The Borel-Ritt problem in Beurling ultraholomorphic classes

TL;DR

This paper completely characterizes the conditions under which the asymptotic Borel map is surjective or admits a continuous right inverse in Beurling ultraholomorphic classes, advancing understanding of ultraholomorphic function spaces.

Contribution

It provides a full solution to the Borel-Ritt problem in non-uniform Beurling ultraholomorphic classes, extending previous results by Thilliez and Schmets-Valdivia.

Findings

Characterization of surjectivity of the Borel map

Conditions for existence of a continuous right inverse

Improvement over previous results by Thilliez and Schmets-Valdivia

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 [10] and Schmets and Valdivia [9].