GENERALIZED DUNKL-SOBOLEV SPACES OF EXPONENTIAL TYPE AND APPLICATIONS

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.


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 In this paper we consider the Dunkl-Bessel Laplace operator k,β defined by where k is the Dunkl Laplacian on R d , and L β is the Bessel operator on ]0, +∞[.We introduce the generalized Dunkl-Sobolev space of exponential type W s,p G * ,k,β (R d+1 + ) by replacing Thanks to the referee for his suggestions and comments.

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(1 + ||ξ|| 2 ) s p by an exponential weight function defined as follows where L p k,β (R d+1 + ) it is the Lebesgue space associated with the Dunkl-Bessel transform and G * 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 for s > 0, u ∈ W s,p G * ,k,β (R d+1 + ) can be analytically continued to the set {z ∈ C d+1 / | Im z| < s}.For the structure theorems we prove that for s > 0, u ∈ W −s,2 G * ,k,β (R d+1 + ) can be represented as an infinite sum of fractional Dunkl-Bessel Laplace operators of square integrable functions g, in other words, 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 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 Section 2 we 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 * .We study the action of the Dunkl-Bessel transform on these spaces.Next we prove two structure theorems for the space G * .We define in Section 4 the generalized Dunkl-Sobolev spaces of exponential type W s,p G * ,k,β (R d+1 + ) and we give their properties.In Section 5 we 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.

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.Let R d be the Euclidean space equipped with a scalar product •, • and let ||x|| = x, x .For α in R d \{0}, let σ α be the reflection in the hyperplane called the reflection group associated with R. We fix a β ∈ R d \ α∈R H α and define a positive Throughout this paper, we will assume that k(α) ≥ 0 for all α ∈ R. We denote by ω k the weight function on R d given by which is invariant and homogeneous of degree 2γ, and by c k the Mehta-type constant defined by .
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, on R d associated with the positive root system R + and the multiplicity function k are given by We define the Dunkl-Laplace operator where and ∇ are the usual Euclidean Laplacian and nabla operators on R d respectively.Then for each y ∈ R d , the system T j u(x, y) = y j u(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 to C d × C d , (cf.[17] for the basic properties of K).

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 ) the space of continuous functions on R d+1 , even with respect to the last variable.
) 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 on R d+1 , even with respect to the last variable.
• D * (R d+1 ) the space of C ∞ -functions on R d+1 which are of compact support, even with respect to the last variable.
• S * (R d+1 ) the space of temperate distributions on R d+1 , even with respect to the last variable.It is the topological dual of S * (R d+1 ).
We consider the Dunkl-Bessel Laplace operator k,β defined by where k is the Dunkl-Laplace operator on R d , and L β the Bessel operator on ]0, +∞[ given by The Dunkl-Bessel kernel Λ is given by where K(ix , z ) is the Dunkl kernel and j β (x d+1 z d+1 ) is the normalized Bessel function.The Dunkl-Bessel kernel satisfies the following properties: i) For all z, t ∈ C d+1 , we have ii) For all ν ∈ N d+1 , x ∈ R d+1 and z ∈ C d+1 , we have The Dunkl-Bessel intertwining operator is the operator R k,β defined on C * (R d+1 ) by where V k is 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 where d+1 = d+1 j=1 ∂ 2 j is the Laplacian on R d+1 .The dual of the Dunkl-Bessel intertwining operator R k,β is the operator t R k,β defined on where t V k is the dual Dunkl intertwining operator (cf.[20]).t R k,β is a topological isomorphism from S * (R d+1 ) onto itself satisfying the following transmutation relation We denote by L p k,β (R d+1 + ) the space of measurable functions on R d+1 where dµ k,β is the measure on R d+1
Some basic properties of this transform are the following: ii) For f in S * (R d+1 ) we have (2.12) iii) For all f ∈ S(R d+1 * ), we have where F o is the transform defined by: ∀ y ∈ R d+1 + , (2.14) where (2.16) i) The Dunkl-Bessel transform F D,B is a topological isomorphism from S * (R d+1 ) onto itself and for all f in S * (R d+1 ), (2.18) By using the Dunkl-Bessel kernel, we introduce a generalized translation and a convolution structure.For a function f ∈ S * (R d+1 ) and y ∈ R d+1 + the Dunkl-Bessel translation τ y f is defined by the following relation: By using the Dunkl-Bessel translation, we define the Dunkl-Bessel convolution product f * D,B g of functions f, g ∈ S * (R d+1 ) as follows: (2.20) This convolution is commutative and associative and satisfies the following:

