On a problem concerning quasianalytic local rings
Volume 111 / 2014
Abstract
Let $(\mathcal{C}_{n})_{n}$ be a quasianalytic differentiable system. Let $m\in\mathbb{N}$. We consider the following problem: let $f\in\mathcal{C}_{m}$ and $\widehat{f}$ be its Taylor series at $0\in\mathbb{ R}^{m}$. Split the set $\mathbb{N}^{m}$ of exponents into two disjoint subsets $A$ and $B$, $\mathbb{N}^{m}= A\cup B$, and decompose the formal series $\widehat{f}$ into the sum of two formal series $G$ and $H$, supported by $A$ and $B$, respectively. Do there exist $g, h\in \mathcal{C}_{m}$ with Taylor series at zero $G$ and $H$, respectively? The main result of this paper is the following: if we have a positive answer to the above problem for some $m\geq2$, then the system $(\mathcal{C}_{n})_{n}$ is contained in the system of analytic germs. As an application of this result, we give a simple proof of Carleman's theorem (on the non-surjectivity of the Borel map in the quasianalytic case), under the condition that the quasianalytic classes considered are closed under differentiation, for $n\geq2$.
