() CUBO A Mathematical Journal Vol.13, No¯ 03, (91–115). October 2011 Uncertainty principle for the Riemann-Liouville operator Hleili Khaled Faculty of Applied Mathematics,Département de Mathématiques et d’Informatique, Institut national des sciences appliquées et de Thechnologie, Centre Urbain Nord BP 676 - 1080 Tunis cedex, Tunisia, email: khaled.hleili@gmail.com Omri Slim Département de Mathématiques Appliquées, Institut préparatoire aux études d’ingénieurs, Campus universitaire Mrezka - 8000 Nabeul, Tunisia. email: slim.omri@issig.rnu.tn and Lakhdar T. Rachdi Département de Mathématiques, Faculté des Sciences de Tunis, 2092 El Manar II, Tunisia. email: lakhdartannech.rachdi@fst.rnu.tn ABSTRACT A Beurling-Hörmander theorem’s is proved for the Fourier transform connected with the Riemann-Liouville operator. Nextly, Gelfand-Shilov and Cowling-Price type theorems are established. RESUMEN Se demuestra el teorema de Beurling-Hörmander por la transformada de Fourier conec- tada con el operador de Riemann-Liouville. Además, se establecen teoremas tipo de Gelfand-Shilov y Cowling-Price. 92 Hleili Khaled, Omri Slim and Lakhdar T. Rachdi CUBO 13, 3 (2011) Keywords: Beurling-Hörmander theorem, Gelfand-Shilov theorem, Cowling- Price theorem, Fourier transform, Riemann-Liouville operator. Mathematics Subject Classification: 43A32; 42B10. 1 Introduction The uncertainty principles play an important role in harmonic analysis and have been studied by many authors, and from many points of view [12, 15]. These principles state that a function f and its Fourier transform f̂ cannot be simultaneously sharply localized. Theorems of Hardy, Morgan, Gelfand-Shilov, or Cowlong-Price,... are established for several Fourier transforms [8, 14, 19, 20, 21], the most recent being the well known Beurling-Hörmander theorem’s which has been proved by Hörmander [16], who took an idea of Beurling [4]. This theorem states that if f is an integrable function on R with respect to the Lebesgue measure, and if ∫∫ R2 |f(x)||f̂(y)|e|xy| dxdy < +∞, then f = 0 almost everywhere. Later, Bonami, Demange and Jaming [5] have generalized the above theorem and have established a strong multidimensional version of this uncertainty principle [15], by showing the following result if f is a square integrable function on Rn with respect to the Lebesgue measure, then ∫ Rn ∫ Rn |f(x)||f̂(y)| (1 + |x| + |y|)d e |〈x/y〉| dxdy < +∞, if and only if f may be written as f(x) = P(x)e−〈Ax/x〉, where A is a real positive definite symmetric matrix and P is a polynomial with degree(P) < d − n 2 . In particular for d 6 n, f is identically zero. The Beurling-Hörmander uncertainty principle in its weak and strong forms has been studied by many authors, and for various Fourier transforms. In particular, Bouattour and Trimèche [6] have showed this theorem for the hypergroup of Chébli-Trimèche, Kamoun and Trimèche [17] have proved an analogue of the Beurling-Hörmander theorem for some singular partial differential op- erators, Trimèche [22] has showed this uncertainty principle for the Dunkl transform, we cite also Yakubovich [26], who has established the same result for the Kontorovich-Lebedev transform. The Beurling-Hörmander uncertainty principle implies many other known quantitative uncertainty principles as those of Gelfand-Shilov [13], Cowling-Price [8], Morgan [3, 19] or also the one of Hardy [14]. In [2], the third author with the others have considered the singular partial differential oper- CUBO 13, 3 (2011) Uncertainty principle for the Riemann-Liouville operator 93 ators defined by    ∆1 = ∂ ∂x , ∆2 = ∂2 ∂r2 + 2α + 1 r ∂ ∂r − ∂2 ∂x2 ; (r,x) ∈]0, +∞[×R ; α ≥ 0, and they associated to ∆1 and ∆2 the following integral transform, called the Riemann-Liouville operator which is defined on C∗(R 2) ( The space of continuous functions on R2, even with respect to he first variable ) by Rα(f)(r,x) =    α π ∫ 1 −1 ∫ 1 −1 f(rs √ 1 − t2,x + rt)(1 − t2)α− 1 2 (1 − s2)α−1 dtds, if α > 0, 1 π ∫ 1 −1 f(r √ 1 − t2,x + rt) dt√ (1 − t2) ; if α = 0. The Fourier transform connected with the operator Rα is defined by Fα(f)(µ,λ) = ∫ +∞ 0 ∫ R f(r,x)ϕµ,λ(r,x)dνα(r,x), where ϕµ,λ(r,x) = Rα ( cos(µ.)e−iλ. ) (r,x). dνα is the measure defined on [0, +∞[×R by, dνα(r,x) = r2α+1 2αΓ(α + 1) √ 2π