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Integral Transforms and Special Functions
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L
p–L
q-version of Morgan's theorem for the Jacobi–Dunkl transform
Radouan Daher a & Sidi Lafdal Hamad a
a Faculty of Sciences of Ain Chock, University Hassan II, Casablanca, Morocco
Version of record first published: 28 Oct 2008.
To cite this article: Radouan Daher & Sidi Lafdal Hamad (2008): L p –L q -version of Morgan's theorem for the Jacobi–Dunkl transform, Integral Transforms and Special Functions, 19:3, 165-169 To link to this article: http://dx.doi.org/10.1080/10652460701699643
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Vol. 19, No. 3, March 2008, 165–169
L
p–L
q-version of Morgan’s theorem for the Jacobi–Dunkl transform
Radouan Daher* and Sidi Lafdal Hamad
Faculty of Sciences of Ain Chock, University Hassan II, Casablanca, Morocco (Received 01 June 2007)
In this paper, we give anLp–Lq-version of Morgan’s theorem for the Jacobi–Dunkl transformFJDonR. More precisely, we prove that for all 1≤p, q≤ +∞, such that 1/p+1/q=1, α >2,β=α/(α−1), a >0, andb >0, then for all measurable functions onR, the conditions ea|x|αf ∈Lpα,β(R), eb|y|βFJD∈ Lqα,β(R), and(aα)1/α(bβ)1/β> (sin(π/2(β−1)))1/βimplyf=0, whereLpα,β(R)is the Lebesgue space associated with the Jacobi–Dunkl transform.
Keywords: Jacobi–Dunkl transform; Morgan’s theorem
1. Introduction
A famous theorem of Morgan (see [4]) asserts that for allα >2,β =α/(α−1),a >0, andb >0 and for all measurable functionsf onR, the conditions ea|x|αf ∈Lp(R)and eb|y|βf∈Lq(R) implyf =0, if and only if(aα)1/α(bβ)1/β> (sin((π/2)(β−1)))1/β, wherefis the classical Fourier transform off onR.
In this paper, we study an analogue ofLp–Lq-version of the Morgan’s theorem for the Jacobi–
Dunkl transformFJDonR.
The contents of the paper are as follows: in Section 2, we recall some basic facts about the Jacobi–Dunkl transform; Section 3 gives a lemma of complex variable theory, which will be used in the sequel; and Section 4 is devoted to theLp–Lq-version of Morgan’s theorem forFJD.
2. Harmonic analysis associated with the Jacobi–Dunkl operator
In this section, we collect relevant material from the harmonic analysis associated with the Jacobi–
Dunkl operator, which was developed recently in ref. [2, 3].
*Corresponding author. Email: ra_daher@yahoo.fr
ISSN 1065-2469 print/ISSN 1476-8291 online
© 2008 Taylor & Francis
DOI: 10.1080/10652460701699643 http://www.informaworld.com
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166 R. Daher and S.L. Hamad
2.1. Jacobi–Dunkl kernel
From now on, letα, β ≥ −1/2;andα= −1/2. We consider the differential-difference operator α,βonRgiven by:
α,βf (x)=df (x)
dx +[(2α+1)coth(x)+(2β+1)tanh(x)]f (x)−f (−x)
2 . (1)
The Jacobi–Dunkl kernelλ(α,β) is the uniqueC∞-solution on Rof the differential-difference equation:
α,βu= −iλu, λ∈C,
u(0)=1. (2)
It has the Laplace integral representation:
∀λ∈C, ∀x ∈R {0}, λ(α,β)(x)= |x|
−|x|K(x, y)e−iλydy, (3) whereK(x, .)is a positive function onR, continuous on(−|x|,|x|), and satisfies:
∀x∈R {0},
RK(x, y)dy =1. (4)
The functionλ(α,β)satisfies the following elementary estimate.
LEMMA1 For alln∈N,x∈R,andλ∈C,we have dn
dλnλ(α,β)(x)
≤ |x|ne|Im(λ)||x|. (5)
In particular forn=0:
λ(α,β)(x)≤e|Im(λ)||x|
Proof See [2, Proposition 1.3.5].
2.2. Jacobi–Dunkl transform
We denote byD(R)the space of compactly supportedC∞-functions onR. DEFINITION2 The Jacobi–Dunkl transform of a functionf ∈D(R)is defined by
∀λ∈C, FJD(λ)=
R
f (x)λ(α,β)(x)α,β(x)dx, (6) where
α,β(x)=(2 sinh(x))2α+1(2 cosh(x))2β+1. (7) Notation 3 We denote byHa(C),a >0, the space of entire functions onC, rapidly decreasing of exponential type, that isg∈Ha(C)if and only ifgis entire onCand, for alln∈N,
Qn(g)=sup
λ∈C(1+ |λ|2)n|g(λ)|ea|Im(λ)|<∞.
