HEISENBERG-PAULI-WEYL UNCERTAINTY PRINCIPLE FOR THE RIEMANN-LIOUVILLE OPERATOR

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Heisenberg-Pauli-Weyl Uncertainty Principle S. Omri and L.T. Rachdi vol. 9, iss. 3, art. 88, 2008

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HEISENBERG-PAULI-WEYL UNCERTAINTY PRINCIPLE FOR THE RIEMANN-LIOUVILLE

OPERATOR

S. OMRI AND L.T. RACHDI

Department of Mathematics Faculty of Sciences of Tunis 2092 Manar 2 Tunis, Tunisia.

EMail:lakhdartannech.rachdi@fst.rnu.tn

Received: 29 March, 2008

Accepted: 08 August, 2008

Communicated by: J.M. Rassias 2000 AMS Sub. Class.: 42B10, 33C45.

Key words: Heisenberg-Pauli-Weyl Inequality, Riemann-Liouville operator, Fourier transform, local uncertainty principle.

Abstract: The Heisenberg-Pauli-Weyl inequality is established for the Fourier transform connected with the Riemann-Liouville operator.Also, a generalization of this inequality is proved. Lastly, a local uncertainty principle is studied.

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1. Introduction

Uncertainty principles play an important role in harmonic analysis and have been studied by many authors and from many points of view [8].These principles state that a function f and its Fourier transform fbcannot be simultaneously sharply lo- calized. The theorems of Hardy, Morgan, Beurling, ... are established for several Fourier transforms in [4], [9], [13] and [14]. In this context, a remarkable Heisen- berg uncertainty principle [10] states, according to Weyl [25] who assigned the result to Pauli, that for all square integrable functionsfonRⁿwith respect to the Lebesgue measure, we have

Z

Rⁿ

x²_j|f(x)|²dx Z

Rⁿ

ξ_j²|fb(ξ)|²dξ

≥ 1 4

Z

Rⁿ

|f(x)|²dx 2

, j ∈ {1, . . . , n}.

This inequality is called the Heisenberg-Pauli-Weyl inequality for the classical Fourier transform.

Recently, many works have been devoted to establishing the Heisenberg-Pauli- Weyl inequality for various Fourier transforms, Rösler [21] and Shimeno [22] have proved this inequality for the Dunkl transform, in [20] Rösler and Voit have established an analogue of the Heisenberg-Pauli-Weyl inequality for the generalized Han- kel transform. In the same context, Battle [3] has proved this inequality for wavelet states, and Wolf [26], has studied this uncertainty principle for Gelfand pairs. We cite also De Bruijn [5] who has established the same result for the classical Fourier transform by using Hermite Polynomials, and Rassias [17,18,19] who gave several generalized forms for the Heisenberg-Pauli-Weyl inequality.

In [2], the second author with others considered the singular partial differential

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operators defined by







∆₁ = _∂x^∂ ,

∆₂ = _∂r^∂²2 +^2α+1_r _∂r^∂ − _∂x^∂²2; (r, x)∈]0,+∞[×R; α≥0.

and they associated to∆1and∆2the following integral transform, called the Riemann- Liouville operator, defined onC∗(R²)(the space of continuous functions onR², even with respect to the first variable) by

Rα(f)(r, x)

=







α π

R1

−1

R1

−1f rs√

1−t², x+rt

(1−t²)^α−¹²(1−s²)^α−1dtds; ifα >0,

1 π

R1

−1f r√

1−t², x+rt√ _dt

(1−t²); ifα = 0.

In addition, a convolution product and a Fourier transformFα connected with the mappingRαhave been studied and many harmonic analysis results have been established for the Fourier transformFα (Inversion formula, Plancherel formula, Paley- Winer and Plancherel theorems, ...).

Our purpose in this work is to study the Heisenberg-Pauli-Weyl uncertainty principle for the Fourier transform Fα connected with Rα.More precisely, using La- guerre and Hermite polynomials we establish firstly the Heisenberg-Pauli-Weyl inequality for the Fourier transformFα, that is

• For allf ∈L²(dνα), we have Z +∞

0

Z

R

(r²+x²)|f(r, x)|²dν_α(r, x) ¹₂

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× Z Z

Γ+

(µ²+ 2λ²)|Fα(f)(µ, λ)|²dγ_α(µ, λ) ¹₂

≥ 2α+ 3 2

Z +∞

0

Z

R

|f(r, x)|²dν_α(r, x)

, with equality if and only if

f(r, x) = Ce⁻

r2+x2 2t2

0 ; C ∈C, t₀ >0, where

• dν_α(r, x)is the measure defined onR+×Rby dνα(r, x) = r^2α+1

2^αΓ(α+ 1)√

2πdr⊗dx.

