Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 6, Issue 3, Article 84, 2005
LITTLEWOOD-PALEY g-FUNCTION IN THE DUNKL ANALYSIS ON Rd
FETHI SOLTANI
D EPARTMENT OF M ATHEMATICS
FACULTY OF S CIENCES OF T UNIS
U NIVERSITY OF EL-M ANAR T UNIS 2092 T UNIS , T UNISIA
Fethi.Soltani@fst.rnu.tn
Received 11 October, 2004; accepted 22 July, 2005
Communicated by S.S. Dragomir
A BSTRACT. We prove Lp -inequality for the Littlewood-Paley g-function in the Dunkl case on
Rd .
Key words and phrases: Dunkl operators, Generalized Poisson integral, g-function.
2000 Mathematics Subject Classification. 42B15, 42B25.
1. I NTRODUCTION
In the Euclidean case, the Littlewood-Paley g-function is given by
!
# 12
"Z
2
∞ ∂
u(x, t) + |∇x u(x, t)|2 t dt ,
g(f )(x) :=
∂t
0
x ∈ Rd ,
where u is the Poisson integral of f and ∇ is the usual gradient. The Lp -norm of this operator
is comparable with the Lp -norm of f for p ∈ ]1, ∞[ (see [19]). Next, this operator plays an
important role in questions related to multipliers, Sobolev spaces and Hardy spaces (see [19]).
Over the past twenty years considerable effort has been made to extend the Littlewood-Paley
g-function on generalized hypergroups [20, 1, 2], and complete Riemannian manifolds [4].
In this paper we consider the differential-difference operators Tj ; j = 1, . . . , d, on Rd introduced by Dunkl in [5] and aptly called Dunkl operators in the literature. These operators
extend the usual partial derivatives by additional reflection terms and give generalizations of
many multi-variable analytic structures like the exponential function, the Fourier transform, the
convolution product and the Poisson integral (see [12, 23, 16] and [13]).
During the last years, these operators have gained considerable interest in various fields of
mathematics and in certain parts of quantum mechanics; one expects that the results in this paper
will be useful when discussing the boundedness property of the Littlewood-Paley g-function in
ISSN (electronic): 1443-5756
c 2005 Victoria University. All rights reserved.
The author is very grateful to the referee for many comments on this paper.
183-04
2
F ETHI S OLTANI
the Dunkl analysis on Rd . Moreover they are naturally connected with certain Schrödinger
operators for Calogero-Sutherland-type quantum many body systems [3, 9].
The main purpose of this paper is to give the Lp -inequality for the Littlewood-Paley gfunction in the Dunkl case on Rd by using continuity properties of the Dunkl transform Fk ,
the Dunkl translation operators of radial functions and the generalized convolution product ∗k .
We will adapt to this case techniques Stein used in [18, 19].
The paper is organized as follows. In Section 2 we recall some basic harmonic analysis
results related to the Dunkl operators on Rd . In particular, we list some basic properties of the
Dunkl transform Fk and the generalized convolution product ∗k (see [8, 23, 15]).
In Section 3 we study the Littlewood-Paley g-function:
"Z
!
# 12
2
∞ ∂
uk (x, t) + |∇x uk (x, t)|2 t dt , x ∈ Rd ,
g(f )(x) :=
∂t
0
where uk (·, t) is the generalized Poisson integral of f .
We prove that g is Lp -boundedness for p ∈ ]1, 2].
Throughout the paper c denotes a positive constant whose value may vary from line to line.
2. T HE D UNKL A NALYSIS ON Rd
p
We consider Rd with the Euclidean inner product h·, ·i and norm kxk = hx, xi.
For α ∈ Rd \{0}, let σα be the reflection in the hyperplane Hα ⊂ Rd orthogonal to α:
2hα, xi
σα x := x −
α.
kαk2
A finite set R ⊂ Rd \{0} is called a root system, if R ∩ R, α = {−α, α} and σα R = R for
all α ∈ R. We assume that it is normalized by kαk2 = 2 for all α ∈ R.
For a root system R, the reflections σα , α ∈ R generate a finite group G ⊂ O(d), the
reflection group associatedS
with R. All reflections in G, correspond to suitable pairs of roots.
d
For a given β ∈ H := R
α∈R Hα , we fix the positive subsystem:
R+ := {α ∈ R / hα, βi > 0}.
Then for each α ∈ R either α ∈ R+ or −α ∈ R+ .
