Lp1 - Lp2 version of miyachi's theorem for the q-dunkl transform on the real line

L

− L

Version of Miyachi’s Theorem for the q -Dunkl Transform on the Real Line

Radouan Daher

D´epartement de Math´ematiques et d I’nformatique, Facult´e des Siences, Universit´e Hassan II, BP. 5366, Maarif, Casablanca, Morocco.

ra daher@yahoo.fr

Abdelmajid Khadari

D´epartement de Math´ematiques et d’Informatique, Facult´e des Siences, Universit´e Hassan II, BP. 5366, Maarif, Casablanca, Morocco.

khadariabd@gmail.com

Abstract

This paper deals with Miyachi’s theorem for theq-Dunkl Transform introduced and studied in [3].

Mathematics Subject Classification: 33D15, 33D60, 39A12, 44A20 Keywords: Hardy uncertainty principle, Miyachi’s theorem, q-Dunkl Trans- form, q-Fourier analysis

1 Introduction

In harmonic analysis, the uncertainty principle states that a function and its Fourier transform can not simultaneously decrease very quickyl. In the literature, this fact is in general given by the way of some inequalities involving a functionf and its Fourier transform ˆf. One of the famous formulations of the uncertainty principle is stated by the so-called Hardy’s theorem (see [8, 12, 11]), which assert that if

sup

x∈^Ê|e^ax2f(x)|<∞, sup

λ∈^Ê|e^bλ2f(λ)ˆ |<∞ and ab > 1 4, then f ≡0.

In [4], M. G. Cowling and J. F. Price obtained and L^p version of Hardy’s theorem by showing that for p, n∈[1,+∞] with at least one of them is finite, if e^ax2f(x)_p <+∞, e^aλ2fˆ(λ)_n<+∞ and ab > ¹₄, then f = 0.

Generalization of these results in both classical and quantum analysis have been revealed (see [2, 6, 9, 5, 17, 18]) and many versions of Hardy uncertainty principles were obtained for several generalized Fourier transforms.

In [3], Bettaibi et al. introduced and studied a q-analogue of the classical Bessel-Dunkl transform (q-Dunkl transform). In particular they provided, for this transform, a Plancherel formula and proved an invertion theorem. In this paper, we state an L^p1 +L^p2 version of Miyachi’s Theorem for the q-Dunkl transform F_D^α,q(f).

This paper is oganized as follows: in Section 2, we present some preliminaries notions and notations useful in the sequel. In Section 3, we recall and state some results and properties from the theory of the q-Dunkl operator and the q-Dunkl transform (see [3]). Section 4 is devoted to give anL^p1+L^p2 version of Miyachi’s Theorem for the q-Dunkl transform F_D^α,q. Section 5 by the applica- tion of the of Miyachi’s Theorem proved in section 4, we deduce the Miyachi’s theorem for the q²-analogue Fourier transform and a corollay for the q-Bessel transform.

2 Notations and preliminaries

Throughtout this paper, we assume q∈]0,1[, we refer to the general reference [10] for the definitions, notations and properties of the q-shifted factorials and the q-hypergeometric functions.

We write Rq={±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. (1) The q²-analogue differential operator is (see [15, 16])

∂_q(f)(z) =

f(q⁻¹z)+f(−q⁻¹z)−f(qz)+f(−qz)+2f(−z)

2(1−q)z if z = 0

lim_x→0∂_q(f)(x) (in Rq) if z = 0 (2) Remark that iff is differentiable atz, then lim

q→1∂_q(f)(x) =f(z). A repeated application of the q²-analogue diffential 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 function f onRq,∂_qf(z) = ^fe(q−1_(1−q)z^z−fe(z))+^fo(z−f_(1−q)z^o(qz)). 2. For two functions f and g on R_q, we have

• if f is even and g 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);

• if f and g are even,

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

Here, for a function f defined on Rq, f_e and f_o are its even and odd parts respectively.

