Green function and Poisson kernel associated to root systems for annular regions

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Green function and Poisson kernel associated to root systems for annular regions

Chaabane Rejeb

To cite this version:

Chaabane Rejeb. Green function and Poisson kernel associated to root systems for annular regions.

2019. �hal-02005375�

Green function and Poisson kernel associated to root systems for annular regions

Chaabane REJEB^∗

Abstract

Let ∆k be the Dunkl Laplacian relative to a fixed root system Rin R^d, d ≥2, and to a nonnegative multiplicity function k on R. Our first purpose in this paper is to solve the ∆k-Dirichlet problem for annular regions. Secondly, we introduce and study the ∆_k-Green function of the annulus and we prove that it can be expressed by means of ∆_k-spherical harmonics. As applications, we obtain a Poisson-Jensen formula for ∆_k-subharmonic functions and we study positive continuous solutions for a ∆_k-semilinear problem.

MSC (2010) primary: 31B05, 35J08, 35J65; secondary: 31C45, 46F10, 47B39.

Key words: Dunkl-Laplace operator, Poisson kernel, Green function, Dirichlet problem, spherical harmonics, Newton kernel.

1 Introduction

Since the 90’s, extensive studies have been carried out on analysis associated with Dunkl operators. These are commuting differential-difference operators onR^d introduced by C.

F. Dunkl (see [6]). The Dunkl analysis includes especially a generalization of the Fourier transform (called the Dunkl transform) and the Laplace operator known as the Dunkl Laplacian (and denoted by ∆_k).

The Dunkl theory has many applications as well in mathematical physics and probability theory. In particular, it has been used in the study of the Calogero-Moser-Sutherland and other integrable systems (see [4, 10]) and in the study of Markov processes generalizing Brownian motion (see [22]).

Recently, a special interest has been devoted to potential theory associated with the Dunkl Laplacian. The study focused on ∆_k-harmonic functions (see [2, 11, 12, 17, 19, 20]), on

∆_k-Newton potential theory (including ∆_k-subharmonic functions) (see [13]) and on ∆_k- Riesz potentials of Radon measures (see [14]). More recently, by means of the ∆_k-Newton kernel, the Green function of the open unit ball has been studied in [15]. Note that finding

∗Universit´e de Tunis El Manar, Facult´e des Sciences de Tunis, Laboratoire d’Analyse Math´ematques et Applications LR11ES11, 2092 El Manar I, Tunis, TUNISIA and Laboratoire de Math´ematiques et Physique Th´eorique CNRS-UMR 7350, Universit´e de Tours, Campus de Grandmont, 37200 Tours, FRANCE; Email:

chaabane.rejeb@gmail.com

∆_k-Green functions for other open sets is a rather difficult problem already in the case of the classical Laplace operator. The aim of this paper is to show that we can determine the

∆_k-Green function for annular regions inR^dby using ∆_k-spherical harmonics as a crucial tool.

Let us assume throughout the paper that d≥2. Let Abe the annulus A:={x∈R^d, ρ <∥x∥<1} with ρ∈]0,1[.

After giving some properties of the ∆_k-Green functionG_k,A of A, we will use it to study the semilinear problem

{ ∆_k(uω_k) =ϕ(., u)ω_k, in the sense of distributions

u=f, on ∂A,

whereωk is a precise weight function (see (2.6) for its expression).

More precisely, under some assumptions on the functionϕ, we will show that iff ∈ C(∂A) is nonnegative, this boundary problem has only and only one positive continuous solution on Awhich satisfies (see Theorem 5.2)

∀ x∈A, u(x) +

∫

A

G_k,A(x, y)ϕ(y, u(y))ω_k(y)dy=P_k,A[f](x).

Here P_k,A[f] is the unique solution in C²(A)∩ C(A) of the boundary Dirichlet problem { ∆ku= 0, onA,

u=f, on∂A, that will be given explicitly in Section 3.

This paper is organized as follows. In Section 2, we recall some basics from Dunkl theory that will be used throughout the paper. In Section 3, we give an explicit solution of the boundary Dirichlet problem for the annulus. The Green function G_k,A will be introduced and studied in Section 4. Some applications will be given in the last Section.

Precisely, we will obtain a Poisson-Jensen formula for ∆k-subharmonic functions in the annulus and we will study positive solutions of the above semilinear problem.

2 Basics from Dunkl theory

We start by recalling some useful facts in the Dunkl theory. Let R be a root system in the Euclidian spaceR^d, in the sense thatR is a finite set inR^d\ {0} such that for every α∈ R,R ∩Rα={±α}and σα(R) =R(where σα is the reflection w.r.t. the hyperplane H_α orthogonal to α). The subgroup W ⊂O(R^d) generated by the reflections σ_α,α ∈ R, is called the Coxeter-Weyl group associated to R. We refer to ([18]) for more details on root systems and their Coxeter-Weyl groups.

