of ${\mathcal}{C}^s$ functions bounded with all their derivatives up to the order $s$.

If $s<0$, we define the “negative Hölder space” 