• dγ_α(µ, λ)is the measure defined on the set Γ₊=R+×R∪

(it, x); (t, x)∈R+×R; t≤ |x| , by

Z Z

Γ+

g(µ, λ)dγ_α(µ, λ)

= 1

2^αΓ(α+ 1)√ 2π

Z +∞

0

Z

R

g(µ, λ)(µ²+λ²)^αµdµdλ +

Z

R

Z |λ|

0

g(iµ, λ)(λ²−µ²)^αµdµdλ

! .

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Next, we give a generalization of the Heisenberg-Pauli-Weyl inequality, that is

• For allf ∈L²(dν_α),a, b∈R;a, b≥1andη ∈Rsuch thatηa= (1−η)b, we have

Z +∞

0

Z

R

(r²+x²)^a|f(r, x)|²dν_α(r, x) ^η₂

× Z Z

Γ+

(µ²+ 2λ²)^b|Fα(f)(µ, λ)|²dγ_α(µ, λ) ^1−η₂

≥

2α+ 3 2

aηZ +∞

0

Z

R

|f(r, x)|²dνα(r, x) ¹₂

, with equality if and only if

a=b = 1 and f(r, x) =Ce⁻

r2+x2 2t2

0 ; C ∈C; t₀ >0.

In the last section of this paper, building on the ideas of Faris [7], and Price [15,16], we develop a family of inequalities in their sharpest forms, which constitute the principle of local uncertainty.

Namely, we have established the following main results

• For all real numbers ξ; 0 < ξ < ^2α+3₂ , there exists a positive constant K_α,ξ such that for all f ∈ L²(dν_α), and for all measurable subsetsE ⊂ Γ₊; 0 <

γ_α(E)<+∞, we have Z Z

E

|Fα(f)(µ, λ)|²dγ_α(µ, λ)

< K_α,ξ γ_α(E)_2α+3^2ξ Z +∞

0

Z

R

(r²+x²)^ξ|f(r, x)|²dν_α(r, x).

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• For all real numberξ;ξ > ^2α+3₂ , there exists a positive constantM_α,ξ such that for all f ∈ L²(dν_α), and for all measurable subsets E ⊂ Γ₊; 0 < γ_α(E) <

+∞, we have Z Z

E

|Fα(f)(µ, λ)|²dγ_α(µ, λ)

< M_α,ξγ_α(E)

Z +∞

0

Z

R

|f(r, x)|²dν_α(r, x)

2ξ−2α−3 2ξ

×

Z +∞

0

Z

R

(r²+x²)^ξ|f(r, x)|²dν_α(r, x) ^2α+3_2ξ

, whereM_α,ξ is the best (the smallest) constant satisfying this inequality.

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2. The Fourier Transform Associated with the Riemann-Liouville Operator

It is well known [2] that for all(µ, λ)∈C², the system











∆₁u(r, x) =−iλu(r, x),

∆2u(r, x) =−µ²u(r, x),

u(0,0) = 1, ^∂u_∂r(0, x) = 0, ∀x∈R, admits a unique solutionϕ_µ,λ, given by

(2.1) ∀(r, x)∈R²; ϕ_µ,λ(r, x) = j_α rp

µ² +λ² e^−iλx, where

j_α(x) = 2^αΓ(α+ 1)J_α(x)

x^α = Γ(α+ 1)

+∞

X

n=0

(−1)ⁿ n!Γ(α+n+ 1)

x 2

2n

,

andJ_αis the Bessel function of the first kind and indexα[6,11,12,24].

The modified Bessel functionj_αhas the following integral representation [1,11], for allµ∈C, andr ∈Rwe have

j_α(rµ) =







2Γ(α+1)

√πΓ(^α+¹₂) R1

0(1−t²)^α−¹² cos(rµt)dt, ifα >−¹₂;

cos(rµ), ifα=−¹₂.

