LITTLEWOOD-PALEY DECOMPOSITION ASSOCIATED WITH THE DUNKL OPERATORS AND PARAPRODUCT OPERATORS

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Littlewood-Paley Decomposition Hatem Mejjaoli vol. 9, iss. 4, art. 95, 2008

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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:hatem.mejjaoli@ipest.rnu.tn

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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2.1 Reflection Groups, Root System and Multiplicity Functions . . . 5 2.2 Dunkl operators-Dunkl kernel and Dunkl intertwining operator . . . 6 2.3 The Dunkl Transform . . . 8 2.4 The Dunkl Convolution Operator . . . 10 3 Littlewood-Paley Theory Associated with Dunkl Operators 13 3.1 Dyadic Decomposition . . . 13 3.2 The Generalized Sobolev Spaces . . . 15 3.3 The Generalized Hölder Spaces. . . 22

4 Applications 27

4.1 Estimates of the Product of Two Functions . . . 27 4.2 Sobolev Embedding Theorem . . . 29 4.3 Gagliardo-Nirenberg Inequality. . . 35 5 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 defini- tions. There are the Lebesgue spacesL^p(R^d), the Sobolev spacesH^s(R^d), the Besov spacesB_p,q^s (R^d), 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 function space 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 functions φ and χ on R^d satisfying suppχb ⊂ B(0,2), suppφb ⊂ ξ ∈R^d, ¹₂ ≤ kξk ≤2 ,and the nondegeneracy condition|χ(ξ)|,b |φ(ξ)| ≥b C >0.

Forj ∈Z, letφj(x) = 2^jdφ(2^jx). In 1967 Peetre [10] proved that (1.1) kfk_Hs(R^d) ' kχ∗fk_L2(R^d)+ X

j≥1

2^2sjkφ_j∗fk²_L2(R^d)

!¹₂ .

Independently, Triebel [15] in 1973 and Lizorkin [8] in 1972 introduced F_p,q^s (the Triebel-Lizorkin spaces) defined originally for1≤p <∞and1≤q≤ ∞by the norm

(1.2) kfk_F_p,q^s =kχ∗fk_L^p₍_R^d₎+

X

j≥1

(2^sj|φ_j∗f|)^q

!¹_q L^p(R^d)

.

For the special caseq= 1ands= 0, Triebel [16] proved that

(1.3) L^p(R^d)'F_p,2⁰ .

Thus by the Littlewood-Paley decomposition we characterize the functional spaces L^p(R^d), Sobolev spaces H^s(R^d), Hölder spaces C^s(R^d) 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 H_k^s(R^d), the Hölder spaces C_k^s(R^d) and the BM O_k(R^d) 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 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 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 Section2we recall the main results about the harmonic analysis associated with the Dunkl operators. We study in Section 3the Littlewood-Paley decomposition associated with the Dunkl operators, we give the sufficient condition onu_pso thatu:=P

u_pbelongs to Sobolev or Hölder spaces associated with the Dunkl operators. We finish this section by the Littlewood-Paley decomposition of the Lebesgue spacesL^p_k(R^d)associated with the Dunkl operators.

In Section4 we give some applications. More precisely we establish the Sobolev embedding theorems and the Gagliardo-Nirenberg inequality. Section5is devoted to defining the paraproduct operators associated with the Dunkl operators and to giving the paraproduct algorithm.

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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 We consider R^d with the euclidean scalar product h·,·i and kxk = p

hx, xi. On C^d, k · kdenotes also the standard Hermitian norm, whilehz, wi=Pd

j=1z_jw_j. Forα ∈R^d\{0}, letσ_α be the reflection in the hyperplaneH_α ⊂ R^dorthogonal toα, i.e.

(2.1) σ_α(x) = x−2hα, xi

kαk² α.

A finite setR⊂R^d\{0}is called a root system ifR∩R·α={α,−α}andσ_αR =R for allα∈R. For a given root systemRthe reflectionsσ_α, 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 ∈ R^d\∪α∈RH_α, 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 thathα, αi= 2for allα∈R₊.

A functionk : R −→Con a root systemRis called a multiplicity function if it is invariant under the action of the associated reflection groupW. If one regardsk as a function on the corresponding reflections, this means that k is constant on the conjugacy classes of reflections inW. For brevity, we introduce the index

(2.2) γ =γ(k) = X

α∈R₊

k(α).

