Volume 9 (2008), Issue 4, Article 95, 25 pp.
LITTLEWOOD-PALEY DECOMPOSITION ASSOCIATED WITH THE DUNKL
OPERATORS AND PARAPRODUCT OPERATORS
HATEM MEJJAOLI
D EPARTMENT OF M ATHEMATICS
FACULTY OF S CIENCES OF T UNIS
CAMPUS 1060 T UNIS , TUNISIA
hatem.mejjaoli@ipest.rnu.tn
Received 11 July, 2007; accepted 25 May, 2008
Communicated by S.S. Dragomir
Dedicated to Khalifa Trimeche.
A BSTRACT. 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 LittlewoodPaley decomposition we establish the Sobolev embedding, Gagliardo-Nirenberg inequality and
we present the paraproduct algorithm.
Key words and phrases: Dunkl operators, Littlewood-Paley decomposition, Paraproduct.
2000 Mathematics Subject Classification. Primary 35L05. Secondary 22E30.
1. I NTRODUCTION
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
s
Lebesgue spaces Lp (Rd ), the Sobolev spaces H s (Rd ), the Besov 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
I am thankful to anonymous referee for his deep and helpful comments.
232-07
2
H ATEM M EJJAOLI
a Fourier transform approach. Pick Schwartz functions φ and χ on Rd satisfying supp χ
b ⊂
1
b
B(0, 2), supp φb ⊂ ξ ∈ Rd , 2 ≤ kξk ≤ 2 , and the nondegeneracy condition |b
χ(ξ)|, |φ(ξ)|
≥
jd
j
C > 0. 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 )
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
(1.3)
0
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 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 Gagliardo-Nirenberg 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 the sufficient condition on up so that
P
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
J. Inequal. Pure and Appl. Math., 9(4) (2008), Art. 95, 25 pp.
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L ITTLEWOOD -PALEY D ECOMPOSITION
3
to defining the paraproduct operators associated with the Dunkl operators and to giving the
paraproduct algorithm.
2. T HE E IGENFUNCTION
OF THE
D UNKL O PERATORS
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. We consider Rd with the
p
euclidean scalar product h·, ·i and kxk = hx, xi. On Cd , k · k denotes also the standard
P
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.
hα, xi
α.
(2.1)
σα (x) = x − 2
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
γ = γ(k) =
(2.2)
X
k(α).
α∈R+
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. Dunkl operators-Dunkl kernel and Dunkl intertwining operator.
Notations. We denote by
– 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 .
– S(Rd ) the space of C ∞ -functions on Rd which are rapidly decreasing as their derivatives.
– D(Rd ) the space of C ∞ -functions on Rd which are of compact support.
J. Inequal. Pure and Appl. Math., 9(4) (2008), Art. 95, 25 pp.
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4
H ATEM M EJJAOLI
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
X
f (x) − f (σα (x))
∂
(2.5)
Tj f (x) =
f (x) +
k(α)αj
,
∂xj
hα,
xi
α∈R
f ∈ C 1 (Rd ).
+
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,
u(0, y) = 1,
for all y ∈ Rd
admits a unique analytic solution on Rd , 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 .
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
|Dzν K(x, z)| ≤ kxk|ν| exp(kxkk Re zk),
(2.7)
and for all x, y ∈ Rd :
|K(ix, y)| ≤ 1,
(2.8)
with Dzν =
∂ν
ν
ν
∂z1 1 ···∂zdd
d
and |ν| = ν1 + · · · + νd .
iii) For all x, y ∈ R 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 ,
Rd
where dµx is a probability measure given on Rd , with support in the closed ball B(0, kxk) of
center 0 and radius kxk.
J. Inequal. Pure and Appl. Math., 9(4) (2008), Art. 95, 25 pp.
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L ITTLEWOOD -PALEY D ECOMPOSITION
5
2.3. The Dunkl Transform. The results of this subsection are given in [7] and [18].
Notations. We denote by
– Lpk (Rd ) the space of measurable functions on Rd such that
Z
p1
p
kf kLpk (Rd ) =
|f (x)| ωk (x)dx
< ∞, if 1 ≤ p < ∞,
Rd
kf kL∞
d = ess sup |f (x)| < ∞.
k (R )
x∈Rd
– 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
(2.11)
FD (f )(y) =
f (x)K(−iy, x)ωk (x)dx,
c k Rd
It satisfies the following properties:
for all y ∈ Rd .
i) For f in L1k (Rd ) we have
kFD (f )kL∞
d ≤
k (R )
(2.12)
1
kf kL1k (Rd ) .
ck
ii) For f in S(Rd ) we have
∀y ∈ Rd ,
(2.13)
FD (Tj f )(y) = iyj FD (f )y),
j = 1, . . . , d.
iii) For all f in L1k (Rd ) such that FD (f ) is in L1k (Rd ), we have the inversion formula
Z
(2.14)
f (y) =
FD (f )(x)K(ix, y)ωk (x) dx, a.e.