Let k : R → C be a multiplicity function on R (i.e. a function which is constant on the orbits
under the action of G). For brevity, we introduce the index:
X
γ = γ(k) :=
k(α).
α∈R+
Moreover, let wk denote the weight function:
Y
wk (x) :=
|hα, xi|2k(α) ,
x ∈ Rd ,
α∈R+
which is G-invariant and homogeneous of degree 2γ.
We introduce the Mehta-type constant ck , by
Z
−1
−kxk2
e
, where dµk (x) := wk (x)dx.
(2.1)
ck :=
dµk (x)
Rd
J. Inequal. Pure and Appl. Math., 6(3) Art. 84, 2005
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L ITTLEWOOD -PALEY G - FUNCTION IN THE D UNKL A NALYSIS
3
The Dunkl operators Tj ; j = 1, . . . , d, on Rd associated with the finite reflection group G and
multiplicity function k are given for a function f of class C 1 on Rd , by
X
∂
f (x) − f (σα x)
Tj f (x) :=
f (x) +
k(α)αj
.
∂xj
hα,
xi
α∈R+
P
The generalized Laplacian ∆k associated with G and k, is defined by ∆k := dj=1 Tj2 . It is
given explicitly by
X
f (x) − f (σα x)
(2.2)
∆k f (x) := Lk f (x) − 2
k(α)
,
2
hα,
xi
α∈R
+
with the singular elliptic operator:
Lk f (x) := ∆f (x) + 2
(2.3)
X
α∈R+
k(α)
h∇f (x), αi
,
hα, xi
where ∆ denotes the usual Laplacian.
The operator Lk can also be written in divergence form:
d
∂
1 X ∂
wk (x)
.
(2.4)
Lk f (x) =
wk (x) i=1 ∂xi
∂xi
This is a canonical multi-variable generalization of the Sturm-Liouville operator for the classical
spherical Bessel function [1, 2, 20].
For y ∈ Rd , the initial value problem Tj u(x, ·)(y) = xj u(x, y); j = 1, . . . , d, with u(0, y) =
1 admits a unique analytic solution on Rd , which will be denoted by Ek (x, y) and called a Dunkl
kernel [6, 14, 16, 23].
This kernel has the Bochner-type representation (see [12]):
Z
(2.5)
Ek (x, z) =
ehy,zi dΓx (y); x ∈ Rd , z ∈ Cd ,
Rd
Pd
where hy, zi := i=1 yi zi and Γx is a probability measure on Rd with support in the closed ball
Bd (o, kxk) of center o and radius kxk.
Example 2.1 (see [23, p. 21]). If G = Z2 , the Dunkl kernel is given by
Z
Γ γ + 12 sgn(x) |x| yz 2
Eγ (x, z) = √
·
e (x − y 2 )γ−1 (x + y)dy.
|x|2γ −|x|
π Γ(γ)
Notation. We denote by D(Rd ) the space of C ∞ −functions on Rd with compact support.
The Dunkl kernel gives an integral transform, called the Dunkl transform on Rd , which was
studied by de Jeu in [8]. The Dunkl transform of a function f in D(Rd ) is given by
Z
Fk (f )(x) :=
Ek (−ix, y)f (y)dµk (y), x ∈ Rd .
Rd
Note that F0 agrees with the Fourier transform F on Rd :
Z
F(f )(x) :=
e−ihx,yi f (y)dy, x ∈ Rd .
Rd
The Dunkl transform of a function f ∈ D(Rd ) which is radial is again radial, and could be
B
computed via the associated Fourier-Bessel transform Fγ+d/2−1
[11, p. 586] that is:
B
Fk (f )(x) = 2γ+d/2 c−1
k Fγ+d/2−1 (F )(kxk),
J. Inequal. Pure and Appl. Math., 6(3) Art. 84, 2005
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4
F ETHI S OLTANI
where f (x) = F (kxk), and
B
Fγ+d/2−1
(F )(kxk)
Z
∞
:=
F (r)
0
jγ+d/2−1 (kxkr) 2γ+d−1
r
dr.
γ + d2
2γ+d/2−1 Γ
Here jγ is the spherical Bessel function [24].
Notations. We denote by Lpk (Rd ), p ∈ [1, ∞], the space of measurable functions f on Rd , such
that
Z
p1
p
kf kLpk :=
|f (x)| dµk (x) < ∞, p ∈ [1, ∞[,
Rd
kf kL∞
:= ess sup |f (x)| < ∞,
k
x∈Rd
where µk is the measure given by (2.1).
