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

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 {Yδ,j:δ ∈ K^0,1 ≤ j ≤ dδ} of L2(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 ht be the heat kernel associated to the Laplace-Beltrami operator and let Qδ(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(kar)| ≤ Cht(r). Further assume that for every δ and j the functions Fδ,j(λ)=Qδ(iλ+ϱ)-1∫K/Mƒ˜(λ,b)Y δj(b)db Satisfy the estimates |Fδ,j(λ) ≤ Cδje-tλ2 for λ ∈ R. Then ƒ is a constant multiple of heat kernel ht