Sectorial extensions, via Laplace transforms, in ultraholomorphic classes defined by weight functions

TL;DR

This paper extends ultraholomorphic function classes defined by weight functions using Laplace transform techniques, generalizing previous results and providing new extension theorems in sectorial domains.

Contribution

It introduces a novel approach to extension theorems for ultraholomorphic classes via Laplace transforms, broadening the framework beyond weight sequences.

Findings

Established extension theorems for classes defined by weight functions

Constructed kernels for Laplace-like integral transforms

Achieved minimal assumptions in mixed weight-sequence settings

Abstract

We prove several extension theorems for Roumieu ultraholomorphic classes of functions in sectors of the Riemann surface of the logarithm which are defined by means of a weight function or weight matrix. Our main aim is to transfer the results of V. Thilliez from the weight sequence case to these different, or more general, frameworks. The technique rests on the construction of suitable kernels for a truncated Laplace-like integral transform, which provides the solution without resorting to Whitney-type extension results for ultradifferentiable classes. As a byproduct, we obtain an extension in a mixed weight-sequence setting in which assumptions on the sequence are minimal.