On Paley-Wiener and Hardy theorems for NA groups

Abstract

Let N be a H-type group and let S=NA be an one dimensional solvable extension of N. For the Helgason Fourier transform on S we prove the following analogue of Hardy's theorem. Let ƒ^ (λ, Y, Z) stand for the Helgason Fourier transform of ƒ and let h_α denote the heat kernel associated to the Laplace-Beltrami operator. Suppose a function ƒ on S satisfies the conditions │ ƒ (x) │ ≤ c h_α (x) and ∫_N │ ƒ^ (λ, Y, Z) │² (1 + │Z│²)^γ dY dZ ≤ c e ^-2βλ2 for all x ∈ S, λ ∈ R where γ > k-1/2, k being the dimension of the centre on N. Then ƒ = 0 of ƒ = ch_α depending on whether α < β or α = β. We also establish a stronger version of Hardy's theorem and a Paley-Wiener theorem. These are generalisations of the corresponding results for rank one symmetric spaces of noncompact type.