Revisiting Hardy's theorem for the Heisenberg group |
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Authors: | S Thangavelu |
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Institution: | (1) Stat-Math Division, Indian Statistical Institute, 8th Mile Mysore Road, Bangalore, 560 059, India (e-mail: veluma@isibang.ac.in) , IN |
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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 .
Received: 9 January 2001 / in final form: 17 April 2001 Published online: 1 February 2002 |
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