LITTLEWOOD-PALEY DECOMPOSITION
ASSOCIATED WITH THE DUNKL OPERATORS AND
PARAPRODUCT OPERATORS
Littlewood-Paley Decomposition
Hatem Mejjaoli
vol. 9, iss. 4, art. 95, 2008
HATEM MEJJAOLI
Department of Mathematics
Faculty of Sciences of Tunis
CAMPUS 1060 Tunis, TUNISIA
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EMail: hatem.mejjaoli@ipest.rnu.tn
Contents
Received:
11 July, 2007
Accepted:
25 May, 2008
Communicated by:
S.S. Dragomir
2000 AMS Sub. Class.:
Primary 35L05. Secondary 22E30.
Key words:
Dunkl operators, Littlewood-Paley decomposition, Paraproduct.
Abstract:
We define the Littlewood-Paley decomposition associated with the Dunkl operators; from this decomposition we give the characterization of the Sobolev,
Hölder and Lebesgue spaces associated with the Dunkl operators. We construct
the paraproduct operators associated with the Dunkl operators similar to those
defined by J.M. Bony in [1]. Using the Littlewood-Paley decomposition we establish the Sobolev embedding, Gagliardo-Nirenberg inequality and we present
the paraproduct algorithm.
Acknowledgement:
I am thankful to anonymous referee for his deep and helpful comments.
Dedicatory:
Dedicated to Khalifa Trimeche.
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Contents
1
Introduction
2
The Eigenfunction of the Dunkl Operators
2.1 Reflection Groups, Root System and Multiplicity Functions . .
2.2 Dunkl operators-Dunkl kernel and Dunkl intertwining operator
2.3 The Dunkl Transform . . . . . . . . . . . . . . . . . . . . . .
2.4 The Dunkl Convolution Operator . . . . . . . . . . . . . . . .
3
.
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5
5
6
8
10
Littlewood-Paley Decomposition
Hatem Mejjaoli
vol. 9, iss. 4, art. 95, 2008
3
4
5
Littlewood-Paley Theory Associated with Dunkl Operators
3.1 Dyadic Decomposition . . . . . . . . . . . . . . . . . . . . . . . .
3.2 The Generalized Sobolev Spaces . . . . . . . . . . . . . . . . . . .
3.3 The Generalized Hölder Spaces . . . . . . . . . . . . . . . . . . . .
13
13
15
22
Applications
4.1 Estimates of the Product of Two Functions . . . . . . . . . . . . . .
4.2 Sobolev Embedding Theorem . . . . . . . . . . . . . . . . . . . .
4.3 Gagliardo-Nirenberg Inequality . . . . . . . . . . . . . . . . . . . .
27
27
29
35
Paraproduct Associated with the Dunkl Operators
38
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1.
Introduction
The theory of function spaces appears at first to be a disconnected subject, because
of the variety of spaces and the different considerations involved in their definitions. There are the Lebesgue spaces Lp (Rd ), the Sobolev spaces H s (Rd ), the Besov
s
spaces Bp,q
(Rd ), the BMO spaces (bounded mean oscillation) and others.
Nevertheless, several approaches lead to a unified viewpoint on these spaces,
for example, approximation theory or interpolation theory. One of the most successful approaches is the Littlewood-Paley theory. This approach has been developed by the European school, which reached a similar unification of function
space theory by a different path. Motivated by the methods of Hörmander in studying partial differential equations (see [6]), they used a Fourier transform approach.
Pick Schwartz functions φ and χ on Rd satisfying supp χ
b ⊂ B(0, 2), supp φb ⊂
b
ξ ∈ Rd , 12 ≤ kξk ≤ 2 , and the nondegeneracy condition |b
χ(ξ)|, |φ(ξ)|
≥ C > 0.
jd
j
For j ∈ Z, let φj (x) = 2 φ(2 x). In 1967 Peetre [10] proved that
! 12
X
.
(1.1)
kf kH s (Rd ) ' kχ ∗ f kL2 (Rd ) +
22sj kφj ∗ f k2L2 (Rd )
Littlewood-Paley Decomposition
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vol. 9, iss. 4, art. 95, 2008
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j≥1
s
Independently, Triebel [15] in 1973 and Lizorkin [8] in 1972 introduced Fp,q
(the
Triebel-Lizorkin spaces) defined originally for 1 ≤ p < ∞ and 1 ≤ q ≤ ∞ by the norm
! 1q X
sj
q
s
(1.2)
kf kFp,q
= kχ ∗ f kLp (Rd ) + (2 |φj ∗ f |)
.
j≥1
p d
L (R )
For the special case q = 1 and s = 0, Triebel [16] proved that
0
(1.3)
Lp (Rd ) ' Fp,2
.
Thus by the Littlewood-Paley decomposition we characterize the functional spaces
Lp (Rd ), Sobolev spaces H s (Rd ), Hölder spaces C s (Rd ) and others. Using the
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Littlewood-Paley decomposition J.M. Bony in [1], built the paraproduct operators
which have been later successfully employed in various settings.
The purpose of this paper is to generalize the Littlewood-Paley theory, to unify
and extend the paraproduct operators which allow the analysis of solutions to more
general partial differential equations arising in applied mathematics and other fields.
More precisely, we define the Littlewood-Paley decomposition associated with the
Dunkl operators. We introduce the new spaces associated with the Dunkl operators, the Sobolev spaces Hks (Rd ), the Hölder spaces Cks (Rd ) and the BM Ok (Rd )
that generalizes the corresponding classical spaces. The Dunkl operators are the
differential-difference operators introduced by C.F. Dunkl in [3] and which played
an important role in pure Mathematics and in Physics. For example they were a main
tool in the study of special functions with root systems (see [4]).
As applications of the Littlewood-Paley decomposition we establish results analogous to (1.1) and (1.3), we prove the Sobolev embedding theorems, and the GagliardoNirenberg inequality. Another tool of the Littlewood-Paley decomposition associated with the Dunkl operators is to generalize the paraproduct operators defined by
J.M. Bony. We prove results similar to [2].
The paper is organized as follows. In Section 2 we recall the main results about
the harmonic analysis associated with the Dunkl operators. We study in Section
3 the Littlewood-Paley decomposition associated
with the Dunkl operators, we give
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the sufficient condition on up so that u := up belongs to Sobolev or Hölder spaces
associated with the Dunkl operators. We finish this section by the Littlewood-Paley
decomposition of the Lebesgue spaces Lpk (Rd ) associated with the Dunkl operators.
In Section 4 we give some applications. More precisely we establish the Sobolev
embedding theorems and the Gagliardo-Nirenberg inequality. Section 5 is devoted
to defining the paraproduct operators associated with the Dunkl operators and to
giving the paraproduct algorithm.
Littlewood-Paley Decomposition
Hatem Mejjaoli
vol. 9, iss. 4, art. 95, 2008
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2.
The Eigenfunction of the Dunkl Operators
In this section we collect some notations and results on Dunkl operators and the
Dunkl kernel (see [3], [4] and [5]).
2.1.
Reflection Groups, Root System and Multiplicity Functions
p
We consider Rd with the euclidean scalar product h·, ·i and kxk = hx, xi. On
P
Cd , k · k denotes also the standard Hermitian norm, while hz, wi = dj=1 zj wj .
For α ∈ Rd \{0}, let σα be the reflection in the hyperplane Hα ⊂ Rd orthogonal
to α, i.e.
(2.1)
hα, xi
σα (x) = x − 2
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. For a given root system R the reflections σα , α ∈ R, generate a finite
group W ⊂ O(d), called the reflection group associated with R. All reflections
in W correspond to suitable pairs of roots. For a given β ∈ Rd \∪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+ . We will assume that hα, αi = 2 for all α ∈ R+ .
A function k : R −→ C on a root system R is called a multiplicity function if it
is invariant under the action of the associated reflection group W . If one regards k
as a function on the corresponding reflections, this means that k is constant on the
conjugacy classes of reflections in W . For brevity, we introduce the index
X
(2.2)
γ = γ(k) =
k(α).
α∈R+
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vol. 9, iss. 4, art. 95, 2008
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Moreover, let ωk denote the weight function
Y
(2.3)
ωk (x) =
| hα, xi |2k(α) ,
α∈R+
which is invariant and homogeneous of degree 2γ. We introduce the Mehta-type
constant
Z
kxk2
(2.4)
ck =
e− 2 ωk (x)dx.
Rd
2.2.
