The Range of the Spectral Projection Associated with the Dunkl Laplacian

For s ∈R, denote by Pk f the “projection” of a function f inDðRdÞ into the eigenspaces of the Dunkl Laplacian Δk corresponding to the eigenvalue −s2: The parameter k comes from Dunkl’s theory of differential-difference operators. We shall characterize the range of Pk on the space of functions f ∈DðRdÞ supported inside the closed ball BðO, RÞ: As an application, we provide a spectral version of the Paley-Wiener theorem for the Dunkl transform.


Introduction
Analysis of the Dunkl Laplacian operator Δ k on ℝ d commenced in the early 90's, inspired by numerous results in the Euclidean setting, as well as some extensions of this to flat symmetric spaces. Here, the parameter k comes from Dunkl's theory of differential-difference operators [1]. In recent years, there have been increasing interests in the study of problems involving the Dunkl Laplacian and have received a lot of attention, see for instance [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. The purpose of this paper is to study a family of eigenfunctions for the Dunkl Laplacian derived through the use of the inversion formula for the Dunkl transform. Our main result may be interpreted as a contribution to the spectral theory of the Dunkl Laplacian.
To state our main result, we need to introduce some notation. Writing the inversion formula for the Dunkl transform in polar coordinates, we obtain where f s k are "projections" of f into the eigenspaces of Δ k corresponding to the eigenvalue −s 2 (see (35)). We may also write the projection operators f ↦ f s k as Dunkl-convolution with a normalized Bessel function of the first kind (see (43)). In this paper, we discuss on Dðℝ d Þ how properties of f are related to properties of the eigenfunctions f s k : Essentially, we prove a theorem characterizing f s k for f ∈ Dðℝ d Þ with supp ð f Þ ⊂ BðO, RÞ, involving analytic continuation to s ∈ ℂ and growth estimates of type, for all N ∈ ℕ and for all multi-index m ∈ ℕ d , where C k,m,N is a positive continuous increasing function on ℝ + (see Theorem 9). Several contributions have been dedicated to this subject, see for instance [16][17][18][19][20][21][22].
As an application of the main result, we prove a spectral version of the complex Paley-Wiener theorem for the Dunkl transform F k given in [23]. More precisely, we characterize the set of functions φðs, ηÞ defined on ℝ × S d−1 for which there exists a compactly supported smooth function f with support in BðO, RÞ so that φðs, ηÞ = F k ð f ÞðsηÞ (see Theorem 10).

Background
For x, y ∈ ℝ d , we let hx, yi denote the usual Euclidean inner product of ℝ d and ∥x∥≔ ffiffiffiffiffiffiffiffiffiffi ffi hx, xi p the Euclidean norm. Let S d−1 be the unit sphere in ℝ d : We denote by dσ as the Lebesgue surface measure on S d−1 : For a nonzero vector α ∈ ℝ d define the reflection r α by A root system is a finite set R of nonzero vectors in ℝ d such that α, β ∈ R implies r α ðβÞ ∈ R: If, in addition, α, β ∈ R and α = cβ for some scalar c implies c = ±1, then R is called reduced. Henceforth, we will assume that R is a reduced root system. Fix a set of positive roots R + , so that The finite reflection group G generated by the root system R is the subgroup of the orthogonal group OðdÞ generated by the reflections fr α : α ∈ R + g: For a given root system R, a multiplicity function Given a reduced root system R on ℝ d and a multiplicity function k = ðk α Þ α∈R , we define the weight function ϑ k by Then, ϑ k is a positively homogeneous G-invariant function of degree 2hki, where The main ingredient of the Dunkl theory is a family of commuting first-order differential-difference operators, T ξ ðkÞ (called the Dunkl operators [1]), defined by where ∂ ξ is the ordinary partial derivative with respect to ξ: The Dunkl operators are akin to the partial derivatives and they can be used to define the Dunkl Laplacian Δ k , which plays the role similar to that of the ordinary Laplacian Δ, where fξ 1 , ⋯, ξ d g is an orthonormal basis of ðℝ d , h·, · iÞ: The above explicit expression of Δ k has been proved in [24].
For arbitrary finite reflection group G, and for any nonnegative multiplicity function k, there is a unique linear operator V k on the space of algebraic polynomials on ℝ d that intertwines between the Dunkl operators and the partial derivatives, It has been proved in [25] that V k has a Laplace type representation which allows to extend V k to larger function spaces: with a unique probability measure μ k x ∈ M 1 ðℝ d Þ: In fact, V k induces a homeomorphism of Cðℝ d Þ and also that of C ∞ ðℝ d Þ ; see [23,26].
(1) For all z, w ∈ ℂ d and λ ∈ ℂ, we have E k ðz, wÞ = E k ðw, zÞ and E k ðλz, wÞ = E k ðz, λwÞ: where where c k is the constant The Dunkl transform was introduced in [28] where the L 2 -isometry (or the Plancherel theorem) was proved, while the main results of the L 1 -theory were established in [11]. In particular, it has been proved that if f and F k f are in L 1

