GENERALIZED DUNKL-SOBOLEV SPACES OF EXPONENTIAL TYPE AND APPLICATIONS

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Dunkl-Sobolev Spaces Hatem Mejjaoli vol. 10, iss. 2, art. 55, 2009

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GENERALIZED DUNKL-SOBOLEV SPACES OF EXPONENTIAL TYPE AND APPLICATIONS

HATEM MEJJAOLI

Department of Mathematics Faculty of Sciences of Tunis Campus-1060. Tunis, Tunisia.

EMail:hatem.mejjaoli@ipest.rnu.tn

Received: 22 March, 2008

Accepted: 23 May, 2009

Communicated by: S.S. Dragomir

2000 AMS Sub. Class.: Primary 46F15. Secondary 46F12.

Key words: Dunkl operators, Dunkl-Bessel-Laplace operator, Generalized Dunkl-Sobolev spaces of exponential type, Pseudo differential-difference operators, Reproduc- ing kernels.

Abstract: We study the Sobolev spaces of exponential type associated with the Dunkl- Bessel Laplace operator. Some properties including completeness and the imbedding theorem are proved. We next introduce a class of symbols of exponential type and the associated pseudo-differential-difference operators, which naturally act on the generalized Dunkl-Sobolev spaces of exponential type. Finally, using the theory of reproducing kernels, some applications are given for these spaces.

Acknowledgement: Thanks to the referee for his suggestions and comments.

Dedicatory: Dedicated to Khalifa Trimèche.

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1. Introduction

The Sobolev space W^s,p(R^d) serves as a very useful tool in the theory of partial differential equations, which is defined as follows

W^s,p(R^d) = n

u∈ S⁰(R^d), (1 +||ξ||²)^p^sF(u)∈L^p(R^d)o . In this paper we consider the Dunkl-Bessel Laplace operator4_k,β defined by

∀x= (x⁰, x_d+1) ∈ R^d×]0,+∞[, 4_k,β =4_k,x⁰ +L_β,x_d+1, β ≥ −1 2, where4_k is the Dunkl Laplacian onR^d, and L_β is the Bessel operator on]0,+∞[.

We introduce the generalized Dunkl-Sobolev space of exponential typeW_G^s,p_∗_,k,β(R^d+1+ ) by replacing(1 +||ξ||²)^s^p by an exponential weight function defined as follows

W_G^s,p

∗,k,β(R^d+1+ ) = n

u∈ G_∗⁰, e^s||ξ||F_D,B(u)∈L^p_k,β(R^d+1+ )o ,

whereL^p_k,β(R^d+1+ )it is the Lebesgue space associated with the Dunkl-Bessel transform andG_∗⁰ is the topological dual of the Silva space. We investigate their properties such as the imbedding theorems and the structure theorems. In fact, the imbedding theorems mean that fors > 0,u ∈ W_G^s,p_∗_,k,β(R^d+1+ )can be analytically continued to the set{z∈C^d+1/|Imz|< s}. For the structure theorems we prove that fors >0, u∈W_G^−s,2

∗,k,β(R^d+1+ )can be represented as an infinite sum of fractional Dunkl-Bessel Laplace operators of square integrable functionsg, in other words,

u=X

m∈N

s^m

m!(−4_k,β)^m²g.

We prove also that the generalized Dunkl-Sobolev spaces are stable by multiplication of the functions of the Silva spaces. As applications on these spaces, we study

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the action for the class of pseudo differential-difference operators and we apply the theory of reproducing kernels on these spaces. We note that special cases include:

the classical Sobolev spaces of exponential type, the Sobolev spaces of exponential types associated with the Weinstein operator and the Sobolev spaces of exponential type associated with the Dunkl operators.

We conclude this introduction with a summary of the contents of this paper. In Section2we recall the harmonic analysis associated with the Dunkl-Bessel Laplace operator which we need in the sequel. In Section 3 we consider the Silva space G∗ and its dual G_∗⁰. We study the action of the Dunkl-Bessel transform on these spaces. Next we prove two structure theorems for the spaceG_∗⁰. We define in Section 4the generalized Dunkl-Sobolev spaces of exponential typeW_G^s,p_∗_,k,β(R^d+1+ )and we give their properties. In Section5we give two applications on these spaces. More precisely, in the first application we introduce certain classes of symbols of exponential type and the associated pseudo-differential-difference operators of exponential type. We show that these pseudo-differential-difference operators naturally act on the generalized Sobolev spaces of exponential type. In the second, using the theory of reproducing kernels, some applications are given for these spaces.

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2. Preliminaries

In order to establish some basic and standard notations we briefly overview the theory of Dunkl operators and its relation to harmonic analysis. Main references are [3,4,5,8,16,17,19,20,21].