Proof: The proof is straightforward (we refer to [3])

The operator ∂_q induces a q-analogue of the classical exponential function (see [15, 16])

e(z;q²) = +∞

n=0

a_n zⁿ

[n]_q!, with a_2n=a_2n+1 =qⁿ⁽ⁿ⁺¹⁾. (3) The q-jackson integrals are defined by (see [13])

_a

0

f(x)d_qx= (1−q)a +∞

n=0

qⁿf(aqⁿ), _b

a

f(x)d_qx= _b

0

f(x)d_qx− _a

0

f(x)d_qx,

_+∞

0

f(x)d_q = (1−q) +∞

n=−∞

qⁿf(qⁿ),

and _+∞

−∞

f(x)d_q = (1−q) +∞

n=−∞

qⁿf(qⁿ) + (1−q) +∞

n=−∞

qⁿf(−qⁿ) provided the sums converge absolutely.

The q-gamma function is given by (see [13]) Γq(x) = (q;q)_∞

(q^x, q)_∞(1−q)^1−x, x= 0,−1,−2, ...

In what follows, we will need the following sets and spaces:

• L^∞_q (R_q) =

f :f_∞,q = sup

x∈^Êq

|f(x)|<+∞

.

• L^p_α,q(Rq) =

f :fp,α,q =_+∞

−∞ |f(x)|^p|x|^2α+1d_qx

1p

|<+∞

.

3 The q -Dunkl operator and the q -Dunkl trans- form

In this section we collect some basic properties of the q-Dunkl operator and the q-Dunkl transform introduced in [3], useful in the sequel.

Forα ≥ −¹₂, the q-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_o →f_e+q^2α+1f_o. It was shown in [3] that for each λ∈C, the function

ψ_λ^α,q :→j_α(λx;q²) + iλx

[2α+ 2]_qj_α+1(λx;q²) (4) is the unique solution of the q-diffentiel equation:

Λ_α,q(f) = iλf f(0) = 1,

wherej_α(.;q²) is the normalized third Jackson’sq-Bessel function given by

j_α(x;q²) = +∞

n=0

(−1)ⁿ qⁿ⁽ⁿ⁺¹⁾

(q²;q²)_n(q^2(α+1);q²)_n((1−q)x)²ⁿ. (5) The function ψ_λ^α,q, has an unique extension to C× C and we have the following properties.

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

• For α=−¹₂,ψ_aλ^α,q(x) = e(iλx;q²) and for α >−¹₂, ψ_aλ^α,q has the following q-integral representation of Mehler type

ψ_aλ^α,q(x) = (1 +q)Γ_q2(α+ 1) 2Γ_q2(¹₂)Γ_q2(α+ ¹₂)

₁

−1

(t²q²;q²)_∞

(t²q^2α+1;q²)_∞(1 +t)e(iλxt;q²)d_qt.

(6)

• For all x, λ∈R_q,

|ψ_λ^α,q(x)|< 4

(q;q)_∞. (7)

The q-Dunkl transform F_D^α,q is defined onL¹_α,q(R_q) (see [3]) by F_D^α,q(f)(λ) = c_α,q

2

_+∞

−∞

f(x)ψ_−λ^α,q(x)|x|^2α+1d_qx, where

c_α,q

2 = (1 +q)^−α Γ_q2(α+ 1). It satisfies the following properties:

• Forα =−¹₂,F_D^α,q is the q²-analogue Fourier transform ˆf(.;q²) given (see [15, 16])

f(λ;ˆ q²) = (1 +q)¹² 2Γ_q2(¹₂)

_+∞

−∞

f(x)e(−iλx;q²)d_qx.

• On the even functions space, F_D^α,q coincides with theq-Bessel transform given by (see [3])

Fα,q(f)(λ) =c_{α,q +∞}

0

f(x)j_α(λx;q²)x^2α+1d_qx.