Let k be a fixed nonnegative multiplicity function on R (i.e. k is W-invariant). For ξ∈R^d, theξ-directional Dunkl operator associated to (W, k) is defined by

D_ξf(x) :=∂_ξf(x) + ∑

α∈R+

k(α)⟨α, ξ⟩f(x)−f(σ_α.x)

⟨α, x⟩ , f ∈ C¹(R^d),

where∂_ξ is the usual ξ-directional partial derivative andR+ is a positive subsystem.

Let us denote by P(R^d) (resp. Pn(R^d)) the space of polynomial functions on R^d (resp.

the space of homogeneous polynomials of degreen∈N).

It was shown that there is a unique linear isomorphism V_k from P(R^d) onto itself such thatV_k(Pn(R^d)) =Pn(R^d) for everyn∈N,V_k(1) = 1 and

∀ ξ∈R^d, D_ξV_k=∂_ξV_k. (2.1) The operator V_k is known as the Dunkl intertwining operator (see [7, 8]). It has been extended to a topological isomorphism fromC^∞(R^d) onto itself satisfying (2.1) (see [26]).

Furthermore, according to [23], for eachx∈R^d, there is a compactly supported probability measure µ_x on R^dsuch that

∀ f ∈ C^∞(R^d), Vk(f)(x) =

∫

R^df(y)dµx(y). (2.2) IfW.x denotes the orbit ofx under theW-action andCo(x) its convex hull, then

supp µx ⊂Co(x)⊂B(0,∥x∥). (2.3) The Dunkl-Laplacian is defined as ∆k =∑_d

j=1D²_ej, where (ej)1≤j≤d is the canonical basis of R^d. It can be expressed as follows

∆kf(x) = ∆f(x)+ ∑

α∈R+

k(α) (

2⟨∇f(x), α⟩

⟨α, x⟩ −∥α∥²f(x)−f(σα(x))

⟨α, x⟩² )

, f ∈ C²(R^d), (2.4) where ∆ (resp. ∇) is the usual Laplace (resp. gradient) operator (see [6, 8]). Note that if kis the zero function, the Dunkl Laplacian reduces to the classical one which is commutes with the action of O(R^d). For general k≥0, ∆_k commutes with theW-action (see [24]) i.e.

∀g∈W, g◦∆k= ∆k◦g. (2.5)

Let L²_k(S^d−1), d≥2, be the Hilbert space endowed with the inner product

⟨p, q⟩_k:= 1 dk

∫

S^d−1

p(ξ)q(ξ)ωk(ξ)dσ(ξ).

We denote by∥.∥_L²

k(S^d−1) the associated Euclidean norm. Here,dσ is the surface measure on the unit sphere S^d−1,ω_k is the weight function given by

ω_k(x) =∏

α∈R⁺| ⟨α, x⟩ |^2k(α) (2.6)

and dk is the constant

d_k=∫

S^d−1ω_k(ξ)dσ(ξ). (2.7)

The function ωk is W-invariant and homogeneous of degree 2γ := 2∑

α∈R+k(α).

To simplify, we introduce the constant λk:= d

2+γ−1≥0. (2.8)

Let H∆k,n(R^d) := Pn(R^d)∩Ker∆_k be the space of ∆_k-harmonic polynomials, homoge- neous of degree nonR^d. From [8], we know that ifn̸=m, thenH∆k,n(R^d)⊥ H∆k,m(R^d) inL²_k(S^d−1). Moreover, for every n∈N, we have

Pn(R^d) =⊕_⌊n/2⌋

j=0 ∥x∥^2jH∆k,n−2j(R^d). (2.9) The restriction to the sphereS^d−1of an element ofH∆k,n(R^d) is called a ∆k-spherical har- monic of degreen. The space of ∆_k-spherical harmonics will be denoted byH∆k,n(S^d−1).

