On the maximal extension in the mixed ultradifferentiable weight sequence setting

TL;DR

This paper investigates the maximal extension of ultradifferentiable weight sequences where the Borel map remains surjective, analyzing the controlled loss of regularity and comparing optimal sequences for different nonquasianalyticity conditions.

Contribution

It determines the maximal sequences allowing mixed ultradifferentiable classes with controlled surjectivity loss and compares these optimal sequences across different nonquasianalyticity conditions.

Findings

Identifies the maximal weight sequences for mixed ultradifferentiable classes.

Provides a comparison of optimal sequences for strong and non-strong nonquasianalyticity.

Analyzes the controlled loss of surjectivity in the Borel map.

Abstract

For the ultradifferentiable weight sequence setting it is known that the Borel map which assigns to each function the infinite jet of derivatives (at 0) is surjective onto the corresponding weighted sequence class if and only if the sequence is strongly nonquasianalytic for both the Roumieu- and Beurling-type classes. Sequences which are nonquasianalytic but not strongly nonquasianalytic admit a controlled loss of regularity and we determine the maximal sequence for which such a mixed setting is possible for both types, hence get information on the controlled loss of surjectivity in this situation. Moreover, we compare the optimal sequences for both mixed strong nonquasianalyticity conditions arising in the literature.