Littlewood-Paley Decomposition
Hatem Mejjaoli
vol. 9, iss. 4, art. 95, 2008
Dunkl operators-Dunkl kernel and Dunkl intertwining operator
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Notations. We denote by
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– C(Rd ) (resp. Cc (Rd )) the space of continuous functions on Rd (resp. with
compact support).
– E(Rd ) the space of C ∞ -functions on Rd .
d
∞
d
– S(R ) the space of C -functions on R which are rapidly decreasing as their
derivatives.
– D(Rd ) the space of C ∞ -functions on Rd which are of compact support.
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We provide these spaces with the classical topology.
Consider also the following spaces
– E 0 (Rd ) the space of distributions on Rd with compact support. It is the topological dual of E(Rd ).
– S 0 (Rd ) the space of temperate distributions on Rd . It is the topological dual of
S(Rd ).
The Dunkl operators Tj , j = 1, . . . , d, on Rd associated with the finite reflection
group W and multiplicity function k are given by
(2.5)
Tj f (x) =
X
f (x) − f (σα (x))
∂
f (x) +
k(α)αj
,
∂xj
hα, xi
α∈R
f ∈ C 1 (Rd ).
Littlewood-Paley Decomposition
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Hatem Mejjaoli
In the case k = 0, the Tj , j = 1, . . . , d, reduce to the corresponding partial derivatives. In this paper, we will assume throughout that k ≥ 0.
For y ∈ Rd , the system
(
Tj u(x, y) = yj u(x, y), j = 1, . . . , d,
for all y ∈ Rd
u(0, y) = 1,
d
admits a unique analytic solution on R , which will be denoted by K(x, y) and called
the Dunkl kernel. This kernel has a unique holomorphic extension to Cd × Cd . The
Dunkl kernel possesses the following properties.
Proposition 2.1. Let z, w ∈ Cd , and x, y ∈ Rd .
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i)
(2.6)
K(z, w) = K(w, z), K(z, 0) = 1 and
K(λz, w) = K(z, λw), for all λ ∈ C.
ii) For all ν ∈ Nd , x ∈ Rd and z ∈ Cd , we have
(2.7)
|Dzν K(x, z)| ≤ kxk|ν| exp(kxkk Re zk),
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and for all x, y ∈ Rd :
|K(ix, y)| ≤ 1,
(2.8)
with Dzν =
∂ν
ν
ν
∂z1 1 ···∂zdd
and |ν| = ν1 + · · · + νd .
iii) For all x, y ∈ Rd and w ∈ W we have
(2.9)
K(−ix, y) = K(ix, y) and
K(wx, wy) = K(x, y).
The Dunkl intertwining operator Vk is defined on C(Rd ) by
Z
(2.10)
Vk f (x) =
f (y)dµx (y), for all x ∈ Rd ,
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vol. 9, iss. 4, art. 95, 2008
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Rd
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where dµx is a probability measure given on Rd , with support in the closed ball
B(0, kxk) of center 0 and radius kxk.
2.3.
The Dunkl Transform
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The results of this subsection are given in [7] and [18].
Notations. We denote by
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– Lpk (Rd ) the space of measurable functions on Rd such that
Z
kf kLpk (Rd ) =
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|f (x)|p ωk (x)dx
Rd
kf kL∞
d = ess sup |f (x)| < ∞.
k (R )
x∈Rd
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p1
< ∞,
if 1 ≤ p < ∞,
– H(Cd ) the space of entire functions on Cd , rapidly decreasing of exponential
type.
– H(Cd ) the space of entire functions on Cd , slowly increasing of exponential
type.
We provide these spaces with the classical topology.
The Dunkl transform of a function f in D(Rd ) is given by
Z
1
f (x)K(−iy, x)ωk (x)dx, for all y ∈ Rd .
(2.11)
FD (f )(y) =
c k Rd
It satisfies the following properties:
i) For f in
L1k (Rd )
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kFD (f )kL∞
d
k (R )
1
≤ kf kL1k (Rd ) .
ck
d
ii) For f in S(R ) we have
FD (Tj f )(y) = iyj FD (f )y),
j = 1, . . . , d.
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FD (f )(x)K(ix, y)ωk (x) dx,
f (y) =
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Z
a.e.
Rd
Theorem 2.2. The Dunkl transform FD is a topological isomorphism.
i) From S(Rd ) onto itself.
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∀y ∈ Rd ,
iii) For all f in L1k (Rd ) such that FD (f ) is in L1k (Rd ), we have the inversion formula
(2.14)
Hatem Mejjaoli
vol. 9, iss. 4, art. 95, 2008
we have
(2.12)
(2.13)
Littlewood-Paley Decomposition
ii) From D(Rd ) onto H(Cd ).
The inverse transform FD−1 is given by
∀y ∈ Rd ,
(2.15)
FD−1 (f )(y) = FD (f )(−y),
f ∈ S(Rd ).
Theorem 2.3. The Dunkl transform FD is a topological isomorphism.
i) From S 0 (Rd ) onto itself.
Littlewood-Paley Decomposition
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ii) From E 0 (Rd ) onto H(Cd ).
vol. 9, iss. 4, art. 95, 2008
Theorem 2.4.
i) Plancherel formula for FD . For all f in S(Rd ) we have
Z
Z
2
(2.16)
|f (x)| ωk (x)dx =
|FD (f )(ξ)|2 ωk (ξ)dξ.
Rd
Rd
ii) Plancherel theorem for FD . The Dunkl transform f → FD (f ) can be uniquely
extended to an isometric isomorphism on L2k (Rd ).
Definition 2.5. Let y be in Rd . The Dunkl translation operator f 7→ τy f is defined
on S(Rd ) by
FD (τy f )(x) = K(ix, y)FD (f )(x),
for all x ∈ Rd .
Example 2.1. Let t > 0, we have
2
τx (e−tkξk )(y) = e−t(kxk
2 +kyk2 )
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2.4. The Dunkl Convolution Operator
(2.17)
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K(2tx, y),
for all x ∈ Rd .
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Remark 1. The operator τy , y ∈ Rd , can also be defined on E(Rd ) by
τy f (x) = (Vk )x (Vk )y [(Vk )−1 (f )(x + y)],
(2.18)
for all x ∈ Rd
(see [18]).
At the moment an explicit formula for the Dunkl translation operators is known
only in the following two cases. (See [11] and [13]).
Littlewood-Paley Decomposition
st
• 1 case: d = 1 and W = Z2 .
Hatem Mejjaoli
• 2nd case: For all f in E(Rd ) radial we have
h p
i
(2.19) τy f (x) = Vk f0
kxk2 + kyk2 + 2hx, ·i (x),
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for all x ∈ Rd ,
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with f0 the function on [0, ∞[ given by
f (x) = f0 (kxk).
Using the Dunkl translation operator, we define the Dunkl convolution product of
functions as follows (see [11] and [18]).
Definition 2.6. The Dunkl convolution product of f and g in D(Rd ) is the function
f ∗D g defined by
Z
(2.20)
f ∗D g(x) =
τx f (−y)g(y)ωk (y)dy, for all x ∈ Rd .
Rd
This convolution is commutative, associative and satisfies the following properties. (See [13]).
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Proposition 2.7.
i) For f and g in D(Rd ) (resp. S(Rd )) the function f ∗D g belongs to D(Rd )
(resp. S(Rd )) and we have
FD (f ∗D g)(y) = FD (f )(y)FD (g)(y),
for all y ∈ Rd .
ii) Let 1 ≤ p, q, r ≤ ∞, such that p1 + 1q − 1r = 1. If f is in Lpk (Rd ) and g is a
radial element of Lqk (Rd ), then f ∗D g ∈ Lrk (Rd ) and we have
Littlewood-Paley Decomposition
Hatem Mejjaoli
vol. 9, iss. 4, art. 95, 2008
(2.21)
kf ∗D gkLr (Rd ) ≤ kf kLp (Rd ) kgkLq (Rd ) .
k
k
k
iii) Let W = Zd2 . We have the same result for all f ∈ Lpk (Rd ) and g ∈ Lqk (Rd ).
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3.
Littlewood-Paley Theory Associated with Dunkl Operators
We consider now a dyadic decomposition of Rd .
3.1.
Dyadic Decomposition
For p ≥ 0 be a natural integer, we set
(3.1)
Cp = {ξ ∈ Rd ; 2p−1 ≤ kξk ≤ 2p+1 } = 2p C0
C−1 = B(0, 1) = {ξ ∈ Rd ; kξk ≤ 1}.
S
Clearly Rd = ∞
p=−1 Cp .
