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

Chaabane Rejeb
email: chaabane.rejeb@gmail.com

Radial mollifiers, mean value operators and harmonic functions in Dunkl theory

Keywords: MSC (2010) primary: 31B05, 31C45, 47B39; secondary:33C52, 43A32, 51F15, Dunkl-Laplacian operator, Dunkl convolution product, Generalized volume mean value operator, Dunkl harmonic functions, Weyl's lemma

In this paper we show how to use mollifiers to regularise functions relative to a set of Dunkl operators in R d with Coxeter-Weyl group W , multiplicity function k and weight function ω k . In particular for Ω a W -invariant open subset of R d , for ϕ ∈ D(R d ) a radial function and u ∈ L 1 loc (Ω, ω k (x)dx), we study the Dunkl-convolution product u * k ϕ and the action of the Dunkl-Laplacian and the volume mean operators on these functions. The results are then applied to obtain an analog of the Weyl lemma for Dunkl-harmonic functions and to characterize them by invariance properties relative to mean value and convolution operators.

Introduction

Let R be a normalized root system in R d and k ≥ 0 a multiplicity function on R. The Dunkl-Laplacian operator associated to R and k, acting on C 2 (R d )-functions is a differential-difference operator of the form

∆ k f (x) = ∆f (x) + 2 ∑ α∈R + k(α) ( ⟨∇f (x), α⟩ ⟨α, x⟩ - f (x) -f (σ α (x)) ⟨α, x⟩ 2 ) , ( 1.1)
where ∆ (resp. ∇ ) is the usual Laplace (resp. gradient) operator, R + is a fixed positive subsystem of R and σ α is the reflexion directed by the root α (see [START_REF] Dunkl | Orthogonal Polynomials of Several variables[END_REF]). We recall that R is normalized if ||α|| 2 = 2 for all α ∈ R and that k is invariant under the action of the Coxeter-Weyl group W generated by the reflections σ α , α ∈ R.

We know that the Dunkl-Laplace operator can be written ∆ k = ∑ d j=1 D 2 ξ j , for ξ j , j = 1, . . . , d, an orthonormal basis of R d and where for every ξ ∈ R d , D ξ is the Dunkl operator defined by

D ξ f (x) = ∂ ξ f (x) + ∑ α∈R + k(α) ⟨α, ξ⟩ f (x) -f (σ α (x)) ⟨α, x⟩ , (1.2)
with ∂ ξ f denoting the usual ξ-directional derivative of f (see [START_REF] Dunkl | Differential-difference operators associated to reflection groups[END_REF] and [START_REF] Dunkl | Orthogonal Polynomials of Several variables[END_REF]). These operators are related to partial derivatives by means of the so-called Dunkl intertwining operator V k (see [START_REF] Dunkl | Hankel transforms associated to finite reflection groups[END_REF] or [START_REF] Dunkl | Orthogonal Polynomials of Several variables[END_REF]) as follows

∀ ξ ∈ R d , D ξ V k = V k ∂ ξ . (1.3)
The operator V k is a topological isomorphism from the space C ∞ (R d )1 onto itself satisfying (1.3) and V k (1) = 1 (see [START_REF] Trimèche | The Dunkl intertwining operator on spaces of functions and distributions and integral representation of its dual[END_REF]). Furthermore, according to [START_REF] Rösler | Positivity of Dunkl's intertwining operator[END_REF] or [START_REF] Rösler | Dunkl Operators: Theory and Applications[END_REF], for every x ∈ R d there exists a unique probability measure µ x on R d with compact support contained in the convex hull of the orbit of x under the group W , such that

∀ f ∈ C ∞ (R d ), V k (f )(x) = ∫ R d f (y)dµ x (y). (1.4)
For abbreviation and later use, we introduce the weight function

ω k (x) := ∏ α∈R + | ⟨α, x⟩ | 2k(α) (1.5)
and the set D + r (R d ), r > 0, of nonnegative radial C ∞ -functions with compact support contained in the Euclidean closed ball B(0, r).

where τ x , x ∈ R d , are the Dunkl translation operators (see Annex A.2). Note also that a very useful formula for the Dunkl translation has been obtained by M. Rösler ( [START_REF] Rösler | A positive radial product formula for the Dunkl kernel[END_REF]) when f ∈ C ∞ (R d ) is a radial function. In such case, the operator τ x is given by

∀ y ∈ R d , τ x f (y) = ∫ R d f ( √ ∥x∥ 2 + ∥y∥ 2 + 2 ⟨x, z⟩)dµ y (z), (1.8)
where f is the profile function of f defined by f (x) = f (∥x∥). This formula shows that the Dunkl translation operators are positivity preserving on the set of radial functions whereas this is not true in general ( [START_REF] Rösler | Bessel-type signed hypergroups on R. Probability measures on groups and related structures XI[END_REF] or [START_REF] Thangavelu | Convolution operator and maximal function for Dunkl transform[END_REF]).

We turn now to the content and the organization of this paper. In section 2, we recall the properties of the volume mean value operator M r B , r > 0, acting on continuous functions as follows

M r B (f )(x) := 1 m k (B(0, r)) ∫ R d f (y)h k (r, x, y)ω k (y)dy.
(1.9)

where y → h k (r, x, y) is a compactly supported measurable function (called harmonic kernel, see [START_REF] Gallardo | A new mean value property for harmonic functions relative to the Dunkl-Laplacian operator and applications[END_REF]) given by

h k (r, x, y) := ∫ R d 1 [0,r] ( √ ∥x∥ 2 + ∥y∥ 2 -2 ⟨x, z⟩)dµ y (z).
(1.10)

In section 3, we study the Dunkl convolution product (1.7). In particular, We will prove that this function is well defined and is of class C ∞ on Ω r . Moreover, we will show the commutativity relations

∆ k (u * k ϕ) = u * k ∆ k ϕ and M r B (u * k ϕ) = M r B (u) * k ϕ
and the following associativity property

(u * k ϕ) * k ψ = (u * k ψ) * k ϕ
In section 4, we apply the previous results to give some new properties of ∆ k -harmonic functions (i.e. fonctions u ∈ C 2 (Ω) such that ∆ k u = 0). As a first result the show that any ∆ k -harmonic function on Ω is in fact of class C ∞ . Then we prove that ∆ k -harmonic functions can be characterized by the local-volume mean value property i.e.

∀ x ∈ Ω, ∃ r x > 0, ∀ r < r x , u(x) = M r B (u)(x).
Finally we will establish the following Weyl's lemma: If u ∈ L 1 loc (Ω, m k ) satisfies ∆ k (uω k ) = 0 in distributional sense, then u coincides almost everywhere with a ∆ k -harmonic function on Ω.

Notations: Let us introduce the following functional spaces and notations which will be used throughout the paper. For Ω a (nonempty) W -invariant open subset of R d , we denote by:

• L 1 k,loc (Ω) = L 1 loc (Ω, m k ) the space of measurable functions f : Ω -→ C such that ∫ K |f (x)|ω k (x)dx < +∞ for any compact set K ⊂ Ω. • D(Ω) the space of C ∞ -functions on Ω with compact support. • D ′ (Ω)

The mean value operators

In this section, we will recall some facts about the volume and the spherical mean operators in Dunkl setting.

Let (r, x, y) → h k (r, x, y) be the harmonic kernel defined by (1.10). We note that when the multiplicity function is the zero function, we have µ y = δ y , h 0 (r, x, y) = 1 [0,r] (∥x-y∥) = 1 B(x,r) (y) and then our volume mean operator given by (1.9) coincides with the classical one. We note also that if f ∈ C ∞ (R d ), the volume mean of f at (x, r) can also be written by means of the Dunkl translation as follows (see [START_REF] Gallardo | A new mean value property for harmonic functions relative to the Dunkl-Laplacian operator and applications[END_REF]):

M r B (f )(x) = 1 m k (B(0, r)) ∫ B(0,r) τ x f (y)ω k (y)dy.
(2.1)

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 non decreasing on ]0, +∞[.

3

. For all fixed r > 0 and

x ∈ R d , supp h k (r, x, . ) ⊂ B W (x, r) := ∪ g∈W B(gx, r). ( 2.2)
4. Let r > 0 and x ∈ R d . For any sequence (χ ε ) ⊂ D(R d ) of radial functions such that for every ε > 0,

0 ≤ χ ε ≤ 1, χ ε = 1 on B(0, r) and ∀ y ∈ R d , lim ε→0 χ ε (y) = 1 B(0,r) (y), (2.3)
we have

∀ y ∈ R d , h k (r, x, y) = lim ε→0 τ -x χ ε (y).
(2.4)

5. For all r > 0, all x, y ∈ R d and all g ∈ W , we have h k (r, x, y) = h k (r, y, x) and h k (r, gx, y) = h k (r, x, g -1 y).

