Josef Finsterer

Sinda Zarrouk-Mahjoub

Keywords:

Letter to the Editor

With interest we read the article by Thomas et al. about a 44 year old female with MELAS syndrome due to the m.3243A N G mutation, which manifested cardiologically as hypertrophic cardiomyopathy and episodes of supraventricular tachycardia requiring atenolol [START_REF] Thomas | Arrhythmia as a cardiac manifestation in MELAS syndrome[END_REF]. We have the following comments and concerns.

MELAS may be associated with sudden cardiac death [START_REF] Taniguchi | MELAS sudden death due to paroxysmal arrhythmia[END_REF]. Did the authors consider implantation of a loop-recoder to monitor if there were also ventricular arrhythmias requiring implantation of an ICD?

The authors describe hypertrophic cardiomyopathy as progressive [START_REF] Thomas | Arrhythmia as a cardiac manifestation in MELAS syndrome[END_REF]. What was the systolic function and wall thickness at the last follow-up? Figure 1B suggests that there is noncompaction of the left ventricular myocardium [START_REF] Thomas | Arrhythmia as a cardiac manifestation in MELAS syndrome[END_REF]. Did the authors look for this unclassified cardiomyopathy, which may occur in addition to hypertrophic cardiomyopathy, goes frequently along with late gadolinium enhancement [START_REF] Cheng | Comparison of cardiovascular magnetic resonance characteristics and clinical consequences in children and adolescents with isolated left ventricular non-compaction with and without late gadolinium enhancement[END_REF], and is most frequently associated with mitochondrial disorders? [START_REF] Finsterer | Cardiogenetics, neurogenetics, and pathogenetics of left ventricular hypertrabeculation/noncompaction[END_REF] A heteroplasmy rate of 25% of the m.3243ANG mutation is low. In which tissue was it determined? Blood, muscle, saliva, or urine? Were other tissues investigated for the heteroplasmy rate as well? Was a muscle biopsy carried out?

Recurrent stroke-like episodes (SLEs) are reported during the 11 years since diagnosis [START_REF] Thomas | Arrhythmia as a cardiac manifestation in MELAS syndrome[END_REF], which are a clinical manifestation of cerebral stroke-like lesions [START_REF] Finsterer | Stroke and stroke-like episodes in muscle disease[END_REF]. How many SLEs did the patient experience, Molecular Genetics and Metabolism Reports 7 (2016) 54 in which cerebral region, and why could the remnants not be seen on cerebral MRI, which only showed atrophy and leucencephalopathy?

Concerning the initial fall, did the patient loose consciousness, did she experience a tongue bite, or did she experience a secessus urinae/alvi? What were the results of the EEG? Did the patient always experience seizures during SLEs? Were antiepileptic drugs administered? Were other causes of the fall excluded?

Overall, this interesting case should be supplemented by results about the heteroplasmy in more than a single tissue, long-term ECG results, and the medication she was taking at last follow-up.


Léonard Gallardo
email: leonard.gallardo@lmpt.univ-tours.fr

Chaabane Rejeb
email: chaabane.rejeb@gmail.com

Mohamed Sifi
email: mohamed.sifi@fst.rnu.tn

Riesz potentials of Radon measures associated to reflection groups

Keywords: MSC (2010) primary: 31B05, 31B10, 31C45, 47B34; secondary: 28C05, 43A32, 46F10, 51F15 Reflection groups, Dunkl-Laplace operator, Dunkl heat kernel, Generalized volume mean operator, Dunkl subharmonic functions, Riesz kernel and potentials, Hedberg's inequality

For a root system R on R d and a nonnegative multiplicity function k on R, we consider the heat kernel p k (t, x, y)

According to the values of β, we study the ∆ k -superharmonicity of these functions and we give some applications like the ∆ k -Riesz measure of I k,β [µ], the uniqueness principle and a pointwise Hedberg's inequality.

Introduction

Let R be a normalized root system in R d . That is, for every α ∈ R, ∥α∥ 2 = 2, R ∩ Rα = {±α} and σ α (R) = R, where σ α is the reflection with respect to the hyperplane H α orthogonal to α (see [START_REF] Humphreys | Reflection groups and Coxeter groups[END_REF] and [START_REF] Kane | Reflection Groups and Invaraint Theory[END_REF]). We fix k ≥ 0 a multiplicity function (i.e. k : R → [0, +∞[ invariant under the action of the Coxeter-Weyl group W associated to R) and we consider the associated Dunkl-Laplacian operator ∆ k given by

∆ k f (x) = ∆f (x) + 2 ∑ α∈R + k(α) ( ⟨∇f (x), α⟩ ⟨x, α⟩ - f (x) -f (σ α (x) ⟨x, α⟩ 2 ) , f ∈ C 2 (R d ), (1.1)
1 with R + a positive subsystem (see [START_REF] Dunkl | Orthogonal Polynomials of Several variables[END_REF]). Acting on C ∞ (R d ), it is related to the classical Laplacian operator ∆ by means of the so-called Dunkl intertwining operator V k (see [START_REF] Dunkl | Hankel transform associated to finite reflection groups[END_REF], [START_REF] Dunkl | Orthogonal Polynomials of Several variables[END_REF], [START_REF] Trimèche | The Dunkl intertwining operator on spaces of functions and distributions and integral representation of its dual[END_REF]) as follows:

∆ k V k = V k ∆. (1.2)
In [START_REF] Rösler | Positivity of Dunkl's intertwining operator[END_REF], M. Rösler has proved that for any x ∈ R d , there exists a compactly supported probability measure µ x on R d (which we call Rösler's measure at point x) such that

∀ f ∈ C ∞ (R d ), V k (f )(x) = ∫ R d f (y)dµ x (y), (1.3)
with supp µ x ⊂ C(x) = co{gx, g ∈ W } (1.4)
(the convex hull of the orbit of x under the group W ). We note that, according to [START_REF] Gallardo | Support properties of the intertwining and the mean value operators in Dunkl theory[END_REF], the support of µ x contains the point x and it is W -invariant under the hypothesis that the multiplicity function is positive.

Let p k (t, x, y) (t > 0, x, y ∈ R d ) be the heat kernel of the Dunkl Laplacian ∆ k which is given by (see [START_REF] Rösler | Generalized Hermite polynomials and the heat equation for Dunkl operators[END_REF] and [START_REF] Rösler | Dunkl Operators: Theory and Applications[END_REF])

p k (t, x, y) := 1 (2t) d/2+γ c k e -(∥x∥ 2 +∥y∥ 2 )/4t E k ( x √ 2t , y √ 2t ), (1.5)
where E k (x, y) = V k (e ⟨.,y⟩ )(x) (x, y ∈ R d ) (1.6) is the Dunkl kernel (see [START_REF] Dunkl | Integral kernels with reflection group invariance[END_REF] and [START_REF] Dunkl | Orthogonal Polynomials of Several variables[END_REF]), c k is the Macdonald-Mehta constant (see [START_REF] Opdam | Dunkl operators, Bessel functions and the discriminant of a finite Coxeter group[END_REF]) given by

c k := ∫ R d e -∥x∥ 2 2 ω k (x)dx, (1.7)
and ω k is the Dunkl weight function

ω k (x) = ∏ α∈R + | ⟨α, x⟩ | 2k(α) (1.8)
which is homogeneous of degree 2γ.

It is also known (see [START_REF] Rösler | Generalized Hermite polynomials and the heat equation for Dunkl operators[END_REF]) that for all fixed x ∈ R d , the function p k (t, x, .) solves the Dunkl heat equation (∆ k -∂ t )p k (t, x, .) = 0.

(1.9)

Let γ = ∑ α∈R + k(α) and suppose that d + 2γ > 2. For β ∈]0, d + 2γ[, we define the ∆ k -Riesz kernel of index β as follows

R k,β (x, y) := 1 Γ(β/2) ∫ +∞ 0 t β 2 -1 p k (t, x, y)dt.
We note that when β = 2, we obtain the Dunkl-Newton kernel which has been introduced and studied in [START_REF] Gallardo | Newtonian Potentials and subharmonic functions associated to root systems[END_REF].

Let x ∈ R d , x ̸ = 0, be fixed and W.x be its W -orbit. If y / ∈ W.x, R k,β (x, y) is finite. But when y ∈ W.x, it seams hard, except in the case y = x, to decide in general if R k,β (x, y) is finite or infinite. These difficulties are illustrated by the particular case of the root system of type

A 1 × A 1 × • • • × A 1 (m times, 1 ≤ m ≤ d),
where we manage to give a complete description of the singularities of the function R k,β (x, .). The aim of this paper is the study, when d + 2γ > 2, of the ∆ k -Riesz kernel R k,β and the corresponding potential

I k,β [µ](x) = ∫ R d R k,β (x, y)dµ(y)
of a signed Radon measure µ on R d . In particular, we will study the sub-or-superharmonicity of these functions in the sense of the Dunkl-Laplace operator and we will describe explicitly their ∆ k -Riesz measures. This notion of subharmonicity, which generalizes the classical one1 has been introduced and studied in some details in [START_REF] Gallardo | Newtonian Potentials and subharmonic functions associated to root systems[END_REF]. More precisely, let Ω be a W -invariant open subset of R d . A function u : Ω -→ [-∞, +∞[ is called ∆ k -subharmonic (D-subharmonic) on Ω if • u is upper semi-continuous (u.s.c.) on Ω, • u is not identically -∞ on each connected component of Ω,

• it satisfies the sub-mean volume property: for every closed ball B(x, r) ⊂ Ω, we have

u(x) ≤ M r B (u)(x) := 1 m k [B(0, r)] ∫ R d u(y)h k (r, x, y)ω k (y)dy, (1.10)
where m k is the measure ω k (x)dx and h k (r, x, y) is a kernel of the form

h k (r, x, y) := ∫ R d
1 [0,r] ( √ ∥x∥ 2 + ∥y∥ 2 -2 ⟨x, z⟩)dµ y (z) (1.11) with µ y Rösler's measure at point y. The function y → h k (r, x, y) is a generalized translate of the indicator function 1 B(0,r) of the ball B(0, r) called harmonic kernel, introduced and studied in [START_REF] Gallardo | A new mean value property for harmonic functions relative to the Dunkl-Laplacian operator and applications[END_REF] and which properties will be recalled in the next section. Moreover the harmonic kernel is a crucial tool to get quite explicit expressions of the ∆ k -Riesz kernel (see section 3). Naturally, a function u is D-superharmonic on Ω if -u is D-subharmonic on Ω.

Finally, we study some applications. The main one is the following version of the uniqueness principle: if µ and ν are finite and nonnegative Radon measures on R d and if

I k,β [µ] = I k,β [ν] a.e. on R d , then µ = ν.
We also prove a pointwise Hedberg's inequality in the sense of the operator ∆ k and we deduce L p -boundedness properties of the ∆ k -Riesz potentials.

Generalities in Dunkl Theory

In order to help the reader, we have collected in this section some basics from Dunkl theory which will be used in the sequel.

Notations: Let us introduce the following functional spaces which are present throughout the paper:

• Ω a W -invariant open subset of R d . • L p k (Ω) (resp. L p k,loc (Ω)), 1 ≤ p < +∞ the space of measurable functions f : Ω -→ C such that ∥f ∥ p L p k (Ω) := ∫ Ω |f (x)| p ω k (x)dx < +∞ (resp. ∫ K |f (x)| p ω k (x)dx < +∞ for any compact set K ⊂ Ω). • L ∞ k (Ω)
the space of measurable and essentially bounded functions on Ω.

• When Ω = R d , the norm of the space L p k (R d ), 1 ≤ p ≤ +∞, will be denoted ∥.∥ k,p instead of ∥.∥ L p k (R d ) . • D(Ω) the space of C ∞ -functions on Ω with compact support.
• D ′ (Ω) the space of distributions on Ω (i.e. the topological dual of D(Ω) carrying the Fréchet topology).

• S(R d ) the Schwartz space of C ∞ -functions on R d which are rapidly decreasing together with their derivatives.

• S ′ (R d ) the space of tempered distributions.

The Dunkl transform

The Dunkl transform of a function f ∈ L 1 k (R d ) is defined by (see [START_REF] De Jeu | The Dunkl transform[END_REF] and [START_REF] Rösler | Dunkl Operators: Theory and Applications[END_REF])

F k (f )(λ) := ∫ R d f (x)E k (-iλ, x)ω k (x)dx, λ ∈ R d , ( 2.1)
where E k (x, y) is the Dunkl kernel (1.6) which is analytically extendable to C d × C d and satisfies the following properties (see [START_REF] Dunkl | Integral kernels with reflection group invariance[END_REF], [START_REF] Dunkl | Orthogonal Polynomials of Several variables[END_REF], [START_REF] De Jeu | The Dunkl transform[END_REF])

1. for all x, y ∈ R d , we have |E k (-ix, y)| ≤ 1. (2.2)
2. for all a ∈ C, x, y ∈ C d and all g ∈ W , we have

E k (ax, y) = E k (x, ay), E k (x, y) = E k (y, x) and E k (gx, gy) = E k (x, y).
It is well known (see [START_REF] De Jeu | The Dunkl transform[END_REF]) that the Dunkl transform F k is an isomorphism of S(R d ) onto itself and its inverse is given by

F -1 k (f )(x) = c -2 k ∫ R d f (λ)E k (ix, λ)ω k (λ)dλ, x ∈ R d , (2.3)
where c k is the constant given by (1.7). We note that for f, g ∈ S(R d ) the following relation holds ∫

R d F k (f )(x)g(x)ω k (x)dx = ∫ R d f (x)F k (g)(x)ω k (x)dx. (2.4)
Moreover, the transformation c -1 k F k extends uniquely to an isometric isomorphism of L 2 k (R d ) (Plancherel theorem, see [START_REF] De Jeu | The Dunkl transform[END_REF]).