Remark 2. We remark that
(3.3)
Hatem Mejjaoli
vol. 9, iss. 4, art. 95, 2008
and
(3.2)
Littlewood-Paley Decomposition
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n
o
\
card q; Cp Cq 6= ∅ ≤ 2.
Now, let us define a dyadic partition of unity that we shall use throughout this
paper.
Lemma 3.1. There exist positive functions ϕ and ψ in D(Rd ), radial with supp
ψ ⊂ C−1 , and supp ϕ ⊂ C0 , such that for any ξ ∈ Rd and n ∈ N, we have
ψ(ξ) +
∞
X
ϕ(2−p ξ) = 1
p=0
and
ψ(ξ) +
n
X
p=0
ϕ(2−p ξ) = ψ(2−n ξ).
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Remark 3. It is not hard to see that for any ξ ∈ Rd
∞
X
1
≤ ψ 2 (ξ) +
ϕ2 (2−p ξ) ≤ 2.
2
p=0
(3.4)
Definition 3.2. Let λ ∈ R. For χ in S(Rd ), we define the pseudo-differentialdifference operator χ(λT ) by
Littlewood-Paley Decomposition
0
FD (χ(λT )u) = χ(λξ)FD (u),
d
u ∈ S (R ).
Hatem Mejjaoli
Definition 3.3. For u in S 0 (Rd ), we define its Littlewood-Paley decomposition associated with the Dunkl operators (or dyadic decomposition) {∆p u}∞
p=−1 as ∆−1 u =
ψ(T )u and for q ≥ 0, ∆q u = ϕ(2−q T )u.
Now we go to see in which case we can have the identity
X
Id =
∆p .
This is described by the following proposition.
d
Proposition 3.4. For u in S (R ), we have u =
hu, f i = hFD (u), FD (f )i =
p=−1
=
∞
X
p=−1
hFD (∆p u), FD (f )i
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P∞
0
d
∆p u, in the sense of S (R ).
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Proof. For any f in S(Rd ), it is easy to see that FD (f ) = ∞
p=−1 FD (∆p f ) in the
d
0
d
sense of S(R ). Then for any u in S (R ), we have
∞
X
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p≥−1
0
vol. 9, iss. 4, art. 95, 2008
p=−1
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hFD (u), FD (∆p f )i
*
=
∞
X
+
FD (∆p u), FD (f )
p=−1
*
=
∞
X
+
∆p u, f
.
p=−1
The proof is finished.
3.2.
The Generalized Sobolev Spaces
Littlewood-Paley Decomposition
In this subsection we will give a characterization of Sobolev spaces associated with
the Dunkl operators by a Littlewood-Paley decomposition. First, we recall the definition of these spaces (see [9]).
Definition 3.5. Let s be in R, we define the space Hks (Rd ) by
s
u ∈ S 0 (Rd ) : (1 + kξk2 ) 2 FD (u) ∈ L2k (Rd ) .
We provide this space by the scalar product
Z
(3.5)
hu, viHks (Rd ) =
(1 + kξk2 )s FD (u)(ξ) F D (v)(ξ)ωk (ξ)dξ,
Rd
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and the norm
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kuk2H s (Rd ) = hu, uiHks (Rd ) .
k
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Another proposition will be useful. Let Sq u = p≤q−1 ∆p u.
(3.6)
Proposition 3.6. For all s in R and for all distributions u in Hks (Rd ), we have
lim Sn u = u.
n→∞
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Proof. For all ξ in Rd , we have
FD (Sn u − u)(ξ) = (ψ(2−n ξ) − 1)FD (u)(ξ).
Hence
lim FD (Sn u − u)(ξ) = 0.
n→∞
On the other hand
(1 + kξk2 )s |FD (Sn u − u)(ξ)|2 ≤ 2(1 + kξk2 )s |FD (u)(ξ)|2 .
Thus the result follows from the dominated convergence theorem.
The first application of the Littlewood-Paley decomposition associated with the
Dunkl operators is the characterization of the Sobolev spaces associated with these
operators through the behavior on q of k∆q ukL2k (Rd ) . More precisely, we now define
a norm equivalent to the norm k · kHks (Rd ) in terms of the dyadic decomposition.
Proposition 3.7. There exists a positive constant C such that for all s in R, we have
X
1
2
kuk
22qs k∆q uk2L2 (Rd ) ≤ C |s|+1 kuk2H s (Rd ) .
s (Rd ) ≤
H
k
k
k
C |s|+1
q≥−1
Proof. Since supp FD (∆q u) ⊂ Cq , from the definition of the norm k · kHks (Rd ) , there
exists a positive constant C such that we have
1
(3.7)
2qs k∆q ukL2k (Rd ) ≤ k∆q ukHks (Rd ) ≤ C |s|+1 2qs k∆q ukL2k (Rd ) .
|s|+1
C
From (3.4) we deduce that
#
Z "
∞
X
1
kuk2H s (Rd ) ≤
ψ 2 (ξ) +
ϕ2 (2−q ξ) (1 + kξk2 )s |FD (u)(ξ)|2 ωk (ξ)dξ
k
2
d
R
q=0
≤ 2kuk2H s (Rd ) .
k
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Hence
X
1
kuk2H s (Rd ) ≤
k∆q uk2H s (Rd ) ≤ 2kuk2H s (Rd ) .
k
k
k
2
q≥−1
Thus from this and (3.7) we deduce the result.
The following theorem is a consequence of Proposition 3.7.
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Theorem 3.8. Let u be in S 0 (Rd ) and u = q≥−1 ∆q u its Littlewood-Paley decomposition. The following are equivalent:
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i) u ∈ Hks (Rd ).
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ii) q≥−1 22qs k∆q uk2L2 (Rd ) < ∞.
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k
iii) k∆q ukL2k (Rd ) ≤ cq 2−qs , with {cq } ∈ l2 .
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Remark 4. Since for u in S 0 (Rd ) we have ∆p u in S 0 (Rd ) and supp FD (∆p u) ⊂ Cp ,
from Theorem 2.3 ii) we deduce that ∆p u is in E(Rd ).
The following propositions will be very useful.
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d
e be an annulus in R and s in R. Let (up )p∈N be a sequence
Proposition 3.9. Let C
of smooth functions. If the sequence (up )p∈N satisfies
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e
supp FD (up ) ⊂ 2p C
and kup kL2k (Rd ) ≤ Ccp 2−ps , {cp } ∈ l2 ,
Close
then we have
! 21
u=
X
p≥0
up ∈ Hks (Rd ) and
kukHks (Rd ) ≤ C(s)
X
22ps kup k2L2 (Rd )
k
p≥0
.
e and C0 are two annuli, there exists an integer N0 so that
Proof. Since C
\
e = ∅.
|p − q| > N0 =⇒ 2p C0 2q C
It is clear that
|p − q| > N0 =⇒ FD (∆q up ) = 0.
Then
∆q u =
X
∆q up .
Littlewood-Paley Decomposition
Hatem Mejjaoli
|p−q|≤N0
vol. 9, iss. 4, art. 95, 2008
By the triangle inequality and definition of ∆q up we deduce that
X
kup kL2k (Rd ) .
k∆q ukL2k (Rd ) ≤
Title Page
|p−q|≤N0
Contents
Thus the Cauchy-Schwartz inequality implies that


!
X
X
X
22(q−p)s 
22qs k∆q uk2L2 (Rd ) ≤ C 
22ps kup k2L2 (Rd ) .
k
k
q≥0
q/|p−q|≤N0
kup kL2k (Rd ) ≤ Ccp 2−ps , {cp } ∈ l2 ,
p≥0
and
kukHks (Rd ) ≤ C(s)
X
22ps kup k2L2 (Rd )
k
q≥0
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Close
! 21
up ∈ Hks (Rd )
I
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then we have
u=
J
Page 18 of 46
Proposition 3.10. Let K > 0 and s > 0. Let (up )p∈N be a sequence of smooth
functions. If the sequence (up )p∈N satisfies
X
II
p≥0
From Theorem 3.8 we deduce that if kup kL2k (Rd ) ≤ Ccp 2−ps then u ∈ Hks (Rd ).
supp FD (up ) ⊂ B(0, K2p ) and
JJ
.
Proof. Since supp FD (up ) ⊂ B(0, K2p ), there exists N1 such that
X
∆q u =
∆q up .
p≥q−N1
So, we get that
2qs k∆q ukL2k (Rd ) ≤
X
2qs kup kL2k (Rd )
Littlewood-Paley Decomposition
p≥q−N1
=
X
Hatem Mejjaoli
(q−p)s ps
2
vol. 9, iss. 4, art. 95, 2008
2 kup kL2k (Rd ) .
p≥q−N1
Title Page
Since s > 0, the Cauchy-Schwartz inequality implies
X
22qs k∆q uk2L2 (Rd ) ≤
k
q
Contents
22N1 s X 2ps
2 kup k2L2 (Rd ) .
k
1 − 2−s p
From Theorem 3.8 we deduce the result.
