() CUBO A Mathematical Journal Vol.17, No¯ 02, (15–30). June 2015 An other uncertainty principle for the Hankel transform Chirine Chettaoui Département de Mathématiques et d’Informatique, Institut national des sciences appliquées et de Thechnologie, Centre Urbain Nord BP 676 - 1080 Tunis cedex, Tunisia. chirine.chettaoui@insat.rnu.tn ABSTRACT We use the Hausdorff-Young inequality for the Hankel transform to prove the uncer- tainly principle in terms of entropy. Next, we show that we can obtain the Heisenberg- Pauli-Weyl inequality related to the same Hankel transform. RESUMEN Usamos la desigualdad de Hausdorff-Young para la transformada de Hankel para pro- bar el principio de incertidumbre en términos de la entroṕıa. Además probamos que podemos obtener la desigualdad de Heisenberg-Pauli-Weyl relacionada con la misma transformada de Hankel. Keywords and Phrases: Uncertainty principle, Hausdorff-Young inequality, entropy, Hankel transform 2010 AMS Mathematics Subject Classification: 43A32, 42B25. 16 Chirine Chettaoui CUBO 17, 2 (2015) 1 Introduction: The uncertainly principles play an import role in harmonic analysis. They state that a function f and its Fourier transform f̂ can not be simultaneously sharply localized in the sense that it is impossible for a nonzero function and its Fourier transform to be simultaneously small. Many mathematical formulations of this fact can be found in [2, 5, 6, 11, 16, 17]. For a probability density function f on Rn, the entropy of f is defined according to [18] by E(f) = − ∫ Rn f(x) ln(f(x))dx. The entropy E(f) is closely related to quantum mechanics [4] and constitutes one of the important way to measure the concentration of f. The uncertainly principle in terms of entropy consists to compare the entropy of |f|2 with that of |f̂|2. A first result has been given in [13], where Hirschman established a weak version of this uncertainly principle by showing that for every square integrable function f on Rn with respect to the Lebesgue measure, such that ||f||2 = 1, we have E(|f|2) + E(|f̂|2) > 0. Later, in [1, 2], Beckner proved the following stronger inequality, that is for every square integrable function f on Rn; ||f||2 = 1, E(|f|2) + E(|f̂|2) > n(1 − ln2). The last inequality constituted a very powerful result which implies in particular the well known Heisenberg-Pauli-Weyl uncertainly principle [17]. In this paper, we consider the singular differential operator defined on ]0,+∞[ by ℓα = d2 dr2 + 2α + 1 r d dr = 1 r2α+1 d dr [r2α+1 d dr ]; α > −1 2 . The Hankel transform associated with the operator ℓα is defined by Hα(f)(λ) = ∫+∞ o f(r)jα(λr)dµα(r), where . dµα(r) is the measure defined on [0,+∞[ by dµα(r) = r2α+1dr 2α Γ(α + 1) . . jα is the modified Bessel-function that will be defined