(1)HEISENBERG-PAULI-WEYL UNCERTAINTY PRINCIPLE FOR THE RIEMANN-LIOUVILLE OPERATOR S

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

S. OMRI AND L.T. RACHDI DEPARTMENT OFMATHEMATICS

FACULTY OFSCIENCES OFTUNIS

2092 MANAR2 TUNIS

TUNISIA.

lakhdartannech.rachdi@fst.rnu.tn

Received 29 March, 2008; accepted 08 August, 2008 Communicated by J.M. Rassias

ABSTRACT. The Heisenberg-Pauli-Weyl inequality is established for the Fourier transform con- nected with the Riemann-Liouville operator.Also, a generalization of this inequality is proved.

Lastly, a local uncertainty principle is studied.

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

2000 Mathematics Subject Classification. 42B10, 33C45.

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 localized. The theorems of Hardy, Morgan, Beurling, ... are established for several Fourier transforms in [4], [9], [13] and [14]. In this context, a remarkable Heisenberg uncertainty principle [10] states, according to Weyl [25]

who assigned the result to Pauli, that for all square integrable functionsf onRⁿwith respect to the Lebesgue measure, we have

Z

Rⁿ

x²_j|f(x)|²dx Z

Rⁿ

ξ_j²|f(ξ)|b 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 trans- form.

Recently, many works have been devoted to establishing the Heisenberg-Pauli-Weyl inequal- ity 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 Hankel transform. In the same context, Battle [3] has proved this inequality for wavelet states, and Wolf [26], has studied this uncertainty principle

100-08

for Gelfand pairs. We cite also De Bruijn [5] who has established the same result for the classi- cal 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 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 mappingRα

have been studied and many harmonic analysis results have been established for the Fourier transform Fα (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 transformFα connected withRα.More precisely, using Laguerre and Hermite polyno- mials we establish firstly the Heisenberg-Pauli-Weyl inequality for the Fourier transform Fα, that is

• For allf ∈L²(dν_α), we have Z +∞

0

Z

R

(r²+x²)|f(r, x)|²dν_α(r, x) ¹₂

× 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 dν_α(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

! .

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

• For allf ∈L²(dνα),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)|²dν_α(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 numbersξ;0< ξ < ^2α+3₂ , there exists a positive constantK_α,ξ such that for allf ∈L²(dν_α), and for all measurable subsetsE ⊂Γ₊; 0< γ_α(E)<+∞, we have Z Z

E

|Fα(f)(µ, λ)|²dγα(µ, λ)< Kα,ξ γα(E)_2α+3^2ξ Z +∞

0

Z

R

(r²+x²)^ξ|f(r, x)|²dνα(r, x).

• 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 subsetsE ⊂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.

2. THEFOURIERTRANSFORMASSOCIATED WITH THERIEMANN-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 function j_α has the following integral representation [1, 11], for all µ∈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.

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 eigenfunction ϕ_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 ϕ_µ,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.

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

• dναthe 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,να =

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_να the inner product defined onL²(dν_α)by hf /gi_να =

Z +∞

0

Z

R

f(r, x)g(r, x)dν_α(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)}, whereθis the bijective function defined on the setΓ₊by

(2.4) θ(µ, λ) =p

µ²+λ², λ

.

• dγ_αthe 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 kfk_p,γα =Z Z

Γ+

|f(µ, λ)|^pdγ_α(µ, λ)¹_p

<∞, ifp∈[1,+∞[, kfk∞,γα = ess sup

(µ,λ)∈Γ+

|f(µ, λ)|<∞, ifp= +∞.

• h / iγα the inner product defined onL²(dγα)by hf /gi_γα =

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)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

! .

In particular

(2.7) dγ_α+1(µ, λ) = µ²+λ²

2(α+ 1)dγ_α(µ, λ).

ii) For all measurable functions f on R+ ×R, the function f ◦θ is measurable on Γ₊. Furthermore, if f is non negative or an integrable function onR+×Rwith 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 transformFα associated with the operatorRα

and we give some properties that we use in the sequel.

Definition 2.1. The Fourier transformFα associated with the Riemann-liouville operatorRα

is 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 transformFα is 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,να.

