Journal of Inequalities in Pure and
Applied Mathematics
LITTLEWOOD-PALEY g-FUNCTION IN THE DUNKL
ANALYSIS ON Rd
volume 6, issue 3, article 84,
2005.
FETHI SOLTANI
Department of Mathematics
Faculty of Sciences of Tunis
University of EL-Manar Tunis 2092 Tunis, Tunisia.
Received 11 October, 2004;
accepted 22 July, 2005.
Communicated by: S.S. Dragomir
EMail: Fethi.Soltani@fst.rnu.tn
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
183-04
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Abstract
We prove Lp -inequality for the Littlewood-Paley g-function in the Dunkl case on
Rd .
2000 Mathematics Subject Classification: 42B15, 42B25.
Key words: Dunkl operators, Generalized Poisson integral, g-function.
The author is very grateful to the referee for many comments on this paper.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
The Dunkl Analysis on Rd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3
The Littlewood-Paley g-Function . . . . . . . . . . . . . . . . . . . . . . . 15
References
Littlewood-Paley g-Function in
the Dunkl Analysis on Rd
Fethi Soltani
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1.
Introduction
In the Euclidean case, the Littlewood-Paley g-function is given by
"Z
g(f )(x) :=
0
∞
!
# 12
2
∂
u(x, t) + |∇x u(x, t)|2 t dt ,
∂t
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 the Dunkl analysis on Rd .
Moreover they are naturally connected with certain Schrödinger operators for
Calogero-Sutherland-type quantum many body systems [3, 9].
Littlewood-Paley g-Function in
the Dunkl Analysis on Rd
Fethi Soltani
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The main purpose of this paper is to give the Lp -inequality for the LittlewoodPaley g-function 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:
Littlewood-Paley g-Function in
the Dunkl Analysis on Rd
Fethi Soltani
"Z
g(f )(x) :=
0
∞
!
# 12
2
∂
uk (x, t) + |∇x uk (x, t)|2 t dt ,
∂t
x ∈ Rd ,
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.
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2.
The Dunkl Analysis 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
α.
σα x := x −
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 associated with R. Allreflections
in G, correspond to
S
d
suitable pairs of roots. 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+
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α∈R+
Moreover, let wk denote the weight function:
Y
|hα, xi|2k(α) ,
wk (x) :=
Littlewood-Paley g-Function in
the Dunkl Analysis on Rd
Page 5 of 28
x ∈ Rd ,
J. Ineq. Pure and Appl. Math. 6(3) Art. 84, 2005
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which is G-invariant and homogeneous of degree 2γ.
We introduce the Mehta-type constant ck , by
−1
Z
−kxk2
, where dµk (x) := wk (x)dx.
(2.1)
ck :=
e
dµk (x)
Rd
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
+
Laplacian ∆k associated with G and k, is defined by ∆k :=
PdThe generalized
2
T
.
It
is
given
explicitly by
j=1 j
(2.2)
∆k f (x) := Lk f (x) − 2
X
k(α)
α∈R+
f (x) − f (σα x)
,
hα, xi2
with the singular elliptic operator:
(2.3)
Lk f (x) := ∆f (x) + 2
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 ∂
∂
(2.4)
Lk f (x) =
wk (x)
.
wk (x) i=1 ∂xi
∂xi
Littlewood-Paley g-Function in
the Dunkl Analysis on Rd
Fethi Soltani
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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
ehy,zi dΓx (y); x ∈ Rd , z ∈ Cd ,
(2.5)
Ek (x, z) =
Rd
P
where hy, zi := di=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
·
e (x − y 2 )γ−1 (x + y)dy.
Eγ (x, z) = √
|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
Littlewood-Paley g-Function in
the Dunkl Analysis on Rd
Fethi Soltani
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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,
B
and could be 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),
Fethi Soltani
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.
2γ+d/2−1 Γ γ + d2
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
p1
|f (x)| dµk (x) < ∞,
Z
kf kLpk :=
Littlewood-Paley g-Function in
the Dunkl Analysis on Rd
p
Rd
kf kL∞
:= ess sup |f (x)| < ∞,
k
x∈Rd
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where µk is the measure given by (2.1).
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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),
d
a.e. x ∈ R .
Littlewood-Paley g-Function in
the Dunkl Analysis on Rd
Fethi Soltani
In [6], Dunkl defines the intertwining operator Vk on P := C[Rd ] (the Calgebra 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 ).
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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:
(2.6)
τ0 f = f,
Fethi Soltani
τx f (y) = τy f (x),
Ek (x, z)Ek (y, z) = τx (Ek (·, z))(x),
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and
(2.7)
Littlewood-Paley g-Function in
the Dunkl Analysis on Rd
Fk (τx f )(y) = Ek (ix, y)Fk (f )(y),
f ∈ D(Rd ).
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),
Rd
x ∈ Rd .
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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.
Littlewood-Paley g-Function in
the Dunkl Analysis on Rd
Fethi Soltani
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Proposition 2.5.
i) If f (x) = F (kxk) in E(Rd ), then we have
Z
p
τx f (y) =
F
kxk2 + kyk2 + 2hy, ξi dΓx (ξ);
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).
d
ii) For all x ∈ R and for f ∈
Lpk (Rd ),
Littlewood-Paley g-Function in
the Dunkl Analysis on Rd
Fethi Soltani
radial, p ∈ [1, ∞],
kτx f kLpk ≤ kf kLpk .
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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 .
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
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|τx f (y)| ≤ τx (|f |)(y).
Page 12 of 28
Hence, it follows readily from (2.6) that
J. Ineq. Pure and Appl. Math. 6(3) Art. 84, 2005
kτx f kL1k ≤ kf kL1k .
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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
σx,y,z