(2.5)

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 , ( 2.6)
where γ := ∑ α∈R + k(α) and d k is the constant

d k := ∫ S d-1 ω k (ξ)dσ(ξ). (2.7)
Here dσ(ξ) is the surface measure of the unit sphere S d-1 of R d .

7. Let r > 0 and x ∈ R d . Then the function h k (r, x, .) is upper semi-continuous on R d .

8. The harmonic kernel satisfies the following geometric inequality: if ∥a -b∥ ≤ 2r with r > 0, then

∀ ξ ∈ R d , h k (r, a, ξ) ≤ h k (4r, b, ξ) (2.8)
Note that in the classical case (i.e. k = 0), this inequality says that if ∥a -b∥ ≤ 2r, then B(a, r) ⊂ B(b, 4r). 9. Let x ∈ R d . Then the family of probability measures

dη k x,r (y) = 1 m k [B(0, r)] h k (r, x, y)ω k (y)dy (2.9)
is an approximation of the Dirac measure δ x as r -→ 0. That is

∀ α > 0, lim r→0 ∫ ∥x-y∥>α dη k x,r (y) = 0 (2.10) and if f ∈ C(Ω), then ∀ x ∈ Ω, lim r→0 ∫ R d f (y)dη k x,r = lim r→0 M r B (f )(x) = f (x). (2.11)
According to [START_REF] Mejjaoli | On a mean value property associated with the Dunkl Laplacian operator and applications[END_REF], the spherical mean of a C ∞ -function f defined on whole R d is given by

M r S (f )(x) := 1 d k ∫ S d-1 τ x f (ry)ω k (y)dσ(y), (2.12)
where d k is the constant (2.7) and τ x is the Dunkl translation (see Annex A.2). Moreover, M. Rösler has proved that there exists a compactly supported probability measure σ k x,r on R d which represents the spherical mean operator (see [START_REF] Rösler | A positive radial product formula for the Dunkl kernel[END_REF]). More precisely, for

f ∈ C ∞ (R d ), the spherical mean of f at (x, r) ∈ R d × R + is given by M r S (f )(x) = ∫ R d f (y)dσ k x,r (y), (2.13)
with supp σ k x,r ⊂ B W (x, r) = ∪ g∈W B(gx, r). (2.14)
Clearly formula (2.13) shows that we can define the spherical mean at (x, r) of any continuous function on B W (x, r).

As a first link between the spherical and the volume operators, we have Proposition 2.1 Let f be a continuous function on Ω. Then the formula

M r B (f )(x) = d + 2γ r d+2γ ∫ r 0 M t S (f )(x)t d+2γ-1 dt (2.15)
holds whenever B(x, r) ⊂ Ω. (2.15) has been proved in [START_REF] Gallardo | A new mean value property for harmonic functions relative to the Dunkl-Laplacian operator and applications[END_REF]. Now if f ∈ C(Ω), x ∈ Ω and r > 0 such that B(x, r) ⊂ Ω, the result follows by using (2.2), (2.13) and an uniform approximation of f by polynomials on the compact set B W (x, r).

Proof: When f ∈ C ∞ (R d ),
At the end of this section, we will give some useful properties of the volume mean operator when it acts on L 1 k,loc (Ω)-functions. Firstly, note that thanks to (2.2) and to the boundedness of the kernel h k , the function (x, r) → M r B (f )(x) is well defined for any f ∈ L 1 k,loc (Ω) whenever B(x, r) ⊂ Ω. Secondly, we will need the following notations which will be used frequently in this paper:

r Ω := sup{r > 0; Ω r ̸ = ∅}, ( 2.16)
with Ω r the open set defined by (1.6). Clearly, we have Ω

r 1 ⊂ Ω r 2 whenever r 2 ≤ r 1 and Ω = ∪ r>0 Ω r = ∪ r<r Ω Ω r (note that, since Ω ̸ = ∅, we have r Ω > 0). Moreover, since Ω r = { x ∈ Ω; B(x, r) ⊂ Ω } , (2.17) the open set Ω r , 0 < r < r Ω , is W -invariant. Proposition 2.2 Let f ∈ L 1 k,loc (Ω). 1) Let 0 < r < r Ω . Then the function M r B (f ) belongs to L 1 k,loc (Ω r ). 2) Let x ∈ Ω. Then the function r → M r B (f )(x) is continuous on ]0, ϱ x [ with ϱ x := dist(x, ∂Ω).
(2.18)

Proof: 1) By compactness, it suffices to prove that M r B (f )ω k ∈ L 1 (B(x 0 , R))
where B(x 0 , R) is an arbitrary closed ball of center x 0 and radius R included in Ω r . We have

I := ∫ B(x 0 ,R) M r B (f )(x) ω k (x)dx ≤ 1 m k (B(0,r)) ∫ B(x 0 ,R) ( ∫ B W (x,r) |f (y)|h k (r, x, y)ω k (y)dy ) ω k (x)dx ≤ 1 m k (B(0,r)) ∫ B(x 0 ,R) ( ∫ B W (x 0 ,R+r) |f (y)|ω k (y)dy ) ω k (x)dx ≤ m k (B(x 0 ,R)) m k (B(0,r)) ∫ B W (x 0 ,R+r) |f (y)|ω k (y)dy < +∞,
where the second inequality follows from the relation h k (r, x, y) ≤ 1 and from the fact that for every x ∈ B(x 0 , R) and every g ∈ W , B(gx, r) ⊂ B(gx 0 , R + r) ⊂ Ω.

2) By (2.6), it suffices to show that

ϕ : r → ∫ R d f (y)h k (r, x, y)ω k (y)dy is continuous on ]0, ϱ x [. Since r → h k (r, x, y) is right-continuous, by the dominated convergence theorem, we can see that ϕ is also right-continuous on ]0, ϱ x [. Now, fix r ∈]0, ϱ x [ and η > 0 such that ]r -η, r + η[⊂]0, ϱ x [. Let (r n ) be a sequence of nonnegative real number such that r n -→ 0 as n -→ +∞.
Using (2.5), (1.10) and applying Fubini's theorem, we obtain

|ϕ(r) -ϕ(r -r n )| ≤ ∫ R d (∫ Ω |f (y)|1 ]r-rn,r] ( √ ∥y∥ 2 + ∥x∥ 2 -2 ⟨y, z⟩)ω k (y)dy ) dµ x (z) = ∫ R d (∫ An |f (y)|ω k (y)dy ) dµ x (z),
where

A n = A n (x, z) := { y ∈ Ω, r -r n < √ ∥y∥ 2 + ∥x∥ 2 -2 ⟨y, z⟩ ≤ r } . Since ∩ n A n is a hypersurface, by the dominated convergence theorem, we get lim n→+∞ ∫ R d ( ∫ An |f (y)|ω k (y)dy)dµ x (z) = 0.
Hence, by the previous relations, we conclude that ϕ is also left continuous.

Dunkl convolution Product

The Dunkl convolution product has been defined by means of the Dunkl translation operators (see [START_REF] Trimèche | Paley-Wiener theorem for the Dunkl transform and Dunkl translation operators[END_REF] and [START_REF] Rösler | Dunkl Operators: Theory and Applications[END_REF]). So that it has been considered only in some particular cases. Here, we will prove that we can define the Dunkl convolution product of a function u ∈ L 1 k,loc (Ω) with a nonnegative and radial function f ∈ D(R d ) and we will study some properties of this product.

For f, g ∈ S(R d ), the Dunkl convolution product is defined by

∀ x ∈ R d , f * k g(x) := ∫ R d τ x f (-y)g(y)ω k (y)dy. (3.1)
We note that it is commutative and satisfies the following property:

F k (f * k g) = F k (f )F k (g), (3.2)
where F k is the Dunkl transform (see Annex A.1).