Theorem 2.2 (Inversion formula). Let f ∈ L¹(dνα) such that Fα(f) ∈ L¹(dγα), 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).

Theorem 2.3 (Plancherel). The Fourier transformFα can be extended to an isometric isomor- phism 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).

3. HILBERT BASIS OF THE SPACESL²(dν_α), ANDL²(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 Hn

)

n∈N

are respectively a Hilbert basis of the Hilbert spaces L²

R+,₂α^rΓ(α+1)^2α+1 dr

and L² R,^√dx

2π

, 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)Hn(x), is a Hilbert basis of the spaceL²(dν_α).

Using the relation (2.8), we deduce that the familyn 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(dγ_α), 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).

(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)Hn+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, 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 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)jα

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),

whereJ_αdenotes the Bessel function of the first kind and indexαdefined for allx >0by J_α(x) =

+∞

X

n=0

(−1)ⁿ n!Γ(α+n+ 1)

x 2

2n+α

.

Proposition 3.3. Let f ∈ 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_να+1 = s

α+m+ 1

2(α+ 1) hf /e^α_m,ni_να− s

m+ 1

2(α+ 1)hf /e^α_m+1,ni_να, and

(3.7) hFα(f)/ξ_m,n^α+1i_γα+1 =

sα+m+ 1

2(α+ 1) (−i)^2m+nhf /e^α_m,ni_να +

s

m+ 1

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

hf /e^α+1_m,ni_να+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_να, hence by using the relation (3.3), we deduce that

hf /e^α+1_m,ni_να+1 = s

α+m+ 1

2(α+ 1) hf /e^α_m,ni_να− s

m+ 1

2(α+ 1)hf /e^α_m+1,ni_να.

In the same manner, and by virtue of the relation (2.7), we have hFα(f)/ξ_m,n^α+1i_γα+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)dγ_α(µ, λ), using the relations (3.4) and (3.5), we deduce that

hFα(f)/ξ_m,n^α+1i_γα+1 = 1 p2(α+ 1)

Z Z

Γ+

Fα(f)(µ, λ)√

α+m+ 1ξ_m,n^α (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_γα

−

s m+ 1

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

hFα(f)/ξ_m,n^α+1i_γα+1 = s

α+m+ 1

2(α+ 1) (−i)^2m+nhf /e^α_m,ni_να +

s m+ 1

2(α+ 1)(−i)^2m+nhf /e^α_m+1,ni_να. 4. HEISENBERG-PAULI-WEYLINEQUALITY FOR THEFOURIERTRANSFORMFα

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 operator Rα. Next we give a generalization of this result, for this we need the following important lemma.

Lemma 4.1. Letf ∈L²(dνα), such that

k|(r, x)|fk2,να <+∞ and k(µ²+ 2λ²)¹²Fα(f)k2,γα <+∞, then

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

+∞

X

m,n=0

2α+ 4m+ 2n+ 3

|am,n|²,

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

∀(r, x)∈R⁺×R; f(r, x) =

+∞

X

m,n=0

am,ne^α_m,n(r, x), and assume that

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

then the functions(r, x) 7−→rf(r, x)and(r, x) 7−→ xf(r, x)belong to the spaceL²(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)

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_να+1

2,

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

+∞

X

m,n=0

√α+m+ 1a_m,n−√

m+ 1a_m+1,n

2.

Similarly, we have

kxfk²_2,να = Z +∞

0

Z

R

x²|f(r, x)|²dν_α(r, x)

=

+∞

X

m,n=0

hxf /e^α_m,ni_να

2 =

+∞

X

m,n=0

hf /xe^α_m,ni_να

2,

and by the relation (3.1), we get

(4.3) kxfk²_2,να =

+∞

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

(4.4) k(µ²+λ²)¹²Fα(f)k²_2,γα =

+∞

X

m,n=0

√α+m+ 1a_m,n +√

m+ 1a_m+1,n

2,

and

(4.5) kλFα(f)k²_2,γα =

+∞

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,να +k(µ²+ 2λ²)¹²Fα(f)k²_2,γα

=krfk²_2,να +k(µ²+λ²)¹²Fα(f)k²_2,γα +kxfk²_2,να +kλFα(f)k²_2,γα

= 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)|am,n|²+ 2

+∞

X

m,n=0

m|am,n|²+ 2

+∞

X

m,n=0

n

2|am,n|²+ 2

+∞

X

m,n=0

n+ 1

2 |am,n|²

=

+∞

X

m,n=0

2α+ 4m+ 2n+ 3

|a_m,n|².