dr ⊗ dx. Many harmonic analysis results are established for the Fourier transform Fα (Inversion formula, Plancherel’s formula, Paley-Winer and Plancherel’s theorems...). The aim of this work is to establish the Beurling-Hörmander theorem for the fourier transform Fα and to deduce the analogues of the Gelfand-Shilov and the Cowling-Price theorems for this transform. More precisely, in the second section, we give some basic harmonic analysis results related to the Fourier transform Fα. The third section is devoted to establish the main result of this paper, that is the the Beurling-Hörmander theorem 94 Hleili Khaled, Omri Slim and Lakhdar T. Rachdi CUBO 13, 3 (2011) . Let f be a square integrable function on [0, +∞[×R with respect to the measure dνα. Let d be a real number, d > 0. If ∫ ∫ Γ+ ∫ +∞ 0 ∫ R |f(r,x)||Fα(f)(µ,λ)| (1 + |(r,x)| + |θ(µ,λ)|)d e |(r,x)||θ(µ,λ)| dνα(r,x) dγ̃α(µ,λ) < +∞. Then i) For d 6 2, f = 0. ii) For d > 2, there exist a positive constant a and a polynomial P on R2 even with respect to the first variable, such that f(r,x) = P(r,x)e−a(r 2 +x 2 ) , with degree(P) < d 2 − 1, where Γ+ = [0, +∞[×R ∪ { (it,x) | (t,x) ∈ [0, +∞[×R , t 6 |x| } . θ is the function defined on the set Γ+ by θ(µ,λ) = ( √ µ2 + λ2,λ). dγ̃α the measure defined on the set Γ+ by ∫ ∫ Γ+ g(µ,λ) dγ̃α(µ,λ) = 1 π ( ∫ +∞ 0 ∫ R g(µ,λ)(µ2 + λ2)− 1 2 µdµdλ + ∫ R ∫ |λ| 0 g(iµ,λ)(λ2 − µ2)− 1 2 µdµdλ ) . The last section of this paper contains the following results that are respectively the Gelfand-Shilov and the Cowling-Price theorems for Fα . Let p,q be two conjugate exponents, p,q ∈]1, +∞[. Let d,ξ,η be non negative real numbers such that ξη > 1. Let f be a measurable function on R2, even with respect to the first variable, such that f ∈ L2(dνα). If ∫ +∞ 0 ∫ R |f(r,x)|e ξp |(r,x)|p p (1 + |(r,x)|)d dνα(r,x) < +∞, and ∫∫ Γ+ |Fα(f)(µ,λ)|e ηq|θ(µ,λ)|q q (1 + |θ(µ,λ)|)d dγ̃α(µ,λ) < +∞, then i) For d 6 1, f = 0. 2i) For d > 1, we have CUBO 13, 3 (2011) Uncertainty principle for the Riemann-Liouville operator 95 a) f = 0 for ξη > 1. b) f = 0 for ξη = 1, and p 6= 2. c) f(r,x) = P(r,x)e−a(r 2 +x 2 ), for ξη = 1, and p = q = 2, where a > 0, and P is a polynomial on R2 even with respect to the first variable, with degree(P) < d − 1. . Let ξ,η,ω1,ω2 be non negative real numbers such that ξη > 1 4 . Let p,q be two exponents, p,q ∈ [1, +∞], and let f be a measurable function on R2, even with respect to the first variable such that f ∈ L2(dνα). If ∥∥∥ eξ|(.,.)| 2 (1 + |(., .)|)ω1 f ∥∥∥ p,να < +∞, and ∥∥∥ eη|θ(.,.)| 2 (1 + |θ(., .)|)ω2 Fα(f) ∥∥∥ q,γ̃α < +∞, then i) For ξη > 1 4 , f = 0. ii) For ξη = 1 4 , there exist a positive constant a and a polynomial P on R2, even with respect to the first variable, such that f(r,x) = P(r,x)e−a(r 2 +x 2 ) . 2 The Fourier transform associated with the Riemann-Liouville operator It’s well known [2] that for all (µ,λ) ∈ C2, the system    ∆1u(r,x) = −iλu(r,x), ∆2u(r,x) = −µ 2 u(r,x), u(0,0) = 1 , ∂u ∂r (0,x) = 0 , ∀x ∈ R, admits a unique solution ϕµ,λ, given by ∀(r,x) ∈ R2; ϕµ,λ(r,x) = jα(r √ µ2 + λ2)e−iλx, where jα(z) = 2αΓ(α + 1) zα Jα(z) = Γ(α + 1) +∞∑ n=0 (−1)n n!Γ(α + n + 1) ( z 2 )2n, z ∈ C, (2.1) 96 Hleili Khaled, Omri Slim and Lakhdar T. Rachdi CUBO 13, 3 (2011) and Jα is the Bessel function of the first kind and index α [9, 10, 18, 25]. The modified Bessel function jα has the following integral representation [18, 25], for all z ∈ C, we have jα(z) =    2Γ(α + 1) √ πΓ(α + 1 2 ) ∫ 1 0 (1 − t2)α− 1 2 cos(zt)dt, if α > − 1 2 ; cos(z), if α = − 1 2 . (2.2) From the relation (2.2), we deduce that for all z ∈ C, we have ∣∣jα(z) ∣∣ 6 e|Im(z)|. (2.3) From the properties of the modified Bessel function jα, we deduce that the eigenfunction ϕµ,λ satisfies the following properties sup (r,x)∈R2 |ϕµ,λ(r,x)| = 1, (2.4) if and only if (µ,λ) belongs to the set Γ = R2 ∪ { (it,x) | (t,x) ∈ R2 , |t| ≤ |x| } . The eigenfunction ϕµ,λ has the following Mehler integral representation ϕµ,λ(r,x) =    α π ∫ 1 −1 ∫ 1 −1 cos(µrs √ 1 − t2)e−iλ(x+rt)(1 − t2)α− 1 2 (1 − s2)α−1 dtds; if α > 0, 1 π ∫ 1 −1 cos(rµ √ 1 − t2)e−iλ(x+rt) dt√ 1 − t2 ; if α = 0. This integral representation allows to define the so-called Riemann-Liouville operator associated with ∆1,∆2 by Rα(f)(r,x) =    α π ∫ 1 −1 ∫ 1 −1 f(rs √ 1 − t2,x + rt)(1 − t2)α− 