Sonine Transform Associated to the Dunkl Kernel on the Real Line

We consider the Dunkl intertwining operator $V_\alpha$ and its dual ${}^tV_\alpha$, we define and study the Dunkl Sonine operator and its dual on $\mathbb{R}$. Next, we introduce complex powers of the Dunkl Laplacian $\Delta_\alpha$ and establish inversion formulas for the Dunkl Sonine operator $S_{\alpha,\beta}$ and its dual ${}^tS_{\alpha,\beta}$. Also, we give a Plancherel formula for the operator ${}^tS_{\alpha,\beta}$.


Introduction
In this paper, we consider the Dunkl operator Λ α , α > −1/2, associated with the reflection group Z 2 on R. The operators were in general dimension introduced by Dunkl in [2] in connection with a generalization of the classical theory of spherical harmonics; they play a major role in various fields of mathematics [3,4,5] and also in physical applications [6].
Also we study the complex powers of the Dunkl Laplacian (−∆ α ) λ , for some complex number λ. In the classical case when α = −1/2, the complex powers of the usual Laplacian are given in [16].
In Section 4, we give the following inversion formulas: where Next, we give the following Plancherel formula for the operator t S α,β :

The Dunkl intertwining operator and its dual
We consider the Dunkl operator Λ α , α ≥ −1/2, associated with the reflection group Z 2 on R: For α ≥ −1/2 and λ ∈ C, the initial problem: has a unique analytic solution E α (λx) called Dunkl kernel [3,5] given by is the modified spherical Bessel function of order α.
Let α > −1/2 and we define the Dunkl intertwining operator which can be written as: Proposition 1 (see [18], Theorem 6.3). The operator V α is a topological automorphism of E(R), and satisfies the transmutation relation: Let α > −1/2 and we define the dual Dunkl intertwining operator t V α on S(R) (the Schwartz space on R), by which can be written as: Proposition 2 (see [19], Theorems 3.2, 3.3).
(i) The operator t V α is a topological automorphism of S(R), and satisfies the transmutation relation: (ii) For all f ∈ E(R) and g ∈ S(R), we have Remark 2 (see [15]).
(i) For α > −1/2 and f ∈ E(R), we can write and ℜ α is the Riemann-Liouville transform (see [17], page 75) given by Thus, we obtain Therefore (see also [20], Proposition 2.2), we get where r = [α + 1/2] denote the integer part of α + 1/2, and d α = and W α is the Weyl integral transform (see [17, page 85]) given by Thus, we obtain The Dunkl kernel gives rise to an integral transform, called Dunkl transform on R, which was introduced by Dunkl in [4], where already many basic properties were established. Dunkl's results were completed and extended later on by de Jeu in [5].
The Dunkl transform of a function f ∈ S(R), is given by We notice that F −1/2 agrees with the Fourier transform F that is given by: Proposition 3 (see [5]).
(ii) F α possesses on S(R) the following decomposition: (iv) The normalized Dunkl transform √ c α F α extends uniquely to an isometric isomorphism Thus the transform F α extends to a topological automorphism on S ′ (R).
In [19], the author defines: These operators satisfy for x, y ∈ R and λ ∈ C the following properties: Proposition 4 (see [11]). If f ∈ C(R) (the space of continuous functions on R) and x, y ∈ R such that (x, y) = (0, 0), then • The Dunkl convolution product * α of two functions f and g in S(R), by This convolution is associative, commutative in S(R) and satisfies (see [19,Theorem 7.2]): For T ∈ S ′ (R) and f ∈ S(R), we define the Dunkl convolution product T * α f , by Note that * −1/2 agrees with the standard convolution * :

The Dunkl Sonine transform
In this section we study the Dunkl Sonine transform, which also studied by Y. Xu on polynomials in [20]. For thus we consider the following identity, which is a consequence of Xu's result when we extend the result of Lemma 2.1 on E(R).
which gives the desired result.
Remark 3. We can write the formula (5) by the following which can be written as: Definition 2. Let α, β ∈ ]−1/2, ∞[, such that β > α. We define the dual Dunkl Sonine transform t S α,β on S(R), by which can be written as:

Proposition 6.
(i) For all f ∈ E(R) and g ∈ S(R), we have (ii) F β possesses on S(R) the following decomposition: Proof . Part (i) follows from Definition 1 by Fubini's theorem. Then part (ii) follows from (i) and (6) by taking f = E α (−iλ.).
In [20, Lemma 2.1] Y. Xu proves the identity S α,β = V β • V −1 α on polynomials. As the intertwiner is a homeomorphism on E(R) and polynomials are dense in E(R), this gives the identity also on E(R). In the following we give a second method to prove this identity.

Theorem 1.
(i) The operator t S α,β is a topological automorphism of S(R), and satisfies the following relations: (ii) The operator S α,β is a topological automorphism of E(R), and satisfies the following relations: Proof . (i) From Proposition 6 (ii), we have On other hand, from (8), Proposition 2 (ii) and Proposition 6 (i) we have Hence from Proposition 1, Using the fact that which completes the proof of the theorem.

Complex powers of ∆ α
For λ ∈ C, Re(λ) > −1, we denote by |x| λ the tempered distribution defined by We write then from [1], we obtain the following result.
(iv) From (iii) we have Using the fact that By applying (9) and Proposition 3 (iii), we obtain Then which gives the result.
Note 1. We denote by • Ψ the subspace of S(R) consisting of functions f , such that • Φ α the subspace of S(R) consisting of functions f , such that The spaces Ψ and Φ −1/2 are well-known in the literature as Lizorkin spaces (see [1,9,13]).
The operator t S α,β is a topological isomorphism from Φ β onto Φ α .
Then the result follows from this identity by applying Proposition 3 (iv).