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

Javier Jiménez-Garrido, Ignacio Miguel-Cantero, Javier Sanz, Gerhard Schindl

TL;DR

The paper broadens the surjectivity theory for the asymptotic Borel map in Carleman ultraholomorphic classes by introducing the shifted moment condition (sm), which is strictly weaker than derivation-closedness yet suffices for Roumieu surjectivity on sectors with angle up to $0<\gamma<\gamma({\boldsymbol{M}})$. The authors develop a constructive approach based on optimal ${\boldsymbol{M}}$-flat functions and ramified Laplace/Borel transforms to produce extension operators, linking moment sequences to kernel-induced moments. In addition to Roumieu results under (sm), the paper advances Beurling-case surjectivity under both derivation-closedness and (sm), utilizing Chaumat-Chollet tools to handle non-uniform growth and to derive local and global extension operators. Overall, the work widens the class of weight sequences for which surjectivity holds and provides explicit, ramified-transform-based methods to obtain right inverses, with implications for both theory and explicit construction of ultraholomorphic functions. The results thus offer a more flexible framework for Borel-type problems in ultraholomorphic classes and suggest further exploration of shift-related kernels in summability contexts.

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.

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

TL;DR

The paper broadens the surjectivity theory for the asymptotic Borel map in Carleman ultraholomorphic classes by introducing the shifted moment condition (sm), which is strictly weaker than derivation-closedness yet suffices for Roumieu surjectivity on sectors with angle up to . The authors develop a constructive approach based on optimal -flat functions and ramified Laplace/Borel transforms to produce extension operators, linking moment sequences to kernel-induced moments. In addition to Roumieu results under (sm), the paper advances Beurling-case surjectivity under both derivation-closedness and (sm), utilizing Chaumat-Chollet tools to handle non-uniform growth and to derive local and global extension operators. Overall, the work widens the class of weight sequences for which surjectivity holds and provides explicit, ramified-transform-based methods to obtain right inverses, with implications for both theory and explicit construction of ultraholomorphic functions. The results thus offer a more flexible framework for Borel-type problems in ultraholomorphic classes and suggest further exploration of shift-related kernels in summability contexts.

Abstract

We prove several improved versions of the Borel-Ritt theorem about the surjectivity of the asymptotic Borel mapping in classes of functions with -uniform asymptotic expansion on an unbounded sector of the Riemann surface of the logarithm. While in previous results the weight sequence 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.
Paper Structure (11 sections, 21 theorems, 91 equations)

This paper contains 11 sections, 21 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Weight sequences and their properties
  4. Associated function with a sequence
  5. Index $\gamma({\boldsymbol{M}})$ for weight sequences
  6. Asymptotic expansions, ultraholomorphic classes and the asymptotic Borel map
  7. Optimal flat functions and surjectivity of the Borel map in the Roumieu case and under $\operatorname{(dc)}$
  8. Improved Borel-Ritt-Gevrey theorem in the Roumieu case
  9. New surjectivity results in the Beurling Case
  10. Continuous right inverses under derivation closedness
  11. Surjectivity for Beurling classes under condition $\operatorname{(sm)}$

Key Result

Lemma 2.3

Let ${\boldsymbol{M}}$ be a sequence such that $a_0:=\inf_{p\in\mathbb{N}_0}m_p>0$ (in particular, this holds if ${\boldsymbol{M}}$ is $\operatorname{(lc)}$). Then, $\operatorname{(dc)}$ implies $\operatorname{(sm)}$.

Theorems & Definitions (35)

  • Remark 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Lemma 2.6
  • proof
  • Lemma 2.7: JimenezSanzSchindlIndex, Remark 3.15
  • Proposition 2.8
  • Lemma 2.9
  • ...and 25 more