We define the Littlewood-Paley decomposition associated with the Dunkl opera- tors

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

HATEM MEJJAOLI DEPARTMENT OFMATHEMATICS

FACULTY OFSCIENCES OFTUNIS

CAMPUS 1060 TUNIS, TUNISIA hatem.mejjaoli@ipest.rnu.tn

Received 11 July, 2007; accepted 25 May, 2008 Communicated by S.S. Dragomir

Dedicated to Khalifa Trimeche.

ABSTRACT. We define the Littlewood-Paley decomposition associated with the Dunkl opera- tors; 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.

Key words and phrases: Dunkl operators, Littlewood-Paley decomposition, Paraproduct.

2000 Mathematics Subject Classification. Primary 35L05. Secondary 22E30.

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 L^p(R^d), the Sobolev spaces H^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 exam- ple, 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

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 |bφ(ξ)| ≥ C >0. Forj ∈Z, letφ_j(x) = 2^jdφ(2^jx). In 1967 Peetre [10] proved that

(1.1) kfk_H^s_(R^d₎ ' kχ∗fk_L²_(R^d₎+ X

j≥1

2^2sjkφj∗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_Fp,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 spacesH^s(R^d), Hölder spacesC^s(R^d)and others. Using the Littlewood-Paley decom- position 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 differ- ential 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 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 de- composition associated with the Dunkl operators, we give the sufficient condition onu_p so that u:=P

u_p belongs 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 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.

2. THEEIGENFUNCTION OF THEDUNKL 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 considerR^dwith the euclidean scalar product h·,·i and kxk = p

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

j=1z_jw_j.

Forα∈R^d\{0}, letσ_αbe the reflection in the hyperplaneH_α ⊂R^dorthogonal toα, i.e.

(2.1) σ_α(x) =x−2hα, xi

kαk² α.

A finite set R ⊂ R^d\{0} is called a root system if R ∩R ·α = {α,−α} and σ_αR = R for all α ∈ R. For a given root system R the reflections σ_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 givenβ ∈ R^d\∪α∈RH_α, we fix the positive subsystem R₊ = {α∈R :hα, βi>0}, then for eachα ∈ Reitherα ∈ R₊or−α ∈ R₊. We will assume that hα, 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 group W. If one regards k as a function on the corresponding reflections, this means thatk 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).

Moreover, letωkdenote the weight function

(2.3) ω_k(x) = Y

α∈R₊

| hα, xi |^2k(α),

which is invariant and homogeneous of degree2γ. 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 of C^∞-functions on R^d which are rapidly decreasing as their deriva- tives.

– 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).

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

The Dunkl operatorsT_j, j = 1, . . . , d, onR^d associated with the finite reflection groupW and multiplicity functionkare 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)

(2.6) K(z, w) =K(w, z), K(z,0) = 1 and K(λz, w) = K(z, λw), for allλ∈C. ii) For allν ∈N^d, x∈R^dandz ∈C^d, we have

(2.7) |D_z^νK(x, z)| ≤ kxk^|ν| exp(kxkkRezk), 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,

wheredµ_x is a probability measure given onR^d, with support in the closed ballB(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)|<∞.

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

– H(C^d)the space of entire functions onC^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 allf inL¹_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.

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 2.1. Lety be 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. 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 2.2. The Dunkl convolution product of f and g in D(R^d) is the function f ∗D g defined by

(2.20) f ∗D g(x) =

Z

R^d

τxf(−y)g(y)ωk(y)dy, for allx∈R^d.

This convolution is commutative, associative and satisfies the following properties. (See [13]).

Proposition 2.5.

i) Forf andginD(R^d)(resp.S(R^d)) the functionf∗_Dgbelongs toD(R^d)(resp.S(R^d)) and we have

F_D(f ∗_Dg)(y) = F_D(f)(y)F_D(g)(y), for ally∈R^d.

ii) Let 1 ≤ p, q, r ≤ ∞, such that ¹_p + ¹_q − ¹_r = 1. Iff is inL^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 ∗Dgk_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).