Theorem 2.1 (see [7]).
i) Plancherel theorem: the normalized Dunkl transform 2−γ−d/2 ck Fk is an isometric automorphism on L2k (Rd ). In particular,
kf kL2k = 2−γ−d/2 ck kFk (f )kL2k .
ii) Inversion formula: let f be a function in L1k (Rd ), such that Fk (f ) ∈ L1k (Rd ). Then
Fk−1 (f )(x) = 2−2γ−d c2k Fk (f )(−x),
a.e. x ∈ Rd .
In [6], Dunkl defines the intertwining operator Vk on P := C[Rd ] (the C-algebra of polynomial functions on Rd ), by
Z
Vk (p)(x) :=
p(y)dΓx (y), x ∈ Rd ,
Rd
where Γx is the representing measure on Rd given by (2.5).
Next, Rösler proved the positivity properties of this operator (see [12]).
Notation. We denote by E(Rd ) and by E 0 (Rd ) the spaces of C ∞ −functions on Rd and of
distributions on Rd with compact support respectively.
In [22, Theorem 6.3], Trimèche has proved the following results:
Proposition 2.2.
i) The operator Vk can be extended to a topological automorphism on E(Rd ).
ii) For all x ∈ Rd , there exists a unique distribution ηk,x in E 0 (Rd ) with supp(ηk,x ) ⊂ {y ∈
Rd / kyk ≤ kxk}, such that
(Vk )−1 (f )(x) = hηk,x , f i,
f ∈ E(Rd ).
Next in [23], the author defines:
• The Dunkl translation operators τx , x ∈ Rd , on E(Rd ), by
τx f (y) := (Vk )x ⊗ (Vk )y [(Vk )−1 (f )(x + y)],
y ∈ Rd .
These operators satisfy for x, y and z ∈ Rd the following properties:
τ0 f = f,
(2.6)
τx f (y) = τy f (x),
Ek (x, z)Ek (y, z) = τx (Ek (·, z))(x),
and
(2.7)
Fk (τx f )(y) = Ek (ix, y)Fk (f )(y),
J. Inequal. Pure and Appl. Math., 6(3) Art. 84, 2005
f ∈ D(Rd ).
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L ITTLEWOOD -PALEY G - FUNCTION IN THE D UNKL A NALYSIS
5
Thus by (2.7), the Dunkl translation operators can be extended on L2k (Rd ), and for
x ∈ Rd we have
kτx f kL2k ≤ kf kL2k , f ∈ L2k (Rd ).
• The generalized convolution product ∗k of two functions f and g in L2k (Rd ), by
Z
f ∗k g(x) :=
τx f (−y)g(y)dµk (y), x ∈ Rd .
Rd
Note that ∗0 agrees with the standard convolution ∗ on Rd :
Z
f ∗ g(x) :=
f (x − y)g(y)dy, x ∈ Rd .
Rd
The generalized convolution ∗k satisfies the following properties:
Proposition 2.3.
i) Let f, g ∈ D(Rd ). Then
Fk (f ∗k g) = Fk (f )Fk (g).
ii) Let f, g ∈ L2k (Rd ). Then f ∗k g belongs to L2k (Rd ) if and only if Fk (f )Fk (g) belongs to
L2k (Rd ) and we have
Fk (f ∗k g) = Fk (f )Fk (g),
in the L2k − case.
Proof. The assertion i) is shown in [23, Theorem 7.2]. We can prove ii) in the same manner
demonstrated in [21, p. 101–103].
Theorem 2.4. Let p, q, r ∈ [1, ∞] satisfy the Young’s condition: 1/p + 1/q = 1 + 1/r. Assume
that f ∈ Lpk (Rd ) and g ∈ Lqk (Rd ). If kτx f kLqk ≤ c kf kLqk for all x ∈ Rd , then
kf ∗k gkLrk ≤ c kf kLpk kgkLqk .
Proof. The assumption that τx is a bounded operator on Lpk (Rd ) ensures that the usual proof of
Young’s inequality (see [25, p. 37]) works.
Proposition 2.5.
i) If f (x) = F (kxk) in E(Rd ), then we have
Z
p
kxk2 + kyk2 + 2hy, ξi dΓx (ξ);
τx f (y) =
F
x, y ∈ Rd ,
Ax,y
where
Ax,y
d
= ξ ∈ R / min kx + gyk ≤ kξk ≤ max kx + gyk ,
g∈G
g∈G
and Γx the representing measure given by (2.5).
ii) For all x ∈ Rd and for f ∈ Lpk (Rd ), radial, p ∈ [1, ∞],
kτx f kLpk ≤ kf kLpk .
iii) Let p, q, r ∈ [1, ∞] satisfy the Young’s condition: 1/p + 1/q = 1 + 1/r. Assume that
f ∈ Lpk (Rd ), radial, and g ∈ Lqk (Rd ), then
kf ∗k gkLrk ≤ kf kLpk kgkLqk .