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Moreover, letω_kdenote the weight function

(2.3) ωk(x) = Y

α∈R+

| hα, xi |^2k(α),

which is invariant and homogeneous of degree 2γ. We introduce the Mehta-type constant

(2.4) c_k=

Z

R^d

e⁻^kxk

2

2 ω_k(x)dx.

2.2. Dunkl operators-Dunkl kernel and Dunkl intertwining operator

Notations. We denote by

– C(R^d) (resp. C_c(R^d)) the space of continuous functions on R^d (resp. with compact support).

– E(R^d)the space ofC^∞-functions onR^d.

– S(R^d)the space ofC^∞-functions onR^dwhich are rapidly decreasing as their derivatives.

– D(R^d)the space ofC^∞-functions onR^dwhich are of compact support.

We provide these spaces with the classical topology.

Consider also the following spaces

– E⁰(R^d)the space of distributions onR^dwith compact support. It is the topological dual ofE(R^d).

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– S⁰(R^d)the space of temperate distributions onR^d. It is the topological dual of S(R^d).

The Dunkl operatorsT_j, j = 1, . . . , d, onR^dassociated with the finite reflection groupW and multiplicity functionk are given by

(2.5) T_jf(x) = ∂

∂x_jf(x) + X

α∈R₊

k(α)α_jf(x)−f(σα(x))

hα, xi , f ∈C¹(R^d).

In the casek = 0, theT_j, j = 1, . . . , d,reduce to the corresponding partial derivatives. In this paper, we will assume throughout thatk≥0.

Fory ∈R^d, the system

( T_ju(x, y) =y_ju(x, y), j = 1, . . . , d, u(0, y) = 1, for ally∈ R^d

admits a unique analytic solution onR^d, which will be denoted byK(x, y)and called the Dunkl kernel. This kernel has a unique holomorphic extension toC^d×C^d. The Dunkl kernel possesses the following properties.

Proposition 2.1. Letz, w ∈C^d, andx, y ∈R^d. i)

K(z, w) = K(w, z), K(z,0) = 1 and (2.6)

K(λz, w) = K(z, λw), for all λ∈C. ii) For allν ∈N^d, x∈R^dandz ∈C^d, we have

(2.7) |D^ν_zK(x, z)| ≤ kxk^|ν| exp(kxkkRezk),

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and for allx, y ∈R^d:

(2.8) |K(ix, y)| ≤1,

withD_z^ν = ^∂^ν

∂z₁^ν¹···∂z_d^νd and|ν|=ν₁+· · ·+ν_d. iii) For allx, y ∈R^dandw∈W we have

(2.9) K(−ix, y) =K(ix, y) and K(wx, wy) =K(x, y).

The Dunkl intertwining operatorV_kis defined onC(R^d)by

(2.10) V_kf(x) =

Z

R^d

f(y)dµ_x(y), for allx∈R^d,

where dµ_x is a probability measure given on R^d, with support in the closed ball B(0,kxk)of center0and radiuskxk.

2.3. The Dunkl Transform

The results of this subsection are given in [7] and [18].

Notations. We denote by

– L^p_k(R^d)the space of measurable functions onR^dsuch that kfk_L^p

k(R^d) = Z

R^d

|f(x)|^pω_k(x)dx ¹_p

<∞, if 1≤p <∞, kfk_L^∞

k (R^d) = ess sup

x∈R^d

|f(x)|<∞.

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– H(C^d) the space of entire functions onC^d, rapidly decreasing of exponential type.

– H(C^d) the space of entire functions on C^d, slowly increasing of exponential type.

We provide these spaces with the classical topology.

The Dunkl transform of a functionf inD(R^d)is given by (2.11) F_D(f)(y) = 1

c_k Z

R^d

f(x)K(−iy, x)ω_k(x)dx, for ally∈R^d. It satisfies the following properties:

i) Forf inL¹_k(R^d)we have

(2.12) kF_D(f)k_L^∞

k (R^d) ≤ 1 c_kkfk_L¹

k(R^d). ii) Forf inS(R^d)we have

(2.13) ∀y∈R^d, F_D(T_jf)(y) =iy_jF_D(f)y), j = 1, . . . , d.

iii) For allfinL¹_k(R^d)such thatF_D(f)is inL¹_k(R^d), we have the inversion formula

(2.14) f(y) =

Z

R^d

F_D(f)(x)K(ix, y)ω_k(x)dx, a.e.