1 2 (1 − s2)α−1 dtds; if α > 0, 1 π ∫ 1 −1 f(r √ 1 − t2,x + rt) dt√ (1 − t2) ; if α = 0. where f is a continuous function on R2, even with respect to the first variable. The transform Rα generalizes the ”mean operator” defined by R0(f)(r,x) = 1 2π ∫ 2π 0 f(r sin θ,x + r cos θ) dθ. In the following, we denote by CUBO 13, 3 (2011) Uncertainty principle for the Riemann-Liouville operator 97 dmn+1 the measure defined on [0, +∞[×Rn by, dmn+1(r,x) = √ 2 π 1 (2π) n 2 dr ⊗ dx. Lp(dmn+1) the space of measurable functions f on [0, +∞[×Rn, such that ‖f‖p,mn+1 = ( ∫ +∞ 0 ∫ Rn |f(r,x)|p dmn+1(r,x) ) 1 p < +∞, if p ∈ [1, +∞[, ‖f‖∞ ,mn+1 = ess sup(r,x)∈[0,+∞ [×Rn |f(r,x)| < +∞, if p = +∞. dνα the measure defined on [0, +∞[×R, by dνα(r,x) = r2α+1 2αΓ(α + 1) √ 2π dr ⊗ dx. Lp(dνα) the space of measurable functions f on [0, +∞[×R such that ‖f‖p,να < +∞. Γ+ = [0, +∞[×R ∪ { (it,x) | (t,x) ∈ [0, +∞[×R , t 6 |x| } . BΓ+ the σ-algebra defined on Γ+ by BΓ+ = {θ −1(B) , B ∈ B([0, +∞[×R)}, where θ is the bijective function defined on the set Γ+ by θ(µ,λ) = ( √ µ2 + λ2,λ). dγα the measure defined on BΓ+ by ∀ A ∈ BΓ+ ; γα(A) = να(θ(A)). Lp(dγα) the space of measurable functions f on Γ+, such that ‖f‖p,γα < +∞. dγ̃α the measure defined on BΓ+ by dγ̃α(µ,λ) = 2α+ 1 2 Γ(α + 1) √ π(µ2 + λ2)α+ 1 2 dγα(µ,λ). S∗(R 2) the Shwartz’s space formed by the infinitely differentiable functions on R2, rapidly decreas- ing together with all their derivatives, and even with respect to the first variable. Then we have the following properties. 98 Hleili Khaled, Omri Slim and Lakhdar T. Rachdi CUBO 13, 3 (2011) Proposition 2.1. i) For all non negative measurable function g on Γ+, we have ∫ ∫ Γ+ g(µ,λ) dγα(µ,λ) = 1 2αΓ(α + 1) √ 2π ( ∫ +∞ 0 ∫ R g(µ,λ)(µ2 + λ2)αµdµdλ + ∫ R ∫ |λ| 0 g(iµ,λ)(λ2 − µ2)αµdµdλ ) . ii) For all measurable function f on [0, +∞[×R, the function foθ is measurable on Γ+. Furthermore if f is non negative or integrable function on [0, +∞[×R with respect to the measure dνα, then we have ∫ ∫ Γ+ (f ◦ θ)(µ,λ) dγα(µ,λ) = ∫ +∞ 0 ∫ R f(r,x) dνα(r,x). iii) For all non negative measurable function f, respectively integrable on [0, +∞[×R with respect to the measure dm2, we have ∫ ∫ Γ+ (f ◦ θ)(µ,λ) dγ̃α(µ,λ) = ∫ +∞ 0 ∫ R f(r,x) dm2(r,x). (2.5) In the following we shall define the Fourier transform Fα associated with the operator Rα, and we shall give some properties that we use in the sequel. Definition 2.1. The Fourier transform Fα associated with the Riemann-Liouville operator Rα is defined on L1(dνα) by ∀(µ,λ) ∈ Γ ; Fα(f)(µ,λ) = ∫ +∞ 0 ∫ R f(r,x)ϕµ,λ(r,x) dνα(r,x). Then, for all (µ,λ) ∈ Γ, Fα(f)(µ,λ) = F̃α(f) ◦ θ(µ,λ), (2.6) where for all (µ,λ) ∈ [0, +∞[×R, F̃α(f)(µ,λ) = ∫ +∞ 0 ∫ R f(r,x)jα(rµ)e −iλx dνα(r,x). (2.7) Moreover, the relation (2.4) implies that the Fourier transform Fα is a bounded linear operator from L1(dνα) into L ∞ (dγα), and that for all f ∈ L1(dνα), we have ‖Fα(f)‖∞ ,γα 6 ‖f‖1,να. (2.8) Theorem 2.1 (Inversion formula). Let f ∈ L1(dνα) such that Fα(f) ∈ L1(dγα), then for almost every (r,x) ∈ [0, +∞[×R, we have f(r,x) = ∫ ∫ Γ+ Fα(f)(µ,λ)ϕµ,λ(r,x) dγα(µ,λ) = ∫ +∞ 0 ∫ R F̃α(f)(µ,λ)jα(rµ)e iλx dνα(µ,λ). CUBO 13, 3 (2011) Uncertainty principle for the Riemann-Liouville operator 99 Lemma 2.2. Let Rα be the mapping defined for all non negative measurable function g on [0, +∞[×R by Rα(g)(r,x) = 2Γ(α + 1) √ πΓ(α + 1 2 ) ∫ 1 0 (1 − s2)α− 1 2 g(rs,x) ds = 2Γ(α + 1)r−2α √ πΓ(α + 1 2 ) ∫ r 0 (r2 − s2)α− 1 2 f(s,x) ds, r > 0. (2.9) Then for all non negative measurable functions f,g on [0, +∞[×R, we have ∫ +∞ 0 ∫ R f(r,x)Rα(g)(r,x) dνα(r,x) = ∫ +∞ 0 ∫ R Wα(f)(r,x)g(r,x) dm2(r,x), (2.10) where Wα is the classical Weyl transform defined for all non negative measurable function on [0, +∞[×R by Wα(f)(r,x) = 1 2α+ 1 2 Γ(α + 1 2 ) ∫ +∞ r (t2 − r2)α− 1 2 f(t,x)2tdt. (2.11) Proposition 2.2. For all f ∈ L1(dνα), the function Wα(f) belongs to L1(dm2), and we have ‖Wα(f)‖1,m2 6 ‖f‖1,να. (2.12) Moreover, for all (µ,λ) ∈ [0, +∞[×R, we have F̃α(f)(µ,λ) = (Λ2 ◦ Wα)(f)(µ,λ), (2.13) where Λ2 is the usual Fourier transform defined on L 1(dm2) by Λ2(g)(µ,λ) = ∫ +∞ 0 ∫ R g(r,x) cos(rµ)e−iλx dm2(r,x). Remark 2.1. It’s well known [23, 24] that the transforms F̃α and Λ2 are topological isomorphisms from S∗(R 2) onto itself. Then by the relation (2.13), we deduce that the classical Weyl transform Wα is also a topological isomorphism from S∗(R 2) onto itself. Proposition 2.3. For all f ∈ S∗(R2), we have W −1 α (f) = (−1) 1+[α+ 1 2 ] W[α+ 1 2 ]−α+ 1 2 (( ∂ ∂t2 )1+[α+ 1 2 ] (f) ) , (2.14) where ( ∂ ∂t2 ) (f)(t,x) = 1 t ∂f ∂t (t,x). Proof. For σ ∈ R, σ > 0, let us define the so-called fractional transform Hσ, defined on S∗(R2) by Hσ(f)(r,x) = 1 2σΓ(σ) ∫ +∞ r (t2 − r2)σ−1f(t,x)2tdt = Wσ− 1 2 (f)(r,x). 