Uncertainty Principles for Heisenberg Motion Group

Abstract

In this article, we will recall the main properties of the Fourier transform on the Heisenberg motion group , where and denote the Heisenberg group. Then, we will present some uncertainty principles associated to this transform as Beurling, Hardy, and Gelfand-Shilov.

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 satisfying then, almost everywhere.

The above theorem of Hörmander was further generalized by Bonami, Demange, and Jaming [4], as follows:

Theorem 1. Let and let satisfying Then, almost everywhere whenever , and if , then where is a positive real number and is a polynomial on of degree .

This last theorem admits another modified version proved by Parui and Sarkar [5]. It is of the following form.

Theorem 2. Let and be such that where is a polynomial of degree . Then, , where is a positive real number and is a polynomial with .

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.

2. Heisenberg Motion Group

Let be the Heisenberg group with the group law where .

Let be the unitary group , we define the Heisenberg motion group to be the semidirect product of and , with the group law where .

The Haar measure on is given by , where and are the normalized Haar measures on and , respectively.

For , we define the Schrödinger representation of on by where and .

Let be any irreducible, unitary representation of . For each , we consider the representations of on the tensor product space defined by where are the metaplectic representations [9], satisfying

Proposition 1 [9]. Each is unitary and irreducible.
For , consider the group Fourier transform where and the partial Fourier transform is defined by

For , we have and the Plancherel formula for the Fourier transform on G reads as where is the measure defined on , is the dimension of the space , and denote the Hilbert-Schmidt norm of [9].

At the end of this paragraph, we introduce an orthonormal basis for [10]. Let be the Hermite polynomials defined by

The normalized Hemite functions are defined by

The -dimensional Hermite functions are defined on by taking the tensor products; that is, where .

It is well known that form an orthonormal basis for [2]. Then, an orthonormal basis for is given by , where is an orthonormal basis for and .

Define the Fourier-Wigner transform of on by

Lemma 1 [10]. For , the following identity holds. In particular, , for .

Set , then, by Lemma 1, we infer that the set is an orthonormal basis for

Proposition 2 [10]. The family is an orthonormal basis for

Lemma 2. The function is a bounded function.

Proof. Let and , we have We know that (see [9] p.21); then Since , , and are unitary representations, then, where .

3. An Analogue of Beurling’s Theorem

In this section, we prove an analogue of Beurling’s theorem on the Heisenberg motion group whose statement is as follows:

Theorem 3. Let and . Suppose that Then, where , and .

Lemma 3. If satisfies the hypotheses of Theorem 3, then, .

Proof. As is not identically zero, then, there exists such that By (22), we have On the other hand, the function is bounded, so there exists a constant such that From where and using (25), so we have .

Proof of Theorem 3. For any , the Schwartz space of , consider the function Since , then, is integrable on , and for any , the Fourier transform of is given by then by (11) As a result, In particular, Note that the previous calculations are generalized to a bounded function , in particular for the bounded functions in the basis of defined in (18).
According to Beurling’s theorem in the Euclidean case, modified version (Theorem 2), for every function , there exists a polynomial function with and a real such that from where Let , since then by Lemma 2.2 in [5], we obtain that are independent of .
Let , then where .
The proof of Theorem 3 is completed.

We will finish this section with a modified version of previous Theorem 3 as follows:

Proposition 3.3. Let and . Suppose that Then, where , and

Proof. By replacing by 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 for .

Theorem 5 (Hardy type). Suppose is a measurable function on satisfying (i), where and .(ii), for some and such that .Then, (1)If , almost everywhere on .(2)If , , for all .

Proof. From (i) and (ii), we have where Then, if , the expression above for is finite.
According to Theorem 3, there exist and such that where . En particular for , we have and From condition (i), we have where is a positive constant.
We have Then, From (11), we have From condition (ii), we obtain where is a positive constant. (1)Case :(i)Suppose that ; then, and , by (42). We conclude that .(ii)Suppose that ; we have and . From (46), we have , and finally .(2)Case :(i)Suppose that ; then (42) and (46) hold if and only if . We conclude that .

Theorem 6 (Gelfand-Shilov type). Let and satisfy for some positive constants and .
Then, implies .

Proof. Suppose that , and consider the following numerical sequences and defined by (i); therefore, there exists such that for .(ii); therefore, there exists such that for .(iii); therefore, there exists such that for .Let , then, for all . We have where . The function is a bounded function, then there exists a positive constant such that According to the hypotheses of the theorem, the integral is finite. According to Theorem 3, there exist and such that where .
We have Then , where is a polynomial function, and from (11), we have From the hypotheses of the theorem, we obtain (i)Suppose that ; then , so implies By Plancherel formula (12), we conclude that . (ii)Suppose that ;, according to the hypotheses of the theorem, we haveSince and for almost every , then, for all and .

Corollary 1 (Another version of Hardy). Suppose satisfy (i), where and .(ii), for some and such that .Then: (1)If , almost everywhere on .(2)If , there exist an infinite number of linearly independent functions meeting hypotheses (i) and (ii).

Proof. (1)Case : if and , then, from the hypothesis of the theorem, we havewhere . Since , then, the Gelfand-Shilov theorem implies that . (2)Case : If , then, any function of the form where are the one-dimensional Hermite functions satisfies the hypotheses of the theorem..

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflict of interest.