• For all f ∈L¹_α,q(R_q), we have:

F_D^α,q_∞,q ≤ 2c_α,q

(q;q)_∞f_1,α,q. (8)

• the q-Dunkl transform F_D^α,q is an isomorphism from L²_α,q(R_q) onto itself and satisfies the following Plancherel formula and the inversion theorem:

∀f ∈L²_α,q(Rq),F_D^α,q(f)2,α,q =f2,α,q (9) and

(F_D^α,q)⁻¹(f)(λ) =F_D^α,q(f)(−λ), λ∈C.

4 L

− L

-version of Miyachi’s Theorem for the q -Dunkl Transform on the Real Line

In this section, we shall state an L^p1 −L^p2-version of Miyachi’s Theorem for the q-Dunkl TransformF_D^α,q. We begin by the tow following lemmas. The first is a deduction from the Mehler type relation (6) and ([2], Propostion 1) and the second proved by A. Miyachi in [14] (Proposition 4) .

Lemma 4.1 for all λ∈C and all x∈R, we have

|ψ_λ^α,q(x)|<2e^k|xλ|, with k= 1 +√q.

Lemma 4.2 Suppose f is an entire function on C and suppose there exist constants A, B >0 such that:

|f(z)| ≤Ae^{B( (z))2} for all z ∈C.

Also suppose _+∞

−∞

log⁺|f(t)|dt <+∞.

Where log⁺x = logx if x > 1 and log⁺x = 0 if x ≤ 1 then f is a constant function.

Lemma 4.3 Let h be an entire function on C and α≥ −¹₂, such that:

∀z ∈C,|h(z)| ≤Ae^{B( (z))2}, for some constantsA, B >0 .

and

_+∞

−∞

log⁺|h(t)||t|^2α+1dt <+∞. Then h is a constant function.

Proof: we have _+∞

−∞

log⁺|h(t)|dt= ₊₁

−1

log⁺|h(t)|dt+

|t|>1

log⁺|h(t)|dt (10)

and

|t|>1

log⁺|h(t)|dt ≤

|t|>1

log⁺|h(t)||t|^2α+1dt <+∞ (11) h is entire, particularly is continuous, then ₊₁

−1 log⁺|h(t)|dt < +∞, we deduce from (10) and (11) that_+∞

−∞ log⁺|h(t)|dt <+∞ the Lemma 4.2 finishes the proof.

Theorem 4.4 Let a > 0, p₁, p₂ ∈ [1,+∞] and f be a function defined on Rq such that

e^ax2f ∈L^p_α,q¹ (Rq) +L^p_α,q² (Rq) (12) Then F_D^α,q(f) is entire on C and for all b∈]0, a[, we have:

∀z ∈C,|F_D^α,q(f)(z)| ≤Ce^k

2|z|2

4b (13)

for some positive constant C.

Proof: Since |ψ_z^α,q(x)| ≤ 2e^1+√q|xz|, for all x ∈ C and t ∈ R, then from the hypothesis, the H¨older’s inequality and the analyticity theorem one deduce that F_D^α,q(f) is entire on C, (12) implies that there exists u₁, u₂ ∈ L^p_α,q^j (Rq) such that e^ax2f(x) =u₁(x) +u₂(x) and for all z ∈C, we have for j = 1,2

|F_D^α,q(f)(z)| ≤ c_α,q 2

_−∞

+∞ |ψ^α,q_z (x)||f(x)||x|^2α+1d_qx

≤ c_{α,q +∞}

−∞

e^{k|z||x|−ax2}e^ax2|f(x)||x|^2α+1d_qx

≤ c_α,q l

j=1

_−∞

+∞

e^{k|z||x|−ax2}|u_j(x)||x|^2α+1d_qx

≤ c_α,q l

j=1

_−∞

+∞

e^nj(k|z||x|−ax²)|x|^2α+1d_qx _nj¹

u_jpj,α,q, where n₁, n₂ is the real satisfying _p¹

1 +_n¹

1 = _p¹

2 +_n¹

1 = 1.

