A linear topological invariant for spaces of quasianalytic functions of Roumieu type

We show that the spaces ${\mathcal{E}}_{\{\omega \}}(\mathrm{\Omega })$ of ultradifferentiable functions of Roumieu type satisfy the dual interpolation estimate for small theta, where $\omega$ is a quasianalytic weight function and $\mathrm{\Omega }$ is an arbitrary open subset of ${\mathbb{R}}^d$. This result was previously shown by Bonet and Doma'nski [2] under the additional assumptions that $\mathrm{\Omega }$ is convex and $\omega$ satisfies the condition $({\alpha }_1)$. In particular, our work solves Problem 9.7 in [1].