From (3.2), (A.8) and the injectivity of the F k transform, we obtain the following relations

Lemma 3.1 Let f, g ∈ S(R d ).
Then, for every x ∈ R d , we have

(τ x f ) * k g = f * k (τ x g) = τ x (f * k g). (3.3) Theorem 3.1 Let u ∈ L 1 k,loc (Ω) and ϕ ∈ D + ρ (R d ) with 0 < ρ < r Ω . Let u * k ϕ(x) := ∫ R d u(y)τ -x ϕ(y)ω k (y)dy. (3.4)
Then 1) the function u * k ϕ is well defined on Ω ρ and can be written

∀ x ∈ Ω ρ , u * k ϕ(x) = ∫ R d u(y)τ -y ϕ(x)ω k (y)dy (3.5) = ∫ R d u(y)τ x ϕ(-y)ω k (y)dy, (3.6) 2) u * k ϕ belongs to C ∞ (Ω ρ ) and we have ∆ k (u * k ϕ) = u * k ∆ k ϕ, (3.7)
3) for all B(x, r) ⊂ Ω ρ , we have

M r B (u * k ϕ)(x) = M r B (u) * k ϕ(x). (3.8)
Proof: 1) • For every ε > 0, we see that

∀ y ∈ R d , 0 ≤ ϕ(y) ≤ ∥ϕ∥ ∞ 1 B(0,ρ) (y) ≤ ∥ϕ∥ ∞ φ ε (y),
where (φ ε ) is a sequence satisfying (2.3) (with r = ρ). Using the positivity of the Dunkl translation operators on radial functions, we deduce that

∀ y ∈ R d , 0 ≤ τ -x ϕ(y) ≤ ∥ϕ∥ ∞ τ -x φ ε (y).
Letting ε → 0 and using (2.4), we obtain

∀ y ∈ R d , 0 ≤ τ -x ϕ(y) ≤ ∥ϕ∥ ∞ h k (ρ, x, y). (3.9)
Consequently, from the relations (2.2) and (3.9), we get that

supp τ -x ϕ ⊂ B W (x, ρ). (3.10)
This implies that for all x ∈ Ω ρ , the function y → u(y)τ -x ϕ(y)ω k (y) is integrable on Ω.

• The relation (3.5) follows from (A.13) and the relation (3.6) follows from (3.5) and (A.10).

2) Let x 0 ∈ Ω ρ and R > 0 such that B(x 0 , R) ⊂ Ω ρ . We shall prove that the function

u * k ϕ is of class C ∞ on • B(x 0 , R). Define the function Φ on R d × R d by (see (A.9)) Φ(x, y) := τ -x ϕ(y) = c -2 k ∫ R d F k (ϕ)(ξ)E k (-ix, ξ)E k (iy, ξ)ω k (ξ)dξ. We see that Φ is in C ∞ (R d × R d )

and by (A.3) and the inequality |E

k (iy, ξ)| ≤ 1, for every multi-indices υ ∈ N d we get ∀ (x, y) ∈ R d × R d , ∂ υ ∂x υ Φ(x, y) ≤ c -2 k ∫ R d |F k (ϕ)(ξ)| ∥ξ∥ |υ| ω k (ξ)dξ := C υ < +∞.
On the other hand, from (3.10) we have

∀ x ∈ • B(x 0 , R), supp Φ(x, .) ⊂ B W (x, ρ) ⊂ B W (x 0 , R + ρ) ⊂ Ω.
This implies that we can write

∀ x ∈ • B(x 0 , R), ∀ y ∈ R d , Φ(x, y) = Φ(x, y)1 B W (x 0 ,R+ρ) (y).
Thus, for every multi-indices υ ∈ N d , we deduce that

∀ x ∈ • B(x 0 , R), ∀ y ∈ R d , ∂ υ ∂x υ Φ(x, y) ≤ C υ 1 B W (x 0 ,R+ρ) (y).
Now, since uω k is locally integrable, this proves that we can differentiate under the integral sign in (3.4) and we obtain the desired result. Furthermore, using respectively (3.5), (A.11) and (A.13) (here note that we can use the relation (A.13) because ∆ k ϕ is also a radial function2 (see [START_REF] Mejjaoli | On a mean value property associated with the Dunkl Laplacian operator and applications[END_REF])), we obtain

∆ k (u * k ϕ)(x) = ∫ R d u(y)∆ k [τ -y ϕ](x)ω k (y)dy = ∫ R d u(y)τ -y [∆ k ϕ](x)ω k (y)dy = ∫ R d u(y)τ -x [∆ k ϕ](y)ω k (y)dy = u * k ∆ k ϕ(x).
This completes the proof of 2).

3) We need the following lemma:

Lemma 3.2 Let f ∈ S(R d ) be radial.
Then, for all r > 0 and a, b ∈ R d , we have • Let r > 0 and a, b ∈ R d . We have

τ a τ b f = τ b τ a f (3.11) and M r B (τ -a f )(b) = M r B (τ -b f )(a). ( 3
M r B (τ -a f )(b) = 1 m k (B(0, r)) ∫ R d τ a f (-y)h k (r, b, y)ω k (y)dy = 1 m k (B(0, r)) ∫ R d τ b τ a f (-y)1 B(0,r) (y)ω k (y)dy = 1 m k (B(0, r)) ∫ R d τ b f (-y)h k (r, a, y)ω k (y)dy = M r B (τ -b f )(a),
where we have used respectively the relations (A.13) and (A.10) in the first equality, the relation (2.1) in the second equality, the relations (3.11) and (2.1) in the third equality and the relations (A.10), (A.13) in the last equality. Now, we turn to the proof of (3.8). Let B(x, r) ⊂ Ω ρ . By Proposition 2.2-1), the function M r B (u) belongs to L 1 k,loc (Ω r ). This proves, by assertion 1), that the function M r B (u) * k ϕ is well defined on Ω ρ+r .

By 2), the function u * k ϕ is clearly in L 1 k,loc (Ω ρ ) and for x ∈ Ω ρ+r we have3

M r B (u * k ϕ)(x) = 1 m k (B(0, r)) ∫ B W (x,r) ( ∫ B W (x,r+ρ) u(z)τ -y ϕ(z)ω k (z)dz) h k (r, x, y)ω k (y)dy = 1 m k (B(0, r)) ∫ B W (x,r+ρ) u(z)( ∫ B W (x,r) τ -z ϕ(y)h k (r, x, y)ω k (y)dy) ω k (z)dz = ∫ B W (x,r+ρ) u(z)M r B ( τ -z ϕ ) (x)ω k (z)dz = ∫ B W (x,r+ρ) u(z)M r B ( τ -x ϕ ) (z)ω k (z)dz = 1 m k (B(0, r)) ∫ B W (x,r+ρ) u(z)( ∫ B W (x,ρ) τ -x ϕ(y)h k (r, z, y)ω k (y)dy) ω k (z)dz = 1 m k (B(0, r)) ∫ B W (x,ρ) τ -x ϕ(y)( ∫ B W (x,r+ρ) u(z)h k (r, y, z)ω k (z)dz) ω k (y)dy = M r B (u) * k ϕ(x),
where, -the first equality follows from the relations (2.2), (3.5), (3.10) and from the fact that

∀ y ∈ B W (x, r), B W (y, ρ) ⊂ B W (x, r + ρ) ⊂ Ω, (3.13)
-the second equality follows from Fubini's theorem and (A.13), -the third equality comes from the relation (1.9), -the forth equality follows from (3.12), -in the the fifth equality we have used the relations (1.9) and (3.10), -in the sixth equality we have used (2.5) and Fubini's theorem. Finally, using (2.2) and (3.13), we obtain the last equality. This completes the proof of the theorem.

Remark 3.1 If u is of class C 2 on Ω, then the relation (3.7) can be also written

∀ x ∈ Ω ρ , ∆ k (u * k ϕ)(x) = (∆ k u) * k ϕ(x). (3.14)
Indeed, using respectively the relations (3.7), (3.4) and (A.11), we get

∆ k (u * k ϕ)(x) = ∫ Ω u(y)τ -x [∆ k ϕ](y)ω k (y)dy = ∫ Ω u(y)∆ k [τ -x ϕ](y)ω k (y)dy. (3.15)
Now, we obtain the desired result by using the following integration by parts formula (see [START_REF] Dunkl | Hankel transforms associated to finite reflection groups[END_REF] or [START_REF] Rösler | Dunkl Operators: Theory and Applications[END_REF]): Let f, g ∈ C 1 (Ω) such that g has compact support. Then, for all ξ ∈ R d , we have

∫ Ω D ξ f (x)g(x)ω k (x)dx = - ∫ Ω f (x)D ξ g(x)ω k (x)dx, (3.16)
with D ξ the ξ-directional Dunkl operator (see (1.2)).

