Inversion Formulas for the Dunkl Intertwining Operator and Its Dual on Spaces of Functions and Distributions

In this paper we prove inversion formulas for the Dunkl intertwining operator $V_k$ and for its dual ${}^tV_k$ and we deduce the expression of the representing distributions of the inverse operators $V_k^{-1}$ and ${}^tV_k^{-1}$, and we give some applications.


Introduction
We consider the differential-difference operators T j , j = 1, 2, . . . , d, on R d associated to a root system R and a multiplicity function k, introduced by C.F. Dunkl in [3] and called the Dunkl operators in the literature. These operators are very important in pure mathematics and in physics. They provide a useful tool in the study of special functions related to root systems [4,6,2]. Moreover the commutative algebra generated by these operators has been used in the study of certain exactly solvable models of quantum mechanics, namely the Calogero-Sutherland-Moser models, which deal with systems of identical particles in a one dimensional space (see [8,11,12]).
C.F. Dunkl proved in [4] that there exists a unique isomorphism V k from the space of homogeneous polynomials P n on R d of degree n onto itself satisfying the transmutation relations We have shown in [15] that for each x ∈ R d , there exists a unique distribution η x in E ′ (R d ) (the space of distributions on R d of compact support) with support in B(0, x ) such that . (1.4) We have studied also in [15] the transposed operator t V k of the operator V k , satisfying for f in S(R d ) (the space of C ∞ -functions on R d which are rapidly decreasing together with their derivatives) and g in E(R d ), the relation where ω k is a positive weight function on R d which will be defined in the following section. It has the integral representation (1.5) where ν y is a positive measure on R d with support in the set {x ∈ R d ; x ≥ y }. This operator is called the dual Dunkl intertwining operator.
We have proved in [15] that the operator t V k is an isomorphism from D(R d ) (the space of C ∞functions on R d with compact support) (resp. S(R d )) onto itself, satisfying the transmutation relations Moreover for each y ∈ R d , there exists a unique distribution Z y in S ′ (R d ) (the space of tempered distributions on R d ) with support in the set {x ∈ R d ; x ≥ y } such that t V −1 k (f )(y) = Z y , f , f ∈ S(R d ). (1.6) Using the operator V k , C.F. Dunkl has defined in [5] the Dunkl kernel K by (1.7) Using this kernel C.F. Dunkl has introduced in [5] a Fourier transform F D called the Dunkl transform.
In this paper we establish the following inversion formulas for the operators V k and t V k : where P is a pseudo-differential operator on R d . When the multiplicity function takes integer values, the formula (1.8) can also be written in the form where Q is a differential-difference operator. Also we give another expression of the operator t V −1 k on the space E ′ (R d ). From these relations we deduce the expressions of the representing distributions η x and Z x of the inverse operators V −1 k and t V −1 k by using the representing measures µ x and ν x of V k and t V k . They are given by the following formulas where t P and t Q are the transposed operators of P and Q respectively.
The contents of the paper are as follows. In Section 2 we recall some basic facts from Dunkl's theory, and describe the Dunkl operators and the Dunkl kernel. We define in Section 3 the Dunkl transform introduced in [5] by C.F. Dunkl, and we give the main theorems proved for this transform, which will be used in this paper. We study in Section 4 the Dunkl convolution product and the Dunkl transform of distributions which will be useful in the sequel, and when the multiplicity function takes integer values, we give another proof of the geometrical form of Paley-Wiener-Schwartz theorem for the Dunkl transform. We prove in Section 5 some inversion formulas for the Dunkl intertwining operator V k and its dual t V k on spaces of functions and distributions. Section 6 is devoted to proving under the condition that the multiplicity function takes integer values an inversion formula for the Dunkl intertwining operator V k , and we deduce the expression of the representing distributions of the inverse operators V −1 k and t V −1 k . In Section 7 we give some applications of the preceding inversion formulas.

