Pathological phenomena in Denjoy-Carleman classes (Q2786454)

The author considers properties of the Denjoy-Carleman \({\mathcal C}^{M}(U, {\mathbb F})\) classes, i.e., classes of \({\mathcal C}^{\infty}\)-function on an open set \(U\subset {\mathbb R}^p\) with values in \({\mathbb F} = {\mathbb R}\) or \({\mathbb C}\) such that for any \(x\in U\) there exists an open set \(V \subset U\) containing \(x\) and constants \(A, B > 0\) such that NEWLINE\[NEWLINE |D^{\alpha} f(y)| \leq A B^{|\alpha|} |\alpha|! M_{|\alpha|} NEWLINE\]NEWLINE NEWLINEfor any multi-index \(\alpha =(\alpha_1,\ldots,\alpha_p)\), \(\alpha_j\in {\mathbb N}\), any \(y \in V\) and a given non-decreasing sequence of positive real numbers \(M = (M_n)\).NEWLINENEWLINEThe following results are obtained: NEWLINE{\parindent=6mm NEWLINE\begin{itemize}\item[--] For any \(\mathcal{C}^M\), there exists \(f\in {\mathcal C}^{\infty}((-1,1), {\mathbb F})\) such that NEWLINE\[NEWLINE f\in {\mathcal C}^{M}((-1,1), {\mathbb F}), NEWLINE\]NEWLINE but for any \({\mathcal C}^{N} \varsubsetneq {\mathcal C}^{M}\) and any open set \(U\subset (-1,1)\), NEWLINE\[NEWLINEf\not\in {\mathcal C}^{N}(U).NEWLINE\]NEWLINE NEWLINE\item[--] Let \(\mathcal{C}^M\) be any Denjoy-Carleman class and \({\mathcal F}^{M}(x,{\mathbb F})\) denote a set of functions formally in \(\mathcal{C}^M\) at \(x\). Then, there exists \(f\in {\mathcal C}^{\infty}({\mathbb R}, {\mathbb F})\) such that NEWLINE\[NEWLINE f\in {\mathcal C}^{M}({\mathbb R}\setminus \{0\}, {\mathbb F}), \quad f\in {\mathcal F}^{M}(0,{\mathbb F}), \text{ and } f\not\in {\mathcal C}^{M}({\mathbb R}, {\mathbb F}). NEWLINE\]NEWLINE \item[--] For any \(p > 2\) and any quasianalytic Denjoy-Carleman class \({\mathcal C}^{M}\) which is not the class of analytic functions, there exists \(f\in {\mathcal C}^{\infty}({\mathbb R}^p, {\mathbb F})\) such that for any curve \(\gamma\in {\mathcal C}^{M}(U,{\mathbb R}^p)\) NEWLINE\[NEWLINEf\circ \gamma \in {\mathcal C}^{M}(U, {\mathbb F}),\text{ but }f\not\in {\mathcal C}^{M}({\mathbb R}^p, {\mathbb F}).NEWLINE\]NEWLINENEWLINENEWLINE\end{itemize}}