How far is the Borel map from being surjective in quasianalytic ultradifferentiable classes?

TL;DR

This paper investigates the size and structure of the image of the Borel map in quasianalytic ultradifferentiable classes, revealing how far it is from being surjective across a broad class of function spaces.

Contribution

It provides a comprehensive analysis of the Borel map's image in quasianalytic classes defined by weight matrices, extending previous results to more general settings.

Findings

The Borel map is not surjective in these classes.

The image of the Borel map has a large, complex structure.

The results apply to classes defined by weight matrices, sequences, or functions.

Abstract

The Borel map takes germs at 0 of smooth functions to the sequence of iterated partial derivatives at 0. In the literature, it is well known that the restriction of this mapping to the germs of quasianalytic ultradifferentiable classes which are strictly containing the real analytic functions can never be onto the corresponding sequence space. In this paper, we are interested in studying how large the image of the Borel map is and we investigate the size and the structure of this image by using different approaches (Baire residuality, prevalence and lineability). We give an answer to this question in the very general setting of quasianalytic ultradifferentiable classes defined by weight matrices, which contains as particular cases the classes defined by a single weight sequence or by a weight function.