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

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 M of positive real numbers. Injectivity had been solved in two of these cases by S. Mandelbrojt and B. Rodríguez-Salinas, respectively, and we completely solve the third one by means of the theory of proximate orders. A growth index ω(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 γ(M) (generally different from ω(M)) for this problem, whose optimality was not established except for the Gevrey case. We considerably extend here their results, proving that γ(M) is indeed optimal in some standard situations (for example, as far as M is strongly regular) and puts apart the values of the opening of the sector for which surjectivity holds or not.2010 MSC: Primary 30D60; secondary 30E05, 47A57, 34E05.According to V. Thilliez [29], if M is (lc), has (mg) and satisfies (snq), we say that M is strongly regular.Obviously, (mg) implies (dc), and (snq) implies (nq). Definition 2.2. For a sequence M we define the sequence of quotients m = (m p ) p∈N 0 by m p := M p+1 M p p ∈ N 0 .

The surjectivity of the Borel mapping in the mixed setting for ultradifferentiable ramification spaces

“…We refer to Sections 2.9 and 5.3 for recalling the definitions of the mentioned spaces in the present work. The property (γ r ), crucially appearing in our results in [9], was introduced by Schmets and Valdivia. It was shown to characterize the surjectivity of the Borel map, and even the existence of an extension map, in Beurling ultradifferentiable r-ramified classes [22,Proposition 4.3 and Theorem 4.4], and to be necessary for the surjectivity of the Borel map in the Roumieu case [22,Proposition 5.2] (in [22,Theorem 5.4] they also show that the existence of an extension map in this case amounts to (γ r ) and the restrictive condition (β 2 ) from [14]).…”

“…Closely related are classes of functions admitting asymptotic expansion (again on unbounded sectors of the Riemann surface of the logarithm). For more details and the historic development we refer to the introduction of [9] and the references therein.…”

“…The main aim of this article is to transfer the mixed-setting results from [23] to these (non-standard) r-ramified classes, and the motivation of this question was arising when inspecting the proof of the surjectivity of the asymptotic Borel map in ultraholomorphic classes [9,Thm. 4.14 (i)] for sequences with the property of derivation closedness.…”

“…We expect that a characterization of such situation in terms of some precise growth condition involving M , N and r, as contained in this paper, will be helpful to obtain mixed results for the (asymptotic) Borel map in ultraholomorphic classes defined by weight sequences, i.e. to transfer the results from [9] to a mixed framework with a control on the loss of regularity. Related to this aim is the following: In [10] and [7] the authors have introduced ultraholomorphic classes defined in terms of weight functions ω and shown partial extension results in this setting.…”

Solution to the Stieltjes moment problem in Gelfand–Shilov spaces

“…In [5], Theorem 1.1 is shown by reducing it to the Borel-Ritt problem [18,19,20,11] in spaces of ultraholomorphic functions on the upper half-plane and then using solutions to this problem from [20,11]; a technique that goes back to A. L. Durán and Estrada [8]. Up until now, this seems to be the only known method to study the Stieltjes moment problem in Gelfand-Shilov spaces.…”