Surjectivity of the asymptotic Borel map in Carleman ultraholomorphic classes defined by sequences with shifted moments

TL;DR

This paper establishes new surjectivity results for the asymptotic Borel map in Carleman ultraholomorphic classes, weakening previous conditions on the weight sequence and extending results to smaller sectors.

Contribution

It introduces a weaker condition on the weight sequence for surjectivity in Roumieu classes and improves classical results for Beurling classes, expanding the applicability of the Borel-Ritt theorem.

Findings

Surjectivity achieved under weaker conditions on the weight sequence.

Extension of surjectivity results to smaller sectors for Beurling classes.

Reproving classical results with improved assumptions.

Abstract

We prove several improved versions of the Borel-Ritt theorem about the surjectivity of the asymptotic Borel mapping in classes of functions with $\boldsymbol{M}$-uniform asymptotic expansion on an unbounded sector of the Riemann surface of the logarithm. While in previous results the weight sequence $\boldsymbol{M}$ of positive numbers is supposed to be derivation closed, a much weaker condition is shown to be sufficient to obtain the result in the case of Roumieu classes. Regarding Beurling classes, we are able to slightly improve a classical result of J. Schmets and M. Valdivia and reprove a result of A. Debrouwere, both under derivation closedness. Our new condition also allows us to obtain surjectivity results for Beurling classes in suitably small sectors, but the technique is now adapted from a classical procedure already appearing in the work of V. Thilliez, in its turn inspired by that of J. Chaumat and A.-M. Chollet.