Hardy's theorem for the helgason Fourier transform on noncompact rank one symmetric spaces

Abstract

Let G be a semisimple Lie group with Iwasawa decomposition G = KAN. Let X=G/K be the associated symmetric space and assume that X is of rank one . Let M be the centraliser of A in K and consider an orthonormal basis {Y δ,j :δ ∈ K^ 0 ,1 ≤ j ≤ d δ } of L 2 (K/M) consisting of K-finite functions of type δ on K/M. For a function ƒ on X let ƒ˜ (λ,b), λ ∈ C, be the Helgason Fourior transform. Let h t be the heat kernel associated to the Laplace-Beltrami operator and let Q δ (iλ+ϱ) bethe Kostant polynomials. We establish the following version of Hardy's theorem for the Helgason Fourier transform: Let ƒ be a function on G/K whcih satisfies |ƒ(ka r )| ≤ Ch t (r). Further assume that for every δ and j the functions F δ,j (λ)=Q δ (iλ+ϱ) -1 ∫ K/M ƒ˜(λ,b)Y δj (b)db Satisfy the estimates |F δ,j (λ) ≤ C δj e -tλ 2 for λ ∈ R. Then ƒ is a constant multiple of heat kernel h t .