2.1. The Dunkl Operators

LetR^d be the Euclidean space equipped with a scalar product h·,·iand let ||x|| = phx, xi. For α in R^d\{0}, let σ_α be the reflection in the hyperplane H_α ⊂ R^d orthogonal toα, i.e. forx∈R^d,

σ_α(x) = x−2hα, xi

||α||²α.

A finite setR ⊂R^d\{0}is called a root system ifR∩Rα ={α,−α}andσ_αR =R for all α ∈ R. For a given root system R, reflectionsσ_α, α ∈ R, generate a finite group W ⊂ O(d), called the reflection group associated with R. We fix a β ∈ R^d\S

α∈RHα and define a positive root systemR+ = {α ∈ R | hα, βi > 0}. We normalize each α ∈ R+ as hα, αi = 2. A functionk : R −→ Con R is called a multiplicity function if it is invariant under the action ofW. We introduce the index γas

γ =γ(k) = X

α∈R+

k(α).

Throughout this paper, we will assume thatk(α)≥ 0for allα ∈ R. We denote by ω_kthe weight function onR^dgiven by

ω_k(x) = Y

α∈R+

|hα, xi|^2k(α),

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which is invariant and homogeneous of degree2γ, and byc_kthe Mehta-type constant defined by

c_k = Z

R^d

exp(−||x||²)ω_k(x)dx −1

.

We note that Etingof (cf. [6]) has given a derivation of the Mehta-type constant valid for all finite reflection group.

The Dunkl operators T_j, j = 1,2, . . . , d, onR^dassociated with the positive root systemR₊ and the multiplicity functionkare given by

T_jf(x) = ∂f

∂x_j(x) + X

α∈R+

k(α)α_jf(x)−f(σ_α(x))

hα, xi , f ∈C¹(R^d).

We define the Dunkl-Laplace operator4konR^dforf ∈C²(R^d)by 4_kf(x) =

d

X

j=1

T_j²f(x)

=4f(x) + 2 X

α∈R+

k(α)

h∇f(x), αi

hα, xi −f(x)−f(σ_α(x)) hα, xi²

,

where4and∇are the usual Euclidean Laplacian and nabla operators onR^drespec- tively. Then for eachy∈R^d, the system

(T_ju(x, y) =y_ju(x, y), j = 1, . . . , d, u(0, y) = 1

admits a unique analytic solution K(x, y), x ∈ R^d, called the Dunkl kernel. This kernel has a holomorphic extension toC^d×C^d, (cf. [17] for the basic properties of K).

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2.2. Harmonic Analysis Associated with the Dunkl-Bessel Laplace Operator In this subsection we collect some notations and results on the Dunkl-Bessel kernel, the Dunkl-Bessel intertwining operator and its dual, the Dunkl-Bessel transform, and the Dunkl-Bessel convolution (cf. [12]).

In the following we denote by

• R^d+1+ =R^d×[0,+∞[.

• x= (x₁, . . . , x_d, x_d+1) = (x⁰, x_d+1)∈R^d+1+ .

• C∗(R^d+1) the space of continuous functions onR^d+1, even with respect to the last variable.

• C_∗^p(R^d+1) the space of functions of class C^p on R^d+1, even with respect to the last variable.

• E∗(R^d+1) the space of C^∞-functions on R^d+1, even with respect to the last variable.

• S∗(R^d+1) the Schwartz space of rapidly decreasing functions onR^d+1, even with respect to the last variable.

• D∗(R^d+1) the space ofC^∞-functions onR^d+1which are of compact support, even with respect to the last variable.

• S⁰_∗(R^d+1) the space of temperate distributions onR^d+1, even with respect to the last variable. It is the topological dual ofS∗(R^d+1).

We consider the Dunkl-Bessel Laplace operator4_k,β defined by

∀x= (x⁰, x_d+1) ∈ R^d×]0,+∞[, (2.1)

4_k,βf(x) =4_k,x⁰f(x⁰, x_d+1) +L_β,x_d+1f(x⁰, x_d+1), f ∈C_∗²(R^d+1),

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where 4_k is the Dunkl-Laplace operator on R^d, and L_β the Bessel operator on ]0,+∞[given by

Lβ = d²

dx²_d+1 +2β+ 1 x_d+1

d

dx_d+1, β >−1 2. The Dunkl-Bessel kernelΛis given by

(2.2) Λ(x, z) = K(ix⁰, z⁰)jβ(xd+1zd+1), (x, z)∈R^d+1×C^d+1,

whereK(ix⁰, z⁰)is the Dunkl kernel andj_β(x_d+1z_d+1)is the normalized Bessel function. The Dunkl-Bessel kernel satisfies the following properties:

i) For allz, t∈C^d+1, we have

(2.3) Λ(z, t) = Λ(t, z); Λ(z,0) = 1 and Λ(λz, t) = Λ(z, λt), for allλ ∈C. ii) For allν∈N^d+1, x∈R^d+1 andz ∈C^d+1, we have

(2.4) |D_z^νΛ(x, z)| ≤ ||x||^|ν| exp(||x|| ||Imz||), whereD_z^ν = ^∂^ν

∂z₁^ν¹···∂z^νd+1_d+1 and|ν|=ν₁+· · ·+ν_d+1.In particular (2.5) |Λ(x, y)| ≤1, for all x, y ∈R^d+1.

