Extension operators for some ultraholomorphic classes defined by sequences of rapid growth

TL;DR

This paper constructs extension operators for specific ultraholomorphic classes with rapid growth sequences, addressing a gap in the theory of asymptotic Borel mappings and their surjectivity.

Contribution

It provides explicit extension operators for certain ultraholomorphic classes beyond Gevrey regularity, using Borel- and Laplace-like transforms.

Findings

Extension operators are constructed for particular ultraholomorphic classes.

The classes considered are stable under differentiation, enabling the technique.

The approach applies to classes considered 'beyond Gevrey regularity'.

Abstract

While the asymptotic Borel mapping, sending a function into its series of asymptotic expansion in a sector, is known to be surjective for arbitrary openings in the framework of ultraholomorphic classes associated with sequences of rapid growth, there is no general procedure to construct extension operators in this case. We do provide such operators in complex sectors for some particular classes considered by S.~Pilipovi{\'c}, N.~Teofanov and F.~Tomi{\'c} in the ultradifferentiable setting. Although these classes are, in their words, "beyond Gevrey regularity", in some cases they keep the property of stability under differentiation, which is crucial for our technique, based on formal Borel- and truncated Laplace-like transforms with suitable kernels.