Pathological phenomena in Denjoy-Carleman classes

Abstract

Let $\mathcal C^M$ denote a Denjoy-Carleman class of $\mathcal C^\infty$ functions (for a given logarithmically-convex sequence $M = (M_n)$). We construct: (1) a function in $\mathcal C^M((-1,1))$ which is nowhere in any smaller class; (2) a function on $\mathbb R$ which is formally $\mathcal C^M$ at every point, but not in $\mathcal C^M(\mathbb R)$; (3) (under the assumption of quasianalyticity) a smooth function on $\mathbb R^p$ ($p \geq 2$) which is $\mathcal C^M$ on every $\mathcal C^M$ curve, but not in $\mathcal C^M(\mathbb R^p)$.