In this paper, we provide, for theq-Dunkl transform studied in [2], a Heisenberg uncertainty principle and two local uncertainty principles leading to a new Heisenberg-Weyl type inequality

ON HEISENBERG AND LOCAL UNCERTAINTY PRINCIPLES FOR THE q-DUNKL TRANSFORM

AHMED FITOUHI, FERJANI NOURI, AND SANA GUESMI FACULTÉ DESSCIENCES DETUNIS

1060 TUNIS, TUNISIA. Ahmed.Fitouhi@fst.rnu.tn

INSTITUTPRÉPARATOIRE AUXÉTUDES D’INGÉNIEUR DENABEUL

TUNISIA.

nouri.ferjani@yahoo.fr FACULTÉ DESSCIENCES DETUNIS

1060 TUNIS, TUNISIA. guesmisana@yahoo.fr

Received 17 January, 2009; accepted 02 May, 2009 Communicated by S.S. Dragomir

ABSTRACT. In this paper, we provide, for theq-Dunkl transform studied in [2], a Heisenberg uncertainty principle and two local uncertainty principles leading to a new Heisenberg-Weyl type inequality.

Key words and phrases: q-Dunkl transform, Heisenberg-Weyl inequality, Local uncertainty principles.

2000 Mathematics Subject Classification. 33D15, 26D10, 26D20, 39A13, 42A38.

1. INTRODUCTION

In harmonic analysis, the uncertainty principle states that a function and its Fourier transform cannot be simultaneously sharply localized. A quantitative formulation of this fact is provided by the Heisenberg uncertainty principle, which asserts that every square integrable function f onRverifies the following inequality

(1.1)

Z +∞

−∞

x²|f(x)|²dx

Z +∞

−∞

λ²|f(λ)|b ²dλ

≥ 1 4

Z +∞

−∞

x²|f(x)|²dx 2

, where

fb(λ) = 1

√2π Z +∞

−∞

f(x)e^−iλxdx is the classical Fourier transform.

Generalizations of this result in both classical and quantum analysis have been treated and many versions of Heisenberg-Weyl type uncertainty inequalities were obtained for several gen- eralized Fourier transforms (see [1], [14], [10]).

019-09

In [2], by the use of the q²-analogue differential operator studied in [11], Bettaibi et al.

introduced a new q-analogue of the classical Dunkl operator and studied its related Fourier transform, which is a q-analogue of the classical Bessel-Dunkl one and called the q-Dunkl transform.

The aim of this paper is twofold: first, we prove a Heisenberg uncertainty principle for the q-Dunkl transform and next, we state for this transform two local uncertainty principles leading to a newq-Heisenberg-Weyl type inequality.

This paper is organized as follows: in Section 2, we present some preliminary notions and notations useful in the sequel. In Section 3, we recall some results and properties from the theory of the q-Dunkl operator and the q-Dunkl transform (see [2]). Section 4 is devoted to proving a Heisenberg uncertainty principle for theq-Dunkl transform and as consequences, we obtain Heisenberg uncertainty principles for the q²-analogue Fourier transform [12, 11] and for the q-Bessel transform [2]. Finally, in Section 5, we state, for theq-Dunkl transform, two local uncertainty principles, which give a new Heisenberg-Weyl type inequality for theq-Dunkl transform.

2. NOTATIONS AND PRELIMINARIES

Throughout this paper, we assumeq ∈]0,1[, and refer to the general reference [6] for the def- initions, notations and properties of theq-shifted factorials and theq-hypergeometric functions.

We writeRq ={±qⁿ:n ∈Z},Rq,+ ={qⁿ:n ∈Z}, [x]_q = 1−q^x

1−q , x∈C and [n]_q! = (q;q)n

(1−q)ⁿ, n∈N. Theq²-analogue differential operator is (see [11, 12])

(2.1) ∂_q(f)(z) =







f(^q−1z)^+f(^−q−1z)−f(qz)+f(−qz)−2f(−z)

2(1−q)z ifz 6= 0

limx→0∂q(f)(x) (inR^q) ifz = 0.

We remark that iff is differentiable atz, thenlimq→1∂q(f)(z) = f⁰(z).

