THE HARMONIC ANALYSIS ASSOCIATED TO THE CHEREDNIK-TRIM\`{E}CHE'S TRANSMUTATION OPERATORS ON $\mathbb{R}^d$

Volume 31 No. 6 2018, 709-725

ISSN: 1311-1728 (printed version); ISSN: 1314-8060 (on-line version) doi:_{http://dx.doi.org/10.12732/ijam.v31i6.2}

THE HARMONIC ANALYSIS ASSOCIATED TO THE CHEREDNIK-TRIM`ECHE’S

TRANSMUTATION OPERATORS ON Rd

Khalifa Trim`eche

Faculty of Sciences of Tunis Department of Mathematics CAMPUS, 2092 Tunis, TUNISIA

Abstract: We consider in this paper two Cherednik operators T_jk, T_jl, j = 1,2,3, ..., d, on Rd_{, associated to the multiplicity functionsk, l. First we define} and study in this paper the Cherednik-Trim`eche’s transmutation operator Ukl

and its dual tU_kl. Next we study the Harmonic Analysis associated to these operators.

AMS Subject Classification: 33E30; 51F15; 33C67; 43A32; 44A15

Key Words: Cherednik operators; Opdam-Cherednik’s kernel; intertwining operators; heat kernel; Cherednik-Trim`eche’s transmutation operators; Chered-nik-Trim`eche’s heat kernel; CheredChered-nik-Trim`eche’s translation operator

1. Introduction

In [1] I. Cherednik has introduced a family of differential-difference operators that nowadays bear his name. These operators play a crucial role in the ory of Heckman-Opdam’s hypergeometric functions, which generalize the the-ory of Harish-Chandra’s spherical functions on Riemmann symetric spaces (see [2,3,4]).

We consider in this paper two Cherednik operatorsT_jkandT_jl, j= 1,2, ..., d, on Rd_{, associated to the multiplicity functionsk, l∈[0,∞).}

By using the Harmonic Analysis associated to the Cherednik operators (see [2,3,4,5,6,7]), given in Sections 2,3,4,5,6, we define and study in the other

sections the Cherednik-Trim`eche’s transmutation operator U_kl and its dual

t_U

kl, the Cherednik-Trim`eche’s translation operator τxkl and its dualtτxkl, the

Cherednik-Trim`eche’s convolution product, the Cherednik-Trim`eche’s heat ker-nelpkl_t (x, y).

2. The Cherednik operators and their eigenfunctions

We considerRd _{with the standard basis{ej;j= 1,2, ..., d} and the inner} prod-ucth., .i for which this basis is orthonormal.

2.1. The root system

Letα∈Rd_{\{0} and ˇα=} 2

kαk2α. We denote by

rα(x) =x− hˇα, xiα, x∈Rd, (2.1)

the reflection on the hyperplan Hα ⊂Rd orthogonal to α. For d= 1, we take α= 2.

A finite setR ⊂ Rd_{\{0} is called a root system if}r_αR=R, for all α∈ R. For a given β ∈ Rd_{\ ∪α∈R Hα, we fix the positive subsystem R+ = {α ∈} R,hα, βi>0}, then for eachα∈ R eitherα∈ R+ or −α∈ R+.

The reflections rα, α ∈ R, generate a finite group W ⊂ O(d), called the

reflection group associated with R. Let Rd

reg = Rd\∪α∈RHα be the set of

regular elements inRd_.

A functionk:R →[0,+∞[ is called a multiplicity function, if it is invariant under the action of the reflection group W. We introduce the index

γ =γ(R) = X

α∈R+

k(α). (2.2)

2.2. The Cherednik operators

The Cherednik operatorsT_jk, j = 1,2, ..., d, onRd_{associated with the reflection} group W and the multiplicity function k, are defined for f of class C1 on Rd and x∈Rd

reg by

T_jkf(x) = ∂

∂x_jf(x) +

X

α∈R+

k(α)αj

1−e−hα,xi{f(x)−f(rαx)} −ρ k

where

ρk_j = 1 2

X

α∈R+

k(α)αj, andαj =hα, e_ji. (2.4)

The Cherednik operators form a commutative system of differential-diffe-rence operators.