STRUCTURE THEOREMS ON THE SILVA SPACE AND ITS DUAL
Definition 3.1.We denote by G * or G * (R d+1 ) the set of all functions ϕ in E * (R d+1 ) such that for any h, p > 0 is finite.The topology in G * is defined by the above seminorms.
Lemma 3.1.Let φ be in G * .Then for every h, p > 0 Proof.We proceed as in Proposition 5.1 of [13], and by a simple calculation we obtain the result.
Theorem 3.2.The transform F D,B is a topological isomorphism from G * onto itself.
Proof.From the relations (2.8), (2.9) and Lemma 3.1 we see that t R k,β is 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 that ), and F D,B is an isomorphism from S * (R d+1 ) onto itself, by (2.17) we obtain the result.
We denote by G * or G * (R d+1 ) the strong dual of the space G * .
Definition 3.2.The Dunkl-Bessel transform of a distribution S in G * is defined by The result below follows immediately from Theorem 3.2.

Corollary 3.3. The transform
Let τ be in G * .We define k,β τ , by This functional satisfies the following property Definition 3.3.The generalized heat kernel Γ k,β is given by The generalized heat kernel Γ k,β has the following properties: + and t > 0. Then we have: ii) iii) For fixed y in R d+1 + , the function u(x, t) := Γ k,β (t, x, y) solves the generalized heat equation: Definition 3.4.The generalized heat semigroup (H k,β (t)) t≥0 is the integral operator given for From the properties of the generalized heat kernel we have where e − ||y|| 2 4t .

Definition 3.5. A function f defined on R d+1
+ is said to be of exponential type if there are constants k, C > 0 such that for every x ∈ R d+1 The following lemma will be useful later.For the details of the proof we refer to Komatsu [9]: Lemma 3.5.For any L > 0 and ε > 0 there exist a function v ∈ D(R) and a differential operator P ( d dt ) of infinite order such that where Here we note that P ( k,β ) is a local operator where 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 3.6.If u ∈ G * then there exists a differential operator P ( d dt ) such that for some C > 0 and L > 0, and there are a continuous function g of exponential type and an entire function h of exponential Proof.Let U (x, t) = u y , Γ k,β (t, x, y) .Since p t belongs to G * for each t > 0, U (x, t) is well defined and real analytic in (R d+1 + ) x for each t > 0. Furthermore, U (x, t) satisfies Here u ∈ G * means that for some k > 0 and h > 0 By Cauchy's inequality and relations (2.4) and (3.8) we obtain, for t > 0 for some C > 0 and k > 0. If we restrict this inequality on the strip 0 < t < ε then it follows that for some constants C > 0 and k > 0. Now let Moreover, we can easily see that Then it follows from (3.9) and (3.10) that (3.11) lim t→0 + U (x, t) = u in G * .Now choose functions u, w and a differential operator of infinite order as in Lemma 3.5.Let Then by taking ε = 2 and L > k in Lemma 3.5 we have Therefore, Ũ (x, t) is a continuous function of exponential type in Hence if we set g(x) = Ũ (x, 0) then g is also a continuous function of exponential type, so that g belongs to G * .
Using (3.5) in Lemma 3.5, we obtain for t > 0 which completes the proof by replacing the coefficients a n of P by (−1) n a n .Theorem 3.7.Let U (x, t) be an infinitely differentiable function in for some ε > 0. Then there exists a unique element u ∈ G * such that U (x, t) = u y , Γ k,β (t, x, y) , t > 0 and Proof.Consider a function, as in (3.12) Then it follows from (3.13), (3.14) and (3.15) that Ũ (x, t) and H(x, t) are continuous on where Furthermore, g(x) = Ũ (x, 0) and h(x) = H(x, 0) are continuous functions of exponential type on R d+1 + .Define u as Then since P (− k,β ) is a local operator, u belongs to G * and lim t→0 + U (x, t) = u in G * .Now define the generalized heat kernels for t > 0 as Then it is easy to show that A(x, t) and B(x, t) converge locally uniformly to g(x) and h(x) respectively so that they are continuous on , and B(x, 0) = h(x).Now let V (x, t) = Ũ (x, t) − A(x, t) and W (x, t) = H(x, t) − B(x, t).Then, since g and h are of exponential type, V (•, t) and W (•, t) are continuous functions of exponential type and V (x, 0) = 0, W (x, 0) = 0. Then by the uniqueness theorem of the generalized heat equations we obtain that It follows from these facts and (3.17) that Now to prove the uniqueness of existence of such u ∈ G * we assume that there exist u, v ∈ G * such that However, since the Dunkl-Bessel transformation is an isomorphism we have u = v, which completes the proof.