Proposition 3.11. Let s > 0 and (up )p∈N be a sequence of smooth functions. If the
sequence (up )p∈N satisfies
up ∈ E(Rd )
and
for all µ ∈ Nd , kT µ up kL2k (Rd ) ≤ Ccp,µ 2−p(s−|µ|) , {cp,µ } ∈ l2 ,
! 21
u=
p≥0
II
J
I
Page 19 of 46
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then we have
X
JJ
up ∈ Hks (Rd )
and
kukHks (Rd ) ≤ C(s)
X
22ps kup k2L2 (Rd )
k
p≥0
.
P
Proof. By the assumption we first have u =
up ∈ L2k (Rd ). Take µ ∈ Nd with
−p
d
|µ| = s0 > s > 0, and χp (ξ) = χ(2 ξ) ∈ D(R ) with supp χ ⊂ B(0, 2), χ(ξ) = 1,
kξk ≤ 1 and 0 ≤ χ ≤ 1, then
supp χp (1 − χp ) ⊂ ξ ∈ Rd ; 2p ≤ kξk ≤ 2p+2 .
Set
FD (up )(ξ) = χp (ξ)FD (up )(ξ) + (1 − χp (ξ))FD (up )(ξ)
(2)
= FD (u(1)
p )(ξ) + FD (up )(ξ),
Littlewood-Paley Decomposition
Hatem Mejjaoli
vol. 9, iss. 4, art. 95, 2008
and we have
Title Page
kup k2L2 (Rd ) = kFD (up )k2L2 (Rd )
k
k
Z
Z
(1)
2
2
=
|FD (up )(ξ)| ωk (ξ)dξ +
|FD (u(2)
p )(ξ)| ωk (ξ)dξ
d
d
R
R
Z
2
+2
|FD (up )(ξ)| χp (ξ)(1 − χp (ξ))ωk (ξ)dξ .
Rd
k
P
p
(1)
up , u(2) =
k
P
p
(2)
II
J
I
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k
k
Similarly, using Theorem 3.1 of [9], we obtain
(1) 2
2
up s0 d + u(2)
s0 d ≤ kup k2 s0 d ≤ c2p 2−2p(s−s0 ) .
p
H (R )
H (R )
H (R )
Set u(1) =
JJ
Page 20 of 46
Since 0 ≤ χp (ξ)(1 − χp (ξ)) ≤ 1, we deduce that
(1) 2
2
up 2 d + u(2)
2 d ≤ kup k2 2 d ≤ c2p 2−2ps .
p
L (R )
L (R )
L (R )
k
Contents
k
up , then u = u(1) + u(2) , and from Proposition 3.10
Close
(2)
we deduce that u(1) belongs to Hks (Rd ). For u(2) the definition of up gives that
2
Z X
k∆q (u(2) )k2L2 (Rd ) =
ϕ(2−q ξ)FD (u(2)
)(ξ)
ωk (ξ)dξ.
p
k
d
R
p≤q+1
Thus by the Cauchy-Schwartz inequality we have
k∆q (u(2) )k2L2 (Rd )
Littlewood-Paley Decomposition
k
≤
X
2−2p(s−s0 )
! Z
X
Rd p≤q+1
p≤q+1
≤
!
2
22p(s−s0 ) |ϕ(2−q ξ)FD (u(2)
p )(ξ)| ωk (ξ)dξ
2
1 − 2−2(q+2)(s−s0 ) −2qs0 X 2p(s−s0 ) ∆q u(2)
s0 d .
2
2
p
−(s−s
)
Hk (R )
0
1−2
p≤q+1
Hatem Mejjaoli
vol. 9, iss. 4, art. 95, 2008
Title Page
Contents
Moreover, since s0 > s > 0,
1 − 2−2(q+2)(s−s0 ) −2qs0
2
≤ C2−2qs ,
−(s−s
)
0
1−2
and C is independent of q. Now set
X
2
s0 d ,
c2q =
22p(s−s0 ) ∆q u(2)
p
H (R )
II
J
I
Page 21 of 46
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k
p≤q+1
JJ
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then
X
22qs k∆q (u(2) )k2L2 (Rd ) ≤
X
c2q ≤
k
q≥−1
Thus by Theorem 3.8 we deduce that u
=
2
s0 d < ∞.
22p(s−s0 ) u(2)
p
H (R )
k
p
q≥−1
(2)
X
P
q
∆q (u(2) ) belongs to Hks (Rd ).
Corollary 3.12. The spaces Hks (Rd ) do not depend on the choice of the function ϕ
and ψ used in the Definition 3.3.
Close
3.3.
The Generalized Hölder Spaces
Definition 3.13. For α in R, we define the Hölder space Ckα (Rd ) associated with the
Dunkl operators as the set of u ∈ S 0 (Rd ) satisfying
kukCkα (Rd ) = sup 2pα k∆p ukL∞
d < ∞,
k (R )
p≥−1
where u =
P
p≥−1
∆p u is its Littlewood-Paley decomposition.
In the following proposition we give sufficient conditions so that the series
belongs to the Hölder spaces associated with the Dunkl operators.
Littlewood-Paley Decomposition
P
q
uq
Hatem Mejjaoli
vol. 9, iss. 4, art. 95, 2008
Proposition 3.14.
Title Page
e be an annulus in Rd and α ∈ R. Let (up )p∈N be a sequence of smooth
i) Let C
functions. If the sequence (up )p∈N satisfies
e and kup kL∞ (Rd ) ≤ C2−pα ,
supp FD (up ) ⊂ 2p C
k
then we have
X
up ∈ Ckα (Rd ) and
u=
kukCkα (Rd ) ≤ C(α) sup 2pα kup kL∞
d .
k (R )
p≥0
p≥0
ii) Let K > 0 and α > 0. Let (up )p∈N be a sequence of smooth functions. If the
sequence (up )p∈N satisfies
supp FD (up ) ⊂ B(0, K2p )
then we have
X
up ∈ Ckα (Rd ) and
u=
p≥0
and
−pα
kup kL∞
,
d ≤ C2
k (R )
kukCkα (Rd ) ≤ C(α) sup 2pα kup kL∞
d .
k (R )
p≥0
Contents
JJ
II
J
I
Page 22 of 46
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Proof. The proof uses the same idea as for Propositions 3.9 and 3.10.
Proposition 3.15. The distribution defined by
X
g(x) =
K(ix, 2p e), with e = (1, . . . , 1),
p≥0
d
belongs to Ck0 (Rd ) and does not belong to L∞
k (R ).
Proposition 3.16. Let ε ∈]0, 1[ and f in Ckε (Rd ), then there exists a positive constant
!
C such that
kf kCkε (Rd )
C
kf kL∞
kf kCk0 (Rd ) log e +
.
d ≤
k (R )
ε
kf kCk0 (Rd )
P
Proof. Since f = p≥−1 ∆p f ,
X
X
∞
kf kL∞
≤
k∆
f
k
+
k∆p f kL∞
d
d
d ,
p
Lk (R )
k (R )
k (R )
p≤N −1
p≥N
with N is a positive integer that will be chosen later. Since f ∈ Ckε (Rd ), using the
definition of generalized Hölderien norms, we deduce that
We take
we obtain
2−(N −1)ε
0
kf kL∞
≤
(N
+
1)kf
k
+
kf kCkε (Rd ) .
d
d
Ck (R )
k (R )
2ε − 1
"
#
kf kCkε (Rd )
1
N =1+
log2
,
ε
kf kCk0 (Rd )
"
!#
kf kCkε (Rd )
C
kf kL∞
kf kCk0 (Rd ) 1 + log
.
d ≤
k (R )
ε
kf kCk0 (Rd )
This implies the result.
Littlewood-Paley Decomposition
Hatem Mejjaoli
vol. 9, iss. 4, art. 95, 2008
Title Page
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JJ
II
J
I
Page 23 of 46
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Now we give the characterization of Lpk (Rd ) spaces by using the dyadic decomposition.
If (fj )j∈N is a sequence of Lpk (Rd )-functions, we set
! 12 X
2
k(fj )kLpk (l2 ) = ,
|f
(x)|
j
j∈N
p d
Lk (R )
the norm in
Littlewood-Paley Decomposition
Lpk (Rd , l2 (N)).
