Lemma 7

Let $k\in \mathbf{N}$ and $\alpha \in \mathbf{N}^n$. For every function $\varphi \in \mathscr{S}(\mathbf{R}^n)$ and every real number $T>0$, one has

$_k\alpha$ 1

(3) $\widetilde{q}_{\alpha ,k}(\varphi )\leqslant T$ sup $|\partial \varphi (x)|+\widetilde{q}_{\alpha ,k+1}(\varphi )$.

$\|x\|\leqslant T$ T

In fact, one has

$\widetilde{q}_{\alpha ,k}(\varphi )\leqslant$ sup$_{\|x\|\leqslant T}\|x\|^k|\partial^{\alpha}\varphi (x)|+$ sup$_{\|x\|>T}\|x\|^k|\partial^{\alpha}\varphi (x)|$

$\leqslant T^k$ sup$_{\|x\|\leqslant T}|\partial^{\alpha}\varphi (x)|+\frac{1}{T}$ sup$_{\|x\|>T}\|x\|^{k+1}|\partial^{\alpha}\varphi (x)|$.