Integral operators on the Oshima compactification of a Riemannian symmetric space of non-compact type. Microlocal analysis and kernel asymptotics

TL;DR

This paper analyzes integral operators on the Oshima compactification of non-compact Riemannian symmetric spaces, characterizing their kernels using microlocal analysis and describing asymptotic behaviors of associated semigroups and resolvents.

Contribution

It introduces a pseudodifferential operator framework for integral operators on the Oshima compactification, detailing their kernel singularities and asymptotic properties.

Findings

Characterization of integral operators as pseudodifferential operators.

Description of kernel singularities and asymptotics.

Analysis of semigroup and resolvent kernels for elliptic operators.

Abstract

Let $\X\simeq G/K$ be a Riemannian symmetric space of non-compact type, $\widetilde \X$ its Oshima compactification, and $(\pi,\mathrm{C}(\widetilde \X))$ the regular representation of $G$ on $\widetilde \X$. We study integral operators on $\widetilde \X$ of the form $\pi(f)$, where $f$ is a rapidly falling function on $G$, and characterize them within the framework of pseudodifferential operators, describing the singular nature of their kernels. In particular, we consider the holomorphic semigroup generated by a strongly elliptic operator associated to the representation $\pi$, as well as its resolvent, and describe the asymptotic behavior of the corresponding semigroup and resolvent kernels.