The Borel map in the mixed Beurling setting

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 Fréchet spaces, a Phragmén–Lindelöf theorem, and Hörmander’s solution of the $$\overline{\partial }$$ ∂ ¯ -problem.

Optimal Flat Functions in Carleman–Roumieu Ultraholomorphic Classes in Sectors

“…We can characterize the previous condition in terms of the index γ(M). This connection has very recently appeared for the first time in a work of Nenning et al [22]. Although the additional hypothesis of (dc) appears in their (indirect) arguments, it can be removed as long as the sequence satisfies (snq), as we now show.…”

“…The key idea comes from a recent paper by Nenning et al [22], where they have studied the mixed Borel problem in Beurling ultradifferentiable classes. They consider a mixed condition inspired by a related one (see (3.7) in this paper) appearing in a work of Langenbruch [17].…”