We will also need the Dunkl transform F k (S) of a tempered distribution S ∈ S ′ (R d ) which is the distribution defined by

⟨F k (S), ϕ⟩ := ⟨S, F k (ϕ)⟩, ϕ ∈ S(R d ).
It is known that F k is a topological isomorphism of S ′ (R d ) onto itself (see [START_REF] Trimèche | Paley-Wiener theorem for the Dunkl transform and Dunkl translation operators[END_REF]). Note that if µ is a bounded Radon measure on R d , µ ∈ S ′ (R d ) and its distributional Dunkl transform can be identified to the continuous function

ξ → ∫ R d E k (-ix, ξ)dµ(x)ω k (ξ).
In the literature, the function

F k (µ) : ξ → ∫ R d E k (-ix, ξ)dµ(x) (2.5)
is called the Dunkl transform of the measure µ. This transformation is injective on the space of bounded Radon measures on R d (see [START_REF] Rösler | Markov processes related with Dunkl operators[END_REF]).

We recall also that the Dunkl-Laplace operator ∆ k leaves the spaces

D ′ (R d ) and S ′ (R d ) invariant where the ∆ k -action on S in D ′ (R d )) (resp. in S ′ (R d )) is defined as in the classical case by ⟨∆ k S, ϕ⟩ = ⟨S, ∆ k ϕ⟩, ϕ ∈ D(R d ) (resp. ϕ ∈ S(R d )). (2.6)

Dunkl's translation operators and heat kernel properties

• The Dunkl translation operators τ x , x ∈ R d , are defined on C ∞ (R d ) by (see [START_REF] Trimèche | Paley-Wiener theorem for the Dunkl transform and Dunkl translation operators[END_REF])

∀ y ∈ R d , τ x f (y) = ∫ R d V k • T z • V -1 k (f )(y)dµ x (z), (2.7)
where T x is the classical translation operator given by T x f (y) = f (x + y). The operators τ x , x ∈ R d , satisfy the following properties:

1) For all x ∈ R d , the operator τ x is continuous from C ∞ (R d ) into itself.
2) For all f ∈ C ∞ (R d ) and all x, y ∈ R d , we have

τ x f (0) = f (x), τ x f (y) = τ y f (x).
3) The Dunkl-Laplace operator ∆ k commutes with the Dunkl translations, i.e. [START_REF] Rösler | A positive radial product formula for the Dunkl kernel[END_REF]) has proved the useful formula

τ x (∆ k f ) = ∆ k (τ x f ), x ∈ R d , f ∈ C ∞ (R d ). 4) If f ∈ C ∞ (R d ) is radial, M. Rösler ([
∀ x ∈ R d , τ x f (y) = ∫ R d f ( √ ∥x∥ 2 + ∥y∥ 2 + 2 ⟨x, z⟩)dµ y (z), (2.8)
where f is the profile of f and µ y is the measure defined by (1.3).

In the particular case when f ∈ S(R d ), τ x f ∈ S(R d ) and using the Dunkl transform we have (see [START_REF] Trimèche | Paley-Wiener theorem for the Dunkl transform and Dunkl translation operators[END_REF]):

τ x f (y) = F -1 k [E k (ix, .)F k (f )](y) = c -2 k ∫ R d F k (f )(λ)E k (ix, λ)E k (iy, λ)ω k (λ)dλ, y ∈ R d .
• Using (2.8), the Dunkl heat kernel can also be written

p k (t, x, y) = 1 (2t) d/2+γ c k τ -x ( e -∥.∥ 2 4t ) (y) (2.9) = 1 (2t) d/2+γ c k ∫ R d e -1
4t (∥x∥ 2 +∥y∥ 2 -2 ⟨x,z⟩) dµ y (z).

(2.10)

For later use, we record also the following properties of the heat kernel (see [START_REF] Rösler | Generalized Hermite polynomials and the heat equation for Dunkl operators[END_REF] and [START_REF] Rösler | Dunkl Operators: Theory and Applications[END_REF])

1. The Dunkl heat kernel is symmetric in x and y i.e. p k (t, x, y) = p k (t, y, x), t > 0.

2. For every t > 0 and x ∈ R d , we have

∥p k (t, x, .)∥ k,1 = ∫ R d p k (t, x, y)ω k (y)dy = 1.
(2.11)

3.

For every t > 0 and x, y ∈ R d ,

p k (t, x, y) = F -1 k ( E k (-ix, .)e -t∥.∥ 2 ) (y) (2.12) = c -2 k ∫ R d e -t∥ξ∥ 2 E k (-ix, ξ)E k (iy, ξ)ω k (ξ)dξ.
(2.13)

4.

For every t > 0, the following inequality holds

∀ x, y ∈ R d , p k (t, x, y) ≤ 1 (2t) d/2+γ c k e -1 4t min g∈W ∥x-gy∥ 2 .
(2.14)

5.

For all t, s > 0, the Dunkl heat kernel satisfies the semi-group property

∀ x, y ∈ R d , p k (t + s, x, y) = ∫ R d p k (t, x, z)p k (s, y, z)ω k (z)dz. (2.15)

The harmonic kernel and ∆ k -subharmonic functions

For r > 0 and x, y ∈ R d , let h k (r, x, y) be the harmonic kernel defined by (1.11). In the classical case (i.e. k = 0), we have µ y = δ y (the Dirac measure at y) and then h 0 (r, x, y) = 1 [0,r] (∥x -y∥) = 1 B(x,r) (y). This implies, in particular, that the Dunklvolume operator defined by (1.10) generalizes the usual one.

The harmonic kernel has the following properties (see [START_REF] Gallardo | A new mean value property for harmonic functions relative to the Dunkl-Laplacian operator and applications[END_REF]):

1) For all r > 0 and x, y ∈ R d , 0 ≤ h k (r, x, y) ≤ 1.

2) For all fixed x, y ∈ R d , the function r -→ h k (r, x, y) is right-continuous and nondecreasing.

3) Let r > 0 and

x ∈ R d . If k(α) > 0 for every α ∈ R, then supp h k (r, x, . ) = B W (x, r) := ∪ g∈W B(gx, r)
and if the function k vanishes somewhere then

B(x, r) ⊂ supp h k (r, x, . ) ⊂ B W (x, r)
(see [START_REF] Gallardo | A new mean value property for harmonic functions relative to the Dunkl-Laplacian operator and applications[END_REF] and [START_REF] Gallardo | Support properties of the intertwining and the mean value operators in Dunkl theory[END_REF]).

4)

For all r > 0 and x, y ∈ R d , we have

h k (r, x, y) = h k (r, y, x).
(2.16)

5)

Let r > 0 and x, y ∈ R d . Then, for all g ∈ W , we have

h k (r, gx, gy) = h k (r, x, y) and h k (r, gx, y) = h k (r, x, g -1 y).
(2.17)

6)

For all r > 0 and x ∈ R d , we have

∥h k (r, x, .)∥ k,1 := ∫ R d h k (r, x, y)ω k (y)dy = m k (B(0, r)) = d k r d+2γ d + 2γ , ( 2.18)
where we recall that dm k (y) = ω k (y)dy and d k is the constant

d k = ∫ S d-1 ω k (ξ)dσ(ξ) = c k 2 d/2+γ-1 Γ(d/2 + γ) . (2.19)
Here dσ(ξ) is the surface measure of the unit sphere S d-1 of R d .

Finally, we recall that • a function u of class C 2 on Ω is D-subharmonic in the sense of (1.10) if and only if ∆ k u ≥ 0 on Ω (see [START_REF] Gallardo | Newtonian Potentials and subharmonic functions associated to root systems[END_REF]).

• if u is D-subharmonic on Ω, then uω k ∈ L 1 loc (Ω) (that is u ∈ L 1 k,loc (Ω)
) and its distributional Dunkl-Laplacian ∆ k (uω k ) is a nonnegative distribution on Ω in the sense that for any nonnegative function ϕ ∈ D(Ω) we have

⟨∆ k (uω k ), ϕ⟩ := ⟨uω k , ∆ k ϕ⟩ = ∫ R d u(x)∆ k ϕ(x)ω k (x)dx ≥ 0.
(2.20)

The nonnegative distribution ∆ k (uω k ) is then a nonnegative Radon measure on Ω called the ∆ k -Riesz measure of the D-subharmonic function u (see [START_REF] Gallardo | Newtonian Potentials and subharmonic functions associated to root systems[END_REF]). In particular, if u ∈ C 2 (Ω) its ∆ k -Riesz measure is equal to ∆ k u(x)ω k (x)dx.

The ∆ k -Riesz kernel

In this section, we will study some properties of the ∆ k -Riesz kernel. Recalling that for x, y ∈ R d and 0 < β < d + 2γ, the ∆ k -Riesz kernel is defined by

R k,β (x, y) := 1 Γ(β/2) ∫ +∞ 0 t β 2 -1 p k (t, x, y)dt.
(3.1)

Remark 3.1 1) Since the Dunkl heat kernel is positive, we have 0 < R k,β (x, y) ≤ +∞ for all x, y ∈ R d . 2) Let x ∈ R d be fixed. From (2.14), we can see that if y / ∈ R d \W.x, then for any β ∈] -∞, d + 2γ[ the function t → t β 2 -1 p k (t, x, y) is integrable on ]0, +∞[. Thus, using the properties of the Gamma function, the function y → 1 Γ(β/2) ∫ +∞ 0 t β 2 -1 p k (t, x, y)dt is well defined on R d \ W.x whenever β ∈] -∞, d + 2γ[\ -2N.
In this case, we will continue denoting it y → R k,β (x, y).

In the following result, we will show that the ∆ k -Riesz kernel can be expressed in terms of the harmonic kernel. This new formula will be a crucial tool in the sequel of the paper.

Proposition 3.1 For every x, y ∈ R d , we have R k,β (x, y) = κ ∫ R d ( ∥x∥ 2 + ∥y∥ 2 -2 ⟨x, z⟩ ) β-(d+2γ) 2 dµ y (z) (3.2) = κ d + 2γ -β ∫ +∞ 0 t β-d-2γ h k (t, x, y) dt t , ( 3.3)
where

κ = κ(d, γ, β) = 2 1-β Γ( d+2γ-β 2 ) d k Γ(β/2)Γ(d/2 + γ) = 2 d 2 +γ-β Γ( d+2γ-β 2 ) c k Γ(β/2) , ( 3.4)
c k and d k being the constants given by (1.7) and (2.19) respectively.

Proof: Using the change of variables 1/4t ↔ t, the relation (2.10) can be rewritten

R k,β (x, y) = 2 d 2 +γ-β Γ(β/2)c k ∫ +∞ 0 t d+2γ-β 2 -1 ∫ R d e -t(∥x∥ 2 +∥y∥ 2 -2 ⟨x,z⟩) dµ x (z)dt.
Now, by Fubuni's theorem and the identity

∀ a ≥ 0, ∀ θ > 0, a -θ/2 = 1 Γ(θ/2) ∫ +∞ 0 s θ 2 -1 e -sa ds
(notice that if we take a = 0, the both terms are equal +∞), we deduce that (3.2) holds.

• Let us now prove (3.3). Starting from (3.2) and applying again Fubini's theorem, we get

R k,β (x, y) = κ ∫ R d ( ∥x∥ 2 + ∥y∥ 2 -2 ⟨x, z⟩ ) β-(d+2γ) 2 dµ y (z) = κ d + 2γ -β ∫ R d ∫ +∞ √ ∥x∥ 2 +∥y∥ 2 -2 ⟨x,z⟩ t β-d-2γ dt t dµ y (z) = κ d + 2γ -β ∫ +∞ 0 t β-d-2γ (∫ R d 1 [0,t] ( √ ∥x∥ 2 + ∥y∥ 2 -2 ⟨x, z⟩)dµ y (z) ) dt t = κ d + 2γ -β ∫ +∞ 0 t β-d-2γ h k (t, x, y) dt t .
This gives the desired relation. [START_REF] Landkof | Foundations of Modern Potential Theory[END_REF]).

Example 3.1 1) When k = 0, as µ y = δ y we have R 0,β (x, y) = κ(d, 0, β)∥x -y∥ β-d the classical Riesz kernel (see
2) Since µ 0 = δ 0 , for any choice of the Coxeter-Weyl group and of a nonnegative multiplicity function, we have

R k,β (x, 0) = κ(d, γ, β)∥x∥ β-d-2γ . 3) We consider R d (d ≥ 1) with the root system R m := {±e 1 , . . . , ±e m }, where m is a fixed integer in {1, . . . , d} and (e j ) 1≤j≤d is the canonical basis of R d . For ξ ∈ R d , we will denote ξ = (ξ (m) , ξ ′ ) ∈ R m × R d-m . We note that the Coxeter-Weyl group is W = Z m 2 and the Z m 2 -orbit of a point ξ ∈ R d is as follows Z m 2 .ξ := { ε.ξ := (ε 1 ξ 1 , . . . , ε m ξ m , ξ ′ ), ε = (ε i ) 1≤i≤m ∈ {±1} m } .
The multiplicity function can be represented by the m-multidimensional parameter k = (k 1 , . . . , k m ) with k j = k(e j ) > 0. Moreover, the Rösler measure is of the form

µ y = µ (y (m) ,y ′ ) = µ y 1 ⊗ • • • ⊗ µ ym ⊗ δ y ′ with µ y i the Z 2 -Rösler measure at point y i . If y i = 0, we know that µ 0 = δ 0 and if y i ̸ = 0, we have ⟨µ y i , f ⟩ := ∫ 1 -1 f (ty i )ϕ k i (t)dt, f ∈ C(R),
where ϕ k i is the Z 2 -Dunkl density function of parameter k i given by (see [START_REF] Dunkl | Integral kernels with reflection group invariance[END_REF] or [START_REF] Rösler | Dunkl Operators: Theory and Applications[END_REF] p.104)

ϕ k i (t) := Γ(k i + 1/2) √ πΓ(k i ) (1 -t) k i -1 (1 + t) k i 1 [-1,1] (t). (3.5) In this case, the ∆ k -Riesz kernel is of the form R k,β (x, y) = κ ∫ [-1,1] m ( ∥x (m) ∥ 2 + ∥y (m) ∥ 2 -2 m ∑ j=1 t j x j y j + ∥x ′ -y ′ ∥ 2 ) β-d-2γ 2 × m ∏ i=1 ϕ k i (t i )dt 1 . . . dt m . (3.6) Proposition 3.2 Suppose that γ > 0. Let 0 < β < d + 2γ and x, y ∈ R d . 1) If y / ∈ W.x, then R k,β (x, y) < +∞. 2) Assume that x ∈ R d \ ∪ α∈R H α . Then R k,β (x, x) = +∞ if and only if d ≥ β. 3) If x ∈ ∪ α∈R H α and β ≤ d, then R k,β (x, x) = +∞.
Proof: At first we note that

∀ x, y ∈ R d , ∀ t > 0, t β 2 -1 p k (t, x, y) ≤ Ct β-d-2γ 2 -1
Hence, as

β < d + 2γ, the function t → t β 2 -1 p k (t, x, y) is integrable on [1, +∞[ for every x, y ∈ R d .

