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

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

Léonard Gallardo, Chaabane Rejeb

To cite this version:

Léonard Gallardo, Chaabane Rejeb. Radial mollifiers, mean value operators and harmonic functions in Dunkl theory. 2016. �hal-01332001�

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

L´eonard GALLARDO^∗ and Chaabane REJEB^†

Abstract

In this paper we show how to use mollifiers to regularise functions relative to a set of Dunkl operators inR^d with Coxeter-Weyl groupW, multiplicity functionkand weight functionωk. In particular for Ω aW-invariant open subset ofR^d, forϕ∈ D(R^d) a radial function and u∈L¹_loc(Ω, ω_k(x)dx), we study the Dunkl-convolution product u∗kϕand 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.

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

1 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 C²(R^d)-functions is a differential-difference operator of the form

∆_kf(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 [6]). 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 σα,α∈R.

∗Laboratoire de Math´ematiques et Physique Th´eorique CNRS-UMR 7350, Universit´e de Tours, Campus de Grandmont, 37200 Tours, FRANCE; Email: [email protected]

†D´epartement de Math´ematiques, Facult´e des Sciences de Tunis, 1060 Tunis, TUNISIE and Laboratoire de Math´ematiques et Physique Th´eorique CNRS-UMR 7350, Universit´e de Tours, Campus de Grandmont, 37200 Tours, FRANCE; Email: [email protected]

We know that the Dunkl-Laplace operator can be written ∆_k = ∑_d

j=1D_ξ²

j, for ξ_j, j = 1, . . . , d, an orthonormal basis ofR^dand where for everyξ∈R^d,D_ξ is the Dunkl operator defined by

D_ξf(x) =∂_ξf(x) + ∑

α∈R+

k(α)⟨α, ξ⟩f(x)−f(σα(x))

⟨α, x⟩ , (1.2)

with∂_ξf denoting the usualξ-directional derivative off (see [3] and [6]). These operators are related to partial derivatives by means of the so-called Dunkl intertwining operatorV_k (see [5] or [6]) as follows

∀ ξ∈R^d, D_ξV_k=V_k∂_ξ. (1.3) The operatorVkis a topological isomorphism from the spaceC^∞(R^d)¹ onto itself satisfying (1.3) and V_k(1) = 1 (see [17]). Furthermore, according to [13] or [14], 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 groupW, such that

∀ f ∈ C^∞(R^d), Vk(f)(x) =

∫

R^df(y)dµx(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 ballB(0, r).

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

Ωr :={

x∈Ω; dist(x, ∂Ω)> r}

(1.6) is nonempty.

For everyu∈L¹_loc(Ω, mk), withdmk=ωk(x)dx, we define the Dunkl-convolution product u∗kϕ(x) :=

∫

R^du(y)τ₋xϕ(y)ωk(y)dy, (1.7) 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. R¨osler ([15]) when f ∈ C^∞(R^d) is a radial function. In such case, the operatorτx is given by

∀ y∈R^d, τxf(y) =

∫

R^d

fe(√

∥x∥²+∥y∥²+ 2⟨x, z⟩)dµy(z), (1.8) where feis the profile function off defined by f(x) =fe(∥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 ([12] or [16]).

1carrying its usual Fr´echet topology.

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_B^r,r >0, acting on continuous functions as follows

M_B^r(f)(x) := 1 m_k(B(0, r))

∫

R^df(y)h_k(r, x, y)ω_k(y)dy. (1.9) where y 7→ hk(r, x, y) is a compactly supported measurable function (called harmonic kernel, see [8]) given by

h_k(r, x, y) :=

∫

R^d1_[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 Ωr. Moreover, we will show the commutativity relations

∆_k(u∗kϕ) =u∗k∆_kϕ and M_B^r(u∗kϕ) =M_B^r(u)∗kϕ and the following associativity property

(u∗kϕ)∗kψ= (u∗kψ)∗kϕ

In section 4, we apply the previous results to give some new properties of ∆_k-harmonic functions (i.e. fonctions u ∈C²(Ω) such that ∆ku = 0). As a first result the show that any ∆_k-harmonic function on Ω is in fact of classC^∞. Then we prove that ∆_k-harmonic functions can be characterized by the local-volume mean value property i.e.