100 Hleili Khaled, Omri Slim and Lakhdar T. Rachdi CUBO 13, 3 (2011) From the remark 2.1, it follows that for all real number σ > 0, the mapping Hσ is a topological isomorphism from S∗(R 2) onto itself. Moreover, we have the following properties For all σ,δ ∈ R; σ,δ > 0 and for every f ∈ S∗(R2), we have ( Hσ ◦ Hδ ) (f) = Hσ+δ(f). For all σ ∈ R, σ > 0, and for every integer k, we have Hσ(f) = (−1) k Hσ+k (( ∂ ∂t2 )k (f) ) . (2.15) where ∂ ∂t2 is the linear continuous operator defined on S∗(R 2) by ∂ ∂t2 (f)(t,x) = 1 t ∂f ∂t (t,x). The relation (2.15) allows us to extend the mapping Hσ on R, by setting Hσ(f)(r,x) = (−1) k Hσ+k (( ∂ ∂t2 )k (f) ) , where k is any integer such that σ + k > 0, σ ∈ R. The extension Hσ, σ ∈ R satisfies ( Hσ ◦ Hδ ) (f) = Hσ+δ(f), σ,δ ∈ R, f ∈ S∗(R2), and H0(f) = f, for all f ∈ S∗(R2). In particular, for all σ ∈ R, the transform Hσ is a topological isomorphism from S∗(R2) onto itself, and the isomorphism inverse is given by H −1 σ = H−σ. Thus, for all real number σ, we have H −1 σ (f) = (−1) 1+[σ] H1+[σ]−σ (( ∂ ∂t2 )1+[σ] (f) ) . In particular W −1 α (f) = H −1 α+ 1 2 (f) = (−1)1+[α+ 1 2 ] H[α+ 1 2 ]−α+ 1 2 (( ∂ ∂t2 )1+[α+ 1 2 ] (f) ) . 3 The Beurling-Hörmander theorem for the Riemann-Liouville operator In this section, we shall establish the main result of this paper, that is the Beurling-Hörmander theorem for the Fourier transform Fα. We recall firstly the following result that has been established by Bonami, Demange and Jaming [5]. CUBO 13, 3 (2011) Uncertainty principle for the Riemann-Liouville operator 101 Theorem 3.1. Let f be a measurable function on R × Rn, even with respect to the first variable such that f ∈ L2(dmn+1), and let d be a real number, d ≥ 0. If ∫ +∞ 0 ∫ Rn ∫ +∞ 0 ∫ Rn |f(r,x)||Λn+1(f)(s,y)| (1 + |(r,x)| + |(s,y)|)d e |(r,x)||(s,y)| dmn+1(r,x) dmn+1(s,y) < +∞, then there exist a positive constant a and a polynomial P on R × Rn, even with respect to the first variable, such that f(r,x) = P(r,x)e−a(r 2 +|x| 2 ) , with degree(P) < d − (n + 1) 2 . In the following, we will establish some intermediary results that we use nextly. Lemma 3.2. Let f ∈ L2(dνα) such that ∫ ∫ Γ+ ∫ +∞ 0 ∫ R |f(r,x)||Fα(f)(µ,λ)| (1 + |(r,x)| + |θ(µ,λ)|)d e |(r,x)||θ(µ,λ)| dνα(r,x) dγ̃α(µ,λ) < +∞, (3.1) then the function f belongs to the space L1(dνα). Proof. From the hypothesis, and the relations (2.5) and (2.6), we have ∫ ∫ Γ+ ∫ +∞ 0 ∫ R |f(r,x)||Fα(f)(µ,λ)| (1 + |(r,x)| + |θ(µ,λ)|)d e |(r,x)||θ(µ,λ)| dνα(r,x) dγ̃α(µ,λ) = ∫ +∞ 0 ∫ R ∫ +∞ 0 ∫ R |f(r,x)||F̃α(f)(µ,λ)| (1 + |(r,x)| + |(µ,λ)|)d e |(r,x)||(µ,λ)| dνα(r,x)dm2(µ,λ) < +∞. We assume of course that f 6= 0. Then, there exists (µ0,λ0) ∈ [0, +∞[×R, such that (µ0,λ0) 6= (0,0), F̃α(f)(µ0,λ0) 6= 0, and |F̃α(f)(µ0,λ0)| ∫ +∞ 0 ∫ R |f(r,x)| e|(r,x)||(µ0,λ0)| (1 + |(r,x)| + |(µ0,λ0)|) d dνα(r,x) < +∞, hence ∫ +∞ 0 ∫ R |f(r,x)| e|(r,x)||(µ0,λ0)| (1 + |(r,x)| + |(µ0,λ0)|) d dνα(r,x) < +∞. Let h be the function defined on [0, +∞[ by h(s) = es|(µ0,λ0)| (1 + s + |(µ0,λ0)|) d , then the function h admits a minimum attained at s0 =    d |(µ0,λ0)| − 1 − |(µ0,λ0)|, if d |(µ0,λ0)| > 1 + |(µ0,λ0)|; 0, if d |(µ0,λ0)| 6 1 + |(µ0,λ0)|. 102 Hleili Khaled, Omri Slim and Lakhdar T. Rachdi CUBO 13, 3 (2011) Consequently, ∫ +∞ 0 ∫ R |f(r,x)| dνα(r,x) 6 1 h(s0) ∫ +∞ 0 ∫ R |f(r,x)|e|(r,x)||(µ0,λ0)| (1 + |(r,x)| + |(µ0,λ0)|) d dνα(r,x) < +∞. Lemma 3.3. Let f ∈ L2(dνα) such that ∫ ∫ Γ+ ∫ +∞ 0 ∫ R |f(r,x)||Fα(f)(µ,λ)| (1 + |(r,x)| + |θ(µ,λ)|)d e |(r,x)||θ(µ,λ)| dνα(r,x) dγ̃α(µ,λ) < +∞. Then, there exists a > 0 such that the function F̃α(f) is analytic on the set Ba = { (µ,λ) ∈ C2 | ∣∣Im(µ) ∣∣ < a , ∣∣Im(λ) ∣∣ < a } . Proof. From the proof of the lemma 3.2, there exists (µ0,λ0) 6= (0,0), such that ∫ +∞ 0 ∫ R |f(r,x)|e|(r,x)||(µ0,λ0)| (1 + |(r,x)| + |(µ0,λ0)|) d dνα(r,x) < +∞. Let a > 0, such that 0 < 2a < |(µ0,λ0)|. Then we