with values in $X$. For $p\in[1,+\infty]$, the notation $L^{p}(0,T,X)$ or $L^{p}_{T}(X)$ stands for the set of measurable functions on $(0,T)$ with values in $X$ such that $t\rightarrow\|f(t)\|_{X}$ belongs to $L^{p}(0,T)$. Littlewood-Paley decomposition corresponds to a dyadic decomposition of the space in Fourier variables. We can use for instance any $\varphi\in C^{\infty}({\mathbb{R}}^{N})$, supported in ${\cal{C}}=\{\xi\in{\mathbb{R}}^{N}/\frac{3}{4}\leq|\xi|\leq\frac{8}{3}\}$ such that: $$\sum_{l\in\mathbb{Z}}\varphi(2^{-l}\xi)=1\,\,\,\,\mbox{if}\,\,\,\,\xi\ne 0.$$ Denoting $h={\cal{F}}^{-1}\varphi$, we then define the dyadic blocks by: $${\Delta}_{l}u=\varphi(2^{-l}D)u=2^{lN}\int_{{\mathbb{R}}^{N}}h(2^{l}y)u(x-y)dy\,\,\,\,\mbox{and}\,\,\,S_{l}u=\sum_{k\leq
l-1}{\Delta}_{k}u\,.$$ Formally, one can write that: $$u=\sum_{k\in\mathbb{Z}}{\Delta}_{k}u\,.$$ This decomposition is called homogeneous Littlewood-Paley decomposition. Let us observe that the above formal equality does not hold in ${\cal{S}}^{'}({\mathbb{R}}^{N})$ for two reasons:

1.  The right hand-side does not necessarily converge in ${\cal{S}}^{'}({\mathbb{R}}^{N})$.

2.  Even if it does, the equality is not always true in ${\cal{S}}^{'}({\mathbb{R}}^{N})$ (consider the case of the polynomials).

Homogeneous Besov spaces and first properties
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For $s\in{\mathbb{R}},\,\,p\in[1,+\infty],\,\,q\in[1,+\infty],\,\,\mbox{and}\,\,u\in{\cal{S}}^{'}({\mathbb{R}}^{N})$ we set: $$\|u\|_{B^{s}_{p,q}}=(\sum_{l\in\mathbb{Z}}(2^{ls}\|{\Delta}_{l}u\|_{L^{p}})^{q})^{\frac{1}{q}}.$$ The Besov space $B^{s}_{p,q}$ is the set of temperate distribution $u$ such that $\|u\|_{B^{s}_{p,q}}<+\infty$.

The above definition is a natural generalization of the nonhomogeneous Sobolev and H$\ddot{\mbox{o}}$lder spaces: one can show that $B^{s}_{\infty,\infty}$ is the nonhomogeneous H$\ddot{\mbox{o}}$lder space $C^{s}$ and that $B^{s}_{2,2}$ is the nonhomogeneous space $H^{s}$.

\[derivation,interpolation\] The following properties holds:

1.  there exists a constant universal $C$ such that:\
    