J. Inequal. Pure and Appl. Math., 6(3) Art. 84, 2005
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6
F ETHI S OLTANI
Proof. The assertion i) is shown by Rösler in [13, Theorem 5.1].
ii) Since f is a radial function, the explicit formula of τx f shows that
|τx f (y)| ≤ τx (|f |)(y).
Hence, it follows readily from (2.6) that
kτx f kL1k ≤ kf kL1k .
By duality the same inequality holds for p = ∞.
Thus by interpolation we obtain the result for p ∈ ]1, ∞[.
iii) follows directly from Theorem 2.4.
Notation. For all x, y, z ∈ R, we put
Wγ (x, y, z) := [1 − σx,y,z + σz,x,y + σz,y,x ] Bγ (|x|, |y|, |z|),
where
 2
2
2
 x + y − z , if x, y ∈ R\{0}
2xy
σx,y,z :=

0,
otherwise
and Bγ is the Bessel kernel given by

γ−1

[((|x| + |y|)2 − z 2 ) (z 2 − (|x| − |y|)2 )]

 dγ
, if |z| ∈ Ax,y
|xyz|2γ−1
Bγ (|x|, |y|, |z|) :=



0,
otherwise,
h
i
2−2γ+1 Γ γ + 12
√
dγ =
, Ax,y = |x| − |y|, |x| + |y| .
π Γ(γ)
Remark 2.6 (see [10]). The signed kernel Wγ is even and satisfies:
Wγ (x, y, z) = Wγ (y, x, z) = Wγ (−x, z, y),
Wγ (x, y, z) = Wγ (−z, y, −x) = Wγ (−x, −y, −z),
and
Z
|Wγ (x, y, z)| dz ≤ 4.
R
We consider the signed measures νx,y (see [10]) defined by

Wγ (x, y, z)|z|2γ dz, if x, y ∈ R\{0}



dδx (z),
if y = 0
dνx,y (z) :=



dδy (z),
if x = 0.
The measures νx,y have the following properties:
Z
supp (νx,y ) = Ax,y ∪ (−Ax,y ),
kνx,y k :=
d|νx,y | ≤ 4.
R
Proposition 2.7 (see [10, 15]). If d = 1 and G = Z2 , then
i) For all x, y ∈ R and for f a continuous function on R, we have
Z
Z
τx f (y) =
f (ξ)dνx,y (ξ) +
f (ξ)dνx,y (ξ).
Ax,y
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L ITTLEWOOD -PALEY G - FUNCTION IN THE D UNKL A NALYSIS
7
ii) For all x ∈ R and for f ∈ Lpγ (R), p ∈ [1, ∞],
kτx f kLpγ ≤ 4 kf kLpγ .
iii) Assume that p, q, r ∈ [1, ∞] satisfy the Young’s condition: 1/p + 1/q = 1 + 1/r. Then
the map (f, g) → f ∗γ g extends to a continuous map from Lpγ (R) × Lqγ (R) to Lrγ (R)
and we have
kf ∗γ gkLrγ ≤ 4 kf kLpγ kgkLqγ .
3. T HE L ITTLEWOOD -PALEY g-F UNCTION
By analogy with the case of Euclidean space [19, p. 61] we define, for t > 0, the functions
Wt and Pt on Rd , by
Z
2
−2γ−d 2
e−tkξk Ek (ix, ξ)dµk (ξ), x ∈ Rd ,
Wt (x) := 2
ck
Rd
and
Pt (x) :=
2−2γ−d c2k
Z
e−tkξk Ek (ix, ξ)dµk (ξ),
x ∈ Rd .
Rd
The function Wt , may be called the generalized heat kernel and the function Pt , the generalized
Poisson kernel respectively.
From [23, p. 37] we have
ck
2
Wt (x) =
e−kxk /4t , x ∈ Rd .
γ+d/2
(4t)
Writing
Z ∞ −s
1
e
√ Wt2 /4s (x)ds, x ∈ Rd ,
(3.1)
Pt (x) = √
π 0
s
we obtain
ck Γ γ + d+1
ak t
2
√
(3.2)
Pt (x) = 2
, ak :=
.