have ∫ +∞ 0 ∫ R |f(r,x)|e|(r,x)||(µ0,λ0)| (1 + |(r,x)| + |(µ0,λ0)|) d dνα(r,x) = ∫ +∞ 0 ∫ R |f(r,x)|e2a|(r,x)| e|(r,x)|(|(µ0,λ0)|−2a) (1 + |(r,x)| + |(µ0,λ0)|) d dνα(r,x) < +∞. Let g be the function defined on [0, +∞[ by g(s) = es(|(µ0,λ0)|−2a) (1 + s + |(µ0,λ0)|) d , then g admits a minimum attained at s0 =    d |(µ0,λ0)| − 2a − 1 − |(µ0,λ0)|, if d |(µ0,λ0)| − 2a > 1 + |(µ0,λ0)|; 0, if d |(µ0,λ0)| − 2a 6 1 + |(µ0,λ0)|. Consequently, ∫ +∞ 0 ∫ R |f(r,x)|e2a|(r,x)| dνα(r,x) 6 1 g(s0) ∫ +∞ 0 ∫ R |f(r,x)|e|(r,x)||(µ0,λ0)| (1 + |(r,x)| + |(µ0,λ0)|) d dνα(r,x) < +∞. (3.2) CUBO 13, 3 (2011) Uncertainty principle for the Riemann-Liouville operator 103 On the other hand, from the relation (2.1) we deduce that for all (r,x) ∈ [0, +∞[×R, the function (µ,λ) 7−→ jα(rµ)e−ixλ is analytic on C2 [7], even with respect to the first variable, and by the relation (2.3) we have ∣∣jα(rµ)e−iλx ∣∣ 6 e|(r,x)|(|Im(µ)|+|Im(λ)|). (3.3) From the relations (2.7), (3.2), and (3.3), it follows that the function F̃α(f) is analytic on Ba, even with respect to the first variable. Corollary 3.1. Let f ∈ L2(dνα); f 6= 0; and let d be a real number, d > 0. If ∫ ∫ Γ+ ∫ +∞ 0 ∫ R |f(r,x)||Fα(f)(µ,λ)| (1 + |(r,x)| + |θ(µ,λ)|)d e |(r,x)||θ(µ,λ)| dνα(r,x) dγ̃α(µ,λ) < +∞. then for all real number a; a > 0, we have να ({ (r,x) ∈ R2 | F̃α(f)(r,x) 6= 0 and |(r,x)| > a }) > 0. Proof. From lemma 3.2, the function f belongs to L1(dνα), and consequently the function F̃α(f) is continuous on R2, even with respect to the first variable. Then for all a > 0, the set { (r,x) ∈ R2 | F̃α(f)(r,x) 6= 0 and |(r,x)| > a } , is on open subset of R2. Assume that να ({ (r,x) ∈ R2 | F̃α(f)(r,x) 6= 0 and |(r,x)| > a }) = 0, then for all (r,x) ∈ R2; |(r,x)| > a, we have F̃α(f)(r,x) = 0. Applying lemma 3.3 and analytic continuation, we deduce that F̃α(f) vanishes on R 2, and by theorem 2.1, it follows that f = 0. Lemma 3.4. Let f ∈ L2(dνα) and let d be a real number d > 0. If ∫ ∫ Γ+ ∫ +∞ 0 ∫ R |f(r,x)||Fα(f)(µ,λ)| (1 + |(r,x)| + |θ(µ,λ)|)d e |(r,x)||θ(µ,λ)| dνα(r,x) dγ̃α(µ,λ) < +∞, then the function Wα(f), belongs to L 2(dm2), where Wα is the mapping defined by the relation (2.11). Proof. From the hypothesis and the relations (2.5) and (2.6), we have ∫ ∫ Γ+ ∫ +∞ 0 ∫ R |f(r,x)||Fα(f)(µ,λ)| (1 + |(r,x)| + |θ(µ,λ)|)d e |(r,x)||θ(µ,λ)| dνα(r,x) dγ̃α(µ,λ) = ∫ +∞ 0 ∫ R ∫ +∞ 0 ∫ R |f(r,x)||F̃α(f)(µ,λ)| (1 + |(r,x)| + |(µ,λ)|)d e |(r,x)||(µ,λ)| dνα(r,x)dm2(µ,λ) < +∞. 104 Hleili Khaled, Omri Slim and Lakhdar T. Rachdi CUBO 13, 3 (2011) By the same way as inequality (3.2) of the lemma 3.3, there exists b ∈ R, b > 0, such that ∫ +∞ 0 ∫ R |F̃α(f)(µ,λ)|e b|(µ,λ)| dm2(µ,λ) < +∞. (3.4) Consequently, the function F̃α(f) lies in L 1(dνα) and by theorem 2.1, we get f(r,x) = ∫ +∞ 0 ∫ R F̃α(f)(µ,λ)jα(rµ)e iλx dνα(µ,λ); a.e. In particular the function f is bounded and ‖f‖∞ ,να 6 ‖F̃α(f)‖1,να. (3.5) Now, we have |Wα(f)(r,x)| 6 1 2α+ 1 2 Γ(α + 1 2 ) ∫ +∞ r (t2 − r2)α− 1 2 |f(t,x)|2tdt = r2α+1 2α+ 1 2 Γ(α + 1 2 ) ∫ +∞ 1 (u2 − 1)α− 1 2 |f(ru,x)|2udu. Using Minkowski’s inequality for integrals [11], we get ( ∫ +∞ 0 ∫ R |Wα(f)(r,x)| 2 dm2(r,x) ) 1 2 6 1 2α+ 1 2 Γ(α + 1 2 ) ( ∫ +∞ 0 ∫ R ( ∫ +∞ 1 r 2α+1(u2 − 1)α− 1 2 |f(ru,x)|2udu )2 dm2(r,x) ) 1 2 6 1 2α+ 1 2 Γ(α + 1 2 ) ∫ +∞ 1 (u2 − 1)α− 1 2 ( ∫ +∞ 0 ∫ R r 4α+2 |f(ru,x)|2 dm2(r,x) ) 1 2 2udu = Γ(α + 1) 1 2 2 α 2 − 3 4 π 1 4 Γ(α + 1 2 ) ( ∫ +∞ 1 (u2 − 1)α− 1 2 u −2α− 1 2 du )( ∫ +∞ 0 ∫ R |f(t,x)|2t2α+1dνα(t,x) ) 1 2 = Γ(α + 1) 1 2 2 α 2 − 7 4 π 1 4 Γ(α + 1 2 ) ( ∫ 1 0 (1 − s)α− 1 2 s 9 4 ds )( ∫ +∞ 0 ∫ R |f(t,x)|2t2α+1dνα(t,x) ) 1 2 = Cα ( ∫ +∞ 0 ∫ R |f(t,x)|2t2α+1dνα(t,x) ) 1 2 and by the relations (3.2) and (3.5), we get ( ∫ +∞ 0 ∫ R |Wα(f)(r,x)| 2 dm2(r,x) ) 1 2 6 Mα‖f‖ 1 2 ∞ ,να ( ∫ +∞ 0 ∫ R |f(t,x)|e2a|(t,x)|dνα(t,x) ) 1 2 < +∞. Remark 3.1. Let f be a function satisfying the hypothesis (3.1), then from the relations (3.2) and (3.4), we can prove that the function f belongs to the Schwartz’s space S∗(R 2). Since the Weyl transform Wα is an isomorphism from S∗(R 2) onto itself, then the function Wα(f) belongs to S∗(R 2), in particular Wα(f) ∈ L2(dm2). CUBO 13, 3 (2011) Uncertainty principle for the Riemann-Liouville operator 105 Remark 3.2. Let σ be a positive real number such that σ + σ2 > d > 0. Then, the function t 7−→ eσt (1 + t + σ)d , is