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 $\bar{\partial }$-problem.

33 pages; some typos corrected

References in corpus (4)

• On subadditivity-like conditions for associated weight functions
• Equality of ultradifferentiable classes by means of indices of mixed O-regular variation
• Ultradifferentiable extension theorems: a survey
• On the maximal extension in the mixed ultradifferentiable weight sequence setting

Cited by in corpus (3)

• On generalized definitions of ultradifferentiable classes
• Optimal flat functions in Carleman-Roumieu ultraholomorphic classes in sectors
• On inclusion relations of weighted $L^p$-type spaces defined in terms of weight function matrices