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In particular, for allr, s∈R, we have

|jα(rs)| ≤ 2Γ(α+ 1)

√πΓ α+¹₂ Z 1

0

(1−t²)^α−¹²|cos(rst)|dt (2.2)

≤ 2Γ(α+ 1)

√πΓ(α+¹₂) Z 1

0

(1−t²)^α−¹²dt = 1.

From the properties of the Bessel function, we deduce that the eigenfunctionϕ_µ,λ satisfies the following properties

•

(2.3) sup

(r,x)∈R²

|ϕ_µ,λ(r, x)|= 1, if and only if(µ, λ)belongs to the set

Γ =R²∪ {(it, x); (t, x)∈R²; |t| ≤ |x|}.

• The eigenfunctionϕ_µ,λhas the following Mehler integral representation ϕµ,λ(r, x) =







α π

R1

−1

R1

−1cos(µrs√

1−t²)e^−iλ(x+rt)(1−t²)^α−¹²(1−s²)^α−1dtds; ifα >0,

1 π

R1

−1cos(rµ√

1−t²)e^−iλ(x+rt)^√_1−t^dt ₂; ifα = 0.

In [2], using this integral representation, the authors have defined the Riemann- Liouville integral transform associated with∆₁,∆₂ by

Rα(f)(r, x) =







α π

R1

−1

R1

−1f rs√

1−t², x+rt

(1−t²)^α−¹²(1−s²)^α−1dtds; ifα >0,

1 π

R1

−1f r√

1−t², x+rt√ _dt

(1−t²); ifα = 0.

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wheref is a continuous function onR², even with respect to the first variable.

The transformRαgeneralizes the "mean operator" defined by R0(f)(r, x) = 1

2π Z 2π

0

f(rsinθ, x+rcosθ)dθ.

In the following we denote by

• dν_αthe measure defined onR+×R, by dν_α(r, x) = r^2α+1

2^αΓ(α+ 1)√

2πdr⊗dx.

• L^p(dν_α)the space of measurable functionsf onR⁺×Rsuch that kfk_p,ν_α =

Z +∞

0

Z

R

|f(r, x)|^pdν_α(r, x) ¹_p

<∞, ifp∈[1,+∞[, kfk∞,να = ess sup_(r,x)∈_R₊_×_R|f(r, x)|<∞, ifp= +∞.

• h / i_ν_α the inner product defined onL²(dν_α)by hf /giνα =

Z +∞

0

Z

R

f(r, x)g(r, x)dνα(r, x).

• Γ₊ =R⁺×R∪

(it, x); (t, x)∈R⁺×R; t ≤ |x| .

• BΓ+ theσ-algebra defined onΓ₊by

BΓ+ ={θ⁻¹(B), B ∈B(R+×R)}, whereθ is the bijective function defined on the setΓ₊by

(2.4) θ(µ, λ) =p

µ² +λ², λ .

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• dγ_α the measure defined onBΓ+ by

(2.5) ∀A∈BΓ+; γ_α(A) = ν_α(θ(A))

• L^p(dγ_α)the space of measurable functionsf onΓ₊, such that kfkp,γα =

Z Z

Γ+

|f(µ, λ)|^pdγα(µ, λ) ¹_p

<∞, ifp∈[1,+∞[, kfk∞,γ_α = ess sup_(µ,λ)∈Γ₊|f(µ, λ)|<∞, ifp= +∞.

• h / i_γ_α the inner product defined onL²(dγ_α)by hf /gi_γ_α =

Z Z

Γ+

f(µ, λ)g(µ, λ)dγ_α(µ, λ).

Then, we have the following properties.

Proposition 2.1.

i) For all non negative measurable functionsg onΓ₊, we have (2.6)

Z Z

Γ+

g(µ, λ)dγα(µ, λ)

= 1

2^αΓ(α+ 1)√ 2π

Z +∞

0

Z

R

g(µ, λ)(µ²+λ²)^αµdµdλ +

Z

R

Z |λ|

0

g(iµ, λ)(λ²−µ²)^αµdµdλ

! .

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In particular

(2.7) dγ_α+1(µ, λ) = µ²+λ²

2(α+ 1)dγ_α(µ, λ).

ii) For all measurable functionsf on R⁺ ×R, the function f ◦θ is measurable onΓ₊. Furthermore, iff is non negative or an integrable function onR⁺×R with respect to the measuredν_α, then we have

(2.8)

Z Z

Γ+

(f ◦θ)(µ, λ)dγα(µ, λ) = Z +∞

0

Z

R

f(r, x)dνα(r, x).

