Generalized Oscillatory Integrals and Fourier Integral Operators

TL;DR

This paper develops a comprehensive theory of generalized oscillatory integrals and Fourier integral operators within Colombeau algebras, enabling analysis of PDEs with non-smooth coefficients and distribution data.

Contribution

It introduces a new framework for generalized FIOs with Colombeau-type phase functions and amplitudes, extending microlocal analysis to non-smooth contexts.

Findings

Defined generalized FIOs acting on Colombeau algebras

Studied mapping properties and microlocal regularity of these operators

Analyzed the impact on generalized wave front sets

Abstract

In this article, a theory of generalized oscillatory integrals (OIs) is developed whose phase functions as well as amplitudes may be generalized functions of Colombeau type. Based on this, generalized Fourier integral operators (FIOs) acting on Colombeau algebras are defined. This is motivated by the need of a general framework for partial differential operators with non-smooth coefficients and distribution data. The mapping properties of these FIOs are studied, as is microlocal Colombeau regularity for OIs and the influence of the FIO action on generalized wave front sets.