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?



Specific estimation of the norm for a linearly transformed function in $\mathcal{S}_0^{\beta}(\mathbb{R}^n)$
$\newcommand\be\beta\newcommand\A{\mathfrak A}\newcommand\R{\mathbb R}\newcommand\N{\mathbb N}$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](https://en.wikipedia.org/wiki/Matrix_norm#Matrix_norms_induced_by_vector_p-norms) 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 $q\in\N\_0^n$.