This space has a reproducing kernelZ_k,n uniquely determined by the properties (see [5, 8]) i) for each x∈S^d−1,Z_k,n(x, .)∈ H∆k,n(S^d−1),

ii) for everyf ∈ H∆_k,n(S^d−1), we have f(x) =⟨f, Z_k,n(x, .)⟩_k= 1

d_k

∫

S^d−1

f(ξ)Z_k,n(x, ξ)ω_k(ξ)dσ(ξ), x∈S^d−1. (2.10) From this formula, we can see that

∀g∈W, ∀ x, y∈S^d−1, Zk,n(gx, gy) =Zk,n(x, y). (2.11) In classical case (i.e. k = 0), Z_0,n(x, .) is known as the zonal harmonic of degree n (see [1, 5]). Note that if{Yj,n, j = 1, . . . , h(n, d) :=dimH∆k,n(R^d)}is a real-orthonormal basis of H∆k,n(S^d−1) in L²_k(S^d−1), then

Z_k,n(x, y) =

h(n,d)∑

j=1

Y_j,n(x)Y_j,n(y). (2.12)

By means of the Dunkl intertwining operator and Gegenbauer polynomials,Z_k,n is given explicitly by (see [5], Theorem 7.2.6. or [27])

∀x, y∈S^d−1, Z_k,n(x, y) = (n+λ_k)(2λ_k)_n λ_k.n! V_k

( P_n^λk(

⟨., y⟩))

(x), (2.13) where λ_k is the constant given by (2.8), Pn^µ, µ > −1/2, is the normalized Gegenbauer polynomial (see [8] p. 17) defined by

P_n^µ(x) := (−1)ⁿ 2ⁿ(µ+ 1/2)n

(1−x²)^1/2−µ dⁿ

dxⁿ(1−x²)^n+µ−1/2, and (x)_n:=x(x+ 1). . .(x+n−1) is the Pochhammer symbol.

At the end of this section, to simplify, when k = 0 we will use the usual nota- tions L²(S^d−1) for L²₀(S^d−1), H∆,n for H∆0,n, ω_d−1 := d₀ the surface area of S^d−1 and Zn:=Z0,n.

3 ∆

-Dirichlet problem for the annulus

In this section, by introducing a Poisson type kernel, we will solve the Dirichlet problem for the Dunkl Laplacian in annular regions

AR1,R2 :={x∈R^d: R1 <∥x∥< R2}. Note that from the homogeneity property of ∆k:

δ_r◦∆_k =r⁻²∆_k◦δ_r, with δ_r(f)(x) :=f(rx),

it suffices to do this for the annular region A_ρ,1 with ρ ∈]0,1[ fixed. In the sequel, to simplify, we will use the notationA instead ofA_ρ,1.

Recall that the ∆k-Poisson kernel of the unit ball (see [8]) is given by P_k(x, y) =

+∞

∑

n=0

Z_k,n(x, y) =

∫

R^d

1− ∥x∥² (1− ⟨x, z⟩+∥x∥²)^d

2+γdµ_y(z), (x, y)∈B×S^d−1. (3.1) From [8], we know that

1 dk

∫

S^d−1

P_k(x, ξ)ω_k(ξ)dσ(ξ) = 1. (3.2)

We start by two preliminary useful results. For each n∈N, we see that the restriction of the Dunkl intertwining operator

Vk:H∆,n(R^d)−→ H∆_k,n(R^d) is a linear isomorphism.

In the first result, we will estimate the matrix-norms of this operator and of its inverse where the space H∆,n(R^d) (resp. H∆k,n(R^d)) is endowed with theL²(S^d−1)-norm (resp.

theL²_k(S^d−1)-norm). More precisely,

Proposition 3.1 Let n be a nonnegative integer.

1. For every f ∈ H∆,n(R^d), we have

∥Vk(f)∥L²_k(S^d−1) ≤dimH∆,n(R^d)∥f∥L²(S^d−1). (3.3) 2. For every f ∈ H∆k,n(R^d), we have

∥V_k⁻¹(f)∥L²(S^d−1)≤ (γ+^d₂)nωd−1

(^d₂)n

dimH∆,n(R^d)∥f∥L²_k(S^d−1). (3.4)

Proof:1)Letf ∈ H∆,n(R^d). After rewriting the reproducing formula (2.10) in the classical case (i.e. k= 0), applying it tof and using Fubini’s theorem, we get

Vk(f)(x) = 1 ω_d−1

∫

S^d−1

f(ξ)Vk[Zn(., ξ)](x)dσ(ξ), x∈R^d. (3.5) But, from [1], Proposition 5.27, we have

∀ z, ξ∈S^d−1, |Zn(z, ξ)| ≤dimH∆,n(R^d) which implies that

∀(z, ξ)∈R^d×S^d−1, |Z_n(z, ξ)| ≤(

dimH∆,n(R^d)

)∥z∥ⁿ. (3.6)

Thus, using the relations (2.3), (3.5) and (3.6) and Cauchy-Schwarz inequality, we obtain

∀ x∈R^d, |Vk(f)(x)| ≤dimH∆,n(R^d)∥f∥_L²_(S^d−1)∥x∥ⁿ. (3.7) This implies that

∥Vk(f)∥_L²

k(S^d−1)≤dimH∆,n(R^d)∥f∥_L²_(S^d−1).