The eigenfunction of the Dunkl operators
In this section we collect some notations and results on the Dunkl operators and the Dunkl kernel (see [3,4,5,7,9,10]).

Reflection groups, root systems and multiplicity functions
We consider R d with the Euclidean scalar product ·, · and x = x, x . On C d , · denotes also the standard Hermitian norm, while z, For a given root system R the reflections σ α , α ∈ R, generate a finite group W ⊂ O(d), the reflection group associated with R. All reflections in W correspond to suitable pairs of roots. For a given β ∈ R d \ ∪ α∈R H α , we fix the positive subsystem R + = {α ∈ R; α, β > 0}, then for each α ∈ R either α ∈ R + or −α ∈ R + . A function k : R → C on a root system R is 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 that k is constant on the conjugacy classes of reflections in W . For abbreviation, we introduce the index Moreover, let ω k denotes the weight function which is W -invariant and homogeneous of degree 2γ.
For d = 1 and W = Z 2 , the multiplicity function k is a single parameter denoted γ and We introduce the Mehta-type constant which is known for all Coxeter groups W (see [3,6]).

The Dunkl operators and the Dunkl kernel
The Dunkl operators T j , j = 1, . . . , d, on R d , associated with the finite reflection group W and the multiplicity function k, are given for a function f of class C 1 on R d by In the case k ≡ 0, the T j , j = 1, 2, . . . , d, reduce to the corresponding partial derivatives. In this paper, we will assume throughout that k ≥ 0 and γ > 0. For f of class C 1 on R d with compact support and g of class For y ∈ R d , the system T j u(x, y) = y j u(x, y), j = 1, 2, . . . , d, admits a unique analytic solution on R d , denoted by K(x, y) and called the Dunkl kernel. This kernel has a unique holomorphic extension to C d × C d .
Example 2.1. From [5], if d = 1 and W = Z 2 , the Dunkl kernel is given by where for α ≥ −1/2, j α is the normalized Bessel function defined by with J α being the Bessel function of first kind and index α (see [16]).
The Dunkl kernel possesses the following properties.
(ii) For all ν ∈ Z d + , x ∈ R d , and z ∈ C d we have In particular (iii) For all x, y ∈ R d and w ∈ W we have K(−ix, y) = K(ix, y) and K(wx, wy) = K(x, y).
(iv) The function K(x, z) admits for all x ∈ R d and z ∈ C d the following Laplace type integral representation where µ x is the measure given by the relation (1.3) (see [14]).
Remark 2.1. When d = 1 and W = Z 2 , the relation (2.6) is of the form Then in this case the measure µ x is given for all x ∈ R\{0} by dµ x (y) = K(x, y)dy with where 1 ]−|x|,|x|[ is the characteristic function of the interval ]−|x|, |x|[.

The Dunkl transform
In this section we define the Dunkl transform and we give the main results satisfied by this transform which will be used in the following sections (see [5,9,10]).
Notation. We denote by H(C d ) the space of entire functions on C d which are rapidly decreasing and of exponential type. We equip this space with the classical topology.
The Dunkl transform of a function f in S(R d ) is given by This transform satisfies the relation where F is the classical Fourier transform on R d given by The following theorems are proved in [9,10].
The inverse transform is given by Remark 3.1. Another proof of Theorem 3.1 is given in [17].
When the multiplicity function satisfies k(α) ∈ N for all α ∈ R + , M.F.E. de Jeu has proved in [10] the following geometrical form of Paley-Wiener theorem for functions.
where I E is the gauge associated to the polar of E, given by For f in S(R d ) the function τ x f can also be written in the form Using properties of the operators V k and t V k we deduce that for f in D(R d ) (resp. S(R d )) and x ∈ R d , the function y → τ x f (y) belongs to D(R d ) (resp. S(R d )) and we have The Dunkl convolution product of f and

