Ultradifferentiable extension theorems: a survey

TL;DR

This survey reviews ultradifferentiable extension theorems, focusing on Denjoy-Carleman classes, their properties, and the existence of continuous linear extension operators, providing an accessible introduction to the subject.

Contribution

It offers a comprehensive overview of ultradifferentiable extension theorems, emphasizing the development of the theory for Denjoy-Carleman classes and related concepts.

Findings

Development of the theory for Denjoy-Carleman classes

Discussion of (non-)quasianalyticity in ultradifferentiable classes

Introduction to Braun-Meise-Taylor classes via intersections and unions

Abstract

We survey ultradifferentiable extension theorems, i.e., quantitative versions of Whitney's classical extension theorem, with special emphasis on the existence of continuous linear extension operators. The focus is on Denjoy-Carleman classes for which we develop the theory from scratch and discuss important related concepts such as (non-)quasianalyticity. It allows us to give an efficient and, to a fair extent, elementary introduction to Braun-Meise-Taylor classes based on their representation as intersections and unions of Denjoy-Carleman classes.