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The essential properties of this Jacobi–Dunkl transform is given by the following theorem, for its proof see [2 Theorem 1.5.3].
THEOREM4 (Paley–Wiener theorem) The Jacobi–Dunkl transformFJDis a topological isomor- phism fromD(R)ontoH (C), whereH (C)= ∪a>0Ha(C).
3. Phragmen–Lindelöf type result
LEMMA5 Suppose thatρ∈(1,2), q∈ [1,+∞], σ >0,andB > σsin(π/2)(ρ−1). Ifgis an entire function onCsatisfying the conditions:
|g(x+iy)| ≤const eσ|y|ρ, for anyx, y ∈R, (8) and
eB|x|ρg
R∈Lq(R), (9)
theng=0.
Proof See [1].
4. Lp–Lq-version of Morgan’s theorem forFJD
LEMMA6 Letp∈ [1,+∞],α >2, andfbe a measurable function onRsuch that for alla >0, ea|x|αf ∈Lpα,β(R),
whereLpα,β(R)is the space of measurable functionsf onRsuch that f p,α,β=
R|f (x)|pα,β(x)dx 1/p
<∞.
Then the function defined onCby FJD(λ)=
Rf (x)λ(α,β)(x)α,β(x)dx
is well defined and entire onC.
Proof By using inequality (5) and the theorem of differentiation under the integral sign.
THEOREM7 Let1≤p, q≤ ∞, such that1/p+1/q =1,α >2,β =α/(α−1),a >0, and b >0, then for all measurable functionsf onR, the conditions
ea|x|αf ∈Lpα,β(R) and eb|λ|βFJD(λ)∈Lqα,β(R) implyf =0, if
(aα)1/α(bβ)1/β>
sin
π 2(β−1)
1/β
. (10)
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168 R. Daher and S.L. Hamad
Proof Letf be a measurable function onRsuch that
ea|x|αf ∈Lpα,β(R), (11)
and
eb|λ|βFJD(f )∈Lqα,β(R).
We want to prove that the Jacobi–Dunkl transform satisfies conditions (8) and (9) of Lemma 5, and hence we deduce thatf =0 almost everywhere.
By Lemma 6, the function
FJD(λ)=
Rf (x)λ(α,β)(x)α,β(x)dx is well defined, entire onC, and satisfies the condition
|FJD(f )(λ)| ≤
R|f (x)||λ(α,β)(x)|α,β(x)dx
≤
R|f (x)|e|Im(λ)||x|α,β(x)dx, for anyλ∈C, so by Hölder’s inequality,
≤
R
|f (x)|ea|x|αp
α,β(x)dx 1/p
R
e−a|x|αe|Im(λ)||x|q
α,β(x)dx 1/q
≤C
Req|Im(λ)||x|−aq|x|αα,β(x)dx 1/q
,
where 1/p+1/q =1. Let C ∈I =
(bβ)−1/β
sinπ
2(β−1) 1/β, (aα)1/α
.
Applying the convex inequality
|ty| ≤ 1
α
|t|α+ 1
β
|y|β
to the positive numbersC|t|and|y|/C, we obtain
|ty| ≤Cα
α |t|α+ 1 βCβ |y|β, and thus
Rexp(q|Im(λ)| |x| −aq|x|α)α,β(x)dx ≤exp
q|Im(λ)|β βCβ
×
Rexp−q
a−Cα α
|x|αα,β(x)dx.
SinceC∈I, it follows thata > Cα/α, and thus the integral
Re−q(a−(Cα/α))|x|αα,β(x)dx
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is finite. Moreover,
|FJD(f )(λ)| ≤Const e|Im(μ)|β/βCβ,for anyλ∈C, (12) condition (11) and inequality (12) imply that the function
g(z)=FJD(f )(z)
satisfies assumptions (8) and (9) of Lemma 5 with ρ=β, σ = 1
βCβ, and B =b.
The conditionC ∈I implies the inequality b > 1
βCβsinπ
2(β−1),
which givesFJD(f )=0 by Lemma 5, thenf =0 by Theorem 4.
References
[1] S. Ben Farah and K. Mokni,Uncertainty principle and theLp–Lq-version of Morgan’s theorem on some groups, Russian J. Math. Phys. 10(3) (2003), pp. 1–16.
[2] F. Chouchane, M. Mili and K. Trimèche,Positivity of the intertwining operator and harmonic analysis associated with the Jacobi-Dunkl operator onR, J. Anal. Appl. 1(4) (2003), pp. 387–412.
[3] ———,An Lp version of Hardy’s theorem for the Jacobi–Dunkl transform, Integral Transforms Spec. Funct. 15(3) (2004), pp. 225–237.
[4] G.W. Morgan,A note on Fourier transforms, J. London Math. Soc. 9 (1934), pp. 178–192.
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