Rd
Theorem 2.2. The Dunkl transform FD is a topological isomorphism.
i) From S(Rd ) onto itself.
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.
ii) From E 0 (Rd ) onto H(Cd ).
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 ).
J. Inequal. Pure and Appl. Math., 9(4) (2008), Art. 95, 25 pp.
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6
H ATEM M EJJAOLI
2.4. The Dunkl Convolution Operator.
Definition 2.1. Let y be in Rd . The Dunkl translation operator f 7→ τy f is defined on S(Rd )
by
for all x ∈ Rd .
FD (τy f )(x) = K(ix, y)FD (f )(x),
(2.17)
Example 2.1. Let t > 0, we have
2
τx (e−tkξk )(y) = e−t(kxk
2 +kyk2 )
for all x ∈ Rd .
K(2tx, y),
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]).
• 1st case: d = 1 and W = Z2 .
• 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),
for all x ∈ Rd ,
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 2.2. 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]).
Proposition 2.5.
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),
ii) Let 1 ≤ p, q, r ≤ ∞, such that
1
p
+
1
q
−
1
r
for all y ∈ Rd .
= 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
kf ∗D gkLr (Rd ) ≤ kf kLp (Rd ) kgkLq (Rd ) .
(2.21)
iii) Let W =
k
Zd2 .
k
We have the same result for all f ∈
J. Inequal. Pure and Appl. Math., 9(4) (2008), Art. 95, 25 pp.
k
Lpk (Rd )
and g ∈ Lqk (Rd ).
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L ITTLEWOOD -PALEY D ECOMPOSITION
3. L ITTLEWOOD -PALEY T HEORY A SSOCIATED
WITH
7
D UNKL O PERATORS
We consider now a dyadic decomposition of Rd .
3.1. Dyadic Decomposition. For p ≥ 0 be a natural integer, we set
Cp = {ξ ∈ Rd ; 2p−1 ≤ kξk ≤ 2p+1 } = 2p C0
(3.1)
and
C−1 = B(0, 1) = {ξ ∈ Rd ; kξk ≤ 1}.
(3.2)
Clearly Rd =
S∞
p=−1
Cp .
Remark 2. We remark that
(3.3)
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
∞
X
ψ(ξ) +
ϕ(2−p ξ) = 1
p=0
and
ψ(ξ) +
n
X
ϕ(2−p ξ) = ψ(2−n ξ).
p=0
Remark 3. It is not hard to see that for any ξ ∈ Rd
∞
X
1
2
(3.4)
≤ ψ (ξ) +
ϕ2 (2−p ξ) ≤ 2.
2
p=0
Definition 3.1. Let λ ∈ R. For χ in S(Rd ), we define the pseudo-differential-difference operator χ(λT ) by
FD (χ(λT )u) = χ(λξ)FD (u),
u ∈ S 0 (Rd ).
Definition 3.2. 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 .
p≥−1
This is described by the following proposition.
Proposition 3.2. For u in S 0 (Rd ), we have u =
J. Inequal. Pure and Appl. Math., 9(4) (2008), Art. 95, 25 pp.
P∞
p=−1
∆p u, in the sense of S 0 (Rd ).
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8
H ATEM M EJJAOLI
P∞
Proof. For any f in S(Rd ), it is easy to see that FD (f ) =
p=−1
FD (∆p f ) in the sense of
S(Rd ). Then for any u in S 0 (Rd ), we have
hu, f i = hFD (u), FD (f )i
∞
X
=
hFD (u), FD (∆p f )i
=
p=−1
∞
X
hFD (∆p u), FD (f )i
p=−1
*
=
∞
X
+
FD (∆p u), FD (f )
*
=
p=−1
∞
X
+
∆p u, f
.
p=−1
The proof is finished.
3.2. The Generalized Sobolev Spaces. 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 3.3. 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
and the norm
kuk2H s (Rd ) = hu, uiHks (Rd ) .
(3.6)
k
Another proposition will be useful. Let Sq u =
P
p≤q−1
∆p u.
Proposition 3.3. For all s in R and for all distributions u in Hks (Rd ), we have
lim Sn u = u.
n→∞
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.