(t + kxk2 )γ+(d+1)/2
π
However, for t > 0 and for all f ∈ Lpk (Rd ), p ∈ [1, ∞], we put:
uk (x, t) := Pt ∗k f (x),
x ∈ Rd .
The function uk is called the generalized Poisson integral of f , which was studied by Rösler in
[11, 13].
Let us consider the Littlewood-Paley g-function (in the Dunkl case). This auxiliary operator
is defined initially for f ∈ D(Rd ), by
"Z
!
# 12
2
∞ ∂
uk (x, t) + |∇x uk (x, t)|2 t dt , x ∈ Rd ,
g(f )(x) :=
∂t
0
where uk is the generalized Poisson integral.
The main result of the paper is:
Theorem 3.1. For p ∈ ]1, 2], there exists a constant Ap > 0 such that, for f ∈ Lpk (Rd ),
kg(f )kLpk ≤ Ap kf kLp .
For the proof of this theorem we need the following lemmas:
Lemma 3.2. Let f ∈ D(Rd ) be a positive function.
N
∂ uk
c
i) uk (x, t) ≥ 0 and ∂tN (x, t) ≤ t2γ+d+N
; k ∈ N and x ∈ Rd .
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8
F ETHI S OLTANI
ii) For kxk large we have
∂uk
c
c
≤
uk (x, t) ≤ 2
and
(x,
t)
.
2
γ+d/2
2
(t + kxk )
∂xi
(t + kxk2 )γ+(d+1)/2
Proof. i) If the generalized Poisson kernel Pt is a positive radial function, then from Proposition
2.5 i) we obtain uk (x, t) ≥ 0.
On the other hand from Proposition 2.5 iii) we have
N
N ∂ uk
≤ kf kL1 ∂ Pt .
(x,
t)
∂tN
k ∂tN ∞
Lk
Then we obtain the result from the fact that
N ∂ Pt ∂tN L∞
k
≤
c
t2γ+d+N
.
ii) From Proposition 2.5 i) we can write
Z
t dΓx (ξ)
τx Pt (−y) = ak
; x, y ∈ R,
2
2
2
γ+(d+1)/2
Rd [t + kxk + kyk − 2hy, ξi]
where ak is the constant given by (3.2).
Since f ∈ D(Rd ), there exists a > 0, such that supp(f ) ⊂ Bd (o, a). Then
Z
Z
t f (y)dΓx (ξ)dµk (y)
uk (x, t) = ak
.
2
2
2
γ+(d+1)/2
Bd (o,a) Ax,y [t + kxk + kyk − 2hy, ξi]
It is easily verified for kxk large and y ∈ Bd (o, a) that
1
c
≤ 2
.
2
2
2
γ+(d+1)/2
[t + kxk + kyk − 2hy, ξi]
(t + kxk2 )γ+(d+1)/2
Therefore and using the fact that t ≤ (t2 + kxk2 )1/2 , we obtain
c
ct
≤
.
uk (x, t) ≤ 2
(t + kxk2 )γ+(d+1)/2
(t2 + kxk2 )γ+d/2
Thus the first inequality is proven.
From (2.6) we can write
Z
Z
t f (−y)dΓy (ξ)dµk (y)
uk (x, t) = ak
.
2
2
2
γ+(d+1)/2
Bd (o,a) Ax,y [t + kxk + kyk + 2hx, ξi]
By derivation under the integral sign we obtain
Z
Z
∂uk
−t(2xi + ξi )f (−y)dΓy (ξ)dµk (y)
(x, t) = ak
.
2
2
2
γ+(d+3)/2
∂xi
Bd (o,a) Ax,y [t + kxk + kyk + 2hx, ξi]
But for kxk large and y ∈ Bd (o, a) we have
[t2
+
kxk2
t|2xi + ξi |
t(2|xi | + |ξi |)
≤ 2
.
2
γ+(d+3)/2
+ kyk + 2hx, ξi]
(t + kxk2 )γ+(d+3)/2
Using the fact that t(2|xi | + |ξi |) ≤ (1 + |ξi |)(t2 + kxk2 ) when kxk large, we obtain
∂uk
c
≤
(x,
t)
∂xi
(t2 + kxk2 )γ+(d+1)/2 ,
which proves the second inequality.
Lemma 3.3. Let f ∈ D(Rd ) be a positive function and p ∈ ]1, ∞[.
