# Uncertainty Principles for Heisenberg Motion Group

Authors: Walid Amghar
Journal: Abstract and Applied Analysis
 (2021)

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## Abstract

In this article, we will recall the main properties of the Fourier transform on the Heisenberg motion groupG=ℍn⋊K, where K=Un and ℍn=ℂn×ℝ denote the Heisenberg group. Then, we will present some uncertainty principles associated to this transform as Beurling, Hardy, and Gelfand-Shilov.


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## Body

## 1. 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∈L2ℝ satisfying
(1)∫ℝ∫ℝfxf^yexydxdy<∞,then, f=0 almost everywhere.The above theorem of Hörmander was further generalized by Bonami, Demange, and Jaming [4], as follows:Theorem 1.
LetN≥0 and let f∈L2ℝn satisfying
(2)∫ℝn∫ℝnfxf^y1+∣x∣+∣y∣Nexydxdy<∞.
Then,f=0 almost everywhere whenever N≤n, and if N>n, then fx=Pxe−ax2, where a is a positive real number and P is a polynomial on ℝn of degree <N−n/2.This last theorem admits another modified version proved by Parui and Sarkar [5]. It is of the following form.Theorem 2.
Letδ≥0 and f∈L2ℝn be such that
(3)∫ℝn∫ℝnfxf^yQyδ1+∣x∣+∣y∣Nexydxdy<∞,where Q is a polynomial of degree m. Then, fx=Pxe−ax2, where a is a positive real number and P is a polynomial with degP<N−n−mδ/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 Section2, 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.


## 2. Heisenberg Motion Group

Letℍn≔ℂn×ℝ be the Heisenberg group with the group law
(4)z,t.w,s=z+w,t+s+12Imz.w¯,where z,w∈ℂn,t,s∈ℝ.LetK be the unitary group Un, we define the Heisenberg motion group G to be the semidirect product of ℍn and K, with the group law
(5)z,t,kw,s,h=z,t.kw,s,kh,where z,t,w,s∈ℍn,k,h∈K.The Haar measure onG is given by dg=dzdtdk, where dzdt and dk are the normalized Haar measures on ℍn and K, respectively.Forλ∈ℝ\0, we define the Schrödinger representation of ℍn on L2ℝn by
(6)πλz,tφξ=eiλteiλx.ξ+12x.yφξ+y,where z=x+iy and φ∈L2ℝn.Letσ,Hσ be any irreducible, unitary representation of K. For each λ≠0, we consider the representations ρσλ of G on the tensor product space L2ℝn⊗Hσ defined by
(7)ρσλz,t,k=πλz,tμλk⊗σk,where μλ are the metaplectic representations [9], satisfying
(8)πλkz,t=μλkπλz,tμλk∗,forallz,t,k∈G.Proposition 1 [9].
Eachρσλ is unitary and irreducible.
Forf∈L1∩L2G, consider the group Fourier transform
(9)f^λ,σ=∫K∫ℝ∫ℂnfz,t,kρσλz,t,kdzdtdk=∫K∫ℂnfλz,kρσλz,kdzdk,where ρσλz,k=ρσλz,0,k and the partial Fourier transform fλz,k is defined by
(10)fλz,k=∫ℝfz,t,keiλtdt.Forf∈L1∩L2G, we have
(11)∫K∫ℂnfλz,k2dzdk=2π−nλn∑σ∈K^dσf^λ,σHS2,and the Plancherel formula for the Fourier transform on G reads as
(12)∫K∫ℍnfz,t,k2dzdtdk=∑σ∈K^dσ∫ℝ\0f^λ,σHS2dτλ,where dτλ=2π−n−1λndλ is the measure defined on ℝ\0, dσ is the dimension of the space Hσ, and f^λ,σHS2 denote the Hilbert-Schmidt norm of f^λ,σ [9].At the end of this paragraph, we introduce an orthonormal basis forL2ℂn×K [10]. Let Hkt be the Hermite polynomials defined by
(13)Hkt=−1ket2dkdtke−t2.The normalized Hemite functions are defined by(14)hkt=2kπk!−1/2Hkte−1/2t2.Then-dimensional Hermite functions Φα are defined on ℝn by taking the tensor products; that is,
(15)Φαx=∏j=1nhαjxj,where α=α1,⋯,αn∈ℕn.It is well known thatΦα,α∈ℕn form an orthonormal basis for L2ℝn [2]. Then, an orthonormal basis for L2ℝn⊗Hσ is given by Bσ=Φα⊗eiσ:α∈ℕn,1≤i≤dσ, where eiσ:1≤i≤dσ is an orthonormal basis for Hσ and dσ=dimHσ.Define the Fourier-Wigner transformVfg of f,g∈L2ℝn⊗Hσ on ℂn×K by
(16)Vfgz,k=2π−n/2ρσ1z,kf,g.Lemma 1 [10].
Forfl,gl∈L2ℝn⊗Hσ,l=1,2, the following identity holds.
(17)∫K∫ℂnVf1g1z,kVf2g2z,k¯dzdk=f1,f2g1,g2¯.
In particular,Vfg∈L2ℂn×K, for f,g∈L2ℝn⊗Hσ.SetΨα,iσ=Φα⊗eiσ, then, by Lemma 1, we infer that the set
(18)VBσ=VΨα,iσΨβ,jσ:Ψα,iσ,Ψβ,jσ∈Bσis an orthonormal basis for Vσ=span¯Vfg:f,g∈L2ℝn⊗Hσ.Proposition 2 [10].
The familyB=VBσ:σ∈K^ is an orthonormal basis for L2ℂn×K.Lemma 2.
The functionz,k↦VΨα,iσΨβ,jσz,k is a bounded function.Proof.
Letz,k∈ℂn×K and Ψα,iσ,Ψβ,jσ∈Bσ, we have
(19)VΨα,iσΨβ,jσz,k=2π−n/2ρσ1z,kΦα⊗eiσ,Φβ⊗ejσ=2π−n/2π1zμ1k⊗σkΦα⊗eiσ,Φβ⊗ejσ=2π−n/2π1zμ1kΦα,Φβσkeiσ,ejσ.
We know thatμ1kΦα=∑∣γ∣=∣α∣μ1kΦα,ΦγΦγ (see [9] p.21); then
(20)VΨα,iσΨβ,jσz,k≤2π−n/2∑∣γ∣=∣α∣μ1kΦα,Φγπ1zΦγ,Φβσkeiσ,ejσ.
Sinceπ1, μ1, and σ are unitary representations, then,
(21)∣VΨα,iσΨβ,jσz,k∣≤2π−n/2m+n−1!m!n−1!,where m=∣α∣.