The Dunkl-Bessel intertwining operator is the operatorR_k,βdefined onC∗(R^d+1) by

(2.6) R_k,βf(x⁰, x_d+1)

=







2Γ(β+ 1)

√πΓ(β+¹₂)x^−2β_d+1 Z x_d+1

0

(x²_d+1−t²)^β−¹²V_kf(x⁰, t)dt, x_d+1 >0,

f(x⁰,0), x_d+1 = 0,

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whereV_kis the Dunkl intertwining operator (cf. [16]).

R_k,β is a topological isomorphism from E∗(R^d+1)onto itself satisfying the following transmutation relation

(2.7) 4k,β(Rk,βf) = Rk,β(4d+1f), for all f ∈ E∗(R^d+1), where4d+1 =

d+1

P

j=1

∂_j² is the Laplacian onR^d+1.

The dual of the Dunkl-Bessel intertwining operator R_k,β is the operator ^tR_k,β defined onD∗(R^d+1)by:∀y= (y⁰, y_d+1)∈R^d×[0,∞[,

(2.8) ^tR_k,β(f)(y⁰, y_d+1) = 2Γ(β+ 1)

√πΓ β+ ¹₂ Z ∞

yd+1

(s²−y_d+1² )^β−¹² ^tV_kf(y⁰, s)sds, where^tV_kis the dual Dunkl intertwining operator (cf. [20]).

tR_k,β is a topological isomorphism fromS_∗(R^d+1)onto itself satisfying the following transmutation relation

(2.9) ^tRk,β(4k,βf) =4d+1(^tRk,βf), for all f ∈ S∗(R^d+1).

We denote byL^p_k,β(R^d+1+ )the space of measurable functions onR^d+1+ such that

||f||_L^p

k,β(R^d+1₊ ) = Z

R^d+1₊

|f(x)|^pdµ_k,β(x)dx

!¹_p

<+∞, if 1≤p < +∞,

||f||_L∞

k,β(R^d+1₊ ) = ess sup

x∈R^d+1+

|f(x)|<+∞, wheredµ_k,βis the measure onR^d+1+ given by

dµk,β(x⁰, xd+1) = ωk(x⁰)x^2β+1_d+1 dx⁰dxd+1.

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The Dunkl-Bessel transform is given forf inL¹_k,β(R^d+1+ )by (2.10) F_D,B(f)(y⁰, y_d+1) =

Z

R^d+1+

f(x⁰, x_d+1)Λ(−x, y)dµ_k,β(x), for all y= (y⁰, y_d+1)∈R^d+1+ . Some basic properties of this transform are the following:

i) Forf inL¹_k,β(R^d+1+ ),

(2.11) ||F_D,B(f)||_L∞

k,β(R^d+1₊ )≤ ||f||_L1

k,β(R^d+1₊ ). ii) Forf inS∗(R^d+1)we have

(2.12) F_D,B(4_k,βf)(y) =−||y||²F_D,B(f)(y), for all y∈R^d+1+ . iii) For allf ∈ S(R^d+1∗ ), we have

(2.13) F_D,B(f)(y) = F_o◦ ^tR_k,β(f)(y), for all y∈R^d+1+ , whereF_o is the transform defined by: ∀y ∈R^d+1+ ,

(2.14) F_o(f)(y) = Z

R^d+1+

f(x)e^−ihy⁰^,x⁰ⁱcos(x_d+1y_d+1)dx, f ∈D∗(R^d+1).

iv) For allf inL¹_k,β(R^d+1+ ), ifF_D,B(f)belongs toL¹_k,β(R^d+1+ ), then (2.15) f(y) =m_k,β

Z

R^d+1+

F_D,B(f)(x)Λ(x, y)dµ_k,β(x), a.e.

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where

(2.16) m_k,β = c²_k

4^γ+β+^d²(Γ(β+ 1))². v) Forf ∈ S∗(R^d+1), if we define

F_D,B(f)(y) =F_D,B(f)(−y), then

(2.17) F_D,BF_D,B =F_D,BF_D =m_k,βId.