J. Inequal. Pure and Appl. Math., 9(4) (2008), Art. 95, 25 pp.
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L ITTLEWOOD -PALEY D ECOMPOSITION
9
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 3.4. 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
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 3.4.
Theorem 3.5. Let u be in S 0 (Rd ) and u =
P
q≥−1
∆q u its Littlewood-Paley decomposition.
The following are equivalent:
i) u ∈ Hks (Rd ).
P
ii) q≥−1 22qs k∆q uk2L2 (Rd ) < ∞.
k
iii) k∆q ukL2k (Rd ) ≤ cq 2−qs , with {cq } ∈ l2 .
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.
e be an annulus in Rd and s in R. Let (up )p∈N be a sequence of smooth
Proposition 3.6. Let C
functions. If the sequence (up )p∈N satisfies
e
supp FD (up ) ⊂ 2p C
and kup kL2k (Rd ) ≤ Ccp 2−ps , {cp } ∈ l2 ,
then we have
! 21
u=
X
up ∈ Hks (Rd ) and
kukHks (Rd ) ≤ C(s)
p≥0
J. Inequal. Pure and Appl. Math., 9(4) (2008), Art. 95, 25 pp.
X
22ps kup k2L2 (Rd )
k
.
p≥0
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10
H ATEM M EJJAOLI
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
X
∆q u =
∆q up .
|p−q|≤N0
By the triangle inequality and definition of ∆q up we deduce that
X
k∆q ukL2k (Rd ) ≤
kup kL2k (Rd ) .
|p−q|≤N0
Thus the Cauchy-Schwartz inequality implies that


!
X
X
X
22qs k∆q uk2L2 (Rd ) ≤ C 
22(q−p)s 
22ps kup k2L2 (Rd ) .
k
k
q≥0
p≥0
q/|p−q|≤N0
From Theorem 3.5 we deduce that if kup kL2k (Rd ) ≤ Ccp 2−ps then u ∈ Hks (Rd ).
Proposition 3.7. Let K > 0 and s > 0. Let (up )p∈N be a sequence of smooth functions. If the
sequence (up )p∈N satisfies
supp FD (up ) ⊂ B(0, K2p )
and
kup kL2k (Rd ) ≤ Ccp 2−ps , {cp } ∈ l2 ,
then we have
! 12
u=
X
up ∈ Hks (Rd ) and
kukHks (Rd ) ≤ C(s)
p≥0
X
22ps kup k2L2 (Rd )
k
.
q≥0
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 )
p≥q−N1
=
X
2(q−p)s 2ps kup kL2k (Rd ) .
p≥q−N1
Since s > 0, the Cauchy-Schwartz inequality implies
X
22qs k∆q uk2L2 (Rd ) ≤
k
q
From Theorem 3.5 we deduce the result.
J. Inequal. Pure and Appl. Math., 9(4) (2008), Art. 95, 25 pp.
22N1 s X 2ps
2 kup k2L2 (Rd ) .
k
1 − 2−s p
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L ITTLEWOOD -PALEY D ECOMPOSITION
11
Proposition 3.8. 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 ,
then we have
! 21
u=
X
up ∈ Hks (Rd )
X
kukHks (Rd ) ≤ C(s)
and
22ps kup k2L2 (Rd )
k
p≥0
.
p≥0
Proof. By the assumption we first have u =
P
up ∈ L2k (Rd ). Take µ ∈ Nd with |µ| = s0 > s >
0, and χp (ξ) = χ(2−p ξ) ∈ D(Rd ) 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),
and we have
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
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
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
Hk (R )
Hk (R )
Hk (R )
P (1)
P (2)
Set u(1) = p up , u(2) = p up , then u = u(1) + u(2) , and from Proposition 3.7 we deduce
(2)
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 )
k
≤
X
p≤q+1
−2p(s−s0 )
2
! Z
!
X
Rd p≤q+1
2p(s−s0 )
2
−q
|ϕ(2
2
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
Hk (R )
1 − 2−(s−s0 )
p≤q+1
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12
H ATEM M EJJAOLI
Moreover, since s0 > s > 0,
1 − 2−2(q+2)(s−s0 ) −2qs0
2
≤ C2−2qs ,
1 − 2−(s−s0 )
and C is independent of q. Now set
X
2
s0 d ,
c2q =
22p(s−s0 ) ∆q u(2)
p
H (R )
k
p≤q+1
then
X
22qs k∆q (u(2) )k2L2 (Rd ) ≤
X
c2q ≤
k
q≥−1
X
k
p
q≥−1
Thus by Theorem 3.5 we deduce that u(2) =
2
s0 d < ∞.