Forf of class C1 on Rd _{with compact support andg of class C}1 _on Rd_{, we} have for j= 1,2, .., d:

Z

Rd

T_jkf(x)g(x)Ak(x)dx=−

Z

Rd

f(x)(T_jk+S_jk)g(x)Ak(x)dx, (2.5)

withAk the weight function given by

∀x∈Rd_{, Ak(x) =} Y

α∈R+

|2 sinhhα 2, xi|

2k(α)_{, (2.6)}

which isW-invariant and

∀x∈Rd_{, S}k

jg(x) =

X

α∈R+

k(α)αjg(r_αx). (2.7)

Example 2.1. We consider for d= 1, the root systemR ={±α,±2α}, with α = 2. Here R+ = {α,2α}, and the reflection group is W = Z_{2. We} denote by k the multiplicity function. The Cherednik operator T₁k is defined forf of class C1 onR_{, and x in}R_{\{0} by}

T₁kf(x) = d

dxf(x) +

2k(α) 1−e−2x +

4k(2α) 1−e−4x

(f(x)−f(−x))−ρkf(x) (2.8)

withρk=k(α) + 2k(2α).

If we putk1 =k(α)+k(2α),k2 =k(2α), the operatorT1ktakes the following form

T₁kf(x) = d

dxf(x)+(k1coth(x) +k2tanh(x)) (f(x)−f(−x))−ρ

k_{f(−x), (2.9)}

withρk=k1+k2.

2.3. The Opdam-Cherednik’s kernel We denote by Gk_λ, λ∈Cd_{, the eigenfunction of the operators T}k

j, j = 1,2, .., d.

system

T_jkGk_λ(x) =iλ_jG_λk(x), j= 1,2, .., d, x∈Rd_,

Gk_λ(0) = 1. (2.10)

It is called the Opdam-Cherednik’s kernel.

Remarks 2.1. Fork= 0, we have for allx∈Rd_,Gk

λ(x) =eihλ,xi.

The functionsGk_λ possess the following properties:

i) For allλ∈Cd_{, the function}x7→G_λk(x) is of class C∞ onRd_. ii) For all x∈Rd_{, the function} λ7→Gk_λ(x) is entire onCd_. iii) For allx∈Rd _and λ∈Cd_{, we have}

G_λk(x) =Gk_−λ¯(x). (2.11) iv) For all x∈Rd _{and λ∈}Cd_{, we have}

|Gk_λ(x)| ≤Gk_iℑm(λ)(x). (2.12)

v) For allx∈Rd _{and λ∈}Rd_{, we have}

|Gk_λ(x)| ≤ |W|1/2. (2.13) vi) Let p and q be polynomials of degree m and n. Then, there exists a

positive constantM such that for allλ∈Cd _{and x∈}Rd_{, we have} |p( ∂

∂λ)q( ∂ ∂x)G

k

λ(x)| ≤M(1 +kxk)m(1 +kλk)nF0k(x)e−maxw∈WImhwλ,xi, (2.14) where

∀x∈Rd, F₀k(x) = 1 |W|

X

w∈W

Gk₀(wx). (2.15)

Example 2.2. We consider for d= 1, the root systemR ={±α,±2α}, with α = 2, the reflection group W = Z_{2, the multiplicity function k, the} parameters k₁,k₂, and the Cherednik operator T₁k, given in Example 2.1.