Proposition 2.1.

i) The Dunkl-Bessel transformF_D,Bis a topological isomorphism fromS∗(R^d+1) onto itself and for allf inS∗(R^d+1),

(2.18)

Z

R^d+1+

|f(x)|²dµ_k,β(x) = m_k,β Z

R^d+1+

|F_D,B(f)(ξ)|²dµ_k,β(ξ).

ii) In particular, the renormalized Dunkl-Bessel transformf →m

1 2

k,βF_D,B(f)can be uniquely extended to an isometric isomorphism onL²_k,β(R^d+1+ ).

By using the Dunkl-Bessel kernel, we introduce a generalized translation and a convolution structure. For a functionf ∈ S_∗(R^d+1)andy∈ R^d+1+ the Dunkl-Bessel translationτ_yf is defined by the following relation:

F_D,B(τ_yf)(x) = Λ(x, y)F_D,B(f)(x).

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Iff ∈ E_∗(R^d+1)is radial with respect to thedfirst variables, i.e.f(x) = F(||x⁰||, x_d+1), then it follows that

(2.19) τ_yf(x) =R_k,β

Fp

||x⁰||²+||y⁰||²+ 2hy⁰,·i, q

x²_d+1+y_d+1² + 2y_d+1

(x⁰, x_d+1).

By using the Dunkl-Bessel translation, we define the Dunkl-Bessel convolution prod- uctf ∗_D,Bgof functionsf, g ∈ S∗(R^d+1)as follows:

(2.20) f∗_D,B g(x) =

Z

R^d+1+

τ_xf(−y)g(y)dµ_k,β(y).

This convolution is commutative and associative and satisfies the following:

i) For allf, g∈ S∗(R^d+1+ ),f ∗D,Bgbelongs toS∗(R^d+1+ )and (2.21) F_D,B(f ∗_D,B g)(y) =F_D,B(f)(y)F_D,B(g)(y).

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

(2.22) kf ∗_D,Bgk_Lr

k,β(R^d+1₊ ) ≤ kfk_L^p

k,β(R^d+1+ )kgk_L^q

k,β(R^d+1+ ).

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3. Structure Theorems on the Silva Space and its Dual

Definition 3.1. We denote byG∗ orG∗(R^d+1)the set of all functionsϕinE∗(R^d+1) such that for anyh, p >0

N_p,h(ϕ) = sup

x∈Rd+1 µ∈Nd+1

e^p||x|||∂^µϕ(x)|

h^|µ|µ!

is finite. The topology inG∗ is defined by the above seminorms.

Lemma 3.2. Letφbe inG∗. Then for everyh, p > 0 N_p,h(ϕ) = sup

x∈Rd+1 m∈N

e^p||x|||4^m_k,βϕ(x)|

h^mm!

! .

Proof. We proceed as in Proposition 5.1 of [13], and by a simple calculation we obtain the result.

Theorem 3.3. The transformF_D,Bis a topological isomorphism fromG∗ onto itself.

Proof. From the relations (2.8), (2.9) and Lemma3.2we see that^tR_k,βis continuous from G∗ onto itself. On the other hand, J. Chung et al. [1] have proved that the classical Fourier transform is an isomorphism from G∗ onto itself. Thus from the relation (2.13) we deduce thatF_D,B is continuous fromG∗ onto itself. Finally since G∗ is included inS∗(R^d+1), andF_D,B is an isomorphism fromS∗(R^d+1)onto itself, by (2.17) we obtain the result.

We denote byG_∗⁰ orG_∗⁰(R^d+1)the strong dual of the spaceG∗.

Definition 3.4. The Dunkl-Bessel transform of a distributionSinG_∗⁰ is defined by hFD,B(S), ψi=hS,FD,B(ψ)i, ψ ∈ G∗.

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The result below follows immediately from Theorem3.3.

Corollary 3.5. The transformF_D,Bis a topological isomorphism fromG_∗⁰ onto itself.

Letτ be inG_∗⁰. We define4_k,βτ, by

h4_k,βτ, ψi=hτ,4_k,βψi, for all ψ ∈ G∗. This functional satisfies the following property

(3.1) F_D,B(4_k,βτ) =−||y||²F_D,B(τ).

Definition 3.6. The generalized heat kernelΓ_k,βis given by Γ_k,β(t, x, y) := 2c_k

Γ(β+ 1)(4t)^γ+β+^d²⁺¹e⁻||x||2+||y||2

4t Λ

−i x

√2t, y

√2t

; x, y ∈R^d+1+ , t >0.