A repeated application of theq²-analogue differential operator is denoted by:

∂_q⁰f =f, ∂_qⁿ⁺¹f =∂_q(∂_qⁿf).

The following lemma lists some useful computational properties of∂_q. Lemma 2.1.

(1) For all functionsf onRq,

∂_qf(z) = f_e(q⁻¹z)−f_e(z)

(1−q)z +f_o(z)−f_o(qz) (1−q)z , where,f_eandf_oare, respectively, the even and the odd parts off.

(2) For two functionsf andgonRq, we have

• iff is even andg is odd,

∂_q(f g)(z) =q∂_q(f)(qz)g(z) +f(qz)∂_q(g)(z)

=∂_q(g)(z)f(z) +qg(qz)∂_q(f)(qz);

• iff andgare even,

∂_q(f g)(z) =∂_q(f)(z)g(q⁻¹z) +f(z)∂_q(g)(z).

The operator∂_qinduces aq-analogue of the classical exponential function (see [11, 12])

(2.2) e(z;q²) =

∞

X

n=0

a_n zⁿ

[n]q!, with a_2n=a_2n+1 =qⁿ⁽ⁿ⁺¹⁾. Theq-Jackson integrals are defined by (see [8])

Z a 0

f(x)d_qx= (1−q)a

∞

X

n=0

qⁿf(aqⁿ),

Z b a

f(x)d_qx= Z b

0

f(x)d_qx− Z a

0

f(x)d_qx,

Z ∞ 0

f(x)dqx= (1−q)

∞

X

n=−∞

qⁿf(qⁿ), and

Z ∞

−∞

f(x)d_qx= (1−q)

∞

X

n=−∞

qⁿf(qⁿ) + (1−q)

∞

X

n=−∞

qⁿf(−qⁿ), provided the sums converge absolutely.

Theq-Gamma function is given by (see [8]) Γ_q(x) = (q;q)_∞

(q^x;q)∞

(1−q)^1−x, x6= 0,−1,−2, ...

• S_q(Rq)the space of functionsf defined onRqsatisfying

∀n, m∈N, P_n,m,q(f) = sup

x∈Rq

x^m∂_qⁿf(x)

<+∞

and

x→0lim∂_qⁿf(x) (in Rq) exists;

• L^∞_q (R^q) = n

f :kfk_∞,q = sup_x∈Rq|f(x)|<∞o

;

• L^p_α,q(Rq) =

f :kfk_p,α,q = R∞

−∞|f(x)|^p|x|^2α+1d_qx¹_p

<∞

;

• L^p_α,q([−a, a]) =

f :kfk_p,α,q = Ra

−a|f(x)|^p|x|^2α+1d_qx¹_p

<∞

.

For the particular case p = 2, we denote by h·;·i the inner product of the Hilbert space L²_α,q(Rq).

3. THEq-DUNKLOPERATOR AND THE q-DUNKL TRANSFORM

In this section, we collect some basic properties of the q-Dunkl operator and the q-Dunkl transform introduced in [2] which will useful in the sequel.

Forα≥ −¹₂, theq-Dunkl operator is defined by

Λ_α,q(f)(x) =∂_q[H_α,q(f)] (x) + [2α+ 1]_qf(x)−f(−x)

2x ,

where

H_α,q :f =f_e+f_o7−→f_e+q^2α+1f_o. It satisfies the following relations:

• Forα =−¹₂,Λα,q =∂q.

• Λ_α,q livesS_q(Rq)invariant.

• If f is odd then Λ_α,q(f)(x) = q^2α+1∂_qf(x) + [2α + 1]_q^f(x)_x and if f is even then Λα,q(f)(x) =∂qf(x).

• For alla∈C,Λα,q[f(ax)] =aΛα,q(f)(ax).

• For allf andg such thatR+∞

−∞ Λ_α,q(f)(x)g(x)|x|^2α+1d_qxexists, we have (3.1)

Z +∞

−∞

Λ_α,q(f)(x)g(x)|x|^2α+1d_qx=− Z +∞

−∞

Λ_α,q(g)(x)f(x)|x|^2α+1d_qx.