In the following, we shall define the Fourier transform Fα associated with the operatorRα and we give some properties that we use in the sequel.

Definition 2.2. The Fourier transform Fα associated with the Riemann-liouville operatorRαis defined onL¹(dν_α)by

(2.9) ∀(µ, λ)∈Γ; Fα(f)(µ, λ) = Z +∞

0

Z

R

f(r, x)ϕ_µ,λ(r, x)dν_α(r, x).

By the relation (2.3), we deduce that the Fourier transformFαis a bounded linear operator fromL¹(dν_α)intoL^∞(dγ_α), and that for allf ∈L¹(dν_α), we have

(2.10) kFα(f)k∞,γα ≤ kfk_1,ν_α.

Theorem 2.3 (Inversion formula). Letf ∈ L¹(dν_α)such thatFα(f) ∈ L¹(dγ_α), then for almost every(r, x)∈R+×R, we have

f(r, x) = Z Z

Γ+

Fα(f)(µ, λ)ϕµ,λ(r, x)dγα(µ, λ).

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Theorem 2.4 (Plancherel). The Fourier transformFα can be extended to an iso- metric isomorphism fromL²(dν_α)ontoL²(dγ_α).

In particular, for allf, g ∈L²(dν_α), we have the following Parseval’s equality (2.11)

Z Z

Γ+

Fα(f)(µ, λ)Fα(g)(µ, λ)dγ_α(µ, λ)

= Z +∞

0

Z

R

f(r, x)g(r, x)dν_α(r, x).

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3. Hilbert Basis of the Spaces L

²

(dν

_α

), and L

²

(dγ

_α

)

In this section, using Laguerre and Hermite polynomials, we construct a Hilbert basis of the spacesL²(dν_α)andL²(dγ_α), and establish some intermediate results that we need in the next section.

It is well known [11, 23] that for everyα ≥0, the Laguerre polynomialsL^α_m are defined by the following Rodriguez formula

L^α_m(r) = 1

m!e^rr^−α d^m

dr^m r^m+αe^−r

; m∈N. Also, the Hermite polynomials are defined by the Rodriguez formula

H_n(x) = (−1)ⁿe^x² dⁿ

dxⁿ e^−x²

; n∈N. Moreover, the families

(s m!

Γ(α+m+ 1)L^α_m )

m∈N

and

(s 1 2ⁿn!√

πH_n )

n∈N

are respectively a Hilbert basis of the Hilbert spacesL²(R+, e^−rr^αdr)andL²(R, e^−x²dx).

Therefore the families (s2^α+1Γ(α+ 1)m!

Γ(α+m+ 1) e⁻^r

2

2 L^α_m(r²) )

m∈N

and

(s 1 2ⁿn!√

πe⁻^x

2 2 H_n

)

n∈N

are respectively a Hilbert basis of the Hilbert spacesL²

R+,₂α^rΓ(α+1)^2α+1 dr

andL² R,^√^dx

2π

,

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hence the familyn e^α_m,no

(m,n)∈N²

defined by

e^α_m,n(r, x) = 2^α+1Γ(α+ 1)m!

2ⁿ⁻¹²n!Γ(m+α+ 1)

!¹₂

e⁻^r2+x

2

2 L^α_m(r²)H_n(x), is a Hilbert basis of the spaceL²(dν_α).

Using the relation (2.8), we deduce that the family n

ξ_m,n^α o

(m,n)∈N²

,defined by ξ_m,n^α (µ, λ) = (e^α_m,n◦θ)(µ, λ)

= 2^α+1Γ(α+ 1)m!

2ⁿ⁻¹²n!Γ(m+α+ 1)

!¹₂

e⁻^µ2+2λ

2

2 L^α_m(µ²+λ²)H_n(λ), is a Hilbert basis of the spaceL²(dγα), whereθis the function defined by the relation (2.4).

In the following, we agree that the Laguerre and Hermite polynomials with negative index are zero.

Proposition 3.1. For all(m, n)∈N²,(r, x)∈R⁺×Rand(µ, λ)∈Γ+, we have (3.1) xe^α_m,n(r, x) =

rn+ 1

2 e^α_m,n+1(r, x) + rn

2e^α_m,n−1(r, x).

