International Journal of Analysis and Applications Volume 17, Number 1 (2019), 64-75 URL: https://doi.org/10.28924/2291-8639 DOI: 10.28924/2291-8639-17-2019-64 L2-UNCERTAINTY PRINCIPLE FOR THE WEINSTEIN-MULTIPLIER OPERATORS AHMED SAOUDI1,2,∗ AND IMEN ALI KALLEL1 1Northern Border University, College of Science, P.O. Box 1231, Arar 91431, Saudi Arabia 2Tunis El Manar University, Faculty of Science of Tunis, Campus Universitaire, 2092, Tunisia ∗Corresponding author: ahmed.saoudi@ipeim.rnu.tn Abstract. The aim of this paper is establish the Heisenberg-Pauli-Weyl uncertainty principle and Donho- Stark’s uncertainty principle for the Weinstein L2-multiplier operators. 1. Introduction The Weinstein operator ∆dW,α defined on R d+1 + = R d × (0,∞), by ∆dW,α = d+1∑ j=1 ∂2 ∂x2j + 2α + 1 xd+1 ∂ ∂xd+1 = ∆d + Lα, α > −1/2, where ∆d is the Laplacian operator for the d first variables and Lα is the Bessel operator for the last variable defined on (0,∞) by Lαu = ∂2u ∂x2d+1 + 2α + 1 xd+1 ∂u ∂xd+1 . The Weinstein operator ∆dW,α has several applications in pure and applied mathematics, especially in fluid mechanics [4]. Received 2018-09-14; accepted 2018-10-26; published 2019-01-04. 2010 Mathematics Subject Classification. 43A32, 44A15. Key words and phrases. Weinstein operator; L2-multiplier operators; Heisenberg-Pauli-Weyl uncertainty principle; Donho- Stark’s uncertainty principle. c©2019 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 64 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-64 Int. J. Anal. Appl. 17 (1) (2019) 65 The Weinstein transform generalizing the usual Fourier transform, is given for ϕ ∈ L1α(R d+1 + ) and λ ∈ Rd+1+ , by FW,α(ϕ)(λ) = ∫ Rd+1 + ϕ(x)Λdα(x,λ)dµα(x), where dµα(x) is the measure on Rd+1+ = R d × (0,∞) and Λdα is the Weinstein kernel given respectively later by (2.1) and (2.4). Let m be a function in L2α(R d+1 + ) and let σ be a positive real number. The Weinstein L 2-multiplier operators is defined for smooth functions ϕ on Rd+1+ , in [14] as Tw,m,σϕ(x) := F−1W,α (mσFW,α(ϕ)) (x), x ∈ R d+1 + , (1.1) where the function mσ is given by mσ(x) = m(σx). These operators are a generalization of the multiplier operators Tm associated with a bounded function m and given by Tm(ϕ) = F−1(mF(ϕ)), where F(ϕ) denotes the ordinary Fourier transform on Rn. These operators gained the interest of several Mathematicians and they were generalized in many settings in [1, 3, 6, 13, 14, 16–18]. In this work we are interested the L2 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 FW,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 L2α(R d+1 + ) ‖ϕ‖α,2 ≤ 2 2α + d + 2 ‖|x|ϕ‖α,2‖|y|FW,α(ϕ)‖α,2. (1.2) In the present paper we are interested in