It was shown in [2] that for eachλ ∈C, the function (3.2) ψ^α,q_λ :x7−→j_α(λx;q²) + iλx

[2α+ 2]_qj_α+1(λx;q²) is the unique solution of theq-differential-difference equation:

( Λ_α,q(f) =iλf f(0) = 1,

wherej_α(·;q²)is the normalized third Jackson’sq-Bessel function given by (3.3) j_α(x;q²) =

∞

X

n=0

(−1)ⁿ qⁿ⁽ⁿ⁺¹⁾

(q²;q²)n(q^2(α+1);q²)n

((1−q)x)²ⁿ.

The functionψ_λ^α,q(x),has a unique extension toC×Cand verifies the following properties.

• ψ^α,q_aλ(x) =ψ^α,q_λ (ax) =ψ^α,q_ax(λ), ∀a, x, λ∈C.

• For allx, λ∈Rq,

(3.4) |ψ_λ^α,q(x)| ≤ 4

(q;q)_∞.

Theq-Dunkl transformF_D^α,q is defined onL¹_α,q(R^q)(see [2]) by F_D^α,q(f)(λ) = c_α,q

2

Z +∞

−∞

f(x)ψ_−λ^α,q(x)|x|^2α+1d_qx,

where

c_α,q = (1 +q)^−α Γ_q²(α+ 1). It satisfies the following properties:

• Forα =−¹₂,F_D^α,q is theq²-analogue Fourier transformf(·;b q²)given by (see [12, 11]) f(λ;b q²) = (1 +q)^1/2

2Γ_q^{2 1}₂

Z +∞

−∞

f(x)e(−iλx;q²)d_qx.

• On the even functions space, F_D^α,q coincides with theq-Bessel transform given by (see [2])

Fα,q(f)(λ) =cα,q

Z +∞

0

f(x)jα(λx;q²)x^2α+1dqx.

• For allf ∈L¹_α,q(Rq), we have:

(3.5) kF_D^α,q(f)k∞,q ≤ 2c_α,q

(q;q)∞

kfk_1,α,q.

• For allf ∈L¹_α,q(Rq), such thatxf ∈L¹_α,q(Rq),

(3.6) F_D^α,q(Λα,qf)(λ) =iλF_D^α,q(f)(λ) and

(3.7) Λα,q(F_D^α,q(f)) = −iF_D^α,q(xf).

• Theq-Dunkl transformF_D^α,qis an isomorphism fromL²_α,q(R^q)(resp.S_q(R^q)) onto itself and satisfies the following Plancherel formula:

(3.8) kF_D^α,q(f)k_2,α,q =kfk_2,α,q, f ∈L²_α,q(Rq).

4. q-ANALOGUE OF THEHEISENBERGINEQUALITY

In this section, we provide a Heisenberg uncertainty principle for the q-Dunkl transform.

For this purpose, inspired by the approach given in [10], we follow the steps of [1], using the operatorΛ_α,q instead of the operator∂_q, and consider the operators

L_α,q(f)(x) = f_e(x) +q^2α+2f_o(qx) and Qf(x) =xf(x), and theq-commutator:

[D_α,q, Q]_q=D_α,qQ−qQD_α,q, where

D_α,q =L_α,qΛ_α,q.

The following theorem gives a Heisenberg uncertainty principle for the q-Dunkl transform F_D^α,q.

Theorem 4.1. Forf ∈ S_q(Rq), we have

(4.1) q^2α+1 1 +q+q^α−1+q^α

qkfk²_2,α,q+

1−q− [2α+ 1]_q q^2α

kf_ok²_2,α,q

≤ kxfk2,α,qkxF_D^α,q(f)(x)k2,α,q. Proof. By Lemma 2.1 and simple calculus, we obtain

[D_α,q, Q]_qf =q^2α+2f_e+q^2α+1

1− [2α+ 1]_q q^2α

f_o.