The generalized heat kernelΓ_k,β has the following properties:

Proposition 3.7. Letx, y inR^d+1+ andt >0. Then we have:

i) Γ_k,β(t, x, y) = Z

R^d+1+

exp(−t||ξ||²)Λ(x, ξ)Λ(−y, ξ)dµ_k,β(ξ).

ii) Z

R^d+1+

Γ_k,β(t, x, y)dµ_k,β(x) = 1.

iii) For fixedyinR^d+1+ , the functionu(x, t) := Γ_k,β(t, x, y)solves the generalized heat equation:

4_k,βu(x, t) = ∂

∂tu(x, t) on R^d+1+ ×]0,+∞[.

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Definition 3.8. The generalized heat semigroup(H_k,β(t))_t≥0is the integral operator given forf inL²_k,β(R^d+1+ )by

H_k,β(t)f(x) :=





 Z

R^d+1+

Γ_k,β(t, x, y)f(y)dµ_k,β(y) if t >0,

f(x) if t= 0.

From the properties of the generalized heat kernel we have

(3.2) H_k,β(t)f(x) :=

( f ∗D,Bpt(x) if t >0, f(x) if t= 0, where

p_t(y) = 2ck

Γ(β+ 1)(4t)^γ+β+^d²⁺¹e⁻^||y||

2 4t .

Definition 3.9. A functionfdefined onR^d+1+ is said to be of exponential type if there are constantsk, C >0such that for everyx∈R^d+1+

|f(x)| ≤expk||x||.

The following lemma will be useful later. For the details of the proof we refer to Komatsu [9]:

Lemma 3.10. For any L > 0 and ε > 0 there exist a functionv ∈ D(R) and a differential operatorP(_dt^d)of infinite order such that

(3.3) suppv ⊂[0, ε], |v(t)| ≤Cexp

−L t

, 0< t <∞;

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(3.4) P

d dt

=

∞

X

k=0

a_k d

dt k

, |a_k| ≤C₁L^k₁

k!², 0< L₁ < L;

(3.5) P

d dt

v(t) = δ+ω(t), whereω ∈D(R),suppω ⊂[₂^ε, ε].

Here we note that P(4_k,β) is a local operator where 4_k,β is a Dunkl-Bessel Laplace operator.

Now we are in a position to state and prove one of the main theorems in this section.

Theorem 3.11. Ifu∈ G_∗⁰ then there exists a differential operatorP(_dt^d)such that for someC > 0andL >0,

P d

dt

=

∞

X

n=0

a_n d

dt n

, |a_n| ≤CLⁿ n!²,

and there are a continuous functiongof exponential type and an entire functionhof exponential type inR^d+1+ such that

(3.6) u=P(4_k,β)g(x) +h(x).

Proof. Let U(x, t) = hu_y,Γ_k,β(t, x, y)i. Since p_t belongs to G_∗ for each t > 0, U(x, t)is well defined and real analytic in(R^d+1+ )_xfor eacht >0.

Furthermore,U(x, t)satisfies

(3.7) (∂t− 4k,β)U(x, t) = 0 for (x, t)∈R^d+1+ ×]0,∞[.

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Hereu∈ G_∗⁰ means that for somek >0andh >0 (3.8) |hu, φi| ≤Csup

x,α

|∂^αφ(x)|expk||x||

h^|α|α! , φ∈ G∗. By Cauchy’s inequality and relations (2.4) and (3.8) we obtain, fort >0

|U(x, t)| ≤C⁰expk⁰

||x||+t+1 t

for someC⁰ >0andk⁰ >0. If we restrict this inequality on the strip0< t < εthen it follows that

|U(x, t)| ≤Cexpk

||x||+1 t

, 0< t < ε for some constantsC >0andk >0. Now let

G(y, t) = Z

R^d+1₊

Γ_k,β(t, x, y)φ(x)dµ_k,β(x), φ∈ G_∗. Moreover, we can easily see that

(3.9) G(·, t)→φ in G∗ ast→0⁺

and (3.10)

Z

R^d+1+

U(x, t)φ(x)dµ_k,β(x) = hu_y, G(y, t)i.

Then it follows from (3.9) and (3.10) that

(3.11) lim

t→0⁺U(x, t) =u inG_∗⁰.

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Now choose functionsu, w and a differential operator of infinite order as in Lemma 3.10. Let

(3.12) U˜(x, t) =

Z ∞ 0

U(x, t+s)v(s)ds.

Then by takingε= 2andL > kin Lemma3.10we have (3.13) |U˜(x, t)| ≤C⁰expk ||x||+t

, t ≥0.

Therefore,U˜(x, t)is a continuous function of exponential type in R^d+1+ ×[0,∞[=n

(x, t) :x∈R^d+1+ , t≥0o . Moreover,U˜ satisfies

(3.14) (∂_t− 4_k,β) ˜U(x, t) = 0 in R^d+1+ ×]0,∞[.

Hence if we setg(x) = ˜U(x,0)theng is also a continuous function of exponential type, so thatg belongs toG_∗⁰.