Hatem Mejjaoli
Lpk (Rd )).
Theorem 3.17 (Littlewood-Paley decomposition of
and 1 < p < ∞. Then the following assertions are equivalent
Let f be in S 0 (Rd )
Title Page
i) f ∈ Lpk (Rd ),
ii) S0 f ∈ Lpk (Rd ) and
vol. 9, iss. 4, art. 95, 2008
P
2
j∈N |∆j f (x)|
21
Contents
∈ Lpk (Rd ).
Moreover, the following norms are equivalent :
! 12 X
2
kf kLpk (Rd ) and kS0 f kLpk (Rd ) + |∆j f (x)|
j∈N
Lpk (Rd )
Proof. If f is in L2k (Rd ), then from Proposition 3.7 we have
! 12 X
2
|∆j f (x)|
≤ kf k2L2 (Rd ) .
k
j∈N
2 d
Lk (R )
Thus the mapping
Λ1 : f 7→ (∆j f )j∈N ,
.
JJ
II
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I
Page 24 of 46
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is bounded from L2k (Rd ) into L2k (Rd , l2 (N)).
On the other hand, from properties of ϕ we see that
k(ϕ
ej (x))j kl2 ≤ Ckxk−(d+2γ) ,
k(∂yi ϕ
ej (x))j kl2 ≤ Ckxk−(d+2γ) ,
for x 6= 0,
for x 6= 0, i = 1, . . . , d,
where
ϕ
ej (x) = 2j(d+2γ) FD−1 (ϕ)(2j x).
We may then apply the theory of singular integrals to this mapping Λ1 (see [14]).
Thus we deduce that
k∆j f kLpk (l2 ) ≤ Cp,k kf kLpk (Rd ) ,
for 1 < p < ∞.
Littlewood-Paley Decomposition
Hatem Mejjaoli
vol. 9, iss. 4, art. 95, 2008
Title Page
The converse uses the same idea. Indeed we put
φej =
1
X
ϕ
ej+i .
i=−1
JJ
II
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I
Page 25 of 46
From Proposition 3.7 the mapping
Λ2 : (fj )j∈N 7→
Contents
X
fj ∗D φej ,
Full Screen
j∈N
is bounded from L2k (Rd , l2 (N)) into L2k (Rd ).
On the other hand, from properties of ϕ we see that
k(φej (x))j kl2 ≤ Ckxk−(d+2γ) ,
k(∂yi φej (x))j kl2 ≤ Ckxk−(d+2γ) ,
Go Back
for x 6= 0,
for x 6= 0, i = 1, . . . , d.
Close
We may then apply the theory of singular integrals to this mapping Λ2 (see [14]).
Thus we obtain
X
∆j f ≤ Cp,k k∆j f kLpk (l2 ) .
p
j∈N
Lk (Rd )
Littlewood-Paley Decomposition
Hatem Mejjaoli
vol. 9, iss. 4, art. 95, 2008
Title Page
Contents
JJ
II
J
I
Page 26 of 46
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4.
Applications
4.1.
Estimates of the Product of Two Functions
Proposition 4.1.
i) Let u, v ∈ Ckα (Rd ) and α > 0 then uv ∈ Ckα (Rd ), and
i
h
α
∞
α
kuvkCkα (Rd ) ≤ C kukL∞
kvk
+
kvk
kuk
d
d
d
d
Ck (R )
Lk (R )
Ck (R ) .
k (R )
Littlewood-Paley Decomposition
Hatem Mejjaoli
vol. 9, iss. 4, art. 95, 2008
ii) Let u, v ∈ Hks (Rd )
kuvkHks (Rd )
d
s
d
L∞
k (R ) and s > 0 then uv ∈ Hk (R ), and
i
h
s
∞
s
∞
≤ C kukLk (Rd ) kvkHk (Rd ) + kvkLk (Rd ) kukHk (Rd ) .
T
Title Page
Contents
P
P
Proof. Let u =
p ∆p u and v =
q ∆q v be their Littlewood-Paley decompositions. Then we have
X
uv =
∆p u∆q v
=
=
X X
=
X
q
∆p u∆q v +
XX
∆p u∆q v +
XX
q
p≤q−1
p
p≤q−1
Sq u∆q v +
q
=
X
1
II
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I
Page 27 of 46
p,q
X X
q
JJ
X
p
+
X
2
.
∆p u∆q v
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p≥q
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∆p u∆q v
q≤p
Sp+1 v∆p u
Close
We have
supp (FD (Sq u∆q v)) = supp (FD (∆q v) ∗D FD (Sq u)).
Hence from Theorem 2.2 we deduce that supp (FD (Sq u∆q v)) ⊂ B(0, C2q ).
i) If u and v are in Ckα (Rd ), then we have
kSq u∆q vkL∞
d ≤ kSq ukL∞ (Rd ) k∆q vkL∞ (Rd ) ,
k (R )
k
k
−qα
≤ CkukL∞
.
d kvkC α (Rd ) 2
k (R )
k
From Proposition 3.14 ii) we deduce
X
≤ CkukL∞
d kvkC α (Rd ) .
k (R )
k
α d
1
Hatem Mejjaoli
vol. 9, iss. 4, art. 95, 2008
Title Page
Ck (R )
Contents
Similarly we prove that
X
2
Littlewood-Paley Decomposition
≤ CkvkL∞
d kukC α (Rd ) ,
k (R )
k
Ckα (Rd )
kSq u∆q vkL2k (Rd ) ≤ kSq ukL∞
d k∆q vkL2 (Rd ) ,
k (R )
k
−qs
≤ CkukL∞
.
d kvkH s (Rd ) cq 2
k (R )
k
Hks (Rd )
II
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I
Page 28 of 46
and this implies the result.
ii) If u and v are in Hks (Rd ), then we have
Thus Proposition 3.10 gives
X
1 JJ
≤ CkukL∞
d kvkH s (Rd ) .
k (R )
k
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Close
Similarly, we prove that
X
2 ≤ CkvkL∞
d kukH s (Rd ) ,
k (R )
k
Hks (Rd )
and this implies the result.
Corollary 4.2. For s >
4.2.
d
2
+ γ, Hks (Rd ) is an algebra.
Littlewood-Paley Decomposition
Hatem Mejjaoli
vol. 9, iss. 4, art. 95, 2008
Sobolev Embedding Theorem
Using the Littlewood-Paley decomposition, we have a very simple proof of Sobolev
embedding theorems:
Theorem 4.3. For any s > γ + d2 , we have the continuous embedding
Hks (Rd )
,→
s−γ− d2
Ck
(Rd ).
Proof. Let u be in Hks (Rd ), u = p≥−1 ∆p u the Littlewood-Paley decomposition.
Take φ in D(Rd ) such that φ(ξ) = 1 on C0 , and
0
d 1
supp φ ⊂ C0 = ξ ∈ R , ≤ kξk ≤ 3 .
3
P
Setting φp (ξ) = φ(2−p ξ), we obtain
FD (∆p u)(ξ) = FD (∆p u)(ξ)φ(2−p ξ).
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II
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Hence
Z
FD (∆p u)(ξ)φ(2−p ξ)K(ix, ξ)ωk (ξ)dξ,
∆p u(x) =
Rd
Z
|FD (∆p u)(ξ)kφ(2−p ξ)|ωk (ξ)dξ.
|∆p u(x)| ≤
Rd
The Cauchy-Schwartz inequality and Theorem 3.8 give that
Z
12 Z
12
|φ(2−p ξ)|2 ωk (ξ)dξ
k∆p ukL∞
|FD (∆p u)(ξ)|2 ωk (ξ)dξ
d ≤
k (R )
Rd
≤ C2
k∆p ukL2k (Rd )
−p(s−γ− d2 )
≤ C2
Title Page
cp .
Contents
s−γ− d2
Then from Definition 3.13 we deduce that u ∈ Ck
(Rd ).
Theorem 4.4. For any 0 < s < γ + d2 , we have the continuous embedding
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Page 30 of 46
Hks (Rd ) ,→ Lpk (Rd ),
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2(2γ+d)
.
2γ+d−2s
Proof. Let f be in S(Rd ), we have, due to Fubini’s theorem,
Z ∞
n
o
p
p−1
(4.1)
kf kLp (Rd ) = p
λ mk |f | ≥ λ dλ,
k
where
Hatem Mejjaoli
vol. 9, iss. 4, art. 95, 2008
Rd
p(γ+ d2 )
where p =
Littlewood-Paley Decomposition
0
n
o Z
mk |f | ≥ λ =
{x; |f (x)|≥λ}
ωk (x)dx.
