On evolution equations for Lie groupoids

TL;DR

This paper develops a framework using Fourier integral operators on Lie groupoids to analyze the fundamental solutions of certain evolution equations involving elliptic pseudodifferential operators, extending previous distribution studies.

Contribution

It introduces a novel approach combining Fourier integral operators and Lie groupoid calculus to study evolution equations and their fundamental solutions.

Findings

Constructed fundamental solutions for evolution equations on Lie groupoids.

Extended the analysis of distributions on Lie groupoids with C*-algebra considerations.

Investigated the local properties of regularizing operators in this context.

Abstract

Using the calculus of Fourier integral operators on Lie groupoids developped in [18], we study the fundamental solution of the evolution equation ($\partial$ $\partial$t + iP)u = 0 where P is a self adjoint elliptic order one G-pseudodifferential operator on the Lie groupoid G. Along the way, we continue the study of distributions on Lie groupoids done in [17] by adding the reduced C *-algebra of G in the picture and we investigate the local nature of the regularizing operators of [32].