22p(s−s0 ) u(2)
p
H (R )
P
q
∆q (u(2) ) belongs to Hks (Rd ).
Corollary 3.9. The spaces Hks (Rd ) do not depend on the choice of the function ϕ and ψ used
in the Definition 3.2.
3.3. The Generalized Hölder Spaces.
Definition 3.4. 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
P
q
uq belongs to
the Hölder spaces associated with the Dunkl operators.
Proposition 3.10.
e be an annulus in Rd and α ∈ R. Let (up )p∈N be a sequence of smooth functions.
i) Let C
If the sequence (up )p∈N satisfies
e
supp FD (up ) ⊂ 2p C
then we have
X
u=
up ∈ Ckα (Rd )
and
−pα
and kup kL∞
,
d ≤ C2
k (R )
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
u=
up ∈ Ckα (Rd )
and
p≥0
and
−pα
kup kL∞
,
d ≤ C2
k (R )
kukCkα (Rd ) ≤ C(α) sup 2pα kup kL∞
d .
k (R )
p≥0
Proof. The proof uses the same idea as for Propositions 3.6 and 3.7.
J. Inequal. Pure and Appl. Math., 9(4) (2008), Art. 95, 25 pp.
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L ITTLEWOOD -PALEY D ECOMPOSITION
Proposition 3.11. The distribution defined by
X
g(x) =
K(ix, 2p e),
13
with e = (1, . . . , 1),
p≥0
belongs to
Ck0 (Rd )
d
and does not belong to L∞
k (R ).
Proposition 3.12. Let ε ∈]0, 1[ and f in Ckε (Rd ), then there exists a positive constant C such
that
kf kL∞
d
k (R )
Proof. Since f =
P
p≥−1
kf kCkε (Rd )
C
≤ kf kCk0 (Rd ) log e +
ε
kf kCk0 (Rd )
!
.
∆p f ,
kf kL∞
d ≤
k (R )
X
k∆p f kL∞
d +
k (R )
p≤N −1
X
k∆p f kL∞
d ,
k (R )
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
kf kL∞
d ≤ (N + 1)kf kC 0 (Rd ) +
k (R )
k
We take
2−(N −1)ε
kf kCkε (Rd ) .
2ε − 1
#
kf kCkε (Rd )
1
N =1+
log2
,
ε
kf kCk0 (Rd )
"
we obtain
kf kL∞
d
k (R )
"
C
≤ kf kCk0 (Rd ) 1 + log
ε
kf kCkε (Rd )
!#
.
kf kCk0 (Rd )
This implies the result.
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
! 21 X
2
k(fj )kLpk (l2 ) = |f
(x)|
j
j∈N
,
Lpk (Rd )
the norm in Lpk (Rd , l2 (N)).
Theorem 3.13 (Littlewood-Paley decomposition of Lpk (Rd )). Let f be in S 0 (Rd ) and 1 < p <
∞. Then the following assertions are equivalent
i) f ∈ Lpk (Rd ),
ii) S0 f ∈ Lpk (Rd ) and
P
2
j∈N |∆j f (x)|
12
∈ Lpk (Rd ).
Moreover, the following norms are equivalent :
kf kLpk (Rd )
! 21 X
2
and kS0 f kLpk (Rd ) + |∆j f (x)|
j∈N
.
Lpk (Rd )
J. Inequal. Pure and Appl. Math., 9(4) (2008), Art. 95, 25 pp.
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14
H ATEM M EJJAOLI
Proof. If f is in L2k (Rd ), then from Proposition 3.4 we have
! 12 X
2
|∆
f
(x)|
≤ kf k2L2 (Rd ) .
j
k
j∈N
2 d
Lk (R )
Thus the mapping
Λ1 : f 7→ (∆j f )j∈N ,
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 < ∞.
The converse uses the same idea. Indeed we put
φej =
1
X
ϕ
ej+i .
i=−1
From Proposition 3.4 the mapping
Λ2 : (fj )j∈N 7→
X
fj ∗D φej ,
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) ,
for x 6= 0,
for x 6= 0, i = 1, . . . , d.
We may then apply the theory of singular integrals to this mapping Λ2 (see [14]).
Thus we obtain
X
∆j f j∈N
≤ Cp,k k∆j f kLpk (l2 ) .