Proposition 3.1 Let u ∈ L 1 k,loc (Ω) and ϕ ∈ D + ρ (R d ) with 0 < ρ < r Ω . If u is with compact support, then u * k ϕ is also with compact support and supp (u * k ϕ) ⊂ B(0, ρ) + W.supp u ⊂ Ω, (3.17)
with W.supp u := {gx, (g, x) ∈ W × supp u}.

Proof: If x / ∈ B(0, ρ)+W.supp u, then x-gy / ∈ B(0, ρ) for every (g, y) ∈ W ×supp u. That is ∥gx -y∥ > ρ for all y ∈ supp u and all g ∈ W . In other words supp u ∩ B W (x, ρ) = ∅. Hence, by (3.10), we obtain u * k ϕ(x) = 0.

It is interesting to note that when u is a continuous function, we can write the Dunkl convolution product in spherical coordinates as follows: Proposition 3.2 Let u be a continuous function on Ω and let ϕ ∈ D + ρ (R d ) with 0 < ρ < r Ω (i.e. Ω ρ is nonempty). Then, for all x ∈ Ω ρ , we have

u * k ϕ(x) = d k ∫ ρ 0 ϕ(t)t d+2γ-1 M t S (u)(x)dt, (3.18)
where ϕ is the profile function of ϕ and d k is the constant given by (2.7).

Proof: At first we suppose that u ∈ C ∞ (R d ). By (A.14), we have

u * k ϕ(x) = ∫ R d ϕ(y)τ x u(y)ω k (y)dy.
Then, using spherical coordinates, we can write

u * k ϕ(x) = ∫ ρ 0 ϕ(t)t d+2γ-1 ∫ S d-1 τ x u(tξ)ω k (ξ)dσ(ξ)dt.
Therefore, from (2.12) we deduce that the relation (3.18) holds in this case.

Let us now suppose only that u is a continuous function on Ω. Let (p n ) a sequence of polynomial functions such that p n -→ u as n -→ +∞ uniformly on the compact set K := B W (x, ρ). Since τ -x ϕ ≥ 0, by (A.12) we conclude that

u * k ϕ(x) -p n * k ϕ(x) ≤ ∥τ -x ϕ∥ L 1 k (R d ) sup K |p n (y) -u(y)| = ∥ϕ∥ L 1 k (R d ) sup K |p n (y) -u(y)|. Hence u * k ϕ(x) = lim n→+∞ p n * k ϕ(x). (3.19)
Furthermore, as the probability measures σ k x,t , 0

< t ≤ ρ, have compact support contained in B W (x, t) ⊂ K = B W (x, ρ) (see (2.14)), we deduce ∀ t ≤ ρ, |M t S (p n -u)(x)| ≤ sup K |p n (y) -u(y)|.
This implies We have the following associativity result for the Dunkl convolution product:

lim n→+∞ d k ∫ ρ 0 ϕ(t)t d+2γ-1 M t S (p n )(x)dt = d k ∫ ρ 0 ϕ(t)t d+2γ-1 M t S (u)(x)dt. ( 3
Proposition 3.3 Let u ∈ L 1 k,loc (Ω), ϕ ∈ D + ρ (R d ) and ψ ∈ D + r (R d ) such that Ω r+ρ is nonempty (i.e. r + ρ < r Ω ). Then ∀ x ∈ Ω r+ρ , ( u * k ϕ ) * k ψ(x) = u * k (ϕ * k ψ)(x) = ( u * k ψ ) * k ϕ(x). ( 3

.21)

Proof: • From Theorem 3.1, the functions

( u * k ϕ ) * k ψ and ( u * k ψ ) * k ϕ are well defined on Ω r+ρ . • We claim that ϕ * k ψ is a nonnegative C ∞ -radial function on R d with compact support contained in B(0, r + ρ) which implies that u * k (ϕ * k ψ
) is also well defined on Ω r+ρ . Indeed, again by Theorem 3.1 we see that ϕ * k ψ is of class C ∞ on R d and using (3.17) we obtain supp ϕ * k ψ ⊂ B(0, r + ρ). Furthermore, by the positivity of Dunkl translation operators on radial functions, we deduce that the function ϕ * k ψ is nonnegative. Now, using the fact that the Dunkl transform F k is an isomorphism of the Schwartz space onto itself and the relation (3.2), we can write that

ϕ * k ψ = F -1 k (F k (ϕ)F k (ψ)) .
Therefore, since F k preserves the radial property (see the relation (A.6)), we deduce that ϕ * k ψ is radial as claimed.

• For x ∈ Ω r+ρ fixed, we have

( u * k ϕ ) * k ψ(x) = ∫ B W (x,r) ( u * k ϕ ) (y)τ -x ψ(y)ω k (y)dy = ∫ B W (x,r) ( ∫ B W (x,r+ρ) u(z)τ -y ϕ(z)ω k (z)dz ) τ -x ψ(y)ω k (y)dy = ∫ B W (x,r+ρ) u(z) ( ∫ B W (x,r) τ -y ϕ(z)τ -x ψ(y)ω k (y)dy ) ω k (z)dz = ∫ B W (x,r+ρ) u(z) ( ∫ B W (x,r) τ -z ϕ(y)τ -x ψ(y)ω k (y)dy ) ω k (z)dz = ∫ B W (x,r+ρ) u(z) ( ϕ * k τ -x ψ ) (z) ω k (z)dz = ∫ B W (x,r+ρ) u(z)τ -x ( ϕ * k ψ ) (z) ω k (z)dz = u * k (ϕ * k ψ)(x).
where

(u * k ψ) * k ϕ(x) = u * k (ψ * k ϕ)(x).
Finally, by the commutativity of the Dunkl convolution product (see (3.2)), we conclude the last equality in (3.21).

Applications to ∆ k -harmonic functions

In this section, we will give some properties of Dunkl harmonic functions. Let us introduce the space H k (Ω) of ∆ k -harmonic functions on Ω.

In order to give further study of Dunkl harmonic functions, we first need some lemmata. Let us consider the following radial function

φ(x) := a exp (- 1 1 -∥x∥ 2 )1 B(0,1) (x), x ∈ R d , (4.1)
where a is a constant such that x -→ φ(x)ω k (x) is a probability density. For ε > 0, define the function

φ ε (x) = 1 ε d+2γ φ( x ε ). (4.2)
It is well known that φ ε is a radial mollifier i.e. φ ε ∈ D + ε (R d ). We begin by the following preparatory result:

Lemma 4.1 Let u be a continuous function on Ω. For 0 < ε < r Ω , define the function u ε by ∀ x ∈ Ω ε , u ε (x) := u * k φ ε (x) := ∫ R d u(y)τ -x φ ε (y)ω k (y)dy. (4.3)
Then the sequence

(u ε ) 0<ε<r Ω satisfies i) for every 0 < ε < r Ω , the function u ε is in C ∞ (Ω ε ), ii) for every x ∈ Ω, u ε (x) -→ u(x) as ε -→ 0.
iii) for every 0 < r < r Ω and every x ∈ Ω r , we have M r B (u

* k φ ε )(x) -→ M r B (u)(x) and M r S (u * k φ ε )(x) -→ M r S (u)(x) as ε -→ 0.
Proof: The first assertion follows immediately from Theorem 3.1.

ii) Let x ∈ Ω. Applying (3.9) with ϕ = φ ε we get

∀ y ∈ R d , 0 ≤ τ -x φ ε (y) ≤ aε -d-2γ h k (ε, x, y).
Using (2.6), we can write

∀ y ∈ R d , 0 ≤ τ -x φ ε (y) ≤ a d k d + 2γ 1 m k [B(0, ε)] h k (ε, x, y). ( 4.4)
Consequently, for every x ∈ Ω and every ε > 0 small enough, we have by (A.12) and (4.4)

|u ε (x) -u(x)| ≤ ∫ R d τ -x φ ε (y)|u(y) -u(x)|ω k (y)dy ≤ a d k d + 2γ 1 m k [B(0, ε)] ∫ R d |u(y) -u(x)|h k (ε, x, y)ω k (y)dy.
This can be rewritten in the following form

|u ε (x) -u(x)| ≤ a d k d + 2γ M ε B ( |u -u(x)| ) (x). (4.5)
Thus from (2.11), we conclude the result.

iii) Let x ∈ Ω r . There exists

ε 0 > 0 such that x ∈ Ω r+ε 0 . For ε ∈]0, ε 0 [, we have ∀ y ∈ B W (x, r), |u * φ ε (y)| ≤ ∫ B W (y,ε) |u(z)|τ -y φ ε (z)ω k (z)dz ≤ ∫ B W (x,r+ε 0 ) |u(z)|τ -y φ ε (z)ω k (z)dz ≤ sup B W (x,r+ε 0 ) |u(y)|,
where in the last line we have used the relation (A.12). According to (2.2), (2.14) and to the statement ii), the previous inequality implies that we can apply the dominated convergence theorem to obtain the results of assertion iii).