2
2
2

 x +y −z ,
2xy
:=

 0,
if x, y ∈ R\{0}
otherwise
and Bγ is the Bessel kernel given by
Bγ (|x|, |y|, |z|)

2
2
2
2 γ−1

 [((|x| + |y|) − z ) (z − (|x| − |y|) )]
dγ
, if |z| ∈ Ax,y
:=
|xyz|2γ−1

 0,
otherwise,
h
i
2−2γ+1 Γ γ + 21
√
dγ =
, Ax,y = |x| − |y|, |x| + |y| .
π Γ(γ)
Remark 1 (see [10]). The signed kernel Wγ is even and satisfies:
Littlewood-Paley g-Function in
the Dunkl Analysis on Rd
Fethi Soltani
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Wγ (x, y, z) = Wγ (y, x, z) = Wγ (−x, z, y),
J. Ineq. Pure and Appl. Math. 6(3) Art. 84, 2005
Wγ (x, y, z) = Wγ (−z, y, −x) = Wγ (−x, −y, −z),
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and
Z
|Wγ (x, y, z)| dz ≤ 4.
R
We consider the signed measures νx,y (see [10]) defined by

2γ

 Wγ (x, y, z)|z| dz, if x, y ∈ R\{0}


dδx (z),
if y = 0
dνx,y (z) :=



 dδ (z),
if x = 0.
y
The measures νx,y have the following properties:
Fethi Soltani
Z
supp (νx,y ) = Ax,y ∪ (−Ax,y ),
kνx,y k :=
Littlewood-Paley g-Function in
the Dunkl Analysis on Rd
d|νx,y | ≤ 4.
R
Proposition 2.6 (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
(−Ax,y )
ii) For all x ∈ R and for f ∈ Lpγ (R), p ∈ [1, ∞],
kτx f k
Lpγ
≤ 4 kf k .
Lpγ
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 .
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3.
The Littlewood-Paley g-Function
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
Wt (x) := 2
ck
e−tkξk Ek (ix, ξ)dµk (ξ), x ∈ Rd ,
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
e−kxk /4t , x ∈ Rd .
Wt (x) =
γ+d/2
(4t)
(3.1)
Z
∞
0
e−s
√ Wt2 /4s (x)ds,
s
x ∈ Rd ,
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we obtain
(3.2)
Fethi Soltani
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Writing
1
Pt (x) = √
π
Littlewood-Paley g-Function in
the Dunkl Analysis on Rd
Close
Pt (x) =
ak t
,
2
(t + kxk2 )γ+(d+1)/2
However, for t > 0 and for all f ∈
Lpk (Rd ),
ak :=
ck Γ γ +
√
π
d+1
2
.
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p ∈ [1, ∞], we put:
J. Ineq. Pure and Appl. Math. 6(3) Art. 84, 2005
uk (x, t) := Pt ∗k f (x),
x ∈ Rd .
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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
g(f )(x) :=
0
∞
!
# 12
2
∂
uk (x, t) + |∇x uk (x, t)|2 t dt ,
∂t
x ∈ Rd ,
where uk is the generalized Poisson integral.
The main result of the paper is:
Littlewood-Paley g-Function in
the Dunkl Analysis on Rd
Fethi Soltani
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
c
i) uk (x, t) ≥ 0 and ∂∂tNuk (x, t) ≤ t2γ+d+N
; k ∈ N and x ∈ Rd .
ii) For kxk large we have
∂uk
c
c
≤
and
(x,
t)
.
uk (x, t) ≤ 2
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.
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On the other hand from Proposition 2.5 iii) we have
N N
∂ uk
∂ Pt .
1
(x,
t)
≤
kf
k
Lk ∂tN
∂tN L∞
k
Then we obtain the result from the fact that
N ∂ Pt c
≤
.
∂tN ∞
t2γ+d+N
L
k
ii) From Proposition 2.5 i) we can write
Z
t dΓx (ξ)
τx Pt (−y) = ak
;
2
2
2
γ+(d+1)/2
Rd [t + kxk + kyk − 2hy, ξi]
Littlewood-Paley g-Function in
the Dunkl Analysis on Rd
Fethi Soltani
x, y ∈ R,
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
uk (x, t) ≤
(t2
c
ct
≤ 2
.
2
γ+(d+1)/2
+ kxk )
(t + kxk2 )γ+d/2
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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
−t(2xi + ξi )f (−y)dΓy (ξ)dµk (y)
∂uk
(x, t) = ak
.
2
2
2
γ+(d+3)/2
∂xi
Bd (o,a) Ax,y [t + kxk + kyk + 2hx, ξi]
Fethi Soltani
