On the Theorem of Borel for Quasianalytic Classes

Abstract

We investigate the surjectivity of the Borel map in the quasianalytic setting for classes of ultradifferentiable functions defined in terms of the growth of the Fourier-Laplace transform. We deal with both the Roumieu $\mathcal{E}_{\{\omega\}}$ and the Beurling $\mathcal{E}_{(\omega)}$ classes for a weight function $\omega$. In particular, we show that a classical result of Carleman for the quasianalytic classes $\mathcal{E}_{\{M_p\}}$ also holds for the classes defined using weights. We also characterize when the space of quasianalytic germs at the origin coincides with the space of real analytic germs at the origin.