Hello. In example 1.8.2, it is shown that $|t^mp^{m}(t)|\leq C$ for all $t>0$ and finally, the function $m(\xi)$ satisfies the condition 1.68.
In my case, I am playing with the function $\rho(t)=t/(1+t)$ ($\gamma=1$) along with $n=2$ (i.e. $\alpha=(\alpha_1,\alpha_2)$ and $\xi=(\xi_1,\xi_2))$ to simplify the calculations and to understand it well. I have already verified that $|t^mp^{m}(t)|\leq C$, but how can I verify that this function satisfies condition 1.68? It complicates me how to calculate the derivative (multi-index) $\partial_{\xi}^{\alpha}m(\xi)$.
From what I see, $\partial_{\xi}^{\alpha}m(\xi)=\partial_{\xi_2}^{\alpha_2}\partial_{\xi_1}^{\alpha_1}\rho(|\xi |)$, but I don't know if there will be something similar like $\partial_{\xi}^{\alpha}(\rho(|\xi|)=\displaystyle \sum_{\text{something with } \alpha} \rho^{|\alpha|}(|\xi| )\partial_{\xi}^{\alpha}(|\xi|)$? (in other words, a kind of chain rule in its multi-index version?)
Thank you.