in the second section . Many harmonic analysis results have been establish for the Hankel transform Hα [14, 19, 20]. Also, many uncertainly principles have been proved for the transform Hα [17, 21]. CUBO 17, 2 (2015) An other uncertainty principle for the Hankel transform 17 Our investigation in this work consists to establish the uncertainly principle in terms of entropy for the Hankel transform Hα. For a nonnegative measurable function f on [0,+∞[, the entropy of f is defined by Eµα(f) = − ∫+∞ 0 f(r) ln(f(r))dµα(r). Then using the Hausdorff-Young inequality for Hα [9], we establish the main result of this work. . Let f ∈ L2(dµα); ||f||2,µα = 1 such that ∫+∞ o |f(r)|2 ∣∣ ln(|f(r)|) ∣∣dµα(r) < +∞, and ∫+∞ o ∣∣Hα(f)(λ) ∣∣2 ∣∣∣ ln ( |Hα(f)(λ)| )∣∣∣dµα(λ) < +∞. Then Eµα ( |f|2 ) + Eµα (∣∣Hα(f) ∣∣2 ) > (2α + 1)(1 − ln2), where Lp(dµα); p ∈ [1,+∞], is the Lebesgue space of measurable functions on [0,+∞[ such that ||f||p,µα < +∞, with ||f||p,µα =    ( ∫+∞ 0 ∣∣f(r) ∣∣pdµα(r) ) 1 p , if p ∈ [1,+∞[, ess sup r∈ [0,+∞[ ∣∣f(r) ∣∣, if p = +∞. Using this result, we prove that we can derive the Heisenberg -Pauli-Weyl inequality for Hα, that is . For every f ∈ L2(dµα); we have ||rf||2,µα||λHα(f)||2,µα > (α + 1)||f|| 2 2,µα . 2 The Hankel operator In this section, we recall some harmonic analysis results related to the convolution product and the Fourier transform associated with Hankel operator. Let ℓα be the Bessel operator defined on ]0 + ∞[ by ℓαu = d2 dr2 u + 2α + 1 r du dr . 18 Chirine Chettaoui CUBO 17, 2 (2015) Then, for every λ ∈ C, the following system    ℓαu(x) = −λ 2u(x), u(0) = 1, u′(0) = 0, admits a unique solution given by jα(λ.), where jα(z) = 2αΓ(α + 1) zα Jα(z) (2.1) = Γ(α + 1) +∞∑ k=0 (−1)k k!Γ(α + k + 1) (z 2 )2k , with Jα is the Bessel function of first kind and index α [7, 8, 15, 22]. The modified Bessel function jα satisfies the following properties for every α > −1 2 , the modified Bessel function jα has the Mehler integral representation, for every z ∈ C, jα(z) =    2Γ(α + 1) √ πΓ(α + 1 2 ) ∫1 0 (1 − t2)α− 1 2 cosztdt, if α > −1 2 , cosz, if α = −1 2 . Consequently, for every k ∈ N and z ∈ C; we have ∣∣j(k)α (z) ∣∣ 6 exp ( |Imz| ) . (2.2) The eigenfunction jα(λ.) satisfies the following product formula [22], for all r,s ∈ [0,+∞[ jα(λr)jα(λs) =    Γ(α + 1) √ πΓ(α + 1 2 ) ∫π 0 jα ( λ √ r2 + s2 + 2rscosθ ) sin2α θdθ; if α > −1 2 , jα ( λ(r + s) ) + jα ( λ ( (r − s) )) 2 , if α = −1 2 . (2.3) The previous product formula allows us to define the Hankel translation operator and the convo- lution product related to the operator ℓα as follows Definition 2.1. i) For every r ∈ [0,+∞[, the Hankel translation operator ταr is defined on Lp(dµα); p ∈ [1,+∞], by ταr f(s) =    Γ(α + 1) √ πΓ(α + 1 2 ) ∫π 0 f (√ r2 + s2 + 2rscosθ ) sin2α θdθ; if α > −1 2 , f(r + s) + f(|r − s|) 2 , if α = −1 2 . CUBO 17, 2 (2015) An other uncertainty principle for the Hankel transform 19 ii) The convolution product of f,g ∈ L1(dµα) is defined for every r ∈ [0,+∞[, by f ∗α g(r) = ∫+∞ o ταr (f)(s)g(s)dµα(s). (2.4) Then the product formula (2.3) can be written ταr (jα(λ.))