2)Letf ∈ H∆_k,n(R^d). Applying the classical case of the formula (2.10) withf ←V_k⁻¹(f) and using (3.6) and Cauchy-Schwarz inequality, we deduce that

∀ x∈R^d, |V_k⁻¹(f)(x)| ≤dimH∆,n(R^d)∥V_k⁻¹(f)∥L²(S^d−1)∥x∥ⁿ.

Now, using the following result (see [8], Proposition 5.2.8): for p ∈ Pn(R^d) and q ∈ H∆,n(R^d), then

1 ω_d−1

∫

S^d−1

p(ξ)q(ξ)dσ(ξ) = (γ+ ^d₂)nωd−1

(^d₂)ndk

∫

S^d−1

p(ξ)V_k(q)(ξ)ω_k(ξ)dσ(ξ) withp=q =V_k⁻¹(f), we obtain

∥V_k⁻¹(f)∥²_L²_(S^d−1)≤ (γ+^d₂)nωd−1

(^d₂)ndk

∫

S^d−1

|V_k⁻¹(f)(ξ)f(ξ)|ωk(ξ)dσ(ξ)

≤ (γ+^d₂)_nω_d−1

(^d₂)_nd_k dimH∆,n(R^d)∥V_k⁻¹(f)∥L²(S^d−1)

∫

S^d−1

|f(ξ)|ω_k(ξ)dσ(ξ)

≤ (γ+^d₂)_nω_d−1

(^d₂)_n dimH∆,n(R^d)∥V_k⁻¹(f)∥L²(S^d−1)∥f∥_L²

k(S^d−1).

This proves the desired relation.

Corollary 3.1 The following inequality holds:

∀ x, y∈S^d−1, |Z_k,n(x, y)| ≤((γ+ ^d₂)_nω_d−1 (^d₂)n

)2(

dimH∆,n(R^d) )5

. (3.8)

Proof: Let {Y_j,n, j = 1, . . . , h(n, d) = dimH∆k,n(R^d)}, is a real-orthonormal basis of H∆k,n(R^d) in L²_k(S^d−1). Using (3.7) with f =V_k⁻¹(Y_j,n) and (3.4), we deduce that

∀x∈S^d−1,|Yj,n(x)| ≤dimH∆,n(R^d)∥V_k⁻¹(Yj,n)∥L²(S^d−1)

≤ (γ+^d₂)nωd−1

(^d₂)n

(dimH∆,n(R^d) )₂

.

Consequently, we obtain the result from (2.12).

Following the classical casek= 0 (see [1]), we define the kernelP_k,1(., .) on A×S^d−1 by P_k,1(x, ξ) :=

+∞

∑

n=0

a_k,n(x)Z_k,n(x, ξ), with a_k,n(x) = 1−(_∥x∥

ρ

)₋2λk−2n

1−ρ^2λk+2n . (3.9) Proposition 3.2 The kernel P_k,1 satisfies the following properties

i) For each ξ ∈ S^d−1, P_k,1(., ξ) is a ∆_k-harmonic function on A and P_k,1(., ξ) = 0 on S(0, ρ).

ii) For every x∈A and ξ ∈S^d−1,

0≤Pk,1(x, ξ)≤Pk(x, ξ). (3.10) iii) Let x∈A and ξ∈S^d−1 fixed. Then

∀ g∈W, P_k,1(gx, gξ) =P_k,1(x, ξ). (3.11) Proof: i)Clearly P_k,1(., ξ) = 0 on S(0, ρ). On the other hand, for any (x, ξ) ∈A×S^d−1 we can write

a_k,n(x)Z_k,n(x, ξ) =c_1,nZ_k,n(x, ξ)−c_2,nK_k[Z_k,n(., ξ)](x),

wherec_1,n, c_2,n are two nonnegative constants andK_k is the ∆_k-Kelvin transform (see [9]) given by

Kk[f](x) =∥x∥^−2λkf(x/∥x∥²) =∥x∥^2−2γ−df(x/∥x∥²) (3.12) and f is a function defined on R^d\ {0}. As the ∆_k-Kelvin transform preserves the ∆_k- harmonic functions onR^d\{0}(see [9]), we deduce that the functionx7→ak,n(x)Zk,n(x, ξ) is ∆_k-harmonic onA.

According to [8] (see also [1] and [5]), we know that dimH∆,n(R^d) =dimH∆k,n(R^d) =

(n+d−1 n

)

−

(n+d−3 n−2

) . Hence, we have

n→+∞lim n^2−ddimH∆,n(R^d) = 2 (d−2)!.

