The Borel map in the mixed Beurling setting

TL;DR

This paper characterizes when the Borel map's image of ultradifferentiable classes in the Beurling setting covers certain sequence spaces, using advanced functional analysis and complex analysis techniques.

Contribution

It provides a comprehensive characterization of the Borel map's image in the mixed Beurling ultradifferentiable setting, unifying classical and modern classes.

Findings

Characterization of Borel map images in terms of weight sequences

Two independent solutions: reduction to Roumieu case and dualization approach

Application of complex analysis and PDE techniques to ultradifferentiable classes

Abstract

The Borel map takes a smooth function to its infinite jet of derivatives (at zero). We study the restriction of this map to ultradifferentiable classes of Beurling type in a very general setting which encompasses the classical Denjoy-Carleman and Braun-Meise-Taylor classes. More precisely, we characterize when the Borel image of one class covers the sequence space of another class in terms of the two weights that define the classes. We present two independent solutions to this problem, one by reduction to the Roumieu case and the other by dualization of the involved Fr\'echet spaces, a Phragm\'en-Lindel\"of theorem, and H\"ormander's solution of the $\bar{\partial}$-problem.