The Dunkl convolution product of tempered distributions
Proof . We remark first that the topology of S(R d ) is also generated by the seminorms We must prove that (τ x ϕ − τ x 0 ϕ) converges to zero in S(R d ) when x tends to x 0 . Let k, ℓ ∈ N and µ ∈ N d such that |µ| ≤ k. From (4.3), Theorem 3.1 and the rela- j the Dunkl Laplacian and p ∈ N such that p > γ + d 2 + 1. Using (2.4) and (2.5) we deduce that Then the function S * D ϕ is continuous at x 0 , and thus it is continuous on R d . Now we will prove that S * D ϕ admits a partial derivative on R d with respect to the variable x j . Let h ∈ R\{0}. We consider the function f h defined on R d by Using the formula we obtain for all k, ℓ ∈ N and µ ∈ N d such that |µ| ≤ k: By applying the preceding method to the function we deduce from the relation (4.4) that Thus the function S * D ϕ(x) admits a partial derivative at x 0 with respect to x j and we have These results is true on R d . Moreover the partial derivatives are continuous on R d . By proceeding in a similar way for partial derivatives of all order with respect to all variables, we deduce that On the other hand using the definition of the Dunkl operator T j and the relation By iteration we get Notation. We denote by H(C d ) the space of entire functions on C d which are slowly increasing and of exponential type. We equip this space with the classical topology.

The Dunkl transform of distributions
The following theorem is given in [17, page 27].
Theorem 4.2. Let S be in S ′ (R d ) and ϕ in S(R d ). Then, the distribution on R d given by (S * D ϕ)ω k belongs to S ′ (R d ) and we have Proof . i) As S belongs to S ′ (R d ) then there exists a positive constant C 0 and k 0 , ℓ 0 ∈ N such that But by using the inequality the relations (4.2), (1.3) and the properties of the operator t V k (see Theorem 3.2 of [17]), we deduce that there exists a positive constant C 1 and k, ℓ ∈ N such that Thus from (4.7) we obtain where C is a positive constant. This inequality shows that the distribution on R d associated with the function (S * D ϕ)ω k belongs to S ′ (R d ).
ii) Let ψ be in S(R d ). We shall prove first that whereŠ is the distribution in S ′ (R d ) given by We consider the two sequences {ϕ n } n∈N and {ψ m } m∈N in D(R d ) which converge respectively to ϕ and ψ in S(R d ). We have Thus Thus from (4.8) there exist a positive constant M and k, ℓ ∈ N such that Thus On the other hand we have and the limit is in S(R d ).
We prove now the relation (4.6). Using (4.9) we obtain for all ψ in S(R d ) Then This completes the proof of (4.6).
We consider the positive function ϕ in D(R d ) which is radial for d ≥ 2 and even for d = 1, with support in the closed ball of center 0 and radius 1, satisfying and φ the function on [0, +∞[ given by For ε ∈]0, 1], we denote by ϕ ε the function on R d defined by ). (4.14) This function satisfies the following properties: i) Its support is contained in the closed ball B ε of center 0, and radius ε.
ii) From [13, pages 585-586] we have with j γ+ d 2 −1 (λr) the normalized Bessel function. iii) There exists a positive constant M such that where the limit is in S ′ (R d ).
Proof . We deduce (4.18) from (4.6), (4.15), (4.17) and Theorem 4.1.  Theorem 4.4. Let S 1 be in S ′ (R d ) and S 2 in E ′ (R d ). Then the distribution S 1 * D S 2 belongs to S ′ (R d ) and we have Proof . We deduce the result from (4.20), the relation and Theorem 4.2.

Another proof of the geometrical form of the Paley-Wiener-Schwartz theorem for the Dunkl transform
In this subsection we suppose that the multiplicity function satisfies k(α) ∈ N\{0} for all α ∈ R + . The main result is to give another proof of the geometrical form of Paley-Wiener-Schwartz theorem for the transform F D , given in [17, pages 23-33].

