Inversion of Riesz Potentials for Dunkl Transform

Dunkl transform is a generalization of the Fourier transform associated with a family of weight functions, hκ, invariant under a finite reflection group. Many papers devote to study the Dunkl transform; see [1–6] and the references therein. In [7], the Riesz potentials Iκ α for Dunkl transform were defined by the generalized translation operators, τy. The explicit expression and boundedness of τy are known only in some special cases such as when G = Zd2 and the case when the kernel is a suitable radial function.The boundedness of Iκ α was given only in the two cases mentioned above. Gorbachev et al. [8] studied the weighted (Lp, Lq)-boundedness properties of Riesz potentials for Dunkl transform represented by the Stein-Weiss inequality. In this paper, we will study the inversion of Iκ α in the case when G = Zd2 . The paper is organized as follows. In Section 2, some necessary facts in Dunkl’s theory are reviewed. Section 3 is devoted to introduce the Semyanistyi-Lizorkin spaces associated with the reflection-invariant measure h2κ(x)dx. In the final section, the inversion of the Riesz potentials Iκ α will be given by the generalized wavelet transforms defined by the generalized translation operators.


Introduction
Dunkl transform is a generalization of the Fourier transform associated with a family of weight functions, ℎ  , invariant under a finite reflection group.Many papers devote to study the Dunkl transform; see [1][2][3][4][5][6] and the references therein.
In [7], the Riesz potentials    for Dunkl transform were defined by the generalized translation operators,   .The explicit expression and boundedness of   are known only in some special cases such as when  = Z  2 and the case when the kernel is a suitable radial function.The boundedness of    was given only in the two cases mentioned above.Gorbachev et al. [8] studied the weighted (  ,   )-boundedness properties of Riesz potentials for Dunkl transform represented by the Stein-Weiss inequality.In this paper, we will study the inversion of    in the case when  = Z  2 .The paper is organized as follows.In Section 2, some necessary facts in Dunkl's theory are reviewed.Section 3 is devoted to introduce the Semyanistyi-Lizorkin spaces associated with the reflection-invariant measure ℎ 2  ().In the final section, the inversion of the Riesz potentials    will be given by the generalized wavelet transforms defined by the generalized translation operators.

Dunkl Operator and Dunkl Transform.
Let  be a finite reflection group on R  with a fixed positive root system  + , normalized so that ⟨V, V⟩ = 2 for all V ∈  + , where ⟨, ⟩ denotes the usual Euclidean inner product.Let  be a nonnegative multiplicity function defined on  + with the property that   =  V whenever   is conjugate to  V in ; then V  →  V is a -invariant function.The weight function is positive homogeneous of degree   := ∑ V∈ +  V defined by Note that ℎ  is invariant under the reflection group .Let D  be Dunkl's differential-difference operators defined in [1] as where  1 ,  2 , . . .,   are the standard unit vectors of R  and  V denotes the reflection with respect to the hyperplane perpendicular to V,  V :=  − 2(⟨, V⟩/‖V‖ 2

Journal of Function Spaces
The intertwining operator   is a linear operator determined uniquely by Let (, ) =  ()   [ ⟨,⟩ ], ,  ∈ R  , where the superscript means that   is applied to the  variable.For  ∈  1 (R  , ℎ 2  ), the Dunkl transform is defined by where  ℎ is the constant defined by )}, and, for the sake of simplicity, set ⟨, ⟩  = ∫ R  ()()ℎ 2  () whenever the integral exists and denote   = 2  + .
The Dunkl transform shares many of the important properties with the usual Fourier transform, part of which are listed as follows ( [2,3]).

Dunkl Transform of Distributions.
References [5,11,12] study the actions of the Dunkl operators and Dunkl transform on the space S  (R  ).Reference [4] gives the definition of the Dunkl transform for the local integrable functions under the measure ℎ 2  ().
); the generalized function associated with  is defined by The Dunkl transform of  ∈ S  (R  ) is defined as Then, for For  ∈ S  (R  ), the dilation transform  is defined as Let  be the Dirac distribution associated with the measure ℎ 2  (); that is, ⟨, ⟩  = (0),  ∈ S(R  ).
Let Ψ = Ψ(R  ) be the class of functions  in S(R  ) vanishing at the origin 0 with all their derivatives; that is, The space Ψ is a closed linear subspace of S(R  ).It can be regarded as a linear topological space with the induced topology generated by the sequence of norms We claim that D  (0) = 0 for  ∈ Ψ, since   (0) = 0 implies D  (0) = 0 for  = 1, 2, . . ., .
Let Φ = Φ(R  ) be the image of Ψ under the Dunkl transform; that is, Φ = { ∈ Ψ : ψ}.Since the Dunkl transform is an automorphism of S(R  ), the space is a closed linear subspace of S(R  ).We equip Φ with the induced topology of the ambient space S(R  ).Then Φ becomes a linear topology space which is isomorphic to Ψ under the action of the Dunkl transform.According to the definition of Φ, we conclude that the space Φ consists of all functions  which are orthogonal to all polynomials as for the measure ℎ 2  (); that is, In fact, if  ∈ Φ, then ψ ∈ Φ, and for any multi-index , by Proposition 1 (iv), we have Denote that Φ  and Ψ  are the spaces of all semilinear functionals on Φ and Ψ, respectively.Some properties of Φ and Ψ are given in the following proposition.Proposition 6. (i) The spaces Φ and Ψ are not empty.
(ii) The space Φ does not contain compactly supported infinitely differentiable functions, rather than 0.
(iii) The space Φ is invariant under the generalized translations.
The proof of the this proposition is similar to the ones in [13][14][15] except with the reflection-invariant measure ℎ 2  ().Now we sketch it below.
(iii) This conclusion can be obtained by Proposition 2 (iii).

𝛼
Now we can give the main result of this paper, the inversion of the Dunkl Riesz potentials    when the group  = Z  2 .The method follows the idea in [16].Rubin [17] gave simpler proofs to some elementary approximate and explicit inversion formulae for the classical Riesz potentials.
) and the integral where Proof.Denote We claim that the operator   is bounded on   (R  , ℎ (48) Combining the above, we obtain (45) as desired.