On the non-extendability of quasianalytic germs

Abstract

Let ${\mathcal{E}}_1(M{)}^{+}$ be the local ring of germs at 0 of functions belonging to a given Denjoy-Carleman quasianalytic class in a neighborhood of 0 in $[0,+\infty [$. We show that the ring ${\mathcal{E}}_1(M{)}^{+}$ contains elements that cannot be extended quasianalytically in a neighborhood of 0 in $\mathbb{R}$, unless it coincides with the ring of real-analytic germs.

This paper has been withdrawn since the author has found out that a similar result, with a slightly different proof, already appeared in a paper of M. Langenbruch (Manuscripta Math. 83 (1994), 123-143; see the remark after Corollary 2.4)