THE GENERALIZED DUNKL-SOBOLEV SPACES OF EXPONENTIAL TYPE
.
For p = 2 we provide this space with the scalar product and the norm 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,B is an isomorphism from G * onto itself, W s,p G * ,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 4.2.Let 1 ≤ p < +∞, and s 1 , s, s 2 be three real numbers satisfying s 1 < s < s 2 .
Then, for all ε > 0, there exists a nonnegative constant C ε such that for all u in W s,p G * ,k,β (R d+1 + ) Proof.We consider s = (1 − t)s 1 + ts 2 , (with t ∈]0, 1[).Moreover it is easy to see Proof.Let u be in W s,p G * ,k,β (R d+1 + ), and m in N. From (3.1) we have As sup ξ∈R d+1

+
||ξ|| 2mp e −ε||ξ|| < ∞, for every m ∈ N and ε > 0 Then m k,β u belongs to W s−ε,p G * ,k,β (R d+1 + ), and Definition 4.2.We define the operator (− k,β ) m 2 be a fractional Dunkl-Bessel Laplace operators of infinite order such that there exist positive constants C and r such that 2 )u belongs to W s−r,p G * ,k,β (R d+1 + ), and there exists a positive constant C such that Proof.The condition (4.4) gives that Thus the result is immediate.
) defined by , and we have then an isometry: . By the Riesz representation theorem and (4.1) there exists w in W s,2 G * ,k,β (R d+1 + ) such that which Proposition 2.1 ii) implies that Hence, we have This completes the proof.
From (2.4), for each µ in N d+1 we have On the other hand from the Cauchy-Schwartz inequality we have Since the integral in the last part of the above inequality is integrable if ||y|| < s, the result follows by the theorem of holomorphy under the integral sign.
Notations.Let m be in N. We denote by: • E m (R d+1 + ) the space of distributions on R d+1 + with compact support and order less than or equal to m.
• E exp,m (R d+1 + ) the space of distributions u in E m (R d+1 + ) such that there exists a positive constant C such that ii) Let 1 ≤ p < +∞.We have, for s < 0 and m an integer, Proof.The proof uses the same idea as Theorem 3.12 of [13].
Theorem 4.10.Let ψ be in G * .For all s in R, the mapping u Proof.Firstly we assume that s > 0.
It is easy to see that , where + ) designates the norm associated to the homogeneous Dunkl-Bessel-Sobolev space defined by On the other hand, proceeding as in Proposition 4.1 of [14] we prove that if u, v belong to . Thus from this we deduce that for s > 0 For s = 0 the result is immediate.
For s < 0 the result is obtained by duality.
where a(x, ξ) belongs to the class S r exp , r ≥ 0, defined below: Definition 5.1.The function a(x, ξ) is said to be in S r exp if and only if a(x, ξ) belongs to C ∞ (R d+1 × R d+1 ) and for each compact set K ⊂ R d+1 + and each µ, ν in N d+1 , there exists a constant C K = C µ,ν,K such that the estimate Proposition 5.1.Let A(x, k,β ) be the pseudo-differential-difference operator associated with the symbol a(x, ξ) + .On the other hand since u is in G * the Cauchy-Schwartz inequality gives that is integrable for s > r.This prove the existence and the continuity of (A(x, k,β )u)(x) for all + .Finally the result follows by using Leibniz formula.Now we consider the symbol which belongs to the class S r,l exp,rad defined below: Definition 5.2.Let r, l in R be real numbers with l > 0. The function a(x, ξ) is said to be in S r,l exp,rad if and only if a(x, ξ) is in C ∞ (R d+1 × R d+1 ), radial with respect to the first d + 1 variables and for each L > 0, and for each µ, ν in N d+1 , there exists a constant C = C r,µ,ν such that the estimate To obtain some deep and interesting results we need the following alternative form of A(x, k,β ).A(x, k,β ) be the pseudo-differential-difference operator associated with the symbol a(x, ξ)