But for kxk large and y ∈ Bd (o, a) we have
[t2
+
kxk2
Littlewood-Paley g-Function in
the Dunkl Analysis on Rd
t(2|xi | + |ξi |)
t|2xi + ξi |
≤ 2
.
2
γ+(d+3)/2
+ kyk + 2hx, ξi]
(t + kxk2 )γ+(d+3)/2
2
2
Using the fact that t(2|xi | + |ξi |) ≤ (1 + |ξi |)(t + kxk ) when kxk large, we
obtain
∂uk
c
∂xi (x, t) ≤ (t2 + kxk2 )γ+(d+1)/2 ,
which proves the second inequality.
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d
Lemma 3.3. Let f ∈ D(R ) be a positive function and p ∈ ]1, ∞[.
i) lim
R
ii) lim
RN R
RN
N →∞ Bd (o,N )
N →∞ 0
0
Bd (o,N )
∂ 2 upk
∂t2
(x, t) tdtdµk (x) =
R
Rd
f p (x)dµk (x).
Lk upk (·, t)(x) dµk (x)tdt = 0,
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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) −
upk (x, N )dµk (x)
Bd (o,N )
Bd (o,N )
Z
∂uk
+ pN
(x, N )dµk (x).
up−1
k (x, N )
∂t
Bd (o,N )
From Lemma 3.2 i), we easily get
Z
upk (x, N )dµk (x) ≤ c N −(p−1)(2γ+d) ,
Littlewood-Paley g-Function in
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Fethi Soltani
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Bd (o,N )
Contents
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
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Z
N
Z
0
Lk upk (·, t)(x) dµk (x)tdt
Bd (o,N )
=
d
X
Close
Ii,N ,
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i=1
Page 19 of 28
where
Z
N
Z
Ii,N =
0
Bd (o,N )
∂
∂xi
∂upk
wk (x)
(x, t) dxtdt,
∂xi
i = 1, . . . , d.
J. Ineq. Pure and Appl. Math. 6(3) Art. 84, 2005
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Let us study I1,N :
Z NZ
I1,N = p
where x(N )
∂uk (N )
(N )
wk (x ) up−1
, t)
(x , t)
k (x
∂x1
0
Bd−1 (o,N )
∂uk
p−1
(N )
(N )
− uk (−x , t)
(−x , t) dx2 . . . dxd tdt,
∂x1
q
Pd
2
2
=
N − i=2 xi , x2 , . . . , xd .
(N )
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
−p(2γ+d)+2γ−1
≤ cN
dx2 . . . dxd tdt
0
≤ cN
−(p−1)(2γ+d)−(d−1)/2
.
t>0
kMk (f )kLpk ≤ Cp kf kLpk ,
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x ∈ Rd .
Then for p ∈ ]1, ∞[, there exists a constant Cp > 0 such that, for f ∈
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Lemma 3.4. Let f ∈ D(Rd ) be a positive function. Define the maximal function
Mk (f ), by
Mk (f )(x) := sup (uk (x, t)) ,
Fethi Soltani
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Bd−1 (o,N )
The same result holds for Ii,N , i = 2, . . . , d, which proves ii).
(3.3)
Littlewood-Paley g-Function in
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Lpk (Rd ),
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moreover the operator Mk is of weak type (1, 1).
Proof. From (3.1) it follows that
Z ∞
t
2
uk (x, t) = √
Ws ∗k f (x)e−t /4s s−3/2 ds,
8 π 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 ∂ uk (x, t) 2 + |∇x uk (x, t)| 2 .
∂t
Let Hk be the operator:
Littlewood-Paley g-Function in
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Fethi Soltani
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∂2
Hk := Lk + 2 ,
∂t
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where Lk is the singular elliptic operator given by (2.3).
Using the fact that
∆k uk (·, t)(x) +
∂2
uk (x, t) = 0,
∂t2
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we obtain for p ∈ ]1, ∞[,
"
#
2
∂
Hk upk (x, t) = p(p − 1)ukp−2 (x, t) uk (x, t) + |∇x uk (x, t)|2
∂t
X
Uα (x, t)
+p
k(α)
,
2
hα,
xi
α∈R
+
where
Uα (x, t) :=
2up−1
k (x, t) [uk (x, t)
− uk (σα x, t)] ,
α ∈ R+ .
Littlewood-Paley g-Function in
the Dunkl Analysis on Rd
Fethi Soltani
Let A, B ≥ 0, then the inequality
