Generalized Fourier Integral Operators on spaces of Colombeau type

TL;DR

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

Contribution

It introduces a comprehensive framework for generalized FIOs acting on Colombeau algebras, including their mapping properties and effects on wave front sets.

Findings

Defined generalized Fourier integral operators on Colombeau algebras.

Analyzed the composition with generalized pseudodifferential operators.

Investigated microlocal regularity and wave front set influence.

Abstract

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