Quasianalyticity of \(L^p\)-functions on Riemannian symmetric spaces of noncompact type

A result of Chernoff gives sufficient condition for an \(L^2\)-function on \({\mathbb { R}}^n\) to be quasi-analytic, in the sense that the function and all its derivatives cannot vanish at a point. This is a generalization of the classical Denjoy–Carleman theorem on \({\mathbb { R}}\) and of the subsequent works on \({\mathbb { R}}^n\) 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.