Moreover, we have

n→lim+∞n^−γ(γ+^d₂)n

(^d₂)n

= lim

n→+∞n^−γ Γ(d/2) Γ(γ+d/2)

Γ(d/2 +γ+n)

Γ(d/2 +n) = Γ(d/2) Γ(γ+d/2). Consequently, from (3.8), there existsC=C(d, γ)>0 such that

∀x∈R^d, ∀y∈S^d−1, |Z_k,n(x, y)| ≤Cn^5d+2γ−10∥x∥ⁿ. (3.13) This inequality as well as the fact that 0≤a_k,n(x)<1 imply that the series

∑

n≥0a_k,n(x)Z_k,n(x, ξ)

converges uniformly on Aρ,R×S^d−1 for every R ∈]ρ,1[. Then , by Corollary 3.3 in [11], the functionP_k,1(., ξ) is ∆_k-harmonic onA.

ii)For ε >0 small enough and ξ∈S^d−1, consider the function h_ε(x) :=∑

n≥0a_k,n(x)Z_k,n((1−ε)x, ξ).

As above, from the inequality (3.13) and the homogeneity ofZk,n(., ξ), we see that hε de- fines a ∆_k-harmonic function in the annular regionAρ,RwithR= (1−ε)⁻¹. Furthermore, h_ε = 0 onS(0, ρ) and if x∈S^d−1, then

hε(x) =∑

n≥0Zk,n((1−ε)x, ξ) =Pk((1−ε)x, ξ). (3.14) whereP_k is the ∆_k-Poisson kernel of the unit ball (see [8]). In particular,h_ε≥0 on S^d−1. Consequently, by the weak minimum principle for ∆_k-harmonic functions (see [11] or [21]), we deduce that

∀x∈A, h_ε(x)≥0.

On the other hand, for each fixed (x, ξ) in A×S^d−1, we have

|P_k,1(x, ξ)−h_ε(x)| ≤∑

n≥1

(1−(1−ε)ⁿ)a_k,n(x)|Z_k,n(x, ξ)|

≤C∑

n≥1

(1−(1−ε)ⁿ)n^5d+2γ−10∥x∥ⁿ.

Hence, by the monotone convergence theorem we haveP_k,1(x, ξ) = lim_ε→0h_ε(x). Finally, we obtainPk,1 ≥0 on A×S^d−1.

• For ξ ∈ S^d−1 fixed, the function x 7→ P_k((1−ε)x, ξ)−hε(x) is ∆_k-harmonic on A.

Moreover, since P_k is a nonnegative kernel, we have

∀ x∈S(0, ρ), Pk((1−ε)x, ξ)−hε(x) =Pk((1−ε)x, ξ)≥0.

By (3.14),x7→P_k((1−ε)x, ξ)−h_ε(x) is the zero function onS^d−1. So, the weak maximum principle implies that

∀x∈A, P_k((1−ε)x, ξ)≥hε(x).

Lettingε−→0, we obtainPk(., ξ)≥Pk,1(., ξ) onA.

iii)The result follows immediately from (2.11).

Proposition 3.3 Let f be a continuous function onS^d−1. Then the function

P_k,1[f](x) = 1 d_k

∫

S^d−1

P_k,1(x, ξ)f(ξ)ω_k(ξ)dσ(ξ) (3.15) is the unique solution inC²(A)∩ C(A) of the boundary Dirichlet problem





∆_ku= 0, onA;

u=f, onS^d−1 u= 0, onS(0, ρ).

Proof: The uniqueness follows from the weak maximum principle for ∆_k-harmonic func- tions (see [11] or [21]). The inequality (3.13) allowed us to write for anyx∈Athat

P_k,1[f](x) =

+∞

∑

n=0

u_n(x), with u_n(x) = a_k,n(x) dk

∫

S^d−1

Z_k,n(x, ξ)f(ξ)ω_k(ξ)dσ(ξ).

By differentiation theorem under integral sign, the functions u_n are ∆_k-harmonic on A.

Moreover, by (3.13) we have

∀ n, |u_n(x)| ≤C∥f∥_∞n^5d+2γ−10∥x∥ⁿ. This proves that the series ∑

n≥0un converges uniformly on each closed annular region A_ρ,R whenever R∈]ρ,1[. Then, we conclude that P_k,1[f] is ∆_k-harmonic onA.

On the other hand, it is easy to see thatPk,1[f] = 0 on S(0, ρ).

It remains to prove that for every ξ∈S^d−1, lim_x→ξP_k,1[f](x) =f(ξ).