proving an analogue of Heisenberg-Pauli-Weyl uncertainty principle For the operators Tw,m,σ. More precisely, we will show, for ϕ ∈ L2α(R d+1 + ) ‖ϕ‖α,2 ≤ 2‖|y|FW,α(ϕ)‖α,2 2α + d + 2 (∫ Rd+1 + ∫ ∞ 0 |x|2|Tw,m,σϕ(x)|2α,2 dσ σ dµα(x) )1 2 , provided m be a function in L2α(R d+1 + ) satisfying the admissibility condition∫ ∞ 0 |mσ(x)| dσ σ = 1, a.e. x ∈ Rd+1+ . (1.3) Moreover, for β,δ ∈ [1,∞) and ε ∈ R, such that βε = (1 −ε)δ, we will show ‖ϕ‖α,2 ≤ ( 2 2α + d + 2 )βε ∥∥|x|βTw,m,σϕ∥∥εα,2 ∥∥|y|δFW,α(ϕ)∥∥1−εα,2 . Int. J. Anal. Appl. 17 (1) (2019) 66 Using the techniques of Donoho and Stark [5], we show uncertainty principle of concentration type for the L2 theory. Let ϕ be a function in L2α(R d+1 + ) and m ∈ L1α(R d+1 + ) ∩L2α(R d+1 + ) satisfying the admissibility condition (1.3). If ϕ is �-concentrated on Ω and Tw,m,σϕ is ν-concentrated on Σ, then ‖m‖α,1 (µα(Ω)) 1 2 (∫ ∫ Σ 1 σ2(2α+d+2) dΘα(σ,x) )1 2 ≥ 1 − (� + ν), where Θα is the measure on (0,∞) ×Rd+1+ given by dΘα(σ,x) := (dσ/σ)dαµ(x). This paper is organized as follows. In section 2, we recall some basic harmonic analysis results related with the Weinstein operator ∆dW,α and we introduce preliminary facts that will be used later. In section 3, we establish Heisenberg-Pauli-Weyl uncertainty principle For the operators Tw,m,σ. The last section of this paper is devoted to Donoho-Stark’s uncertainty principle for the Weinstein L2- multiplier operators. 2. 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 ∆dW,α. Main references are [10–12]. In the following we denote by • Rd+1+ = Rd × (0,∞). • x = (x1, ...,xd,xd+1) = (x′,xd+1). • −x = (−x′,xd+1). • C∗(Rd+1), the space of continuous functions on Rd+1, even with respect to the last variable. • S∗(Rd+1), the space of the C∞ functions, even with respect to the last variable, and rapidly decreasing together with their derivatives. • Lpα(R d+1 + ), 1 ≤ p ≤∞, the space of measurable functions f on R d+1 + such that ‖f‖α,p = (∫ Rd+1 + |f(x)|p dµα(x) )1/p < ∞, p ∈ [1,∞), ‖f‖α,∞ = ess sup x∈Rd+1 + |f(x)| < ∞, where dµα(x) = x2α+1d+1 (2π)d22αΓ2(α + 1) dx. (2.1) • Aα(Rd+1) = { ϕ ∈ L1α(R d+1 + ); FW,αϕ ∈ L1α(R d+1 + ) } the Wiener algebra space. We consider the Weinstein operator ∆dW,α defined on R d+1 + by ∆dW,α = d+1∑ j=1 ∂2 ∂x2j + 2α + 1 xd+1 ∂ ∂xd+1 = ∆d + Lα, α > −1/2, (2.2) Int. J. Anal. Appl. 17 (1) (2019) 67 where ∆d is the Laplacian operator for the d first variables and Lα is the Bessel operator for the last variable defined on (0,∞) by Lαu = ∂2u ∂x2d+1 + 2α + 1 xd+1 ∂u ∂xd+1 . The Weinstein operator ∆dW,α have remarkable applications in diffrerent branches of mathematics. For instance, they play a role in Fluid Mechanics [4]. 