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Close
P
For
A
>
0,
we
set
f
=
f
+
f
with
f
=
1,A
2,A
1,A
2j <A ∆j f and f2,A =
P
∆
f
.
j
j
2 ≥A
We have
X
X
∞ (Rd ) ≤
kf1,A kL∞
k∆
f
k
kFD (∆j f )kL1k (Rd ) .
d) ≤
j
(R
L
k
k
2j <A
2j <A
Using the Cauchy-Schwartz inequality, the Parseval’s identity associated with the
Dunkl operators and Theorem 3.8, we obtain
X
γ+ d2 −s
j(γ+ d2 −s)
s (Rd ) ≤ CA
kf1,A kL∞
2
c
kf
k
kf kHks (Rd ) .
d) ≤
j
(R
H
k
k
Littlewood-Paley Decomposition
Hatem Mejjaoli
vol. 9, iss. 4, art. 95, 2008
2j <A
Title Page
On the other hand for all λ > 0, we have
n
o
λ
λ
(4.2)
mk |f | ≥ λ ≤ mk |f1,A | ≥
+ mk |f2,A | ≥
.
2
2
JJ
II
From (4.2) we infer that if we take
J
I
A = Aλ =
λ
4Ckf kHks (Rd )
!
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Page 31 of 46
1
γ+ d
2 −s
,
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then
kf1,Aλ kL∞
d ≤
k (R )
Hence
mk
λ
|f1,Aλ | ≥
2
λ
.
4
= 0.
Close
From (4.1) and (4.2) we deduce that
Z ∞
n
o
p
kf kLp (Rd ) ≤ p
λp−1 mk 2|f2,Aλ | ≥ λ dλ.
k
0
Moreover the Bienaymé-Tchebytchev inequality yields
n
o
4
mk 2|f2,Aλ | ≥ λ ≤ 2 kf2,Aλ k2L2 (Rd ) .
k
λ
Thus we obtain
kf kpLp (Rd ) ≤ p
k
Z
Littlewood-Paley Decomposition
Hatem Mejjaoli
vol. 9, iss. 4, art. 95, 2008
∞
λp−3 kf2,Aλ k2L2 (Rd ) dλ.
k
0
On the other hand, by using the Cauchy-Schwartz inequality for all ε > 0, we have
2
Z X
2
ωk (x)dx
kf2,Aλ kL2 (Rd ) =
∆
f
(x)
j
k
Rd j
2 ≥Aλ



Z X
X
≤
22jε |∆j f (x)|2 ωk (x)dx 
2−2jε 
Rd
≤ A−2ε
λ
2j ≥Aλ
X
2j ≥Aλ
22jε k∆j f k2L2 (Rd ) .
k
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2j ≥Aλ
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So by using the definition of Aλ and the Fubini theorem, we can write
Z ∞
X
p
kf kLp (Rd ) ≤ p
λp−3 A−2ε
22jε k∆j f k2L2 (Rd ) dλ
λ
k
0
k
2j ≥Aλ
d
≤C
XZ
4C2j(γ+ 2 −s) kf kH s (Rd )
k
λ
p−3−
2ε
γ+ d
2 −s
k
0
j≥−1
≤ Ckf kp−2
H s (Rd )
X
2εd
dλ 4Ckf kHks (Rd ) γ+ 2 −s 22jε k∆j f k2L2 (Rd )
d
2j(p−2)(γ+ 2 −s) k∆j f k2L2 (Rd )
k
k
j≥−1
≤ Ckf kp−2
H s (Rd )
X
22js k∆j f k2L2 (Rd ) ≤ Ckf kpH s (Rd ) .
k
k
k
Littlewood-Paley Decomposition
j≥−1
Hatem Mejjaoli
This implies the result.
vol. 9, iss. 4, art. 95, 2008
Definition 4.5. We define the space BM Ok as the set of functions u ∈ L1loc,k (Rd )
satisfying
Z
1
sup
|u(x) − uB |ωk (x)dx < ∞,
B mesk (B) B
where
Z
1
B = B(x0 , R), uB =
u(x)ωk (x)dx
mesk (B) B
Z
denote the average of u on B and mesk (B) =
ωk (x)dx.
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Page 33 of 46
B
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Theorem 4.6. We have the continuous embedding
d
Hk2
+γ
Full Screen
(Rd ) ,→ BM Ok .
N
Proof. For R > 0 small enough, let N be such that 2 =
Set u = u(1) + u(2) with
u(1) =
N
−1
X
p=−1
∆p u
and
u(2) =
[ R1 ].
X
p≥N
d
+γ
2
Let u be in Hk
∆p u.
Close
d
(R ).
From the Cauchy-Schwartz inequality we have
2
Z
Z
1
1
|u(x) − uB |ωk (x)dx ≤
|u(x) − uB |2 ωk (x)dx.
mesk (B) B
mesk (B) B
It is easy to see that this implies
2
Z
1
|u(x) − uB |ωk (x)dx
mesk (B) B
Z Z 2
2
2
(1)
(2)
(1) (2) ≤
u (x) − uB ωk (x)dx +
u (x) − uB ωk (x)dx .
mesk (B) B
B
Moreover, from the mean value theorem, we have
Z Z
2
(1) 2
R2
1
(1)
(1) Du (x) ωk (x)dx,
u (x) − uB ωk (x)dx ≤
mesk (B) B
mesk (B) B
≤ R2 kDu(1) k2L∞ (Rd ) .
k
kDu kL∞
d ≤
k (R )
Hatem Mejjaoli
vol. 9, iss. 4, art. 95, 2008
Title Page
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Page 34 of 46
By (2.7) we deduce that
(1)
Littlewood-Paley Decomposition
Z
kξk|FD (u(1) )(ξ)|ωk (ξ)dξ.
Rd
By recalling that supp FD (∆p u) ⊂ Cp and |FD (∆p u)(ξ)| ≤ |FD (u)(ξ)|, we apply
the Parseval identity associated with the Dunkl operators and the Cauchy-Schwartz
inequality. We deduce that
!2
Z
N
−1 Z
X
1
(1)
|u(1) (x) − uB |2 ωk (x)dx ≤ R2
kξk |FD (∆p u)(ξ)|ωk (ξ)dξ
mesk (B) B
C
p
p=−1
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≤R
2
Z
kξk
2−2γ−d
ωk (ξ)dξ kuk2
d +γ
Hk2 (Rd )
B(0,2N )
≤ C22N R2 kuk2
d +γ
Hk2
.
(Rd )
For the second term, we have
Z
Z
2
1
(2) 2
(2)
|u (x) − uB | ωk (x)dx ≤
|u(2) (x)|2 ωk (x)dx
mesk (B) B
mesk (B) B
Z
−d−2γ
|FD (u)(ξ)|2 ωk (ξ)dξ
≤ CR
kξk≥2N
−d−2γ
2
≤ C(2N R)
kuk
d +γ
Hk2
.
Hatem Mejjaoli
vol. 9, iss. 4, art. 95, 2008
(Rd )
Hence,
Title Page
(4.3)
Littlewood-Paley Decomposition
1
mesk (B)
2
Z
|u(x) − uB |ωk (x)dx
≤ Ckuk2
d +γ
Hk2 (Rd )
B
d
Contents
.
+γ
We have proved (4.3) for small R, since u ∈ Hk2 (Rd ) ⊂ L2k (Rd ), (4.3) is evident
for R > R0 with constant C = C(R0 ). This implies the continuous embedding
d
+γ
2
Hk
(Rd ) ,→ BM Ok .
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4.3.
Gagliardo-Nirenberg Inequality
We will use the generalized Sobolev space Wks,r (Rd ) associated with the Dunkl operators defined as
s
Wks,r (Rd ) = u ∈ S 0 (Rd ) : (−4k ) 2 u ∈ Lrk (Rd ) ,
with
4k u =
d
X
Tj2 u.
j=1
The main result of this subsection is the following theorem.
Theorem 4.7. Let f be in Wks,r (Rd ) ∩ Lqk (Rd ) with q, r ∈ [1, ∞] and s ≥ 0. Then f
belongs to Wkt,p (Rd ), and we have
t
s 1−θ
(−4k ) 2 f p d ≤ Ckf kθLqk (Rd ) (−4k ) 2 f Lr (Rd ) ,
Lk (R )
where
1
p
=
θ
q
+
1−θ
,
r
Hatem Mejjaoli
vol. 9, iss. 4, art. 95, 2008
k
t = (1 − θ)s and θ ∈]0, 1[.