(s) = jα(λr)jα(λs). (2.5) We have the properties Proposition 2.2. i. For every f ∈ Lp(dµα); 1 6 p 6 +∞, and for every r ∈ [0,+∞[, the function ταr (f) belongs to L p(dµα) and we have ∣∣∣∣ταr (f) ∣∣∣∣ p,µα 6 ||f||p,µα. (2.6) ii. For f, g ∈ L1(dµα), the function f ∗α g belongs to L1(dµα); the convolution product is commutative, associative and we have ||f ∗α g||1,µα 6 ||f||1,µα||g||1,µα. (2.7) Moreover, if 1 6 p,q,r 6 +∞ are such that 1 r = 1 p + 1 q − 1 and if f ∈ Lp(dµα),g ∈ Lq(dµα), then the function f ∗α g belongs to Lr(dµα), and we have the Young’s inequality ||f ∗α g||r,µα 6 ||f||p,µα||g||q,µα. (2.8) iii. For every f ∈ L1(dµα), and r ∈ [0,+∞[ the function ταr (f) belongs to L1(dµα) and we have ∫+∞ o ταr (f)(s)dµα(s) = ∫+∞ o f(r)dµα(r). (2.9) We denoted by . C∗,0(R) the space of even continuous functions f on R such that lim |r|→+∞ f(r) = 0. . Se(R) the space of even infinitely differentiable functions on R; rapidly decreasing together with all their derivatives. Now, we shall define the Hankel transform and we give the most important properties. Definition 2.3. The Hankel transform Hα is defined on L 1(dµα) by ∀λ ∈ R; Hα(f)(λ) = ∫+∞ o f(r)jα(λr)dµα(r), (2.10) where jα is the modified Bessel function defined by the relation(2.1). 20 Chirine Chettaoui CUBO 17, 2 (2015) Proposition 2.4. i. For every f ∈ L1(dµα), the function Hα(f) belongs to the space C∗,0(R) and ∣∣∣∣Hα(f) ∣∣∣∣ ∞,µα 6 ||f||1,µα. (2.11) ii. For every f ∈ L1(dµα) and r ∈ [0,+∞[, Hα(τ α r (f))(λ) = jα(λr)Hα(f)(λ). (2.12) iii. For all f,g ∈ L1(dµα), Hα(f ∗α g)(λ) = Hα(f)(λ)Hα(g)(λ). (2.13) Theorem 2.5. (Inversion formula) Let f ∈ L1(dµα) such that Hα(f) ∈ L1(dµα), then for almost every r ∈ [0,+∞[, we have f(r) = ∫+∞ o Hα(f)(λ)jα(λr)dµα(λ) = Hα ( Hα(f) ) (r). (2.14) Theorem 2.6. (Plancherel) The Hankel transform Hα can be extented to an isometric isomor- phism from L2(dµα) onto itself. In particular, for all f and g ∈ L2(dµα), we have (Parseval equality) ∫+∞ o f(r)g(r)dµα(r) = ∫+∞ o Hα(f)(λ)Hα(g)(λ)dµα(λ). (2.15) Proposition 2.7. i. Let f ∈ L1(dµα)and g ∈ L2(dµα); by the relation (2.8), the function f∗αg belongs to L2(dµα), moreover Hα(f ∗α g)(λ) = Hα(f)(λ)Hα(g)(λ). (2.16) ii. For all f and g ∈ L2(dµα), the function f ∗α g belongs to C∗,0(R) and we have f ∗α g = Hα ( Hα(f).Hα(g) ) . (2.17) iii. The Hankel transform Hα is a topological isomorphism from Se(R) onto itself and we have H −1 α = Hα. (2.18) iv. For every f ∈ S(R) and g ∈ L2(dµα), we have Hα(fg)(λ) = Hα(f)(λ) ∗α Hα(g)(λ). (2.19) Definition 2.8. The Gaussian kernel associated with the Hankel operator is defined by ∀t > 0, gt(r) = e −r2 2t2 t2α+2 . (2.20) Thus, one can easily see that the family (gt)t>0 is an approximation of the identity, in par- ticular, for every f ∈ L2(dµα) we have lim t→0+ ∣∣∣∣gt ∗α f − f ∣∣∣∣ 2,µα = 0. (2.21) CUBO 17, 2 (2015) An other uncertainty principle for the Hankel transform 21 3 Uncertainty principle in terms of entropy for the Hankel transform This section contains the main result of this paper that is the uncertainty principle in terms of entropy for the Hankel transform Hα. We start this section by the following Hausdorff-Young inequality. Theorem 3.1. [9] Let p ∈ ]1,2], for every f ∈ Lp(dµα), the function Hα(f) belongs to Lp ′ (dµα); p ′ = p p−1 , and we have ||Hα(f)||p′,µα 6 Ap,α||f||p,µα, (3.1) where Ap,α is the Babenko-Beckner constant, Ap,α = ( p 1 p ( p p−1 ) p−1 p )α+1 . Lemma 3.2. Let x be a positive real number. For every p ∈ [1,2[, x2 − x 6 xp − x2 p − 2 6 x2 lnx. (3.2) Proof. Let x > 0 and let us put g(p) = xp − x2 p − 2 . g is differentiable on [1,2[ and we have g′(p) = h(p) (p − 2)2 , where h(p) = (p − 2) ln(x).xp − xp + x2. On the other hand, ∀p ∈ [1,2[; h′(p) = (p − 2)xp(ln(x))2 < 0 and h(2) = 0. Thus, for every p ∈ [1,2], h(p) > 0 and the function g is increasing on [1,2[, hence g(1) 6 g(p) 6 lim p→2− g(p). In the following, we shall prove the uncertainty principle in terms of entropy for f ∈ L1(dµα)∩ L2(dµα) such that ||f||2,µα = 1. 22 Chirine Chettaoui CUBO 17, 2 (2015) Theorem 3.3. Let f ∈ L1(dµα) ∩ L2(dµα); ||f||2,µα = 1, such that ∫ ∞ 0 |f(r)|2 ∣∣ ln(|f(r)|) ∣∣ dµα(r) < +∞, and ∫ ∞ 0 |H (f)(λ)|2 ∣∣ ln(|H (f)(λ)|) ∣∣ dµα(λ) < +∞. Then, Eµα(|f| 2) + Eµα(|Hα(f)| 2) > (2α + 2)(1 − ln2). (3.3) Proof. Let f ∈ L1(dµα) ∩ L2(dµα); ||f||2,µα = 1 and let ϕ be the function defined on ]1,2] by ϕ(p) = ∫+∞ o |Hα(f)(λ)| p p−1 dµα(λ) − ( 1 p 1 p ( p p−1 ) p−1 p ) p(α+1) p−1 ( ∫+∞ o |f(x)|pdµα(x) ) 1 p−1 . Then, by relation (3.1), ∀p ∈]1,2]; ϕ(p) 6 0. On the other hand, Theorem 2.6 means that ϕ(2) = 0. This implies that dϕ dp (2−) > 0. (3.4) Since f ∈ L1(dµα)∩L2(dµα), then by a standard interpolation argument, f belongs to Lp(dµα); p ∈ [1,2]. Using Lemma 3.2, the hypothesis and Lebesgue dominated convergence theorem, we deduce that d dp [ ∫+∞ o |f(r)|pdµα(r) ] (2−) = ∫+∞ o lim p→2− |f(r)|p − |f(r)|2 p − 2 dµα(r). (3.5) Thus d dp [ ∫+∞ o |f(r)|pdµα(r) ] (2−) = 1 2 ∫+∞ o ln |f(r)|2|f(r)|2dµα(r)]. (3.6) Now, since f ∈ L1(dµα) ∩ L2(dµα), by using again the Lebesgue dominated convergence theorem, we get lim p→2 ∫+∞ o |f(r)|pdµα(r) = ∫+∞ o |f(r)|2dµα(r) = 1. (3.7) Combining (3.6) and (3.7), we get d dp [( ∫+∞ o |f(r)|pdµα(r) ) 1 p−1 ] (2−) = − 1 2 Eµα(|f| 2 ). (3.8) CUBO 17, 2 (2015) An other uncertainty principle for the Hankel transform 23 As the same way, one can see that d dp [ ∫+∞ o |Hα(f)(λ)| p p−1 dµα(λ) ] (2−) = − 1 2 Eµα(|Hα(f)| 2). (3.9) Finally, basic calculations show that d dp [ ( 1 p 1 p ( p p−1 ) p−1 p ) p p−1 (α+1) ] (2−) = (α + 1)(1 − ln2). (3.10) Thus according to relations(3.8), (3.9) and (3.10), it follows that dϕ dp (2−) = 1 2 Eµα(|f| 2) + 1 2 Eµα(|Hα(f)| 2) − (α + 1)(1 − ln2). (3.11) The proof is complete by using (3.4). Lemma 3.4. Let f be a measurable function on [0,+∞[ and let ω : [0,+∞[7−→ [0,+∞[ be a convex function such that ω(|f|) belongs to L1(dµα). Let (fk)k∈N be a sequence of nonnegative measurable functions on R+ such that for every k ∈ N; ||fk||1,µα = 1 and the sequence (fk ∗α f)k∈N converges pointwise to f. Then, for every k ∈ N, the function ω(|fk ∗α f| belongs to L1(dµα), and we have lim k→+∞ ∫+∞ o ω(|fk ∗α f|)(r)dµα(r) = ∫+∞ o ω(|f(r)|)dµα(r). (3.12) Proof. We have ||ω ◦ |f|||1,µα = ∫+∞ o lim inf k→+∞ ω(|fk ∗α f|)(r)dµα(r), (3.13) and by using Fatou’s lemma, we get ||ω ◦ |f|||1,α 6 lim inf p→+∞ ∫+∞ o ω(|fk ∗α f(r)|)dµα(r). (3.14) Conversely, according to relation (2.9), we have ∀k ∈ N; ∀λ ∈ R+, ∫+∞ o ταλ (fk)(r)dµα(r) = ||fk||1,µα = 1, (3.15) which means that for every λ ∈ R+, ταλ (fk)(r)dµα(r) is a probability measure on R+. Therefore, by using Jensen’s inequality for convex functions, we get 24 Chirine Chettaoui CUBO 17, 2 (2015) ∀r ∈ R+, ω(|fk ∗α f(r)|) = ω(| ∫+∞ o f(x)ταr (fk)(x)dµα(x)) 6 ω( ∫+∞ o |f(x)ταr (fk)(x)|dµα(x) 6 ∫+∞ o ω(|f(x)|)ταr (fk)(x)dµα(x) = fk ∗α ( ω ◦ |f| ) (r). (3.16) In particular, ω ◦ |fk ∗α f| ∈ L1(dµα). Hence by relations (2.9) and (3.16), we deduce that lim sup k→+∞ ∫+∞ o ω(|fk ∗α f(r)|)dµα(r) 6 lim sup k→+∞ ∫+∞ o fk ∗α ( ω ◦ |f(r)| ) dµα(r) = lim sup k→+∞ ||fk ∗α ( ω ◦ |f| ) ||1,α 6 ||ω ◦ |f|||1,α. (3.17) The relations (3.14) and (3.17) show that lim k→+∞ ∫+∞ o ω(|fk ∗α f|)(r)dµα(r) = ∫+∞ o ω(|f(r)|)dµα(r). Now, e are able to prove the main result. Theorem 3.5. Let f ∈ L2(dµα); ||f||2,µα = 1, such that ∫ ∞ 0 |f(r)|2 ∣∣ ln(|f(r)|) ∣∣ dµα(r) < +∞, and ∫ ∞ 0 |Hα(f)(λ)| 2 ∣∣ ln(|Hα(f)(λ)|) ∣∣ dµα(λ) < +∞. Then, we have Eµα(|f| 2 ) + Eµα(|Hα(f)| 2 ) > (2α + 2)(1 − ln2). (3.18) Proof. The main idea of this proof is to combine Theorem 3.3 with the standard density argument. Indeed, we will show that for every f ∈ L2(dµα); there exists a sequence (fk)k∈N ∈ L1(dµα) ∩ L2(dµα) such that lim k→+∞ ||fk||2,µα = ||f||2,µα, (3.19) CUBO 17, 2 (2015) An other uncertainty principle for the Hankel transform 25 lim k→+∞ Eµα(|fk| 2) = Eµα(|f| 2), (3.20) lim k→+∞ Eα(|Hα(fk)| 2) = Eα(|Hα(f)| 2). (3.21) Let (hk)k∈N be the sequence of functions defined by hk(r) = 2 α+1k2α+2e−k 2 r 2 = g 1 k √ 2 (r). (3.22) Then, by relation(2.21), we