Uncertainty Principles for Heisenberg Motion Group

In Harmonic analysis, the uncertainty principle states that a nonzero function and its Fourier transform cannot simultaneously decay very rapidly. This fact is expressed by several versions which were proved by Hardy, Cowling-Price, Morgan, and Gelfand-Shilov [1, 2]. In more recent times, Beurling gave a different approach to expressing this uncertainty principle. The proof of the theorem was given by Hörmander [3], and it states that if f ∈ L2ðRÞ satisfying ð


Introduction
In Harmonic analysis, the uncertainty principle states that a nonzero function and its Fourier transform cannot simultaneously decay very rapidly. This fact is expressed by several versions which were proved by Hardy, Cowling-Price, Morgan, and Gelfand-Shilov [1,2].
In more recent times, Beurling gave a different approach to expressing this uncertainty principle. The proof of the theorem was given by Hörmander [3], and it states that if f ∈ L 2 ðℝÞ satisfying then, f = 0 almost everywhere. The above theorem of Hörmander was further generalized by Bonami, Demange, and Jaming [4], as follows: Then, f = 0 almost everywhere whenever N ≤ n, and if N > n, then f ðxÞ = PðxÞe −ajxj 2 , where a is a positive real number and P is a polynomial on ℝ n of degree < ðN − nÞ/2. This last theorem admits another modified version proved by Parui and Sarkar [5]. It is of the following form.

Theorem 2.
Let δ ≥ 0 and f ∈ L 2 ðℝ n Þ be such that where Q is a polynomial of degree m. Then, f ðxÞ = PðxÞe −ajxj 2 , where a is a positive real number and P is a polynomial with deg ðPÞ < ðN − n − mδÞ/2.
Beurling's theorem has been extended to different settings. Huang and Liu established an analogue of Beurling's theorem on the Heisenberg group [6]. An analogue of Beurling's theorem for Euclidean motion groups was also formulated by Sarkar and Thangavelu [7].
In [8], Baklouti and Thangavelu gave an analogue of Hardy's theorem for the Heisenberg motion group by means of the heat kernel and also proved an analogue of Miyachi's theorem and Cowling-Price uncertainty principle. In my paper, we would like to establish other uncertainty principles such as Beurling's theorem and Gelfand-Shilov and prove Hardy's theorem as a consequence of Beurling's theorem.
This paper is organized as follows. In Section 2, we present the group G and the Fourier transform on G, and we will cite some of its fundamental properties. Section 3 is devoted to formulate and prove an analogue of Beurling's theorem associated to the group Fourier transform on the Heisenberg motion group and prove a modified version of this principle. Finally, we derive some other versions of uncertainty principles such as Hardy uncertainty principle and Gelfand-Shilov.

Heisenberg Motion Group
Let ℍ n ≔ ℂ n × ℝ be the Heisenberg group with the group law where z, w ∈ ℂ n , t, s ∈ ℝ.
Let K be the unitary group UðnÞ, we define the Heisenberg motion group G to be the semidirect product of ℍ n and K, with the group law where ðz, tÞ, ðw, sÞ ∈ ℍ n , k, h ∈ K.
The Haar measure on G is given by dg = dzdtdk, where dzdt and dk are the normalized Haar measures on ℍ n and K , respectively.
Let ðσ, H σ Þ be any irreducible, unitary representation of K. For each λ ≠ 0, we consider the representations ρ λ σ of G on the tensor product space L 2 ðℝ n Þ ⊗ H σ defined by where μ λ are the metaplectic representations [9], satisfying Proposition 1 [9]. Each ρ λ σ is unitary and irreducible. For f ∈ L 1 ∩ L 2 ðGÞ, consider the group Fourier transform where ρ λ σ ðz, kÞ = ρ λ σ ðz, 0, kÞ and the partial Fourier transform f λ ðz, kÞ is defined by and the Plancherel formula for the Fourier transform on G reads as where dτðλÞ = ð2πÞ −n−1 jλj n dλ is the measure defined on ℝ \ f0g, d σ is the dimension of the space H σ , and kf ðλ, σÞk 2 HS denote the Hilbert-Schmidt norm off ðλ, σÞ [9]. At the end of this paragraph, we introduce an orthonormal basis for L 2 ðℂ n × KÞ [10]. Let H k ðtÞ be the Hermite polynomials defined by The normalized Hemite functions are defined by The n-dimensional Hermite functions Φ α are defined on ℝ n by taking the tensor products; that is, where α = ðα 1 , ⋯, α n Þ ∈ ℕ n . It is well known that fΦ α , α ∈ ℕ n g form an orthonormal basis for L 2 ðℝ n Þ [2]. Then, an orthonormal basis for L 2 ðℝ n Þ ⊗ H σ is given by Lemma 1 [10]. For f l , g l ∈ L 2 ðℝ n Þ ⊗ H σ , l = 1, 2, the following identity holds.

An Analogue of Beurling's Theorem
In this section, we prove an analogue of Beurling's theorem on the Heisenberg motion group G = ℍ n ⋊K whose statement is as follows: Theorem 3. Let f ∈ L 2 ðGÞ and d ≥ 0. Suppose that Then, where a > 0, φ j ∈ L 1 ∩ L 2 ðℂ n × KÞ and m < ððd − n/2 − 1Þ/2Þ.
Proof of Theorem 3. For any φ ∈ Sðℂ n × KÞ, the Schwartz space of ℂ n × K, consider the function Since f ∈ L 1 ðGÞ, then, F φ is integrable on ℝ, and for any λ ∈ ℝ \ f0g, the Fourier transform of F φ is given by 3 Abstract and Applied Analysis then by (11) As a result, In particular, Note that the previous calculations are generalized to a bounded function φ ∈ L 2 ðℂ n × KÞ, in particular for the bounded functions φ in the basis B of L 2 ðℂ n × KÞ defined in (18).
According to Beurling's theorem in the Euclidean case, modified version (Theorem 2), for every function φ ∈ B, there exists a polynomial function P φ with deg ðP φ Þ < ðd − ðn/2Þ − 1Þ/2 and a real a φ > 0 such that from where Let φ, ψ ∈ B, since then by Lemma 2.2 in [5], we obtain that a φ = a ψ = a are independent of φ.
The proof of Theorem 3 is completed.
We will finish this section with a modified version of previous Theorem 3 as follows: Then, where a > 0, φ j ∈ L 1 ∩ L 2 ðℂ n × KÞ and m < ðd − ðn/2Þ − 1 − δÞ/2: Proof. By replacing f ðz, t, kÞ by f ðz, t, kÞ/ð1 + kzkÞ p and proceeding as in the proof of Theorem 3, one can apply Theorem 3 to get the result.

Applications to Other Uncertainty Principles
Let us first state and prove the following analogue of Hardy's theorem for G.
Proof. From (i) and (ii), we have