same notations– one can see that it suffices to prove (\[II2\]) with $R_p(\sigma)$ replaced by $R_p(p_k{\langle\xi\rangle}^m)$, provided that the estimate is uniform in $k\in{{\mathbb Z}}^d$. Without loss of generality, we can also assume that $m=0$.\
Now, remark that $$\label{II3}
    2^{r(s+t-d/2)}\Big\vert \varphi_r(D)\sum_{p\geq r-n_0}
    R_p(p_k)u\Big\vert_2
    \leq {\mbox{\textnormal{Cst }}}2^{r(s+t)}\Big\vert\sum_{p\geq r-n_0}R_p(p_k)
    u\Big\vert_{L^1}$$ and that $$\label{II4}
    \left\vert R_p(p_k)u\right\vert_{L^1}=
    \Big\vert \sum_{\vert p-q\vert\leq N}\varphi_q(D_x)p_k(x,D)
    (1-\psi(D))\varphi_p(D) u\Big\vert_{L^1}.$$ Now, using the expression of $p_k(x,\xi)$ given in Prop. \[propA2\], one can write $$\begin{aligned}
    \lefteqn{\varphi_q(D_x)p_k(x,D)(1-\psi(D))
    \varphi_p(D)u
    =}\\
    & &\sum_{l\geq -1}\varphi_q(D_x)c_{k,l}(x)\lambda_k(2^{-l}D)
    (1-\psi(D))\varphi_p(D) u,\end{aligned}$$ and since $\lambda_k(2^{-l}\xi)\varphi_p(\xi)=0$ if $\vert p-l\vert>n_1$, for some $n_1\in{{\mathbb N}}$, one deduces that the summation in the r.h.s. of the above inequality is over a finite number of integers $l$; therefore, by Cauchy-Schwartz’s inequality and Prop. \[propA2\], $$\label{II5}
    \left\vert\varphi_q(D_x)p_k(x,D)(1-\psi(D))
    \varphi_p(D)u\right\vert_{L^1}\lesssim N^m_{{{2[\frac{d}{2}]+2}},0}(\sigma_{(q)})
    \left\vert
    \varphi_p(D)u\right\vert_{L^2}.$$ From (\[II4\]) and (\[II5\]) one obtains $$\left\vert R_p(p_k)u\right\vert_{L^1}\lesssim
    2^{-p(s+t)}\sum_{\vert p-q\vert\leq N}
    2^{qs}N^m_{{{2[\frac{d}{2}]+2}},0}(\sigma_{(q)})
    2^{pt}\left\vert 
    \varphi_p(D)u\right\vert_2,$$ and the l.h.s. of (\[II3\]) is therefore bounded from above by $$\begin{aligned}
     \sum_{p\geq r-n_0} 2^{(r-p)(s+t)}
    \sum_{\vert p-q\vert\leq N}
    2^{qs}N^m_{{{2[\frac{d}{2}]+2}},0}(\sigma_{(q)})
    2^{pt}\left\vert  
    \varphi_p(D)u\right\vert_2. \end{aligned}$$ Since $s+t>0$, Hölder’s inequality yields that the l.h.s. of (\[II2\]) is bounded from above by $$\left\vert\left(
    2^{ps}N^m_{{{2[\frac{d}{2}]+2}},0}(\sigma_{(p)})
    2^{pt}\Big\vert  
    \varphi_p(D)u\right\vert_2\right)_{p\geq -1}\Big\vert_{l^1}.$$ By Cauchy-Schwartz’s inequality, Lemma \[lemmLP1\] and an argument similar to the one used in (\[truc\]), one obtains (\[II2\]), which concludes 