2.1. The eigenfunction of the Weinstein operator. For all λ = (λ1, ...,λd+1) ∈ Cd+1, the system ∂2u ∂x2j (x) = −λ2ju(x), if 1 ≤ j ≤ d Lαu(x) = −λ2d+1u(x), u(0) = 1, ∂u ∂xd+1 (0) = 0, ∂u ∂xj (0) = −iλj, if 1 ≤ j ≤ d (2.3) has a unique solution denoted by Λdα(λ,.), and given by Λdα(λ,x) = e −ijα(xd+1λd+1) (2.4) where x = (x′,xd+1), λ = (λ ′,λd+1) and jα is is the normalized Bessel function of index α defined by jα(x) = Γ(α + 1) ∞∑ k=0 (−1)kx2k 2kk!Γ(α + k + 1) . The function (λ,x) 7→ Λdα(λ,x) has a unique extension to Cd+1×Cd+1, and satisfied the following properties. Proposition 2.1. i). For all (λ,x) ∈ Cd+1 ×Cd+1 we have Λdα(λ,x) = Λ d α(x,λ). (2.5) ii). For all (λ,x) ∈ Cd+1 ×Cd+1 we have Λdα(λ,−x) = Λ d α(−λ,x). (2.6) iii). For all (λ,x) ∈ Cd+1 ×Cd+1 we get Λdα(λ, 0) = 1. (2.7) vi). For all ν ∈ Nd+1, x ∈ Rd+1 and λ ∈ Cd+1 we have ∣∣DνλΛdα(λ,x)∣∣ ≤‖x‖|ν|e‖x‖‖=λ‖ (2.8) where Dνλ = ∂ ν/(∂λν11 ...∂λ νd+1 d+1 ) and |ν| = ν1 + ... + νd+1. In particular, for all (λ,x) ∈ R d+1 ×Rd+1, we have ∣∣Λdα(λ,x)∣∣ ≤ 1. (2.9) Int. J. Anal. Appl. 17 (1) (2019) 68 2.2. The Weinstein transform. Definition 2.1. The Weinstein transform is given for ϕ ∈ L1α(R d+1 + ) by FW,α(ϕ)(λ) = ∫ Rd+1 + ϕ(x)Λdα(λ,x)dµα(x), λ ∈ R d+1 + , (2.10) where µα is the measure on Rd+1+ given by the relation (2.1). Some basic properties of this transform are as follows. For the proofs, we refer [11, 12]. Proposition 2.2. (1) For all ϕ ∈ L1α(R d+1 + ), the function FW,α(ϕ) is continuous on R d+1 + and we have ‖FW,αϕ‖α,∞ ≤‖ϕ‖α,1 . (2.11) (2) The Weinstein transform is a topological isomorphism from S∗(Rd+1+ ) onto itself. The inverse trans- form is given by F−1W,αϕ(λ) = FW,αϕ(−λ), for all λ ∈ R d+1 + . (2.12) (3) Parseval formula: For all ϕ,φ ∈S∗(Rd+1+ ), we have∫ Rd+1 + ϕ(x)φ(x)dµα(x) = ∫ Rd+1 + FW,α(ϕ)(x)FW,α(φ)(x)dµα(x). (2.13) (4) Plancherel formula: For all ϕ ∈S∗(Rd+1+ ), we have ‖FW,αϕ‖α,2 = ‖ϕ‖α,2 . (2.14) (5) Plancherel Theorem: The Weinstein transform FW,α extends uniquely to an isometric isomor- phism on L2α(R d+1 + ). (6) Inversion formula: Let ϕ ∈ L1α(R d+1 + ) such that FW,αϕ ∈ L1α(R d+1 + ), then we have ϕ(λ) = ∫ Rd+1 + FW,αϕ(x)Λdα(−λ,x)dµα(x), a.e. λ ∈ R d+1 + . (2.15) 2.3. The translation operator associated with the Weinstein operator. Definition 2.2. The translation operator ταx , x ∈ R d+1 + associated with the Weinstein operator ∆ d W,α, is defined for a continuous function ϕ on Rd+1+ which is even with respect to the last variable and for all y ∈ Rd+1+ by ταx ϕ(y) = Cα ∫ π 0 ϕ ( x′ + y′, √ x2d+1 + y 2 d+1 + 2xd+1yd+1 cos θ ) (sin θ) 2α dθ, with Cα = Γ(α + 1) √ πΓ(α + 1/2) . Int. J. Anal. Appl. 17 (1) (2019) 69 By using the Weinstein kernel, we can also define a generalized translation, for a function ϕ ∈ S∗(Rd+1) and y ∈ Rd+1+ the generalized translation ταx ϕ is defined by the following relation FW,α(ταx ϕ)(y) = Λ d α(x,y)FW,α(ϕ)(y). (2.16) The following proposition summarizes some properties of the Weinstein translation operator. Proposition 2.3. The translation operator ταx , x ∈ R d+1 + satisfies the following properties. i). For ϕ ∈ C∗(Rd+1), we have for all x,y ∈ Rd+1+ ταx ϕ(y) = τ α y ϕ(x) and τ α 0 ϕ = ϕ. ii). Let ϕ ∈ Lpα(R d+1 + ), 1 ≤ p ≤∞ and x ∈ R d+1 + . Then τ α x ϕ belongs to L p α(R d+1 + ) and we have ‖ταx ϕ‖α,p ≤‖ϕ‖α,p . (2.17) Note that the Aα(Rd+1+ ) is contained in the intersection of L1α(R d+1 + ) and L ∞ α (R d+1 + ) and hence is a subspace of L2α(R d+1 + ). For ϕ ∈Aα(R d+1 + ) we have ταx ϕ(y) = Cα,d ∫ Rd+1 + Λdα(x,z)Λ d α(−y,z)FW,αϕ(z)dµα(z). (2.18) By using the generalized translation, we define the generalized convolution product ϕ∗W ψ of the functions ϕ, ψ ∈ L1α(R d+1 + ) as follows ϕ∗W ψ(x) = ∫ Rd+1 + ταx ϕ(−y)ψ(y)dµα(y). (2.19) This convolution is commutative and associative, and it satisfies the following properties. Proposition 2.4. i) For all ϕ,ψ ∈ L1α(R d+1 + ), (resp. ϕ,ψ ∈ S∗(R d+1 + )), then ϕ ∗W ψ ∈ L1α(R d+1 + ), (resp. ϕ∗W ψ ∈S∗(Rd+1+ )) and we have FW,α(ϕ∗W ψ) = FW,α(ϕ)FW,α(ψ). (2.20) ii) Let p,q,r ∈ [1,∞], such that 1 p + 1 q − 1 r = 1. Then for all ϕ ∈ Lpα(R d+1 + ) and ψ ∈ Lqα(R d+1 + ) the function ϕ∗W ψ belongs to Lrα(R d+1 + ) and we have ‖ϕ∗W ψ‖α,r ≤‖ϕ‖α,p‖ψ‖α,q . (2.21) iii) Let ϕ,ψ ∈ L2α(R d+1 + ). Then ϕ∗W ψ = F−1W,α (FW,α(ϕ)FW,α(ψ)) . (2.22) iv) Let ϕ,ψ ∈ L2α(R d+1 + ). Then ϕ ∗W ψ belongs to L2α(R d+1 + ) if and only if FW,α(ϕ)FW,α(ψ) belongs to L2α(R d+1 + ) and we have FW,α(ϕ∗W ψ) = FW,α(ϕ)FW,α(ψ). (2.23) Int. J. Anal. Appl. 17 (1) (2019) 70 v) Let ϕ,ψ ∈ L2α(R d+1 + ). Then ‖ϕ∗W ψ)‖α,2 = ‖FW,α(ϕ)FW,α(ψ)‖α,2, (2.24) where both sides are finite or infinite. 3. Heisenberg-Pauli-Weyl uncertainty principle In this section we establish Heisenberg-Pauli-Weyl uncertainty principle for the operator Tw,m,σ. Theorem 3.1. Let m be a function in L2α(R d+1 + ) satisfying the admissibility condition (1.3). Then, for ϕ ∈ L2α(R d+1 + ), we have ‖ϕ‖α,2 ≤ 2‖|y|FW,α(ϕ)‖α,2 2α + d + 2 (∫ Rd+1 + ∫ ∞ 0 |x|2|Tw,m,σϕ(x)|2α,2 dσ σ dµα(x) )1 2 . (3.1) Proof. Let ϕ ∈ L2α(R d+1 + ). The inequality (3.1) holds if ‖|y|FW,α(ϕ)‖α,2 = +∞ or ∫ Rd+1 + ∫ ∞ 0 |x|2|Tw,m,σϕ(x)|2α,2 dσ σ dµα(x) = +∞. Let us now assume that ‖|y|FW,α(ϕ)‖α,2 + ∫ Rd+1 + ∫ ∞ 0 |x|2|Tw,m,σϕ(x)|2α,2 dσ σ dµα(x) < +∞. Inequality (1.2) leads to ∫ Rd+1 + |Tw,m,σϕ(x)|2α,2dµα(x) < (∫ Rd+1 + |x|2|Tw,m,σϕ(x)|2α,2dµα(x) )1 2 × (∫ Rd+1 + |y|2|FW,α(Tw,m,σϕ(.))(y)|2α,2dµα(y) )1 2 . Integrating with respect to dσ/σ, we get ‖Tw,m,σϕ)‖2α,2 < ∫ ∞ 0 (∫ Rd+1 + |x|2|Tw,m,σϕ(x)|2α,2dµα(x) )1 2 × (∫ Rd+1 + |y|2|FW,α(Tw,m,σϕ(.))