# Microlocal Asymptotic Analysis in Algebras of Generalized Functions

## TL;DR

This paper develops a novel microlocal asymptotic analysis framework within algebras of generalized functions, enabling better understanding of singularities and their propagation through nonlinear operations, distinct from classical Fourier-based methods.

## Contribution

It introduces a new microlocal analysis approach based on presheaf properties and regularizing parameters, expanding the tools for studying singularities in nonlinear contexts.

## Key findings

- Defined a singular asymptotic spectrum with favorable nonlinear properties
- Demonstrated propagation of singularities through nonlinear operators
- Provided examples illustrating the new analysis framework

## Abstract

We introduce a new type of local and microlocal asymptotic analysis in algebras of generalized functions, based on the presheaf properties of those algebras and on the properties of their elements with respect to a regularizing parameter. Contrary to the more classical frequential analysis based on the Fourier transform, we can describe a singular asymptotic spectrum which has good properties with respect to nonlinear operations. In this spirit we give several examples of propagation of singularities through nonlinear operators.