- If f ∈ H∆k,m(R^d), then u_n = 0 if n ̸= m and u_m(x) = a_k,m(x)f(x) = P_k,1[f](x).

Therefore, Pk,1[f] =f on S^d−1.

- Iff ∈ Pm(R^d), then by (2.9), there existf₁, . . . , f_m, withf_j ∈ H∆k,n−2j(R^d) such that f(x) =∑[m/2]

j=0 ∥x∥^2jf_j(x).

This implies thatP_k,1[f] =f onS^d−1.

- Iff is an arbitrary polynomial function, the result also holds.

- Suppose that f is a continuous function on S^d−1 and letp be a polynomial function.

By (3.10) and (3.2) we have

|P_k,1[f](x)−f(x)| ≤ |P_k,1[f](x)−P_k,1[p](x)|+|P_k,1[p](x)−p(x)|+|p(x)−f(x)|

≤2∥f−p∥_∞+|P_k,1[p](x)−p(x)|.

This inequality as well as the Stone-Weierstrass theorem show that limx→ξPk,1[f](x) =

f(ξ) for everyξ ∈S^d−1. This completes the proof.

Now, for x∈Aand ξ ∈S^d−1, consider the functions b_k,n(x) =∥x∥⁻ⁿ(∥x∥

ρ

)₋2λk−n1− ∥x∥^2λk+2n

1−ρ^2λk+2n =ρ⁻ⁿ(1−a_k,n(x))

and

P_k,2(x, ξ) :=

+∞

∑

n=0

b_k,n(x)Z_k,n(x, ξ). (3.16) By means of the Poisson kernel of the unit ball, we can write

P_k,2(x, ρξ) =P_k(x, ξ)−P_k,1(x, ξ). (3.17) This relation as well as the properties ofP_kandP_k,1prove thatP_k,2(., ρξ) is a nonnegative

∆_k-harmonic function onA withP_k,2(., ρξ) = 0 on S^d−1.

Let f be a continuous function on S(0, ρ) and define the function Pk,2[f](x) = 1

d_k

∫

S^d−1

Pk,2(x, ρξ)f(ρξ)ωk(ξ)dσ(ξ), x∈A. (3.18) Using (3.17), we can write

P_k,2[f](x) = 1 d_k

∫

S^d−1

(P_k(x, ξ)−P_k,1(x, ξ))

f(ρξ)ω_k(ξ)dσ(ξ)

=P_k[δρ.f](x)−P_k,1[δρ.f](x).

Here,Pk[ϕ] denotes the Poisson integral ofϕ andδρ.f(x) =f(ρx).

Then, using Proposition 3.3 and theorem A in [19], we obtain immediately the following result:

Proposition 3.4 Let f be a continuous function on S(0, ρ). Then P_k,2[f]is the unique solution in C²(A)∩ C(A) of the boundary Dirichlet problem





∆_ku= 0, onA;

u= 0, onS^d−1 u=f, onS(0, ρ).

Definition 3.1 Letf be a continuous function on∂A. We define the∆_k-Poisson integral of f for the annulusA by

P_k,A[f](x) :=P_k,1[f](x) +P_k,2[f](x) (3.19) Remark 3.1 1. We can see that P_k,A[1] = 1.

2. Using (3.11) and a similar relation for the kernel P_k,2, we obtain

g.P_k,A[f] =P_k,A[g.f], with g.f(x) :=f(g⁻¹x). (3.20) From Propositions 3.3 and 3.4, we deduce the following main result:

Theorem 3.1 Letf ∈ C(∂A). Then the functionP_k,A[f]is the unique solution inC²(A)∩ C(A) of the boundary Dirichlet problem





∆_ku= 0, on A;

u=f on ∂A.

From this theorem and the weak maximum principle for ∆_k-harmonic function (see [11]), we obtain the following result:

Corollary 3.2 Let h be a ∆k-harmonic function on A and continuous onA. Then,

∀ x∈A, h(x) =P_k,A[h](x).

4 ∆

-Green function of the annulus

Our aim in this section is to introduce and study the Green function of the annular region A={x∈R^d, ρ <∥x∥<1}for the Dunkl-Laplace operator. In the sequel, we will assume thatd+ 2γ >2 i.e. λ_k>0 with λ_k the constant (2.8).

Let us first recall that the ∆_k-Newton kernel, introduced in [13], is given by N_k(x, y) :=

∫ _+∞

0

p_k(t, x, y)dt, (4.1)

withp_k the Dunkl heat kernel (see [21, 24]) p_k(t, x, y) = 1

(2t)^d/2+γc_k

∫

R^de^{−(∥x∥2+∥y∥2−2⟨x,z⟩)/4t}dµ_y(z) (4.2) and c_k the Macdonald-Mehta constant given by

ck:=

∫

R^dexp(−∥x∥²

2 )ωk(x)dx.