(y)|2α,2dµα(y) )1 2 dσ σ . From [14, Theorem 2.3] and Schwarz’s inequality, we obtain Int. J. Anal. Appl. 17 (1) (2019) 71 ‖ϕ‖2α,2 < (∫ ∞ 0 ∫ Rd+1 + |x|2|Tw,m,σϕ(x)|2α,2dµα(x) dσ σ )1 2 × (∫ ∞ 0 ∫ Rd+1 + |y|2|FW,α(Tw,m,σϕ(.))(y)|2α,2dµα(y) dσ σ )1 2 . From (1.1), Fubini-Tonnelli’s theorem and the admissibility condition (1.3), we have ∫ ∞ 0 ∫ Rd+1 + |y|2|FW,α(Tw,m,σϕ(.))(y)|2α,2dµα(y) dσ σ = ∫ ∞ 0 ∫ Rd+1 + |y|2|mσ(y)|2|FW,α(ϕ)(y)|2α,2dµα(y) dσ σ = ∫ Rd+1 + |y|2|FW,α(ϕ)(y)|2α,2dµα(y). This gives the result and completes the proof of the theorem. � Theorem 3.2. Let m be a function in L2α(R d+1 + ) satisfying the admissibility condition (1.3) and β,δ ∈ [1,∞). Let ε ∈ R, such that βε = (1 −ε)δ then, for ϕ ∈ L2α(R d+1 + ), we have ‖ϕ‖α,2 ≤ ( 2 2α + d + 2 )βε ∥∥|x|βTw,m,σϕ∥∥εα,2 ∥∥|y|δFW,α(ϕ)∥∥1−εα,2 . (3.2) Proof. Let ϕ ∈ L2α(R d+1 + ). The inequality (3.1) holds if ∥∥|x|βTw,m,σϕ∥∥εα,2 = +∞ or ∥∥|y|δFW,α(ϕ)∥∥1−εα,2 = +∞. Let us now assume that ϕ ∈ L2α(R d+1 + ) with ϕ 6= 0 such that ∥∥|x|βTw,m,σϕ∥∥εα,2 + ∥∥|y|δFW,α(ϕ)∥∥1−εα,2 < +∞, therefore, for all δ > 1, we have ∥∥|x|βTw,m,σϕ∥∥ 1βα,2 ‖Tw,m,σϕ‖ 1β′α,2 = ∥∥∥|x|2|Tw,m,σϕ|2β ∥∥∥12α,β ∥∥∥|Tw,m,σϕ| 2β′ ∥∥∥12 α,β′ , with β′ = β β−1. Applying the Hölder’s inequality, we get ‖|x|Tw,m,σϕ‖α,2 ≤ ∥∥|x|βTw,m,σϕ∥∥ 1βα,2 ‖Tw,m,σϕ‖ 1β′α,2 . According to [14, Theorem 2.3], we have for all β ≥ 1 ‖|x|Tw,m,σϕ‖α,2 ≤ ∥∥|x|βTw,m,σϕ∥∥ 1βα,2 ‖ϕ‖ 1β′α,2 , (3.3) Int. J. Anal. Appl. 17 (1) (2019) 72 with equality if β = 1. In the same manner, for all δ ≥ and using Plancherel formula (2.14), we get ‖|y|FW,α(ϕ)‖α,2 ≤ ∥∥|y|δFW,α(ϕ)∥∥1δα,2 ‖ϕ‖ 1δ′α,2 , (3.4) with equality if δ = 1. By using the fact that βε = (1−ε)δ and according to inequalities (3.3) and (3.4), we have  ‖|x|Tw,m,σϕ‖α,2 ‖|y|FW,α(ϕ)‖α,2 ‖ϕ‖ 1 β′ + 1 δ′ α,2  βδ ≤ ∥∥|x|βTw,m,σϕ∥∥εα,2 ∥∥|y|δFW,α(ϕ)∥∥1−εα,2 , with equality if β = δ = 1. Next by Theorem 3.1, we obtain ‖ϕ‖α,2 ≤ ( 2 2α + d + 2 )βε ∥∥|x|βTw,m,σϕ∥∥εα,2 ∥∥|y|δFW,α(ϕ)∥∥1−εα,2 , which completes the proof of the theorem. � 4. Donoho-Stark’s uncertainty principle Definition 4.1. (i) Let Ω be a measurable subset of Rd+1+ , we say that the function ϕ ∈ L2α(R d+1 + ) is �-concentrated on Ω, if ‖ϕ−χΩϕ‖α,2 ≤ �‖ϕ‖α,2, (4.1) where χΩ is the indicator function of the set Ω. (ii) Let Σ be a measurable subset of (0,∞) × Rd+1+ and let ϕ ∈ L2α(R d+1 + ). We say that Tw,m,σϕ is ν-concentrated on Σ, if ‖Tw,m,σϕ−χΣTw,m,σϕ‖2,α ≤ ν‖Tw,m,σ‖2,α, (4.2) where χΣ is the indicator function of the set Σ. We need the following Lemma for the proof of Donoho-Stark’s uncertainty principle. Lemma 4.1. Let m,ϕ ∈ L1α(R d+1 + ) ∩ L2α(R d+1 + ). Then the operators Tw,m,σ satisfy the following integral