1)

We obtain the result by using (2.14).

2)

Fix x ∈ R d such that x is not in any hyperplane H α , α ∈ R (i.e.
x lives in a Weyl chamber). We will use the following short-time asymptotic result of the Dunkl type heat kernel which has been established in ([24], Corollary 2): Let C be a fixed Weyl chamber. If x, y ∈ C, then

p k (t, x, y) ∼ t→0 ( ω k (x)ω k (y) ) -1/2 (4πt) -d/2 e -∥x-y∥ 2 4t . ( 3.7)
Taking y = x, we deduce that the function

t → t β 2 -1 p k (t, x, x) is not integrable near 0 if and only if d ≥ β. 3) Let x ∈ H α for some α ∈ R. One can see that the function ψ : ξ -→ R k,β (ξ, ξ) is the increasing limit of the sequence of continuous functions ξ -→ ∫ n 1/n t β 2 -1 p k (t, ξ, ξ)dt. This implies that ψ is lower semi-continuous on R d . Consequently, when β ≤ d we have R k,β (x, x) = lim inf ξ→x R k,β (ξ, ξ) = +∞.
As already mentioned, for g ̸ = id, it is much more difficult to see if R k,β (x, gx) is finite or infinite. This new phenomena will be illustrated by the following complete characterization of the singularities of the ∆ k -Riesz kernel in the case of the Z m 2 -Coxeter-Weyl group acting on R d . More precisely, we have:

Proposition 3.3 Let x ∈ R d \{0}.
Using the same notations of Example 3.1, 3), denoting H i the hyperplane orthogonal to e i and recalling ε.

x = (ε 1 x 1 , . . . , ε m x m , x ′ ) ∈ Z m 2 .
x, we have

1. If x ∈ ∩ m i=1 H i , then x = ε.x and R k,β (x, x) = +∞. 2. Assume that x / ∈ ∩ m i=1 H i . Set A := {i ∈ {1, . . . , m}, x i ̸ = 0} and ε (n) .x = (ε 1 x 1 , . . . , ε m x m , x ′ ) the point of Z m 2 -orbit of x such that {j ∈ A, ε j = 1} = n i.e the point ε (n) .
x has exactly n among the nonzero coordinates (x j ) j∈A that have not been changed under the action of Z m 2 . Then,

R k,β (x, ε (n) .x) = +∞ ⇐⇒ d ≥ 2 ( |A| -n + ∑ j∈A k j -γ ) + β. (3.8) 3. Assume that x / ∈ ∪ m i=1 H i . Then, R k,β (x, ε (n) .x) = +∞ ⇐⇒ d ≥ 2(m -n) + β. (3.9)
In this case, we have

∑ m n=max(0,⌊m-d 2 + β 2 ⌋) ( m n ) singularities living in R d \ ∪ m i=1 H i .
Proof: For abbreviation, we will use the following constants

C 1 := 2 β-d-2γ 2 κ, C(k) := Γ(k + 1/2) √ πΓ(k) . (3.10) From (3.6), it is easy to see that R k,β (x, ε.x) = C 1 ∫ [-1,1] m ( m ∑ j=1 (1 -ε j t j )x 2 j ) β-d-2γ 2 m ∏ j=1 ϕ k j (t j ) ⊗ m j=1 dt j . (3.11) 1) Clearly, from (3.11), the condition x ∈ ∩ m i=1 H i i.e. x (m) = 0 implies that x = ε.x = (0, x ′ ) and R k,β (x, ε.x) = +∞. 2) Suppose that x / ∈ ∩ m i=1 H i .
Using the notations of the Proposition, Fubini's theorem and the fact that ϕ k j are probability densities, (3.11) can be written in the following form

R k,β (x, ε.x) = C 1 ∫ [-1,1] |A| ( ∑ j∈A (1 -ε j t j )x 2 j ) β-d-2γ 2 ∏ j∈A ϕ k j (t j ) ⊗ j∈A dt j .
(3.12)

We will distinguish two cases: First case |A| = 1. Let i ∈ {1, . . . , m} such that x i ̸ = 0. In this case, using (3.5) and (3.10), we deduce that (3.12) takes the form

R k,β (x, ε.x) = C 1 ∫ 1 -1 ( (1 -ε i s)x 2 i ) β-d-2γ 2 ϕ k i (s)ds = C(k i )C 1 |x i | β-d-2γ ∫ 1 -1 (1 -ε i s) β-d-2γ 2 (1 -s) k i -1 (1 + s) k i ds.
• If ε i = 1, then according to our notations, we have n = |A| = 1, ε.x = ε (1) .x = x and

R k,β (x, ε (1) .x) = C(k i )C 1 |x i | β-d-2γ ∫ 1 -1 (1 -s) k i + β-d-2γ 2 -1 (1 + s) k i ds.
Consequently, R k,β (x, ε (1) .x) = +∞ if and only if d ≥ β + 2k i -2γ. Then, the result is proved in this case.

• When ε i = -1, we have n = 0, ε.x = ε (0) .x and R k,β (x, ε (0) .x) = C(k i )|x i | β-d-2γ ∫ 1 -1 (1 + s) k i + β-d-2γ 2 (1 -s) k i -1 ds.
Thus, as

k i > 0 we have R k,β (x, ε (0) .x) = +∞ if and only if d ≥ 2(1 + k i -γ) + β.
Second case |A| = r ≥ 2. Using (3.12) and the change of variables t j ↔ 1 -ε j t j , we obtain

R k,β (x, ε.x) = C 1 ∫ ]0,2[ |A| ( ∑ j∈A t j x 2 j ) β-d-2γ 2 ∏ j∈A ϕ k j (ε j -ε j t j ) ⊗ j∈A dt j = C 1 ∫ ]0,2[ |A| ∩Br + C 1 ∫ ]0,2[ |A| \Br = C 1 I(x, ε.x) + C 1 J(x, ε.x),
where B r is the open unit ball in

R |A| = R r .
The singularities of these integrals being at point 0 and thus it is clear that J(x, ε.x) < +∞. Thus, we need to know when the integral I(x, ε.x) diverges. To do this, we will identify (t j ) j∈A with v = (v 1 , . . . , v r ) ∈ R r and use the spherical coordinates in R r :

ρ = ∥v∥, v 1 = ρa 1 , . . . , v r-1 = ρa r-1 and v r = ρa r ,
where

a 1 = cos θ 1 , . . . , a r-1 = r-2 ∏ i=1 sin θ i cos θ r-1 , a r = r-1 ∏ i=1 sin θ i .
Notice that all a j are positive.

I(x, ε.x) = ∫ S r-1 + ψ(a (r) , x (r) ) ( ∫ 1 0 ∏ j∈A ϕ k j (ε j -ε j a j ρ)ρ r+ β-d-2γ 2 -1 dρ ) dσ r (a (r) ), (3.13)
where S r-1

+ :=]0, 2[ r ∩S r-1
, dσ r is the surface measure of the unit sphere S r-1 of R r , a (r) = (a j ) j∈A , x (r) = (x j ) j∈A and ψ(a (r) , x (r)

) := ( ∑ j∈A a j x 2 j ) β-d-2γ 2 .
We have

ϕ k j (ε j -ε j a j ρ) = C(k j )(1 -ε j + ε j a j ρ) k j -1 (1 + ε j -ε j a j ρ) k j .
Hence,

ϕ k j (ε j -ε j a j ρ) =      C(k j )a k j -1 j ρ k j -1 (2 -a j ρ) k j , if ε j = 1 C(k j )a k j j ρ k j (2 -a j ρ) k j -1 , if ε j = -1. (3.14) Define A 1 := { j ∈ A, ε j = 1 } , A 2 = A\A 1 .
According to our notations, we have

|A 1 | = |{j, ε j = 1}| = n.
Then, from (3.13), (3.14) and recalling the definition of the vector ε (n) .x, we deduce that

I(x, ε (n) .x) = ∫ S r-1 + ψ(a (r) , x (r) ) ( ∫ 1 0 f (a (r) , ρ)ρ λ+r+ β-d-2γ 2 -1 dρ ) dσ r (a (r) ), (3.15)
with

f (a (r) , ρ) := ∏ j∈A 1 C(k j )a k j -1 j (2 -a j ρ) k j ∏ j∈A 2 C(k j )a k j j (2 -a j ρ) k j -1 .
and λ :=

∑ j∈A 1 (k j -1) + ∑ j∈A 2 k j = ∑ j∈A k j -n.
The function ρ -→ f (a (r) , ρ) is continuous and does not vanish on the compact set [0, 1]. So that the singularity in the dρ-integral is only in the term of

ρ λ+r+ β-d-2γ 2 -1 = ρ ( ∑ j∈A k j )-n+r+ β-d-2γ 2 -1 .
Finally, we conclude that

R k,β (x, ε (n) .x) = +∞ ⇔ I(x, ε (n) .x) = +∞ ⇔ d ≥ 2(|A| -n + ∑ j∈A k j -γ) + β.
This completes the proof of the assertion 2).

3) When x / ∈ ∪ m i=1 H i , we have A = {1, . . . , m} and then the result is a particular case of the statement 2). Proposition 3. [START_REF]Two results on the Dunkl maximal function[END_REF] The Riesz kernel R k,β (., .) satisfies the following properties 1) For every x, y ∈ R d and g ∈ W , we have

R k,β (x, y) = R k,β (y, x), R k,β (gx, y) = R k,β (x, g -1 y).
(3.16)

2) Let β, θ > 0 such that β + θ < d + 2γ. Then we have the following generalized Riesz composition formula ∫

R d R k,β (x, z)R k,θ (y, z)ω k (z)dz = R k,β+θ (x, y).
(3.17)

3) Let x ∈ R d . Then, for every y ∈ R d \W.x, we have κ min g∈W ( ∥x -gy∥ β-d-2γ ) ≤ R k,β (x, y) ≤ κ max g∈W ( ∥x -gy∥ β-d-2γ ) (3.18) 4) Let y ∈ R d . Then, the function x → R k,β (x, y) is -lower semi-continuous (l.s.c.) on R d . -of class C ∞ on R d \W.
x and we have

∂ j R k,β (x, y) = (β -d -2γ)κ ∫ R d (x j -z j ) ( ∥x∥ 2 + ∥y∥ 2 -2 ⟨x, z⟩ ) β-2-d-2γ
2 dµ y (z).

(

3.19)

Proof: 1) The result follows from (3.3), (2.16) and (2.17).

2) The result follows from the Fubini's theorem and the semi-group property of the Dunklheat kernel (2.15).

3) Let y ∈ R d . From (1.4) for any z ∈ supp µ y , we can write z = ∑ g∈W λ g (z)gy, where λ g (z) ∈ [0, 1] are such that ∑ g∈W λ g (z) = 1. Then, we have

∥x∥ 2 + ∥y∥ 2 -2 ⟨x, z⟩ = ∑ g∈W λ g (z)∥x -gy∥ 2 . (3.20) As ψ : t -→ t β-d-2γ 2
is a convex function on ]0, +∞[, by (3.20) we have

( ∥x∥ 2 + ∥y∥ 2 -2 ⟨x, z⟩ ) β-d-2γ 2 = ( ∑ g∈W λ g (z)∥x -gy∥ 2 ) β-d-2γ 2 ≤ max g∈W ( ∥x -gy∥ β-d-2γ
) .

This implies the right inequality. Again by convexity, Jensen's inequality and (3.20), we get

R k,β (x, y) ≥ κ (∫ R d (∥x∥ 2 + ∥y∥ 2 -2 ⟨x, z⟩)dµ y (z) ) β-d-2γ 2 ≥ κ   ∑ g∈W (∫ R d λ g (z)dµ y (z) ) ∥x -gy∥ 2   β-d-2γ 2 ≥ κ ( max g∈W ∥x -gy∥ 2 ) β-d-2γ 2 = κ min g∈W ( ∥x -gy∥ β-(d+2γ)
) ,

where in the last line we have used the fact that ψ is a decreasing function.

