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

Abstract

We prove several improved versions of the Borel-Ritt theorem about the surjectivity of the asymptotic Borel mapping in classes of functions with $\mathbf{M}$-uniform asymptotic expansion on an unbounded sector of the Riemann surface of the logarithm. While in previous results the weight sequence $\mathbf{M}$ 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.

arXiv comment25 pages

References in corpus (5)

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• Injectivity and surjectivity of the asymptotic Borel map in Carleman ultraholomorphic classes
• Indices of O-regular variation for weight functions and weight sequences
• On generalized definitions of ultradifferentiable classes
• Optimal flat functions in Carleman-Roumieu ultraholomorphic classes in sectors