The Opdam-Cherednik’s kernel is given by

∀x∈R_,∀λ∈C_{, G}k

λ(x) =ϕ

(k1−1₂,k2−1₂)

λ (x)+

1

iλ−ρk d dxϕ

(k1−1₂,k2−1₂)

whereϕ(_λa,b)(x) is the Jacobi function of index (a, b) given by

ϕ(_λa,b)(x) = 2F1( 1

2(ρ+iλ), 1

2(ρ−iλ);α+ 1;−(sinh(x)) 2_), with2F1 the hypergeometric function of Gauss and ρ=a+b+ 1.

3. The intertwining operator V_k

and its dual tVk

Notation. We denote by

- E(Rd_{) the space of C}∞_{-functions on} Rd_{. Its topology is defined by the} semi-norms

qn,K(ϕ) = sup |µ|≤n

x∈K

|Dµϕ(x)|,

whereK is a compact ofRd_,n∈N_{, and}

Dµ= ∂

|µ|

∂xµ1

1 ...∂xµ d

d

, µ= (µ1, ..., µd)∈Nd, |µ|= d

X

i=1

µi.

- D(Rd_{) the space of C}∞_{-functions on} Rd_{, with compact support. We have} D(Rd_{) =} [

a>0

Da(Rd),

where Da(Rd) is the space of C∞-functions on Rd with support in the closed

ballB(0, a) of center 0 and radius a. The topology of Da(Rd) is defined by the

semi-norms

pn(ψ) = sup

0≤|µ|≤n x∈B(0,a)

|Dµϕ(x)|, n∈N_.

The spaceD(Rd_{) is equipped with the inductive limit topology.}

- S(Rd_{) the classical Schwartz space on} Rd_{. Its topology is defined by the} semi-norms

Qℓ,n(f) = sup

0≤|µ|≤n x∈Rd

(1 +kxk)ℓ|Dµf(x)|, n, l∈N_.

- S2(Rd) the generalized Schwartz space of C∞-functions on Rd such that forℓ, n∈N_{, we have}

Pn,l(f) = sup

0≤|µ|≤n x∈Rd

where F₀k(x) is the function given by the relation (2.15). It is topologized by means of the semi-normsPn,l, n, l∈N.

- D′_(Rd_{) the space of distributions on R}d_{. It is the topological dual of}

D(Rd_).

- E′(Rd_{) the space of distributions on} Rd _{with compact support. It is the} topological dual ofE(Rd_).

Definition 3.1. i) The intertwining operatorVkis the unique linear

topo-logical isomorphism from E(Rd_{) onto itself satisfying the transmutations} rela-tions

∀x∈Rd, T_jkV_k(g)(x) =V_k( ∂

∂y_jg)(x), j= 1,2..., d, (3.1)

and the relation

V_k(g)(0) =g(0). (3.2)

ii) The dualtVk of the operator Vk is defined by the following duality relation

Z

Rd

t_V

k(f)(y)g(y)dy =

Z

Rd

Vk(g)(x)f(x)Ak(x)dx. (3.3)

withf inD(Rd_{) and g inE(}Rd_).

Proposition 3.1. i) The operatort_V

k is a linear topological isomorphism from

-D(Rd_{) onto itself,}

-S2(Rd) ontoS(Rd),

satisfying the transmutation relations

∀y∈Rd_, t_Vk((Tk

j +Sjk)f)(y) = ∂ ∂y

t_V

k(f)(y), (3.4)

where S_jk is the operator on D(Rd_{) (resp. S2(}Rd_{) ) given by the relation}

(2.7).

ii) The dualtV_k−1of the operatorV_k−1satisfies the following duality relation

Z

Rd

t_V−1

k (f)(y)g(y)Ak(y)dy =

Z

Rd

V_k−1(g)(x)f(x)dx, (3.5)

withf inD(Rd_{) (resp. S2(}Rd_{)) and} g inE(Rd_).

iii) For allf inD(Rd_{) we have}

Suppf ⊂B(0, a)⇒SupptVk(f)⊂B(0, a), (3.6)

Remarks 3.1. From the relations (2.3),(3.1),(3.4) we deduce that the operators V0 and tV0 are the identity operators.

