A Volterra Calculus for Lie Groupoids

TL;DR

This paper develops a pseudodifferential Volterra calculus tailored for Lie groupoids, facilitating the analysis of heat kernels and fundamental solutions on singular manifolds with corners, advancing geometric analysis in complex settings.

Contribution

It introduces a novel Volterra calculus for Lie groupoids, enabling the study of heat flows and fundamental solutions on singular manifolds with corners.

Findings

Established short time asymptotic expansion for heat kernels on Lie groupoids.

Developed a calculus for inverting parabolic differential equations on Lie groupoids.

Applied the calculus to analyze heat flows on manifolds with corners.

Abstract

A pseudodifferential Volterra calculus for inverting parabolic differential equations on Lie groupoids is introduced. This enables the study of fundamental solutions of various cases of heat flows on singular manifolds with corners with non-resonant boundary indicial symbols, such as the $b$-manifolds, as well as other geometric bisection covariant heat flows. We also establish the short time asymptotic expansion for the heat kernel of a positive, elliptic differential operator on a Lie groupoid that acts on suitable Sobolev Hilbert modules and is positive definite with respect to the appropriate $L^2$ inner product.