∀ x∈Ω, ∃rx>0, ∀ r < rx, u(x) =M_B^r(u)(x).

Finally we will establish the following Weyl’s lemma: Ifu∈L¹_loc(Ω, mk) satisfies ∆k(uωk) = 0 in distributional sense, thenucoincides 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 ofR^d, we denote by:

• L¹_k,loc(Ω) = L¹_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 space of distributions on Ω (i.e. the topological dual of D(Ω) carrying the Fr´echet topology).

• S(R^d) the Schwartz space ofC^∞-functions onR^dwhich are rapidly decreasing together with their derivatives.

•B(a, ρ) (resp. B^◦(a, ρ)) the closed (resp. the open) Euclidean ball centered at aand with radiusρ >0.

• D⁺r(R^d),r >0, the set of radial mollifiers with support contained inB(0, r).

2 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)7→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 haveµy =δy,h0(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 off at (x, r) can also be written by means of the Dunkl translation as follows (see [8]):

M_B^r(f)(x) = 1 m_k(B(0, r))

∫

B(0,r)

τxf(y)ωk(y)dy. (2.1) The harmonic kernel has the following properties (see [8]):

1. For allr >0 andx, y∈R^d, 0≤h_k(r, x, y)≤1.

2. For all fixed x, y ∈ R^d, the function r 7−→ h_k(r, x, y) is right-continuous and non decreasing on ]0,+∞[.

3. For all fixed r >0 and x∈R^d,

supph_k(r, x, . )⊂B^W(x, r) :=∪g∈WB(gx, r). (2.2) 4. Letr >0 and x∈R^d. For any sequence (χε)⊂ D(R^d) of radial functions such that

for everyε >0,

0≤χε ≤1, χε= 1 on B(0, r) and∀y∈R^d, lim

ε→0χε(y) =1_B(0,r)(y), (2.3) we have

∀y∈R^d, h_k(r, x, y) = lim

ε→0τ_−xχ_ε(y). (2.4) 5. For allr >0, all x, y∈R^d and allg∈W, we have

h_k(r, x, y) =h_k(r, y, x) and h_k(r, gx, y) =h_k(r, x, g⁻¹y). (2.5) 6. For allr >0 andx∈R^d, we have

∥h_k(r, x, .)∥k,1 :=

∫

R^dh_k(r, x, y)ω_k(y)dy=m_k(B(0, r)) = d_kr^d+2γ

d+ 2γ , (2.6) whereγ :=∑

α∈R+k(α) and d_k is the constant d_k:=∫

S^d−1ω_k(ξ)dσ(ξ). (2.7)

Here dσ(ξ) is the surface measure of the unit sphereS^d−1 of R^d.

7. Letr >0 andx∈R^d. Then the function h_k(r, x, .) is upper semi-continuous onR^d. 8. The harmonic kernel satisfies the following geometric inequality: if∥a−b∥ ≤2rwith

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, thenB(a, r)⊂B(b,4r).

9. Letx∈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 measureδx asr −→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^df(y)dη^k_x,r = lim

r→0M_B^r(f)(x) =f(x). (2.11) According to [10], the spherical mean of aC^∞-functionf defined on wholeR^dis given by

M_S^r(f)(x) := 1 d_k

∫

S^d−1

τxf(ry)ωk(y)dσ(y), (2.12) whered_k is the constant (2.7) andτ_xis the Dunkl translation (see Annex A.2). Moreover, M. R¨osler has proved that there exists a compactly supported probability measureσ_x,r^k on R^dwhich represents the spherical mean operator (see [15]). More precisely, forf ∈ C^∞(R^d), the spherical mean of f at (x, r)∈R^d×R+ is given by

M_S^r(f)(x) =

∫

R^df(y)dσ^k_x,r(y), (2.13) with

suppσ^k_x,r ⊂B^W(x, r) =∪g∈WB(gx, r). (2.14) Clearly formula (2.13) shows that we can define the spherical mean at (x, r) of any con- tinuous 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_B^r(f)(x) = d+ 2γ r^d+2γ

∫ _r

0

M_S^t(f)(x)t^d+2γ−1dt (2.15) holds whenever B(x, r)⊂Ω.

Proof: When f ∈ C^∞(R^d), (2.15) has been proved in [8]. 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).

