Littlewood-Paley Decomposition Hatem Mejjaoli vol. 9, iss. 4, art. 95, 2008 Title Page Contents Page 1 of 46 Go Back Full Screen Close LITTLEWOOD-PALEY DECOMPOSITION ASSOCIATED WITH THE DUNKL OPERATORS AND PARAPRODUCT OPERATORS HATEM MEJJAOLI Department of Mathematics Faculty of Sciences of Tunis CAMPUS 1060 Tunis, TUNISIA EMail: [email protected]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 op- erators; 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 es- tablish 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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Littlewood-Paley Decomposition
Hatem Mejjaoli
vol. 9, iss. 4, art. 95, 2008
Title Page
Contents
JJ II
J I
Page 1 of 46
Go Back
Full Screen
Close
LITTLEWOOD-PALEY DECOMPOSITIONASSOCIATED WITH THE DUNKL OPERATORS AND
PARAPRODUCT OPERATORS
HATEM MEJJAOLIDepartment of MathematicsFaculty of Sciences of TunisCAMPUS 1060 Tunis, TUNISIA
Abstract: We define the Littlewood-Paley decomposition associated with the Dunkl op-erators; from this decomposition we give the characterization of the Sobolev,Hölder and Lebesgue spaces associated with the Dunkl operators. We constructthe paraproduct operators associated with the Dunkl operators similar to thosedefined by J.M. Bony in [1]. Using the Littlewood-Paley decomposition we es-tablish the Sobolev embedding, Gagliardo-Nirenberg inequality and we presentthe paraproduct algorithm.
Acknowledgement: I am thankful to anonymous referee for his deep and helpful comments.
1. IntroductionThe theory of function spaces appears at first to be a disconnected subject, becauseof the variety of spaces and the different considerations involved in their defini-tions. There are the Lebesgue spacesLp(Rd), the Sobolev spacesHs(Rd), the BesovspacesBs
p,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 suc-cessful approaches is the Littlewood-Paley theory. This approach has been de-veloped by the European school, which reached a similar unification of functionspace theory by a different path. Motivated by the methods of Hörmander in study-ing partial differential equations (see [6]), they used a Fourier transform approach.Pick Schwartz functionsφ andχ on Rd satisfyingsupp χ̂ ⊂ B(0, 2), supp φ̂ ⊂{ξ ∈ Rd, 1
2≤ ‖ξ‖ ≤ 2
}, and the nondegeneracy condition|χ̂(ξ)|, |φ̂(ξ)| ≥ C > 0.
For j ∈ Z, letφj(x) = 2jdφ(2jx). In 1967 Peetre [10] proved that
(1.1) ‖f‖Hs(Rd) ' ‖χ ∗ f‖L2(Rd) +
(∑j≥1
22sj‖φj ∗ f‖2L2(Rd)
) 12
.
Independently, Triebel [15] in 1973 and Lizorkin [8] in 1972 introducedF sp,q (the
Triebel-Lizorkin spaces) defined originally for1≤p<∞ and1≤q≤∞ by the norm
(1.2) ‖f‖F sp,q
= ‖χ ∗ f‖Lp(Rd) +
∥∥∥∥∥∥(∑
j≥1
(2sj|φj ∗ f |)q
) 1q
∥∥∥∥∥∥Lp(Rd)
.
For the special caseq = 1 ands = 0, Triebel [16] proved that(1.3) Lp(Rd) ' F 0
p,2.
Thus by the Littlewood-Paley decomposition we characterize the functional spacesLp(Rd), Sobolev spacesHs(Rd), Hölder spacesCs(Rd) and others. Using the
Littlewood-Paley decomposition J.M. Bony in [1], built the paraproduct operatorswhich have been later successfully employed in various settings.
The purpose of this paper is to generalize the Littlewood-Paley theory, to unifyand extend the paraproduct operators which allow the analysis of solutions to moregeneral partial differential equations arising in applied mathematics and other fields.More precisely, we define the Littlewood-Paley decomposition associated with theDunkl operators. We introduce the new spaces associated with the Dunkl opera-tors, the Sobolev spacesHs
k(Rd), the Hölder spacesCsk(Rd) and theBMOk(Rd)
that generalizes the corresponding classical spaces. The Dunkl operators are thedifferential-difference operators introduced by C.F. Dunkl in [3] and which playedan important role in pure Mathematics and in Physics. For example they were a maintool in the study of special functions with root systems (see [4]).
As applications of the Littlewood-Paley decomposition we establish results analo-gous to (1.1) and (1.3), we prove the Sobolev embedding theorems, and the Gagliardo-Nirenberg inequality. Another tool of the Littlewood-Paley decomposition associ-ated with the Dunkl operators is to generalize the paraproduct operators defined byJ.M. Bony. We prove results similar to [2].