Now, for b∈]0, a[, we have _+∞

−∞

e^nj(k|z||x|−ax²)|x|^2α+1d_qx _nj¹

=

_+∞

−∞

e^nj(k|z||x|−bx²)e^{−nj(a−b)x2}|x|^2α+1d_qx _nj¹

≤

sup

x∈[0,+∞]

e^{nj(k|z|x−bx2) 1}

nj _+∞

−∞

e^{−nj(a−b)x2}|x|^2α+1d_qx ¹

nj

= C_je^k4b2|z|2, with C_j =_+∞

−∞ e^{−nj(a−b)x2}|x|^2α+1d_qx

nj1 . Then we have |F_D^α,q(f)(z)| ≤Ce^k

2|z|2

4b with C =c_α,q(C₁+C₂).

Theorem 4.5 Let a, b >0such that ab < ^k₄² where k =√

q+ 1 andλ∈C, for a < t < ^k_4b² we put

h^α,q_t (x) = c_α,q 2

_+∞

−∞

e^−k4t2λ2ψ_λ^α,q(x)|λ|^2α+1d_qλ then we have :

• There exists D >0 and E_t>0 depend on t such that:

|e^ax2h^α,q_t (x)| ≤E_t e^−Dx2

• For p∈[1,+∞]

e^ax2h^α,q_t (x)∈L^p_α,q(Rq)

Proof: -We have by the use of Lemma (4.1) and we not lose generality if we assume that x≥0, for a very sall ε >0 :

|h^α,q_t (x)| ≤ c_α,q 2

_+∞

−∞

e^−k4t2λ2|ψ_λ^α,q(x)||λ|^2α+1d_qλ

≤ c_{α,q +∞}

−∞

e^−k4t2λ2e^k|λx||λ|^2α+1d_qλ

≤ 2c_{α,q +∞}

0

e^{(kλx−k4t2λ2)}λ^2α+1d_qλ

≤ 2c_α,q sup

λ∈[0,+∞]

e^{(kλx+(ε−k4t2)λ2)}

+∞

0

e^−ελ2λ^2α+1d_qλ

≤ 2c_α,q e

3k2 4(ε−k2

4t)x²

_+∞

0

e^−ελ2λ^2α+1d_qλ

we put M =_+∞

0 e^−ελ2λ^2α+1d_qλ then we have

|e^ax2h^α,q_t (x)| ≤2M c_α,q e ^a+

3k2 4(ε−k2

4t)

x²

for a very small ε >0 we have a+ ^3k2

4(ε−^k_4t²) <0 becausea < t, which finish the proof of the first statement.

-We have |e^ax2h^α,q_t (x)| ≤E_t e^−Dx2 for D, E_t>0 then we conclude that e^ax2h^α,q_t (x)∈L^p_α,q(R_q)

for p∈[1,+∞]

Theorem 4.6 Let a, b > 0, p₁, p₂ ∈ [1,+∞] where i = 1, ..., l and f be a function defined on Rq such that:

e^ax2f ∈L^p_α,q¹ (R_q) +L^p_α,q¹ (R_q) (14) and

_+∞

−∞

log⁺ |F_D^α,q(f)(x)e^bx2|

λ |x|^2α+1dx <+∞ (15) for some λ >0, then

• if ab > ^k₄² then f = 0 on Rq.

• if ab= ^k₄² then f =N h^α,q_a where N a constant and |N| ≤λ.

• if ab < ^k₄² then there exists many functions.