According to ([13]), the positive and symmetric kernelNk takes the following form N_k(x, y) = 1

2d_kλ_k

∫

R^d

(∥x∥²+∥y∥²−2⟨x, z⟩)₋λk

dµ_y(z). (4.3) Note that ify= 0, µ_y =δ₀ (withδ_x0 the Dirac measure atx₀∈R^d) and then

N_k(x,0) = 1

2dkλk∥x∥^−2λk.

In addition, for each fixed x∈R^d, the functionNk(x, .) is ∆k-harmonic and of classC^∞ on R^d\W.x (where W.x is the W-orbit of x), ∆_k-superharmonic (see below for precise definition) on wholeR^d and satisfies

−∆_k[N_k(x, .)ω_k] =δ_x, in D^′(R^d), where

-for Ω ⊂ R^d a W-invariant open set, D(Ω) and D^′(Ω) denote respectively the space of C^∞-functions on Ω with compact support and the space of Schwartz distributions on Ω.

-forf ∈L¹_loc(Ω, ω_k(x)dx), ∆_k(f ω_k) is the Schwartz distribution on Ω defined by

⟨∆k(f ωk), φ⟩=⟨f ωk,∆kφ⟩, φ∈ D(Ω).

Moreover, for any x ∈ R^d, N_k(x, x) = +∞. For more details on the properties of the

∆_k-Newton kernel one can see Section 6 in [13].

Let Ω be a W-invariant open subset of R^d. Recall that a functionu: Ω−→[−∞,+∞[ is

∆_k-subharmonic if (see [13])

1. u is upper semi-continuous on Ω,

2. u is not identically−∞ on each connected component of Ω,

3. usatisfies the volume sub-mean property i.e. for each closed ballB(x, r)⊂Ω, we have u(x)≤M_B^r(u)(x) := 1

m_k[B(0, r)]

∫

R^du(y)hk(r, x, y)ωk(y)dy. (4.4) Here m_k is the measure dm_k(x) :=ω_k(x)dx and y 7→ h_k(r, x, y) is the nonnegative com- pactly supported measurable function given by

hk(r, x, y) :=

∫

R^d1_[0,r](√

∥x∥²+∥y∥²−2⟨x, z⟩)dµy(z). (4.5) We refer to [11] for more details on the kernelhk.

The following result gives some useful facts about the Poisson integral of the ∆_k- Newton kernel:

Proposition 4.1

i) For each x∈A, the functionP_k,A[N_k(x, .)] is the solution of the Dirichlet problem { ∆_ku= 0, on A;

u=N_k(x, .) on ∂A.

ii) The function (x, y)7→Pk,A[Nk(x, .)](y) is continuous on A×A.

iii) For each fixed y∈A, the function x7→P_k,A[N_k(x, .)](y) is∆_k-harmonic in A.

Proof: i)If x ∈ A, then the function N_k(x, .) is continuous on ∂A and by Theorem 3.1, we obtain the first assertion.

ii)At first, we shall prove the following result

Lemma 4.1 The function (x, y) 7→ N_k(x, y) is continuous on R^d×R^d\ {(x, gx), x ∈ R^d, g∈W}.

Proof: Using the following inequality (see [24] Lemma 4.2)

∀t >0, p_k(t, x, y)≤ 1

c_k(2t)^d/2+γ max

g∈We^{−∥x−gy∥2/4t},

we can apply the dominated convergence theorem in formula (4.1) to obtain the result of

the lemma.

From the first assertion, for each x ∈ A, P_k,A[N_k(x, .)] is extendable to a continuous function on A withP_k,A[N_k(x, .)] =N_k(x, .) on ∂A.

Let (x₀, y₀)∈A×A. For every (x, y)∈A×A we have

Pk,A[Nk(x, .)](y)−Pk,A[Nk(x0, .)](y0)≤Pk,A[Nk(x, .)](y)−Pk,A[Nk(x0, .)](y) +P_k,A[N_k(x0, .)](y)−P_k,A[N_k(x0, .)](y0)

≤P_k,1[K_x0(x, .)](y) +P_k,2[K_x0(x, .)](y) +P_k,A[N_k(x0, .)](y)−P_k,A[N_k(x0, .)](y0), whereKx0(x, y) :=N_k(x, y)−N_k(x0, y).

We already know that

ylim→y0

P_k,A[N_k(x₀, .)](y) =P_k,A[N_k(x₀, .)](y₀).