Lpk (Rd )
J. Inequal. Pure and Appl. Math., 9(4) (2008), Art. 95, 25 pp.
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L ITTLEWOOD -PALEY D ECOMPOSITION
15
4. A PPLICATIONS
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
kuvkCkα (Rd ) ≤ C kukL∞
d ) kvkC α (Rd ) + kvkL∞ (Rd ) kukC α (Rd ) .
(R
k
k
k
k
T
d
s
d
ii) Let u, v ∈ Hks (Rd ) L∞
k (R ) and s > 0 then uv ∈ Hk (R ), and
i
h
s
∞
s
kuvkHks (Rd ) ≤ C kukL∞
kvk
+
kvk
kuk
d
d
d
d
Hk (R )
Lk (R )
Hk (R ) .
k (R )
Proof. Let u =
P
p
∆p u and v =
P
q
∆q v be their Littlewood-Paley decompositions. Then we
have
uv =
X
∆p u∆q v
p,q
=
X X
=
X X
q
∆p u∆q v +
XX
∆p u∆q v +
XX
q
p≤q−1
q
p
p≤q−1
=
X
=
X
Sq u∆q v +
q
X
∆p u∆q v
p≥q
∆p u∆q v
q≤p
Sp+1 v∆p u
p
+
1
X
.
2
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 3.10 ii) we deduce
X
α
1
≤ CkukL∞
d kvkC α (Rd ) .
k (R )
k
Ck (Rd )
Similarly we prove that
X
2
≤ CkvkL∞
d kukC α (Rd ) ,
k (R )
k
Ckα (Rd )
and this implies the result.
ii) If u and v are in Hks (Rd ), then we have
J. Inequal. Pure and Appl. Math., 9(4) (2008), Art. 95, 25 pp.
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16
H ATEM M EJJAOLI
kSq u∆q vkL2k (Rd ) ≤ kSq ukL∞
d k∆q vkL2 (Rd ) ,
k (R )
k
−qs
.
≤ CkukL∞
d kvkH s (Rd ) cq 2
k
k (R )
Thus Proposition 3.7 gives
X
1 ≤ CkukL∞
d kvkH s (Rd ) .
k (R )
k
X
2 ≤ CkvkL∞
d kukH s (Rd ) ,
k (R )
k
Hks (Rd )
Similarly, we prove that
Hks (Rd )
and this implies the result.
Corollary 4.2. For s >
d
2
+ γ, Hks (Rd ) is an algebra.
4.2. 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
s−γ− d2
Hks (Rd ) ,→ Ck
Proof. Let u be in Hks (Rd ), u =
P
p≥−1
(Rd ).
∆p u the Littlewood-Paley decomposition. Take φ in
d
D(R ) such that φ(ξ) = 1 on C0 , and
supp φ ⊂
C00
d 1
= ξ ∈ R , ≤ kξk ≤ 3 .
3
Setting φp (ξ) = φ(2−p ξ), we obtain
FD (∆p u)(ξ) = FD (∆p u)(ξ)φ(2−p ξ).
Hence
Z
FD (∆p u)(ξ)φ(2−p ξ)K(ix, ξ)ωk (ξ)dξ,
∆p u(x) =
Rd
Z
|∆p u(x)| ≤
|FD (∆p u)(ξ)kφ(2−p ξ)|ωk (ξ)dξ.
Rd
The Cauchy-Schwartz inequality and Theorem 3.5 give that
Z
12 Z
2
k∆p ukL∞
|FD (∆p u)(ξ)| ωk (ξ)dξ
d ≤
k (R )
−p
2
12
|φ(2 ξ)| ωk (ξ)dξ
Rd
Rd
d
≤ C2p(γ+ 2 ) k∆p ukL2k (Rd )
d
≤ C2−p(s−γ− 2 ) cp .
s−γ− d2
Then from Definition 3.4 we deduce that u ∈ Ck
J. Inequal. Pure and Appl. Math., 9(4) (2008), Art. 95, 25 pp.
(Rd ).
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L ITTLEWOOD -PALEY D ECOMPOSITION
17
Theorem 4.4. For any 0 < s < γ + d2 , we have the continuous embedding
Hks (Rd ) ,→ Lpk (Rd ),
where p =
2(2γ+d)
.
2γ+d−2s
Proof. Let f be in S(Rd ), we have, due to Fubini’s theorem,
Z ∞
n
o
p
(4.1)
kf kLp (Rd ) = p
λp−1 mk |f | ≥ λ dλ,
k
0
where
n
o Z
mk |f | ≥ λ =
ωk (x)dx.
{x; |f (x)|≥λ}
For A > 0, we set f = f1,A + f2,A with f1,A =
P
2j <A
∆j f and f2,A =
P
2j ≥A
∆j f .
