Let $g:\mathbb{R}^n\rightarrow \mathbb{R}^m$ be a continuous bijective map and suppose that $g^{-1}$ is well-defined, of class $C^1$, and satisfies $$ \sup_{x\in \mathbb{R}^n}\, |\operatorname{det}J_{g^{-1}}(x)|=:M<\infty. $$
Fix a positive integer $n$.
A perturbation of the argument of this article shows that the associated composition operator
$$
\begin{aligned}
C_g: L^2(\mathbb{R}^n,\mathbb{R}^n) & \rightarrow L^2(\mathbb{R}^n,\mathbb{R}^n)\\
f &\mapsto f\circ g
\end{aligned}
$$
is a well-defined bounded-linear operator. Actually this follows from the change-of-variable formulate from multivariate calculus since we have assumed that $g^{-1}$ is of class $C^1$.
If I am not mistaken then the operator norm of $C_g$ can be estimated as follows. Suppose that $f\in L^2(\mathbb{R}^n,\mathbb{R}^n)$ and the $\|f\|_{L^2}=1$. Then, $$ \begin{aligned} C_g(f) := & \int_{x \in \mathbb{R}^n}\, \|f\circ g(x)\|^2 dx \\ = & \int_{u \in g^{-1}(\mathbb{R}^n)}\, |\det J_g^{-1}(u)| \|f\circ g\circ g^{-1}(u)\|^2 du \\ = & \int_{u \in g^{-1}(\mathbb{R}^n)}\, |\det J_g^{-1}(u)| \|f(u)\|^2 du \\ \leq & \sup_{v\in \mathbb{R}^n}\, |\det (J_g^{-1}(v))| \, \int_{u \in g^{-1}(\mathbb{R}^n)}\, \|f(u)\|^2 du \\ \leq & M \|f\|_{L^2}. \end{aligned} $$
Whence we conclude that $$ \|C_g\|_{op} := \sup_{f\in L^2(\mathbb{R}^n,\mathbb{R}^n);\, \|f\|_{L^2}=1}\, \|C_g(f)\|_{L^2} \leq M \|f\| = M . $$
Is this proof correct?
If so does anyone know of sharper estimates?
Interesting related note: I have found several results on the operator norm of composition operators when the involved spaces are not $L^2$ but rather the complex Hardy space (e.g. this 2016 paper or this old Annals of Math paper)
Another cute reference: https://arxiv.org/pdf/1705.00325.pdf