Now, let ε > 0 and R > 0 be such that B(x₀, R) ⊂ A. Since (x, ξ) 7−→ N_k(x, ξ) is uniformly continuous onB(x0, R)×S^d−1, we deduce that there existsη >0 such that

∀(x, ξ)∈B(x0, η)×S^d−1, |Kx0(x, ξ)|=Nk(x, ξ)−Nk(x0, ξ)< ε.

Then, using (3.15) as well as the inequalities (3.2) and (3.10), we get for everyx∈B(x0, η) and everyy∈A

Pk,1[Kx0(x, .)](y)≤ 1 d_k

∫

S^d−1

Pk,1(y, ξ)|Kx0(x, ξ)|ωk(ξ)dσ(ξ)≤ε.

The same idea works if we replace the kernel P_k,1 byP_k,2. Finally, we obtain lim

(x,y)→(x0,y0)P_k,A[N_k(x, .)](y) =P_k,A[N_k(x0, .)](y0).

That is the function (x, y)7→P_k,A[N_k(x, .)](y) is continuous on A×A as desired.

iii)According to Corollary 4.6 in [12], it is enough to show that the functionsx7→u_y(x) :=

P_k,1[N_k(x, .)](y) and x7→v_y(x) :=P_k,2[N_k(x, .)](y) satisfy the volume-mean property.

Let then x0 ∈A and R > 0 such thatB(x0, R)⊂A. As the kernels N_k,h_k and P_k,1 are nonnegative, we can use Fubini’s theorem to obtain

M_B^R(uy)(x0) = 1 d_k

∫

S^d−1

Pk,1(y, ξ)M_B^R[Nk(., ξ)](x0)ωk(ξ)dσ(ξ)

But for any ξ ∈S^d−1, the function N_k(., ξ) is ∆_k-harmonic on A. Hence, it satisfies the volume-mean property i.e. M_B^R[N_k(., ξ)](x₀) =N_k(x₀, ξ). Therefore, we obtain

M_B^R(u_y)(x₀) = 1 dk

∫

S^d−1

P_k,1(y, ξ)N_k(x₀, ξ)(x₀)ω_k(ξ)dσ(ξ)

=Pk,1[Nk(x0, .)](y) =uy(x0).

By the same way, we get that x 7→ v_y(x) := P_k,2[N_k(x, .)](y) is also a ∆_k-harmonic function inA. This proves the desired result.

Definition 4.1 For x∈A, the function G_k,A(x, .) defined by

G_k,A(x, y) :=N_k(x, y)−P_k,A[N_k(x, .)](y), y∈A, (4.6) is called the ∆k-Green function of A with pole x.

The ∆_k-Green function G_k,A has the following properties:

Proposition 4.2 Let x∈A. Then

1. The function G_k,A(x, .) is ∆_k-harmonic on A\W.x, is ∆_k-superharmonic on A and satisfies

−∆_k[G_k,A(x, .)ω_k] =δ_x in D^′(A). (4.7) 2. G_k,A(x, x) = +∞ and G_k,A(x, y)<+∞ whenever y /∈W.x .

3. For every ξ∈∂A, limy→ξGk,A(x, y) = 0.

4. For every y∈A,G_k,A(x, y)>0.

5. For every x, y∈A, G_k,A(x, y) =G_k,A(y, x).

6. For every x, y∈A and g∈W, G_k,A(gx, gy) =G_k,A(x, y).

7. The zero function is the greatest ∆k-subharmonic minorant of Gk,A(x, .) on A.

8. The function (x, y)7→G_k,A(x, y) is continuous on A×A\ {(x, gx) : x∈A, g∈W}. Proof: The first and the second assertions follow from the properties of the ∆_k-Newton kernel previously mentioned. In addition, by Proposition 4.1, we easily obtain the third statement.

4)As lim_{y→ξ∈∂A}G_k,A(x, y) = 0, the weak minimum principle for ∆_k-superharmonic func- tions (see [13], Theorem 3.1) implies thatG_k,A(x, .)≥0 onA.

IfG_k,A(x, y0) = 0 for somey0∈A, from the strong maximum principle (see [13]), we must have G_k,A(x, .) is the zero function on A which is impossible because G_k,A(x, x) = +∞. Thus,G_k,A is a positive kernel on A×A.

5)Since Nk is a symmetric kernel, we have to prove that

∀ x, y∈A, P_k,A[N_k(x, .)](y) =P_k,A[N_k(y, .)](x).

Lety∈A and consider the function

H_y(x) :=P_k,A[N_k(x, .)](y)−P_k,A[N_k(y, .)](x).