Theorem 2.2. The Dunkl transformF_D is a topological isomorphism.

i) FromS(R^d)onto itself.

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ii) FromD(R^d)ontoH(C^d).

The inverse transformF_D⁻¹ is given by

(2.15) ∀y∈R^d, F_D⁻¹(f)(y) = F_D(f)(−y), f ∈S(R^d).

Theorem 2.3. The Dunkl transformF_D is a topological isomorphism.

i) FromS⁰(R^d)onto itself.

ii) FromE⁰(R^d)ontoH(C^d).

Theorem 2.4.

i) Plancherel formula forF_D. For all f inS(R^d)we have (2.16)

Z

R^d

|f(x)|²ω_k(x)dx= Z

R^d

|F_D(f)(ξ)|²ω_k(ξ)dξ.

ii) Plancherel theorem forF_D. The Dunkl transformf → F_D(f)can be uniquely extended to an isometric isomorphism onL²_k(R^d).

2.4. The Dunkl Convolution Operator

Definition 2.5. Letybe inR^d. The Dunkl translation operatorf 7→τ_yf is defined onS(R^d)by

(2.17) F_D(τ_yf)(x) = K(ix, y)F_D(f)(x), for allx∈R^d. Example 2.1. Lett >0, we have

τ_x(e^−tkξk²)(y) =e^−t(kxk²^+kyk²⁾K(2tx, y), for allx∈R^d.

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Remark 1. The operatorτ_y,y∈R^d, can also be defined onE(R^d)by (2.18) τ_yf(x) = (V_k)_x(V_k)_y[(V_k)⁻¹(f)(x+y)], for allx∈R^d (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]).

• 1^st case:d= 1andW =Z2.

• 2^nd case: For allf inE(R^d)radial we have (2.19) τ_yf(x) = V_kh

f₀p

kxk²+kyk² + 2hx,·ii

(x), for allx∈R^d, withf₀ 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 off andg inD(R^d)is the function f∗_D gdefined by

(2.20) f∗_D g(x) = Z

R^d

τ_xf(−y)g(y)ω_k(y)dy, for allx∈R^d.

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(R^d) (resp. S(R^d)) the function f ∗_D g belongs to D(R^d) (resp.S(R^d)) and we have

FD(f ∗Dg)(y) = FD(f)(y)FD(g)(y), for ally∈R^d.

ii) Let 1 ≤ p, q, r ≤ ∞, such that ¹_p + ¹_q − ¹_r = 1. If f is in L^p_k(R^d)andg is a radial element ofL^q_k(R^d),thenf∗_D g ∈L^r_k(R^d) and we have

(2.21) kf∗_D gk_Lr

k(R^d) ≤ kfk_L^p

k(R^d)kgk_L^q

k(R^d).

iii) LetW =Z^d2. We have the same result for allf ∈L^p_k(R^d)andg ∈L^q_k(R^d).

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3. Littlewood-Paley Theory Associated with Dunkl Operators

We consider now a dyadic decomposition ofR^d. 3.1. Dyadic Decomposition

Forp≥0be a natural integer, we set

(3.1) C_p ={ξ ∈R^d; 2^p−1 ≤ kξk ≤2^p+1}= 2^pC₀ and

(3.2) C−1 =B(0,1) = {ξ∈R^d; kξk ≤1}.

ClearlyR^d=S∞ p=−1C_p. Remark 2. We remark that

(3.3) card

n q; Cp

\Cq6=∅ o≤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(R^d), radial with supp ψ ⊂C₋₁, and suppϕ ⊂C₀, such that for anyξ ∈R^dandn∈N, we have

ψ(ξ) +

∞

X

p=0

ϕ(2^−pξ) = 1

and

ψ(ξ) +

n

X

p=0

ϕ(2^−pξ) =ψ(2⁻ⁿξ).

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Remark 3. It is not hard to see that for anyξ∈R^d

(3.4) 1

2 ≤ψ²(ξ) +

∞

X

p=0

ϕ²(2^−pξ)≤2.