Lemma 5.2. Let
) is given by: for all u ∈ G * where all the involved integrals are absolutely convergent.
Proof.When we proceed as [10] we see that for all L > 0 there exist C > 0 and 0 < τ < (Ld) −1 such that On the other hand since u in G * , we have Thus from the positivity for the Dunkl-Bessel translation operator for the radial functions we obtain Moreover it is clear that the function η → exp(−(t − r) ||η||)τ −η (exp(−τ • ))(ξ) belongs to The right-hand side is a Dunkl-Bessel convolution product of two integrable functions and hence is an integrable function on R d+1 + .Therefore the function Proof.We consider the function Then invoking (5.2) and (2.19) we deduce that The integral of (5. . This proves (5.3).

Extremal Function for the Generalized Heat Semigroup Transform.
In this subsection, we prove for a given function is attained at some unique function denoted by f * r,g , called the extremal function.We start with the following fundamental theorem (cf.[11,18]).Theorem 5.7.Let H K be a Hilbert space admitting the reproducing kernel K(p, q) on a set E and H a Hilbert space.Let L : H K → H be a bounded linear operator on H K into H.For r > 0, introduce the inner product in H K and call it H Kr as Then i) H Kr is the Hilbert space with the reproducing kernel K r (p, q) on E which satisfies the equation K(•, q) = (rI + L * L)K r (•, q), where L * is the adjoint operator of L : H K → H. ii) For any r > 0 and for any g in H, the infimum is attained by a unique function f * r,g in H K and this extremal function is given by (5.9) f * r,g (p) = g, LK r (•, p) H .
We can now state the main result of this subsection.Proof.By Proposition 5.6 and Theorem 5.7 ii), the infimum given by (5.10) is attained by a unique function f * r,g , and from (5.9) the extremal function f * r,g is represented by f * r,g (y) = g, H k,β (t)(P r (•, y)) L 2 k,β (R d+1 + ) , y ∈ R d+1 + , where P r is the kernel given by Proposition 5.6.On the other hand we have This gives (5.12).Proof.From (5.12) and Fubini's theorem we have

Theorem 5 . 3 .
Applying the inverse Dunkl-Bessel transform we get the result.Now we prove the fundamental result Let A(x, k,β ) be the pseudo-differential-difference operator where the symbol a(x, ξ) := A(x, −||ξ|| 2 ) belongs to S r,l exp,rad .Then for all u in G * and all s in R (5.3) ||A(x, k,β )u|| W s,p G * ,k,β ≤ C s ||u|| W r+s,p G * ,k,β .
,k,β, and thus we obtain the result.
which completes the proof.Let s > 0. Then every u in W −s,2 G * ,k,β (R d+1 + ) can be represented as an infinite sum of fractional Dunkl-Bessel Laplace operators of square integrable function g, in other words,