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

Abstract

The Borel map 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 ∞ 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 ∞ 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 ∞ 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.

On the maximal extension in the mixed ultradifferentiable weight sequence setting

“…see [22,Theorem 3]). In [4] and [5] it has been shown that the image of B is small and the "size of the failure" has been measured with different concepts, for more details we refer to the introductions and citations in these papers. However, it is still an open question to give a full characterization of the image of B in this setting.…”

“…This gives also a first information to attack the following problem: How far is the Borel map from being surjective in the nonquasianalytic but not strongly nonquasianalytic setting? Unfortunately, it seems that for the proofs from [4] and [5] the quasianalyticity for the weight sequence is indispensable but can we transfer the results to this situation? This question has been asked by Prof. Javier Sanz after a talk of the author about the results from [5] during a research stay at the Universidad de Valladolid.…”

On the maximal extension in the mixed ultradifferentiable weight sequence setting

“…see [27,Theorem 3] and [20]). In [7] and [8] it has been shown that the image of B is small and the "size of the failure" has been measured by using different concepts; for more details we refer to the introductions and citations in those papers. However, it is still an open question to give a full characterization of the image of B in this setting, i.e.…”

“…This also gives a first piece of information related to the following problem: How far is the Borel map from being surjective in the nonquasianalytic but not strongly nonquasianalytic setting? Unfortunately, it seems that for the proofs from [7] and [8] the quasianalyticity of the weight sequence is indispensable; but can we transfer the results to this situation? This question has been asked by Prof. Javier Sanz after a talk of the author about the results from [8] during a research stay at the Universidad de Valladolid.…”

“…Unfortunately, it seems that for the proofs from [7] and [8] the quasianalyticity of the weight sequence is indispensable; but can we transfer the results to this situation? This question has been asked by Prof. Javier Sanz after a talk of the author about the results from [8] during a research stay at the Universidad de Valladolid.…”