Remark 1. From the relation (4.1), we deduce that for allf ∈L²(dν_α), we have

(4.6) k|(r, x)|fk²_2,να +k(µ²+ 2λ²)¹²Fα(f)k²_2,γα ≥(2α+ 3)kfk²_2,να, 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,να <+∞ and k(µ²+ 2λ²)¹²Fα(f)k_2,γα <+∞, then

1)For allt >0, 1

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

i) k|(r, x)|fk2,ναk(µ²+ 2λ²)¹²Fα(f)k2,γα = 2α+ 3

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

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

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

(4.7) kf_tk²_2,να = 1

t^2α+3kfk²_2,να, and

(4.8) k|(r, x)|f_tk²_2,να = 1

t^2α+5k|(r, x)|fk²_2,να. 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(dνα);f 6= 0.

• Assume that

k|(r, x)|fk2,ναk(µ²+ 2λ²)¹²Fα(f)k2,γα = 2α+ 3

2 kfk²_2,να. By Theorem 2.2, we havek(µ²+ 2λ²)¹²Fα(f)k_2,γα 6= 0, then for

t₀ =

s k|(r, x)|fk_2,να k(µ²+ 2λ²)¹²Fα(f)k2,γα

, we have

1

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

k|(r, x)|f_t0k²_2,να +k(µ²+ 2λ²)¹²Fα(f_t0)k²_2,γα = (2α+ 3)kf_t0k²_2,να.

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

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

(4.11) 1

t²₁k|(r, x)|fk²_2,να +t²₁k(µ²+ 2λ²)¹²Fα(f)k²_2,γα = (2α+ 3)kfk²_2,να. 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,να k(µ² + 2λ²)¹²Fα(f)k_2,γα and

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

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

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

Theorem 4.3 (Heisenbeg-Pauli-Weyl inequality). For allf ∈L²(dν_α), we have

(4.12) k|(r, x)|fk_2,ναk(µ²+ 2λ²)¹²Fα(f)k_2,γα ≥ (2α+ 3)

2 kfk²_2,να with equality if and only if

∀(r, x)∈R+×R; f(r, x) = Ce⁻

r2+x2 2t2

0 ; t₀ >0, C ∈C.

Proof. It is obvious that iff = 0, or if k|(r, x)|fk2,να = +∞, ork(µ² + 2λ²)¹²Fα(f)k2,γα = +∞, then the inequality (4.12) holds.

Let us suppose thatk|(r, x)|fk_2,να +k(µ²+ 2λ²)¹²Fα(f)k_2,γα <+∞, andf 6= 0.

By1)of Lemma 4.2, we have for allt >0 1

t²k|(r, x)|fk²_2,να +t²k(µ²+ 2λ²)¹²Fα(f)k²_2,γα ≥(2α+ 3)kfk²_2,να, and the result follows if we pick

t =

s k|(r, x)|fk_2,να k(µ² + 2λ²)¹²Fα(f)k_2,γα. By2)of Lemma 4.2, we have

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

2 kfk²_2,να, if and only if there existst₀, such that

k|(r, x)|f_t0k²_2,να +k(µ²+ 2λ²)¹²Fα(f_t0)k²_2,γα = (2α+ 3)kf_t0k²_2,να, and according to Remark 1, this is equivalent to

f_t0(r, x) =Ce^−r2+x

2

2 ; C ∈C,

which means that

f(r, x) = Ce⁻

r2+x2 2t2

0 ; C ∈C.

The following result gives a generalization of the Heisenberg-Pauli-Weyl inequality.

