$f\in\widetilde{L}^{\rho_{2}}_{T}
(\widetilde{B}^{s-2+2/\rho_{2}}_{p,r})$. Let u be a solution of: $$\begin{cases}
\begin{aligned}
&{\partial}_{t}u-\mu{\Delta}u=f\\
&u_{t=0}=u_{0}\,.
\end{aligned}
\end{cases}$$ Then there exists $C>0$ depending only on $N,\mu,\rho_{1}$ and $\rho_{2}$ such that: $$\|u\|_{\widetilde{L}^{\rho_{1}}_{T}(\widetilde{B}^{s+2/\rho_{1}}_{p,r})}\leq C\big(
 \|u_{0}\|_{B^{s}_{p,r}}+\mu^{\frac{1}{\rho_{2}}-1}\|f\|_{\widetilde{L}^{\rho_{2}}_{T}
 (\widetilde{B}^{s-2+2/\rho_{2}}_{p,r})}\big)\,.$$ If in addition $r$ is finite then $u$ belongs to $C([0,T],B^{s}_{p,r})$.

Hybrid Besov spaces
-------------------

The homogeneous Besov spaces fail to have nice inclusion properties: owing to the low frequencies, the embedding $B^{s}_{p,1}\hookrightarrow B^{t}_{p,1}$ does not hold for $s>t$. Still, the functions of $B^{s}_{p,1}$ are locally more regular than those of $B^{t}_{p,1}$: for any $\phi\in C^{\infty}_{0}$ and $u\in B^{s}_{p,1}$, the function $\phi u\in B^{t}_{p,1}$. This motivates the definition of Hybrid Besov spaces introduced by R. Danchin in [@DG] where the growth conditions satisfied by the dyadic blocks and the coefficient of integrability are not the same for low and high frequencies. Hybrid Besov spaces have been used in [@DG1] to prove global well-posedness for compressible gases in critical spaces. We generalize here a little bit the definition by distinguishing the coefficients of integrability.

\[def1.9\] Let $s,t\in{\mathbb{R}}$ and $(p,q)\in[1,+\infty]$. We set: $$\|u\|_{\widetilde{B}^{s,t}_{p,q,1}}=\sum_{l\leq 0}2^{ls}\|{\Delta}_{l}u\|_{L^{p}}+\sum_{l\leq 0}2^{lt}\|{\Delta}_{l}u\|_{L^{q}}.$$

We will often use the following notation: $$u_{BF}=\sum_{l\leq0}{\Delta}_{l}u\;\;\;\mbox{and}\;\;\;u_{HF}=\sum_{l>0}{\Delta}_{l}u.$$

We have the following properties:

-   We have $\widetilde{B}^{s,s}_{p,p,1}=B^{s}_{p,1}$.

-   If $s_{1}\geq s_{3}$ and $s_{2}\geq s_{4}$ then $\widetilde{B}^{s_{3},s_{2}}_{p,q,1}{\hookrightarrow}\widetilde{B}^{s_{1},s_{4}}_{p,q,1}$.

We 