We have
kf1,A kL∞
d ≤
k (R )
X
k∆j f kL∞
d ≤
k (R )
2j <A
X
kFD (∆j f )kL1k (Rd ) .
2j <A
Using the Cauchy-Schwartz inequality, the Parseval’s identity associated with the Dunkl operators and Theorem 3.5, we obtain
X
d
d
kf1,A kL∞
2j(γ+ 2 −s) cj kf kHks (Rd ) ≤ CAγ+ 2 −s kf kHks (Rd ) .
d) ≤
(R
k
2j <A
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
From (4.2) we infer that if we take
λ
4Ckf kHks (Rd )
A = Aλ =
!
1
γ+ d
2 −s
,
then
kf1,Aλ kL∞
d ≤
k (R )
λ
.
4
Hence
λ
mk |f1,Aλ | ≥
= 0.
2
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
Z
∞
kf kpLp (Rd ) ≤ p
k
J. Inequal. Pure and Appl. Math., 9(4) (2008), Art. 95, 25 pp.
λp−3 kf2,Aλ k2L2 (Rd ) dλ.
0
k
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18
H ATEM M EJJAOLI
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
2j ≥Aλ
X
≤ A−2ε
λ
2j ≥Aλ
22jε k∆j f k2L2 (Rd ) .
k
2j ≥Aλ
So by using the definition of Aλ and the Fubini theorem, we can write
kf kpLp (Rd )
Zk ∞
X
≤p
λp−3 A−2ε
22jε k∆j f k2L2 (Rd ) dλ
λ
k
0
2j ≥Aλ
d
≤C
XZ
4C2j(γ+ 2 −s) kf kH s (Rd )
k
λ
p−3−
2ε
γ+ d
2 −s
0
j≥−1
X
≤ Ckf kp−2
H s (Rd )
dλ 4Ckf kHks (Rd )
2ε
γ+ d
2 −s
22jε k∆j f k2L2 (Rd )
k
d
2j(p−2)(γ+ 2 −s) k∆j f k2L2 (Rd )
k
k
j≥−1
X
≤ Ckf kp−2
H s (Rd )
22js k∆j f k2L2 (Rd )
k
k
j≥−1
≤ Ckf kpH s (Rd ) .
k
This implies the result.
Definition 4.1. 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.
B
Theorem 4.5. We have the continuous embedding
d
Hk2
+γ
(Rd ) ,→ BM Ok .
d
Proof. For R > 0 small enough, let N be such that 2N = [ R1 ]. Let u be in Hk2
+γ
(Rd ). Set
u = u(1) + u(2) with
u
(1)
=
N
−1
X
∆p u
p=−1
J. Inequal. Pure and Appl. Math., 9(4) (2008), Art. 95, 25 pp.
and
u(2) =
X
∆p u.
p≥N
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L ITTLEWOOD -PALEY D ECOMPOSITION
19
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
1
R2
(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
By (2.7) we deduce that
Z
(1)
kDu kL∞
d ≤
k (R )
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
1
mesk (B)
Z
N
−1
X
(1)
|u(1) (x) − uB |2 ωk (x)dx ≤ R2
B
p=−1
≤R
!2
Z
kξk |FD (∆p u)(ξ)|ωk (ξ)dξ
Cp
Z
2
kξk
2−2γ−d
R kuk
d +γ
Hk2
ωk (ξ)dξ kuk2
d +γ
Hk2
B(0,2N )
2N 2
2
≤ C2
(Rd )
.
(Rd )
For the second term, we have
Z
Z
1
2
(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
≤ CR
|FD (u)(ξ)|2 ωk (ξ)dξ
kξk≥2N
−d−2γ
N
≤ C(2 R)
kuk2
d +γ
Hk2
.
(Rd )
Hence,
(4.3)
1
mesk (B)
2
Z
|u(x) − uB |ωk (x)dx
≤ Ckuk2
B
d
We have proved (4.3) for small R, since u ∈ Hk2
+γ
d +γ
Hk2
.
(Rd )
(Rd ) ⊂ L2k (Rd ), (4.3) is evident for R > R0
with constant C = C(R0 ). This implies the continuous embedding
d
Hk2
+γ
(Rd ) ,→ BM Ok .
J. Inequal. Pure and Appl. Math., 9(4) (2008), Art. 95, 25 pp.
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20
H ATEM M EJJAOLI
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 4.6. 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
s 1−θ
t
≤ Ckf kθLq (Rd ) (−4k ) 2 f Lr (Rd ) ,
(−4k ) 2 f p
Lk (Rd )
where
1
p
=
θ
q
+
1−θ
,
r
k
k
t = (1 − θ)s and θ ∈]0, 1[.