3. LITTLEWOOD-PALEYTHEORYASSOCIATED WITH DUNKLOPERATORS

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=−1Cp. Remark 2. We remark that

(3.3) card n

q; C_p\

C_q 6=∅ o≤2.

Now, let us define a dyadic partition of unity that we shall use throughout this paper.

Lemma 3.1. There exist positive functionsϕandψinD(R^d), radial with suppψ ⊂ 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).

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 3.1. Letλ∈R. ForχinS(R^d), we define the pseudo-differential-difference opera- torχ(λT)by

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

Definition 3.2. ForuinS⁰(R^d), we define its Littlewood-Paley decomposition associated 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 3.2. ForuinS⁰(R^d), we haveu=P∞

p=−1∆_pu, in the sense ofS⁰(R^d).

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

p=−1F_D(∆_pf) in the sense of S(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

=

* _∞ 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 3.3. 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 3.3. For allsinRand for all distributionsuinH_k^s(R^d), we have

n→∞lim Snu=u.

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 oper- ators is the characterization of the Sobolev spaces associated with these operators through the behavior onq of k∆_quk_L²

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

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

Proposition 3.4. 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. Since suppF_D(∆_qu)⊂C_q, from the definition of the norm k · 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). 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 Proposition 3.4.

Theorem 3.5. Let u be in S⁰(R^d) and u = P

q≥−1∆_qu its Littlewood-Paley decomposition.

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) ≤cq2^−qs,with{cq} ∈l².

Remark 4. Since for u in S⁰(R^d) we have ∆_pu in S⁰(R^d) and supp F_D(∆_pu) ⊂ C_p, from Theorem 2.3 ii) we deduce that∆_puis inE(R^d).

The following propositions will be very useful.

Proposition 3.6. LetCe be an annulus inR^dandsinR. Let(u_p)p∈N be a sequence of smooth functions. If the sequence(up)p∈Nsatisfies

suppF_D(u_p)⊂2^pCe and ku_pk_L²

k(R^d)≤Cc_p2^−ps,{c_p} ∈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 .

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

∆_qu_p.

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

2^2psku_pk²_L2 k(R^d)

! .

From Theorem 3.5 we deduce that ifku_pk_L²

k(R^d)≤Cc_p2^−ps thenu∈H_k^s(R^d).

Proposition 3.7. LetK > 0ands >0. Let(up)p∈Nbe a sequence of smooth functions. If the sequence(u_p)_p∈Nsatisfies

suppF_D(u_p)⊂B(0, K2^p) and ku_pk_L²

k(R^d) ≤Cc_p2^−ps,{c_p} ∈l², then we have

u=X

p≥0

up ∈H_k^s(R^d) and kuk_H^s

k(R^d) ≤C(s) X

q≥0

2^2pskupk²_L2 k(R^d)

!¹₂ .

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^qsku_pk_L²

k(R^d)

= X

p≥q−N₁

2^(q−p)s2^psku_pk_L²

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

X

q

2^2qsk∆_quk²_L2

k(R^d)≤ 2^2N1s 1−2^−s

X

p

2^2psku_pk²_L2 k(R^d).

From Theorem 3.5 we deduce the result.

Proposition 3.8. Let s > 0 and (u_p)p∈N be a sequence of smooth functions. If the sequence (u_p)p∈Nsatisfies

up ∈ E(R^d) and for allµ ∈N^d, kT^µupk_L²

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

u=X

p≥0

up ∈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 .

Proof. By the assumption we first haveu=P

u_p ∈L²_k(R^d). Takeµ∈N^dwith|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)

=FD(u⁽¹⁾_p )(ξ) +FD(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)≤ ku_pk²_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_k^s0(R^d)≤ ku_pk²_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 Proposition 3.7 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)F_D(u⁽²⁾_p )(ξ)

2

ω_k(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)FD(u⁽²⁾_p )(ξ)|²ωk(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 Theorem 3.5 we deduce thatu⁽²⁾ =P

q∆_q(u⁽²⁾)belongs toH_k^s(R^d).

