Beurling’s Theorem and $L^p$−$L^q$ Morgan’s Theorem for Step Two Nilpotent Lie Groups

Abstract

We prove Beurling's theorem and L p − L q Morgan's theorem for step two nilpotent Lie groups. These two theorems together imply a group of uncertainty theorems.

A Beurling–Hörmander theorem associated with the Riemann–Liouville operator

“…R. P. Sarkar and J. Sengupta [2007b] established the analogue of the Beurling theorem on the full group SL(2, ‫. )ޒ‬ As for the noncompact semisimple Lie group case, S. Thangavelu [2004] first gave the analogue on rank 1 symmetric spaces with an additional condition like the one required in the Cowling-Price theorem, so he called it the Cowbeurling Theorem; then R. P. Sarkar and J. Sengupta [2007a] removed this additional condition and gave the analogue in rank 1 symmetric spaces; recently, L. Bouattour [2008] generalized this result and gave the analogue for real symmetric spaces of rank d. For more Beurling theorems in different settings, refer to [Kamoun and Trimèche 2005;Parui and Sarkar 2008].…”

Heisenberg uniqueness pairs for the Fourier transform on the Heisenberg group

“…Hence G can be identified with b ⊕ z. Thus, we can write X + T ∈ b ⊕ z for exp(X + T ) and denote it by (X, For more details, please refer to [8,18,19].…”

“…If f ∈ L 1 ∩ L 2 (G), then f (λ, µ) is a Hilbert Schmidt operator on L 2 (η λ ) and satisfies (see [22]) P f (λ) f (λ, µ) 2 HS = (2π) n η λ ξ λ |f λ,µ (x, y)| 2 dxdy. (7.1) For f ∈ L 2 (G) we have the Plancherel formula (see [18])…”

Qualitative uncertainty principles for the Opdam–Cherednik transform

“…The above theorem of Hörmander was further generalized by Parui and Sarkar [20] which also accommodates the optimal point of this trade-off between the function and its Fourier transform.…”