Lemma 4.2 Let u be a C 2 -function on Ω. Then, for any 0 < r < r Ω and any

x ∈ Ω r , we have M r S (u)(x) = u(x) + 1 d + 2γ ∫ r 0 M t B (∆ k u)(x) tdt, ( 4.6
)

and M r B (u)(x) = u(x) + 1 r d+2γ ∫ r 0 ∫ ρ 0 M t B (∆ k u)(x) t dt ρ d+2γ-1 dρ. ( 4.7)
Remark 4.1 We have obtained in [START_REF] Gallardo | A new mean value property for harmonic functions relative to the Dunkl-Laplacian operator and applications[END_REF] that the relations (4.6) and (4.7) hold when u ∈ C ∞ (R d ). Furthermore, by polynomial approximation, we have extended them to any function u of class C 2 on an arbitrary W -invariant open set Ω ⊂ R d and any x ∈ Ω but with the condition r ∈]0, ϱ x /3[, ϱ x being defined by (2.18).

Proof:

Step 1: Suppose that u is of class C ∞ on Ω. Let x ∈ Ω and r ∈]0, ϱ x [ and let ϵ > 0 such that B(x, r + ϵ) ⊂ Ω. We can find ϕ ∈ D(R d ) such that 1. ϕ = 1 on the compact set B W (x, r + ϵ/2), 2. supp ϕ ⊂ B W (x, r + ϵ), 3. 0 ≤ ϕ ≤ 1. Therefore, the function f = uϕ is in C ∞ (R d ), supp f ⊂ B W (x, r + ϵ) and f = 1 on B W (x, r + ϵ/2).
According to Remark 4.1 and noting that the relations (4.6) and (4.7) only involve the compact set B W (x, r) (through the supports of h k (r, x, .) and σ k x,r ), we can replace f by u in theses formulas.

Step 2: Here, we will suppose that u ∈ C 2 (Ω). By (2.15), it is enough to prove (4.6). Fix 0 < r < r Ω , x ∈ Ω r and ε 0 > 0 such that x ∈ Ω r+ε 0 . By step 1, for every 0

< ε < ε 0 we have M r S (u * k φ ε )(x) = u * k φ ε (x) + 1 d + 2γ ∫ r 0 M t B (∆ k [u * k φ ε ])(x)tdt. ( 4
M t B (∆ k [u * k φ ε ])(x) = lim ε→0 M t B ([∆ k u] * k φ ε )(x) = M t B (∆ k u)(x),
and following the proof of the statement iii) of Lemma 4.1, for every 0 < ε < ε 0 we obtain

∀ t ≤ r, |M t B (∆ k [u * k φ ε ])(x)| ≤ sup B W (x,r+ε 0 ) |∆ k u(y)|.
Therefore, we can use the dominated convergence theorem in the integral of (4.8) and using again Lemma 4.1, we obtain the result by letting ε -→ 0.

In the following result, we will characterize the ∆ k -harmonicity by the global mean value property.

i) u ∈ H k (Ω), ii) u(x) = M r S (u)(x) whenever B(x, r) ⊂ Ω, iii) u(x) = M r B (u)(x) whenever B(x, r) ⊂ Ω, iv) M r B (u)(x) = M r S (u)(x) whenever B(x, r) ⊂ Ω. Proof: i) ⇒ ii) It
d dr M r B (f )(x) = d + 2γ r ( M r S (f )(x) -M r B (f )(x)
) . (4.9)

Now, from (4.9) clearly iii) implies iv). iv) ⇒ i) Let x ∈ Ω. Using (4.9) and (2.11), we deduce that

∀ r ∈ [0, ρ x [, M r B (u)(x) = u(x).
Now, if we use (4.7) et we differentiate two times with respect to r, we get M r B (∆ k u)(x) = 0 for every r ∈]0, ρ x [. Finally, by (2.11), we obtain ∆ k u(x) = 0. Proof: Let u be a ∆ k -harmonic function on Ω. As u satisfies the spherical mean property, by (3.18), we see that u = u * k φ ε on Ω ε . Therefore, u is of class C ∞ on Ω ε for every ε > 0 arbitrary small.

Corollary 4.2 Every

∆ k -harmonic function on R d is real analytic. Proof: Let f be a ∆ k -harmonic function on R d . Since f ∈ C ∞ (R d ) (by Corollary 4.1) and V k : C ∞ (R d ) -→ C ∞ (R d ) is a topological isomorphism, the function g := V -1
k (f ) is harmonic on R d in the usual sense (i.e. ∆g = 0) as an immediate consequence of the intertwining relation 4

∆ k V k = V k ∆.
It is well known that g is real analytic (see [START_REF] Axler | Harmonic Function Theory[END_REF]) and thus, using multi-indices υ = (υ 1 , ..., υ d ) ∈ N d , can be written

g(x) = ∑ υ a υ x υ , x ∈ R d , where a υ are real coefficients. If g N (N ∈ N) denotes the partial sum g N (x) := ∑ |υ|≤N a υ x υ (with |υ| = υ 1 + • • • + υ d ), then g N -→ g as N → +∞ in the Fréchet topology of C ∞ (R d ). Therefore V k (g N ) -→ V k (g) = f in the Fréchet topology.
In particular, f is real analytic as being the uniform limit of the polynomials5 V k (g N ) on each compact subset of R d . whenever 0 < r < r Ω and ϕ ∈ D + r (R d ) is such that ϕω k is a probability density. Proof: Using the spherical mean value property and the relation (3.18) we see that any ∆ k -harmonic function on Ω satisfies (4.10). Conversely, let u ∈ L 1 k,loc (Ω) satisfying (4.10). At first, by Theorem 3.1, clearly the function u is of class C ∞ on Ω. On the other hand, let x ∈ Ω and 0 < r < ρ x be fixed. Let (φ ε ) be a sequence as in (2.3) (i.e. such that τ -x φ ε (y) -→ h k (r, x, y) as ε → 0). So, by the dominated convergence theorem, the hypothesis

u * k φ ε (x) ∥φ ε ∥ L 1 (R d ,m k ) = u(x)
implies that M r B (u)(x) = u(x). Thus, u ∈ H k (Ω) by the volume mean value property. In the following result, we will characterize de ∆ k -harmonicity by the local mean value property: is well defined and is of class C ∞ on Ω ε . Furthermore, using (3.15) and the fact that uω k is weakly ∆ k -harmonic on Ω, we can see that the function u ε is (strongly) ∆ k -harmonic in Ω ε .

• Let us now fix r such that 0 < r < r Ω . We claim that

u ε 1 (x) = u ε 2 (x) whenever x ∈ Ω r and ε 1 + ε 2 < r (4.12)
Indeed, since u ε 1 ∈ H k (Ω r ), by the spherical mean property and (3.18), we have

u ε 1 = u ε 1 * k φ ε 2 = (u * k φ ε 1 ) * k φ ε 2 on Ω r .
If we change the role of ε 1 and ε 2 , we also obtain

u ε 2 = u ε 2 * k φ ε 1 = (u * k φ ε 2 ) * k φ ε 1 on Ω r .
Thus the relation (4.12) follows from the associativity property (3.21).

• Now, we will use the following lemma which follows from (4.5) and Lebesgue's differentiation theorem (see [START_REF] Rejeb | Lebesgue's differentiation theorem in Dunkl setting[END_REF]):

Lemma 4.3 For almost every x ∈ Ω, u ε (x) -→ u(x) as ε -→ 0.
Letting ε 2 -→ 0 in the relation (4.12) and using Lemma 4.3, we deduce that u coincides almost everywhere with the ∆ k -harmonic function h := u ε 1 on Ω r . Since r can be taken arbitrarily small, the proof of the theorem is complete. [START_REF] Daher | Weyl's lemma and converse mean value for Dunkl operators[END_REF] for ∆ k -harmonic functions f on whole R d and under the additional assumption that the function f is locally bounded. Now, we will characterize the strong ∆ k -harmonicity by means of weak ∆ k -harmonicity.