21)
where I E is the function given by (3.4).
Proof . Necessity condition. We consider a distribution S in E ′ (R d ) with support in E. Let X be in D(R d ) equal to 1 in a neighborhood of E, and θ in E(R) such that We put η = Im z, z ∈ C d and we take ε > 0. We denote by ψ z the function defined on R d by This function belongs to D(R d ) and as E is W -invariant, then it is equal to K(−ix, z) in a neighborhood of E. Thus As S is with compact support, then it is of finite order N . Then there exists a positive constant C 0 such that Using the Leibniz rule, we obtain and if M is the estimate of sup From this relation and (4.22) we obtain where C 2 is a positive constant, and the supremum is calculated when z ≥ 1, for because if not we have θ = 0. This inequality implies sup x∈E e max w∈W wx,η ≤ e 2 · e I E (η) . (4.28) From (4.27), (4.28) we deduce that there exists a positive constant C 3 independent from ε such that If we make ε → 0 in this relation we obtain (4.21) for z ≥ 1. But this inequality is also true (with another constant) for z ≤ 1, because in the set {z ∈ C d , z ≤ 1} the function Sufficient condition. Let f be an entire function on C d satisfying the condition (4.21). It is clear that the distribution given by the restriction of f ω k to R d belongs to S ′ (R d ). Thus from Theorem 4.1i there exists a distribution S in S ′ (R d ) such that (4.29) We shall show that the support of S is contained in E. Let ϕ ε be the function given by the relation (4.14). We consider the distribution The properties of the function f and (4.15), (4.16) and (4.17) show that the function f ε can be extended to an entire function on C d which satisfies: for all q ∈ N there exists a positive constant C q such that Then from (4.31), Theorem 3.2 and (4.30), the function (S * ϕ ε )ω k belongs to D(R d ) with support in E + B ε . But from Theorem 4.3, the family (S * ϕ ε )ω k converges to S in S ′ (R d ) when ε tends to zero. Thus for all ε > 0, the support of S is in E + B ε , then it is contained in E. Let E be a W -invariant compact convex set of R d and x ∈ E. The function f (x, ·) defined on C d by is entire on C d and satisfies |f (x, z)| ≤ e I E (Im z) .
Thus from Theorem 4.5 there exists a distributionη x in E ′ (R d ) with support in E such that Applying now the remainder of the proof given in [17, page 32], we deduce that the support of the representing distribution η x of the inverse Dunkl intertwining operator V −1 k is contained in E.

Inversion formulas for the Dunkl intertwining operator
and its dual 5.1 The pseudo-differential operators P Definition 5.1. We define the pseudo-differential operator P on S(R d ) by where * is the classical convolution production of a distribution and a function on R d .
Proof . It is clear that the distribution T ω k given by the function ω k belongs to S ′ (R d ). On the other hand from the relation (5.1) we have Thus With the definition of the classical convolution product of a distribution and a function on R d , the relation (5.2) can also be written in the form Proof . By derivation under the integral sign, and by using the relation we obtain (5.3).

5.2
Inversion formulas for the Dunkl intertwining operator and its dual on the space S(R d ) Proof . From [15,Theorem 4.1] for all f in S(R d ), the function t V −1 k (f ) belongs to S(R d ). Then from Theorem 3.1 we have But from the relations (3.2), (1.7), (1.3), we have whereμ x is the probability measure given for a continuous function f on R d by Thus (5.5) can also be written in the form Then by using (5.1), the properties of the Fourier transform F and Fubini's theorem we obtain Thus Proof . We deduce the relation (5.6) by replacing f by t V k (f ) in (5.4) and by using the fact that the operator V k is an isomorphism from E(R d ) onto itself.