Then, using the Cauchy-Schwarz inequality and the properties of theq-Dunkl operator, one can write

q^2α+2kf_ek²_2,α,q+q^2α+1

1− [2α+ 1]_q q^2α

kf_ok²_2,α,q

=|h[D_α,q, Q]_qf;fi|

=|hDα,qQf−qQDα,qf;fi| ≤ |hDα,qQf;fi|+q|hQDα,qf;fi|

=|hD_α,q(xf_e+xf_o);fi|+q|hD_α,qf;xfi|

=

hΛ_α,q(xf_e) +q^2α+2Λ_α,q(xf_o)(qx);fi +q

hq^2α+2Λ_α,q(f_e)(qx) + Λ_α,q(f_o)(x);xfi

≤ |hΛ_α,q(xf_e);fi|+q^2α+2|hΛ_α,q(xf_o)(qx);fi|

+q^2α+3|hΛ_α,q(f_e)(qx);xfi|+q|hΛ_α,q(f_o)(x);xfi|

=|hΛ_α,q(xf_e);fi|+q^2α+1|hΛ_α,q(xf_o(q.));fi|

+q^2α+2|hΛ_α,q(f_e(q.));xfi|+q|hΛ_α,q(f_o)(x);xfi|

≤ kxf_ek_2,α,qkxF_D^α,q(f)k_2,α,q+q^α−1kxf_ok_2,α,qkxF_D^α,q(f)k_2,α,q

+q^αkxf_ek_2,α,qkxF_D^α,q(f)k_2,α,q+qkxf_ok_2,α,qkxF_D^α,q(f)k_2,α,q

≤(1 +q+q^α−1+q^α)kxfk_2,α,qkxF_D^α,q(f)k_2,α,q,

which achieves the proof.

As a consequence, we obtain a Heisenberg-Weyl uncertainty principle for the q²-analogue Fourier transform (by takingα=−1/2) and theq-Bessel transform (in the even case).

Corollary 4.2.

(1) Forf ∈ S_q(Rq), we have

(4.2) q

1 +q+q^−3/2 +q^−1/2kfk²_2,q ≤ kxfk_2,qkλf(λ;b q²)k_2,q. (2) For an even functionf ∈ S_q(Rq), we have

(4.3) q^2α+2

1 +q+q^α−1+q^αkfk²_2,α,q ≤ kxfk2,α,qkλFα,q(f)(λ)k2,α,q.

We remark that whenqtends to1⁻, (4.2) tends at least formally to the classical Heisenberg uncertainty principle given by (1.1).

5. LOCALUNCERTAINTYPRINCIPLES

In this section, we will state, for the q-Dunkl transform, two local uncertainty principles leading to a new Heisenberg-Weyl type inequality.

Notations: ForE ⊂R^q andf defined onR^q, we write Z

E

f(t)dqt = Z ∞

−∞

f(t)χE(t)dqt and |E|α = Z

E

|t|^2α+1dqt,

whereχE is the characteristic function ofE.

Theorem 5.1. If 0< a < α+ 1, then for all bounded subsetsE ofRq and allf ∈ L²_α,q(Rq), we have

(5.1)

Z

E

|F_D^α,q(f)(λ)|²|λ|^2α+1d_qλ≤K_a,α|E|^α+1a kx^afk²_2,α,q, where

K_a,α = 2ec_α,q p[2(α+ 1−a)]q

α+ 1−a a

!_α+1^2a

α+ 1 α+ 1−a

2

and ecα,q = _(q;q)^2cα,q

∞.

Proof. Forr >0, letχ_r =χ_[−r,r]the characteristic function of[−r, r]andχe_r = 1−χ_r. Then forr >0, we have, sincef ·χ_r ∈L¹_q(Rq),

Z

E

|F_D^α,q(f)(λ;q²)|²|λ|^2α+1d_qλ 1/2

=kF_D^α,q(f)·χ_Ek_2,α,q

≤ kF_D^α,q(f·χ_r)χ_Ek_2,α,q+kF_D^α,q(f·χe_r)χ_Ek_2,α,q

≤ |E|^1/2_α kF_D^α,q(f·χ_r)k∞,q+kF_D^α,q(f·χe_r)k_2,α,q. Now, on the one hand, we have by the relation (3.5) and the Cauchy-Schwartz inequality,

kF_D^α,q(f.χ_r)k∞,q ≤ec_α,qkf ·χ_rk_1,α,q

=ec_α,qkx^−aχ_r·x^afk_1,α,q

≤ec_α,qkx^−aχ_rk_2,α,qkx^afk_2,α,q

≤ 2ec_α,q

p[2(α+ 1−a)]_qr^(α+1)−akx^afk2,α,q.