increasing on [0, +∞[. Theorem 3.5. Let f ∈ L2(dνα), and let d be a real number, d > 0. If ∫ ∫ Γ+ ∫ +∞ 0 ∫ R |f(r,x)||Fα(f)(µ,λ)| (1 + |(r,x)| + |θ(µ,λ)|)d e |(r,x)||θ(µ,λ)| dνα(r,x) dγ̃α(µ,λ) < +∞, then ∫ +∞ 0 ∫ R ∫ +∞ 0 ∫ R |Wα(f)(r,x)||F̃α(f)(µ,λ)| (1 + |(r,x)| + |(µ,λ)|)d e |(r,x)||(µ,λ)| dm2(r,x)dm2(µ,λ) < +∞. Proof. From the hypothesis and the relations (2.5) and (2.6), we have ∫ +∞ 0 ∫ R ∫ +∞ 0 ∫ R |f(r,x)||F̃α(f)(µ,λ)| (1 + |(r,x)| + |(µ,λ)|)d e |(r,x)||(µ,λ)| dνα(r,x)dm2(µ,λ) < +∞. (3.6) i) If d = 0, then by Fubini’s theorem we have ∫ +∞ 0 ∫ R ∫ +∞ 0 ∫ R |Wα(f)(r,x)||F̃α(f)(µ,λ)|e |(r,x)||(µ,λ)| dm2(r,x)dm2(µ,λ) 6 ∫ +∞ 0 ∫ R |F̃α(f)(µ,λ)| ( ∫ +∞ 0 ∫ R |Wα(f)(r,x)|e |(r,x)||(µ,λ)| dm2(r,x) ) dm2(µ,λ) 6 ∫ +∞ 0 ∫ R |F̃α(f)(µ,λ)| ( ∫ +∞ 0 ∫ R Wα(|f|)(r,x)e |(r,x)||(µ,λ)| dm2(r,x) ) dm2(µ,λ). (3.7) Using the relation (2.10), we deduce that ∫ +∞ 0 ∫ R Wα(|f|)(r,x)e |(r,x)||(µ,λ)| dm2(r,x) = ∫ +∞ 0 ∫ R |f(r,x)|Rα(e |(.,.)||(µ,λ)|)(r,x) dνα(r,x), (3.8) but for all (r,x) ∈ [0, +∞[×R Rα(e |(.,.)||(µ,λ)|)(r,x) 6 e|(r,x)||(µ,λ)|. (3.9) Combining the relations (3.6), (3.7), (3.8), and (3.9), we get ∫ +∞ 0 ∫ R ∫ +∞ 0 ∫ R |Wα(f)(r,x)||F̃α(f)(µ,λ)|e |(r,x)||(µ,λ)| dm2(r,x)dm2(µ,λ) 6 ∫ +∞ 0 ∫ R |F̃α(f)(µ,λ)| ( ∫ +∞ 0 ∫ R |f(r,x)|e|(r,x)||(µ,λ)| dνα(r,x) ) dm2(µ,λ) < +∞. 106 Hleili Khaled, Omri Slim and Lakhdar T. Rachdi CUBO 13, 3 (2011) ii) If d > 0, let Bd = { (u,v) ∈ [0, +∞[×R | |(u,v)| 6 d } . . By Fubini’s theorem, we have ∫∫ Bc d ∫ +∞ 0 ∫ R |F̃α(f)(µ,λ)||Wα(f)(r,x)| (1 + |(r,x)| + |(µ,λ)|)d e |(r,x)||(µ,λ)| dm2(r,x)dm2(µ,λ) 6 ∫∫ Bc d |F̃α(f)(µ,λ)| ( ∫ +∞ 0 ∫ R Wα(|f|)(r,x) e|(r,x)||(µ,λ)| (1 + |(r,x)| + |(µ,λ)|)d × dm2(r,x) ) dm2(µ,λ), and by the relation (2.10), we get ∫∫ Bc d ∫ +∞ 0 ∫ R |F̃α(f)(µ,λ)||Wα(f)(r,x)| (1 + |(r,x)| + |(µ,λ)|)d e |(r,x)||(µ,λ)| dm2(r,x)dm2(µ,λ) 6 ∫∫ Bc d |F̃α(f)(µ,λ)| ( ∫ +∞ 0 ∫ R |f(r,x)|Rα ( e|(.,.)||(µ,λ)| (1 + |(., .)| + |(µ,λ)|)d ) (r,x) × dνα(r,x) ) dm2(µ,λ). (3.10) However, by the relation (2.9) and remark 3.2, we have for all (µ,λ) ∈ Bcd Rα ( e|(.,.)||(µ,λ)| (1 + |(., .)| + |(µ,λ)|)d ) (r,x) 6 e|(r,x)||(µ,λ)| (1 + |(r,x)| + |(µ,λ)|)d . (3.11) Combining the relations (3.10) and (3.11), we obtain ∫∫ Bc d ∫ +∞ 0 ∫ R |F̃α(f)(µ,λ)||Wα(f)(r,x)| (1 + |(r,x)| + |(µ,λ)|)d e |(r,x)||(µ,λ)| dm2(r,x)dm2(µ,λ) 6 ∫∫ Bc d |F̃α(f)(µ,λ)| ( ∫ +∞ 0 ∫ R |f(r,x)| e|(r,x)||(µ,λ)| (1 + |(r,x)| + |(µ,λ)|)d ) dνα(r,x) ) dm2(µ,λ) 6 ∫ +∞ 0 ∫ R ∫ +∞ 0 ∫ R |F̃α(f)(µ,λ)||f(r,x)| (1 + |(r,x)| + |(µ,λ)|)d e |(r,x)||(µ,λ)| dνα(r,x)dm2(µ,λ) < +∞. . ∫∫ Bd ∫∫ Bc d |Wα(f)(r,x)||F̃α(f)(µ,λ)| (1 + |(r,x)| + |(µ,λ)|)d e |(r,x)||(µ,λ)| dm2(r,x)dm2(µ,λ) 6 ∫∫ Bd |F̃α(f)(µ,λ)| ( ∫∫ Bc d Wα(|f|)(r,x) e|(r,x)||(µ,λ)| (1 + |(r,x)| + |(µ,λ)|)d dm2(r,x) ) dm2(µ,λ). CUBO 13, 3 (2011) Uncertainty principle for the Riemann-Liouville operator 107 But for (µ,λ) ∈ Bd, ∫∫ Bc d Wα(|f|)(r,x) e|(r,x)||(µ,λ)| (1 + |(r,x)| + |(µ,λ)|)d dm2(r,x) = ∫ +∞ 0 ∫ R |f(r,x)|Rα ( e|(.,.)||(µ,λ)| (1 + |(., .)| + |(µ,λ)|)d 1Bc d ) (r,x) dνα(r,x) 6 ∫∫ Bc d |f(r,x)| ed|(r,x)| (1 + |(r,x)| + d)d dνα(r,x). Hence, ∫∫ Bd ∫∫ Bc d |Wα(f)(r,x)||F̃α(f)(µ,λ)| (1 + |(r,x)| + |(µ,λ)|)d e |(r,x)||(µ,λ)| dm2(r,x)dm2(µ,λ) 6 ( ∫∫ Bd |F̃α(f)(µ,λ)|dm2(µ,λ) )( ∫∫ Bc d |f(r,x)| ed|(r,x)| (1 + |(r,x)| + d)d dνα(r,x) ) . In virtue of the relation (2.8), we have ∫∫ Bd ∫∫ Bc d |Wα(f)(r,x)||F̃α(f)(µ,λ)| (1 + |(r,x)| + |(µ,λ)|)d e |(r,x)||(µ,λ)| dm2(r,x)dm2(µ,λ) 6 ‖f‖1,ναm2(Bd) ( ∫∫ Bc d |f(r,x)| ed|(r,x)| (1 + |(r,x)| + d)d dνα(r,x) ) . (3.12) On he other hand, from corollary 3.1 and the relation (3.6), there exists (µ0,λ0) ∈ [0, +∞[×R, |(µ0,λ0)| > d, F̃α(f)(µ0,λ0) 6= 0, and ∫∫ Bc d |f(r,x)| e|(µ0,λ0)||(r,x)| (1 + |(r,x)| + |(µ0,λ0)|) d dνα(r,x) < +∞, (3.13) so, by remark 3.2, ∫∫ Bc d |f(r,x)| ed|(r,x)| (1 + |(r,x)| + d)d dνα(r,x) 6 ∫∫ Bc d |f(r,x)| e|(µ0,λ0)||(r,x)| (1 + |(r,x)| + |(µ0,λ0)|) d dνα(r,x) < +∞. (3.14) The relations (3.12), (3.13), and (3.14) imply that ∫∫ Bd ∫∫ Bc d |Wα(f)(r,x)||F̃α(f)(µ,λ)| (1 + |(r,x)| + |(µ,λ)|)d e |(r,x)||(µ,λ)| dm2(r,x)dm2(µ,λ) < +∞. 