Surjectivity of the asymptotic Borel map in Carleman–Roumieu ultraholomorphic classes defined by regular sequences

We study the surjectivity of, and the existence of right inverses for, the asymptotic Borel map in Carleman–Roumieu ultraholomorphic classes defined by regular sequences in the sense of E. M. Dyn’kin. We extend previous results by J. Schmets and M. Valdivia, by V. Thilliez, and by the authors, and show the prominent role played by an index, associated with the sequence, that was introduced by V. Thilliez. The techniques involve regular variation, integral transforms and characterization results of A. Debrouwere in a half-plane, stemming from his study of the surjectivity of the moment mapping in general Gelfand–Shilov spaces.

Uniformly factoring weakly compact operators and parametrised dualisation

This article deals with the problem of when, given a collection $\mathcal {C}$ of weakly compact operators between separable Banach spaces, there exists a separable reflexive Banach space Z with a Schauder basis so that every element in $\mathcal {C}$ factors through Z (or through a subspace of Z). In particular, we show that there exists a reflexive space Z with a Schauder basis so that for each separable Banach space X, each weakly compact operator from X to $L_1[0,1]$ factors through Z. We also prove the following descriptive set theoretical result: Let $\mathcal {L}$ be the standard Borel space of bounded operators between separable Banach spaces. We show that if $\mathcal {B}$ is a Borel subset of weakly compact operators between Banach spaces with separable duals, then for $A \in \mathcal {B}$ , the assignment $A \to A^*$ can be realised by a Borel map $\mathcal {B}\to \mathcal {L}$ .

On the Construction of Large Algebras Not Contained in the Image of the Borel Map

The Borel map $$j^{\infty }$$j∞ takes germs at 0 of smooth functions to the sequence of iterated partial derivatives at 0. It is well known that the restriction of $$j^{\infty }$$j∞ to the germs of quasianalytic ultradifferentiable classes which are strictly containing the real analytic functions can never be onto the corresponding sequence space. In a recent paper the authors have studied the size of the image of $$j^{\infty }$$j∞ by using different approaches and worked in the general setting of quasianalytic ultradifferentiable classes defined by weight matrices. The aim of this paper is to show that the image of $$j^{\infty }$$j∞ is also small with respect to the notion of algebrability and we treat both the Cauchy product (convolution) and the pointwise product. In particular, a deep study of the stability of the considered spaces under the pointwise product is developed.

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

We consider r-ramification ultradifferentiable classes, introduced by J. Schmets and M. Valdivia in order to study the surjectivity of the Borel map, and later on also exploited by the authors in the ultraholomorphic context. We characterize quasianalyticity in such classes, extend the results of Schmets and Valdivia about the image of the Borel map in a mixed ultradifferentiable setting, and obtain a version of the Whitney extension theorem in this framework.

On the Theorem of Borel for Quasianalytic Classes

We investigate the surjectivity of the Borel map in the quasianalytic setting for classes of ultradifferentiable functions defined in terms of the growth of the Fourier-Laplace transform. We deal with both the Roumieu $\mathcal{E}_{\{\omega\}}$ and the Beurling $\mathcal{E}_{(\omega)}$ classes for a weight function $\omega$. In particular, we show that a classical result of Carleman for the quasianalytic classes $\mathcal{E}_{\{M_p\}}$ also holds for the classes defined using weights. We also characterize when the space of quasianalytic germs at the origin coincides with the space of real analytic germs at the origin.