2Ap−1 (A − B) ≥ (Ap−1 + B p−1 )(A − B)
Contents
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
(x,
t)
+
u
(σ
x,
t)
[uk (x, t) − uk (σα x, t)] ,
α
k
k
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and therefore we get
(3.4)
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2
∂
uk (x, t) + |∇x uk (x, t)|2
∂t
≤
1
u2−p (x, t) [vk (x, t) + Hk upk (x, t)] ,
p(p − 1) k
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where
vk (x, t) = p
X k(α) p−1
p−1
u
(σ
x,
t)
+
u
(x,
t)
[uk (σα x, t) − uk (x, t)] .
α
k
k
hα, xi2
α∈R
+
Second step: The inequality kg(f )kLpk ≤ Ap kf kLpk , for p ∈ ]1, 2[.
From (3.4), we have
Z ∞
1
2
[g(f )(x)] ≤
uk2−p (x, t) [vk (x, t) + Hk upk (x, t)] tdt
p(p − 1) 0
1
≤
Ik (f )(x) [Mk (f )(x)]2−p , x ∈ Rd ,
p(p − 1)
where
Ik (f )(x) :=
Fethi Soltani
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∞
Z
Littlewood-Paley g-Function in
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[vk (x, t) +
Hk upk (x, t)] tdt,
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0
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and Mk (f ) the maximal function given by (3.3).
Thus it is proven that
kg(f )kpLp
k
≤
1
p(p − 1)
p2 Z
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[Ik (f )(x)]p/2 [Mk (f )(x)](2−p)p/2 dµk (x).
Rd
(3.5)
kg(f )kpLp ≤
k
1
p(p − 1)
p2
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By applying Hölder’s inequality, we obtain
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p/2
(2−p)p/2
kIk (f )kL1 kMk (f )kLp
k
k
.
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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 )
Littlewood-Paley g-Function in
the Dunkl Analysis on Rd
Bd (o,N )
Fethi Soltani
Thus
Z
vk (x, t)dµk (x) = 0.
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Bd (o,N )
Hence from Lemma 3.3, we deduce that
Z
Z N
(3.6)
kIα (f )kL1k = lim
Hk upk (x, t)tdtdµk (x) = kf kpLp .
N →∞
Bd (o,N )
k
0
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On the other hand from Lemma 3.4 we have
Close
(3.7)
kMk (f )kLpk ≤ Cp kf kLpk .
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Finally, from (3.5), (3.6) and (3.7), we obtain
kg(f )kLpk ≤ Ap kf kLpk ,
Ap =
1
p(p − 1)
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Cp(2−p)/2 .
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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[.
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.
Littlewood-Paley g-Function in
the Dunkl Analysis on Rd
Fethi Soltani
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References
[1] A. ACHOUR AND K. TRIMÈCHE, La g-fonction de Littlewood-Paley
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[2] H. ANNABI AND A. FITOUHI, La g-fonction de Littlewood-Paley associée à une classe d’opérateurs différentiels sur ]0, ∞[ contenant l’opérateur
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Littlewood-Paley g-Function in
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[10] M. RÖSLER, Bessel-type signed hypergroups on R , In : H.Heyer,
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d
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the Dunkl Analysis on Rd
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[19] E.M. STEIN, Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton, New Jersey, 1971.
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the Dunkl Analysis on Rd
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