108 Hleili Khaled, Omri Slim and Lakhdar T. Rachdi CUBO 13, 3 (2011) Finally . ∫∫ Bd ∫∫ Bd |Wα(f)(r,x)||F̃α(f)(µ,λ)| (1 + |(r,x)| + |(µ,λ)|)d e |(r,x)||(µ,λ)| dm2(r,x)dm2(µ,λ) 6 e d 2 ( ∫∫ Bd |F̃α(f)(µ,λ)|dm2(µ,λ) )( ∫∫ Bd |Wα(f)(r,x)| dm2(r,x) ) 6 e d 2 m2(Bd)‖Fα(f)‖∞ ,γα ‖Wα(f)‖1,m2, and therefore by the relations (2.8) and (2.12), we deduce that ∫∫ Bd ∫∫ Bd |Wα(f)(r,x)||F̃α(f)(µ,λ)| (1 + |(r,x)| + |(µ,λ)|)d e |(r,x)||(µ,λ)| dm2(r,x)dm2(µ,λ) 6 e d 2 m2(Bd)‖f‖21,να < +∞, and the proof of theorem 3.5 is complete. Theorem 3.6 (Beurling-Hörmander for Rα). Let f ∈ L2(dνα), and let d be a real number, d > 0. If ∫ ∫ Γ+ ∫ +∞ 0 ∫ R |f(r,x)||Fα(f)(µ,λ)| (1 + |(r,x)| + |θ(µ,λ)|)d e |(r,x)||θ(µ,λ)| dνα(r,x) dγ̃α(µ,λ) < +∞. Then i) For d 6 2, f = 0. ii) For d > 2, there exist a positive constant a and a polynomial P, even with respect to the first variable, such that f(r,x) = P(r,x)e−a(r 2 +x 2 ) , with degree(P) < d 2 − 1. Proof. Let f ∈ L2(dνα), satisfying the hypothesis. From proposition 2.2, lemma 3.2, and lemma 3.4, we deduce that the function Wα(f) belongs to the space L1(dm2) ∩ L2(dm2) and that F̃α(f) = Λ2 ◦ Wα(f). Thus from theorem 3.5, we get ∫ +∞ 0 ∫ R ∫ +∞ 0 ∫ R ∣∣Wα(f)(r,x) ∣∣∣∣Λ2 ( Wα(f) ) (µ,λ) ∣∣e|(r,x)||(µ,λ)| (1 + |(r,x)| + |(µ,λ)|)d dm2(r,x)dm2(µ,λ) < +∞. Applying theorem 3.1, when f is replaced by Wα(f), we deduce that If d 6 2, Wα(f) = 0, and by remark 2.1, f = 0. CUBO 13, 3 (2011) Uncertainty principle for the Riemann-Liouville operator 109 If d > 2, then there exist a > 0 and a polynomial Q even with respect to the first variable such that Wα(f)(r,x) = Q(r,x)e −a(r 2 +x 2 ) = ∑ 2p+q6m ap,qr 2p x q e −a(r 2 +x 2 ) . In particular, the function Wα(f) belongs to the space S∗(R 2). From remark 2.1, the function f belongs to S∗(R 2) and from the relation (2.14), we get f(r,x) = H−α− 1 2 ( Q(t,y)e−a(t 2 +y 2 ) ) (r,x) = (−1)[α+ 1 2 ]+1 H[α+ 1 2 ]−α+ 1 2 (( ∂ ∂t2 )[α+ 1 2 ]+1( P(t,y)e−a(t 2 +y 2 ) )) (r,x) = ∑ 2p+q6m ap,q(−1) [α+ 1 2 ]+1 H[α+ 1 2 ]−α+ 1 2 (( ∂ ∂t2 )[α+ 1 2 ]+1( t 2p y q e −a(t 2 +y 2 ) )) (r,x). (3.15) However, for all k ∈ N, ( ∂ ∂t2 )k( t 2p y q e −a(t 2 +y 2 ) ) = ( min(p,k)∑ j=0 C j k 2jp! (p − j)! (−2a)k−jt2(p−j) ) y q e −a(t 2 +y 2 ) , (3.16) and for all σ ∈ R, σ > 0, Hσ ( t 2p y q e −a(t 2 +y 2 ) ) (r,x) = 1 2σΓ(σ) ( p∑ j=0 C j p Γ(σ + p − j) aσ+p−j r 2j ) x q e −a(r 2 +x 2 ) . (3.17) Combining the relations (3.15), (3.16) and (3.17), we deduce that f(r,x) = P(r,x)e−a(r 2 +x 2 ) . Where P is a polynomial, even with respect to the first variable and degree(P) = degree(Q). 4 Applications of Beurling-Hörmander theorem In this section, we shall deduce from the precedent Beurling-Hörmander theorem two most im- portant uncertainty principles for the Fourier transform Fα, that are the Gelfand-Shilov and the Cowling-Price theorems. Lemma 4.1. Let P be a polynomial on R2, P 6= 0, with degree(P) = m. Then there exist two positive constants A and C such that ∀t > A, p(t) = ∫ 2π 0 ∣∣P(t cos(θ),t sin(θ) ∣∣dθ > Ctm. 110 Hleili Khaled, Omri Slim and Lakhdar T. Rachdi CUBO 13, 3 (2011) Proof. Let P be a polynomial on R2, P 6= 0 and with degree(P) = m. We have p(t) = ∫ 2π 0 ∣∣ m∑ j=0 aj(θ)t j ∣∣dθ, where the functions aj, 0 6 j 6 m, are continuous on [0,2π]. It’s clear that the function p is continuous on [0, +∞[, and by dominate convergence theorem’s, we have p(t) ∼ Cmt m (t −→ +∞), (4.1) where Cm = ∫ 2π 0 |am(θ)|dθ > 0. Now the relation (4.1) involves that there exists A > 0 such that ∀t > A, p(t) > Cm 2 t m . Theorem 4.2 (Gelfand-Shilov for Rα). Let p,q be two conjugate exponents, p,q ∈]1, +∞[. Let ξ,η be non negative real numbers such that ξη > 1. Let f be a measurable function on R2, even with respect to the first variable, such that f ∈ L2(dνα). If ∫ +∞ 0 ∫ R |f(r,x)|e ξp |(r,x)|p p (1 + |(r,x)|)d dνα(r,x) < +∞, and ∫∫ Γ+ |Fα(f)(µ,λ)|e ηq |θ(µ,λ)|q q (1 + |θ(µ,λ)|)d dγ̃α(µ,λ) < +∞ ; d > 0. Then i) For d 6 1, f = 0. ii) For d > 1, we have a) f = 0 for ξη > 1. b) f = 0 for ξη = 1, and p 6= 2. c) f(r,x) = P(r,x)e−a(r 2 +x 2 ) for ξη = 1 and p = q = 2, where a > 0 and P is a polynomial on R2 even with respect to the first variable, with degree(P) < d − 1. Proof. Let f be a function satisfying the hypothesis. Since ξη > 1, and by a convexity argument, CUBO 13, 3 (2011) Uncertainty principle for the Riemann-Liouville operator 111 we have ∫∫ Γ+ ∫ +∞ 0 ∫ R |f(r,x)||Fα(f)(µ,λ)| (1 + |(r,x)| + |θ(µ,λ)|)2d e |(r,x)||θ(µ,λ)| dνα(r,x) dγ̃α(µ,λ) 6 ∫∫ Γ+ ∫ +∞ 0 ∫ R |f(r,x)||Fα(f)(µ,λ)| (1 + |(r,x)|)d(1 + |θ(µ,λ)|)d e ξη|(r,x)||θ(µ,λ)| dνα(r,x) dγ̃α(µ,λ) 6 ( ∫∫ Γ+ |Fα(f)(µ,λ)| (1 + |θ(µ,λ)|)d