(3.2) λξ_m,n^α (µ, λ) =

rn+ 1

2 ξ_m,n+1^α (µ, λ) + rn

2ξ_m,n−1^α (µ, λ).

(3.3) r²e^α+1_m,n(r, x) =p

2(α+ 1)(α+m+ 1)e^α_m,n(r, x)

−p

2(α+ 1)(m+ 1)e^α_m+1,n(r, x).

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(3.4) (µ² +λ²)ξ_m,n^α+1(µ, λ) =p

2(α+ 1)(α+m+ 1)ξ_m,n^α (µ, λ)

−p

2(α+ 1)(m+ 1)ξ_m+1,n^α (µ, λ).

Proof. We know [11] that the Hermite polynomials satisfy the following recurrence formula

H_n+1(x)−2xH_n(x) + 2nHn−1(x) = 0; n ∈N, Therefore, for all(r, x)∈R⁺×R, we have

xe^α_m,n(r, x) = 2^α+1Γ(α+ 1)m!

2ⁿ⁻¹²n!Γ(m+α+ 1)

!¹₂

e⁻^r2+x

2

2 L^α_m(r²)xH_n(x)

=

rn+ 1 2

2^α+1Γ(α+ 1)m!

2ⁿ⁺¹²(n+ 1)!Γ(m+α+ 1)

!¹₂

e⁻^r2+x

2

2 L^α_m(r²)H_n+1(x)

+ rn

2

2^α+1Γ(α+ 1)m!

2ⁿ⁻³²(n−1)!Γ(m+α+ 1)

!¹₂

e⁻^|(r,x)|

2

2 L^α_m(r²)Hn−1(x)

=

rn+ 1

2 e^α_m,n+1(r, x) + rn

2e^α_m,n−1(r, x)

and it is obvious that the same relation holds for the elementsξ_m,n^α . On the other hand

r²e^α+1_m,n(r, x) = 2^α+2Γ(α+ 2)m!

2ⁿ⁻¹²n!Γ(m+α+ 2)

!¹₂

e⁻^r2+x

2

2 r²L^α+1_m (r²)H_n(x).

However, the Laguerre polynomials satisfy the following recurrence formulas (m+ 1)L^α_m+1(r) + (r−α−2m−1)L^α_m(r) + (m+α)L^α_m−1(r) = 0; m ∈N,

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and

L^α+1_m (r)−L^α+1_m−1(r) =L^α_m(r); m∈N. Hence, we deduce that

r²e^α+1_m,n(r, x)

= 2^α+2Γ(α+ 2)m!

2ⁿ⁻¹²n!Γ(m+α+ 2)

!¹₂

e⁻^r2+x

2 2 H_n(x)

× (α+ 2m+ 2)L^α+1_m (r²)−(m+ 1)L^α+1_m+1(r²)−(α+m+ 1)L^α+1_m−1(r²)

= 2^α+2Γ(α+ 2)m!

2ⁿ⁻¹²n!Γ(m+α+ 2)

!¹₂

(α+m+ 1)e⁻^r2+x

2

2 L^α_m(r²)H_n(x)

− 2^α+2Γ(α+ 2)m!

2ⁿ⁻¹²n!Γ(m+α+ 2)

!¹₂

(m+ 1)e⁻^r2+x

2

2 L^α_m+1(r²)H_n(x)

=p

2(α+ 1)(α+m+ 1)e^α_m,n(r, x)−p

2(α+ 1)(m+ 1)e^α_m+1,n(r, x).

Proposition 3.2. For all(m, n)∈N², and(µ, λ)∈Γ₊, we have (3.5) Fα(e^α_m,n)(µ, λ) = (−i)^2m+nξ^α_m,n(µ, λ).

Proof. It is clear that for all (m, n) ∈ N², the function e^α_m,n belongs to the space

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L¹(dν_α)∩L²(dν_α), hence by using Fubini’s theorem, we get Fα(e^α_m,n)(µ, λ) = 2^α+1Γ(α+ 1)m!

