On a theorem of Chernoff on rank one Riemannian symmetric spaces

TL;DR

This paper extends Chernoff's 1975 theorem from Euclidean space to all rank one Riemannian symmetric spaces, providing a quasi-analyticity criterion using Laplace-Beltrami operators.

Contribution

It proves an exact analogue of Chernoff's theorem for rank one symmetric spaces, broadening the theorem's applicability beyond Euclidean settings.

Findings

Established a Chernoff-type theorem for rank one symmetric spaces.

Connected iterates of Laplace-Beltrami operators to quasi-analyticity.

Unified treatment for noncompact and compact symmetric spaces.

Abstract

In 1975, P.R. Chernoff used iterates of the Laplacian on $\mathbb{R}^n$ to prove an $L^2$ version of the Denjoy-Carleman theorem which provides a sufficient condition for a smooth function on $\mathbb{R}^n$ to be quasi-analytic. In this paper, we prove an exact analogue of Chernoff's theorem for all rank one Riemannian symmetric spaces (of noncompact and compact types) using iterates of the associated Laplace-Beltrami operators.