Title Page
Proof. First, we prove this theorem for q and r in ]1, ∞]. Let f be in S(Rd ). It is
easy to see that
X
X
t−s
t
t
s (−4k ) 2 f =
(−4k ) 2 ∆j f +
(−4k ) 2 ∆j (−4k ) 2 f ,
j≤A
Littlewood-Paley Decomposition
Contents
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j>A
Page 36 of 46
where A will be chosen later.
On the other hand, by a simple calculation, if a is a homogenous function in
C ∞ (R∗ ) of degree m, we can write
X
1
(4.4)
a (−4k ) 2 ∆j f = 2jm+j(d+2γ) b(δ2j ) ∗D
∆j 0 f,
|j−j 0 |≤1
where δ2j is defined by δ2j x = 2j x, x ∈ Rd and b is in S(Rd ) such that
FD (b)(ξ) = ϕ(ξ)a(kξk).
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We proceed as in [12, p. 21] to obtain
1
2
(4.5)
∆
f
(x)
a
(−4
)
≤ C2jm Mk f (x),
k
j
where Mk (f ) is a maximal function of f associated with the Dunkl operators (see
[13]).
t−s
Hence by applying (4.5) for a(r) = rt and a(r) = r 2 , we get
!
X
X
t
s
2jt Mk f (x) +
2j(t−s) Mk ((−4k f ) 2 )(x)
(−4k ) 2 f (x) ≤ C
j≤A
Littlewood-Paley Decomposition
Hatem Mejjaoli
vol. 9, iss. 4, art. 95, 2008
j>A
s ≤ C2tA Mk f (x) + C2(t−s)A Mk (−4k ) 2 f (x).
We minimize over A to obtain
st
1− st s t
2
2
Mk (−4k ) f (x) .
(−4k ) f (x) ≤ C Mk f (x)
By this inequality and the Hölder inequality, we have
1−θ
s
t
≤ CkMk f kθLq (Rd ) Mk ((−4k ) 2 f )Lr (Rd ) ,
(−4k ) 2 f p
Lk (Rd )
k
k
with θ = 1 − st .
Now, we apply Theorem 6.1 of [13] to deduce the result if q and r ∈]1, ∞].
Now, we assume q = r = 1. Let f be in S(Rd ). We have
X
X
t−s
t
t
s
2
2
2
2
+ (−4k ) ∆j (−4k ) f (−4k ) f 1 d ≤ (−4k ) ∆j f Lk (R )
j≤A
j>A
L1k (Rd )
L1k (Rd )
s ≤ C2(1−θ)sA kf kL1k (Rd ) + C2−θsA (−4k ) 2 f L1 (Rd ) .
k
By minimizing over A, we obtain the result.
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5.
Paraproduct Associated with the Dunkl Operators
In this section, we are going to study how the product acts on Sobolev and Hölder
spaces associated with the Dunkl operators. This could be very useful in nonlinear
partial differential-difference equations. Of course, we shall use the LittlewoodPaley decomposition associated with the Dunkl operators. Let us consider two temperate distributions u and v. We write
X
X
u=
∆p u and v =
∆q v.
p
q
Formally, the product can be written as
X
uv =
∆p u∆q v.
Littlewood-Paley Decomposition
Hatem Mejjaoli
vol. 9, iss. 4, art. 95, 2008
Title Page
Contents
p,q
Now we introduce the paraproduct operator associated with the Dunkl operators.
Definition 5.1. We define the paraproduct operator Πa : S 0 (Rd ) → S 0 (Rd ) by
X
Πa u =
(Sq−2 a)∆q u,
q≥1
where uP∈ S 0 (Rd ); {∆q a} and {∆q u} are the Littlewood-Paley decompositions and
Sq a = p≤q−1 ∆p a.
Let R indicate the following bilinear symmetric operator on S 0 (Rd ) defined by
X
R(u, v) =
∆p u∆q v,
for all u, v ∈ S 0 (Rd ).
|p−q|≤1
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Obviously from Definition 5.1 it is clear that
uv = Πu v + Πv u + R(u, v).
The following theorems describe the action of the paraproduct and remainder on the
Sobolov and the Hölder spaces associated with the Dunkl operators.
Theorem 5.2. There exists a positive constant C such that the operator Π has the
following properties:
s
s
1. kΠkL(L∞
≤C
d
d
d
k (R )×Ck (R ),Ck (R ))
s+1
s
s
2. kΠkL(L∞
≤ C s+1 ,
d
d
d
k (R )×Hk (R ),Hk (R ))
for all s > 0.
Title Page
3. kΠkL(C t (Rd )×H s (Rd ),H s+t (Rd )) ≤
C s+t+1
,
−t
for all s, t with s + t > 0 and t < 0.
4. kΠkL(C t (Rd )×C s (Rd ),C s+t (Rd )) ≤
C s+t+1
for all s, t with s + t > 0 and t < 0.
k
k
5. kΠk
s<
k
k
k
−t
k
,
d
s+t−γ− d
2 (Rd ))
L(Hks (Rd )×Hkt (Rd ),Hk
d
2
≤ C s+t−γ− 2 +1 ,
Hatem Mejjaoli
vol. 9, iss. 4, art. 95, 2008
for all s > 0.
,
Littlewood-Paley Decomposition
with s + t > γ +
+ γ.
Proof. Let u and v be in S 0 (Rd ). We have
supp(FD (Sq−2 u∆q v)) = supp(FD (∆q v) ∗D FD (Sq−2 u)) ⊂ B(0, C2q ).
d
2
and
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d
s
d
1) If u in L∞
k (R ) and v in Ck (R ), then we have
kSq−2 u∆q vkL∞
d ≤ kSq−2 ukL∞ (Rd ) k∆q vkL∞ (Rd )
k (R )
k
k
−qs
≤ CkukL∞
.
d kvkC s (Rd ) 2
k (R )
k
Applying Proposition 3.14 ii), we obtain
kΠu vkCks (Rd ) ≤ CkukL∞
d kvkC s (Rd ) .
k (R )
k
d
s
d
2) If u is in L∞
k (R ) and v is in Hk (R ), then we have
kSq−2 u∆q vkL2k (Rd ) ≤ kSq−2 ukL∞
d k∆q vkL2 (Rd )
k (R )
k
−qs
≤ CkukL∞
.
d kvkH s (Rd ) cq 2
k (R )
k
Littlewood-Paley Decomposition
Hatem Mejjaoli
Thus Proposition 3.10 gives
vol. 9, iss. 4, art. 95, 2008
kΠu vkHks (Rd ) ≤ CkukL∞
d kvkH s (Rd ) ,
k (R )
k
Title Page
this implies the result.
3) Let u be in Ckt (Rd ) and v in Hks (Rd ). We have
Contents
kSq−2 u∆q vkL2k (Rd ) ≤ kSq−2 ukL∞
d k∆q vkL2 (Rd ) .
k (R )
k
Since t < 0, we estimate kSq−2 ukL∞
in a different way. In fact, Sq−2 u =
d
k (R )
P
t
d
p≤q−3 ∆p u. Since u ∈ Ck (R ) and t < 0, we obtain
kSq−2 ukL∞
d ≤ kukC t (Rd )
k (R )
k
X
p≤q−3
−pt
2
C −qt
2 kukCkt (Rd ) .
≤
−t
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Hence
kSq−2 u∆q vkL2k (Rd ) ≤
C −q(t+s)
2
cq kukCkt (Rd ) kvkHks (Rd ) ,
−t
Thus Proposition 3.10 gives the result.
The proof of 4) uses the same idea.
cq ∈ l 2 .
5) We have
kSq−2 ukL∞
d ≤ kFD (Sq−2 u)kL1 (Rd )
k (R )
k
Z
−s
s
−(q−2)
≤
|ψ(2
ξ)|(1 + kξk2 ) 2 |FD (u)(ξ)|(1 + kξk2 ) 2 ωk (ξ)dξ.
Rd
The Cauchy-Schwartz inequality implies that
Z
21
−(q−2)
2
2 −s
kSq−2 ukL∞
|ψ(2
ξ)| (1 + kξk ) ωk (ξ)dξ kukHks (Rd ) ,
d ≤
k (R )
Rd
q( d2 +γ)
Z
2
≤ C2
|ψ(t)| (1 + 2
B(0,1)
2(q−2)
12
ktk ) ωk (t)dt kukHks (Rd ) .
Littlewood-Paley Decomposition
Hatem Mejjaoli
vol. 9, iss. 4, art. 95, 2008
2 −s
Title Page
If s ≥ 0,
Contents
d
q( +γ−s)
kSq−2 ukL∞
d ≤ C2 2
k (R )
q( d2 +γ−s)
≤ C2
12
2
−2s
|ψ(t)| ktk ωk (t)dt
Z
B(0,1)
.