2ⁿ⁻¹²n!Γ(m+α+ 1)

!¹₂

×

Z +∞

0

e⁻^r

2

2 L^α_m(r²)j_α rp

µ²+λ² r^2α+1 2^αΓ(α+ 1)dr

× Z

R

e⁻^x

2 2 −iλx

H_n(x) dx

√2π

, and then the required result follows from the following equalities [11]:

∀m ∈N;

Z +∞

0

e⁻^r²L^α_m(r)J_α(√

ry)r^α²dr= (−1)^m2e⁻^y²y^α²L^α_m(y), and

∀n∈N; Z

R

e^ixye⁻^x

2

2 H_n(x)dx=iⁿ√ 2πe⁻^y

2

2 H_n(y),

where J_α denotes the Bessel function of the first kind and index α defined for all x >0by

J_α(x) =

+∞

X

n=0

(−1)ⁿ n!Γ(α+n+ 1)

x 2

2n+α

.

Proposition 3.3. Letf ∈L²(dν_α)∩L²(dν_α+1)such thatFα(f)∈L²(dγ_α+1), then for all(m, n)∈N², we have

(3.6) hf /e^α+1_m,ni_ν_α+1 =

sα+m+ 1

2(α+ 1) hf /e^α_m,ni_ν_α −

s m+ 1

2(α+ 1)hf /e^α_m+1,ni_ν_α,

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and

(3.7) hFα(f)/ξ_m,n^α+1i_γ_α+1 =

sα+m+ 1

2(α+ 1) (−i)^2m+nhf /e^α_m,ni_ν_α +

s m+ 1

2(α+ 1)(−i)^2m+nhf /e^α_m+1,niνα. Proof. We have

hf /e^α+1_m,ni_ν_α+1 = Z +∞

0

Z

R

f(r, x)e^α+1_m,n(r, x)dν_α+1(r, x)

= 1

2(α+ 1) Z +∞

0

Z

R

f(r, x)r²e^α+1_m,n(r, x)dν_α(r, x)

= 1

2(α+ 1)hf /r²e^α+1_m,ni_ν_α, hence by using the relation (3.3), we deduce that

hf /e^α+1_m,ni_ν_α+1 =

sα+m+ 1

2(α+ 1) hf /e^α_m,ni_ν_α−

s m+ 1

2(α+ 1)hf /e^α_m+1,ni_ν_α. In the same manner, and by virtue of the relation (2.7), we have

hFα(f)/ξ_m,n^α+1i_γ_α+1 = Z Z

Γ+

Fα(f)(µ, λ)ξ_m,n^α+1(µ, λ)dγ_α+1(µ, λ)

= 1

2(α+ 1) Z Z

Γ+

Fα(f)(µ, λ)(µ² +λ²)ξ_m,n^α+1(µ, λ)dγα(µ, λ),

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using the relations (3.4) and (3.5), we deduce that hFα(f)/ξ_m,n^α+1i_γ_α+1 = 1

p2(α+ 1) Z Z

Γ+

Fα(f)(µ, λ)√

α+m+ 1ξ^α_m,n(µ, λ)

−√

m+ 1ξ_m+1,n^α (µ, λ)

dγ_α(µ, λ)

=

sα+m+ 1

2(α+ 1) hFα(f)/(i)^2m+nFα(e^α_m,n)i_γ_α

− s

m+ 1

2(α+ 1)hFα(f)/(i)^2m+2+nFα(e^α_m+1,n)i_γ_α, hence, according to the Parseval’s equality (2.11), we obtain

hFα(f)/ξ_m,n^α+1i_γ_α+1 = s

α+m+ 1

2(α+ 1) (−i)^2m+nhf /e^α_m,ni_ν_α +

s m+ 1

2(α+ 1)(−i)^2m+nhf /e^α_m+1,ni_ν_α.

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4. Heisenberg-Pauli-Weyl Inequality for the Fourier Transform F

α

In this section, we will prove the main result of this work, that is the Heisenberg- Pauli-Weyl inequality for the Fourier transform Fα connected with the Riemann- Liouville operatorRα. Next we give a generalization of this result, for this we need the following important lemma.