Definition 3.2. Let λ ∈ R. For χ in S(R^d), we define the pseudo-differential- difference operatorχ(λT)by

F_D(χ(λT)u) = χ(λξ)F_D(u), u∈ S⁰(R^d).

Definition 3.3. ForuinS⁰(R^d), we define its Littlewood-Paley decomposition asso- ciated with the Dunkl operators (or dyadic decomposition){∆_pu}^∞_p=−1 as∆−1u = ψ(T)uand forq≥0,∆_qu=ϕ(2^−qT)u.

Now we go to see in which case we can have the identity Id= X

p≥−1

∆_p.

This is described by the following proposition.

Proposition 3.4. ForuinS⁰(R^d), we haveu=P∞

p=−1∆_pu, in the sense ofS⁰(R^d).

Proof. For anyf in S(R^d), it is easy to see thatF_D(f) = P∞

p=−1F_D(∆_pf)in the sense ofS(R^d). Then for anyuinS⁰(R^d), we have

hu, fi=hF_D(u),F_D(f)i=

∞

X

p=−1

hF_D(u),F_D(∆_pf)i

=

∞

X

p=−1

hF_D(∆_pu),F_D(f)i

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=

* _∞ X

p=−1

F_D(∆_pu),F_D(f) +

=

* _∞ X

p=−1

∆_pu, f +

.

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 3.5. Letsbe inR, we define the spaceH_k^s(R^d)by u∈ S⁰(R^d) : (1 +kξk²)^s²F_D(u)∈L²_k(R^d) . We provide this space by the scalar product

(3.5) hu, vi_H^s

k(R^d) = Z

R^d

(1 +kξk²)^sF_D(u)(ξ)F_D(v)(ξ)ω_k(ξ)dξ,

and the norm

(3.6) kuk²_Hs

k(R^d) =hu, ui_H^s

k(R^d). Another proposition will be useful. LetS_qu=P

p≤q−1∆_pu.

Proposition 3.6. For allsinRand for all distributionsuinH_k^s(R^d), we have

n→∞lim S_nu=u.

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Proof. For allξinR^d, we have

F_D(S_nu−u)(ξ) = (ψ(2⁻ⁿξ)−1)F_D(u)(ξ).

Hence

n→∞lim F_D(S_nu−u)(ξ) = 0.

On the other hand

(1 +kξk²)^s|F_D(S_nu−u)(ξ)|² ≤2(1 +kξk²)^s|F_D(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 onqofk∆_quk_L²

k(R^d). More precisely, we now define a norm equivalent to the normk · k_H^s

k(R^d) in terms of the dyadic decomposition.

Proposition 3.7. There exists a positive constantCsuch that for allsinR, we have 1

C^|s|+1kuk²_Hs

k(R^d) ≤ X

q≥−1

2^2qsk∆_quk²_L2

k(R^d)≤C^|s|+1kuk²_Hs k(R^d). Proof. SincesuppF_D(∆_qu)⊂C_q, from the definition of the normk · k_H^s

k(R^d), there exists a positive constantCsuch that we have

(3.7) 1

C^|s|+12^qsk∆_quk_L²

k(R^d)≤ k∆_quk_H^s

k(R^d) ≤C^|s|+12^qsk∆_quk_L²

k(R^d). From (3.4) we deduce that

1 2kuk²_Hs

k(R^d)≤ Z

R^d

"

ψ²(ξ) +

∞

X

q=0

ϕ²(2^−qξ)

#

(1 +kξk²)^s|F_D(u)(ξ)|²ω_k(ξ)dξ

≤2kuk²_Hs k(R^d).

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Hence

1 2kuk²_Hs

k(R^d) ≤ X

q≥−1

k∆_quk²_Hs

k(R^d)≤2kuk²_Hs k(R^d). Thus from this and (3.7) we deduce the result.

The following theorem is a consequence of Proposition3.7.

Theorem 3.8. Letube inS⁰(R^d)andu=P

q≥−1∆_quits Littlewood-Paley decom- position. The following are equivalent:

i) u∈H_k^s(R^d).

ii) P

q≥−12^2qsk∆_quk²_L2

k(R^d) <∞.

iii) k∆_quk_L²

k(R^d) ≤c_q2^−qs,with{c_q} ∈l².