2
Journal of Function Spaces It is worth mentioning that the Dunkl transform is a homeomorphism of the Schwartz space Sðℝ d Þ: Further, according to ( [29], Proposition 5. where and H α is the Hankel transform of index α on L 1 ðℝ + , r 2α+1 drÞ, given by Here, j α is the normalized Bessel function of the first kind defined by Let y ∈ ℝ d be given. For f ∈ Sðℝ d Þ, the generalized translation operator is defined by Fact 2 (see [26]). The translation operator has the following properties: (1) For all x, y ∈ ℝ d ,τ y f ðxÞ = τ x f ðyÞ: (2) For fixed y ∈ ℝ d ,τ y extends to a continuous linear The generalized translation operator is used to define a convolution structure: For f , g ∈ Sðℝ d Þ, where gðxÞ ≔ gð−xÞ: We can also write the convolution * k as We refer the reader to [15] for more details on the convolution product * k : It is worth mentioning, for the distributional version of the Dunkl transform, the translation operator and the Dunkl convolution of distributions and properties, we refer the reader to [30].
For n ∈ ℕ, let H n k be the space of k-harmonic polynomials of degree n on ℝ d , where Δ k is the Dunkl Laplacian and P n ðℝ d Þ denotes the space of homogeneous polynomials of degree n on ℝ d : The restriction of elements in H n k on the unit sphere S d−1 in ℝ d are the so-called k-spherical harmonics. We shall not distinguish between Y n k ∈ H n k and its restriction to S d−1 : The space H n k has a reproducing kernel Q n k ð·, · Þ in the sense that Here, d k is the constant where c k and λ k are as defined in (13) and (16), respectively. According to [31], for x, y ≠ 0, the kernel Q n k can be written as where V k is the Dunkl intertwining operator (8), and C α n is the Gegenbauer polynomial of degree n, for α > 0, with 2 F 1 is the Gauss hypergeometric function.
The following analogue of the Funk-Hecke formula for k -spherical harmonics will be used later on; for the proof, the reader is referred to [32]. Let h be a continuous function on ½−1, 1: Then, for any Y n k ∈ H n k , 3 Journal of Function Spaces where Λ n ðhÞ is a constant defined by We summarize some basic properties of Gegenbauer polynomials in a way that we shall use later. . For λ ∈ ℂ such that Re λ > 0, the following two integral formulas hold: Let D R ðℝÞ e denote the space of even compactly supported smooth functions with support in ½−R, R, where R > 0: The Paley-Wiener theorem for the Hankel transform H α (see (17)) states that H α maps D R ðℝÞ e bijectively onto the space H R ðℂÞ e of even entire functions g satisfying, for all N ∈ ℕ, for some positive constant C N ; see for instance [34]. This result has been generalized by de Jeu [23] to the Dunkl transform. To state the (complex) Paley-Wiener theorem for F k , we introduce the following notation. For R > 0, let H R ðℂ d Þ be the space of entire functions F on ℂ d with the property that for all N ∈ ℕ there exists a constant C N > 0 such that We let D R ðℝ d Þ denote the space of smooth compactly supported functions with support contained in the closed ball BðO, RÞ ⊂ ℝ d with radius R > 0 and the origin as center.
Fact 4 (see [23]). The Dunkl transform F k is a linear isomorphism between D R ðℝ d Þ and H R ðℂ d Þ, for all R > 0: An immediate consequence of the above Paley-Wiener theorems can be stated as follows: for all N ∈ ℕ: Proof. The statement follows from the fact that F k f ðξÞ = H λ k f 0 ð∥ξ∥Þ whenever f is a radial function with f ðxÞ = f 0 ð∥ x∥Þ (see (15)), together with the Paley-Wiener theorems stated above for the Hankel and the Dunkl transforms.