Lemma 4.1. Letf ∈L²(dν_α), such that

k|(r, x)|fk_2,ν_α <+∞ and k(µ²+ 2λ²)¹²Fα(f)k_2,γ_α <+∞, then

(4.1) k|(r, x)|fk²_2,ν_α+k(µ²+2λ²)¹²Fα(f)k²_2,γ_α =

+∞

X

m,n=0

2α+4m+2n+3

|am,n|²,

wherea_m,n =hf /e^α_m,ni_ν_α; (m, n)∈N². Proof. Letf ∈L²(dν_α), such that

∀(r, x)∈R+×R; f(r, x) =

+∞

X

m,n=0

a_m,ne^α_m,n(r, x), and assume that

k|(r, x)|fk_2,ν_α <+∞ and k(µ²+ 2λ²)¹²Fα(f)k_2,γ_α <+∞,

then the functions(r, x) 7−→ rf(r, x)and (r, x) 7−→ xf(r, x)belong to the space L²(dν_α), in particularf ∈L²(dν_α)∩L²(dν_α+1). In the same manner, the functions

(µ, λ)7−→(µ²+λ²)¹²Fα(f)(µ, λ), and (µ, λ)7−→λFα(f)(µ, λ)

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belong to the space L²(dγ_α). In particular, by the relation (2.7), we deduce that Fα(f)∈L²(dγ_α)∩L²(dγ_α+1), and we have

krfk²_2,ν_α = Z +∞

0

Z

R

r²|f(r, x)|²dν_α(r, x)

= 2(α+ 1)kfk²_2,ν_α+1

= 2(α+ 1)

+∞

X

m,n=0

hf /e^α+1_m,ni_ν_α+1

2,

hence, according to the relation (3.6), we obtain (4.2) krfk²_2,ν_α =

+∞

X

m,n=0

√α+m+ 1a_m,n−√

m+ 1a_m+1,n

2.

Similarly, we have kxfk²_2,ν_α =

Z +∞

0

Z

R

x²|f(r, x)|²dν_α(r, x)

=

+∞

X

m,n=0

hxf /e^α_m,ni_ν_α

2 =

+∞

X

m,n=0

hf /xe^α_m,ni_ν_α

2,

and by the relation (3.1), we get (4.3) kxfk²_2,ν_α =

+∞

X

m,n=0

rn+ 1

2 a_m,n+1+ rn

2am,n−1

2

.

By the same arguments, and using the relations (3.2), (3.7) and the Parseval’s equality (2.11), we obtain

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(4.4) k(µ² +λ²)¹²Fα(f)k²_2,γ_α =

+∞

X

m,n=0

√α+m+ 1a_m,n+√

m+ 1a_m+1,n

2,

and

(4.5) kλFα(f)k²_2,γ_α =

+∞

X

m,n=0

rn+ 1

2 a_m,n+1− rn

2am,n−1

2

. Combining now the relations (4.2), (4.3), (4.4) and (4.5), we deduce that

k|(r, x)|fk²_2,ν

α +k(µ²+ 2λ²)¹²Fα(f)k²_2,γ

α

=krfk²_2,ν_α+k(µ² +λ²)¹²Fα(f)k²_2,γ_α+kxfk²_2,ν_α+kλFα(f)k²_2,γ_α

= 2

+∞

X

m,n=0

(α+m+ 1)|a_m,n|²+ (m+ 1)|a_m+1,n|²

+ 2

+∞

X

m,n=0

n+ 1

2 |am,n+1|²+ n

2|am,n−1|²

= 2

+∞

X

m,n=0

(α+m+ 1)|a_m,n|²+ 2

+∞

X

m,n=0

m|a_m,n|²

+ 2

+∞

X

m,n=0

n

2|a_m,n|² + 2

+∞

X

m,n=0

n+ 1 2 |a_m,n|²

=

+∞

X

m,n=0

2α+ 4m+ 2n+ 3

|a_m,n|².

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Remark 1. From the relation (4.1), we deduce that for allf ∈L²(dν_α), we have (4.6) k|(r, x)|fk²_2,ν_α +k(µ²+ 2λ²)¹²Fα(f)k²_2,γ_α ≥(2α+ 3)kfk²_2,ν_α, with equality if and only if

∀(r, x)∈R+×R; f(r, x) =Ce⁻^r2+x

2

2 ; C ∈C.