4)

The function x → R k,β (x, y) is l.s.c. on R d as being the increasing limit of the sequence (f n ) of continuous functions defined by

f n : x → ∫ n 1/n t β 2 -1 p k (t, x, y)dt. Fix y ∈ R d .
Using the fact that µ y is with compact support and the fact that the function

(x, z) -→ ( ∥x∥ 2 + ∥y∥ 2 -2 ⟨x, z⟩ ) β-d-2γ 2 is of class C ∞ on R d \W.y × R d ,
∀ x ∈ R d , ∥R k,β (x, .)∥ L p k (B(0,R)) ≤ C. (3.21)
In particular, for every

x ∈ R d , R k,β (x, .) is in L p k,loc (R d ).
Proof: By Jensen's inequality and (3.2), we have

(R k,β (x, y)) p ≤ κ p ∫ R d ( ∥x∥ 2 + ∥y∥ 2 -2 ⟨x, z⟩ ) p(β-d-2γ) 2 dµ y (z).
Using the same idea as in the proof of (3.3), we can write the previous inequality as follows

(R k,β (x, y)) p ≤ κ p p(d + 2γ -β) ∫ +∞ 0 t p(β-d-2γ) h k (t, x, y) dt t = C 1 ∫ 1 0 t p(β-d-2γ) h k (t, x, y) dt t + C 1 ∫ +∞ 1 t p(β-d-2γ) h k (t, x, y) dt t ≤ C 1 ∫ 1 0 t p(β-d-2γ) h k (t, x, y) dt t + C 1 p(d + 2γ -β) ,
where C 1 = κ p p(d+2γ-β) and we have used the fact that h k (t, x, y) ≤ 1 in the last inequality. Let then R > 0. From (2.18), Fubini's theorem and our hypothesis, we deduce that ∫ B(0,R)

∫ 1 0 t p(β-d-2γ) h k (t, x, y) dt t ω k (y)dy ≤ d k d + 2γ ∫ 1 0 t p(β-d-2γ) t d+2γ dt t := C 2 < +∞.
This proves the desired inequality where we can take

C = ( C 1 C 2 + C 1 m k [B(0, R)] p(d + 2γ -β) ) 1/p . Proposition 3.6 Let 0 < β < d + 2γ and x 0 ∈ R d . Then, the function R k,β (x 0 , .) is i) D-superharmonic on R d when β ≥ 2, ii) D-harmonic on R d \W.x 0 when β = 2,
iii) D-subharmonic on R d \W.x 0 when β ≤ 2

Proof: The case β = 2 (i.e. the case of the Dunkl-Newton kernel) has been done in [START_REF] Gallardo | Newtonian Potentials and subharmonic functions associated to root systems[END_REF]. So, we will deal with the case β ̸ = 2. i) Suppose that β > 2. We consider the function S x 0 ,β,r

S x 0 ,β,r (x) := 1 Γ(β/2) ∫ +∞ r t β 2 -1 p k (t, x 0 , x)dt.
By the monotone convergence theorem, we see that the function R k,β (x 0 , .) is the pointwise increasing limit of the sequence

( S x 0 ,β, 1 n ) n
. Hence, by Proposition 3.3 in [START_REF] Gallardo | Newtonian Potentials and subharmonic functions associated to root systems[END_REF], it suffices to prove that for every r > 0, S x 0 ,β,r is D-superharmonic on R d . To do this, we have only to show that S x 0 ,β,r is of class C 2 on R d and ∆ k S x 0 ,β,r ≤ 0 on R d (see [START_REF] Gallardo | Newtonian Potentials and subharmonic functions associated to root systems[END_REF], Proposition 4.1).

The function p k (t, x 0 , .) is of class C ∞ on R d and we can differentiate under the integral sign in the relation (2.10) to obtain

∂ j p k (t, x 0 , .)(x) = - 1 2t 1 (2t) d 2 +γ c k ∫ R d (x j -z j )e -1 4t (∥x∥ 2 +∥x 0 ∥ 2 -2 ⟨x,z⟩) dµ x 0 (z) (3.22)
and

∂ i ∂ j p k (t, x 0 , .)(x) = -δ ij 1 2t p k (t, x 0 , x) + 1 4t 2 1 (2t) d 2 +γ c k ∫ R d (x j -z j )(x i -z i )e -1 4t (∥x∥ 2 +∥x 0 ∥ 2 -2 ⟨x,z⟩) dµ x 0 (z), (3.23)
where δ ij is the Kronecker symbol.

Using the fact that supp µ x 0 ⊂ B(0, ∥x 0 ∥), we deduce from (3.22) and (3.23) that

|∂ j p k (t, x 0 , .)(x)| ≤ ∥x∥ + ∥x 0 ∥ (2t) 1+ d 2 +γ c k , |∂ i ∂ j p k (t, x 0 , .)(x)| ≤ 1 (2t) 1+ d 2 +γ c k + (∥x∥ + ∥x 0 ∥) 2 (2t) 2+ d 2 +γ c k .
Let R > 0. The previous inequalities and the differentiation theorem under the integral sign imply that S x 0 ,β,r is of class C 2 on the open ball • B(0, R) and as x → p k (t, x 0 , x) is a solution of the Dunkl-heat equation (1.9), we deduce that (3.22), (3.23) and (3.20), we can see that

∀ x ∈ • B(0, R), ∆ k S x 0 ,β,r (x) = 1 Γ(β/2) ∫ +∞ r t β 2 -1 ∆ k (p k (t, x 0 , .)) (x)dt = 1 Γ(β/2) ∫ +∞ r t β 2 -1 ∂ t p k (t, x 0 , x)dt = - r β 2 -1 Γ(β/2) p k (r, x 0 , x) - β -2 2Γ(β/2) ∫ +∞ r t β 2 -2 p k (t, x 0 , x)dt < 0. Therefore, S x 0 ,β,r is D-superharmonic on • B(0, R). As R > 0 is arbitrary, we conclude that S x 0 ,β,r is D-superharmonic on R d as desired. iii) Let β ∈]0, 2[. Using
|∂ j p k (t, x 0 , .)(x)| ≤ ∥x∥ + ∥x 0 ∥ (2t) 1+ d 2 +γ c k e - min g∈W (∥x-gx 0 ∥ 2 ) 4t
,

|∂ i ∂ j p k (t, x 0 , .)(x)| ≤ ( 1 (2t) 1+ d 2 +γ c k + (∥x∥ + ∥x 0 ∥) 2 (2t) 2+ d 2 +γ c k ) e - min g∈W (∥x-gx 0 ∥ 2 ) 4t
.

Fix an arbitrary open Dunkl ball O

W (a, R) := ∪ g∈W • B(ga, R) such that its closure is contained in R d \W.x 0 .
The previous inequalities imply that we can differentiate with respect to x ∈ O W (a, R) under the integral sign in the relation (3.1). Furthermore, using the heat equation (1.9) and integrating by parts, we obtain

∀ x ∈ O W (a, R), ∆ k (R k,β (x 0 , .)) (x) = 1 Γ(β/2) ∫ +∞ 0 t β 2 -1 ∂ t p k (t, x 0 , x)dt = - β -2 2Γ(β/2) ∫ +∞ 0 t β 2 -2 p k (t, x 0 , x)dt ≥ 0.
According to Remark 3.1-2), the above relation can be written as

∀ x ∈ O W (a, R), ∆ k (R k,β (x 0 , .)) (x) = -R k,β-2 (x 0 , x) ≥ 0. (3.24)
Therefore, the function R k,β (x 0 , .) is D-subharmonic on O W (a, R) and so on R d \W.x 0 .

Proposition 3.7 Let β ∈]0, d+2γ[ and x 0 ∈ R d . Then, the function x → R k,β (x 0 , x)ω k (x)
defines a tempered distribution and we have

F k (R k,β (x 0 , .)ω k ) = E k (-ix 0 , .)∥.∥ -β ω k in S ′ (R d ). ( 3

.25)

Proof: Let m ∈ N such that m > d + 2γ. We claim that there exists a constant

C m = C(d, γ, β, m) > 0 such that ∀ x 0 ∈ R d , ∫ R d (1 + ∥x∥ 2 ) -m R k,β (x 0 , x)ω k (x)dx ≤ C m . (3.26) From (3.3), we can write R k,β (x 0 , x) = κ d + 2γ -β ( ∫ 1 0 t β-d-2γ-1 h k (t, x 0 , x)dt + ∫ +∞ 1 t β-d-2γ-1 h k (t, x 0 , x)dt ) := A(x 0 , x) + B(x 0 , x).
• Using Fubini's theorem and the relation (2.18), for any

x 0 ∈ R d we obtain ∫ R d (1 + ∥x∥ 2 ) -m A(x 0 , x)ω k (x)dx ≤ ∫ R d A(x 0 , x)ω k (x)dx = κ d + 2γ -β ∫ 1 0 t β-d-2γ-1 ∥h k (t, x 0 , .)∥ k,1 dt = d k κ β(d + 2γ)(d + 2γ -β) := C 1,m .
• Now, using the inequality h k (t, x 0 , x) ≤ 1, we deduce that

∀ x 0 ∈ R d , B(x 0 , x) ≤ κ (d + 2γ -β) 2 .

This relation and the choice of m imply that

∀ x 0 ∈ R d , ∫ R d (1 + ∥x∥ 2 ) -m B(x 0 , x)ω k (x)dx ≤ κ (d + 2γ -β) 2 ∫ R d (1 + ∥x∥ 2 ) -m ω k (x)dx := C 2,m < +∞.
This proves (3.26) and this implies that the function R k,β (x 0 , .)ω k defines a tempered distribution (see [START_REF] Schwartz | Théorie des distributions[END_REF], Theorem VII, p. 242).

Let us now prove (3.25). For ϕ ∈ S(R d ), we have

⟨F k (R k,β (x 0 , .)ω k ) , ϕ⟩ = 1 Γ(β/2) ∫ R d (∫ +∞ 0 t β 2 -1 p k (t, x 0 , x)dt ) F k (ϕ)(x)ω k (x)dx.
Multiplying and dividing by (1 + ∥x∥ 2 ) m (the integer m is chosen as above) and using the fact that F k (ϕ) ∈ S(R d ), we see that we can use Fubini's theorem in the above relation. Moreover, from (2.4) and (2.12), we obtain

⟨F k (R k,β (x 0 , .)ω k ) , ϕ⟩ = 1 Γ(β/2) ∫ +∞ 0 t β 2 -1 (∫ R d F k (p k (t, x 0 , .))(x)ϕ(x)ω k (x)dx ) dt = 1 Γ(β/2) ∫ +∞ 0 t β 2 -1 (∫ R d E k (-ix 0 , x)e -t∥x∥ 2 ϕ(x)ω k (x)dx ) dt.
Applying again Fubini's theorem, we deduce that

⟨F k (R k,β (x 0 , .)ω k ) , ϕ⟩ = ∫ R d E k (-ix 0 , x)∥x∥ -β ϕ(x)ω k (x)dx.
This completes the proof.

Corollary 3.1 For every x 0 ∈ R d , we have

lim β→0 R k,β (x 0 , .)ω k = δ x 0 in S ′ (R d ). ( 3

.27)

Proof: We can see that for every ξ ∈ R d ,

∥ξ∥ -β ≤ 1 R d \B(0,1) (ξ) + ∥ξ∥ -d-2γ 1 B(0,1) (ξ).
Consequently, we can use the dominated convergence theorem to obtain from (3.25)

lim β→0 F k (R k,β (x 0 , .)ω k ) = E k (-ix 0 , .)ω k = F k (δ x 0 ) in S ′ (R d ).
Thus, we deduce the result by using the properties of the Dunkl transform on S ′ (R d ).

From the formula (3.24), we see that the ∆ k -Riesz measure related to the D-subharmonic function R k,β (x 0 , .), β < 2, is given by -R k,β-2 (x 0 , x)ω k (x)dx. In the following result, we will compute the ∆ k -Riesz measure of the D-superharmonic function

R k,β (x 0 , .) with β ∈ [2, d + 2γ[. Corollary 3.2 Let 2 ≤ β < d + 2γ and x 0 ∈ R d . If m ∈ [1, β/2] be an integer, then the function x → R k,β (x 0 , x) satisfies (-∆ k ) m (R k,β (x 0 , .)ω k ) =    R k,β-2m (x 0 , .)ω k in S ′ (R d ) if β > 2m, δ x 0 in S ′ (R d ) if β = 2m, (3.28)
where δ x 0 is the Dirac measure at x 0 .

Proof: At first, we remark that if U ∈ S ′ (R d ), then

F k (∆ k U ) = -∥.∥ 2 F k (U ), (3.29)
as easily follows from the relation

∆ k F k (f ) = -F k (∥.∥ 2 f ) for all f ∈ S(R d ).
From (3.29) and (3.25), we obtain

F k ( (-∆ k ) m (R k,β (x 0 , .)ω k ) ) = E k (-ix 0 , .)∥.∥ β-2m ω k =    F k ( R k,β-2m (x 0 , .)ω k ) in S ′ (R d ) if β > 2m, F k (δ x 0 ) in S ′ (R d ) if β = 2m.
Hence, we deduce the result by the fact that

F k is a topological isomorphism of S ′ (R d ) onto itself. Remark 3.2 Let 1 ≤ m < γ + d/2 an integer. Taking x 0 = 0 in (3.28), we deduce that the function S : y → R k,2m (0, y)ω k (y) = κ∥y∥ 2m-d-2γ ω k (y) is the fundamental solution of the Dunkl-polylaplacian of order m (-∆ k ) m i.e. (-∆ k ) m S = δ 0 in S ′ (R d ).

Riesz potentials of Radon measures

The sets M(R d

I k,β [µ](x) = ∫ R d R k,β (x, y)dµ(y), x ∈ R d . (4.1) Proposition 4.1 Let µ ∈ M + (R d ) and β ∈]0, d + 2γ[. 1. If µ is bounded, then I k,β [µ] ∈ L p k,loc (R d ) whenever p ∈ [1, d+2γ d+2γ-β [. In particular, I k,β [µ] is finite a.e. in R d .

