Surjectivity of the asymptotic Borel map in Carleman-Roumieu ultraholomorphic classes defined by regular sequences

TL;DR

This paper investigates the surjectivity of the asymptotic Borel map within Carleman-Roumieu ultraholomorphic classes, extending previous results and highlighting the importance of a sequence index linked to regular variation and integral transforms.

Contribution

It extends prior work on the surjectivity of the Borel map in ultraholomorphic classes by incorporating a sequence index and advanced techniques like regular variation and integral transforms.

Findings

Extended surjectivity results for the Borel map in ultraholomorphic classes.

Identified the role of Thilliez's sequence index in surjectivity.

Connected the problem to Gelfand-Shilov space techniques.

Abstract

We study the surjectivity of, and the existence of right inverses for, the asymptotic Borel map in Carleman-Roumieu ultraholomorphic classes defined by regular sequences in the sense of E. M. Dyn'kin. We extend previous results by J. Schmets and M. Valdivia, by V. Thilliez, and by the authors, and show the prominent role played by an index associated with the sequence and introduced by Thilliez. The techniques involve regular variation, integral transforms and characterization results of A. Debrouwere in a half-plane, steming from his study of the surjectivity of the moment mapping in general Gelfand-Shilov spaces.