Remark 4.2 A version of Weyl's lemma has been proved in

More precisely, we have Proof: By integration by parts formula (3.16) and (2.11), we obtain the 'only if' part. Let us now prove the 'if' part. By Theorem 4.2, there exists h ∈ H k (Ω) such that u = h a.e. on Ω. Therefore, for all x ∈ Ω, we have ∀ r ∈]0, ϱ x [, M r B (u)(x) = M r B (h)(x) = h(x), where ϱ x is defined by (2.18). We obtain u = h on Ω by letting r → 0. (A.3)

• 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 , (A.4)
where c k is the Macdonald-Mehta constant given by c k = ∫ R d e -∥x∥ 2 2 ω k (x)dx (see [START_REF] Etingof | A uniform proof of the Macdonald-Mehta-Opdam identity for finite Coxeter groups[END_REF]). Moreover, the following Plancherel theorem holds (see [START_REF] De Jeu | The Dunkl transform[END_REF]): The transformation c -1 k F k extends uniquely to an isometric isomorphism of L 2 (R d , m k ) and we have:

∀ f ∈ L 2 (R d , m k ), ∥c -1 k F k (f )∥ L 2 (R d ,m k ) = ∥f ∥ L 2 (R d ,m k ) . (A.5)
• It is useful to note that if f ∈ L 1 (R d , m k ) is radial, F k (f ) is also radial. Precisely, using spherical coordinates and Corollary 2.5 of ( [START_REF] Rösler | A positive radial product formula for the Dunkl kernel[END_REF]), we have

F k (f )(λ) = d k ∫ +∞ 0 f (r)j γ+

A.2 Dunkl's translation operators

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), (A.7)
where T x is the classical translation operator given by T x f (y) = f (x + y). If 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]):

∀ y ∈ R d , τ x f (y) = F -1 k [E k (ix, .)F k (f )](y) (A.8) = 1 c 2 k ∫ R d F k (f )(λ)E k (ix, λ)E k (iy, λ)ω k (λ)dλ. (A.9)
In particular, the relations (A.9) and (A.3) show that (x, y)

-→ τ x f (y) is of class C ∞ on R d × R d .
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.

  Let Ω ⊂ R d be a (nonempty) W -invariant open set and ϕ a radial mollifier i.e. ϕ ∈ D + r (R d ) with r > 0 small enough in order that the open set Ω r := { x ∈ Ω; dist(x, ∂Ω) > r } (1.6) is nonempty. For every u ∈ L 1 loc (Ω, m k ), with dm k = ω k (x)dx, we define the Dunkl-convolution product u * k ϕ(x) := ∫ R d u(y)τ -x ϕ(y)ω k (y)dy, (

  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.• B(a, ρ) (resp. • B(a, ρ)) the closed (resp. the open) Euclidean ball centered at a and with radius ρ > 0. • D + r (R d ), r > 0, the set of radial mollifiers with support contained in B(0, r).

. 12 ) 2 •

 122 Proof of Lemma 3.We obtain (3.11) from (A.8) and the injectivity of the Dunkl transform on S(R d ).

  .20) From(3.19),(3.20) and the first step, we deduce that the relation(3.18) holds when the function u is continuous on Ω.

  we have used -the relations (3.4) and (3.10) in the first line; -the same relations in the second line with (3.13); -Fubini's theorem in the third line: the relation (3.9), the inequality h k (R, a, b) ≤ 1 and the hypothesis u ∈ L 1 k,loc (Ω) imply that we can use Fubini's theorem; -the relation (A.13) in the forth line; -the relation (3.4) in the fifth line; -relation (3.3) in the sixth line; -the above properties of the function ϕ * k ψ and (3.10) in the last line. Now, changing the role of ϕ and ψ, we obtain

Proposition 4 . 1

 41 Let u ∈ C 2 (Ω). The following statements are equivalent

Corollary 4 . 1

 41 Every ∆ k -harmonic function on Ω is of class C ∞ .

Corollary 4 . 3

 43 Let u be a function defined on Ω. Then u ∈ H k (Ω) if and only if u ∈ L 1 k,loc (Ω) and satisfies u * k ϕ = u on Ω r (4.10)

Corollary 4 . 4

 44 Let u be a function defined on Ω. Then u ∈ H k (Ω), if and only if u satisfies : u ∈ L 1 k,loc (Ω), ∆ k (uω k ) = 0 in D ′ (Ω) and u(x) = lim r→0 M r B (u)(x) for every x ∈ Ω.

Corollary 4 . 5

 45 The space H k (Ω) is closed for the L 1 k,loc (Ω)-topology.Proof: Let (u n ) be a sequence of ∆ k -harmonic functions on Ω such that u n -→ u in L 1 k,loc (Ω). As the functions u n ω k and uω k are in L 1 loc (Ω), we have also u n ω k -→ uω k in D ′ (Ω). In particular, ∆ k (u n ω k ) -→ ∆ k (uω) in D ′ (Ω). Now, since u n ∈ H k (Ω), we deduce that ∆ k (uω k ) = 0 in D ′ (Ω).Finally, by Theorem 4.2 there exists a ∆ k -harmonic function h on Ω such that u = h a.e. in Ω. Thus u = h in L 1 k,loc (Ω) and the result is proved. As a last result, we will improve Proposition 4.1 as follows 2. for all x ∈ R d , y ∈ C d and all multi-indices υ ∈ N d , ∂ υ ∂y υ E k (x, y) ≤ ∥x∥ |υ| max g∈W e Re ⟨gx,y⟩ .

d 2 - 1

 21 (r∥λ∥)r 2γ+d-1 dr, λ ∈ R d , (A.6)where d k is defined by the relation (2.7) and for λ ≥ -1/2, j λ is the normalized Bessel function given by j λ (z) = Γ(λ + 1)

carrying its usual Fréchet topology.

More precisely, we have ∆ k ϕ(x) = ( d 2 dr 2 + d+2γ-1 r d dr ) ϕ(r), r = ∥x∥ and ϕ the profile of ϕ.

Note that, in the integrals below, the consideration of the supports permits to justify the correct application of Fubini's theorem.

which follows clearly from (1.3).

V k is a bijection of the space of polynomials of degree ≤ n onto itself ([START_REF] Dunkl | Orthogonal Polynomials of Several variables[END_REF]).

ii) u satisfies the local spherical mean property i.e. for every x ∈ Ω, there exists r x > 0 such that ∀ r < r x , u(x) = M r S (u)(x),

iii) u satisfies the local volume mean property i.e. for every x ∈ Ω, there exists r x > 0 such that ∀ r < r x , u(x) = M r B (u)(x).

Proof: i) =⇒ ii) We know that the ∆ k -harmonic function u satisfies the global spherical mean property and then it satisfies also the local one.

ii

The local volume property and the relation (4.7) imply

If we differentiate two times with respect to r, we obtain

Hence, according to (2.11), we get ∆ k u(x) = 0.

In order to prove a Weyl type lemma for Dunkl harmonic functions, we will clarify some facts about the action of Dunkl operators on distributions. For a distribution T ∈ D ′ (Ω), we define the weak Dunkl ξ-directional derivative of T (ξ ∈ R d ) by

Note that by the intertwining relation (1.3), the operator

) is continuous for the Fréchet topology. Moreover, since D ξ leaves the space D(Ω) invariant, we deduce that D ξ : D(Ω) -→ D(Ω) is also continuous for the Fréchet topology. This justifies that D ξ T is well defined as an element of D ′ (Ω).

Then, there exists a Dunkl harmonic function h on Ω such that u = h a.e. on Ω.

Proof: Let 0 < ε < r Ω and let φ ε the function defined by (4.2). From Theorem 3.1, we know that the function

Corollary 4.6 Let u be a function defined on Ω. The following statements are equivalent

iii) u is continuous on Ω and u(x) = M r B (u)(x) whenever B(x, r) ⊂ Ω.

), clearly we obtain the result.

iii) ⇒ i) Let ε 0 > 0 small enough. Since u satisfies the volume mean value property, from (3.8) we see that the C ∞ -function u * k φ ε (ε < ε 0 ) defined by (4.3) satisfies also

for any r such that 0 < r < r + ε 0 < r Ω with r Ω given by (2.16). Thus, according to Proposition 4.1, u * k φ ε ∈ H k (Ω ε 0 ). On the other hand, let K be a compact subset of Ω ε 0 . Noting that for all 0 < ε < ε 0 and all

0, ε 0 ) and using (A.12) and the fact that τ -x φ ε ≥ 0 , we can see that

From this inequality and from the dominated convergence theorem, we deduce that when ε tends to zero, u * k φ ε -→ u in L 1 (K, m k ). In other words, u * k φ ε -→ u in L 1 k,loc (Ω ε 0 ). Finally, Corollary 4.5 and the continuity of u imply that u coincides with a Dunkl harmonic function h on Ω ε 0 . As ε 0 > 0 can be chosen arbitrary small, the result is proved.

