, \label{a2}$$where $p_\ast$ is the conjugate of $p$. So, does not matter the value of $\theta _{1},$ one may replace it by $\frac{d%
}{p_{\ast }}.$

**Proof** We take $n_{\ast }\in \N$ and we define $f_{n}=0$ for $n\leq
n_{\ast }$ and $f_{n}=\varphi p_t$ for $n>n_{\ast }.$ Notice that $d_{0}(\varphi p_{t},0)\leq m.$ Then (\[reg4\]) with $k=0$ gives ($C$ denoting a positive constant which may change from a line to another) $$\begin{aligned}
\left\Vert \varphi p_{t}\right\Vert _{q,p} 
&\leq &C\Big(m\sum_{n=0}^{n_{\ast }}2^{n(q+\frac{d}{p_{\ast }})}+\left\Vert \varphi p_{t}\right\Vert _{2h+q,2h,p}\sum_{n=n_{\ast }+1}^{\infty }\frac{1%
}{2^{2nh}}\Big) \\
&\leq &C\Big(m2^{n_{\ast }(q+\frac{d}{p_{\ast }})}+\left\Vert \varphi p_{t}\right\Vert _{2h+q,2h,p}\frac{1}{2^{2n_{\ast }h}}\Big).\end{aligned}$$We denote $\rho _{h}=(q+\frac{d}{p_{\ast }})/2h.$ We optimize over $n_{\ast
}$ and we obtain $$\begin{aligned}
\left\Vert \varphi p_{t}\right\Vert _{q,p} &\leq &2C\times
m^{\frac{1}{1+\rho _{h}}}\times \left\Vert \varphi p_{t}\right\Vert _{2h+q,2h,p}^{\frac{\rho _{h}}{1+\rho _{h}}} \\
&\leq &2Cm^{\frac 1{1+\rho_h}}\times Ct^{-\theta _{0}(2h+q+\theta _{1})%
\frac{\rho _{h}}{1+\rho _{h}}}.\end{aligned}$$Since $\lim_{h\rightarrow \infty }\rho _{h}=0$ and $\lim_{h\rightarrow \infty }(2h+q+\theta _{1})\frac{\rho _{h}}{1+\rho
_{h}}=q+\frac{d}{p_{\ast }}$ the proof is completed, we choose $h$ large enough and we obtain (\[a2\]). $\square $

We will also use the following consequence of Lemma \[lemma-inter\].

\[REG\] Let $k,q,h\in {\mathbb{N}}$, with $h\geq 1$, and $p>1$ be given and set $$\rho _{h}:=\frac{k+q+d/p_{\ast }}{2h}.  \label{reg5}$$We consider an increasing sequence $\theta (n)\geq 1,n\in {\mathbb{N}}$such that $\lim_{n}\theta (n)=\infty $ and $\theta (n+1)\leq \Theta \times
\theta (n)$ for some constant $\Theta \geq 1.$ Suppose that we may find a sequence of functions $f_{n}\in C^{2h+q}({\mathbb{R}}^{d}),n\in {\mathbb{N}}$ such that $$\left\Vert f_{n}\right\Vert _{2h+q,2h,p}\leq \theta (n)  \label{reg9}$$and, with $\m