Injectivity and surjectivity of the asymptotic Borel map in Carleman ultraholomorphic classes

TL;DR

This paper investigates the conditions under which the Borel map is injective or surjective in certain ultraholomorphic classes defined by a sequence $ extbf{M}$, using indices $ ext{ω}( extbf{M})$ and $ ext{γ}( extbf{M})$ to characterize these properties.

Contribution

It completely solves the injectivity problem in a specific class using proximate orders and extends the surjectivity results by establishing the optimality of the index $ ext{γ}( extbf{M})$ in standard cases.

Findings

Injectivity characterized by the growth index ω(𝕄) and sector opening.

Surjectivity partially characterized, with γ(𝕄) shown to be optimal in certain cases.

Extended previous results, providing a more complete understanding of Borel map properties.

Abstract

We study the injectivity and surjectivity of the Borel map in three instances: in Roumieu-Carleman ultraholomorphic classes in unbounded sectors of the Riemann surface of the logarithm, and in classes of functions admitting, uniform or nonuniform, asymptotic expansion at the corresponding vertex. These classes are defined in terms of a log-convex sequence $\mathbb{M}$ of positive real numbers. Injectivity had been solved in two of these cases by S. Mandelbrojt and B. Rodr\'iguez-Salinas, respectively, and we completely solve the third one by means of the theory of proximate orders. A growth index $\omega(\mathbb{M})$ turns out to put apart the values of the opening of the sector for which injectivity holds or not. In the case of surjectivity, only some partial results were available by J. Schmets and M. Valdivia and by V. Thilliez, and this last author introduced an index $\gamma(\mathbb{M})$ (generally different from $\omega(\mathbb{M})$) for this problem, whose optimality was not established except for the Gevrey case. We considerably extend here their results, proving that $\gamma(\mathbb{M})$ is indeed optimal in some standard situations (for example, as far as $\mathbb{M}$ is strongly regular) and puts apart the values of the opening of the sector for which surjectivity holds or not.