Revisiting Hardy's theorem for the Heisenberg group

Abstract

We establish several versions of Hardy's theorem for the Fourier transform on the Heisenberg group. Let ƒ^ (λ) be the Fourier transform of a function ƒ on Hⁿ and assume ƒ^ (λ)*ƒ^ (λ) ≤ c p^2b (λ) where p_s is the heat kernel associated to the sublaplacian. We show that if │ ƒ(z, t)│ ≤ c p_a (z, t) then ƒ = 0 whenever a < b. When a ≥ b we replace the condition on ƒ by │ ƒ ^λ (z) │ ≤ c p_a^λ(z) where ƒ^λ (z) is the Fourier transform of ƒ in the t-variable. Under suitable assumptions on the 'spherical harmonic coefficients' of ƒ^ (λ) we prove: (i) ƒ^λ(z) = c(λ) p_a^λ(z) when a=b; (ii) when a > b there are infinitely many linearly independent functions ƒ satisfying both conditions on ƒ^λ and ƒ^(λ).