Remark 4. Since foruinS⁰(R^d)we have∆_puinS⁰(R^d)andsupp F_D(∆_pu)⊂C_p, from Theorem2.3ii) we deduce that∆_puis inE(R^d).

The following propositions will be very useful.

Proposition 3.9. LetCebe an annulus inR^dandsinR. Let(u_p)p∈Nbe a sequence of smooth functions. If the sequence(u_p)p∈Nsatisfies

suppFD(up)⊂2^pCe and kupk_L²

k(R^d)≤Ccp2^−ps,{cp} ∈l², then we have

u=X

p≥0

u_p ∈H_k^s(R^d) and kuk_H^s

k(R^d) ≤C(s) X

p≥0

2^2psku_pk²_L2 k(R^d)

!¹₂ .

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Proof. SinceCeandC₀ are two annuli, there exists an integerN₀ so that

|p−q|> N₀ =⇒ 2^pC₀\

2^qCe =∅. It is clear that

|p−q|> N0 =⇒ FD(∆qup) = 0.

Then

∆qu= X

|p−q|≤N0

∆qup.

By the triangle inequality and definition of∆_qu_pwe deduce that k∆_quk_L²

k(R^d) ≤ X

|p−q|≤N0

ku_pk_L²

k(R^d). Thus the Cauchy-Schwartz inequality implies that

X

q≥0

2^2qsk∆_quk²_L2

k(R^d) ≤C



 X

q/|p−q|≤N0

2^2(q−p)s



 X

p≥0

! .

From Theorem3.8we deduce that ifku_pk_L²

k(R^d) ≤Cc_p2^−ps thenu∈H_k^s(R^d).

Proposition 3.10. Let K > 0 and s > 0. Let (u_p)p∈N be a sequence of smooth functions. If the sequence(u_p)p∈Nsatisfies

suppFD(up)⊂B(0, K2^p) and kupk_L²

k(R^d) ≤Ccp2^−ps,{cp} ∈l², then we have

u=X

p≥0

k(R^d) ≤C(s) X

q≥0

!¹₂ .

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Proof. SincesuppF_D(u_p)⊂B(0, K2^p), there existsN₁ such that

∆_qu= X

p≥q−N₁

∆_qu_p.

So, we get that

2^qsk∆quk_L²

k(R^d)≤ X

p≥q−N1

2^qskupk_L²

k(R^d)

= X

p≥q−N1

2^(q−p)s2^pskupk_L²

k(R^d). Sinces >0, the Cauchy-Schwartz inequality implies

X

q

2^2qsk∆_quk²_L2

k(R^d) ≤ 2^2N¹^s 1−2^−s

X

p

2^2psku_pk²_L2 k(R^d). From Theorem3.8we deduce the result.

Proposition 3.11. Lets >0and(up)p∈N be a sequence of smooth functions. If the sequence(up)p∈Nsatisfies

u_p ∈ E(R^d) and for allµ ∈N^d, kT^µu_pk_L²

k(R^d) ≤Cc_p,µ2^{−p(s−|µ|)}, {c_p,µ} ∈l², then we have

u=X

p≥0

k(R^d) ≤C(s) X

p≥0

!¹₂ .

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Proof. By the assumption we first have u = P

u_p ∈ L²_k(R^d). Take µ ∈ N^d with

|µ|=s₀ > s >0, andχ_p(ξ) =χ(2^−pξ)∈D(R^d)withsuppχ⊂B(0,2), χ(ξ) = 1, kξk ≤1and0≤χ≤1, then

suppχ_p(1−χ_p)⊂

ξ ∈R^d; 2^p ≤ kξk ≤2^p+2 . Set

F_D(u_p)(ξ) =χ_p(ξ)F_D(u_p)(ξ) + (1−χ_p(ξ))F_D(u_p)(ξ)

=F_D(u⁽¹⁾_p )(ξ) +F_D(u⁽²⁾_p )(ξ), and we have

ku_pk²_L2

k(R^d) =kF_D(u_p)k²_L2 k(R^d)

= Z

R^d

|F_D(u⁽¹⁾_p )(ξ)|²ω_k(ξ)dξ+ Z

R^d

|F_D(u⁽²⁾_p )(ξ)|²ω_k(ξ)dξ

+2 Z

R^d

|F_D(u_p)(ξ)|²χ_p(ξ)(1−χ_p(ξ))ω_k(ξ)dξ

.

