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 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 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/Mƒ˜(λ,b)Y _δj(b)db Satisfy the estimates |F_δ,j(λ) ≤ C_δje^-tλ2 for λ ∈ R. Then ƒ is a constant multiple of heat kernel h_t.