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Page 41 of 46
If s ≤ 0,
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d
q( +γ−s)
kSq−2 ukL∞
d ≤ C2 2
k (R )
q( d2 +γ−s)
≤ C2
Z
12
2
2 −s
|ψ(t)| (1 + ktk ) ωk (t)dt
B(0,1)
.
By proceeding as in the previous cases we deduce the result.
Theorem 5.3. There exists a positive constant C such that the operator Π has the
following properties:
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d
1. If a ∈ L∞
k (R ) is radial, then for any s in R, we have
kΠa kL(Cks (Rd ),Cks (Rd )) ≤ CkakL∞
d ,
k (R )
kΠa kL(Hks (Rd ),Hks (Rd )) ≤ CkakL∞
d .
k (R )
2. If a ∈ Ckt (Rd ) is radial with t < 0, then for all s, we have
kΠa kL(H s (Rd ),H s+t (Rd )) ≤ CkakCkt (Rd ) ,
k
kΠa kL(C s (Rd ),C s+t (Rd )) ≤ CkakCkt (Rd ) .
k
k
3. If a ∈ Hkt (Rd ) is radial, then for all s, t with s <
kΠa k
s+t−γ− d
2
L(Hks (Rd ),Hk
d
2
+ γ, we have
≤ Ckak
(Rd ))
k
Hkt (Rd )
.
Proof. From the relation (2.19) and Definition 2.6 we deduce that there exists an
e0 such that supp FD (Sq−2 a∆q u) ⊂ 2q C
e0 . Thus we proceed as in the proof
annulus C
of Theorem 4.3 and using Propositions 3.9 and 3.14 i), we obtain the result.
Littlewood-Paley Decomposition
Hatem Mejjaoli
vol. 9, iss. 4, art. 95, 2008
Title Page
Contents
Remark 5. In the case W = Zd2 the assumption that a is radial is not necessary.
JJ
II
Theorem 5.4. There exists a positive constant C such that the operator R has the
following properties:
J
I
1. kRkL(C t (Rd )×H s (Rd ),H s+t (Rd )) ≤
C s+t+1
,
s+t
2. kRkL(C t (Rd )×C s (Rd ),C s+t (Rd )) ≤
C s+t+1
k
k
3. kRk
k
k
k
s+t
k
s+t−γ− d
2 (Rd ))
L(Hkt (Rd )×Hks (Rd ),Hk
≤
,
for all s, t with s + t > 0.
for all s, t with s + t > 0.
C s+t+1
s+t−γ− d2
,
for all s, t with s + t > γ + d2 .
Proof. By the definition of the remainder operator
R(u, v) =
X
q
Rq
with
Rq =
1
X
i=−1
∆q−i u∆q v.
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By the definition of ∆q , the support of the Dunkl transform of Rq is included in
B(0, C2q ). Then, to prove 1) it is sufficient to estimate kRq kL2k (Rd ) . In fact, we have
kRq k
L2k (Rd )
≤ k∆q vk
1
X
L2k (Rd )
k∆q−i ukL∞
d .
k (R )
i=−1
Using the facts that u ∈ Ckt (Rd ) and v ∈ Hks (Rd ), we obtain
−qs
kRq kL2k (Rd ) ≤ 2
cq kvkHks (Rd )
1
X
Littlewood-Paley Decomposition
Hatem Mejjaoli
−(q−i)t
2
kukCkt (Rd ) ,
cq ∈ l
2
vol. 9, iss. 4, art. 95, 2008
i=−1
≤ Ccq 2−q(s+t) kukCkt (Rd ) kvkHks (Rd ) .
Title Page
Now we apply Proposition 3.10 to conclude the proof. The proof
Pof the second case
uses the same idea. We want to prove 3). We have R(u, v) = q Rq . We proceed
as in 1), so
kRq kL2k (Rd ) ≤ CkFD (∆q v)kL2k (Rd )
1
X
k∆q−i ukL∞
d ,
k (R )
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i=−1
≤ CkFD (∆q v)kL2k (Rd )
1
X
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kFD (∆q−i u)kL1k (Rd ) .
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Using the fact that v ∈ Hks (Rd ), we obtain
−qt
kRq kL2k (Rd ) ≤ Ccq 2
kvkHkt (Rd )
1
X
i=−1
kFD (∆q−i u)kL1k (Rd ) ,
cq ∈ l 2 .
On the other hand, by the Cauchy-Schwartz inequality we have
Z
−s
s
|ϕ(2−(q−i) ξ)|(1 + kξk2 ) 2 (1 + kξk2 ) 2 FD (u)(ξ)ωk (ξ)dξ
kFD (∆q−i u)kL1k (Rd ) ≤
Rd
Z
≤ kukHks (Rd )
q(γ+ d2 )
≤ C2
−(q−i)
|ϕ(2
12
2 −s
2
ξ)| (1 + kξk ) ωk (ξ)dξ
Rd
Z
kukHks (Rd )
2
2(q−i)
|ϕ(t)| (1 + 2
Rd
21
ktk ) ωk (t)dt
2 −s
Littlewood-Paley Decomposition
Hatem Mejjaoli
vol. 9, iss. 4, art. 95, 2008
d
≤ C2q(γ+ 2 −s) .
Hence
q(γ+ d2 −s−t)
kRq kL2k (Rd ) ≤ C2
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cq ,
2
cq ∈ l .
Then we conclude the result by using Proposition 3.10.
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References
[1] J.M. BONY, Calcul symbolique et propogation des singularités pour les
équations aux dérivées partielles non linéaires, Annales de l’École Normale
Supérieure, 14 (1981), 209–246.
[2] J.Y. CHEMIN, Fluides Parfaits Incompressibles, Astérisque, 230 (1995).
[3] C.F. DUNKL, Differential-difference operators associated to reflection groups,
Trans. Am. Math. Soc., 311 (1989), 167–183.
[4] C.F. DUNKL, Integral kernels with reflection group invariance, Canad. J.
Math., 43 (1991), 1213–1227.
Littlewood-Paley Decomposition
Hatem Mejjaoli
vol. 9, iss. 4, art. 95, 2008
Title Page
[5] C.F. DUNKL, Hankel transforms associated to finite reflection groups, Contemp. Math., 138 (1992), 123–138.
[6] L. HÖRMANDER, The Analysis of Linear Partial Differential Operators III,
Springer 1985.
[7] M.F.E. de JEU, The Dunkl transform, Invent. Math., 113 (1993), 147–162.
[8] P.L. LIZORKIN, Operators connected with fractional derivatives and classes of
differentiable functions, Trudy Mat. Ins. Steklov, 117 (1972), 212–243.
[9] H. MEJJAOLI AND K. TRIMÈCHE, Hypoellipticity and hypoanaliticity of the
Dunkl Laplacian operator, Integ. Transf. and Special Funct., 15(6) (2004), 523–
548.
[10] J. PEETRE, Sur les espaces de Besov, C.R. Acad. Sci. Paris Sér. 1 Math., 264
(1967), 281–283.
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II
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[11] M. RÖSLER, A positive radial product formula for the Dunkl kernel, Trans.
Amer. Math. Soc., 355 (2003), 2413–2438.
[12] B. RUBIN, Fractional Iintegrals and Potentials, Addison-Wesley and Longman, 1996.
[13] S. THANGAVELU AND Y. XU, Convolution operator and maximal functions
for Dunkl transform, J. d’Analyse Mathematique, 97 (2005), 25–56.
[14] C. SADOSKY, Interpolation of Operators and Singular Integrals, Marcel
Dekker Inc., New York and Basel, 1979.
[15] H. TRIEBEL, Spaces of distributions of Besov type on Eucledean n−spaces
duality interpolation, Ark. Mat., 11 (1973), 13–64.
[16] H. TRIEBEL, Interpolation Theory, Function Spaces Differential Operators,
North Holland, Amesterdam, 1978.
[17] K. TRIMÈCHE, The Dunkl intertwining operator on spaces of functions and
distributions and integral representation of its dual, Integ. Transf. and Special
Funct., 12(4) (2001), 349–374.
[18] K. TRIMÈCHE, Paley-Wiener theorems for Dunkl transform and Dunkl translation operators, Integ. Transf. and Special Funct., 13 (2002), 17–38.
Littlewood-Paley Decomposition
Hatem Mejjaoli
vol. 9, iss. 4, art. 95, 2008
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