At the end of this section, we will give some useful properties of the volume mean operator when it acts on L¹_k,loc(Ω)-functions. Firstly, note that thanks to (2.2) and to the boundedness of the kernel h_k, the function (x, r) 7→M_B^r(f)(x) is well defined for any f ∈L¹_k,loc(Ω) whenever B(x, r)⊂Ω.

Secondly, we will need the following notations which will be used frequently in this paper:

rΩ:= sup{r >0; Ωr ̸=∅}, (2.16) with Ωr the open set defined by (1.6). Clearly, we have Ωr1 ⊂Ωr2 whenever r2 ≤r1 and Ω =∪r>0Ωr=∪r<rΩΩr (note that, since Ω̸=∅, we have rΩ >0). Moreover, since

Ωr ={

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 ∈L¹_k,loc(Ω).

1) Let 0< r < r_Ω. Then the function M_B^r(f) belongs to L¹_k,loc(Ω_r).

2) Let x∈Ω. Then the functionr 7→M_B^r(f)(x) is continuous on]0, ϱx[with

ϱ_x :=dist(x, ∂Ω). (2.18)

Proof: 1) By compactness, it suffices to prove that M_B^r(f)ωk ∈ L¹(B(x0, R)) where B(x₀, R) is an arbitrary closed ball of centerx₀ and radiusR included in Ω_r . We have

I :=∫

B(x0,R)M_B^r(f)(x)ω_k(x)dx

≤ _mk(B(0,r))¹ ∫

B(x0,R)

( ∫

B^W(x,r)|f(y)|h_k(r, x, y)ω_k(y)dy )

ω_k(x)dx

≤ _mk(B(0,r))¹ ∫

B(x0,R)

( ∫

B^W(x0,R+r)|f(y)|ωk(y)dy )

ωk(x)dx

≤ ^m_m^k_k^(B(x_(B(0,r))^0,R))∫

B^W(x0,R+r)|f(y)|ω_k(y)dy <+∞,

where the second inequality follows from the relationh_k(r, x, y)≤1 and from the fact that for every x∈B(x0, R) and every g∈W,B(gx, r)⊂B(gx0, R+r)⊂Ω.

2) By (2.6), it suffices to show that ϕ :r 7→ ∫

R^df(y)hk(r, x, y)ωk(y)dy is continuous on ]0, ϱ_x[. Since r 7→h_k(r, x, y) is right-continuous, by the dominated convergence theorem, we can see thatϕis also right-continuous on ]0, ϱ_x[.

Now, fix r ∈]0, ϱx[ and η > 0 such that ]r−η, r+η[⊂]0, ϱx[. Let (rn) 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 An=An(x, z) :=

{

y∈Ω, r−rn<√

∥y∥²+∥x∥²−2⟨y, z⟩ ≤r }

. Since∩nAn is a hypersurface, by the dominated convergence theorem, we get

n→lim+∞

∫

R^d(

∫

An

|f(y)|ω_k(y)dy)dµ_x(z) = 0.

Hence, by the previous relations, we conclude thatϕ is also left continuous.

3 Dunkl convolution Product

The Dunkl convolution product has been defined by means of the Dunkl translation opera- tors (see [18] and [14]). 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 functionu∈L¹_k,loc(Ω) with a nonnegative and radial function f ∈ D(R^d) and we will study some properties of this product.

Forf, g∈ S(R^d), the Dunkl convolution product is defined by

∀ x∈R^d, f∗kg(x) :=

∫

R^dτxf(−y)g(y)ωk(y)dy. (3.1) We note that it is commutative and satisfies the following property:

Fk(f∗kg) =Fk(f)Fk(g), (3.2) where Fk is the Dunkl transform (see Annex A.1).

