On theorems of Beurling and Cowling-Price for certain nilpotent Lie groups

Let G be a connected simply connected nilpotent Lie group. In A. Baklouti, N. Ben Salah, The Lp−Lq version of Hardy's Theorem on nilpotent Lie groups, Forum Math. 18 (2006) 245-262], we proved for 2?p,q?+∞ the Lp−Lq version of Hardy's Theorem known as the Cowling-Price Theorem. In the setup where 1?p,q?+∞, the problem is still unsolved and the upshot is known only for few cases. We prove in this paper such a result in the context of 2-NPC nilpotent Lie groups. A proof of the analogue of Beurling's Theorem is also provided in the same context.