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, that was introduced by V. Thilliez. The techniques involve regular variation, integral transforms and characterization results of A. Debrouwere in a half-plane, stemming from his study of the surjectivity of the moment mapping in general Gelfand?Shilov spaces.

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