On a theorem of Chernoff on rank one Riemannian symmetric spaces

In 1975, P.R. Chernoff used iterates of the Laplacian on Rn to prove an L 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 and compact types) using iterates of the associated Laplace-Beltrami operators.

[1]  Generalized Spectral Projections on Symmetric Spaces of Noncompact Type: Paley–Wiener Theorems , 1996 .

[2]  Marcel de Jeu,et al.  Determinate multidimensional measures, the extended Carleman theorem and quasi-analytic weights , 2001, math/0111019.

[4]  S. Bochner,et al.  Some Theorems on Quasi-Analyticity for Functions of Several Variables , 1939 .

[5]  B. Kostant.  On the existence and irreducibility of certain series of representations , 1969 .

[6]  T. Sherman.  The Helgason Fourier transform for compact Riemannian symmetric spaces of rank one , 1990 .

[7]  C. de,et al.  Quatre leçons sur les fonctions quasi-analytiques de variable réelle , 2022 .

[8]  S. Thangavelu,et al.  An uncertainty principle for spectral projections on rank one symmetric spaces of noncompact type , 2020, Annali di Matematica Pura ed Applicata (1923 -).

[9]  S. Mossaheb.  A Note on Fourier Transforms , 1979 .

[10]  T. Koornwinder.  Jacobi Functions and Analysis on Noncompact Semisimple Lie Groups , 1984 .

[11]  Kenneth D. Johnson.  Composition series and intertwining operators for the spherical principal series. II , 1976 .

[12]  Michael E. Taylor,et al.  Differential Geometry I , 1994 .

[13]  M. Pinsky,et al.  Pointwise Fourier Inversion on Rank One Symmetric Spaces and Related Topics , 1997 .

[14]  Fractional integrals on compact Riemannian symmetric spaces of rank one , 2012, 1205.3957.

[15]  Tom H. Koornwinder,et al.  The addition formula for Jacobi polynomials and spherical harmonics : prepublication , 1973 .

[16]  S. K. Ray,et al.  Theorems of Ingham and Chernoff on Riemannian symmetric spaces of noncompact type , 2020 .

[17]  Sigurdur Helgason,et al.  Geometric Analysis on Symmetric Spaces , 1994 .

[18]  Mithun Bhowmik,et al.  A Theorem of Chernoff on Quasi-analytic Functions for Riemannian Symmetric Spaces , 2021, 2103.07667.

[19]  S. Helgason.  Groups and geometric analysis , 1984 .

[20]  A. Nussbaum.  Quasi-analytic vectors , 1965 .

[21]  N. Wallach,et al.  Composition series and intertwining operators for the spherical principal series , 1972 .

[22]  Classes of C∞ vectors and essential self-adjointness☆ , 1972 .

[23]  Quasi-analytic vectors and quasi-analytic functions , 1975 .

[24]  S. Thangavelu,et al.  An uncertainty principle for some eigenfunction expansions with applications , 2020 .