Using (3.5) in Lemma3.10, we obtain fort >0 P(−4_k,β) ˜U(x, t) = P

−d dt

U˜(x, t)

=U(x, t) + Z ∞

0

U(x, t+s)w(s)ds.

(3.15)

If we seth(x) = −R∞

0 U(x, s)w(s)ds then h is an entire function of exponential type. Ast→0⁺, (3.15) becomes

u=P(−4_k,β)g(x) +h(x)

which completes the proof by replacing the coefficientsanofP by(−1)ⁿan.

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Theorem 3.12. Let U(x, t)be an infinitely differentiable function in R^d+1+ ×]0,∞[

satisfying the conditions:

i) (∂_t− 4_k,β)U(x, t) = 0inR^d+1+ ×]0,∞[.

ii) There existk >0andC >0such that (3.16) |U(x, t)| ≤Cexpk

||x||+ 1 t

,0< t < ε, x∈R^d+1+

for someε >0. Then there exists a unique elementu∈ G_∗⁰ such that U(x, t) = hu_y,Γ_k,β(t, x, y)i, t >0

and

t→0lim⁺U(x, t) =u in G_∗⁰. Proof. Consider a function, as in (3.12)

U˜(x, t) = Z ∞

0

U(x, t+s)v(s)ds.

Then it follows from (3.13), (3.14) and (3.15) thatU˜(x, t)andH(x, t)are continuous onR^d+1+ ×[0,∞[and

(3.17) U(x, t) = P(−4_k,β) ˜U(x, t) +H(x, t), where

H(x, t) = − Z ∞

0

U(x, t+s)w(s)ds.

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Furthermore,g(x) = ˜U(x,0)andh(x) =H(x,0)are continuous functions of exponential type onR^d+1+ . Defineuas

u=P(−4_k,β)g(x) +h(x).

Then sinceP(−4k,β)is a local operator,ubelongs toG_∗⁰ and lim

t→0⁺U(x, t) =u in G_∗⁰. Now define the generalized heat kernels fort >0as

A(x, t) = (g ∗D,B pt)(x) = Z

R^d+1+

g(y)Γk,β(t, x, y)dµk,β(y) and

B(x, t) = (h ∗D,B pt)(x) = Z

R^d+1₊

h(y)Γk,β(t, x, y)dµk,β(y).

Then it is easy to show thatA(x, t)andB(x, t)converge locally uniformly tog(x) andh(x)respectively so that they are continuous onR^d+1+ ×[0,∞[, A(x,0) = g(x), andB(x,0) =h(x). Now letV(x, t) = ˜U(x, t)−A(x, t)andW(x, t) =H(x, t)− B(x, t). Then, sinceg and h are of exponential type, V(·, t) andW(·, t) are continuous functions of exponential type andV(x,0) = 0, W(x,0) = 0. Then by the uniqueness theorem of the generalized heat equations we obtain that

U˜(x, t) = g ∗_D,B p_t (x) and

H(x, t) = h ∗_D,B p_t (x).

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It follows from these facts and (3.17) that u ∗_D,B p_t=

h

P(−4_k,β)g+h i

∗_D,B p_t

=P(−4_k,β) ˜U(·, t) +H(·, t)

=U(·, t).

Now to prove the uniqueness of existence of suchu∈ G_∗⁰ we assume that there exist u, v ∈ G_∗⁰ such that

U(x, t) = u ∗_D,B p_t

(x) = v ∗_D,B p_t (x).

Then

FD,B(u)FD,B(pt) =FD,B(v)FD,B(pt)

which implies that F_D,B(u) = F_D,B(v), since F_D,B(p_t) 6= 0. However, since the Dunkl-Bessel transformation is an isomorphism we haveu = v, which completes the proof.

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4. The Generalized Dunkl-Sobolev Spaces of Exponential Type

Definition 4.1. Letsbe inR,1≤p <∞. We define the spaceW_G^s,p_∗_,k,β(R^d+1+ )by n

u∈ G_∗⁰ :e^s||ξ||F_D,B(u)∈L^p_k,β(R^d+1+ )o . The norm onW_G^s,p_∗_,k,β(R^d+1+ )is given by

||u||_W^s,p

G∗,k,β = mk,β

Z

R^d+1₊

e^ps||ξ|||FD,B(u)(ξ)|^pdµk,β(ξ)

!¹_p . Forp= 2we provide this space with the scalar product

(4.1) hu, vi_W^s,2

G∗,k,β =m_k,β Z

R^d+1+

e^2s||ξ||F_D,B(u)(ξ)F_D,B(v)(ξ)dµ_k,β(ξ), and the norm

(4.2) ||u||²_Ws,2

G∗,k,β

=hu, ui_W^s,2

G∗,k,β. Proposition 4.2.

i) Let1≤p < +∞. The spaceW_G^s,p_∗_,k,β(R^d+1+ )provided with the normk·k_W^s,p

G∗,k,β

is a Banach space.

ii) We have

W_G^0,2_∗_,k,β(R^d+1+ ) =L²_k,β(R^d+1+ ).

iii) Let1≤p < +∞ands₁, s₂inRsuch thats₁ ≥s₂ then W_G^s¹^,p

∗,k,β(R^d+1+ ),→W_G^s²^,p

∗,k,β(R^d+1+ ).