On the other hand, since f ∈ L²_α,q(Rq), we have f · χe_r ∈ L²_α,q(Rq) and by the Plancherel formula, we obtain

kF_D^α,q(f.χe_r)k_2,α,q =kf·χe_rk_2,α,q

=kx^−aχer·x^afk2,α,q

≤ kx^−aχe_rk∞,qkx^afk_2,α,q

≤r^−akx^afk_2,α,q. So,

Z

E

|F_D^α,q(f)(λ)|²|λ|^2α+1d_qλ ¹₂

≤ 2ec_α,q

p[2(α+ 1−a)]_q|E|α¹²r^α+1−a+r^−a

!

kx^afk_2,α,q.

The desired result is obtained by minimizing the right hand side of the previous inequality over

r >0.

Corollary 5.2. Forα≥ −¹₂, 0< a < α+ 1andb >0, we have for allf ∈L²_α,q(Rq), (5.2) kfk^(a+b)_2,α,q ≤K_a,b,αkx^afk^b_2,α,qkλ^bF_D^α,q(f)k^a_2,α,q,

with

K_a,b,α =

"

b a

_a+b^a +a

b _a+b^b

#^a+b₂

(2K_a,α)^b2 q−(2α+1)(a+b)

([2α+ 2]_q)^2(α+1)ab whereKa,αis the constant given in Theorem 5.1.

Proof. Forr >0, we putE_r =]−r, r[∩RqandEe_rthe supplementary ofE_rinRq. We haveE_ris a bounded subset ofR^q and|E_r|_α ≤2_[2α+2]^r2α+2

q.Then the Plancherel formula and the previous theorem lead to

kfk²_2,α,q =kF_D^α,q(f)k²_2,α,q

= Z

Er

|F_D^α,q(f)|²(λ)|λ|^2α+1d_qλ+ Z

Eer

|F_D^α,q(f)|²(λ)|λ|^2α+1d_qλ

≤2K_a,α|E_r|

a

αα+1kx^afk²_2,α,q+r^−2bkλ^bF_D^α,q(f)k²_2,α,q

≤2 K_a,α [2α+ 2]

a

qα+1

r^2akx^afk²_2,α,q+r^−2bkλ^bF_D^α,q(f)k²_2,α,q.

The desired result follows by minimizing the right expressions overr >0.

Theorem 5.3. For α ≥ −¹₂ and a > α+ 1, there exists a constant K_a,α,q⁰ such that for all bounded subsetsEof Rqand allf inL²_α,q(Rq), we have

(5.3)

Z

E

|F_D^α,q(f)(λ)|²|λ|^2α+1d_qλ≤K_a,α,q⁰ |E|_α kfk²⁽¹⁻

α+1 a )

2,α,q kx^afk²

α+1 a

2,α,q. The proof of this result needs the following lemmas.

Lemma 5.4. Supposea > α+ 1, then for allf ∈L²_α,q(Rq)such thatx^af ∈L²_α,q(Rq),

(5.4) kfk²_1,α,q ≤K₂

kfk²_2,α,q+kx^afk²_2,α,q , where

K₂ = 2(1−q)(q^2a, q^2a,−q^2α+2,−q^{2(a−α−1)};q^2a)∞

(q^2α+2, q^{2(a−α−1)},−q^2a,−1;q^2a)∞

.

Proof. From ([4, Example 1]) and Hölder’s inequality, we have

kfk²_1,α,q =

Z +∞

−∞

(1 +|x|^2a)¹²|f(x)|(1 +|x|^2a)⁻¹²|x|^2α+1d_qx 2

≤K₂

kfk²_2,α,q+kx^afk²_2,α,q , where

K₂ = 2 Z +∞

0

x^2α+1 1 +x^2ad_qx

= 2(1−q)(q^2a, q^2a,−q^2α+2,−q^{2(a−α−1)};q^2a)∞

(q^2α+2, q^{2(a−α−1)},−q^2a,−1;q^2a)∞

.

