Quasi-analytic $L^p$-functions on Riemannian symmetric spaces of noncompact type, a theorem of Chernoff

TL;DR

This paper extends Chernoff's quasi-analyticity result from $L^2$ functions on Euclidean space to $L^p$ functions on noncompact Riemannian symmetric spaces, without restrictions on rank or symmetry.

Contribution

It provides an exact analogue of Chernoff's theorem for $L^p$ functions on symmetric spaces of noncompact type, broadening the scope beyond Euclidean spaces.

Findings

Generalizes Chernoff's result to $L^p$ spaces with $p eq 2$

Applies to symmetric spaces of any rank without symmetry restrictions

No additional conditions needed for the quasi-analyticity criterion

Abstract

A result of Chernoff gives sufficient condition for an $L^2$-function on $\R^n$ to be quasi-analytic. This is a generalization of the classical Denjoy-Carleman theorem on $\R$ and of the subsequent work on $\R^n$ by Bochner and Taylor. In this note we endeavour to obtain an exact analogue of the result of Chernoff for $L^p, p\in [1,2]$ functions on the Riemannian symmetric spaces of noncompact type. No restriction on the rank of the symmetric spaces and no condition on the symmetry of the functions is assumed.