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Abstract. We establish several versions of Hardy's theorem for the Fourier transform on the Heisenberg group. Let be the Fourier transform of a function f on and assume where is the heat kernel associated to the sublaplacian. We show that if then whenever . When we replace the condition on f by where is the Fourier transform of f in the t -variable. Under suitable assumptions on the ‘spherical harmonic coefficients?of we prove: (i) when a = b ; (ii) when a > b there are infinitely many linearly independent functions f satisfying both conditions on and .
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