Exponential coordinates and regularity of groupoid heat kernels

TL;DR

This paper demonstrates that on asymptotically Euclidean boundary groupoids, the heat kernel associated with the Laplacian is a smooth pseudo-differential operator, extending understanding of heat kernel regularity in complex geometric settings.

Contribution

It establishes the smoothness of the heat kernel as a groupoid pseudo-differential operator on a specific class of boundary groupoids, using exponential coordinates.

Findings

Heat kernel is smooth on asymptotically Euclidean boundary groupoids.

Laplacian's heat kernel belongs to the class of groupoid pseudo-differential operators.

Provides a framework for analyzing heat kernels in non-compact geometric contexts.

Abstract

We prove that on an asymptotically Euclidean boundary groupoid, the heat kernel of the Laplacian is a smooth groupoid pseudo-differential operator.