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L ITTLEWOOD -PALEY G - FUNCTION IN THE D UNKL A NALYSIS
R
∂ 2 upk
p
2 (x, t) tdtdµk (x) = Rd f (x)dµk (x).
∂t
B
(o,N
)
0
d
N →∞
RN R
lim 0 Bd (o,N ) Lk upk (·, t)(x) dµk (x)tdt = 0,
N →∞
i) lim
ii)
9
RN
R
where Lk is the singular elliptic operator given by (2.4).
Proof. i) Integrating by parts, we obtain
Z
Z N 2 p
∂ uk
(x, t) tdtdµk (x)
∂t2
Bd (o,N ) 0
Z
Z
p
=
f (x)dµk (x) −
Bd (o,N )
upk (x, N )dµk (x)
Bd (o,N )
Z
up−1
k (x, N )
+ pN
Bd (o,N )
From Lemma 3.2 i), we easily get
Z
∂uk
(x, N )dµk (x).
∂t
upk (x, N )dµk (x) ≤ c N −(p−1)(2γ+d) ,
Bd (o,N )
and
Z
up−1
k (x, N )
N
Bd (o,N )
∂uk
(x, N )dµk (x) ≤ c N −(p−1)(2γ+d) ,
∂t
which gives i).
ii) We have
N
Z
Z
0
Lk upk (·, t)(x) dµk (x)tdt
=
Bd (o,N )
d
X
Ii,N ,
i=1
where
Z
N
Z
Ii,N =
0
Bd (o,N )
Let us study I1,N :
Z NZ
I1,N = p
0
wk (x
Bd−1 (o,N )
(N )
∂
∂xi
∂upk
wk (x)
(x, t) dxtdt,
∂xi
i = 1, . . . , d.
∂uk (N )
(N )
) up−1
, t)
(x , t)
k (x
∂x1
−
where x
(N )
∂uk
(N )
up−1
, t)
(−x(N ) , t)
k (−x
∂x1
dx2 . . . dxd tdt,
q
Pd
2
2
=
N − i=2 xi , x2 , . . . , xd .
Then, by using Lemma 3.2 ii) and the fact that wk (x(N ) ) ≤ 2γ N 2γ we obtain for N large,
Z NZ
dx2 . . . dxd tdt
2γ
I1,N ≤ c N
2
2 (γ+d/2)p+1/2
0
Bd−1 (o,N ) (t + N )
Z NZ
≤ c N −p(2γ+d)+2γ−1
dx2 . . . dxd tdt
0
≤ cN
−(p−1)(2γ+d)−(d−1)/2
Bd−1 (o,N )
.
The same result holds for Ii,N , i = 2, . . . , d, which proves ii).
J. Inequal. Pure and Appl. Math., 6(3) Art. 84, 2005
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10
F ETHI S OLTANI
Lemma 3.4. Let f ∈ D(Rd ) be a positive function. Define the maximal function Mk (f ), by
(3.3)
x ∈ Rd .
Mk (f )(x) := sup (uk (x, t)) ,
t>0
Then for p ∈ ]1, ∞[, there exists a constant Cp > 0 such that, for f ∈ Lpk (Rd ),
kMk (f )kLpk ≤ Cp kf kLpk ,
moreover the operator Mk is of weak type (1, 1).
Proof. From (3.1) it follows that
t
uk (x, t) = √
8 π
Z
∞
2 /4s
Ws ∗k f (x)e−t
s−3/2 ds,
0
which implies, as in [18, p. 49] that
Z y
1
Mk (f )(x) ≤ c sup
Qs f (x)ds ,
y 0
y>0
x ∈ Rd ,
where Qs f (x) = Ws ∗k f (x), which is a semigroup of operators on Lpk (Rd ). Hence using the
Hopf-Dunford-Schwartz ergodic theorem as in [18, p. 48], we get the boundedness of Mk on
Lpk (Rd ) for p ∈ ]1, ∞] and weak type (1, 1).
Proof of Theorem 3.1. Let f ∈ D(Rd ) be a positive function. From Lemma 3.2 i) the generalized Poisson integral uk of f is positive.
∂
First step: Estimate of the quantity ∂t
uk (x, t) 2 + |∇x uk (x, t)| 2 .
Let Hk be the operator:
∂2
Hk := Lk + 2 ,
∂t
where Lk is the singular elliptic operator given by (2.3).