From (3.2), (A.8) and the injectivity of theFktransform, we obtain the following relations Lemma 3.1 Let f, g ∈ S(R^d). Then, for every x∈R^d, we have

(τxf)∗kg=f ∗k(τxg) =τx(f∗kg). (3.3) Theorem 3.1 Let u∈L¹_k,loc(Ω)and ϕ∈ Dρ⁺(R^d) with 0< ρ < rΩ. Let

u∗kϕ(x) :=

∫

R^du(y)τ_−xϕ(y)ω_k(y)dy. (3.4) Then

1) the function u∗kϕ is well defined onΩ_ρ and can be written

∀ x∈Ωρ, u∗kϕ(x) =

∫

R^du(y)τ₋yϕ(x)ω_k(y)dy (3.5)

=

∫

R^du(y)τ_xϕ(−y)ω_k(y)dy, (3.6) 2) u∗kϕbelongs to C^∞(Ω_ρ) and we have

∆k(u∗kϕ) =u∗k∆kϕ, (3.7)

3) for all B(x, r)⊂Ω_ρ, we have

M_B^r(u∗kϕ)(x) =M_B^r(u)∗kϕ(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∥_∞φε(y),

where (φε) 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₋xϕ(y)≤ ∥ϕ∥_∞τ₋xφε(y).

Letting ε→0 and using (2.4), we obtain

∀y∈R^d, 0≤τ_−xϕ(y)≤ ∥ϕ∥_∞h_k(ρ, x, y). (3.9) Consequently, from the relations (2.2) and (3.9), we get that

suppτ₋xϕ⊂B^W(x, ρ). (3.10)

This implies that for allx∈Ω_ρ, the functiony7→u(y)τ_−xϕ(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 x₀ ∈ Ω_ρ and R > 0 such that B(x₀, R) ⊂ Ω_ρ. We shall prove that the function u∗kϕis of classC^∞ on B(x^◦ ₀, R).

Define the function Φ on R^d×R^d by (see (A.9)) Φ(x, y) :=τ_−xϕ(y) =c⁻_k²

∫

R^dFk(ϕ)(ξ)E_k(−ix, ξ)E_k(iy, ξ)ω_k(ξ)dξ.

We see that Φ is inC^∞(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^υΦ(x, y) ≤c⁻_k²

∫

R^d|Fk(ϕ)(ξ)| ∥ξ∥^|υ|ωk(ξ)dξ:=Cυ<+∞. On the other hand, from (3.10) we have

∀x∈B^◦(x₀, R), supp Φ(x, .)⊂B^W(x, ρ)⊂B^W(x₀, 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_(x0,R+ρ)(y).

Thus, for every multi-indicesυ∈N^d, we deduce that

∀ x∈B(x^◦ 0, R), ∀y ∈R^d, ∂^υ

∂x^υΦ(x, y)≤Cυ1_B^W_(x0,R+ρ)(y).

Now, sinceuω_kis 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 ∆_kϕis also a radial function² (see [10])), we obtain

∆_k(u∗kϕ)(x) =

∫

R^du(y)∆_k[τ_−yϕ](x)ω_k(y)dy=

∫

R^du(y)τ_−y[∆_kϕ](x)ω_k(y)dy

=

∫

R^du(y)τ₋x[∆kϕ](y)ωk(y)dy=u∗k∆kϕ(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

τ_aτ_bf =τ_bτ_af (3.11)

and

M_B^r(τ_−af)(b) =M_B^r(τ_−bf)(a). (3.12) Proof of Lemma 3.2

• We obtain (3.11) from (A.8) and the injectivity of the Dunkl transform onS(R^d).

• Letr >0 anda, b∈R^d. We have M_B^r(τ_−af)(b) = 1

mk(B(0, r))

∫

R^dτ_af(−y)h_k(r, b, y)ω_k(y)dy

= 1

m_k(B(0, r))

∫

R^dτ_bτaf(−y)1_B(0,r)(y)ω_k(y)dy

= 1

mk(B(0, r))

∫

R^dτ_bf(−y)h_k(r, a, y)ω_k(y)dy

=M_B^r(τ_−bf)(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).

LetB(x, r)⊂Ωρ. By Proposition 2.2- 1), the functionM_B^r(u) belongs to L¹_k,loc(Ωr). This proves, by assertion 1), that the function M_B^r(u)∗kϕis well defined on Ω_ρ+r.

2More precisely, we have ∆kϕ(x) = (_dr^d2₂ +^d+2γ_r⁻¹_dr^d)ϕ(r),e r=∥x∥andϕethe profile ofϕ.