## 3. An Analogue of Beurling’s Theorem

In this section, we prove an analogue of Beurling’s theorem on the Heisenberg motion groupG=ℍn⋊K whose statement is as follows:Theorem 3.
Letf∈L2G and d≥0. Suppose that
(22)∫G∫ℝ∣fz,t,k∣∑σ∈K^dσf^λ,σHS21/21+∣t∣+∣λ∣detλdτλdzdtdk<∞.
Then,(23)fz,t,k=e−at2∑j=0mφjz,ktj,where a>0, φj∈L1∩L2ℂn×K and m<d−n/2−1/2.Lemma 3.
Iff satisfies the hypotheses of Theorem 3, then, f∈L1G.Proof.
Asf is not identically zero, then, there exists λ∈ℝ\0 such that
(24)∑σ∈K^dσf^λ,σHS2≠0.
By (22), we have
(25)∫G∣fz,t,k∣1+∣t∣+∣λ∣detλdzdtdk<∞.
On the other hand, the functiont,λ↦1+∣t∣+∣λ∣d/etλ is bounded, so there exists a constant C such that
(26)1+∣t∣+∣λ∣detλ≤C.
From where∫G∣fz,t,k∣dzdtdk≤C∫G∣fz,t,k∣/1+∣t∣+∣λ∣detλdzdtdk and using (25), so we have f∈L1G.Proof of Theorem 3.
For anyφ∈Sℂn×K, the Schwartz space of ℂn×K, consider the function
(27)Fφt=∫ℂn×Kfz,t,kφz,kdzdk.
Sincef∈L1G, then, Fφ is integrable on ℝ, and for any λ∈ℝ\0, the Fourier transform of Fφ is given by
(28)Fφ^λ=2π−1/2∫ℝFφte−iλtdt=2π−1/2∫ℝ∫ℂn×Kfz,t,kφz,ke−iλtdzdkdt=2π−1/2∫ℂn×Kf−λz,kφz,kdzdk,then by (11)
(29)∣Fφ^λ∣≤2π−1/2∥φ∥2∫ℂn×Kf−λz,k2dzdk1/2≤2π−n/2−1∥φ∥2λn/2∑σ∈K^dσf^λ,σHS21/2.
As a result,(30)I=∫ℝ2FψtFφ^λ1+∣t∣+∣λ∣detλλn/2dtdλ≤2πn/2∥φ∥2∥ψ∥∞∫G∫ℝ∣fz,t,k∣∑σ∈K^dσf^λ,σHS21/21+∣t∣+∣λ∣detλdzdtdkdτλ<+∞.
In particular,(31)∫ℝ2FφtFφ^λ1+∣t∣+∣λ∣detλλn/2dtdλ<∞.
Note that the previous calculations are generalized to a bounded functionφ∈L2ℂn×K, in particular for the bounded functions φ in the basis B of L2ℂn×K defined in (18).
According to Beurling’s theorem in the Euclidean case, modified version (Theorem 2), for every functionφ∈B, there exists a polynomial function Pφ with degPφ<d−n/2−1/2 and a real aφ>0 such that
(32)∫ℂn×Kfz,t,kφz,k¯dzdk=Pφte−aφt2,from where
(33)fz,t,k=∑φ∈BPφte−aφt2φz,k.
Letφ,ψ∈B, since
(34)∫ℝ∫ℝFψtFφ^λ1+∣t∣+∣λ∣detλλn/2dtdλ<∞,then by Lemma 2.2 in [5], we obtain that aφ=aψ=a are independent of φ.
LetPφt=∑j=0mcj,φtj, then
(35)fz,t,k=e−at2∑j=0m∑φ∈Bcj,φφz,k·tj=e−at2∑j=0mξjz,k·tj,where ξjz,k=∑j=0mcj,φφαz,k∈L2ℂn×K.
The proof of Theorem3 is completed.We will finish this section with a modified version of previous Theorem3 as follows:Proposition 3.3.
Letf∈L2G and p,d,δ≥0. Suppose that
(36)∫G∫ℝ∣fz,t,k∣∑σ∈K^dσf^λ,σHS21/21+zp1+∣t∣+∣λ∣dλδetλdτλdzdtdk<∞.
Then,(37)fz,t,k=e−at21+zp∑j=0mφjz,ktj,where a>0, φj∈L1∩L2ℂn×K and m<d−n/2−1−δ/2.Proof.
By replacingfz,t,k by fz,t,k/1+zp and proceeding as in the proof of Theorem 3, one can apply Theorem 3 to get the result.