Since0≤χ_p(ξ)(1−χ_p(ξ))≤1, we deduce that u⁽¹⁾_p

2

L²_k(R^d)+ u⁽²⁾_p

2

L²_k(R^d)≤ kupk²_L2

k(R^d) ≤c²_p2^−2ps. Similarly, using Theorem 3.1 of [9], we obtain

u⁽¹⁾_p

2

H_k^s⁰(R^d)+ u⁽²⁾_p

2

H^s_k⁰(R^d)≤ kupk²_H^s0

k (R^d) ≤c²_p2^−2p(s−s⁰⁾. Setu⁽¹⁾ =P

pu⁽¹⁾p ,u⁽²⁾ =P

pu⁽²⁾p , thenu=u⁽¹⁾+u⁽²⁾, and from Proposition3.10

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we deduce thatu⁽¹⁾belongs toH_k^s(R^d). Foru⁽²⁾the definition ofu⁽²⁾p gives that k∆q(u⁽²⁾)k²_L2

k(R^d) = Z

R^d

X

p≤q+1

ϕ(2^−qξ)FD(u⁽²⁾_p )(ξ)

2

ωk(ξ)dξ.

Thus by the Cauchy-Schwartz inequality we have k∆_q(u⁽²⁾)k²_L2

k(R^d)

≤ X

p≤q+1

2^−2p(s−s⁰⁾

! Z

R^d

X

p≤q+1

2^2p(s−s⁰⁾|ϕ(2^−qξ)F_D(u⁽²⁾_p )(ξ)|²ω_k(ξ)dξ

!

≤ 1−2−2(q+2)(s−s0)

1−2^−(s−s⁰⁾ 2^−2qs⁰ X

p≤q+1

2^2p(s−s⁰⁾

∆_q u⁽²⁾_p

2

H_k^s⁰(R^d). Moreover, sinces₀ > s >0,

1−2−2(q+2)(s−s0)

1−2^−(s−s⁰⁾ 2^−2qs⁰ ≤C2^−2qs, andC is independent ofq. Now set

c²_q = X

p≤q+1

2^2p(s−s⁰⁾

∆q u⁽²⁾_p

2

H_k^s⁰(R^d), then

X

q≥−1

2^2qsk∆_q(u⁽²⁾)k²_L2

k(R^d)≤ X

q≥−1

c²_q ≤X

p

2^2p(s−s⁰⁾ u⁽²⁾_p

2

H_k^s⁰(R^d) <∞.

Thus by Theorem3.8we deduce thatu⁽²⁾ =P

q∆_q(u⁽²⁾)belongs toH_k^s(R^d).

Corollary 3.12. The spacesH_k^s(R^d)do not depend on the choice of the functionϕ andψ used in the Definition3.3.

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3.3. The Generalized Hölder Spaces

Definition 3.13. ForαinR, we define the Hölder spaceC_k^α(R^d)associated with the Dunkl operators as the set ofu∈ S⁰(R^d)satisfying

kuk_C^α

k(R^d)= sup

p≥−1

2^pαk∆_puk_L^∞

k (R^d)<∞, whereu=P

p≥−1∆_puis its Littlewood-Paley decomposition.

In the following proposition we give sufficient conditions so that the seriesP

quq

belongs to the Hölder spaces associated with the Dunkl operators.

Proposition 3.14.

i) Let Ce be an annulus inR^d andα ∈ R. Let(u_p)p∈N be a sequence of smooth functions. If the sequence(u_p)p∈Nsatisfies

suppF_D(u_p)⊂2^pCe and ku_pk_L^∞

k (R^d) ≤C2^−pα, then we have

u=X

p≥0

u_p ∈C_k^α(R^d) and kuk_C^α

k(R^d)≤C(α) sup

p≥0

2^pαku_pk_L^∞

k (R^d). ii) LetK > 0andα > 0. Let (u_p)_p∈_N be a sequence of smooth functions. If the

sequence(u_p)_p∈_Nsatisfies

suppF_D(u_p)⊂B(0, K2^p) and ku_pk_L^∞

k (R^d) ≤C2^−pα, then we have

u=X

p≥0

u_p ∈C_k^α(R^d) and kuk_C^α

k(R^d)≤C(α) sup

p≥0

2^pαku_pk_L^∞

k (R^d).

