On a theorem of Chernoff on rank one Riemannian symmetric spaces

Abstract

In 1975, P.R. Chernoff used iterates of the Laplacian on Rn to prove an L2 version of the Denjoy-Carleman theorem which provides a sufficient condition for a smooth function on Rn 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 type using iterates of the associated Laplace-Beltrami operators. Moreover, we also prove an analogue of Chernoff's theorem for the sphere which is a rank one compact symmetric space.