Lp1-Lp2 Version of Miyachi's Theorem for q-Bessel Transform on the Real Line

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Lp1-Lp2 Version of Miyachi's Theorem for q- Bessel Transform on the Real Line

Article in International Journal of Mathematical Analysis · January 2013

DOI: 10.12988/ijma.2013.212329

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http://dx.doi.org/10.12988/ijma.2013.212329

L

− L

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

Abdelmajid Khadari

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

khadariabd@gmail.com

Radouan Daher

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

ra daher@yahoo.fr

Copyright c 2013 Abdelmajid Khadari and Radouan Daher. This is an open access article distributed under the Creative Commons Attribution License, which permits unre- stricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper deals with Miyachi’s theorem for theq-Bessel transform on the Real Line in [3].

Mathematics Subject Classification: 33D15, 33D60, 39A12, 44A20 Keywords: Hardy uncertainty principle, Miyachi’s theorem,q-Bessel trans- form

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

2656 Abdelmajid Khadari and Radouan Daher uncertainty principle is stated by the so-called Hardy’s theorem (see [8, 11, 10]), 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 [5], 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, 7, 9, 6, 12, 13]) 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 Plancheral formula and proved an invertion theorem, in [4]

R. Daher and A. Khadari provedL^p₁−L^p₂ Version of Miyachi’s Theorem for the q-Dunkl Transform on the Real Line and concluded the Miyachi’s Teorem for q-Bessel transform on the Real Line just for the even functions. In this paper, we state and prove an L^p1−L^p2 version of Miyachi’s Theorem for theq-Bessel transform Fα,q(f), for more details about theq-Bessel transform see [3].

2 L

− L

-version of Miyachi’s Theorem for the q-Bessel transform on the Real Line

In this section, we shall state an L^p1 −L^p2-version of Miyachi’s Theorem for the q-Bessel transform Fα,q(f). We begin by the tow following lemmas. The first is a deduction from the ([4], Lemma 4.1) and the second see ([4], Lemma 4.3).

Lemma 2.1 for all λ∈C and all x∈R, we have

|j_α(λx, q²)|< e^k|xλ|, with k= 1 +√

q.

Proof: we have j_α(λx;q²) = ¹₂(ψ_λ^α,q(x) +ψ_−λ^α,q(x)) by the use of ([4], Lemma 4.1) then we have |j_α(λx;q²)| ≤ ¹₂(|ψ_λ^α,q(x)|+|ψ_−λ^α,q(x)|)≤e^k|xλ|

Lemma 2.2 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.

Theorem 2.3 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) (1) Then Fα,q(f) is entire onC and for all b∈]0, a[, we have:

∀z∈C,|Fα,q(f)(z)| ≤Ce^k

2|z|2

4b (2)

for some positive constant C.

Proof: Since |j_α(z;q²)(x)| ≤e^(1+√q)|xz|, for all z ∈C and x ∈R, then from the hypothesis, the H¨older’s inequality and the analycity theorem one deduce that Fα,q(f) is entire onC, (1) implies that there existsu₁, u₂ ∈L^p_α,q^j (Rq) such that e^ax2f(x) =u₁(x) +u₂(x) and for all z ∈C, we have forj = 1,2

|Fα,q(f)(z)| ≤ c_{α,q +∞}

0 |j_α(λ;q²)(x)||f(x)||x|^2α+1d_qx

≤ c_{α,q +∞}

0

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

≤ c_α,q l

j=1

_+∞

0

e^{k|z||x|−ax2}|u_j(x)||x|^2α+1d_qx

≤ c_α,q l

j=1

_+∞

0

e^nj(k|z||x|−ax²)|x|^2α+1d_qx ¹

nj u_jpj,α,q,

where n₁, n₂ is the real satisfying _p¹

1 +_n¹

1 = _p¹

2 +_n¹

2 = 1.

Now, for b∈]0, a[, we have

2658 Abdelmajid Khadari and Radouan Daher

_+∞

0

e^nj(k|z||x|−ax²)|x|^2α+1d_qx _nj¹

=

_+∞

0

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 _+∞

0

e^{−nj(a−b)x2}|x|^2α+1d_qx _nj¹

= C_je^k4b2|z|2,

with C_j = ₀^+∞e^{−nj(a−b)x2}|x|^2α+1d_qx ¹

nj. Then we have |Fα,q(f)(z)| ≤Ce^k

2|z|2

4b with C =c_α,q(C₁+C₂).

