On a theorem of Chernoff on rank one Riemannian symmetric spaces

Abstract

In 1975, P.R. Chernoff used iterates of the Laplacian on R n to prove an L 2 version of the Denjoy-Carleman theorem which provides a sufficient condition for a smooth function on 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 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.

Analogues of theorems of Chernoff and Ingham on the Heisenberg group

“…Proving an exact analogue of Chernoff's theorem is still open though there are some partial results. Recently in [4] the authors have proved an exact analogue of Chernoff's theorem for K-biinvariant functions on the group G. Under the assumption that X is of rank one, we have proved an exact analogue of Chernoff's theorem in a joint work with R. Manna [12]: Theorem 1.3 (Ganguly-Manna-Thangavelu). Let X = G/K be a rank one Riemannian symmetric space of noncompact type.…”

An analogue of Ingham’s theorem on the Heisenberg group

“…Unlike the previous one, this technique has also been proven to be beneficial in the contexts of rank one Reimannian symmetric spaces. See [12] for further details in this regard.…”