Proof: Leta, b >0 and h be the function on C defined by:

h(z) =e^k4a2z2F_D^α,q(f)(z) (16)

• Case 1: (ab > ^k₄²)

from Theorem 4.4 we deduce that the functionhis entire, on other hand, we note that for a very small ε >0 we have a−ε∈]0, a[:

∀z ∈C, |h(z)| ≤Ce ^k

2

2(a−ε)( (z))² (17)

and _+∞

−∞

log⁺|h(x)||x|^2α+1dx =

_+∞

−∞

log⁺|e^k4a2x2F_D^α,q(f)(x)||x|^2α+1dx

=

_+∞

−∞

log⁺

λe^(k4a2−b)x2e^bx2|F_D^α,q(f)(x)| λ

|x|^2α+1dx

≤ _+∞

−∞

log⁺λe^(k4a2−b)x2|x|^2α+1dx+ _+∞

−∞

log⁺ e^bx2|F_D^α,q(f)(x)|

λ |x|^2α+1dx because log⁺(ρ)≤log⁺() +ρ for all , ρ >0, (15) implies that:

_+∞

−∞

log⁺|h(x)||x|^2α+1dx <+∞. (18) Then it follows from (17) and (18) that h satisfies the assumptions in Lemma (4.3), and thus, his a constant and

F_D^α,q(f)(x) = Ce^−k4a2x2.

Since ab > ^k₄², (15) holds whenever C = 0 and the Plancherel formula implies that f = 0 almost everywhere.

• Case 2: (ab= ^k₄²)

As in the previous case, we have the relation (17) and:

_+∞

−∞

log⁺|h(x)

λ ||x|^2α+1dx =

_+∞

−∞

log⁺|e^bx2F_D^α,q(f)(x)

λ ||x|^2α+1dx

< +∞ then h is a constant and we have

F_D^α,q(f)(x) =Ce^−k4a2x2 =Ce^−bx2. The relation (15) holds whenever |N| ≤λ

• Case 3: (ab < ^k₄²)

leta < t < ^k_4b² by Theorem (4.5) we conclude that e^ax2h^α,q_t (x)∈L^p_α,q^j (R_q) for p_j ∈[1,+∞] and j = 1,2. Then h^α,q_t satisfy (14) , on the other hand, we have :

F_D^α,q(h^α,q_t )(x) =e^−k4t2x2 then h^α,q_t satisfy (15) because t < ^k_4b².

5 Applications

By the use of the theorem 4.6 we can easily deduce the L^p1 +L^p2-version of Miyachi’s Theorem for the q²-analogue Fourier transform on the Real Line (see theorem 5.1) and L^p1+L^p2-version of Miyachi’s Theorem for the q-Bessel transform on the Real Line for the even functions (see corollary 5.2)

Theorem 5.1 Let a, b >0, p₁, p₂ ∈[1,+∞]and f be a function defined on R_q such that:

e^ax2f ∈L^p_α,q¹ (R_q) +L^p_α,q² (R_q) (19) and

_+∞

−∞

log⁺|f(x;ˆ q²)e^bx2|

λ |x|^2α+1dx <+∞ (20) for some λ >0, then

• if ab > ^k₄² then f = 0 on R_q.

• if ab= ^k₄² then f =N h^α,q_a where N a constant and |N| ≤λ.

• if ab < ^k₄² then there exists many functions.

Proof: We conclude from Theorem (4.6) forα =−¹₂. Corollary 5.2 Let a, b > 0, p_i ∈ [1,+∞] where i = 1, ..., l and f be an even function defined on Rq such that:

e^ax2f ∈L^p_α,q¹ (Rq) +L^p_α,q² (Rq) (21)

and

_+∞

−∞

log⁺|F_α,q(f)(x)e^bx2|

λ |x|^2α+1dx <+∞ (22) for some λ >0, then

• if ab > ^k₄² then f = 0 on Rq.

• if ab= ^k₄² then f =N h^α,q_a where N a constant and |N| ≤λ.

• if ab < ^k₄² then there exists many functions.

Proof: For the even function we have F_α,q(f) = F_α,q(f), then we conclude

from Theorem (4.6).

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Received: April, 2012