Corollary 3.9. The spacesH_k^s(R^d)do not depend on the choice of the functionϕ andψ used in the Definition 3.2.

3.3. The Generalized Hölder Spaces.

Definition 3.4. For α in R, we define the Hölder space C_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

qu_qbelongs to the Hölder spaces associated with the Dunkl operators.

Proposition 3.10.

i) LetCe be an annulus inR^dandα ∈R. Let(u_p)p∈Nbe a sequence of smooth functions.

If the sequence(up)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∈Nbe a sequence of smooth functions. If the sequence (u_p)p∈Nsatisfies

suppFD(up)⊂B(0, K2^p) and kupk_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).

Proof. The proof uses the same idea as for Propositions 3.6 and 3.7.

Proposition 3.11. 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 3.12. Let ε ∈]0,1[andf inC_k^ε(R^d), then there exists a positive constantC such 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.

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

If(fj)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 3.13 (Littlewood-Paley decomposition ofL^p_k(R^d)). Letf 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 L^p_k(R^d)

.

Proof. Iff is inL²_k(R^d), then from Proposition 3.4 we 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, 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(∂_yiϕe_j(x))_jk_l² ≤Ckxk^−(d+2γ), forx6= 0, i= 1, . . . , d,

where

ϕe_j(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 Proposition 3.4 the 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(∂_yiφe_j(x))_jk_l² ≤Ckxk^−(d+2γ), forx6= 0, i= 1, . . . , d.

We may then apply the theory of singular integrals to this mappingΛ₂(see [14]).

Thus we obtain

X

j∈N

∆jf L^p_k(R^d)

≤Cp,kk∆jfk_L^p

k(l²).

4. APPLICATIONS

4.1. Estimates of the Product of Two Functions.

Proposition 4.1.

i) Letu, v ∈C_k^α(R^d)andα >0thenuv ∈C_k^α(R^d), and kuvk_C^α

k(R^d) ≤Ch kuk_L^∞

k (R^d)kvk_C^α

k(R^d)+kvk_L^∞

k (R^d)kuk_C^α

k(R^d)

i . ii) Letu, v ∈H_k^s(R^d)T

L^∞_k (R^d)ands >0thenuv ∈H_k^s(R^d), and kuvk_H^s

k(R^d)≤C h

kuk_L^∞

k (R^d)kvk_H^s

k(R^d)+kvk_L^∞

k (R^d)kuk_H^s

k(R^d)

i . Proof. Letu =P

p∆_puandv =P

q∆_qv be their Littlewood-Paley decompositions. Then we have

uv =X

p,q

∆_pu∆_qv

=X

q

X

p≤q−1

∆_pu∆_qv+X

q

X

p≥q

∆_pu∆_qv

=X

q

X

p≤q−1

∆pu∆qv+X

p

X

q≤p

∆pu∆qv

=X

q

S_qu∆_qv+X

p

S_p+1v∆_pu

=X

1

+X

2

.

We have

supp (F_D(S_qu∆_qv)) = supp (F_D(∆_qv)∗_DF_D(S_qu)).

Hence from Theorem 2.2 we deduce thatsupp (F_D(S_qu∆_qv))⊂B(0, C2^q).

i) Ifuandv are inC_k^α(R^d), then we have kS_qu∆_qvk_L^∞

k (R^d) ≤ kS_quk_L^∞

k (R^d)k∆_qvk_L^∞

k(R^d),

≤Ckuk_L^∞

k (R^d)kvk_C^α

k(R^d)2^−qα. From Proposition 3.10 ii) we deduce

X

1

C^α_k(R^d)

≤Ckuk_L^∞

k(R^d)kvk_C^α

k(R^d). Similarly we prove that

X

2

C^α_k(R^d)

≤Ckvk_L^∞

k(R^d)kuk_C^α

k(R^d), and this implies the result.

ii) Ifuandv are inH_k^s(R^d), then we have