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Proof. i) It is clear that the space L^p(R^d+1+ , e^ps||ξ||dµ_k,β(ξ)) is complete and since F_D,Bis an isomorphism fromG_∗⁰ onto itself,W_G^s,p_∗_,k,β(R^d+1+ )is then a Banach space.

The results ii) and iii) follow immediately from the definition of the generalized Dunkl-Sobolev space of exponential type.

Proposition 4.3. Let1 ≤ p < +∞, and s₁, s, s₂ be three real numbers satisfying s₁ < s < s₂. Then, for allε > 0,there exists a nonnegative constant C_ε such that for alluinW_G^s,p_∗_,k,β(R^d+1+ )

(4.3) ||u||_W^s,p

G∗,k,β ≤Cε||u||_W^s¹^,p

G∗,k,β +ε||u||_W^s²^,p

G∗,k,β.

Proof. We considers= (1−t)s₁+ts₂,(witht∈]0,1[). Moreover it is easy to see

||u||_W^s,p

G∗,k,β ≤ ||u||^1−t_Ws1,p G∗,k,β

||u||^t_W^s2,p G∗,k,β. Thus

||u||_W^s,p

G∗,k,β ≤

ε⁻^1−t^t ||u||_W^s1,p G∗,k,β

1−t

ε||u||_W^s2,p G∗,k,β

t

≤ε⁻^1−t^t ||u||_W^s¹^,p

G∗,k,β +ε||u||_W^s²^,p

G∗,k,β. Hence the proof is completed forC_ε =ε⁻^1−t^t .

Proposition 4.4. ForsinR,1≤p <∞andminN, the operator4^m_k,βis continuous fromW_G^s,p_∗_,k,β(R^d+1+ )intoW_G^s−ε,p_∗_,k,β(R^d+1+ )for anyε >0.

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Proof. Letube inW_G^s,p_∗_,k,β(R^d+1+ ), andminN. From (3.1) we have Z

R^d+1+

ep(s−ε)||ξ|||F_D,B(4^m_k,βu)(ξ)|^pdµ_k,β(ξ)

= Z

R^d+1+

ep(s−ε)||ξ||||ξ||^2mp|F_D,B(u)(ξ)|^pdµ_k,β(ξ).

Assup_ξ∈

R^d+1₊ ||ξ||^2mpe^−ε||ξ|| <∞, for everym ∈Nandε >0 Z

R^d+1+

ep(s−ε)||ξ|||F_D,B(4^m_k,βu)(ξ)|^pdµ_k,β(ξ)

≤ Z

R^d+1₊

e^ps||ξ|||F_D,B(u)(ξ)|^pdµ_k,β(ξ)<+∞.

Then4^m_k,βubelongs toW_G^s−ε,p_∗_,k,β(R^d+1+ ), and

||4^m_k,βu||_W^s−ε,p

G∗,k,β ≤ ||u||_W^s,p

G∗,k,β.

Definition 4.5. We define the operator(−4_k,β)¹² by∀x∈R^d+1+ , (−4_k,β)¹²u(x) =m_k,β

Z

R^d+1+

Λ(x, ξ)||ξ||F_D,B(u)(ξ)dµ_k,β(ξ), u∈ S∗(R^d+1).

Proposition 4.6. LetP((−4_k,β)¹²) = P

m∈Na_m(−4_k,β)^m² be a fractional Dunkl- Bessel Laplace operators of infinite order such that there exist positive constantsC andrsuch that

(4.4) |a_m| ≤Cr^m

m!, for all m ∈N.

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Let1≤p <+∞andsinR. If an elementuis inW_G^s,p_∗_,k,β(R^d+1+ ), thenP((−4_k,β)¹²)u belongs toW_G^s−r,p_∗_,k,β(R^d+1+ ), and there exists a positive constantCsuch that

||P((−4_k,β)¹²)u||_W^s−r,p

G∗,k,β ≤C||u||_W^s,p

G∗,k,β. Proof. The condition (4.4) gives that

|P(||ξ||)| ≤C exp(r||ξ||).

Thus the result is immediate.