Inversion formulas for the dual Dunkl intertwining operator on the space
The dual Dunkl intertwining operator t V k on E ′ (R d ) is defined by 6 Other expressions of the inversion formulas for the Dunkl intertwining operator and its dual when the multiplicity function is integer In this section we suppose that the multiplicity function satisfies k(α) ∈ N\{0} for all α ∈ R + . The following two Propositions give some other properties of the operator P defined by (5.1).
Proof . From the relation (5.1) we have We consider the function F defined by This function is entire on C d and by using Theorem 2.6 of [1] we deduce that for all q ∈ N, there exists a positive constant C q such that where I E is the function given by (3.4). The relation (6.1) can also be written in the form Proof . For all f in S(R d ), we have From (6.5), (6.6) we obtain This relation, Definition 5.1 and the inversion formula for the Fourier transform F imply (6.4).
Remark 6.1. In this case the operator P is not a pseudo-differential operator but it is a partial differential operator.
6.1 The differential-difference operator Q Definition 6.1. We define the differential-difference operator Q on S(R d ) by i) The operator Q is linear and continuous from S(R d ) into itself.
ii) For all f in S(R d ) we have where T j , j = 1, 2, . . . , d, are the Dunkl operators.
Proof . We deduce the result from the properties of the operator t V k (see Theorem 3.2 of [17]), and Proposition 5.2.
Proof . Using the relations (3.2), (5.1) and the properties of the operator t V k (see Theorem 3.2 of [17]), we deduce from Definition 6.1 that As the function ω k F D (f ) belongs to S(R d ), then by applying the fact that the classical Fourier transform F is bijective from S(R d ) onto itself, we obtain Proposition 6.5. The distribution T ω 2 k given by the function ω 2 k is in S ′ (R d ) and for all f in S(R d ) we have where * D is the Dunkl convolution product of a distribution and a function on R d .
Proof . It is clear that the distribution T ω 2 k given by the function ω 2 k belongs to S ′ (R d ). On the other hand from the relations (6.7), (3.3) and (4.3) we obtain

Other expressions of the inversion formulas for the Dunkl intertwining operator and its dual on spaces of functions and distributions
In this subsection we give other expressions of the inversion formulas for the operators V k and t V k and we deduce the expressions of the representing distributions of the operators V −1 k and t V −1 k .
Proof . We obtain this result by using of Proposition 6.3, Theorem 5.2 and Definition 6.1.
Proof . For all f in D(R d ), we obtain from (3.2) the relations We deduce ( Proof . We obtain (6.13) from Definition 6.1, Propositions 6.1 and 6.7.
where T j , j = 1, 2, . . . , d, are the Dunkl operators defined on S ′ (R d ) by T j S, f = − S, T j f , f ∈ S(R d ).
Proposition 6.10. For all S in S ′ (R d ) we have F −1 ( t P (S)) = π d c 2 k 2 2γ F −1 (S)ω k , Proof . We deduce these relations from (5.1), (6.7) and the definitions of the classical Fourier transform and the Dunkl transform of tempered distributions on R d . and ∀ x ∈ R d , Z x = t P (µ x ), (6.16) where µ x and ν x are the representing measures of the Dunkl intertwining operator V k and its dual t V k .
Proof . From (1.5), for all f in S(R d ) we have On the other hand from (1.4) We obtain (6.15) from this relation, (6.17) and (6.11). Using (1.3), for all f in S(R d ) we can also write the relation (5.4) in the form But from (1.6) we have We deduce (6.16) from this relation and (6.18).
Corollary 6.2. We have and Proof . We deduce these relations from Theorem 6.3 and Proposition 6.9. From the relation given in [17, page 27] and Corollary 6.1, we deduce that the distribution F −1 D (f ) is in E ′ (R d ) and its support is contained in E.

Other expressions of the Dunkl translation operators
We consider the Dunkl translation operators τ x , x ∈ R d , given by the relations (4.1), (4.2). Theorem 7.1.
By using the definition of the classical convolution product of two measures with compact support on R d , we obtain ∀ x, y ∈ R d , τ x (f )(y) = µ x * µ y (V −1 k (f )).
ii) The same proof as for i) and Theorem 6.1 give the relation (7.2).