have ∀f ∈ L2(dµα); lim k→+∞ ||hk ∗α f − f||2,µα = 0. (3.23) Furthermore, according to Weber’s formula [15], we know that for all k ∈ N∗, α > −1 2 , ∫+∞ o e−k 2 r 2 jα(xr)r 2α+1dr = Γ(α + 1) e − x 2 4k2 2k2α+2 . (3.24) Hence, by relation (3.24), we deduce that H −1 α (hk)(λ) = 2k2α+2 Γ(α + 1) ∫+∞ o e−k 2 r 2 jα(λr)r 2α+1dµα(r) = e − λ 2 4k2 . (3.25) Let (ψk)k∈ N be the sequence of functions defined on R+ by ψk(λ) = e − λ 2 4k2 = H −1α (hk)(λ). (3.26) Let f ∈ L2(dµα); ||f||2,µα = 1, then according to relation(3.23), we have lim k→+∞ ||Hα(ψk) ∗α Hα(f) − Hα(f)||2,µα = 0. In particular, there is a subsequence (ψσ(k))k∈N such that Hα(ψσ(k)) ∗α Hα(f) = hσ(k) ∗α Hα(f) converges pointwise to Hα(f). Let (fk)k∈N be the sequence of measurable functions on R+ defined by fk = ψσ(k)f. (3.27) Since ψσ(k) ∈ L2(dµα) ∩ C∗,0(R+), then fk belongs to the space L1(dµα) ∩ L2(dµα). Replacing f by fk ||fk||2,µα in Theorem 3.3 and using the fact that ||f||2,µα = ||Hα(f)||2,µα; f ∈ L 2 (dµα), 26 Chirine Chettaoui CUBO 17, 2 (2015) we deduce that − ∫+∞ o ln ( |fk(r)| 2 ) |fk(r)| 2dµα(r) − ∫+∞ o ln ( |Hα(fk)(λ)| 2 ) |Hα(fk)(λ)| 2dµα(λ) (3.28) > (2α + 2)(1 − ln2)||fk|| 2 2,µα − 2||fk|| 2 2,α ln ( ||fk|| 2 2,µα ) . (3.29) Now, by using the Lebesgue dominated convergence theorem, we have lim k→+∞ ||fk||2,µα = ||f||2,µα. (3.30) On the other hand, one can see that for all k ∈ N, and for almost every r ∈ [0,+∞[, we have ln |fk(r)| 2 |fk(r)| 2 6 C|f(r)|2 + ln |f(r)|2|f(r)|2, (3.31) Hence, using again the Lebesgue dominated convergence theorem, we get − lim k→+∞ ∫+∞ o ln ( |fk(r)| 2 ) |fk(r)| 2dµα(r) = Eµα(|f| 2). (3.32) Now, let us show that lim k→+∞ Eµα(|Hα(fk)| 2 ) = Eµα(Hα(f)| 2 ). For this, we denote by ω1 ,ω2 the functions defined on R by ω1(t)= { t2 ln |t|, if |t| > 1 0, if |t| 6 1, and ω2(t)=    2t2, if |t| > 1 −t2 ln |t| + 2t2, if |t| 6 1 ,t 6= 0 0, if t = 0. Then ω1 and ω2 are both nonnegative and convex, moreover; we have ∀t > 0, t2 ln |t| = ω1(t) − ω2(t) + 2t2. (3.33) From the hypothesis, for each i ∈ {1,2}, the function ωi(|Hα(f)|) belongs to L1(dµα). Now, from Proposition 2.7 iv), for every k ∈ N∗; we have Hα(fk) = hσ(k) ∗α Hα(f) and we know that hσ(k) ∗α Hα(f) converges pointwise to Hα(f) and ||hσ(k)||1,µα = 1. So, the hypothesis of Lemma 3.4 are satisfies and we get lim k→+∞ ∫+∞ o ωi(|Hα(fk)|)(r)dµα(r) = ∫+∞ o ωi(|Hα(f)|)(r)dµα(r), (3.34) and therefore, by relations(3.30) and (3.33) we get lim k→+∞ ∫+∞ o ln |Hα(fk)| 2|Hα(fk)(r)| 2dµα(r) = Eµα(|Hα(f)| 2). (3.35) The proof is complete by combining relations (3.29), (3.30), (3.32)and(3.35). CUBO 17, 2 (2015) An other uncertainty principle for the Hankel transform 27 4 Heisenberg-Pauli-Weyl inequality for the Hankel tran- sorm Lemma 4.1. Let f ∈ L2(dµα) such that ||f||2,µα = 1.Then, for every