Volume growth, temperedness and integrability of matrix coefficients on a real spherical space

TL;DR

This paper investigates the geometric properties of real spherical spaces, focusing on volume growth, Schwartz spaces, and criteria for matrix coefficient integrability, advancing understanding of harmonic analysis on these spaces.

Contribution

It introduces a geometric criterion for $L^p$-integrability of matrix coefficients and applies the local structure theorem and polar decomposition to analyze volume growth.

Findings

Established a geometric criterion for $L^p$-integrability of matrix coefficients.

Controlled volume growth on real spherical spaces using structure theorems.

Defined the Harish-Chandra Schwartz space on these spaces.

Abstract

We apply the local structure theorem and the polar decomposition to a real spherical space Z=G/H and control the volume growth on Z. We define the Harish-Chandra Schwartz space on Z. We give a geometric criterion to ensure $L^p$-integrability of matrix coefficients on Z.