A smooth function $m \in \mathcal C^\infty(\mathbb R^n)$ is said to be slowly increasing if for all $\alpha \in \mathbb N^n_0$ there exists $C_\alpha, k_\alpha$ such that $|\partial_\alpha f(x)| \leq C_\alpha(1+|x|^2)^{k_\alpha}$
1) Is a function satisfying the above also called 'of polynomial growth'? I do not really understand why the RHS of above estimate has the special form it has, e.g. what is the meaning/relevance of the power of 2 on the RHS?
2) I try to prove the following elementary lemma: if $m$ is slowly increasing, then multiplication by $m$ defines an operator $\mathcal S(\mathbb R^n) \to \mathcal S(\mathbb R^n)$ on the Schwartz space of rapidly decreasing functions. So, let $f \in S(\mathbb R^n)$. I want to show that $m f \in S(\mathbb R^n)$. In the following, $\alpha, \beta$ are multiindices and $\partial_\beta$ is the corresponding partial differential operator. My problem: if I estimate $|x^\alpha (\partial_\beta(mf))(x)|$ I get that this is smaller or equal $|x^\alpha f(x)| C_\beta(1+|x|^2)^{k_\beta} + |\partial_\beta f(x)| C_0 (1+|x|^2)^{k_0}$ but terms like $C_\beta(1+|x|^2)^{k_\beta}$ are unbounded. What do I miss here?
The power $2$ has no real significance: $(1+|x|^2)^k$ and $(1+|x|)^{2k}$ are equivalent.