## 4. Applications to Other Uncertainty Principles

Let us first state and prove the following analogue of Hardy’s theorem forG.Theorem 5 (Hardy type).
Supposef is a measurable function on G satisfying
(i)
∣fz,t,k∣≤gz,ke−αt2, where g∈L1∩L2ℂn×K and α>0.(ii)
λn/2∥f^λ,σ∥HS≤cσe−βλ2, for some β>0 and cσ>0 such that ∑σ∈K^dσcσ2<+∞.
Then,(1)
Ifαβ>1/4, f=0 almost everywhere on G.(2)
Ifαβ=1/4, fz,t,k=e−αt2fz,0,k, for all z,t,k∈ℂn×ℝ×K.Proof.
From (i) and (ii), we have(38)∫G∫ℝ∣fz,t,k∣∑σ∈K^dσf^λ,σHS21/21+∣t∣+∣λ∣detλdτλdzdtdk≤Cte×M,where
(39)M=∫ℝ∫ℝλn/21+∣t∣+∣λ∣de−α∣t∣−∣λ∣/2α2e−β−1/4αλ2dtdλ.
Then, ifαβ≥1/4, the expression above for d>n/2+2 is finite.
According to Theorem3, there exist a>0 and φj∈L1∩L2ℂn×K such that
(40)fz,t,k=e−at2∑j=0mφjz,ktj,where m<d−n/2+1/2. En particular for n/2+2<d≤n/2+3, we have m=0 and
(41)fz,t,k=e−at2fz,0,k.
From condition (i), we have(42)∫ℂn×Kfz,0,kdzdk≤K×ea−αt2,where K is a positive constant.
We have(43)fλz,k=∫ℝfz,t,keiλtdt=πae−λ2/4afz,0,k.
Then,(44)∫ℂn×Kfλz,k2dzdk=πae−λ2/2a∫ℂn×Kfz,0,k2dzdk.
From (11), we have
(45)2π−nλn∑σ∈K^dσf^λ,σHS2=πae−λ2/2a∫ℂn×Kfz,0,k2dzdk.
From condition (ii), we obtain(46)∫ℂn×Kfz,0,k2dzdk≤Ce1/2a−2βλ2,where C is a positive constant.
(1)
Caseαβ>1/4:(i)
Suppose thata<α; then, limt⟶+∞ea−αt2=0 and ∫ℂn×K∣fz,0,k∣dzdk=0, by (42). We conclude that f=0.(ii)
Suppose thata≥α; we have 1/2a<2β and limt⟶+∞e1/2a−2βλ2=0. From (46), we have ∫ℂn×Kfz,0,k2dzdk=0, and finally f=0.(2)
Caseαβ=1/4:(i)
Suppose thatf≠0; then (42) and (46) hold if and only if a=α. We conclude that fz,t,k=e−at2fz,0,k.Theorem 6 (Gelfand-Shilov type).
Letf∈L2G and d≥0 satisfy
(47)∫G∣fz,t,k∣eαt21+∣t∣ddzdtdk<+∞,∫ℝ∑σ∈K^dσf^λ,σHS21/2eβλ21+∣λ∣ddτλ<+∞,for some positive constants α and β.
Then,αβ>1/4 implies f=0.Proof.
Suppose thatαβ>1/4, and consider the following numerical sequences αpp∈ℕ∗ and βpp∈ℕ∗ defined by
(48)αp=α−1p,βp=β−1p.(i)
limp⟶+∞αp=α>0; therefore, there exists Nα∈ℕ∗ such that αp>0 for p>Nα.(ii)
limp⟶+∞βp=β>0; therefore, there exists Nβ∈ℕ∗ such that βp>0 for p>Nβ.(iii)
limp⟶+∞αpβp=αβ>1/4; therefore, there exists Nαβ∈ℕ∗ such that αpβp>1/4 for p>Nαβ.
