# Almost analytic extensions of ultradifferentiable functions with applications to microlocal analysis

## TL;DR

This paper explores the use of almost analytic extensions to characterize ultradifferentiable functions and applies these concepts to microlocal analysis, including wave front sets and regularity theorems, in a broad and unified framework.

## Contribution

It extends the theory of ultradifferentiable functions via almost analytic extensions and applies this to develop new microlocal analysis tools and results.

## Key findings

- Defined ultradifferentiable wave front set using almost analytic extensions and FBI transform
- Extended elliptic regularity and Holmgren uniqueness theorems to ultradifferentiable setting
- Provided a unified framework encompassing classical ultradifferentiable classes

## Abstract

We review and extend the description of ultradifferentiable functions by their almost analytic extensions, i.e., extensions to the complex domain with specific vanishing rate of the $\bar \partial$-derivative near the real domain. We work in a general uniform framework which comprises the main classical ultradifferentiable classes but also allows to treat unions and intersections of such. The second part of the paper is devoted to applications in microlocal analysis. The ultradifferentiable wave front set is defined in this general setting and characterized in terms of almost analytic extensions and of the FBI transform. This allows to extend its definition to ultradifferentiable manifolds. We also discuss ultradifferentiable versions of the elliptic regularity theorem and obtain a general quasianalytic Holmgren uniqueness theorem.