Theorem 4.4. Leta, b ≥ 1and η ∈ R such thatηa = (1−η)b,then for all f ∈ L²(dνα)we have

k|(r, x)|^afk^η_2,ν

αk(µ²+ 2λ²)^b2Fα(f)k^1−η_2,γ

α ≥

2α+ 3 2

aη

kfk_2,να with equality if and only ifa=b = 1and

∀(r, x)∈R⁺×R; f(r, x) = Ce⁻

r2+x2 2t2

0 ; t0 >0, C ∈C. Proof. Letf ∈L²(dν_α),f 6= 0, such that

k|(r, x)|^afk_2,να +k(µ²+ 2λ²)^b2Fα(f)k_2,γα <+∞.

Then for alla >1, we have

k|(r, x)|^afk_2,ν^1a _αkfk

1 a0

2,να =k|(r, x)|²|f|^a2ka,ν¹² αk|f|^a20k_a¹²0,να, wherea⁰is defined as usual bya⁰ = _a−1^a .By Hölder’s inequality we get

k|(r, x)|^afk_2,ν^1a _αkfk

1 a0

2,να >k|(r, x)|fk_2,να.

The strict inequality here is justified by the fact that if f 6= 0, then the functions|(r, x)|^2a|f|² and|f|²cannot be proportional.Thus for alla ≥1, we have

(4.13) k|(r, x)|^afk_2,ν^1a _α ≥ k|(r, x)|fk_2,να kfk

1 a0

2,να

.

with equality if and only ifa= 1.

In the same manner and using Plancherel’s Theorem 2.3, we have for allb≥1 k(µ²+ 2λ²)^b2Fα(f)k_2,γ^1b

α ≥ k(µ²+ 2λ²)¹²Fα(f)k_2,γα kFα(f)k

1 b0

2,γα

(4.14)

≥ k(µ²+ 2λ²)¹²Fα(f)k_2,γα kfk

1 b0

2,να

. with equality if and only ifb = 1.

Letη = _a+b^b , then by the relations (4.13), (4.14) and for alla, b≥1,we have k|(r, x)|^afk^η_2,ναk(µ²+ 2λ²)^b2Fα(f)k^1−η_2,γα ≥





k|(r, x)|fk_2,ναk(µ²+ 2λ²)¹²Fα(f)k_2,γα kfk

1 a0+_b¹0

2,να





ηa

,

with equality if and only ifa=b = 1.

Applying Theorem 4.3, we obtain

k|(r, x)|^afk^η_2,ναk(µ²+ 2λ²)^2bFα(f)k^1−η_2,γα ≥

2α+ 3 2

ηa

kfk_2,να, with equality if and only ifa=b = 1and

∀(r, x)∈R+×R; f(r, x) = Ce⁻

r2+x2 2t2

0 ; t₀ >0, C ∈C.

Remark 2. In the particular case whena=b= 2, the previous result gives us the Heisenberg- Pauli-Weyl inequality for the fourth moment of Heisenberg

k|(r, x)|²fk_2,ναk(µ²+ 2λ²)Fα(f)k_2,γα >

2α+ 3 2

2

kfk²_2,να. 5. THELOCALUNCERTAINTYPRINCIPLE

Theorem 5.1. Letξbe a real number such that0< ξ < ^2α+3₂ , then for allf ∈L²(dν_α),f 6= 0, and for all measurable subsets E ⊂Γ₊;0< γ_α(E)<+∞, we have

(5.1)

Z Z

E

|Fα(f)(µ, λ)|²dγ_α(µ, λ)< K_α,ξ γ_α(E)_2α+3^2ξ

k|(r, x)|^ξfk²_2,να, where

K_α,ξ = 2α+ 3−2ξ ξ²2^α+52Γ α+ ³₂

!_2α+3^2ξ

2α+ 3 2α+ 3−2ξ

2

. Proof. For alls >0, we put

B_s =

(r, x)∈R+×R; r²+x² < s² . Letf ∈L²(dν_α). By Minkowski’s inequality, we have

Z Z

E

|Fα(f)(µ, λ)|²dγ_α(µ, λ) ¹₂ (5.2)

=kFα(f)1Ek2,γα

≤ kFα(f1_Bs)1_Ek_2,γα +kFα(f1_Bs^c)1_Ek_2,γα

≤ γ_α(E)¹₂

kFα(f1_Bs)k∞,γα+kFα(f1_B^c_s)k_2,γα.