e ηq|θ(µ,λ)|q q dγ̃α(µ,λ) ) × ( ∫ +∞ 0 ∫ R |f(r,x)| (1 + |(r,x)|)d e ξp|(r,x)|p p dνα(r,x) ) < +∞. (4.2) Then from the Beurling-Hörmander theorem, we deduce that i) For d 6 1, f = 0. ii) For d > 1, there exist a positive constant a, and a polynomial P on R2, even with respect to the first variable such that f(r,x) = P(r,x)e−a(r 2 +x 2 ) , (4.3) with degree(P) < d − 1, and by a standard calculus, we obtain F̃α(f)(µ,λ) = Q(µ,λ)e − 1 4a (µ 2 +λ 2 ) , (4.4) where Q is a polynomial on R2, even with respect to the first variable, with degree(P) = degree(Q). On the other hand, from the relations (2.5), (2.6), (4.2), (4.3) and (4.4), we get ∫ +∞ 0 ∫ R ∫ +∞ 0 ∫ R |P(r,x)||Q(µ,λ)| (1 + |(r,x)|)d(1 + |(µ,λ)|)d e ξη|(r,x)||(µ,λ)|−a(r 2 +x 2 ) × e− 1 4a (µ 2 +λ 2 ) dνα(r,x)dµdλ < +∞, so ∫ +∞ 0 ∫ +∞ 0 ϕ(t) (1 + t)d ψ(ρ) (1 + ρ)d e ξηtρ e −at 2 e − 1 4a ρ 2 t 2α+2 ρdtdρ < +∞, (4.5) where ϕ(t) = ∫ 2π 0 ∣∣P(t cos(θ),t sin(θ)) ∣∣∣∣ cos(θ) ∣∣2α+1dθ, and ψ(ρ) = ∫ 2π 0 ∣∣Q(ρ cos(θ),ρ sin(θ)) ∣∣dθ. . Suppose that ξη > 1. If f 6= 0, then each of the polynomials P and Q is not identically zero, let m = degree(P) = degree(Q). From lemma 4.1, there exist two positive constants A and C such that ∀t > A, ϕ(t) > Ctm, 112 Hleili Khaled, Omri Slim and Lakhdar T. Rachdi CUBO 13, 3 (2011) and ∀ρ > A, ψ(ρ) > Cρm. Then, the inequality (4.5) leads to ∫ +∞ A ∫ +∞ A eξηtρ (1 + t)d(1 + ρ)d e −at 2 e − 1 4a ρ 2 dtdρ < +∞. (4.6) Let ε > 0, such that ξη − ε = σ > 1. The relation (4.6) implies that ∫ +∞ A ∫ +∞ A eεtρ (1 + t)d(1 + ρ)d e σtρ e −at 2 e − 1 4a ρ 2 dtdρ < +∞. (4.7) However, for all t > A > d ε and ρ > A, we have eερt (1 + t)d(1 + ρ)d > eεA 2 (1 + A)2d , and by the relation (4.7) it follows that ∫ +∞ A ∫ +∞ A e σtρ e −at 2 e − 1 4a ρ 2 dtdρ < +∞. (4.8) Let F(t) = ∫ +∞ A e σρt− 1 4a ρ 2 dρ, then F can be written F(t) = eaσ 2 t 2 ( ∫ +∞ A e − 1 4a ρ 2 dρ + 2aσe− A2 4a ∫ t 0 e Aσs−aσ 2 s 2 ds ) , in particular F(t) > eaσ 2 t 2 ∫ +∞ A e − 1 4a ρ 2 dρ. Thus ∫ +∞ A ∫ +∞ A e σtρ e −at 2 e − 1 4a ρ 2 dtdρ = ∫ +∞ A e −at 2 F(t)dt > ∫ +∞ A e − 1 4a ρ 2 dρ ∫ +∞ A e a(σ 2 −1)t 2 dt = +∞, because σ > 1. This contradics the relation (4.8) and shows that f = 0. . Suppose that ξη = 1 and p 6= 2. In this case we have p > 2 or q > 2. Suppose that q > 2, then from the second hypothesis and the relation (4.4), we have ∫ +∞ 0 ψ(ρ)e− ρ2 4a e ηqρq q (1 + ρ)d ρdρ < +∞. (4.9) If f 6= 0, then the polynomial Q is not identically zero, and by lemma 4.1 and the relation (4.9), it follows that ∫ +∞ 0 e− ρ2 4a e ηqρq q (1 + ρ)d dρ < +∞, CUBO 13, 3 (2011) Uncertainty principle for the Riemann-Liouville operator 113 which is impossible because q > 2. The proof of theorem 4.2 is complete. Theorem 4.3 (Cowling-Price for Rα). Let ξ,η,ω1,ω2 be non negative real numbers such that ξη > 1 4 . Let p,q be two exponents, p,q ∈ [1, +∞], and let f be a measurable function on R2, even with respect to the first variable such that f ∈ L2(dνα). If ∥∥∥ eξ|(.,.)| 2 (1 + |(., .)|)ω1 f ∥∥∥ p,να < +∞, (4.10) and ∥∥∥ eη|θ(.,.)| 2 (1 + |θ(., .)|)ω2 Fα(f) ∥∥∥ q,γ̃α < +∞, (4.11) then i) For ξη > 1 4 , f = 0. ii) For ξη = 1 4 , there exist a positive constant a and a polynomial P on R2, even with respect to the first variable, such that f(r,x) = P(r,x)e−a(r 2 +x 2 ) . Proof. Let p′ and q′ be the conjugate exponents of p respectively q. Let us pick d1,d2 ∈ R, such that d1 > 2α + 3 and d2 > 2. Finally, let d be a positive real number such that d > max ( ω1 + d1 p′ ,ω2 + d2 q′ ,1 ) . From Hölder’s inequality and the relations (4.10) and (4.11), we deduce that ∫ +∞ 0 ∫ R |f(r,x)|eξ|(r,x)| 2 (1 + |(r,x)|) ω1+ d1 p ′ dνα(r,x) < +∞, and ∫ ∫ Γ+ |Fα(f)(µ,λ)|e η|θ(µ,λ)| 2 (1 + |θ(µ,λ)|) ω2+ d2 q ′ dγ̃α(µ,λ) < +∞. Consequently we have ∫ +∞ 0 ∫ R |f(r,x)|eξ|(r,x)| 2 (1 + |(r,x)|)d dνα(r,x) < +∞, and ∫ ∫ Γ+ |Fα(f)(µ,λ)|e η|θ(µ,λ)| 2 (1 + |θ(µ,λ)|)d dγ̃α(µ,λ) < +∞. Then, the desired result follows from theorem 4.2. Remark 4.1. The Hardy’s theorem is a special case of theorem 4.3 when p = q = +∞. Received: July 2010. Revised: August 2010. 114 Hleili Khaled, Omri Slim and Lakhdar T. Rachdi CUBO 13, 3 (2011) References [1] G. Andrews, R. Askey and R. Roy, Special Functions, Cambridge University Press, New-York 1999. [2] C. Baccar, N. B. Hamadi and L. T. 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