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The Borel–Ritt Problem in Beurling Ultraholomorphic Classes

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Abstract

We give a complete solution to the Borel–Ritt problem in non-uniform spaces \({\mathscr {A}}^-_{(M)}(S)\) of ultraholomorphic functions of Beurling type, where S is an unbounded sector of the Riemann surface of the logarithm and M is a strongly regular weight sequence. Namely, we characterize the surjectivity and the existence of a continuous linear right inverse of the asymptotic Borel map on \({\mathscr {A}}^-_{(M)}(S)\) in terms of the aperture of the sector S and the weight sequence M. Our work improves previous results by Thilliez (Results Math 44:169–188, 2003) and Schmets and Valdivia (Stud Math 143:221–250, 2000).

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Notes

  1. This result is shown in [5, Corollary 4.19] in the Roumieu case but, as we shall explain in Sect. 5, the proof provided there can be adapted to the Beurling case.

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Funding

This work was supported by the Research Foundation-Flanders (grant number 12T0519N).

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  1. Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Krijgslaan 281, 9000, Gent, Belgium

    A. Debrouwere

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  1. A. Debrouwere
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Correspondence to A. Debrouwere.

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Debrouwere, A. The Borel–Ritt Problem in Beurling Ultraholomorphic Classes. Results Math 76, 151 (2021). https://doi.org/10.1007/s00025-021-01458-7

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