representation. Tw,m,σ = 1 σ2α+d+2 ∫ Rd+1 + Ψα(x,y)ϕ(y)dµα(y), (σ,x) ∈ (0,∞) ×Rd+1+ , where Ψα(x,y) = ∫ Rd+1 + mσ(z)Λ d α(λ,x)Λ d α(λ,−y)dµα(z). Proof. The result follows from the definition of the Weinstein L2-Multiplier operators (1.1) and the inversion formula of the Weinstein transform (2.12) using Fubini-Tonnelli’s theorem. � Int. J. Anal. Appl. 17 (1) (2019) 73 Theorem 4.1. Let ϕ be a function in L2α(R d+1 + ) and m ∈ L1α(R d+1 + )∩L2α(R d+1 + ) satisfying the admissibility condition (1.3). If ϕ is �-concentrated on Ω and Tw,m,σϕ is ν-concentrated on Σ, then ‖m‖α,1 (µα(Ω)) 1 2 (∫ ∫ Σ 1 σ2(2α+d+2) dΘα(σ,x) )1 2 ≥ 1 − (� + ν), where Θα is the measure on (0,∞) ×Rd+1+ given by dΘα(σ,x) := (dσ/σ)dαµ(x). Proof. Let ϕ be a function in L2α(R d+1 + ). Assume that 0 < µα(Ω) < ∞ and∫ ∫ Σ 1 σ2(2α+d+2) dΘα(σ,x) < ∞. According to [14, Theorem 2.3] and inequalities (4.1)-(4.2), we get ‖Tw,m,σϕ−χΣTw,m,σ(χΩϕ)‖2,α ≤ ‖Tw,m,σϕ−χΣTw,m,σϕ‖2,α +‖Tw,m,σϕ−χΣTw,m,σ(ϕ−χΩϕ)‖2,α ≤ ‖Tw,m,σϕ−χΣTw,m,σ(ϕ−χΩϕ)‖2,α +ν‖Tw,m,σϕ‖2,α ≤ (� + ν)‖ϕ‖2,α. By triangle inequality it follows that ‖Tw,m,σϕ‖2,α ≤ ‖Tw,m,σϕ−χΣTw,m,σ(χΩϕ)‖2,α + ‖χΣTw,m,σ(χΩϕ)‖2,α ≤ (� + ν)‖ϕ‖2,α + ‖χΣTw,m,σ(χΩϕ)‖2,α. (4.3) On the other hand, we have ‖χΣTw,m,σ(χΩϕ)‖2,α = (∫ ∫ Σ |Tw,m,σ(χΩϕ)(x)|2dΘα(σ,x) )1 2 and moreover m,χΩϕ ∈ L1α(R d+1 + ) ∩L2α(R d+1 + ), then by Lemma 4.1, we obtain |Tw,m,σ(χΩϕ)(x)| ≤ 1 σ2α+d+2 ‖m‖1,α‖ϕ‖2,α(µα(Ω)) 1 2 . Therefore, thus ‖χΣTw,m,σ(χΩϕ)‖2,α ≤ ‖m‖1,α‖ϕ‖2,α(µα(Ω)) 1 2 × (∫ ∫ Σ 1 σ2(2α+d+2) dΘα(σ,x) )1 2 . Hence, according to last inequality and (4.3) ‖Tw,m,σ(ϕ)‖2,α ≤ ‖m‖1,α‖ϕ‖2,α(µα(Ω)) 1 2 × (∫ ∫ Σ 1 σ2(2α+d+2) dΘα(σ,x) )1 2 + (� + ν)‖ϕ‖2,α. Int. J. Anal. Appl. 17 (1) (2019) 74 Applying Plancherel formula [14, Theorem 2.3], we obtain ‖m‖α,1 (µα(Ω)) 1 2 (∫ ∫ Σ 1 σ2(2α+d+2) dΘα(σ,x) )1 2 ≥ 1 − (� + ν), which completes the proof of the theorem. � Corollary 4.1. If Σ = {(σ,x) ∈ (0,∞) ×Rd+1+ : σ ≥ %} for some % > 0, one assumes that ρ = max { 1/σ : (σ,x) ∈ Σ for some x ∈ Rd+1+ } . Then by the previous Theorem, we deduce that ρ2α+d+2‖m‖α,1 (µα(Ω)) 1 2 (Θα(Σ)) 1 2 ≥ 1 − (� + ν). Acknowledgement 4.1. The authors gratefully acknowledge the approval and the support of this research study by the grant no.7385-SCI-2017-1-8-F from the Deanship of Scientific Research at Northern Boder University, Arar, K.S.A. References [1] J.P. Anker, Lp Fourier multipliers on Riemannian symmetric spaces of the noncompact type, Ann. Math. (2). 132(3) (1990), 597-628. [2] N. Ben Salem and AR. Nasr, Heisenberg-type inequalities for the Weinstein operator, Integral Transforms Spec. Funct. 