Lemma 5.5. Supposea > α+ 1, then for allf ∈L²_α,q(R^q)such thatx^af ∈L²_α,q(R^q), we have (5.5) kfk_1,α,q ≤K3kfk⁽¹⁻_2,α,q^{α+1a )}kx^afk_2,α,q^α+1a ,

where

K₃ =K₃(a, α, q) =

"

q^2(α+1)( a

α+ 1 −1) ^α+1_a

q^−2(α+1)(1 + α+ 1

a−α−1)K₂

#¹₂ .

Proof. Fors∈Rq,define the functionf_sbyf_s(x) = f(sx), x∈Rq. We have

kf_sk_1,α,q =s^−2(α+1)kfk_1,α,q, kx^af_sk²_2,α,q =s^−2(α+a+1)kx^afk²_2,α,q. Replacement off byf_sin Lemma 5.4 gives:

kfk²_1,α,q ≤K₂

s^2(α+1)kfk²_2,α,q+s^2(α−a+1)kx^afk²_2,α,q .

Now, for allr > 0,putα(r) = ^Log(r)_Log(q) −E

Log(r) Log(q)

.We haves = _qα(r)^r ∈ Rq andr ≤ s < ^r_q. Then, for allr >0,

kfk²_1,α,q ≤K₂

"

r q

2(α+1)

kfk²_2,α,q +r^2(α−a+1)kx^afk²_2,α,q

# .

The right hand side of this inequality is minimized by choosing r=

a α+ 1 −1

_2a¹

q^α+1a kfk⁻

1 a

2,α,q kx^afk

1 a

2,α,q.

When this is done we obtain the result.

Proof of Theorem 5.3. LetE be a bounded subset of R^q. When the right hand side of the in- equality is finite, Lemma 5.4 implies thatf ∈ L¹_q(R^q), so, F_D^α,q(f)is defined and bounded on R^q. Using Lemma 5.5, the relation (3.5) and the fact that

Z

E

|F_D^α,q(f)(λ)|²|λ|^2α+1d_qλ≤|E |_α kF_D^α,q(f)k²_∞,q, we obtain the result with

K_a,α,q⁰ = 4(1 +q)^−2α Γ²_q2(α+ 1)(q;q)²_∞K₃²

= 8(1−q)(1 +q)^−2α Γ²_q2(α+ 1)(q;q)²_∞

q^2(α+1)

a α+ 1 −1

^α+1_a

q^−2(α+1)

1 + α+ 1 a−α−1

× (q^2a, q^2a,−q^2α+2,−q^{2(a−α−1)};q^2a)∞

(q^2α+2, q^{2(a−α−1)},−q^2a,−1;q^2a)_∞ .

Corollary 5.6. Forα≥ −¹₂,a > α+ 1andb >0, we have for allf ∈L²_α,q(Rq),

(5.6) kfk^(a+b)_2,α,q ≤K_a,b,α⁰ kx^afk^b_2,α,qkλ^bF_D^α,q(f)k^a_2,α,q, with

K_a,b,α⁰ =

K_a,α,q⁰ [2α+ 2]_q

_2α+2^ab

q^−(4α+2)

"

b α+ 1

_α+b+1^α+1 +

b α+ 1

− ^b

α+b+1

#!

a(α+b+1) 2(α+1)

,

whereK_a,α,q⁰ is the constant given in the previous theorem.

Proof. The same techniques as in Corollary 5.2 give the result.

The following result gives a new Heisenberg-Weyl type inequality for theq-Dunkl transform.

Theorem 5.7. Forα≥ −¹₂, α6= 0, we have for allf ∈L²_α,q(R^q), (5.7) kfk²_2,α,q ≤K_αkxfk_2,α,qkλF_D^α,q(f)k_2,α,q, with

K_α =

( K1,1,α if α >0 K_1,1,α⁰ if α <0.

Proof. The result follows from Corollaries 5.2 and 5.6, by takinga=b= 1.

Remark 1. Note that Theorem 4.1 and Theorem 5.7 are both Heisenberg-Weyl type inequalities for the q-Dunkl transform. However, the constants in the two theorems are different, the first one seems to be more optimal. Moreover, Theorem 4.1 is true for everyα >−¹₂ and uses both f andf₀, in contrast to Theorem 5.7, which is true only forα6= 0and uses onlyf.

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