Heisenberg-Pauli-Weyl and Donoho-Stark's uncertainty principle for the Weinstein $L^2$-multiplier operators

The aim of this paper is establish the Heisenberg-Pauli-Weyl uncertainty principle and Donho-Stark's uncertainty principle for the Weinstein $L^2$-multiplier operators.


Introduction
The Weinstein operator ∆ d W,α defined on R d+1 where ∆ d is the Laplacian operator for the d first variables and L α is the Bessel operator for the last variable defined on (0, ∞) by The Weinstein operator ∆ d W,α has several applications in pure and applied mathematics, especially in fluid mechanics [4].
These operators are a generalization of the multiplier operators T m associated with a bounded function m and given by T m (ϕ) = F −1 (mF (ϕ)), where F (ϕ) denotes the ordinary Fourier transform on R n . These operators gained the interest of several Mathematicians and they were generalized in many settings in [1,3,6,13,14,16,17,18].
In this work we are interested the L 2 uncertainty principles for the Weinstein multiplier operators. The uncertainty principles play an important role in harmonic analysis. These principles state that a function ϕ and its Fourier transform F (ϕ) cannot be simultaneously sharply localized. Many aspects of such principles are studied for several Fourier transforms.
Many uncertainty principles have already been proved for the Weinstein transform F W,α , namely by N. Ben Salem, A. R. Nasr [2] and Mejjaoli H. and Salhi M. [9]. The authors have established in [9] the Heisenberg-Pauli-Weyl inequality for the Weinstein transform, by showing that, for every ϕ in In the present paper we are interested in proving an analogue of Heisenberg-Pauli-Weyl uncertainty principle For the operators T w,m,σ . More precisely, we will show, for ϕ ∈ L 2 provided m be a function in L 2 α (R d+1 + ) satisfying the admissibility condition Moreover, for β, δ ∈ [1, ∞) and ε ∈ R, such that βε = (1 − ε)δ, we will show Using the techniques of Donoho and Stark [5], we show uncertainty principle of concentration type for the L 2 theory. Let ϕ be a function in L 2 α (R d+1 + ) and m ∈ L 1 α (R d+1 + ) ∩ L 2 α (R d+1 + ) satisfying the admissibility condition (1.3). If ϕ is ǫ-concentrated on Ω and T w,m,σ ϕ is ν-concentrated on Σ, then m α,1 (µ α (Ω)) where Θ α is the measure on (0, ∞)×R d+1 + given by dΘ α (σ, x) := (dσ/σ)d α µ(x). This paper is organized as follows. In section 2, we recall some basic harmonic analysis results related with the Weinstein operator ∆ d W,α and we introduce preliminary facts that will be used later.
In section 3, we establish Heisenberg-Pauli-Weyl uncertainty principle For the operators T w,m,σ .
The last section of this paper is devoted to Donoho-Stark's uncertainty principle for the Weinstein L 2 -multiplier operators.

Harmonic analysis Associated with the Weinstein Operator
In this section, we shall collect some results and definitions from the theory of the harmonic analysis associated with the Weinstein operator ∆ d W,α . Main references are [10,11,12].
In the following we denote by , the space of continuous functions on R d+1 , even with respect to the last variable. • S * (R d+1 ), the space of the C ∞ functions, even with respect to the last variable, and rapidly decreasing together with their derivatives.
We consider the Weinstein operator ∆ d W,α defined on R d+1 where ∆ d is the Laplacian operator for the d first variables and L α is the Bessel operator for the last variable defined on (0, ∞) by The Weinstein operator ∆ d W,α have remarkable applications in diffrerent branches of mathematics. For instance, they play a role in Fluid Mechanics [4].
where µ α is the measure on R d+1 + given by the relation (2.1).
Some basic properties of this transform are as follows. For the proofs, we refer [11,12]. (2.11) 2. The Weinstein transform is a topological isomorphism from S * (R d+1 + ) onto itself. The inverse transform is given by (2.14) 5. Plancherel Theorem: The Weinstein transform F W,α extends uniquely to an isometric isomorphism on L 2 α (R d+1 + ). 6. Inversion formula: By using the Weinstein kernel, we can also define a generalized translation, for a function ϕ ∈ S * (R d+1 ) and y ∈ R d+1 + the generalized translation τ α x ϕ is defined by the following relation F W,α (τ α x ϕ)(y) = Λ d α (x, y)F W,α (ϕ)(y). (2.16) The following proposition summarizes some properties of the Weinstein translation operator.
Proposition 2.5. The translation operator τ α x , x ∈ R d+1 + satisfies the following properties. i). For ϕ ∈ C * (R d+1 ), we have for all x, y ∈ R d+1 By using the generalized translation, we define the generalized convolution product ϕ * W ψ of the functions ϕ, ψ ∈ L 1 α (R d+1 + ) as follows This convolution is commutative and associative, and it satisfies the following properties.

24)
where both sides are finite or infinite.

Heisenberg-Pauli-Weyl uncertainty principle
In this section we establish Heisenberg-Pauli-Weyl uncertainty principle for the operator T w,m,σ .

Inequality (1.2) leads to
Integrating with respect to dσ/σ, we get From [14, Theorem 2.3] and Schwarz's inequality, we obtain From (1.1), Fubini-Tonnelli's theorem and the admissibility condition (1.3), we have This gives the result and completes the proof of the theorem.

Donoho-Stark's uncertainty principle
where χ Ω is the indicator function of the set Ω.
We need the following Lemma for the proof of Donoho-Stark's uncertainty principle.
. Then the operators T w,m,σ satisfy the following integral representation.
Proof. The result follows from the definition of the Weinstein L 2 -Multiplier operators (1.1) and the inversion formula of the Weinstein transform (2.12) using Fubini-Tonnelli's theorem.