Microlocal analysis in the dual of a Colombeau algebra: generalized wave front sets and noncharacteristic regularity

TL;DR

This paper develops new microlocal analysis tools for the dual of Colombeau algebras, introducing generalized wave front sets to measure regularity and providing Fourier transform characterizations for specific functionals.

Contribution

It introduces novel notions of wave front sets for dual Colombeau algebra functionals and characterizes their regularity using Fourier transforms for a subclass of these functionals.

Findings

Defined new wave front set concepts for dual Colombeau algebra functionals

Provided Fourier transform characterizations for basic structure functionals

Established results on noncharacteristic regularity in the generalized setting

Abstract

We introduce different notions of wave front set for the functionals in the dual of the Colombeau algebra $\Gc(\Om)$ providing a way to measure the $\G$ and the $\Ginf$- regularity in $\LL(\Gc(\Om),\wt{\C})$. For the smaller family of functionals having a ``basic structure'' we obtain a Fourier transform-characterization for this type of generalized wave front sets and results of noncharacteristic $\G$ and $\Ginf$-regularity.