Proof. First, we prove this theorem for q and r in ]1, ∞]. Let f be in S(Rd ). It is easy to see
that
t
(−4k ) 2 f =
X
t
(−4k ) 2 ∆j f +
j≤A
X
(−4k )
t−s
2
s ∆j (−4k ) 2 f ,
j>A
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).
We proceed as in [12, p. 21] to obtain
1
2
(4.5)
a (−4k ) ∆j f (x) ≤ C2jm Mk f (x),
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
s
t
2jt Mk f (x) +
2j(t−s) Mk ((−4k f ) 2 )(x)
(−4k ) 2 f (x) ≤ C
j≤A
j>A
tA
(t−s)A
≤ C2 Mk f (x) + C2
s Mk (−4k ) 2 f (x).
We minimize over A to obtain
1− st st
s t
Mk (−4k ) 2 f (x) .
(−4k ) 2 f (x) ≤ C Mk f (x)
By this inequality and the Hölder inequality, we have
1−θ
s
t
2
≤ CkMk f kθLq (Rd ) Mk ((−4k ) 2 f )Lr (Rd ) ,
(−4k ) f p
Lk (Rd )
J. Inequal. Pure and Appl. Math., 9(4) (2008), Art. 95, 25 pp.
k
k
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L ITTLEWOOD -PALEY D ECOMPOSITION
21
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
s
t
t
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.
5. PARAPRODUCT A SSOCIATED
WITH THE
D UNKL O PERATORS
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 differentialdifference equations. Of course, we shall use the Littlewood-Paley 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.
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
0
d
where u ∈ S (R ); {∆q a} and {∆q u} are the Littlewood-Paley decompositions and Sq a =
P
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
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 5.1. There exists a positive constant C such that the operator Π has the following
properties:
s
s
(1) kΠkL(L∞
≤ C s+1 ,
d
d
d
k (R )×Ck (R ),Ck (R ))
s
s
(2) kΠkL(L∞
≤C
d
d
d
k (R )×Hk (R ),Hk (R ))
s+1
J. Inequal. Pure and Appl. Math., 9(4) (2008), Art. 95, 25 pp.
,
for all s > 0.
for all s > 0.
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22
H ATEM M EJJAOLI
C s+t+1
, for all s, t with s + t > 0 and t < 0.
−t
s+t+1
kΠkL(C t (Rd )×C s (Rd ),C s+t (Rd )) ≤ C −t , for all s, t with s + t > 0 and t < 0.
k
k
k
d
kΠk s d
≤ C s+t−γ− 2 +1 , with s + t > γ + d2 and s
s+t−γ− d
t
2
d
d
L(H (R )×H (R ),H
(R ))
(3) kΠkL(C t (Rd )×H s (Rd ),H s+t (Rd )) ≤
k
(4)
(5)
k
k
k
k
d
2
+ γ.
p≤q−3
∆p u.
<
k
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
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 3.10 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
Thus Proposition 3.7 gives
kΠu vkHks (Rd ) ≤ CkukL∞
d kvkH s (Rd ) ,
k (R )
k
this implies the result.
3) Let u be in Ckt (Rd ) and v in Hks (Rd ). We have
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∞
d in a different way. In fact, Sq−2 u =
k (R )
Since u ∈
Ckt (Rd )
P
and t < 0, we obtain
kSq−2 ukL∞
d ≤ kukC t (Rd )
k (R )
k
X
p≤q−3
2−pt ≤
C −qt
2 kukCkt (Rd ) .
−t
Hence
C −q(t+s)
2
cq kukCkt (Rd ) kvkHks (Rd ) ,
−t
Thus Proposition 3.7 gives the result.
kSq−2 u∆q vkL2k (Rd ) ≤
cq ∈ l 2 .
The proof of 4) uses the same idea.
5) We have
kSq−2 ukL∞
d ≤ kFD (Sq−2 u)kL1 (Rd )
k (R )
k
Z
−s
s
≤
|ψ(2−(q−2) ξ)|(1 + kξk2 ) 2 |FD (u)(ξ)|(1 + kξk2 ) 2 ωk (ξ)dξ.
Rd
J. Inequal. Pure and Appl. Math., 9(4) (2008), Art. 95, 25 pp.
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L ITTLEWOOD -PALEY D ECOMPOSITION
23
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
2(q−2)
|ψ(t)| (1 + 2
≤ C2
B(0,1)
12
ktk ) ωk (t)dt kukHks (Rd ) .