The Range of the Spectral Projection Associated with Δ k
Recall from (12) that the Dunkl transform of f ∈ Dðℝ d Þ is defined by Using polar coordinates, the Dunkl inversion formula (14) becomes where Notice that Δ x k P s k f ðxÞ = −s 2 P s k f ðxÞ: From (36), we may derive a second formula for P s k f : Indeed, substituting (34) into (36), we obtain According to [[35], page 2424], the inner integral is equal to 4 Journal of Function Spaces where d k is the constant (25), Q n k ð·, · Þ is the reproducing kernel (26), and j α is the normalized Bessel function (18). We now use the well-known addition formula for Bessel functions (see [[36] , p. 215]): for a,b,θ ∈ ℝ, which converges uniformly with respect to θ ∈ ℝ: Using (38) and (26) together with the Laplace representation (9) for V k , we deduce that Define j s,λ k ðyÞ ≔ j λ k ðs∥y∥Þ, then, by [ [35] , p. 2429], we have Consequently, the eigenfunction P s k f can be rewritten as Above, we have used some of the properties of the generalized translation operator listed in Fact 2. The following statement lists the necessary conditions for Theorem 9.
Proposition 6. Assume that f ∈ D R ðℝ d Þ and let P s k f ðxÞ defined either by (36) or (42), with s ∈ ℝ and x ∈ ℝ d : Then the following hold: (5) For any k-spherical harmonic Y ℓ k of degree ℓ and for every r > 0, the map is entire on ℂ: Proof.
(1) In view of properties of the translation operator τ x and the normalized Bessel function j α , the first statement follows from the representation (42) of P s k f ðxÞ: (2) The second property is immediate from (36) We now apply the Funk-Hecke formula (28) to deduce that ð where, by Fact 3, we have Using the above identities, it follows that ð where φ ℓ+λ k ðzÞ ≔ 1/ð2 λ k +ℓ Γðλ k + ℓ + 1ÞÞz ℓ j ℓ+λ k ðzÞ and Above, we have used the fact that d k = c k /f2 λ k Γðλ k + 1Þg ; see (25). In conclusion, The desired result now follows from the fact that s −ℓ φ ℓ+λ k ðszÞ is an entire function of s ∈ ℂ: The following lemma is needed for later use.
(1) In the polar coordinates x = rω, the Dunkl Laplacian operator is expressed as where Δ k,S d−1 is the analogue of the Laplace-Beltrami operator on the sphere S d−1 , which, in particular, has k-spherical harmonics as eigenfunctions, We refer the reader to [37] for more details on Δ k,S d−1 : (2) Obvious Next we will list the sufficient conditions for Theorem 9.
Proposition 8. For s ∈ ℝ and x ∈ ℝ d , let f s ðxÞ be a function satisfying the following conditions: (1) f s ðxÞ is smooth on ℝ × ℝ d (2) f s ðxÞ is an eigenfunction of the Dunkl Laplacian with eigenvalue −s 2 The mapping s ↦ f s ðxÞ extends to an even entire function on ℂ