The following statements are equivalent

i) I k,β [µ] is finite a.e. in R d , ii) the measure µ satisfies ∫ R d (1 + ∥y∥) β-d-2γ dµ(y) < +∞, (4.2)
iii)

I k,β [µ](x 0 ) < +∞ for some x 0 ∈ R d . If ii) holds, then I k,β [µ] ∈ L 1 k,loc (R d ).
Proof: 1) Assume that µ is a probability measure on R d . Let p as in the proposition and R > 0. Using respectively (4.1), Jensen's inequality, Fubini's theorem, the fact that the Riesz kernel is symmetric and (3.21), we get ∫ B(0,R)

(I k,β [µ](x)) p (x)ω k (x)dx ≤ ∫ B(0,R) ( ∫ R d (R k,β (x, y)) p dµ(y) ) ω k (x)dx = ∫ R d ( ∫ B(0,R) (R k,β (x, y)) p ω k (x)dx ) dµ(y) ≤ C < +∞,
where C is the constant in (3.21).

2) ii) ⇒ i) Assume that the condition (4.2) holds. We will prove that

x → I k,β [µ](x) is in L 1 k,loc (R d ). Let R > 1
. By Fubini's theorem, we have

A R := ∫ B(0,R) I k,β [µ](x)ω k (x)dx = ∫ R d ∫ B(0,R) R k,β (x, y)ω k (x)dxdµ(y) = ∫ ∥y∥≤2R ∫ B(0,R) R k,β (x, y)ω k (x)dxdµ(y) + ∫ ∥y∥>2R ∫ B(0,R) R k,β (x, y)ω k (x)dxdµ(y) := A 1,R + A 2,R .
Applying the assertion 1) with the finite measure µ |B(0,R) , we get A 1,R < +∞. Now, from (3.18) we deduce that

A 2,R ≤ κ ∫ ∥y∥>2R ∫ B(0,R) max g∈W ( ∥x -gy∥ β-d-2γ
) ω k (x)dxdµ(y).

But, for every x ∈ B(0, R) and every y ∈ R d \B(0, 2R), we have ∥x-gy∥ ≥ ∥y∥-∥x∥ ≥ ∥y∥ 2 . Moreover, as R > 1, we see that ∥y∥ ≥ 1 2 (1 + ∥y∥) whenever ∥y∥ ≥ 2R. In other words, the inequality max

g∈W ( ∥x -gy∥ β-d-2γ ) ≤ 4 β-d-2γ (1 + ∥y∥) β-d-2γ
holds for every x ∈ B(0, R) and every y ∈ R d \ B(0, 2R). Hence, by our hypothesis we conclude that

A 2,R ≤ 4 β-d-2γ κ m k [B(0, R)] ∫ ∥y∥≥2R (1 + ∥y∥) β-d-2γ dµ(y) < +∞ and thus the function x → I k,β [µ](x)ω k (x) is locally integrable on R d . In particular, I k,β [µ](x) < +∞ a.e. on R d . i) ⇒ iii) It is obvious. iii) ⇒ ii) Let x 0 ∈ R d such that I k,β [µ](x 0 ) < +∞. From (3.
18), we can see that

I k,β [µ](x 0 ) ≥ κ ∫ R d min g∈W ( ∥x 0 -gy∥ β-d-2γ
) dµ(y)

≥ κ ∫ R d (∥x 0 ∥ + ∥y∥) β-d-2γ dµ(y).
I k,β [µ] ≤ 2 d+2γ-β M on R d . (4.6)
Proof: Let x / ∈ W.supp µ and x 0 ∈ W.supp µ such that ∥x -x 0 ∥ = dist(x, W.supp µ). We have

∀ y ∈ supp µ, ∀ g ∈ W, ∥x 0 -gy∥ ≤ ∥x 0 -x∥ + ∥x -gy∥ ≤ 2∥x -gy∥.
Hence, by (3.20) we deduce that

∀ y ∈ supp µ, ∀ z ∈ supp µ y , ∥x 0 ∥ 2 + ∥y∥ 2 -2 ⟨x 0 , z⟩ ≤ 4(∥x∥ 2 + ∥y∥ 2 -2 ⟨x, z⟩).
Now, using (2.10), we obtain

∀ y ∈ supp µ, 4 -d 2 -γ p t/4 (x, y) ≤ p t (x 0 , y).
From (3.1), the above inequality implies that

∀ y ∈ supp µ, 2 -d-2γ+β R k,β (x, y) ≤ R k,β (x 0 , y).
Finally, if we integrate with respect to the measure dµ(y) and use our hypothesis, the inequality (4.6) follows.

In the following result, we will study some continuity properties of the β-∆ k -Riesz potentials:

F n (x) = 1 Γ(β/2) ∫ supp µ ( ∫ n 1/n t β 2 -1 p k (t, x, y)dt ) dµ(y). As t β 2 -1 p k (t, x, y) ≤ 2 -d 2 -γ c -1 k t -β-d-2γ 2 -1
, by the continuity theorem under the integral sign, we see that F n is continuous on R d . Moreover, from the monotone convergence theorem, we deduce that the function I k,β [µ] is l.s.c. on R d as being the pointwise increasing limit of the sequence (F n ).

• Let us prove the second part of 1). Fix a closed ball B(x 0 , R) in R d \ W.supp µ and set η := dist (B(x 0 , R), W.supp µ) > 0.

From (2.14), we deduce that

∀(x, y) ∈ B(x 0 , R) × supp µ, p k (t, x, y) ≤ 1 (2t) d 2 +γ c k e -η 2 4t .
Then, writing

I k,β [µ](x) = 1 Γ(β/2) ∫ supp µ ( ∫ +∞ 0 t β 2 -1 p k (t, x, y)dt ) dµ(y)
and using the continuity theorem under the integral sign, it follows that I k,β [µ] is continuous on B(x 0 , R). As the ball B(x 0 , R) is arbitrary, the result follows.

2) Fix x 0 ∈ W.supp µ and ε > 0. Let (x n ) be a sequence which converges to x 0 . For R > 0 (small), set µ R := µ |B(x 0 ,R) and ν R := µ -µ R . In particular, we note that

I k,β [µ] = I k,β [µ R ] + I k,β [ν R ].
We have

I k,β [µ](x n )-I k,β [µ](x 0 ) ≤ I k,β [µ R ](x n )+I k,β [µ R ](x 0 )+ I k,β [ν R ](x n )-I k,β [ν R ](x 0 ) . (4.7)
• As x 0 / ∈ W.supp ν R , by the assertion 1, the function

I k,β [ν R ] is continuous at x 0 . Hence, there exists N 1 ∈ N such that ∀ n ≥ N 1 , I k,β [ν R ](x n ) -I k,β [ν R ](x 0 ) ≤ ε. (4.8) • For every n, let x ′ n ∈ K := W.supp µ R = W. ( supp µ ∩ B(x 0 , R) ) such that ∥x n -x ′ n ∥ = dist(x n , K) = inf{∥x n -ξ∥, ξ ∈ K}. As x 0 ∈ K, we can see that ∥x n -x 0 ∥ ≥ ∥x n -x ′ n ∥. This implies that ∥x ′ n -x 0 ∥ ≤ ∥x n -x ′ n ∥ + ∥x n -x 0 ∥ ≤ 2∥x n -x 0 ∥
and thus x ′ n -→ x 0 as n → +∞. Using the inequality (4.6), we deduce that

I k,β [µ R ](x n ) ≤ 2 d+2γ-β I k,β [µ R ](x ′ n ) = 2 d+2γ-β ( I k,β [µ](x ′ n ) -I k,β [ν R ](x ′ n )
) . (4.9)

But, x ′ n ∈ W.supp µ and the restriction of I k,β [µ] on W.supp µ is continuous. Thus,

lim n→+∞ I k,β [µ](x ′ n ) = I k,β [µ](x 0 ). Again by continuity of I k,β [ν R ] at x 0 , we also get lim n→+∞ I k,β [ν R ](x ′ n ) = I k,β [ν R ](x 0 ). Therefore, I k,β [µ R ](x ′ n ) -→ I k,β [µ R ](x 0 ) as n → +∞. Let then N 2 ∈ N such that ∀ n ≥ N 2 , I k,β [µ R ](x ′ n ) ≤ I k,β [µ R ](x 0 ) + ε. (4.10)
Finally from (4.7), (4.8), (4.9) and (4.10), the inequality

I k,β [µ](x n ) -I k,β [µ](x 0 ) ≤ ( 2 d+2γ-β + 1 )( ε + I k,β [µ R ](x 0 ) ) (4.11)
holds for every n ≥ max(N 1 , N 2 ) and every R > 0. But, since x 0 ∈ W.supp µ and I k,β [µ] is continuous on W.supp µ, we must have I k,β [µ](x 0 ) < +∞. Furthermore, because x 0 is a singularity of R k,β (x 0 , .), Proposition 4.2 imply that lim

R→0 I k,β [µ R ](x 0 ) = lim R→0 ∫ B(x 0 ,R) R k,β (x 0 , y)dµ(y) = 0. (4.12)
Finally, by the relations (4.11) and (4.12) we deduce that

I k,β [µ] is continuous at x 0 . Theorem 4.1 Let β ∈]0, d + 2γ[ and µ ∈ M + k,β (R d ) with compact support. Then, the function I k,β [µ] is i) D-superharmonic on R d if β ≥ 2, ii) D-harmonic on R d \ W.supp µ if β = 2, iii) D-subharmonic on R d \ W.supp µ if β ≤ 2.
We need the following lemma:

(I k,β [µ]) (x) ≤ I k,β [µ](x). Since I k,β [µ]
is l.s.c and finite a.e., we deduce that the function

I k,β [µ] is D-superharmonic on R d .
ii) If β = 2, we are in the case of the Dunkl-Newton potential and the result has been proved in [START_REF] Gallardo | Newtonian Potentials and subharmonic functions associated to root systems[END_REF].

iii) Let β < 2. From Lemma 4.4, we know that I k,β [µ] is a continuous function on R d \ W.supp µ. Furthermore, by Proposition 3.6 and Fubini's theorem, the sub-mean property is satisfied by the function

I k,β [µ] on R d \ W.supp µ. Thus, I k,β [µ] is D-subharmonic on R d \ W.supp µ. Corollary 4.1 Let β ∈ [2, d + 2γ[ . If µ ∈ M + k,β (R d ), then the function I k,β [µ] is D- superharmonic on R d .
Proof: Let Φ n the function defined by Φ n (x) = ∫ B(0,n) R k,β (x, y)dµ(y). From Theorem 4.1, the function Φ n is D-superharmonic on R d . Thus, as I k,β [µ] is not identically +∞ by hypothesis, the function I k,β [µ] is D-superharmonic on R d as being an increasing pointwise limit of the sequence (Φ n ) n of D-superharmonic functions (see [START_REF] Gallardo | Newtonian Potentials and subharmonic functions associated to root systems[END_REF], Proposition 3.3).

Proposition 4.5 Let

µ ∈ M + k,β (R d ) with β ∈ [2, d + 2γ[ and m ∈ N be such that 1 ≤ m ≤ β/2. Then, the function x → I k,β [µ](x)ω k (x) satisfies (-∆ k ) m (I k,β [µ]ω k ) =    I k,β-2m [µ]ω k in D ′ (R d ) if β > 2m, µ in D ′ (R d ) if β = 2m, . (4.13)
Proof: Let ϕ ∈ D(R d ). We will only prove the result in the case β > 2m and by the same arguments it can be obtained when β = 2m. We have

⟨(-∆ k ) m (I k,β [µ]ω k ) , ϕ⟩ = ∫ R d ( ∫ R d R k,β (x, y)(-∆ k ) m ϕ(x)ω k (x)dx ) dµ(y) = ∫ R d ( ∫ R d R k,β-2m (x, y)ϕ(x)ω k (x)dx ) dµ(y) = ∫ R d I k,β-2m [µ](x)ϕ(x)ω k (x)dx,
where we have used -Fubini's theorem in the first and the last lines (it is possible because

I k,β [µ] ∈ L 1 k,loc (R d ) and by Remark 4.1, I k,β-2m [µ] is also in L 1 k,loc (R d ))
; -the fact that the ∆ k -Riesz kernel is symmetric and the relation (3.28) in the second line.

From the previous proposition, we obtain immediately the uniqueness principle for ∆ k -Riesz potential of index 2m:

Corollary 4.2 Let m ∈]0, d 2 + γ[ be an integer and µ, ν ∈ M + k,2m (R d ). If I k,2m [µ] = I k,2m [ν] a.e, then µ = ν.
For an arbitrary index β ∈]0, d + 2γ[, we have the following version of the uniqueness principle for finite measures:

F k (I k,β [µ]ω k ) = ∥.∥ -β F k (µ)ω k in S ′ (R d ). (4.14)
Here, F k (µ) is the function defined by (2.5).

Proof: Let m > d+2γ an integer and C m as in (3.26). By Fubini's theorem, the symmetric property of the ∆ k -Riesz kernel and the relation (3.26), we get ∫ In order to extend the pointwise Hedbreg inequality in Dunkl setting, in the following result we give the link between the ∆ k -Riesz potential and the volume mean of a nonnegative Radon measure. Proof: The result follows from (3.3), Fubini's theorem, (2.18) and (4.16).