2 −s
If s ≥ 0,
q( d2 +γ−s)
Z
2
|ψ(t)| ktk
kSq−2 ukL∞
d ≤ C2
k (R )
−2s
12
ωk (t)dt
B(0,1)
d
≤ C2q( 2 +γ−s) .
If s ≤ 0,
q( d2 +γ−s)
12
|ψ(t)| (1 + ktk ) ωk (t)dt
Z
kSq−2 ukL∞
d ≤ C2
k (R )
2 −s
2
B(0,1)
q( d2 +γ−s)
≤ C2
.
By proceeding as in the previous cases we deduce the result.
Theorem 5.2. There exists a positive constant C such that the operator Π has the following
properties:
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
(3) If a ∈
Hkt (Rd )
k
kΠa kL(C s (Rd ),C s+t (Rd )) ≤ CkakCkt (Rd ) .
k
is radial, then for all s, t with s <
kΠa k
s+t−γ− d
2
L(Hks (Rd ),Hk
(Rd ))
d
2
k
+ γ, we have
≤ CkakHkt (Rd ) .
e0
Proof. From the relation (2.19) and Definition 2.2 we deduce that there exists an annulus C
e0 . Thus we proceed as in the proof of Theorem 4.3 and
such that supp FD (Sq−2 a∆q u) ⊂ 2q C
using Propositions 3.6 and 3.10 i), we obtain the result.
Remark 5. In the case W = Zd2 the assumption that a is radial is not necessary.
Theorem 5.3. There exists a positive constant C such that the operator R has the following
properties:
C s+t+1
, for all s, t with s + t > 0.
s+t
C s+t+1
kRkL(C t (Rd )×C s (Rd ),C s+t (Rd )) ≤ s+t , for all s, t with s + t > 0.
k
k
k
C s+t+1
kRk t d
≤ s+t−γ−
for all s, t with s +
d ,
s+t−γ− d
s
2
d
d
L(H (R )×H (R ),H
(R ))
2
(1) kRkL(C t (Rd )×H s (Rd ),H s+t (Rd )) ≤
k
(2)
(3)
k
k
k
k
t > γ + d2 .
k
J. Inequal. Pure and Appl. Math., 9(4) (2008), Art. 95, 25 pp.
http://jipam.vu.edu.au/
24
H ATEM M EJJAOLI
Proof. By the definition of the remainder operator
R(u, v) =
X
Rq
1
X
Rq =
with
q
∆q−i u∆q v.
i=−1
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
1
X
kRq kL2k (Rd ) ≤ k∆q vkL2k (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
1
X
cq kvkHks (Rd )
2−(q−i)t kukCkt (Rd ) ,
cq ∈ l 2
i=−1
−q(s+t)
≤ Ccq 2
kukCkt (Rd ) kvkHks (Rd ) .
Now we apply Proposition 3.7 to conclude the proof. The proof of the second case uses the
P
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 )
i=−1
≤ CkFD (∆q v)k
L2k (Rd )
1
X
kFD (∆q−i u)kL1k (Rd ) .
i=−1
Using the fact that v ∈ Hks (Rd ), we obtain
kRq kL2k (Rd ) ≤ Ccq 2−qt kvkHkt (Rd )
1
X
kFD (∆q−i u)kL1k (Rd ) ,
cq ∈ l 2 .
i=−1
On the other hand, by the Cauchy-Schwartz inequality we have
Z
−s
s
kFD (∆q−i u)kL1k (Rd ) ≤
|ϕ(2−(q−i) ξ)|(1 + kξk2 ) 2 (1 + kξk2 ) 2 FD (u)(ξ)ωk (ξ)dξ
Rd
Z
≤ kukHks (Rd )
q(γ+ d2 )
≤ C2
|ϕ(2−(q−i) ξ)|2 (1 + kξk2 )−s ωk (ξ)dξ
12
Rd
Z
kukHks (Rd )
2
2(q−i)
|ϕ(t)| (1 + 2
12
ktk ) ωk (t)dt
2 −s
Rd
d
≤ C2q(γ+ 2 −s) .
Hence
d
kRq kL2k (Rd ) ≤ C2q(γ+ 2 −s−t) cq ,
Then we conclude the result by using Proposition 3.7.
J. Inequal. Pure and Appl. Math., 9(4) (2008), Art. 95, 25 pp.
cq ∈ l 2 .
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L ITTLEWOOD -PALEY D ECOMPOSITION
25
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