26(9) (2015), 700-718. [3] J. J. Betancor, Ó. Ciaurri and J. L. Varona, The multiplier of the interval [−1, 1] for the Dunkl transform on the real line, J. Funct. Anal. 242(1) (2007), 327-336. [4] M. Brelot, Equation de Weinstein et potentiels de Marcel Riesz, Semin. Theor. Potent., Paris, No. 3, Lect. Notes Math. 3 (1978), 18-38. [5] D.L.Donoho and P.B. Stark, Uncertainty principles and signal recovery, SIAM J. Appl. Math. 49(3) (1989), 906931. [6] J. Gosselin and K. Stempak, A weak-type estimate for Fourier-Bessel multipliers, Proc. Amer. Math. Soc. 106(3) (1989), 655-662. [7] G. Kimeldorf and G. Wahba, Some results on Tchebycheffian spline functions and stochastic processes, J. Math. Anal. Appl. 33(1) (1971), 82-95. [8] T. Matsuura, S. Saitoh and D. Trong, Approximate and analytical inversion formulas in heat conduction on multidimen- sional spaces, J. Inverse Ill-Posed Probl. 13(5) (2005), 479-493. [9] H. Mejjaoli and M. Salhi, Uncertainty principles for the Weinstein transform, Czech. Math. J. 61 (2011), 941-974. [10] H. Ben Mohamed and B. Ghribi, Weinstein-Sobolev spaces of exponential type and applications, Acta Math. Sin., Engl. Ser. 29 (3) (2013), 591-608. [11] Z. Ben Nahia and N. Ben Salem, On a mean value property associated with the Weinstein operator, Potential theory - ICPT ’94. Proceedings of the international conference, Kouty, Czech Republic, Berlin: de Gruyter (1996), 243-253. [12] Z. Ben Nahia and N. Ben Salem, Spherical harmonics and applications associated with the Weinstein operator, Potential theory - ICPT ’94. Proceedings of the international conference, Kouty, Czech Republic, Berlin: de Gruyter (1996), 233-241. [13] A. Nowak and K. Stempak, Relating transplantation and multipliers for Dunkl and Hankel transforms, Math. Nachr. 281(11) (2008), 1604-1611. Int. J. Anal. Appl. 17 (1) (2019) 75 [14] A. Saoudi, Calderón’s reproducing formulas for the Weinstein L2-multiplier operators, arXiv:1801.08939 [math.AP]. [15] S.Saitoh, Approximate real inversion formulas of the gaussian convolution, Appl. Anal. 83(7) (2004), 727-733. [16] F. Soltani, Lp-Fourier multipliers for the Dunkl operator on the real line, J. Funct. Anal. 209(1) (2004), 16-35. [17] F. Soltani, Multiplier operators and extremal functions related to the dual Dunkl-Sonine operator, Acta Math. Sci., Ser. B, Engl. Ed. 33(2) (2013), 430-442. [18] F. Soltani, Dunkl multiplier operators on a class of reproducing kernel Hilbert spaces. J. Math. Res. Appl. 36(6) (2016), 689-702. 1. Introduction 2. Harmonic analysis Associated with the Weinstein Operator 2.1. The eigenfunction of the Weinstein operator 2.2. The Weinstein transform 2.3. The translation operator associated with the Weinstein operator 3. Heisenberg-Pauli-Weyl uncertainty principle 4. Donoho-Stark's uncertainty principle References