Using the fact that
∂2
∆k uk (·, t)(x) + 2 uk (x, t) = 0,
∂t
we obtain for p ∈ ]1, ∞[,
"
#
2
X
∂
Uα (x, t)
uk (x, t) + |∇x uk (x, t)|2 + p
Hk upk (x, t) = p(p − 1)up−2
(x,
t)
k(α)
,
k
∂t
2
hα,
xi
α∈R
+
where
Uα (x, t) := 2up−1
k (x, t) [uk (x, t) − uk (σα x, t)] ,
α ∈ R+ .
Let A, B ≥ 0, then the inequality
2Ap−1 (A − B) ≥ (Ap−1 + B p−1 )(A − B)
is equivalent to
(Ap−1 − B p−1 )(A − B) ≥ 0,
which holds if A ≥ B or A < B. Thus we deduce that
p−1
Uα (x, t) ≥ up−1
k (x, t) + uk (σα x, t) [uk (x, t) − uk (σα x, t)] ,
and therefore we get
2
∂
1
p
+ |∇x uk (x, t)|2 ≤
(3.4)
u
(x,
t)
u2−p
k
k (x, t) [vk (x, t) + Hk uk (x, t)] ,
∂t
p(p − 1)
J. Inequal. Pure and Appl. Math., 6(3) Art. 84, 2005
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L ITTLEWOOD -PALEY G - FUNCTION IN
THE
D UNKL A NALYSIS
11
where
vk (x, t) = p
X k(α) p−1
uk (σα x, t) + up−1
k (x, t) [uk (σα x, t) − uk (x, t)] .
2
hα, xi
α∈R
+
Second step: The inequality kg(f )kLpk ≤ Ap kf kLpk , for p ∈ ]1, 2[.
From (3.4), we have
Z ∞
1
p
2
u2−p
[g(f )(x)] ≤
k (x, t) [vk (x, t) + Hk uk (x, t)] tdt
p(p − 1) 0
1
≤
Ik (f )(x) [Mk (f )(x)]2−p , x ∈ Rd ,
p(p − 1)
where
∞
Z
[vk (x, t) + Hk upk (x, t)] tdt,
Ik (f )(x) :=
0
and Mk (f ) the maximal function given by (3.3).
Thus it is proven that
p2 Z
1
p
kg(f )kLp ≤
[Ik (f )(x)]p/2 [Mk (f )(x)](2−p)p/2 dµk (x).
k
p(p − 1)
d
R
By applying Hölder’s inequality, we obtain
p2
1
p/2
(2−p)p/2
p
(3.5)
kg(f )kLp ≤
kIk (f )kL1 kMk (f )kLp
.
k
k
k
p(p − 1)
Since vk (x, t) + Hk upk (x, t) ≥ 0, we can apply Fubini-Tonnelli’s Theorem to obtain
Z NZ
kIk (f )kL1k = lim
[vk (x, t) + Hk upk (x, t)] dµk (x)tdt.
N →∞
0
Bd (o,N )
Putting y = σα x and using the fact that σα2 = id; hσα y, αi = −hy, αi, then as in the argument
of [16, p. 390] we obtain
Z
Z
vk (x, t)dµk (x) = −
vk (y, t)dµk (y).
Bd (o,N )
Thus
Bd (o,N )
Z
vk (x, t)dµk (x) = 0.
Bd (o,N )
Hence from Lemma 3.3, we deduce that
Z
(3.6)
kIα (f )kL1k = lim
N →∞
N
Z
Bd (o,N )
0
Hk upk (x, t)tdtdµk (x) = kf kpLp .
k
On the other hand from Lemma 3.4 we have
(3.7)
kMk (f )kLpk ≤ Cp kf kLpk .
Finally, from (3.5), (3.6) and (3.7), we obtain
kg(f )kLpk ≤ Ap kf kLpk ,
Ap =
1
p(p − 1)
21
Cp(2−p)/2 .
Since the operator g is sub-linear, we obtain the inequality for f ∈ D(Rd ). And by an easy
limiting argument one shows that the result is also true for any f ∈ Lpk (Rd ), p ∈ ]1, 2[.
J. Inequal. Pure and Appl. Math., 6(3) Art. 84, 2005
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12
F ETHI S OLTANI
For the case p = 2, using (3.4) and (3.6) we get
Z Z ∞
1
1
2
kg(f )kL2 ≤
vk (x, t) + Hk u2k (x, t) tdtdµk (x) = kf k2L2 ,
k
k
2 Rd 0
2
which completes the proof of the theorem.