R d (1 + ∥x∥ 2 ) -m I k,β [µ](x)ω k (x)dx = ∫ R d ( ∫ R d (1 + ∥x∥ 2 ) -m R k,β (x, y)ω k (x)dx ) dµ(y) ≤ C m µ(R d ) < +∞. This shows that I k,β [µ]ω k ∈ S ′ (R d ). Let ϕ ∈ S(R d ). We have ⟨F k (I k,β [µ]ω k ) , ϕ⟩ = ∫ R d ( ∫ R d R k,β (x, y)dµ(y) ) F k (ϕ)(x)ω k (x)dx = ∫ R d ( ∫ R d R k,β (x, y)F k (ϕ)(x)ω k (x)dx ) µ(y) = ∫ R d ( ∫ R d E k (-iy, x)∥x∥ -β ω k (x)ϕ(x)dx ) dµ(y) = ∫ R d ∥x∥ -β F k (µ)(x)ω k (x)ϕ(x)
In the following result, we will extend the pointwise Hedberg inequality (see [START_REF] Hedberg | On certain convolution inequalities[END_REF]). We recall that the Dunkl-Hardy-Littlewood maximal operator is defined for f ∈ L 1 k,loc (R d ) by (see [START_REF] Thangavelu | Convolution operator and maximal function for Dunkl transform[END_REF]) According to [START_REF] Rejeb | Volume mean operator and differentiation results associated to root systems[END_REF], we have h k (r, x, .) = τ -x (1 B(0,r) ) a.e. on R d . Thus, we will take this remark into account in the formula (4.17 • Clearly, we see that

M k (f )(x) = sup
I 1 (x) ≤ CA β M k (f )(x). ( 4

.19)

• We have

I 2 (x) = C +∞ ∑ n=0 ∫ 2 n+1 A 2 n A t β-d-2γ-1 ∫ R d |f (y)|h k (t, x, y)ω k (y)dydt ≤ C∥f ∥ k,p +∞ ∑ n=0 ∫ 2 n+1 A 2 n A t β-d-2γ-1 t d+2γ(1-1/p) dt ≤ C∥f ∥ k,p +∞ ∑ n=0 (2 n A) β-d+2γ p ,
where we have used Hölder's inequality and the relation (2.18) in the second line. Therefore, we have Letting ε -→ 0, we get (4.18).

Using the Hedberg inequality (4.18), the L p k -boundedness properties of the Dunkl-Hardy-Littlewood maximal function (see [START_REF]Two results on the Dunkl maximal function[END_REF] or [START_REF] Thangavelu | Convolution operator and maximal function for Dunkl transform[END_REF]) and following the same proof as in the classical case (see Theorem 3.1.4 in [START_REF] Adams | Functions Spaces and Potential Theory[END_REF]), we obtain the Sobolev inequality:

Proposition 3 . 5

 35 we can differentiate under the integral in the relation (3.2) and we obtain the result. In the following result, we study the L p k,loc (R d )-integrability of the function R k,β (x, .), for fixed x ∈ R d . Let 0 < β < d + 2γ and p ∈ [1, d+2γ d+2γ-β [. Then, for every R > 0, there exists a positive constant C = C(R, p, d, γ, β) such that

Definition 4 . 1

 41 ) and M + (R d ) denote respectively the space of signed Radon measures on R d and the convex cone of nonnegative Radon measures on R d . Let µ ∈ M + (R d ) and β ∈]0, d + 2γ[. The β-∆ k -Riesz potential of µ is defined by

Proposition 4 . 3

 43 Let 0 < β < d + 2γ and µ be a compactly supported nonnegative Radon measure on R d . If I k,β [µ] ≤ M holds on W.supp µ, then

Proposition 4 . 4 1 ) 2 )

 4412 Let β ∈]0, d + 2γ[ and µ ∈ M + k,β (R d ) with compact support. The function I k,β [µ] is lower semi-continuous on R d and continuous on R d \ W.supp µ. If the restriction of the function I k,β [µ] on W.supp µ is continuous on W.supp µ, then I k,β [µ] is continuous on R d . Proof: 1) • Consider the function F n given by

  Proof of Theorem 4.1: i) Let β > 2. Using Fubini's theorem and the D-superharmonicty of the ∆ k -Riesz kernel (see Proposition 3.6), we can easily see that I k,β [µ] satisfies the super-mean property i.e. for all x ∈ R d and all r > 0, M r B

Theorem 4 . 2 4 . 1

 4241 Let β ∈]0, d+2γ[ and let µ, ν be two finite and nonnegative Radon measures on R d . If I k,β [µ] = I k,β [ν] a.e. on R d , then µ = ν. We start by the following result Lemma Let µ be a finite and nonnegative Radon measure on R d . Then, I k,β [µ]ω k is a tempered distribution and its distributional Dunkl transform is given by

  dx, where we have used -Fubini's theorem in the first second line: it is possible because F k (ϕ) ∈ S(R d ) and then the function x → (1 + ∥x∥ 2 ) m F k (ϕ)(x) is bounded with m the integer chosen as above; -the relations (3.25) and R k,β (x, y) = R k,β (y, x) in the third line; -the boundedness of the function (x, y) → E k (iy, x) (see (2.2)), Fubini's theorem and (2.5) in the last line. Proof of Theorem 4.2: By our hypothesis and Lemma 4.1, we have I k,β [µ] = I k,β [ν] in S ′ (R d ). Applying Dunkl transform to the both terms and using the relation (4.14), we deduce that∥.∥ -β F k (µ)ω k = ∥.∥ -β F k (ν)ω k in S ′ (R d ).As the functions ∥.∥ -β F k (µ)ω k and ∥.∥ -β F k (ν)ω k are locally integrable on R d , we get ∥.∥ -β F k (µ)ω k = ∥.∥ -β F k (ν)ω k a.e. on R d .Now, by continuity it follows that the functions F k (µ) and F k (ν) coincide everywhere on R d . Finally, by the injectivity of the Dunkl transform on the space of finite Radon measures on R d , we conclude that µ = ν.

Proposition 4 . 6

 46 Let µ be a nonnegative Radon measure on R d . Then, for all β ∈]0, d + 2γ[, we haveI k,β [µ](x) = d k κ (d + 2γ)(d + 2γ -β) k (t,x, y)dµ(y).

  (y)|τ -x (1 B(0,r) )(y)ω k (y)dy, (4.17) where τ -x (1 B(0,r) ) denotes the L 2 k (R d )-function with Dunkl transform ξ → E k (-ix, ξ)F k (

Theorem 4 . 3

 43 ) and in the sequel of the paper.Moreover, when dµ(y) = |f (y)|ω k (y)dy, f ∈ L 1 k,loc (R d ), we will use the notation I k,β [|f |] instead of I k,β [|f (y)|ω k (y)dy]. For 0 < β < d+2γ, 1 ≤ p < d+2γβ , there exists constants C = C(d, γ, β, p) > 0 such that for any measurable function f and any x ∈ R d , we haveI k,β [|f |](x) ≤ C∥f ∥ βp d+2γ k,p (M k (f )(x))For every A > 0, by(4.15) where we take dµ(y) = |f (y)|ω k (y)dy, we can writeI k,β [|f |](x) = I k,β [|f |ω k ](x) = C ∫ A 0 t β-1 M t B (|f |)(x)dt + C ∫ +∞ A t β-1 M t B (|f |)(x)dt := I 1 (x) + I 2 (x).

I 2 (

 2 x) ≤ CA β-d+2γ p ∥f ∥ k,p . (4.20) Now, using (4.19), (4.20) and choosingA = A(x) = ( ∥f ∥ k,p M k (f )(x) + ε ) p d+2γ , we obtain I k,β [|f |](x) ≤ C∥f ∥ βp d+2γ k,p (M k (f )(x) + ε) 1-βp d+2γ .

Corollary 4 . 3 1 )( 1 )

 4311 Let 0 < β < d + 2γ, 1 ≤ p < d+2γ β and p * = p(d+2γ) d+2γ-βp . If p = 1, then I k,β is of weak type (1, p * ) i.e. there exists a constant C = C(β, d, γ) such that ∀ λ > 0, ∀ f ∈ L 1 k (R d ), ∫ {x: I k,β [|f |]>λ} ω k (x)dx ≤ C If p > 1, then I k,β is of strong type (p, p * ) i.e. I k,β : L p k (R d ) -→ L p * k (R d ) is bounded.

see for example[START_REF] Armitage | Classical Potential Theory[END_REF],[START_REF] Hayman | Subharmonic functions[END_REF],[START_REF] Helms | Potential theory[END_REF] and[START_REF] Landkof | Foundations of Modern Potential Theory[END_REF].

If ∥x 0 ∥ ≤ 1, we deduce immediately from the previous inequality that (4.2) holds. If ∥x 0 ∥ > 1, using the fact that ∥x 0 ∥ + ∥y∥ ≤ ∥x 0 ∥(1 + ∥y∥) and using again the above inequality, we obtain that (4.2) holds. This finishes the proof.

) and µ = µ + -µ -its Hahn-Jordan decomposition. If µ + and µ - satisfy (4.2), then the ∆ k -Riesz potential of µ is well defined almost everywhere by setting

• Let β, θ > 0 be such that β + θ < d + 2γ. Then using the generalized Riesz composition formula (3.17) and Fubini's theorem we can see that

In the following result we will establish that any measure µ ∈ M + k,β (R d ) doesn't charge the singularities of the function R k,β (x, .) whenever its β-∆ k -Riesz potential valued at x is finite. More precisely, we have

) is l.s.c. at gx, there exists r > 0 such that R k,β (x, y) ≥ n for all y ∈ B(gx, r). This implies that

This proves the first part. Now, since β ≤ d, we know that x is always a singularity of the function R k,β (x, .). Thus the second part follows from the first one. Now, we establish a boundedness principle for the potential of a compactly supported measure which generalizes the known result in the classical case (i.e. k = 0) (see [START_REF] Landkof | Foundations of Modern Potential Theory[END_REF], Theorem 1.5). [START_REF] Hassani | Riesz Potential and fractional maximal function for the Dunkl transform[END_REF] by another proof using interpolation methods and in the particular case when the Coxeter-Weyl group is Z d 2 in [START_REF] Thangavelu | Riesz transform and Riesz potentials for Dunkl transform[END_REF].

Remark 4.2 The previous result has been obtain in


Alexander Sproedt
email: asproedt@ethz.ch

Johannes Plehn
email: jplehn@ethz.ch

Philipp Hertz
email: phertz@ethz.ch

A Simulation Enabled Procedure for Eco-Efficiency Optimization in Production Systems

Keywords: Eco-Efficiency, Manufacturing, Sustainability, Simulation

Environmental sustainability in manufacturing has been experiencing increasing attention in practice and academia over the last decades. Manufacturing companies strive to improve their eco-efficiency, which is commonly understood as delivering high value products at lower cost and environmental impact. They need tools and methods to translate this strategic goal onto the operational shop floor level. This paper presents a procedure for optimizing the eco-efficiency of production systems, supported by a discrete-event material flow simulation model. Its application is shown in a case with a company from the Swiss consumer goods industry.

Introduction

Sustainability considerations have gained importance in the public over the last decades. Especially for manufacturing companies, responsible for over 36% of worldwide CO2 emissions [START_REF] Hertz | Tracking Industrial Energy Efficiency and CO 2 Emissions[END_REF], increasing the energy and resource efficiency has become a major challenge and success factor. Reasons are steadily rising material and energy prices, increasing customer demand for environmentally friendly goods and legislative pressure holding companies accountable for their environmental impact.

Although measures to improve their environmental impact can be taken along the whole supply chain, SMEs typically lack the bargaining power to influence their supply chain partners. Improvement measures have to be found within their own operations for being the only area they can influence. The main issue for manufacturing SMEs thus is translating a strategic goal to increase their environmental sustainability into concrete improvement measures on a shop floor level, while still ensuring their profitability. This is understood as the optimization of their eco-efficiency [START_REF] Van Berkel | Cleaner production and eco-efficiency[END_REF].

Manufacturing companies need pragmatic analysis and decision support tools to solve this problem. This paper proposes a procedure to assess and optimize the ecoefficiency of production systems. The procedure is supported by a newly developed simulation approach, which couples the static material flow assessment (environmental perspective, MFA) with a dynamic discrete-event simulation (economic perspective, DES) in one model.

Project background and methodology

The findings presented in this paper stem from a two year Swiss national project called "EcoFactory". It is funded by the Swiss Commission for Technology and Innovation and carried out by the BWI Center for Industrial Management (project lead), the Swiss Federal Institute for Material Sciences (EMPA) and the University of Applied Sciences HTW Berlin.

Case study research was chosen as the methodology in order to derive theories from practice [START_REF] Voss | Case research in operations management[END_REF]. As such, the project encompassed four case companies, as shown in Table 1. For the case selection, random companies were first approached for their interest in participating in the project. The four case companies were then selected for the sake of covering different industry sectors and company sizes. The purpose of the research can be categorized as theory building [START_REF] Eisenhardt | Building Theories from Case Study Research[END_REF]. In a first step the state of the art in the field was assessed through exploratory literature reviews. Data from the case companies was further gathered mainly through semi-structured interviews. The research resulted in conceptual models for the simulation approach and the analysis procedure.

3

State of the art

Assessing and monitoring the economic performance of their production system has become a mature process for most manufacturing companies. The system boundaries for the analysis are clear and a broad consensus exists on which performance measures should be used for specific purposes. Further, the availability of data for the economic assessment is usually very good, since they can commonly be derived from data (semi-)automatically collected and updated from ERP or MRP systems. An assessment of the environmental performance of a production system on the other hand is by far more challenging, since the environmental impact of a production system has different dimensions (e.g. air and water pollution, toxicity)and is driven by manifold aspects (e.g. energy and material consumption rates, support processes) [START_REF] Sproedt | The Dimensions of Corporate (QYLURQPHQWDO 6XVWDLQDELOLW\ௗ $ &RQFHSWXDO )UDPHZRUN RI (QYLURQPHQWDO $VSHFWV WR Sustainable Manufacturing[END_REF]. Furthermore, there are different methods available to assess the environmental performance, each well suited for a different purpose (for an overview see [START_REF] Sproedt | Environmental Sustainability Assessment: An Evaluation of Established Methods and Analysis of their Suitable Applications in Industrial Practice[END_REF]), while Life Cycle Assessment (LCA) represents the predominant one. Thiede and Herrmann state that an integrated assessment of environmental and economic performance of a production system requires a system-oriented approach in order to identify the truly relevant fields of action for improvements [START_REF] Thiede | Energy Flow Simulation for Manufacturing Systems[END_REF]. Discreteevent simulation (DES) models have been acknowledged as powerful tools for in-creasing the understanding of complex systems. They allow the user to gain insights into interactions between important variables in the performance of the system through their ability to reflect the dynamics of a system [START_REF] Law | Simulation modeling and analysis[END_REF]. Since commercially available simulation tools do not yet consider environmental aspects, some effort has been undertaken in research during the last years to develop simulation models that aid the environmental assessment of production systems. A comprehensive overview of different approaches can be found in [START_REF] Thiede | Energy efficiency in manufacturing systems[END_REF]. However, most developed approaches are still based on different model representations for the economic and environmental assessments. As such, they first use a DES model to investigate time-sensitivity and stochastic effects in regard to the economic perspective. Although based on the same object of reference, a material-flow assessment model is additionally required as a basis for the LCA of the manufacturing system, resulting in additional modeling efforts and is inconsistent when aiming at an integrated assessment of the production system. For overcoming this gap, a simulation model needed to be developed that integrates both the environmental and the economic perspective in one unified model. The simulation model is preceded by a system analysis procedure reducing modeling efforts and ensuring that environmentally relevant consumption parameters are made available in the required granularity. The results will be explained in the following.

