Specific estimation of the norm for a linearly transformed function in $\mathcal{S}_0^{\beta}(\mathbb{R}^n)$ According to the standard definition, $\mathcal{S}_0^{\beta}(\mathbb{R})$ is a subspace of smooth functions on $\mathbb{R}$ with the property that
\begin{equation}
\lvert x^k f^{(q)}(x) \rvert \leq CA^kB^q q^{\beta q}
\end{equation}
for some constants A, B, C that depend on each $f$. I am aware that $f$ must be compactly supported  and $\beta>1$ is required for a nontrivial $f$ to exist.
Now, if I tensor-product $n$ copies of $f$, that is, define
\begin{equation}
F(x_1, \cdots, x_n):= f(x_1) \times \cdots \times f(x_n)
\end{equation}
it is clear that $F$ belongs to $\mathcal{S}_0^{\beta}(\mathbb{R}^n)$, with the same $A,B,C$ for each coordinate.
Now, let us think of any invertible $n \times n$ matrix $\mathfrak{A}$ and consider the function
\begin{equation}
G(x_1, \cdots, x_n):= F\bigl( \mathfrak{A}(x_1, \cdots, x_n) \bigr).
\end{equation}
Then, I believe that $G$ must be in $\mathcal{S}_0^{\beta}(\mathbb{R}^n)$ as well. However, I cannot figure out further information for the bounds on $G(x_1, \cdots, x_n)$. For example, is it possible to express the constants $A_G^1, B^1_G, C^1_G$ such that
\begin{equation}
\lvert (x_1)^k (\partial_1)^q G(x_1, \cdots, x_n) \rvert \leq C_G^1 (A_G^1)^k (B_G^1)^q q^{\beta q}
\end{equation}
in terms of $A, B, C,$ of the orignal function $f$ and some quantities related to $\mathfrak{A}$, like its operator norm?
This kind of detailed estimate seems quite delicate for me.. Could anyone please help me?
 A: $\newcommand\be\beta\newcommand\A{\mathfrak A}\newcommand\R{\mathbb R}\newcommand\N{\mathbb N}$Suppose that
\begin{equation}
|x_j^k F^{(q)}(x)| \le CA^kB^{|q|} q^{\beta q}
\end{equation}
for some positive real $A,B,C,\be$, all $k\in\N_0:=\{0,1,\dots\}$, all $j\in[n]:=\{1,\dots,n\}$, all $x=[x_1,\dots,x_n]^\top\in\R^n$, and all $q=[q_1,\dots,q_n]^\top\in\N_0^n$, where $|q|:=q_1+\dots+q_n$ and $q^{\be q}:=\prod_{j\in[n]}q_j^{\be q_j}$.
Let
$$G(x):=F(\A x)$$
for some invertible matrix $\A=[A_{i,j}\colon(i,j)\in[n]^2]$ and all $x\in\R^n$.
Then for any $j\in[n]$ and all $x\in\R^n$
$$D_j G(x)=\sum_{i\in[n]}A_{i,j}D_iF(\A x),$$
where $D_i$ is the operator of the partial differentiation with respect to $x_i$.
So, for any $r\in\N_0$, any $j_1,\dots,j_r$ in $[n]$, and all $x\in\R^n$ one has
$$D_{j_r}\cdots D_{j_1}G(x)
=\sum_{i_r\in[n]}A_{i_r,j_r}\cdots \sum_{i_1\in[n]}A_{i_1,j_1}
D_{i_r}\cdots D_{i_1}F(\A x).$$
So,
$$|G^{(q)}(x)|\le m(\A)^{|q|}|F^{(q)}(\A x)|$$
for all $x\in\R^n$ and $q\in\N_0^n$, where $$m(\A):=\max_{j\in[n]}\sum_{i\in[n]}|A_{i,j}|=\|\A\|_{1,1},$$
the $(\ell^1,\ell^1)$ operator norm of the matrix $\A$.
Also, $\|x\|_\infty:=\max_{1\le j\le n}|x_j|\le\|\A^{-1}\|_{\infty,\infty}\|\A x\|_\infty$ for all $x\in\R^n$, where $\|\cdot\|_{\infty,\infty}$ is the $(\ell^\infty,\ell^\infty)$ operator norm.
Thus, as desired,
\begin{equation}
|x_j^k G^{(q)}(x)| \le C(A\|\A^{-1}\|_{\infty,\infty})^k
(B\|\A\|_{1,1})^{|q|} q^{\beta q}
\end{equation}
for all $k\in\N_0$, $j\in[n]$, $x\in\R^n$, and $q\in\N_0^n$.