A Annex

A.1 The Dunkl transform

In this Annex we recall some properties of the Dunkl transform (see [START_REF] De Jeu | The Dunkl transform[END_REF] and [START_REF] Rösler | Dunkl Operators: Theory and Applications[END_REF] ).

where E k (x, y) := V k (e ⟨x,.⟩ )(y), x, y ∈ R d , is the Dunkl kernel 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] and [START_REF] Rösler | Dunkl Operators: Theory and Applications[END_REF])

1. for all x, y ∈ C d , all λ ∈ C and all g ∈ W ,

3) For all f ∈ C ∞ (R d ) and all x, y ∈ R d , we have

4) The Dunkl-Laplace operator ∆ k commutes with the Dunkl translations i.e

f (y)ω k (y)dy, (A.12)

6)

Let f ∈ S(R d ) be radial. Then we have (see [START_REF] Gallardo | A new mean value property for harmonic functions relative to the Dunkl-Laplacian operator and applications[END_REF], Lemme 3.1)

The following duality formula has been established by the authors (see [START_REF] Gallardo | A new mean value property for harmonic functions relative to the Dunkl-Laplacian operator and applications[END_REF], Proposition 2.1):


Alvaro De Andres-Pablo

Antonin Morillon
email: antonin.morillon@curie.fr

Maxime Wery

LncRNAs, lost in translation or licence to regulate?

Over the last decade, advances in transcriptomics have revealed that the pervasive transcription of eukaryotic genomes produces plethora of long noncoding RNAs (lncRNAs), which are now recognized as major regulators of multiple cellular processes. Although they have been thought to lack any protein-coding potential, recent ribosome-profiling data indicate that lncRNAs can interact with the translation machinery, leading to the production of functional peptides in some cases. In this perspective, we have explored the idea that translation can be part of the fate of cytoplasmic lncRNAs, raising the possibility for them to work as bifunctional RNAs, endowed with dual coding and regulatory functions.

Introduction: the arrival on the scene of the non-coding RNAs

With protein-coding genes representing only 2% of the human genome, the other 98% have been considered for a long time as inactive material, regions of several mega-bases without any function, so-called 'junk DNA' [START_REF] Taft | The relationship between non-protein-coding DNA and eukaryotic complexity[END_REF]). However, the overwhelming development of high-density micro-arrays and high-throughput sequencing technologies, as well as of bioinformatics analyses, has enabled to go deeper into transcriptomes, revealing that eukaryotic genomes are pervasively transcribed [START_REF] Berretta | Pervasive transcription constitutes a new level of eukaryotic genome regulation[END_REF]. For instance, the ENCODE project revealed that up to 75% of the human genome is transcribed in at least one cell line or condition [START_REF] Djebali | Landscape of transcription in human cells[END_REF].

The pervasive transcription of eukaryotic genomes produces thousands of non-coding transcripts, that are commonly classified into small and long (l)ncRNAs, and that are now recognized as major regulators involved in multiple cellular processes, including cell differentiation and development, chromosome dosage compensation, imprinting, regulation of gene expression, cell cycle control and adaptation to environment changes [START_REF] Rinn | Genome regulation by long noncoding RNAs[END_REF][START_REF] Sole | A novel role for lncRNAs in cell cycle control during stress adaptation[END_REF][START_REF] Wery | Noncoding RNAs in gene regulation[END_REF]). In addition, lncRNAs show tissuespecificity [START_REF] Djebali | Landscape of transcription in human cells[END_REF], indicating that their expression is tightly regulated. Furthermore, dysregulation of ncRNAs has been associated to human diseases, such as cancer or neurodegenerative disorders [START_REF] Taft | Non-coding RNAs: regulators of disease[END_REF].

In most eukaryotes, small and long regulatory ncRNAs coexist and even cooperate. However, the budding yeast S. cerevisiae constitutes an exception to this paradigm as it has lost the RNA interference system and is therefore devoid of small-interfering (si)RNAs and micro (mi)RNAs [START_REF] Drinnenberg | RNAi in Budding Yeast[END_REF]. In this respect, S. cerevisiae is a unique model to specifically study the regulatory effects of lncRNAs, which in other organisms might be partially hidden by the effects of the small RNAs (such as siRNAs and miRNAs).

Cytoplasmic lncRNAs are targeted by a translation-dependent surveillance pathway

Over the last years, thousands of lncRNAs have been annotated in S. cerevisiae [START_REF] Tisseur | Pervasive transcription -Lessons from yeast[END_REF].

Strikingly, most of them were found to be cryptic due to their rapid and extensive degradation in the nucleus or in the cytoplasm [START_REF] Tisseur | Pervasive transcription -Lessons from yeast[END_REF][START_REF] Tudek | Non-coding transcription by RNA polymerase II in yeast: Hasard or necessite?[END_REF]. Notably, the nuclear and cytoplasmic RNA degradation machineries target distinct types of lncRNAs. For instance, the nuclear exosome-dependent 3'-5' decay pathway degrades the so-called Cryptic Unstable Transcripts (CUTs), a class of lncRNAs that are mainly transcribed from divergent bidirectional promoters [START_REF] Neil | Widespread bidirectional promoters are the major source of cryptic transcripts in yeast[END_REF][START_REF] Xu | Bidirectional promoters generate pervasive transcription in yeast[END_REF]). On the other hand, the cytoplasmic Xrn1-dependent 5'-3' RNA decay pathway is specialized into the degradation of another class of lncRNAs referred to as Xrn1-sensitive Unstable Transcripts (XUTs), most of which are antisense to proteincoding genes [START_REF] Van Dijk | XUTs are a class of Xrn1-sensitive antisense regulatory non coding RNA in yeast[END_REF].

XUTs are synthesized by RNA Polymerase II as capped and poly-adenylated transcripts, similarly to mRNAs [START_REF] Van Dijk | XUTs are a class of Xrn1-sensitive antisense regulatory non coding RNA in yeast[END_REF][START_REF] Wery | Nonsense-Mediated Decay Restricts LncRNA Levels in Yeast Unless Blocked by Double-Stranded RNA Structure[END_REF]. But what determines their instability has remained unclear until the recent finding that the majority of them are specifically targeted by the Nonsense-Mediated Decay (NMD) pathway [START_REF] Malabat | Quality control of transcription start site selection by nonsense-mediated-mRNA decay[END_REF][START_REF] Wery | Nonsense-Mediated Decay Restricts LncRNA Levels in Yeast Unless Blocked by Double-Stranded RNA Structure[END_REF]. NMD is a translation-dependent RNA decay pathway that targets mRNAs with aberrant translation termination, such as mRNAs with premature stop codon [START_REF] Muhlrad | Premature translational termination triggers mRNA decapping[END_REF] and long 3'-UTR [START_REF] Muhlrad | Aberrant mRNAs with extended 3' UTRs are substrates for rapid degradation by mRNA surveillance[END_REF]. The sensitivity of XUTs to NMD suggests that once in the cytoplasm, they associate to the translation machinery and undergo at least a pioneer round of translation. Supporting this hypothesis, ribosome profiling data revealed the presence of small open reading frames (smORFs) on yeast lncRNAs [START_REF] Smith | Translation of small open reading frames within unannotated RNA transcripts in Saccharomyces cerevisiae[END_REF]), including a large set of XUTs [START_REF] Malabat | Quality control of transcription start site selection by nonsense-mediated-mRNA decay[END_REF][START_REF] Wery | Nonsense-Mediated Decay Restricts LncRNA Levels in Yeast Unless Blocked by Double-Stranded RNA Structure[END_REF], and some of them were shown to be translated into peptides in vivo [START_REF] Smith | Translation of small open reading frames within unannotated RNA transcripts in Saccharomyces cerevisiae[END_REF]). Strikingly, ribosomes on NMD-sensitive XUTs are restricted to a short 5'-proximal region, followed by a long ribosome-free 3'-UTR, which probably constitutes the NMD-activating signal [START_REF] Wery | Nonsense-Mediated Decay Restricts LncRNA Levels in Yeast Unless Blocked by Double-Stranded RNA Structure[END_REF].

