Right inverses for the asymptotic Borel map in ultraholomorphic classes on sectors

We present results about the existence of local and global right inverses for the asymptotic Borel map in Carleman–Roumieu ultraholomorphic classes in sectors. By local right inverses we understand those which interpolate for asymptotic power series of a fixed type, while the global ones are defined in the corresponding whole space of Carleman–Roumieu formal power series. We extend previous results by J. Schmets and M. Valdivia and by V. Thilliez, and show the prominent role played by the index $\gamma(\mathbb{M})$ of Thilliez and the condition $(\beta_2)$ of H.-J. Petzsche. The techniques involve different facts from the general theory of regular variation.
This is joint work with J. Jiménez-Garrido (University of Cantabria, Spain) and G. Schindl (University of Vienna, Austria).