Theorem 2.4 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 +∞}

0

e^−k4t2λ2j_α(λx;q²)|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(R_q)

Proof: -We have by the use of Lemma (2.1) and we do not lose generality if we assume that x≥0, for a very sall ε >0 :

|h^α,q_t (x)| ≤ c_{α,q +∞}

0

e^−k4t2λ2|j_α(λx;q²)||λ|^2α+1d_qλ

≤ c_{α,q +∞}

0

e^−k4t2λ2e^k|λx||λ|^2α+1d_qλ

≤ c_{α,q +∞}

0

e^{(kλx−k4t2λ2)}λ^2α+1d_qλ

≤ c_α,q sup

λ∈[0,+∞]

e^{(kλx+(ε−k4t2)λ2)}

+∞

0

e^−ελ2λ^2α+1d_qλ

≤ c_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)| ≤M 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 2.5 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) (3) and

_+∞

−∞

log⁺|Fα,q(f)(x)e^bx2|

λ |x|^2α+1dx <+∞ (4) 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_α,q(f)(z) (5)

• Case 1: (ab > ^k₄²)

from Theorem 2.3 we deduce that the functionhis entire, on other hand, we note that for a very small ε >0 we have a−ε∈]0, a[:

2660 Abdelmajid Khadari and Radouan Daher

∀z ∈C, |h(z)| ≤Ce ^k

2

2(a−ε)( (z))² (6)

and _+∞

−∞

log⁺|h(x)||x|^2α+1dx =

_+∞

−∞

log⁺|e^k4a2x2F_α,q(f)(x)||x|^2α+1dx

=

_+∞

−∞

log⁺

λe^(k4a2−b)x2e^bx2|Fα,q(f)(x)| λ

|x|^2α+1dx

≤ _+∞

−∞

log⁺λe^(k4a2−b)x2|x|^2α+1dx+ _+∞

−∞

log⁺e^bx2|Fα,q(f)(x)|

λ |x|^2α+1dx because log⁺(ρ)≤log⁺() +ρ for all , ρ >0, (4) implies that:

_+∞

−∞

log⁺|h(x)||x|^2α+1dx <+∞. (7) Then it follows from (6) and (7) that h satisfies the assumptions in Lemma (2.2), and thus, his a constant and

F_α,q(f)(x) =Ce^−k4a2x2.

Since ab > ^k₄², (4) 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 (6) and:

_+∞

−∞

log⁺|h(x)

λ ||x|^2α+1dx =

_+∞

−∞

log⁺|e^bx2F_α,q(f)(x)

λ ||x|^2α+1dx

< +∞ then h is a constant and we have

Fα,q(f)(x) = Ce^−k4a2x2 =Ce^−bx2. The relation (4) holds whenever |N| ≤λ

• Case 3: (ab < ^k₄²)

leta < t < ^k_4b² by Theorem (2.4) we conclude that e^ax2h^α,q_t (x)∈L^p_α,q^j (R_q) for p_j ∈[1,+∞] and j = 1,2. Thenh^α,q_t satisfy (3) , on the other hand, we have :

Fα,q(h^α,q_t )(x) = e^−k4t2x2 then h^α,q_t satisfy (4) because t < ^k_4b².

References

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Sci. Res. J.,11(11)(2007), 590-602.

[3] N. Bettaib and R. H. Bettaieb, q-Analogue of the Dunkl transform on the real line, arXiv:0801.0069v1 [math. QA],(2007), to appear in Tamsui Oxford Journal of Matheamatical Sciences.

[4] R. Daher and A. Khadari, L^p1 −L^p2 version of Miyachi’s theorem for the q-Dunkl transform on the real line, Int. Journal of Math. Analysis, Vol.

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[9] L. Gallardo and K. trim`eche, Un analogue d’une theoerem de Hardy pour la transformation de Dunkl,C.R. Acad. Sci. Paris, t. 334, S´erie I, (2002) 84955.

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Received: December 5, 2012

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