Microlocal analysis in generalized function algebras based on generalized points and generalized directions

TL;DR

This paper advances microlocal analysis within Colombeau generalized function algebras by defining wave fronts as sets of generalized points, providing a more refined and consistent framework compared to traditional methods.

Contribution

It introduces a novel approach to microlocal analysis by using generalized points for wave front sets in Colombeau algebras, enhancing the theory's precision and consistency.

Findings

Wave front sets are characterized as sets of generalized points in the cotangent bundle.

The new approach is shown to be consistent with existing non-generalized point methods.

The refined theory improves the analysis of singularities in generalized functions.

Abstract

We develop a refined theory of microlocal analysis in the algebra ${\mathcal G}(\Omega)$ of Colombeau generalized functions. In our approach, the wave front is a set of generalized points in the cotangent bundle of $\Omega$, whereas in the theory developed so far, it is a set of nongeneralized points. We also show consistency between both approaches.