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

Volume 126 / 2021

Abdelhafed El Khadiri Annales Polonici Mathematici 126 (2021), 21-30 MSC: Primary 26E10, 30D60, 58C25, 46E25, 34A34. DOI: 10.4064/ap200714-4-11 Published online: 18 February 2021


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.


  • Abdelhafed El KhadiriFaculty of Sciences
    University Ibn Tofail
    Kenitra, Morocco

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