The paper is organized as follows. In Section2 we recall the main results aboutthe harmonic analysis associated with the Dunkl operators. We study in Section3 the Littlewood-Paley decomposition associated with the Dunkl operators, we givethe sufficient condition onup so thatu :=
∑up belongs to Sobolev or Hölder spaces
associated with the Dunkl operators. We finish this section by the Littlewood-Paleydecomposition of the Lebesgue spacesLp
k(Rd) associated with the Dunkl operators.In Section4 we give some applications. More precisely we establish the Sobolevembedding theorems and the Gagliardo-Nirenberg inequality. Section5 is devotedto defining the paraproduct operators associated with the Dunkl operators and togiving the paraproduct algorithm.
In this section we collect some notations and results on Dunkl operators and theDunkl kernel (see [3], [4] and [5]).
2.1. Reflection Groups, Root System and Multiplicity Functions
We considerRd with the euclidean scalar product〈·, ·〉 and‖x‖ =√〈x, x〉. On
Cd, ‖ · ‖ denotes also the standard Hermitian norm, while〈z, w〉 =∑d
j=1 zjwj.Forα ∈ Rd\{0}, let σα be the reflection in the hyperplaneHα ⊂ Rd orthogonal
to α, i.e.
(2.1) σα(x) = x− 2〈α, x〉‖α‖2
α.
A finite setR ⊂ Rd\{0} is called a root system ifR∩R·α = {α,−α} andσαR = Rfor all α ∈ R. For a given root systemR the reflectionsσα, α ∈ R, generate a finitegroupW ⊂ O(d), called the reflection group associated withR. All reflectionsin W correspond to suitable pairs of roots. For a givenβ ∈ Rd\∪α∈RHα, we fixthe positive subsystemR+ = {α ∈ R : 〈α, β〉 > 0}, then for eachα ∈ R eitherα ∈ R+ or−α ∈ R+. We will assume that〈α, α〉 = 2 for all α ∈ R+.
A function k : R −→ C on a root systemR is called a multiplicity function if itis invariant under the action of the associated reflection groupW . If one regardskas a function on the corresponding reflections, this means thatk is constant on theconjugacy classes of reflections inW . For brevity, we introduce the index
admits a unique analytic solution onRd, which will be denoted byK(x, y) and calledthe Dunkl kernel. This kernel has a unique holomorphic extension toCd × Cd. TheDunkl kernel possesses the following properties.
Definition 3.2. Let λ ∈ R. For χ in S(Rd), we define the pseudo-differential-difference operatorχ(λT ) by
FD(χ(λT )u) = χ(λξ)FD(u), u ∈ S ′(Rd).
Definition 3.3. For u in S ′(Rd), we define its Littlewood-Paley decomposition asso-ciated with the Dunkl operators (or dyadic decomposition){∆pu}∞p=−1 as∆−1u =ψ(T )u and forq ≥ 0, ∆qu = ϕ(2−qT )u.
Now we go to see in which case we can have the identity
Id =∑p≥−1
∆p.
This is described by the following proposition.
Proposition 3.4. For u in S ′(Rd), we haveu =∑∞
p=−1 ∆pu, in the sense ofS ′(Rd).
Proof. For anyf in S(Rd), it is easy to see thatFD(f) =∑∞
p=−1FD(∆pf) in thesense ofS(Rd). Then for anyu in S ′(Rd), we have
In this subsection we will give a characterization of Sobolev spaces associated withthe Dunkl operators by a Littlewood-Paley decomposition. First, we recall the defi-nition of these spaces (see [9]).
Definition 3.5. Lets be inR, we define the spaceHsk(Rd) by{
u ∈ S ′(Rd) : (1 + ‖ξ‖2)s2FD(u) ∈ L2
k(Rd)}.
We provide this space by the scalar product
(3.5) 〈u, v〉Hsk(Rd) =
∫Rd
(1 + ‖ξ‖2)sFD(u)(ξ) FD(v)(ξ)ωk(ξ)dξ,
and the norm
(3.6) ‖u‖2Hs
k(Rd) = 〈u, u〉Hsk(Rd).
Another proposition will be useful. LetSqu =∑
p≤q−1 ∆pu.
Proposition 3.6. For all s in R and for all distributionsu in Hsk(Rd), we have
(1 + ‖ξ‖2)s|FD(Snu− u)(ξ)|2 ≤ 2(1 + ‖ξ‖2)s|FD(u)(ξ)|2.Thus the result follows from the dominated convergence theorem.
The first application of the Littlewood-Paley decomposition associated with theDunkl operators is the characterization of the Sobolev spaces associated with theseoperators through the behavior onq of ‖∆qu‖L2
k(Rd). More precisely, we now definea norm equivalent to the norm‖ · ‖Hs
k(Rd) in terms of the dyadic decomposition.