t > 0, ∫+∞ o |f(r)|2 ln ( |f(r)|2 gt(r) ) dµα(r) > 0, (4.1) where gt(r)is given by(2.20). Proof. Since the function ω(t) = t lnt is convex on ]0,+∞[, and dνα(r) = gt(r)dµα(r) is a probability measure on ]0,+∞[ then, applying Jensen’s inequality, we get ∫+∞ o |f(r)|2 ln( |f(r)|2 gt(r) )dµα(r) = ∫+∞ o ω( |f(r)|2 gt(r) )dνα(r) > ω( ∫+∞ o |f(r)|2 gt(r) dνα(r)) = ω(||f||22,µα) = ω(1) = 0. Theorem 4.2. (Heisenberg-Pauli-Weyl inequality) Let f ∈ L2(dµα), then ||rf||2,µα||λHα(f)||2,µα > (α + 1)||f|| 2 2,µα . (4.2) Proof. Let h ∈ L2(dµα); ||h||2,µα = 1. From Lemma 4.1, we get ∫+∞ o [ |h(r)|2 ln(|h(r)|2) − |h(r)|2 ln(|gt(r)|) ] dµα(r) > 0. (4.3) Then, Eµα(|h| 2) 6 ln ( t2α+2 ) + 1 2t2 ∫+∞ o |h(r)|2r2dµα(r). (4.4) Since ||Hα(h)||2,µα = ||h||2,µα = 1, we get also Eµα(|Hα(h)| 2) 6 ln ( t2α+2 ) + 1 2t2 ∫+∞ o |Hα(h)(λ)| 2λ2dµα(λ), (4.5) adding (4.4) and (4.6), it follows that ||rh||22,µα + ||λHα(h)|| 2 2,µα > 2t2 [ Eµα(|h| 2 ) + Eµα(|Hα(h)| 2 ) − 2 ln(t2α+2) ] . 28 Chirine Chettaoui CUBO 17, 2 (2015) Applying Theorem 3.5, we obtain ||rh||22,µα + ||λHα(h)|| 2 2,µα > 2t2 [ (2α + 2)(1 − 2 ln2) − 2(2α + 2) lnt ] = 2t2(2α + 2)(1 − ln2t2). In particular, for t = 1√ 2 ; we deduce that for every h ∈ L2(dµα);||h||2,µα = 1, ||rh||22,µα + ||λHα(h)|| 2 2,µα > 2α + 2. (4.6) Let f ∈ L2(dµα), replacing h by f ||f||2 2,µα in relation (4.6), we claim that for every f ∈ L2(dµα), ||rf||22,µα + ||λHα(f)|| 2 2,µα > (2α + 2)||f||22,µα. (4.7) On the other hand, for f ∈ L2(dµα) and t > 0, we denote by ft the dilated of f defined by ft(r) = f(tr), then, ft belongs to L 2(dµα) and we have ||ft|| 2 2,µα = ∫+∞ o |ft(r)| 2dµα(r) = 1 t2α+2 ∫+∞ o |f(r)|2dµα(r) = 1 t2α+2 ||f||22,µα. (4.8) Moreover ||rft|| 2 2,µα = ∫+∞ o r2|ft(r)| 2dµα(r) = 1 t2α+4 ||rf||22,µα, (4.9) and ||λHα(ft)|| 2 2,µα = ∫+∞ o λ2|Hα(ft)(λ)| 2dµα(λ) (4.10) Hα(ft)(λ) = ∫+∞ o ft(x)jα(λx)dµα(x) = 1 t2α+2 Hα(f)( λ t ). (4.11) Then ||λHα(ft)|| 2 2,µα = 1 t2α ||λHα(f)|| 2 2,µα . (4.12) CUBO 17, 2 (2015) An other uncertainty principle for the Hankel transform 29 Now, we assume that ||rf||2,µα and ||λHα(f)||2,µα are both non zero and finite, hence the same holds for ft , t > 0 and from relation (4.7), we have ||rft|| 2 2,µα + ||λHα(ft)|| 2 2,µα > (2α + 2)||ft|| 2 2,µα . (4.13) Then, by relations (4.8), (4.9) and (4.12), we get for every t > 0 1 t2α+4 ||rf||22,µα + 1 t2α ||λHα(f)|| 2 2,µα > (2α + 2) 1 t2α+2 ||f||22,µα, or 1 t2 ||rf||22,µα + t 2||λHα(f)|| 2 2,µα > (2α + 2)||f||22,µα. In particular for t = t0 = √ ||rf||2,µα ||λHα(f)||2,µα . We obtain ||λHα(f)||2,µα||rf||2,µα > (α + 1)||f|| 2 2,µα . Received: June 2014. Accepted: March 2015. References [1] W. Beckner, Inequalities in Fourier analysis, Ann. of Math., 102 (1975) 159-182. [2] W. Beckner, Pitt’s inequality and the uncertainty