R EFERENCES
[1] A. ACHOUR AND K. TRIMÈCHE, La g-fonction de Littlewood-Paley associée à un opérateur
différentiel singulier sur (0, ∞), Ann. Inst. Fourier, Grenoble, 33 (1983), 203–226.
[2] H. ANNABI AND A. FITOUHI, La g-fonction de Littlewood-Paley associée à une classe
d’opérateurs différentiels sur ]0, ∞[ contenant l’opérateur de Bessel, C.R. Acad. Sc. Paris, 303
(1986), 411–413.
[3] T.H. BAKER AND P.J. FORRESTER, The Calogero-Sutherland model and generalized classical
polynomials, Comm. Math. Phys., 188 (1997), 175–216.
[4] T. COULHON, X.T. DUONG AND X.D. LI, Littlewood-Paley-Stein functions on manifolds 1 ≤
p ≤ 2, Studia Math., 154 (2003), 37–57.
[5] C.F. DUNKL, Differential-difference operators associated with reflection groups, Trans. Amer.
Math. Soc., 311 (1989), 167–183.
[6] C.F. DUNKL, Integral kernels with reflection group invariance, Can. J. Math., 43 (1991), 1213–
1227.
[7] C.F. DUNKL, Hankel transforms associated to finite reflection groups, Contemp. Math., 138 (1992)
123–138.
[8] M.F.E. de JEU, The Dunkl transform, Inv. Math., 113 (1993), 147–162.
[9] L. LAPOINTE AND L. VINET, Exact operator solution of the Calogero- Sutherland model, Comm.
Math. Phys., 178 (1996), 425–452.
[10] M. RÖSLER, Bessel-type signed hypergroups on R , In : H.Heyer, A.Mukherjea (Eds.), Proc XI,
Probability measures on groups and related structures, Oberwolfach, 1994, World Scientific, Singapore, (1995), p. 292–304.
[11] M. RÖSLER AND M. VOIT, Markov processes related with Dunkl operator, Adv. Appl. Math., 21
(1998), 575–643.
[12] M. RÖSLER, Positivity of Dunkl’s intertwining operator, Duke Math. J., 98 (1999), 445–463.
[13] M. RÖSLER, A positive radial product formula for the Dunkl kernel, Trans. Amer. Math. Soc., 355
(2003) 2413–2438.
[14] M. SIFI AND F. SOLTANI, Generalized Fock spaces and Weyl relations for the Dunkl kernel on
the real line, J. Math. Anal. Appl., 270 (2002), 92–106.
[15] F. SOLTANI, Lp -Fourier multipliers for the Dunkl operator on the real line, J. Functional Analysis,
209 (2004), 16–35.
[16] F. SOLTANI, Generalized Fock spaces and Weyl commutation relations for the Dunkl kernel, Pacific J. Math., 214 (2004), 379–397.
[17] F. SOLTANI, Inversion formulas in the Dunkl-type heat conduction on Rd , Applicable Analysis, 84
(2005), 541–553.
[18] E.M. STEIN, Topics in Harmonic Analysis related to the Littlewood-Paley theory, Annals of Mathematical Studies, 63 (1970), Princeton Univ. Press, Princeton, New Jersey.
J. Inequal. Pure and Appl. Math., 6(3) Art. 84, 2005
http://jipam.vu.edu.au/
L ITTLEWOOD -PALEY G - FUNCTION IN
THE
D UNKL A NALYSIS
13
[19] E.M. STEIN, Singular integrals and differentiability properties of functions, Princeton Univ. Press,
Princeton, New Jersey, 1971.
[20] K. STEMPAK, La théorie de Littlewood-Paley pour la transformation de Fourier-Bessel, C.R.A.S.
Paris, Série I, Math., 303 (1986), 15–18.
[21] K. TRIMÈCHE, Inversion of the Lions transmutation operators using generalized wavelets, App.
Comput. Harm. Anal., 4 (1997), 97–112.
[22] K. TRIMÈCHE, The Dunkl intertwining operator on spaces of functions and distributions and
integral representation of dual, Int. Trans. Spec. Funct., 12 (2001), 349–374.
[23] K. TRIMÈCHE, Paley-Wiener Theorems for the Dunkl transform and Dunkl translation operators,
Int. Trans. Spec. Funct., 13 (2002), 17–38.
[24] G.N. WATSON, A Treatise on Theory of Bessel Functions, Cambridge University Press, 1966.
[25] A. ZYGMUND, Trigonometric Series, Cambridge University Press, 1959.
J. Inequal. Pure and Appl. Math., 6(3) Art. 84, 2005
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