4. The hypergeometric Fourier transform associated with the Cherednik operators

Notation. Fora >0, we denote by P W(Cd_{)a (resp. PW(}Cd_{)a) the spaces of} functions hwhich are entire on Cd _{and satisfying}

∀m∈N_{, sm(h) = sup}

λ∈Cd(1 +kλk)meakImλk|h(λ)|<∞. (resp.∃m∈N_{, σm(h) = sup}

λ∈Cd(1 +kλk)−meakImλk|h(λ)|<∞.) Their topologies is given by the semi-normssm, m∈N, (resp. σm, m∈N).

- We consider the spacesP W(Cd_{) (resp. PW(}Cd_{) ) of the entire functions} onCd_{which are rapidly decreasing (resp. slowly increasing) and of exponential} type. We have

P W(Cd_{) =∪a>0P W(}Cd_)a(resp.PW(Cd_{) =∪a>0PW(}Cd_)a. They are equipped with the inductive limit topology.

Definition 4.1. The hypergeometric Fourier transform Hk is defined for all function f inD(Rd_{) (resp. S2(}Rd_{)) by}

∀λ∈Cd_{, H}k_{(f)(λ) =} Z

Rd

f(x)Gk_λ(x)Ak(x)dx. (4.1)

Theorem 4.1. The hypergeometric Fourier transformHkis a topological isomorphism from

- D(Rd_{) onto P W(}Cd_),

- S2(Rd_{) onto S(}Rd_).

The inverse transform(Hk)−1 is given by

∀x∈Rd_,(Hk₎−1_{(h)(x) =} Z

Rd

h(λ)Gk_λ(−x)Ck(λ)dλ, (4.2)

where for all λ∈Cd_,

- Fork∈(0,∞)

Ck(λ) =c

Y

α∈R+

Γ(−ihλ,αˇi+1₂k(α₂) +k(α))Γ(ihλ,αˇi+k(α₂) +k(α) + 1) Γ(−ihλ,αˇi+1₂Γ(α₂))Γ(ihλ,αˇi+ 1₂Γ(α₂) + 1) ,

withc a normalising constant. - Fork= 0,

Ck(λ) = 1. (4.4)

Definition 4.2. The hypergeometric Fourier transform Hk is defined for

S inE′_(Rd_{) by}

∀λ∈Rd, Hk(S)(λ) =hS, Gk_λi. (4.5)

Theorem 4.2. The transform Hk _{is a topological isomorphism from}

E′(Rd_{) ontoPW(}Cd_).

5. The hypergeometric translation operator and its dual and the hypergeometric convolution product associated with the

Cherednik operators

5.1. The hypergeometric translation operator and its dual Definition 5.1. The hypergeometric translation operator T_xk,x ∈Rd_{, is} defined onE(Rd_{) by}

∀y ∈Rd,T_xk(f)(y) = (V_k)x((Vk)y[Vk−1(f)(x+y)]. (5.1)

Proposition 5.1. The operator Tk

x, x∈Rd, satisfies the following prop-erties:

i) For allx∈Rd_{, the operatorT}k

x, is continuous fromE(Rd) into itself. ii) For allf inE(Rd_{) andx, y∈}Rd_{, we have}

T_xk(f)(0) =f(x), and T_xk(f)(y) =T_yk(f)(x). (5.2)

iii) For allx, y∈Rd_{, and} λ∈Cd_{, we have the product formula}

T_xk(Gk_λ)(y) =Gk_λ(x)Gk_λ(y), (5.3)

whereGk_λ, the opdam-cherednick kernel given by (2.10).

