Optimal flat functions in Carleman-Roumieu ultraholomorphic classes in sectors

TL;DR

This paper develops optimal flat functions within Carleman-Roumieu ultraholomorphic classes for sectors, providing a general method for extension operators and illustrating with explicit examples like q-Gevrey classes.

Contribution

It introduces a general construction of optimal flat functions and extension operators in ultraholomorphic classes associated with strongly nonquasianalytic weight sequences.

Findings

Constructed optimal flat functions in ultraholomorphic classes.

Provided a general procedure for extension operators.

Explicit examples including q-Gevrey classes.

Abstract

We construct optimal flat functions in Carleman-Roumieu ultraholomorphic classes associated to general strongly nonquasianalytic weight sequences, and defined on sectors of suitably restricted opening. A general procedure is presented in order to obtain linear continuous extension operators, right inverses of the Borel map, for the case of regular weight sequences in the sense of Dyn'kin. Finally, we discuss some examples (including the well-known $q$-Gevrey case) where such optimal flat functions can be obtained in a more explicit way.