principle, Proc. Amer. Math. Soc., (1995) 1897-1905. [3] A. Beurling, The collected works of Arne Beurling, Birkhäuser, Vol.1-2, Boston 1989. [4] I. Bialynicki-Birula, Entropic uncertainty relations in quantum mechanics, quantum probabil- ity and applications II, Eds. L. Accardi and W. Von Waldenfels, Lecture Notes in Mathematics, 1136 (1985) 90-103. [5] A. Bonami, B. Demange, and P. Jaming, Hermite functions and uncertainty priciples for the Fourier and the widowed Fourier transforms, Rev. Mat. Iberoamericana, 19 (2003) 23–55. [6] M.G. Cowling and J. F. Price, Generalizations of Heisenberg inequality in Harmonic analysis, (Cortona, 1982), Lecture Notes in Math., 992 (1983) 443–449. [7] A. Erdélyi et al., Tables of integral transforms, Mc Graw-Hill Book Compagny., Vol.2, New York 1954. [8] A. Erdélyi , Asymptotic expansions, Dover publications, New-York 1956. 30 Chirine Chettaoui CUBO 17, 2 (2015) [9] A. Fitouhi, Inégalité de Babenko et inégalité logarithmique de Sobolev pour l’opérateur de Bessel, C. R. Acad. Sci. Paris, t. 305, Srie I, (1987) 877-880. [10] G. B. Folland, Real analysis modern thechniques and their applications, Pure and Applied Mathematics, John Wiley and Sons, New York 1984. [11] G. B. Folland and A. Sitaram, The uncertainty principle: a mathematical survey, J. Fourier Anal. Appl., 3 (1997) 207–238. [12] G. H. Hardy, A theorem concerning Fourier transform, J. London. Math. Soc., 8 (1933) 227- 231. [13] I. I. Hirschman, A note on entropy, Amer. J. Math., 79, 1 (1957) 152-156. [14] I. I. Hirschman, Variation diminishing Hankel transforms, J. Anal. Math., Vol. 8, No. 01, (1960) 307-336. [15] N. N. Lebedev, Special Functions and Their Applications, Dover publications, New-York 1972. [16] G. W. Morgan, A note on Fourier transforms, J. London. Math. Soc., 9 (1934) 178–192. [17] M. Rösler and M. Voit, An uncertaintly principle for the Hankel transforms, American Math- ematical Society, 127 (1) (1999), 183-194. [18] C. Shanon, A mathematical theory of communication, Bell System Tech. J., 27 (1948/1949), 379-423 and 623-656. [19] K. Trimèche, Transformation intégrale de Weyl et théorème de Paley-Wiener associés à un opérateur différentiel singulier sur (0,+∞), J. Math. Pures Appl., 60 (1981) 51–98. [20] K. Trimèche, Inversion of the Lions transmutation operator using genaralized wavelets, Appl. Comput. Harmon. Anal., 4 (1997) 97–112. [21] Vu Kim Tuan, Uncertainty principle for the Hankel transform. Integral Transforms Spec. Funct. Vol. 18, Issue 5 (2007), 369-381. [22] G. N. Watson, A treatise on the theory of Bessel functions, Cambridge univ. Press., 2nd ed., Cambridge 1959. Introduction: The Hankel operator Uncertainty principle in terms of entropy for the Hankel transform Heisenberg-Pauli-Weyl inequality for the Hankel transorm