By 2), the function u∗kϕis clearly in L¹_k,loc(Ω_ρ) and forx∈Ω_ρ+r we have³ M_B^r(u∗kϕ)(x) = 1

m_k(B(0, r))

∫

B^W(x,r)

(

∫

B^W(x,r+ρ)

u(z)τ₋yϕ(z)ωk(z)dz)hk(r, x, y)ωk(y)dy

= 1

m_k(B(0, r))

∫

B^W(x,r+ρ)

u(z)(

∫

B^W(x,r)

τ₋zϕ(y)hk(r, x, y)ωk(y)dy) ωk(z)dz

=

∫

B^W(x,r+ρ)

u(z)M_B^r( τ₋zϕ)

(x)ωk(z)dz

=

∫

B^W(x,r+ρ)

u(z)M_B^r( τ₋xϕ)

(z)ω_k(z)dz

= 1

m_k(B(0, r))

∫

B^W(x,r+ρ)

u(z)(

∫

B^W(x,ρ)

τ₋xϕ(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_−xϕ(y)(

∫

B^W(x,r+ρ)

u(z)h_k(r, y, z)ω_k(z)dz) ω_k(y)dy

=M_B^r(u)∗kϕ(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 C² on Ω, then the relation (3.7) can be also written

∀ x∈Ω_ρ, ∆_k(u∗kϕ)(x) = (∆_ku)∗kϕ(x). (3.14) Indeed, using respectively the relations (3.7), (3.4) and (A.11), we get

∆_k(u∗kϕ)(x) =

∫

Ω

u(y)τ_−x[∆_kϕ](y)ω_k(y)dy=

∫

Ω

u(y)∆_k[τ_−xϕ](y)ω_k(y)dy. (3.15) Now, we obtain the desired result by using the following integration by parts formula (see [5] or [14]): Let f, g∈ C¹(Ω) such that g has compact support. Then, for all ξ ∈R^d, we

have ∫

Ω

D_ξf(x)g(x)ω_k(x)dx=−

∫

Ω

f(x)D_ξg(x)ω_k(x)dx, (3.16) with D_ξ the ξ-directional Dunkl operator (see (1.2)).

3Note that, in the integrals below, the consideration of the supports permits to justify the correct application of Fubini’s theorem.

Proposition 3.1 Let u ∈ L¹_k,loc(Ω) and ϕ ∈ Dρ⁺(R^d) with 0 < ρ < r_Ω. If u is with compact support, then u∗kϕ is also with compact support and

supp (u∗kϕ)⊂B(0, ρ) +W.supp u⊂Ω, (3.17) with W.supp u:={gx, (g, x)∈W ×supp u}.

Proof: Ifx /∈B(0, ρ)+W.suppu, thenx−gy /∈B(0, ρ) for every (g, y)∈W×suppu. That is ∥gx−y∥> ρ for all y∈suppu and all g∈W. In other words supp u∩B^W(x, ρ) =∅.

Hence, by (3.10), we obtain u∗kϕ(x) = 0.

It is interesting to note that whenu 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 Ωand let ϕ∈ D⁺ρ(R^d) with 0< ρ <

rΩ (i.e. Ωρ is nonempty). Then, for all x∈Ωρ, we have u∗kϕ(x) =d_k

∫ _ρ

0

ϕ(t)te ^d+2γ−1M_S^t(u)(x)dt, (3.18)

where ϕeis the profile function of ϕ andd_k is the constant given by (2.7).

Proof: At first we suppose thatu∈ C^∞(R^d). By (A.14), we have u∗kϕ(x) =

∫

R^dϕ(y)τ_xu(y)ω_k(y)dy.

Then, using spherical coordinates, we can write u∗kϕ(x) =

∫ _ρ

0

ϕ(t)te ^d+2γ−1

∫

S^d−1

τxu(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 pn −→ u as n −→ +∞ uniformly on the compact set K :=B^W(x, ρ). Since τ₋xϕ≥0, by (A.12) we conclude that

u∗kϕ(x)−pn∗kϕ(x)≤ ∥τ₋xϕ∥L¹_k(R^d)sup

K |pn(y)−u(y)|=∥ϕ∥L¹_k(R^d)sup

K |pn(y)−u(y)|. Hence

u∗kϕ(x) = lim

n→+∞pn∗kϕ(x). (3.19)

Furthermore, as the probability measuresσ_x,t^k , 0< t≤ρ, have compact support contained inB^W(x, t)⊂K =B^W(x, ρ) (see (2.14)), we deduce

∀ t≤ρ, |M_S^t(pn−u)(x)| ≤sup

K |pn(y)−u(y)|.

