Ultraholomorphic sectorial extensions of Beurling type

TL;DR

This paper establishes sectorial extension theorems for ultraholomorphic functions of Beurling type, utilizing a reduction lemma and growth indices to control sector openings, with implications for extension operators.

Contribution

It introduces new sectorial extension results for Beurling ultraholomorphic classes using a reduction lemma and growth indices, extending previous Roumieu case results.

Findings

Proved sectorial extension theorems for Beurling ultraholomorphic classes.

Derived extension results for classes defined by weight sequences.

Provided conditions for the existence of continuous linear extension operators.

Abstract

We prove sectorial extension theorems for ultraholomorphic function classes of Beurling type defined by weight functions with a controlled loss of regularity. The proofs are based on a reduction lemma, due to the second author, which allows to extract the Beurling from the Roumieu case, which was treated recently by Jim\'{e}nez-Garrido, Sanz, and the third author. In order to have control on the opening of the sectors, where the extensions exist, we use the (mixed) growth index and the order of quasianalyticity of weight functions. As a consequence we obtain corresponding extension results for classes defined by weight sequences. Additionally, we give information on the existence of continuous linear extension operators.