Proposition 4.7. Let1≤p <+∞,t, sinR. The operatorexp(t(−4_k,β)¹²)defined by

∀x∈R^d+1+ , exp(t(−4_k,β)¹²)u(x)

=m_k,β Z

R^d+1+

Λ(x, ξ)e^t||ξ||F_D,B(u)(ξ)dµ_k,β(ξ), u∈ G∗. is an isomorphism fromW_G^s,p_∗_,k,β(R^d+1+ )ontoW_G^s−t,p_∗_,k,β(R^d+1+ ).

Proof. ForuinW_G^s,p_∗_,k,β(R^d+1+ )it is easy to see that kexp(t(−4_k,β)¹²)uk_W^s−t,p

G∗,k,β =||u||_W^s,p

G∗,k,β, and thus we obtain the result.

Proposition 4.8. The dual ofW_G^s,2_∗_,k,β(R^d+1+ )can be identified asW_G^−s,2_∗_,k,β(R^d+1+ ). The relation of the identification is given by

(4.5) hu, vi_k,β =m_k,β Z

R^d+1+

F_D,B(u)(ξ)F_D,B(v)(ξ)dµ_k,β(ξ), withuinW_G^s,2_∗_,k,β(R^d+1+ )andv inW_G^−s,2_∗_,k,β(R^d+1+ ).

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Proof. Let u be in W_G^s,2_∗_,k,β(R^d+1+ ) and v in W_G^−s,2_∗_,k,β(R^d+1+ ). The Cauchy-Schwartz inequality gives

|hu, vi_k,β| ≤ ||u||_W^s,2

G∗,k,β||v||_W^−s,2

G∗,k,β.

Thus forv in W_G^−s,2_∗_,k,β(R^d+1+ ) fixed we see thatu 7−→ hu, vik,β is a continuous lin- ear form on W_G^s,2_∗_,k,β(R^d+1+ ) whose norm does not exceed ||v||_W^−s,2

G∗,k,β. Taking the u₀ = F_D,B⁻¹ e^−2s||ξ||F_D,B(v)

element of W_G^s,2_∗_,k,β(R^d+1+ ), we obtain hu₀, vi_k,β =

||v||_W^−s,2

G∗,k,β.

Thus the norm of u 7−→ hu, vi_k,β is equal to ||v||_W^−s,2

G∗,k,β, and we have then an isometry:

W_G^−s,2_∗_,k,β R^d+1+

−→ W_G^s,2_∗_,k,β(R^d+1+ )⁰ . Conversely, let L be in W_G^s,2_∗_,k,β(R^d+1+ )⁰

. By the Riesz representation theorem and (4.1) there existswinW_G^s,2_∗_,k,β(R^d+1+ )such that

L(u) =hu, wi_W^s,2

G∗,k,β

=m_k,β Z

R^d+1+

e^2s||ξ||F_D,B(u)(ξ)F_D,B(w)(ξ)dµ_k,β(ξ), for allu∈W_G^s,2_∗_,k,β(R^d+1+ ).

If we set v = F_D⁻¹ e^2s||ξ||F_D,B(w)

then v belongs toW_G^−s,2

∗,k,β(R^d+1+ )and L(u) = hu, vi_k,βfor alluinW_G^s,2

∗,k,β(R^d+1+ ), which completes the proof.

Proposition 4.9. Lets > 0. Then everyuin W_G^−s,2_∗_,k,β(R^d+1+ )can be represented as an infinite sum of fractional Dunkl-Bessel Laplace operators of square integrable functiong, in other words,

u=X

m∈N

s^m

m!(−4_k,β)^m²g.

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Proof. IfuinW_G^−s,2_∗_,k,β(R^d+1+ )then by definition

e^−s||ξ||F_D,B(u)(ξ)∈L²_k,β(R^d+1+ ), which Proposition2.1ii) implies that

FD,B(g)(ξ) = F_D,B(u)(ξ) P

m∈N s^m

m!||ξ||^m ∈L²_k,β(R^d+1+ ).

Hence, we have

F_D,B(u) = X

m∈N

s^m

m!||ξ||^mF_D,B(g), in G_∗⁰

=X

m∈N

s^m

m!F_D,B (−4_k,β)^m²g

, in G_∗⁰ This completes the proof.

Proposition 4.10. Let1 ≤ p < +∞. Every uin W_G^s,p_∗_,k,β(R^d+1+ ) is a holomorphic function in the strip{z ∈C^d+1, ||Imz||< s}fors >0.

Proof. Let

u(z) =m_k,β Z

R^d+1+

Λ(z, ξ)F_D,B(u)(ξ)dµ_k,β(ξ), z =x+iy.

From (2.4), for eachµinN^d+1 we have

D^µ_z

Λ(z, ξ)F_D,B(u)(ξ)

≤ ||ξ||^|µ|e||y|| ||ξ|||F_D,B(u)(ξ)|.