This implies

n→lim+∞d_k

∫ _ρ

0

ϕ(t)te ^d+2γ−1M_S^t(pn)(x)dt=d_k

∫ _ρ

0

ϕ(t)te ^d+2γ−1M_S^t(u)(x)dt. (3.20) From (3.19), (3.20) and the first step, we deduce that the relation (3.18) holds when the

functionu is continuous on Ω.

We have the following associativity result for the Dunkl convolution product:

Proposition 3.3 Let u ∈ L¹_k,loc(Ω), ϕ ∈ Dρ⁺(R^d) and ψ ∈ D⁺r(R^d) such that Ω_r+ρ is nonempty (i.e. r+ρ < rΩ). Then

∀ x∈Ω_r+ρ, ( u∗kϕ)

∗kψ(x) =u∗k(ϕ∗kψ)(x) =(

u∗kψ)

∗kϕ(x). (3.21) Proof: • From Theorem 3.1, the functions (

u∗kϕ)

∗kψ and( u∗kψ)

∗kϕare well defined on Ωr+ρ.

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

ϕ∗kψ=F_k⁻¹(Fk(ϕ)Fk(ψ)).

Therefore, since Fk preserves the radial property (see the relation (A.6)), we deduce that ϕ∗kψ is radial as claimed.

• Forx∈Ω_r+ρ fixed, we have (u∗kϕ)

∗kψ(x) =

∫

B^W(x,r)

(u∗kϕ)

(y)τ₋xψ(y)ω_k(y)dy

=

∫

B^W(x,r)

( ∫

B^W(x,r+ρ)

u(z)τ_−yϕ(z)ω_k(z)dz )

τ_−xψ(y)ω_k(y)dy

=

∫

B^W(x,r+ρ)

u(z) ( ∫

B^W(x,r)

τ_−yϕ(z)τ_−xψ(y)ω_k(y)dy )

ω_k(z)dz

=

∫

B^W(x,r+ρ)

u(z) ( ∫

B^W(x,r)

τ_−zϕ(y)τ_−xψ(y)ω_k(y)dy )

ω_k(z)dz

=

∫

B^W(x,r+ρ)

u(z)(

ϕ∗kτ_−xψ)

(z) ω_k(z)dz

=

∫

B^W(x,r+ρ)

u(z)τ_−x(

ϕ∗kψ)

(z) ω_k(z)dz

=u∗k(ϕ∗kψ)(x).

where 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∈L¹_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∗kψand (3.10) in the last line.

Now, changing the role ofϕ andψ, we obtain

(u∗kψ)∗kϕ(x) =u∗k(ψ∗kϕ)(x).

Finally, by the commutativity of the Dunkl convolution product (see (3.2)), we conclude

the last equality in (3.21).

4 Applications to ∆

-harmonic functions

In this section, we will give some properties of Dunkl harmonic functions. Let us introduce the space Hk(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) :=aexp (− 1

1− ∥x∥²)1_B(0,1)(x), x∈R^d, (4.1) where ais a constant such thatx7−→φ(x)ω_k(x) is a probability density.

For ε >0, define the function

φε(x) = 1 ε^d+2γφ(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 uε by

∀ x∈Ωε, uε(x) :=u∗kφε(x) :=

∫

R^du(y)τ₋xφε(y)ωk(y)dy. (4.3) Then the sequence (u_ε)0<ε<rΩ satisfies

i) for every0< ε < rΩ, the function uε is in C^∞(Ωε), 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_B^r(u∗kφ_ε)(x)−→ M_B^r(u)(x) and M_S^r(u∗kφε)(x)−→M_S^r(u)(x) as ε−→0.

Proof: The first assertion follows immediately from Theorem 3.1.

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

∀ y∈R^d, 0≤τ_−xφ_ε(y)≤aε^−d−2γh_k(ε, x, y).

