# On the construction of large Algebras not contained in the image of the Borel map

## TL;DR

This paper investigates the limitations of the Borel map in quasianalytic ultradifferentiable classes, demonstrating that its image is small in terms of algebrability and analyzing the stability of these classes under product operations.

## Contribution

It introduces the concept of algebrability to the image of the Borel map in quasianalytic classes and studies their stability under pointwise and convolution products.

## Key findings

- The image of the Borel map is small in terms of algebrability.
- The classes are stable under convolution product.
- The classes are analyzed for stability under pointwise product.

## Abstract

The Borel map $j^{\infty}$ takes germs at 0 of smooth functions to the sequence of iterated partial derivatives at 0. It is well known that the restriction of $j^{\infty}$ 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 $j^{\infty}$ 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 $j^{\infty}$ 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.