Project findings and results

System analysis

The developed system analysis procedure is based on the standard process of a simulation study, as originally formulated by Nance [START_REF] Nance | The Conical Methodology: A Framework for Development[END_REF]. Figure 1 shows the basic steps of the system analysis, which will be described in the following.

Fig. 1. System analysis procedure

The strategic goal to improve the eco-efficiency of a manufacturing company can be motivated externally or internally. Therefore, as a first step, it is important to understand the area of conflict the company is operating in. An extended stakeholder analysis was deemed a viable tool to identify which stakeholders impose pressure upon the company to improve their eco-efficiency. The most relevant aspects analyzed in this context are customers, suppliers, competitors, government, employees, society, technology and environment.

In addition to external pressures, manufacturing companies may have a proactive attitude towards the environmental sustainability of their business. In such cases, companies search for the activities and aspects that mainly drive the environmental impact of their production. In order to identify the production processes yielding the highest optimization potential, a qualitative analysis is conducted of the whole pro-

Goal & Scope Definition

Stakeholder Analysis

Quick Scan

EVSM

Communicative Model

Data Gathering

Data Model duction facility. For this purpose, the Cleaner Production Quick-Scan was adapted [START_REF] Van Berkel | Comparative evaluation of cleaner production working methods[END_REF][START_REF] Effizienzagentur | Quick Scan[END_REF]. It is based on semi-structured interviews, checklists and a thorough factory walk-through. All production processes are assessed according to their economic and environmental optimization potential. The results of the Quick-Scan help to delimit the system that will later be analyzed in detail and modeled for the simulation. These two first steps lead to the Goal and Scope Definition, which serves as a basis and guidance for the further analysis and the whole simulation study. For the in-depth analysis of the previously defined area of intervention, the Environmental Value Stream Map (EVSM) has been developed. It is based on the traditional Value Stream Mapping technique known from the Lean Production/ Just in Time Toolkit. The purpose of a traditional VSM is to illustrate the material and information flows along the value stream of a product in order to identify the sources of wasted productivity [START_REF] Rother | Learning to see: value stream mapping to create value and eliminate muda[END_REF]. It has been extended to include an input-output perspective on each of the production processes in order to additionally capture their environmentally relevant aspects. The structure of the EVSM is shown in Figure 2.

Fig. 2. Structure of the Environmental Value Stream Map [14][15]

The blue boxes on the top capture all environmentally relevant inputs into a production process, sub-categorized into energy, material and water inputs. The green boxes on the bottom capture all outputs of the process not related to the actual product or component produced. These are categorized into waste and emissions (air and water). As a result, the EVSM delivers a clear and accurate picture of all parameters needed to assess the eco-efficiency of the system. It is thus a suitable basis for the application of the simulation model that will be introduced in section 4.2. Experience has shown that most environmentally relevant parameters, such as consumption rates of electricity, are only seldom available on a process level, so that measurements become necessary. For this purpose, a measuring procedure was developed together with the Institute for Machine Tools and Manufacturing of the ETH Zurich. Details on the measuring procedure can be found in [START_REF] Plehn | From Strategic Goals to focused Ecoefficiency Improvement in Production -Bridging the gap using Environmental Value Stream Mapping[END_REF][START_REF] Gontarz | Resource consumption measurement in manufacturing environments[END_REF].

Simulation model

To in the EcoFactory project. The model is based on a prototype architecture previously developed by HTW [START_REF] Wohlgemuth | Combining discrete event simulation and material flow analysis in a component-based approach to industrial environmental protection[END_REF][START_REF] Page | Advances in Environmental Informatics: Integration of Discrete Event Simulation Methodology with ecological Material Flow Analysis for Modelling eco-efficient Systems[END_REF]. The approach focuses on integrating DES and MFA into one simulation model. The model focuses on a cradle-to-gate perspective considering the environmental impacts of the production system. As such, the production facility is considered as the foreground system, which is explicitly represented and parameterized in the simulation model. The environmental impact of the upstream supply chain of the production is considered as the background system and not explicitly modeled. Within the simulation model, this aspect is accounted for through the integration of LCI data. Figure 3 shows the structural model of the simulation approach.

Fig. 3. Structural model of the simulation approach

As the illustration shows, the production system is modeled as a workflow of interlinked manufacturing operations. Each operation is determined by the workstation it takes place on (e.g. machine, assembly station), the product type being processed and the materials required as inputs (e.g. energy, water, auxiliary) or generated as outputs (emissions, waste). LCI datasets are imported for the definition of the material in-and outputs through an interface to the "ecoinvent" database, which is hosted and managed by EMPA. For this purpose, a search function for the LCI flows is incorporated in the simulation software. This interface also allows for the use of all LCA methods available in "ecoinvent" to support the evaluation of the environmental impacts. Hence, the structure of the simulation model is able to deliver a dynamic, job oriented LCA of the production system. The architecture further entails a definite attribution of workstations, products and materials. This enables the assessment of both the environmental and economic performance. For further details on the simulation software see [START_REF] Sproedt | Integrated Environmental and Economic Assessment of Production Systems using a Material Flow Simulation Model Enabling Manufacturing Competitiveness and Economic Sustainability[END_REF][START_REF] Widok | Achieving sustainability through a combination of LCA and DES integrated in a simulation software for production processes[END_REF].

Case study application

In this section, the application of the before described system analysis procedure and simulation model is shown on the example of case company A. The company counts 55 employees and produces made-to-order eyeglass lenses from specialist plastic billets. The billets are first milled to their required size and geometry before surface finishing and coating processes are applied according to customer specifications. The stakeholder analysis and the Quick Scan showed that, while being an integral part of the company's proactive strategy, there is no significant pressure from stakeholders to reduce environmental impact. Reducing the material waste generation and electrici- ty consumption were targeted as the main objectives. Figure 4 shows the EVSM of the company's production system as well as the simulation model in the graphical user interface of the software.

Fig. 4. EVSM and screenshot of the simulation model

The computed eco-efficiency per workstation for company A is shown in Figure 5. It is evident that the highest optimization potential lies in the antireflection finishing, milling and polishing processes. Figure 5 further shows an analysis of the ecoefficiency of the most relevant materials consumed within the production system. For this purpose, GWP100a (Global Warming Potential, in tons of CO2 equivalents) was chosen as the environmental performance indicator. This analysis shows that electricity consumption is the most relevant environmental impact factor, whereas the coating material use is the biggest cost driver. The simulation results further showed a rather poor capacity utilization of the antireflection finishing and polishing processes with idle times accounting up to 30-40% of total time respectively. An improved job-order planning for these processes, allowing to switch off the machines for longer idle periods, is being tested at the company. Based on simulation experiments, this measure has an estimated potential to reduce the overall electricity consumption by up to 10%.

Discussion and Conclusion

This paper presented a novel simulation approach, which combines discrete-event simulation with LCA in one unified model. Its architecture allows for a definite attribution of materials, workstations and products, delivering a higher degree of transparency to the production processes than current approaches. Further, a system analysis procedure was presented, designed to deliver all relevant data and information for building the simulation model of a production system with minimum efforts.

The main limitation to the presented approach is the commonly limited data availability at a process level, especially concerning environmentally relevant parameters. To unfold its full potential, the simulation model requires product-specific material consumption parameters on a process level. The lack of this data calls for dedicated measurements, which can result in significant efforts for the company. Nevertheless, at the case companies these measurements alone increased the awareness and transparency of consumption patterns within the production system and were by themselves deemed useful by the responsible production managers. In summary, the application of the approach in company A resulted in time efforts of 10 days (6 days for the system analysis, 4 days for modeling activities), that were deemed acceptable by the company representatives. The simulation analysis pointed to an improvement measure estimated to reduce the electricity consumption in production by about 10%, while further measures are currently being evaluated.

Fig. 5 .

 5 Fig. 5. Eco-efficiency of materials and workstations

Table 1 .

 1 Overview of case companies in EcoFactory project

	Case Company	A	B	C	D
	Industry	Consumer Goods	Packaging	Food	Metal Working
	Size	55	115	800	3500

Process outputs Unit process 1 Value added time Lead time …

  overcome the shortcomings of current simulation models, a new model was developed in close cooperation with the University of Applied Sciences HTW Berlin with-

				Unit process n	Multi-machine system
	Supplier	Energy	Electricity, Compressed air	Electricity, Compressed air	Customer	Electricity, Compressed air
	Process	Materials	Auxiliary materials	Auxiliary materials		Auxiliary materials
	inputs
		Water	Cooling water, Process water	Cooling water, Process water		Cooling water, Process water
			Operation	Operation		Production system
			Working center	Working center	Location
			Cycle time	Cycle time		Cumulated cycle time
	Process parameters		Setup time Scrap rate Lot size	Setup time Scrap rate Lot size		Cumulated setup time Scrap rate Maximum lot size
			…	…		…
		Waste	Waste materials	Waste materials		Waste materials
		Emissions	Emissions to air, Emissions to water	Emissions to air, Emissions to water		Emissions to air, Emissions to water
			Processing Setup Standby	Processing Setup	Standby	Processing Setup	Standby
						Overall lead time
			Machine Status			Value added time


Teaching Supply Chain Management: Mixed vs. Pure Strategies in Simulation Based Training

Keywords: Management Education, Simulation Based Training, Competitive, Cooperative 1

We investigate two inter-entity strategies for using SBT by teams-a competitive strategy and a cooperative strategy within the supply chain management domain. We examine how a combination of these inter-entity strategies would work at situations wherein both are used, one following the other. The goal is to investigate from a pedagogical and practical perspective which strategy is the most effective using SBT techniques for business education-pure strategy (strictly cooperative/competitive) or mixed (a combination of both). The results indicate that it is better to teach with mixed strategies when using SBT by teams, than to teach only one (pure) strategy. Moreover, if both strategies are used, then the order of teaching is significant. It is better to start with a competitive strategy and then move to a cooperative strategy rather than the reverse.

INTRODUCTION

Over the past decade, there has been an enormous growth and development in business education. New programs have emerged while existing ones have been extended and enriched with upgraded content. These educational programs include courses that are taught using various pedagogical methods that include frontal lectures, books, recorded presentations, seminars, group learning, etc. by virtue of being confined to a classroom environment traditional methods have struggled to transform expert intuition and theoretical knowledge into practical experience. An emulation of a real physical environment is one way to aid in this transformation. However, efficiently and effectively providing this emulation is an important challenge of management education which has yet to be answered [START_REF] Gunes | Educational tool for design and implementation of an autonomous mobile robot[END_REF][START_REF] Doulgeri | Development, integration and evaluation of a web-based virtual robot task simulator in the teaching of robotics[END_REF]. Nonetheless, over the past decade spectacular advances in infrastructure technologies that enable the use of modern tools to enhance learning have been proposed.

One tool that has been extensively investigated is computerized automated simulators. Simulation creates an artificial environment that reflects and illustrates real-life experiences [START_REF] Bell | Current issues and future directions in simulation-based training in North America[END_REF]. Of particular interest to us is the use of simulation for knowledge transfer. This process is typified by the knowledge a trainee gains during his experience with the artificial environment (simulation) and the subsequent implementation of the knowledge. One characteristic of an efficient simulation tool is its ability to facilitate learning that can be transferred to a real-life environment [START_REF] Thompson | Simulation in medical education[END_REF]. In the context of education, using simulation as a teaching methodology is commonly referred to in the literature as Simulation Based Training SBT.

SBT for business education provides several advantages over traditional techniques, such as supplying hands-on practice and allowing for the development of skills at a faster pace [START_REF] Salas | Using simulation-based training to enhance management education[END_REF]. Salas et al. delineated seven basic stages of SBT develop-ment in business education [START_REF] Salas | Using simulation-based training to improve patient safety: What does it take?[END_REF]. Other researchers provided some practical guidelines as to how best to implement and use SBT in business education-e.g., offer detailed focused feedback, and directly measure outcomes [START_REF] Salas | Using simulation-based training to enhance management education[END_REF]. Some researchers [START_REF] Salas | Using simulation-based training to enhance management education[END_REF][START_REF] Shtub | On the use of simulation based training for management education: Mind the gap as one size does not fit all[END_REF] asserted that SBT can be an efficient tool for business education, but only when the gap between the trainee's a priori knowledge and the difficulty of the simulation exercise are matched. Inasmuch as SBT is important for current and future education, it is essential to identify the circumstances under which SBT is most effective.