Using (2.6), we can write

∀ y∈R^d, 0≤τ_−xφ_ε(y)≤a d_k d+ 2γ

1

m_k[B(0, ε)]h_k(ε, x, y). (4.4) Consequently, for everyx∈Ω and every ε >0 small enough, we have by (A.12) and (4.4)

|u_ε(x)−u(x)| ≤

∫

R^dτ_−xφ_ε(y)|u(y)−u(x)|ω_k(y)dy

≤a d_k d+ 2γ

1 mk[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

|uε(x)−u(x)| ≤a d_k

d+ 2γM_B^ε(

|u−u(x)|)

(x). (4.5)

Thus from (2.11), we conclude the result.

iii) Letx∈Ω_r. There existsε₀ >0 such thatx∈Ω_r+ε0. Forε∈]0, ε₀[, we have

∀ y∈B^W(x, r), |u∗φε(y)| ≤

∫

B^W(y,ε)

|u(z)|τ₋yφε(z)ωk(z)dz

≤

∫

B^W(x,r+ε0)

|u(z)|τ₋yφε(z)ωk(z)dz

≤ sup

B^W(x,r+ε0)

|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 aC²-function on Ω. Then, for any0< r < r_Ω and anyx∈Ω_r, we have

M_S^r(u)(x) =u(x) + 1 d+ 2γ

∫ _r

0

M_B^t(∆ku)(x) tdt, (4.6) and

M_B^r(u)(x) =u(x) + 1 r^d+2γ

∫ _r

0

∫ _ρ

0

M_B^t(∆_ku)(x) t dt ρ^d+2γ−1dρ. (4.7) Remark 4.1 We have obtained in [8] that the relations (4.6) and (4.7) hold when u ∈ C^∞(R^d). Furthermore, by polynomial approximation, we have extended them to any func- tion u of class C² on an arbitrary W-invariant open set Ω⊂R^d and any x∈Ω but with the condition r∈]0, ϱx/3[,ϱx being defined by (2.18).

Proof: Step 1: Suppose thatu is of class C^∞ on Ω.

Let x∈Ω and r∈]0, ϱ_x[ and let ϵ >0 such thatB(x, r+ϵ)⊂Ω. We can find ϕ∈ D(R^d) such that

1. ϕ= 1 on the compact setB^W(x, r+ϵ/2), 2. suppϕ⊂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 hk(r, x, .) and σ^k_x,r), we can replace f by u in theses formulas.

Step 2: Here, we will suppose thatu∈ C²(Ω). By (2.15), it is enough to prove (4.6). Fix 0 < r < rΩ, x∈Ωr and ε0 >0 such that x ∈Ωr+ε0. By step 1, for every 0< ε < ε0 we have

M_S^r(u∗kφ_ε)(x) =u∗kφ_ε(x) + 1 d+ 2γ

∫ _r

0

M_B^t(∆_k[u∗kφ_ε])(x)tdt. (4.8) Now, using (3.14) and Lemma 4.1 , we get

εlim→0M_B^t(∆_k[u∗kφ_ε])(x) = lim

ε→0M_B^t([∆_ku]∗kφ_ε)(x) =M_B^t(∆_ku)(x),

and following the proof of the statement iii) of Lemma 4.1, for every 0< ε < ε0 we obtain

∀t≤r, |M_B^t(∆_k[u∗kφ_ε])(x)| ≤ sup

B^W(x,r+ε0)

|∆_ku(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 lettingε−→0.

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

Proposition 4.1 Let u∈ C²(Ω). The following statements are equivalent i) u∈ Hk(Ω),

ii) u(x) =M_S^r(u)(x) whenever B(x, r)⊂Ω, iii) u(x) =M_B^r(u)(x) whenever B(x, r)⊂Ω, iv) M_B^r(u)(x) =M_S^r(u)(x) whenever B(x, r)⊂Ω.

Proof: i)⇒ii) It follows from (4.6).

ii)⇒iii) By (2.15), we obtain immediately the result.

iii) ⇒ iv) Let f ∈ C²(Ω) and let x ∈ Ω be fixed. By (4.6) and (2.15) the function r 7→M_B^r(f)(x) is of classC² on ]0, ρ_x[ and we have

d

drM_B^r(f)(x) = d+ 2γ r

(

M_S^r(f)(x)−M_B^r(f)(x) )

. (4.9)