Proposition 3.7. There exists a positive constantC such that for alls in R, we have
1
C |s|+1‖u‖2
Hsk(Rd) ≤
∑q≥−1
22qs‖∆qu‖2L2
k(Rd) ≤ C |s|+1‖u‖2Hs
k(Rd).
Proof. SincesuppFD(∆qu) ⊂ Cq, from the definition of the norm‖ · ‖Hsk(Rd), there
By recalling thatsuppFD(∆pu) ⊂ Cp and|FD(∆pu)(ξ)| ≤ |FD(u)(ξ)|, we applythe Parseval identity associated with the Dunkl operators and the Cauchy-Schwartzinequality. We deduce that
5. Paraproduct Associated with the Dunkl Operators
In this section, we are going to study how the product acts on Sobolev and Hölderspaces associated with the Dunkl operators. This could be very useful in nonlinearpartial differential-difference equations. Of course, we shall use the Littlewood-Paley decomposition associated with the Dunkl operators. Let us consider two tem-perate distributionsu andv. We write
u =∑
p
∆pu and v =∑
q
∆qv.
Formally, the product can be written as
uv =∑p,q
∆pu∆qv.
Now we introduce the paraproduct operator associated with the Dunkl operators.
Definition 5.1. We define the paraproduct operatorΠa : S ′(Rd) → S ′(Rd) by
Πau =∑q≥1
(Sq−2 a)∆qu,
whereu ∈ S ′(Rd); {∆qa} and{∆qu} are the Littlewood-Paley decompositions andSqa =
∑p≤q−1 ∆pa.
LetR indicate the following bilinear symmetric operator onS ′(Rd) defined by
1. If a ∈ L∞k (Rd) is radial, then for anys in R, we have
‖Πa‖L(Csk(Rd),Cs
k(Rd)) ≤ C‖a‖L∞k (Rd), ‖Πa‖L(Hsk(Rd),Hs
k(Rd)) ≤ C‖a‖L∞k (Rd).
2. If a ∈ Ctk(Rd) is radial with t < 0, then for alls, we have
‖Πa‖L(Hsk(Rd),Hs+t
k (Rd)) ≤ C‖a‖Ctk(Rd), ‖Πa‖L(Cs
k(Rd),Cs+tk (Rd)) ≤ C‖a‖Ct
k(Rd).
3. If a ∈ H tk(Rd) is radial, then for alls, t with s < d
2+ γ, we have
‖Πa‖L(Hs
k(Rd),Hs+t−γ− d
2k (Rd))
≤ C‖a‖Htk(Rd).
Proof. From the relation (2.19) and Definition2.6 we deduce that there exists anannulusC̃0 such thatsuppFD(Sq−2a∆qu) ⊂ 2qC̃0. Thus we proceed as in the proofof Theorem4.3and using Propositions3.9and3.14i), we obtain the result.
Remark5. In the caseW = Zd2 the assumption thata is radial is not necessary.
Theorem 5.4. There exists a positive constantC such that the operatorR has thefollowing properties:
1. ‖R‖L(Ctk(Rd)×Hs
k(Rd),Hs+tk (Rd)) ≤ Cs+t+1
s+t, for all s, t with s+ t > 0.
2. ‖R‖L(Ctk(Rd)×Cs
k(Rd),Cs+tk (Rd)) ≤ Cs+t+1
s+t, for all s, t with s+ t > 0.
3. ‖R‖L(Ht
k(Rd)×Hsk(Rd),H
s+t−γ− d2
k (Rd))≤ Cs+t+1
s+t−γ− d2
, for all s, t with s+ t > γ + d2.
Proof. By the definition of the remainder operator
[1] J.M. BONY, Calcul symbolique et propogation des singularités pour leséquations aux dérivées partielles non linéaires,Annales de l’École NormaleSupérieure, 14 (1981), 209–246.
[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 ofdifferentiable functions,Trudy Mat. Ins. Steklov, 117(1972), 212–243.
[9] H. MEJJAOLIAND K. TRIMÈCHE, Hypoellipticity and hypoanaliticity of theDunkl 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.
[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 Long-man, 1996.
[13] S. THANGAVELU AND Y. XU, Convolution operator and maximal functionsfor Dunkl transform,J. d’Analyse Mathematique,97 (2005), 25–56.
[14] C. SADOSKY, Interpolation of Operators and Singular Integrals, MarcelDekker Inc., New York and Basel, 1979.
[15] H. TRIEBEL, Spaces of distributions of Besov type on Eucledeann−spacesduality 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 anddistributions and integral representation of its dual,Integ. Transf. and SpecialFunct., 12(4) (2001), 349–374.
[18] K. TRIMÈCHE, Paley-Wiener theorems for Dunkl transform and Dunkl trans-lation operators,Integ. Transf. and Special Funct., 13 (2002), 17–38.