In this paper, we investigate two inter-entity strategies for using SBT by teams-a competitive strategy and a cooperative strategy. Research has attempted to determine which strategy is better for training teams [START_REF] Okebukola | The Relative Effectiveness Of Cooperative And Competitive Interaction Techniques in Strengthening Students' Performance In Science Classes[END_REF][START_REF] Johnson | Comparison Of Computer-Assisted Cooperative, Competitive, And Individualistic Learning[END_REF][START_REF] Tjosvold | Cooperative and Competitive Goal Approach to Conflict: Accomplishments Challenges[END_REF][START_REF] Sherman | Cooperative Versus Competitive Educational Psychology Classrooms: A Comparative Study[END_REF]]. Sherman's study [START_REF] Sherman | Cooperative Versus Competitive Educational Psychology Classrooms: A Comparative Study[END_REF] showed no significant differences in cooperative and competitive learning. A recent study on these SBT inter-entity strategies was carried out within the project management domain [START_REF] Nembhard | Comparing competitive and cooperative strategies for learning project management[END_REF] and indicated that cooperative strategies yield better results in the overall outcome. There are pros and cons for each strategy [START_REF] Sherman | Cooperative Versus Competitive Educational Psychology Classrooms: A Comparative Study[END_REF], yet the cooperative one is considered to be more effective [START_REF] Nembhard | Comparing competitive and cooperative strategies for learning project management[END_REF][START_REF] Johnson | Effects of cooperative, competitive, and individualistic goal structures on achievement: A meta-analysis[END_REF]. Students who were trained using a cooperative strategy, which is better for long term relationships [START_REF] Su | The impact of supply chain relationship quality on cooperative strategy[END_REF], tend to employ this strategy in their professional life. The parties involved are usually more satisfied, and as a result, the overall business profit rises [START_REF] Benton | The Influence of Power Driven Buyer/Seller Relationships on Supply Chain Satisfaction[END_REF]. To the best of our knowledge, no study has examined how a combination of these inter-entity strategies would work. That is to say, research has looked at the results of using one or the other strategy but not at situations wherein both are used, one following the other. This is a lacuna that we come to fill in. Unlike past research that focused on finding the best strategy we look at a combination of the two strategies. We investigate this issue within the supply chain management domain. The goal of this paper is to investigate from a pedagogical and practical perspective which strategy is the most effective using SBT techniques for business education-pure strategy (strictly cooperative/competitive) or mixed (a combination of both).

EXPERIMENT RESEARCH DESIGN

In this section we present our experimental design, starting with our hypothesis and culminating in a detailed experimental design. In Section 3 we report on the results of the experiment and analyze the data.

Hypothesis

Different parties comprise the supply chain, each desiring to meet their own goals. In this pursuit they can adopt omnifarious inter-entity strategies, including competitive and cooperative ones, in regard to the interrelationships with their colleagues. When students are being taught about the interdependence among the supply chain parties, the teacher can choose to teach using one or both of these inter-entity strategies. Our experiment tests the following hypotheses in regard to cooperative and competitive inter-entity strategies in using SBT by teams:

Hypothesis 1: Using a mixed strategy (cooperative and competitive) achieves better learning outcomes than using a pure (single) strategy.

Hypothesis 2: When using a mixed strategy, the order of the two strategies affects the learning outcomes.

Experiment Design

To examine these pedagogical hypotheses we use a new advanced SBT computerized simulation environment. The software, Supply Chain Simulator (SCS), is a web based computer simulation application developed at the Technion. It is designed as an educational platform to enable students to gain better understanding and experience in managing supply chains. SCS is based on the following principles: Ease of use, Scenario based training, Flexibility, and Supportive data. Two classes (I and II) of freshman undergraduates in the Industrial Engineering and Management Faculty of the Technion participated in the experiment. Class I consisted of 134 students and Class II consisted of 121 students. None of the students had theoretical or formal training with supply chain management. Prior to their first task using SCS the students received an introduction to supply chain management and SCS. The introduction included an oral presentation and live demonstration of how to use SCS, instructions on how to run scenarios and an explanation of the performance measure used (i.e., profit) and how it is calculated. The students were motivated to pursue the highest profit possible as this performance measure determined their grade in each task. The students ran four scenarios, two of them were individual tasks (the first and fourth scenarios) and the other two were run in pairs. The nature of the four scenarios was as follows.

Scenario 1: Managing a supply chain with an intermediate level of difficulty. The supply chain scenario comprised seven facilities (two manufacturers, two warehouses and three retailers). In the beginning of each period the students had to make several managerial decisions such as the quantity to produce at each manufacturing facility, the order quantity of each warehouse and retailer, and the supply priority in case of a conflict between external demand (customers) and internal demand (orders from another facility within the supply chain). The students' goal was to maximize the overall profit over twelve time periods. Each student had ten trials and the best performance was recorded. The purpose of this scenario, as far as this research is concerned, was to expose the students to the simulator and allow them to practice so that all groups of students would be aligned in their ability to operate the simulator.

Scenarios 2 and 3: For these scenarios (2 and 3) each class (I and II) was divided into teams of two students. Additional segmentation was done as the teams were divided randomly into two groups (A and B). The teams and the groups remained unchanged. Each scenario included a supply chain with an advanced level of difficulty that needed to be managed by the team. Each supply chain scenario comprised several facilities (manufacturers, warehouses and retailers). One student was in charge of the "manufacturing division," which included the manufacturers and the warehouses, and the second student was in charge of the "marketing division," which included the retailers. Each student had an individual goal to maximize his own facilities' profit over twelve time periods; nonetheless the students had to work together as a team to manage the same scenario.

The nature of the work relationship between the team members was sometimes competitive or cooperative. The different inter-entity strategies were realized in the way the students were graded. Every team had twenty trials for each scenario. In the competitive strategy, the students had to decide by themselves which trial was the one used as their grade. They had to choose the same trial for both team members even though this trial would not necessarily represent the supply chain's best overall performance. The students in the same team were nonetheless individually graded according to their individual performance, regardless of the overall team performance. Clearly, the two students on each team were at odds with each other since each team member wanted to choose the trial that was best for him and this was generally not the best for his partner. When the team members were unable to agree upon a specific trial, the average performance over the last ten trials was used as the grade.

In contrast, in the cooperative strategy the students did not have to decide which trial would be considered for their grade; the trial that produced the maximum overall profit for the entire supply chain was used. The students in the same team were still individually graded according to their individual performance.

The assignment of the teams is as follows. In Class I, the teams in Group A were assigned a competitive strategy for Scenario 2 and a cooperative strategy for Scenario 3. The assignment to Group B was reversed: the teams used a cooperative strategy in Scenario 2 and a competitive strategy in Scenario 3. Both scenarios were actually the same though the students were unaware of this fact (the facilities of the supply chain were organized differently on the map). Consequently, the potential performance (overall profit) of each exercise was the same.

In Class II, the teams in the groups were always assigned the same strategy. Teams in Group A were assigned the competitive strategy and teams in Group B were assigned the cooperative strategy. Scenario 4: An individual task similar to the first one. The scenario had an advanced level of difficulty, requiring each student first to design an efficient supply chain using predetermined facilities and then to manage it. The goal was the same as in Scenarios 1 -3, i.e. to maximize the overall profit over twelve time periods. The goal of the experiment was to measure each student's individual ability after they had learned and gained experience. In addition, this scenario enabled us to compare the two groups' performances. Moreover, Scenario 4 was exactly the same for the two classes, thus enabling us to compare the two classes' performances after handling Scenarios 1 -3.

EXPERIMENT RESULTS AND ANALYSIS

For Class I we had 66 teams in Scenarios 2 and 3, 32 teams in Group A and 34 teams in Group B. In addition, we collected 120 results of individual students in Scenario 1 and 127 results of individual students in Scenario 4. For Class II we collected the results of 61 teams in Scenarios 2 and 3, 30 teams in the 'Comp-Comp' group and 31 teams in the 'Coop-Coop' group. In addition, we collected 111 results of individual students in Scenario 1 and 112 results of individual students in Scenario 4. The reduction in the number of students participating in the various scenarios was caused by students dropping the course. Moreover, we believe that their reasons for not continuing in the course were unrelated to the simulator. We note that the number of students dropping the course was no greater than normal. In addition, in each class there were 2-3 outliers observations that were removed. These outliers were students that achieved very law score in one of the scenarios, usually that occurred when they started the scenario but abandoned it in the middle.

Whereas one may feel that a simple t-test is suitable to check our hypotheses, we found this not to be the case. The main reason is that the within-group variance masks the differences between groups. This can be understood by the bivariate fit analysis of the performances of both classes. When analyzing Class I and II results, a strong correlation between the teams and their performance was revealed (Class I p.value < 0.002, Class II p.value < 0.001). Fig. 1 present the bivariate fit test results of the correlation between the teams and their performance in Scenarios 1 and 4 in Class I. Class II demonstrates a similar behavior. The teams tended to show either high or low performance regardless of strategy. Due to the correlation findings and in order to reduce the within-group variance, in all our statistical tests below we used ANCOVA to test the significance of differences between the group means of the students' performance in Scenario 4 (dependent variable) using students' performance Scenario 1 as a covariate.

Hypothesis 1

In order to test whether a mixed strategy achieves better learning outcomes than using a pure strategy, we checked whether there is a significance difference in the performance of Class I vs. Class II in Scenario 4. We used ANCOVA to test the significance of differences between the means of the students' performance in Scenario 4

(dependent variable). Due to the correlation findings and in order to reduce the within-group variance, we used students' performance in Scenario 1 as a covariate. The results indicate that the performance in Scenario 4 is dependent on the Class. The significance of the model has a p value < 0.001 and the significance of the independent variable (class) has also a p value < 0.001.

Hypothesis 2

In order to test whether the order of the two strategies is significant when using a mixed strategy, we checked whether there is a significant difference in the performances in Scenario 4 between the groups of Class I. We used ANCOVA to test the significance of differences between the mean performances of the two students groups in Scenario 4 (dependent variable). Due to the correlation findings and in order to reduce the within-group variance, we again used the students' performance in Scenario 1 as a covariate. The results indicate that the performance in Scenario 4 is dependent on the Group when correcting for the groups' performance in Scenario 1. The significance of the model has a p value < 0.001 and the significance of the independent variable (Group) has a p value of 0.007.

DISCUSSION AND CONCLUSIONS

This research focused on team learning using SBT techniques in the business management education domain. We utilized a simulator to manage a supply chain. Two different strategies in were investigated-cooperative and competitive-trying to determine whether a pure strategy or a mixed one is the most effective when using SBT. This study, adopting statistical models to assess the effect of the variables on the students' performance, showed significant results. These results indicate that it is better to teach with mixed strategies when using SBT by teams, than to teach only one (pure) strategy. Moreover, if both strategies are used, then the order of teaching is significant. It is better to start with a competitive strategy and then move to a cooperative strategy rather than the reverse. The students from Class I in Scenario 4 outperformed the students from Class II in this scenario, thus it supports our first hypothesis that it is better to teach both strategies than only one. A possible explanation to the observed phenomena is that teaching both strategies when using SBT by teams, exposes the students to a broader experience than students who are being taught by only one strategy. This experience enriches the learning process thus it is more efficient.

It seems that teams of Class I that started with a competitive strategy and then used a cooperative strategy learned from this process. The learning was expressed by their better performance of the teams when using a cooperative strategy than a competitive one; i.e., the second round performance was better than the first one. In contrast, teams that started with a cooperative strategy and then used a competitive one did not learn enough from the first round in order to improve their performance in the second round. This observation is supported by the zero difference in the performance of the first and second rounds for the teams using a cooperative and then a competitive strategy. That being the case, allegedly learning from Scenario 2 to Scenario 3 in Class I did not occur (on average) for a team that started with a cooperative strategy.

An additional finding that strengthens the assertion that it is better to start with competitive strategy and then move to cooperative is the students' performance in Scenario 4, which enabled us to compare all the participants' performance. In Scenario 4 the students of Class I that started with the competitive task significantly outperformed the students who started with the cooperative task.

A potential explanation to the differences in the performances of the two groups in Class I is that a competitive exercise causes the team to experience the negative effects of competition and thus during the next session, the team members will appreciate the opportunity to cooperate. Accordingly, one may conclude that learning occurs when the student starts from the "negative example" (competitive), and not the other way around. That is to say that a "negative example" is a better way to induce learning.

Another possible explanation is that a team that started with a cooperative strategy has already internalized the benefits of cooperation. Thus when the team faced a competitive strategy (scenario 3), its members did not act according to the competitive instructions, but continued to act as if they were in a cooperative exercise. It is possible that trying to teach a competitive strategy after a cooperative strategy has already been taught is not effective, as the participants do not follow the competitive strategy instructions but prefer to act as if using a cooperative strategy. The effect here was intensified because the competitive teams were not motivated to act competitively-there was no penalty for them if they did not do so.

A technical explanation for the above findings may perhaps be that a competitive exercise could be more effectively handled from the simulator point of view. It forces each member of the team to deal with the exercise (each team member must take care of himself if he wants to succeed, whereas in a cooperative exercise, each team member knows he can count on his partner).

After considering the above explanations, we propose some guidelines on how to use SBT for team training:

x Teach using both cooperative and competitive strategies (and not just one of them) when using SBT by teams. x When training teams and trying to encourage a certain behavior and eliminate another behavior considered to be negative, it is better for trainees to first gain experience with the negative behavior and its repercussions; only afterwards should they be exposed to the desired behavior and its benefits because in the latter situation the differences are more obvious and the learning process is intensified. x Even if teams are being trained in the reverse order (i.e., first the desired behavior and only afterwards the negative one), then make sure that the teams follow the instructions and actually act upon the negative behavior, they should be given motivation to do so. x Even though the team acts as one unit, in order to ensure that each member participates fully in the exercise and personally gains from it, each should be given per-

Fig. 1 .

 1 Fig. 1. Bivariate fit graph of Class I Scenarios 1 vs. 4

sonal responsibility and a separate grade. This minimizes the free rider phenomenon.