Isomorphism of spaces of ultradifferentiable functions of Beurling type (Q881199)

Let \(\omega\) be a weight function in the sense of \textit{R. W. Braun, R. Meise} and \textit{B. A. Taylor} [Result. Math. 17, No. 3/4, 207--237 (1990; Zbl 0735.46022)]. Denote by \(\varphi^*\) the Young conjugate of the function \(\varphi (t):=\omega(e^t)\) and define the Fréchet space of \((\omega)\)-ultradifferentiable functions on \(\mathbb{R}^N\) by \[ {\mathcal E}_{(\omega)}(\mathbb{R}^N):=\{f\in C^\infty(\mathbb{R}^N):\forall p\in\mathbb{N}\;\|f\|_p:=\sup_{|x|\leq p}\sup_{\alpha \in\mathbb{N}_0^N}|f^{(\alpha)}(x)| e^{-p\varphi^*(|\alpha|/p)}<\infty\}. \] The author uses Kolmogorov widths to prove the following result Theorem. Let \(\omega\) and \(\sigma\) be weight functions. Then \({\mathcal E}_{(\omega)}(\mathbb{R}^N)\) and \({\mathcal E}_{(\sigma)}(\mathbb{R}^N)\) are linearly topologically isomorphic if and only if \(\omega\) and \(\sigma\) are equivalent in the sense that there exists \(C\geq 1\) such that, for all \(t\geq 0\), \[ \frac 1C\omega (t)-C\leq\sigma(t)\leq C\omega(t)+C. \] This theorem holds even if \({\mathcal E}_{(\omega)}(\mathbb{R}^N)\) (resp., \({\mathcal E}_{(\sigma)}(\mathbb{R}^N))\) is replaced by \({\mathcal E}_{(\omega)}(G_1)\) (resp., \({\mathcal E}_{(\sigma)} (G_2))\), where \(G_1\) and \(G_2\) are open sets in \(\mathbb{R}^N\).