Notably, ribosome-profiling approaches also identified smORFs on transcripts annotated as non-coding in other Eukaryotes, including Drosophila [START_REF] Aspden | Extensive translation of small Open Reading Frames revealed by Poly-Ribo[END_REF], zebrafish [START_REF] Bazzini | Identification of small ORFs in vertebrates using ribosome footprinting and evolutionary conservation[END_REF]) and mouse [START_REF] Ingolia | Ribosome profiling of mouse embryonic stem cells reveals the complexity and dynamics of mammalian proteomes[END_REF], although there is debate on the extent to which ribosome-footprints detection reflects genuine on-going translation [START_REF] Chew | Ribosome profiling reveals resemblance between long non-coding RNAs and 5' leaders of coding RNAs[END_REF][START_REF] Guttman | Ribosome profiling provides evidence that large noncoding RNAs do not encode proteins[END_REF].

The observation of ribosome binding to smORFs on transcripts annotated as lncRNAs challenges the initial assumption that these transcripts are really noncoding and raises the fundamental question of the function of the peptides produced upon translation of such smORFs. In this regard, recent works described lncRNAs producing smORFs peptides that control heart activity in Drosophila [START_REF] Magny | Conserved regulation of cardiac calcium uptake by peptides encoded in small open reading frames[END_REF]) and mammals [START_REF] Nelson | A peptide encoded by a transcript annotated as long noncoding RNA enhances SERCA activity in muscle[END_REF], or cell movement during embryogenesis in zebrafish [START_REF] Pauli | Toddler: an embryonic signal that promotes cell movement via Apelin receptors[END_REF]. In yeast, the evolutionary conservation of a subset of lncRNAs smORFs within yeast species indicates that the encoded peptides might have biological importance [START_REF] Smith | Translation of small open reading frames within unannotated RNA transcripts in Saccharomyces cerevisiae[END_REF]).

Thus, many transcripts initially thought to lack coding potential are likely to bear smORFs, that can be translated and give rise to functional peptides (Figure 1). On the other hand, we speculate that a fraction of smORFs-bearing transcripts, reminiscent of the yeast NMD-sensitive XUTs, will also be targeted by the NMD in other eukaryotic cells (Figure 1). In this respect, NMD inhibition in mouse embryonic stem cells has been shown to result in stabilization of a subset of annotated lncRNAs [START_REF] Smith | Translation of small open reading frames within unannotated RNA transcripts in Saccharomyces cerevisiae[END_REF]). Conceptually, one can also imagine that a cryptic transcript targeted to the NMD in one condition could escape the NMD and be stabilized in another condition (see below), possibly giving rise to a functional smORF peptide. Additional work will be required to define the comprehensive landscape of NMD-sensitive lncRNAs in different eukaryote models, but given the extent of ribosome association to lncRNAs, we anticipate the NMD to be recognized in the future as a major regulator of cytoplasmic lncRNAs.

NMD as an additional layer in lncRNA-mediated regulation of gene expression

In yeast, antisense XUTs can regulate paired-sense gene expression, at the transcriptional level, through histone modifications [START_REF] Berretta | A cryptic unstable transcript mediates transcriptional transsilencing of the Ty1 retrotransposon in S. cerevisiae[END_REF][START_REF] Van Dijk | XUTs are a class of Xrn1-sensitive antisense regulatory non coding RNA in yeast[END_REF]) and constitute to date the only class of lncRNAs for which the associated gene-regulation is thought to depend on the lncRNA per se, rather than its transcription. Interestingly, NMD specifically and exclusively targets this class of regulatory lncRNAs [START_REF] Wery | Nonsense-Mediated Decay Restricts LncRNA Levels in Yeast Unless Blocked by Double-Stranded RNA Structure[END_REF] and might be considered in that way as a novel player in the lncRNA-dependent buffering of genome expression.

NMD not only acts as a surveillance pathway targeting aberrant mRNAs and cryptic lncRNAs to degradation, but it also directly regulates physiological mRNAs in yeast, Drosophila and human [START_REF] Peccarelli | Regulation of natural mRNAs by the nonsense-mediated mRNA decay pathway[END_REF]. NMD itself is tightly regulated, and its activity is modulated in response to multiple stresses, including hypoxia, amino-acid or nutrient deprivation [START_REF] Karam | Regulation of nonsense-mediated mRNA decay: implications for physiology and disease[END_REF][START_REF] Lykke-Andersen | Nonsense-mediated mRNA decay: an intricate machinery that shapes transcriptomes[END_REF].

Interestingly, many stress-related mRNAs are targeted by the NMD under normal physiological conditions but are stabilized upon stress, due to NMD activity inhibition (Lykke-Andersen and Jensen 2015). Note that under stress conditions, global translation also decreases, selectively preserving translation of stress-related mRNAs to the detriment of "housekeeping" mRNAs [START_REF] Yamasaki | Reprogramming mRNA translation during stress[END_REF]. Another consequence of such a stress-mediated reduction of translation is that transcripts (aberrant mRNAs or lncRNAs) that are normally targeted by the NMD will be stabilized, since transcripts evading translation escape NMD.

On this basis, we propose a model where a stress that results in translation inhibition and/or NMD inhibition will lead to stabilization of regulatory antisense NMD-sensitive lncRNAs (such as yeast XUTs), which could in turn repress the transcription of their paired-sense genes (Figure 2). This would prevent the synthesis of mRNAs that could probably not be translated, avoiding the cell to waste an energy that could be crucial to survive the stress. Alternatively but not exclusively, the regulatory antisense lncRNAs could also regulate the paired-sense mRNAs at the post-transcriptional level, potentially through the formation of double-stranded (ds) RNA structures. In yeast, sense mRNAs and antisense XUTs have been shown to form dsRNA in vivo, and this protects XUTs from NMD [START_REF] Wery | Nonsense-Mediated Decay Restricts LncRNA Levels in Yeast Unless Blocked by Double-Stranded RNA Structure[END_REF]. Reciprocally, formation of dsRNA with a stabilized antisense lncRNA might affect sense mRNA stability, as suggested by a recent study of mRNA isoforms half-lives [START_REF] Geisberg | Global analysis of mRNA isoform half-lives reveals stabilizing and destabilizing elements in yeast[END_REF]. Besides RNA stability, regulatory antisense lncRNAs might also interfere with mRNA splicing, localization, or translation. Future work will be needed to decipher these potential lncRNA-mediated regulatory mechanisms and determine whether they can be integrated within larger stress-activated signaling networks [START_REF] Ho | Exploiting the yeast stress-activated signaling network to inform on stress biology and disease signaling[END_REF].

Concluding remarks

Over the years, lncRNAs have been recognized as major regulators of multiple cellular processes.

However, the initial assumption that they are devoid of coding potential is now challenged. Conceptually, coding a peptide/protein in specific circumstances and functioning as a regulatory RNA molecule in others are not exclusive possibilities for a transcript, whatever it has been primarily annotated as a lncRNA or mRNA.

Examples of bifunctional RNAs with dual coding and regulatory functions have been reported [START_REF] Ulveling | When one is better than two: RNA with dual functions[END_REF]. In this respect, in yeast, convergent mRNAs can regulate each other at the RNA level providing additional intriguing cases where mRNAs can switch their initial coding function into regulatory RNAs [START_REF] Sinturel | Cytoplasmic Control of Sense-Antisense mRNA Pairs[END_REF]. But how many among the thousands of lncRNAs annotated in the different eukaryotic models correspond to such bifunctional RNAs remains to be determined. The classical distinction between coding and non-coding RNAs might therefore become less strict in the future, if the possibility to switch between regulatory and coding functions in response to specific stimuli appears to be a common feature of "lncRNAs". NMD contributes to genome expression buffering by restricting the levels of a class of cytoplasmic antisense regulatory lncRNAs (red) that can attenuate the transcription of their paired-sense protein-coding genes (blue). In this model, any stress leading to translation and/or NMD inhibition would result in the stabilization of the lncRNA and transcriptional attenuation of the paired-sense gene.

Figure legends Figure 1 .

 legends1 Figure legends

Figure 2 .

 2 Figure 2. NMD buffers genome expression by controlling levels of regulatory antisense lncRNAs.

Acknowledgements

We thank all the members of our lab for discussions and critical reading of the manuscript. A.

Morillon's lab is supported by the ANR "DNA-Life" and ERC "DARK" consolidator grants.

Translation into functional peptide (A) n

Xrn1-dependent degradation

Heart activity Embryogenesis …

NMD