LetN=maxNα,Nβ,Nαβ, then, for all p>N.
(49)αp>0,βp>0,αpβp>14,tλ≤αpt2+βpλ2.
We have(50)∣fz,t,k∣∑σ∈K^dσf^λ,σHS21/21+∣t∣+∣λ∣detλ≤∣fz,t,k∣eαt21+∣t∣d∑σ∈K^dσf^λ,σHS21/2eβλ21+∣λ∣dht,λ,where ht,λ=1+∣t∣d1+∣λ∣d/1+∣t∣+∣λ∣de−1/pt2e−1/pλ2. The function ht,λ is a bounded function, then there exists a positive constant K such that
(51)∣fz,t,k∣∑σ∈K^dσf^λ,σHS21/21+∣t∣+∣λ∣detλ≤K∣fz,t,k∣eαt21+∣t∣d∑σ∈K^dσf^λ,σHS21/2eβλ21+∣λ∣d.
According to the hypotheses of the theorem, the integral(52)∫G∫ℝ∣fz,t,k∣∑σ∈K^dσf^λ,σHS21/21+∣t∣+∣λ∣detλdτλdzdtdkis finite. According to Theorem 3, there exist a>0 and ψj∈L2ℂn×K such that
(53)fz,t,k=e−at2∑j=0mφjz,ktj, where m<d−n/2+1/2.
We have(54)fλz,k=∑j=0mφjz,k∫ℝtje−at2eiλt=∑j=0mφjz,kPjλe−1/4aλ2wherePjλisapolynomialfunction=∑j=0mφjz,kPjλe−1/4at2.
Thenfλ2=hλe−1/4aλ2, where hλ is a polynomial function, and from (11), we have
(55)∑σ∈K^dσf^λ,σHS21/2=2πn/2λn/2hλe−1/4aλ2.
From the hypotheses of the theorem, we obtain(56)∫ℝλn/21+∣λ∣dhλeβ−1/4aλ2dλ<+∞.(i)
Suppose thata≥α; then β−1/4a>0, so hλ=0 implies f^λ,σ=0
By Plancherel formula (12), we conclude that f=0.
(ii)
Suppose thata<α;, according to the hypotheses of the theorem, we have(57)∫ℝeα−at21+∣t∣d∫ℂn×K∣∑j=0mφjz,ktj∣dzdkdt<+∞.
Sinceα−a>0 and ∑j=0mφjz,ktj∈L1ℂn×K for almost every t∈ℝ, then, φj=0 for all j=0,⋯,m and f=0.Corollary 1 (Another version of Hardy).
Supposef∈L1∩L2G satisfy
(i)
∣fz,t,k∣≤gz,ke−αt2, where g∈L1∩L2ℂn×K and α>0.(ii)
∥f^λ,σ∥HS≤cσe−βλ2, for some β>0 and cσ>0 such that ∑σ∈K^dσcσ2<+∞.
Then:(1)
Ifαβ>1/4, f=0 almost everywhere on G.(2)
Ifαβ<1/4, there exist an infinite number of linearly independent functions meeting hypotheses (i) and (ii).Proof.
(1)
Caseαβ>1/4: if αβ>1/4 and d>n+1, then, from the hypothesis of the theorem, we have(58)∫G∣fz,t,k∣eαt21+∣t∣ddzdtdk≤C∫ℝ11+∣t∣ddt<+∞,∫ℝ∑σ∈K^dσf^λ,σHS21/2eβλ21+∣λ∣ddτλ≤K∫ℝλn1+∣λ∣ddλ<+∞,where C,K>0. Since αβ>1/4, then, the Gelfand-Shilov theorem implies that f=0.
(2)
Caseαβ<1/4: If αβ<1/4, then, any function of the form gz,khkt where hk are the one-dimensional Hermite functions satisfies the hypotheses of the theorem..



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