On the surjectivity of the Borel mapping for some quasi-analytic rings

The Borel mapping takes germs, at the origin in $\mathbb R$, of smooth functions ($\equiv $ elements of the ring of germs $\mathcal {E}$) to the sequence of iterated derivatives at the origin. It is a classical result due to Borel that the Borel mapping is surjective and not injective. A subring $\mathcal {B}\subset \mathcal {E}$ is called quasianalytic if the restriction of the Borel mapping to $\mathcal {B}$ is injective. By a classical result due to Carleman, the Borel mapping restricted to the subring of smooth functions in a quasianalytic Denjoy–Carleman class which strictly contains the analytic class is never onto the corresponding subring of ${\mathbb R}[[t]]$.

In this paper, a necessary and sufficient condition is given for the surjectivity of the restriction of the Borel mapping to a general quasianalytic subring $\mathcal {B}\subset \mathcal {E}$. As a consequence, we show the existence of quasianalytic subrings such that the restriction of the Borel mapping to these subrings is surjective and hence bijective.