Is the multi-dimensional Gevrey-class $G^1$ closed under composition?

2018-04-21 21:37:51

Let $d,n \in \mathbb{N}$, and let $\Omega \subseteq \mathbb{R}^d$ and $\Psi \in \mathbb{R}^n$. We say that $f \in G^1(\Omega,\Psi)$ if all of its coordinate functions $f_1, \dots, f_n$ are elements of the Gevrey-class $G^1(\Omega)$, as defined here. For convenience, I will restate the definition here. $f \in G^1(\Omega,\Psi)$ if and only if:

$f \in C^{\infty}(\Omega,\Psi)$

Let $K \subseteq \Omega$ be compact. Then there exists a $C_K > 0$ such that for all $\mathbf{x} \in K$, $\alpha \in \mathbb{N}^d$ and $j \in \{1, \dots, n\}$, we have:

$$|D^{\alpha}f_j(\mathbf{x})| \leq C^{|\alpha|+1} \cdot |\alpha|!$$

From this question, which links to this paper, we know that if $\Omega,\Psi,\Phi \subseteq \mathbb{R}$, then:

$$f \in G^1(\Omega,\Psi), g \in G^1(\Psi,\Phi) \implies g \circ f \in G^1(\Omega,\Phi)$$

A sketch of the proof can be found in the question linked to.

I am wondering if a similar result holds for higher dimensional choices of $\Omega$, $\Psi$ and $\Phi$. So,