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

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Let G be a connected simply connected nilpotent Lie group. In [A. Baklouti, N. Ben Salah, The L p − L q version of Hardy's Theorem on nilpotent Lie groups, Forum Math. 18 (2006) 245–262], we proved for 2 ⩽ p , q ⩽ + ∞ the L p − L q 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.