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Proof. The proof uses the same idea as for Propositions3.9and3.10.

Proposition 3.15. The distribution defined by g(x) = X

p≥0

K(ix,2^pe), with e= (1, . . . ,1),

belongs toC_k⁰(R^d)and does not belong toL^∞_k (R^d).

Proposition 3.16. Letε∈]0,1[andf inC_k^ε(R^d), then there exists a positive constant Csuch that

kfk_L^∞

k (R^d)≤ C εkfk_C⁰

k(R^d)log e+ kfk_C^ε

k(R^d)

kfk_C⁰

k(R^d)

! .

Proof. Sincef =P

p≥−1∆_pf, kfk_L^∞

k(R^d)≤ X

p≤N−1

k∆_pfk_L^∞

k (R^d)+X

p≥N

k∆_pfk_L^∞

k(R^d),

withN is a positive integer that will be chosen later. Sincef ∈ C_k^ε(R^d), using the definition of generalized Hölderien norms, we deduce that

kfk_L^∞

k (R^d) ≤(N + 1)kfk_C⁰

k(R^d)+ 2^−(N^−1)ε 2^ε−1 kfk_C^ε

k(R^d). We take

N = 1 +

"

1

ε log₂ kfk_C^ε

k(R^d)

kfk_C⁰

k(R^d)

# , we obtain

kfk_L^∞

k (R^d) ≤ C εkfk_C⁰

k(R^d)

"

1 + log kfk_C^ε

k(R^d)

kfk_C⁰

k(R^d)

!#

. This implies the result.

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Now we give the characterization of L^p_k(R^d)spaces by using the dyadic decomposition.

If(f_j)j∈Nis a sequence ofL^p_k(R^d)-functions, we set k(f_j)k_L^p

k(l²) =

X

j∈N

|f_j(x)|²

!¹₂ L^p_k(R^d)

,

the norm inL^p_k(R^d, l²(N)).

Theorem 3.17 (Littlewood-Paley decomposition of L^p_k(R^d)). Let f be inS⁰(R^d) and1< p <∞.Then the following assertions are equivalent

i) f ∈L^p_k(R^d), ii) S₀f ∈L^p_k(R^d)and

P

j∈N|∆_jf(x)|²¹₂

∈L^p_k(R^d).

Moreover, the following norms are equivalent :

kfk_L^p

k(R^d) and kS₀fk_L^p

k(R^d)+

X

j∈N

|∆_jf(x)|²

!¹₂ _Lp

k(R^d)

.

Proof. Iff is inL²_k(R^d), then from Proposition3.7we have

X

j∈N

|∆_jf(x)|²

!¹₂ L²_k(R^d)

≤ kfk²_L2 k(R^d). Thus the mapping

Λ1 :f 7→(∆jf)j∈N,

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is bounded fromL²_k(R^d)intoL²_k(R^d, l²(N)).

On the other hand, from properties ofϕwe see that k(ϕe_j(x))_jk_l² ≤Ckxk^−(d+2γ), forx6= 0,

k(∂_y_iϕe_j(x))_jk_l² ≤Ckxk^−(d+2γ), forx6= 0, i= 1, . . . , d, where

ϕej(x) = 2^j(d+2γ)F_D⁻¹(ϕ)(2^jx).

We may then apply the theory of singular integrals to this mappingΛ₁ (see [14]).

Thus we deduce that k∆_jfk_L^p

k(l²) ≤C_p,kkfk_L^p

k(R^d), for 1< p <∞.

The converse uses the same idea. Indeed we put

φe_j =

1

X

i=−1

ϕe_j+i. From Proposition3.7the mapping

Λ₂ : (f_j)j∈N7→X

j∈N

f_j∗_D φe_j,

is bounded fromL²_k(R^d, l²(N))intoL²_k(R^d).

On the other hand, from properties ofϕwe see that k(φe_j(x))_jk_l² ≤Ckxk^−(d+2γ), forx6= 0,

k(∂_y_iφe_j(x))_jk_l² ≤Ckxk^−(d+2γ), forx6= 0, i= 1, . . . , d.