Lemma 4.2. Letf ∈L²(dνα)such that,

k|(r, x)|fk_2,ν_α <+∞ and k(µ²+ 2λ²)¹²Fα(f)k_2,γ_α <+∞, then

1)For allt >0, 1

t²k|(r, x)|fk²_2,ν_α +t²k(µ²+ 2λ²)¹²Fα(f)k²_2,γ_α ≥(2α+ 3)kfk²_2,ν_α. 2)The following assertions are equivalent

i) k|(r, x)|fk_2,ν_αk(µ²+ 2λ²)¹²Fα(f)k_2,γ_α = 2α+ 3

2 kfk²_2,ν_α. ii)There existst₀ >0, such that

k|(r, x)|f_t₀k²_2,ν_α +k(µ²+ 2λ²)¹²Fα(f_t₀)k²_2,γ_α = (2α+ 3)kf_t₀k²_2,ν_α, wheref_t₀(r, x) =f(t₀r, t₀x).

Proof. 1)Let f ∈ L²(dν_α) satisfy the hypothesis.For allt > 0 we put f_t(r, x) = f(tr, tx), and then by a simple change of variables, we get

(4.7) kf_tk²_2,ν_α = 1

t^2α+3kfk²_2,ν_α,

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and

(4.8) k|(r, x)|f_tk²_2,ν_α = 1

t^2α+5k|(r, x)|fk²_2,ν_α. For all(µ, λ)∈Γ,

(4.9) Fα(f_t)(µ, λ) = 1

t^2α+3Fα(f) µ

t,λ t

, and by using the relation (2.6), we deduce that

(4.10) k(µ² + 2λ²)¹²Fα(f_t)k²_2,γ_α = 1

t^2α+1k(µ²+ 2λ²)¹²Fα(f)k²_2,γ_α. Then, the desired result follows by replacingf byf_tin the relation (4.6).

2)Letf ∈L²(dν_α);f 6= 0.

• Assume that

k|(r, x)|fk_2,ν_αk(µ²+ 2λ²)¹²Fα(f)k_2,γ_α = 2α+ 3

2 kfk²_2,ν_α. By Theorem 2.3, we havek(µ²+ 2λ²)¹²Fα(f)k_2,γ_α 6= 0, then for

t₀ =

s k|(r, x)|fk_2,ν_α k(µ²+ 2λ²)¹²Fα(f)k_2,γ_α, we have

1

t²₀k|(r, x)|fk²_2,ν_α +t²₀k(µ²+ 2λ²)¹²Fα(f)k²_2,γ_α = (2α+ 3)kfk²_2,ν_α, and this is equivalent to

k|(r, x)|f_t₀k²_2,ν

α +k(µ²+ 2λ²)¹²Fα(f_t₀)k²_2,γ

α = (2α+ 3)kf_t₀k²_2,ν

α.

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• Conversely, suppose that there existst₁ >0, such that

k|(r, x)|f_t₁k²_2,ν_α +k(µ²+ 2λ²)¹²Fα(f_t₁)k²_2,γ_α = (2α+ 3)kf_t₁k²_2,ν_α. This is equivalent to

(4.11) 1

t²₁k|(r, x)|fk²_2,ν_α +t²₁k(µ²+ 2λ²)¹²Fα(f)k²_2,γ_α = (2α+ 3)kfk²_2,ν_α. However, lethbe the function defined on]0,+∞[, by

h(t) = 1

t²k|(r, x)|fk²_2,ν_α +t²k(µ²+ 2λ²)¹²Fα(f)k²_2,γ_α, then, the minimum of the functionhis attained at the point

t₀ =

s k|(r, x)|fk_2,ν_α k(µ²+ 2λ²)¹²Fα(f)k_2,γ_α and

h(t₀) = 2k|(r, x)|fk_2,ν_αk(µ²+ 2λ²)¹²Fα(f)k_2,γ_α. Thus by1)of this lemma, we have

h(t₁)≥h(t₀) = 2k|(r, x)|fk_2,ν_αk(µ²+ 2λ²)¹²Fα(f)k_2,γ_α ≥(2α+ 3)kfk²_2,ν_α. According to the relation (4.11), we deduce that

h(t₁) = h(t₀) = 2k|(r, x)|fk_2,ν_αk(µ²+ 2λ²)¹²Fα(f)k_2,γ_α = (2α+ 3)kfk²_2,ν_α.

Theorem 4.3 (Heisenbeg-Pauli-Weyl inequality). For allf ∈L²(dνα), we have (4.12) k|(r, x)|fk_2,ν_αk(µ²+ 2λ²)¹²Fα(f)k_2,γ_α ≥ (2α+ 3)

2 kfk²_2,ν_α