Definition 5.2. For each x∈Rd_{, the dual of the hypergeometric} transla-tion operator Tk

x, is the operator tTxk defined onD(Rd) (resp. S2(Rd)) by ∀y∈Rd_,t_Tk

Proposition 5.2. We give in the following the properties of the operator t_Tk

x:

i) For allx∈Rd_{, the operator}t_Tk

x is continuous from - D(Rd_{) into itself,}

- S2(Rd) into itself.

ii) The operatort_Tk

x,x∈Rd, is related to the operator Txk,x∈Rd, by the following relation

Z

Rd

T_xk(g)(y)f(y)Ak(y)dy =

Z

Rd

g(z)tT_xk(f)(z)Ak(z)dz, (5.5)

withg inE(Rd_{), and f inD(}Rd_{) (resp. S2(}Rd_)).

iii) For allf inD(Rd_{) (resp. S2(}Rd_))and x∈Rd_{, we have}

∀λ∈Rd,Hk(tT_xk(f))(λ) =Gk_λ(x)Hk(f)(λ). (5.6)

Thus from the relation (4.2) we have

∀y∈Rd_,t_Tk

x(f)(y) =

Z

Rd

Gk_λ(x)Gk_λ(−y)Hk(f)(λ)Ck(λ)dλ. (5.7)

iv) For all f in D(Rd_{) with support in the closed ball B(0, a) of center o}

and radiusa >0, andx∈Rd_{, we have}

SupptT_xk(f)⊂B(0, a+kxk). (5.8)

5.2. The hypergeometric convolution product

Definition 5.3. The hypergeometric convolution product f ∗kg of the

functions f, gin D(Rd_{) (resp. S2(}Rd_{)) is defined by}

∀x∈Rd, f ∗kg(x) =

Z

Rd

t_Tk

x(f)(y)g(y)Ak(y)dy. (5.9)

Proposition 5.3. The convolution product∗ksatisfies the following prop-erties:

i) For allf, g inD(Rd_{)(resp. S2(}Rd_{))the function f∗kgbelongs toD(}Rd₎

(resp. S2(Rd)).

ii) For allf, g inD(Rd_{)(resp. S2(}Rd_{)), we have}

iii) This convolution product is commutative and associative. iv) For all f, g inD(Rd_{)(resp. S2(}Rd_{)), we have}

t_V

k(f∗kg) =tVk(f)∗tVk(g), (5.11)

where∗ is the classical convolution product onRd_.

6. The heat kernel associated to the Cherednik operators

Definition 6.1. Let t >0. The heat kernel pk_t(x, y) associated with the Cherednik operators, is defined for allx, y∈Rd_{, by}

pk_t(x, y) = Z

Rd

e−t(kλk2+kρkk2)Gk_λ(x)Gk_λ(−y)Ck(λ)dλ. (6.1)

Notation. We denote by:

- Hk the heat operator associated with the Cherednik operator given by

H_k=Lk− ∂ ∂t − kρ

k_k2_{, (6.2)}

whereLk is the Heckman-Opdam Laplacian defined forf of classC2 on Rdby

Lkf = d

X

j=1

(T_jk)2(f). (6.3)

- E_tk, t >0, the fundamental solution of the operatorHk given by

∀x∈Rd_{, E}k

t(x) =pkt(x,0). (6.4)

Proposition 6.1. i) For allt >0, the functionE_tk belongs to S2(Rd).

ii) For allt >0, we have

∀λ∈Rd_,Hk_(Ek

t)(λ) =e−t(kλk

2_+kρk

k2₎

. (6.5)

iii) The function(x, t)→E_tk(x) is strictly positive on Rd_×(0,∞).

iv) For all t >0, we have

Z

Rd

Ek_t(x)Ak(x)dx= 1. (6.6)

v) We have

Proposition 6.2. i) For allt >0 and x∈Rd_{, the function} y →pk_t(x, y)

belongs to S2(Rd).

ii) For allt >0and x, y∈Rd_{, we have}

pk_t(x, y) =tT_xk(E_tk)(y). (6.8)

iii) The functionpk_t(x, y) is strictly positive on Rd_×Rd_×(0,∞).

iv) For all t >0and x∈Rd_{, we have} Z

Rd

pk_t(x, y)Ak(y)dy = 1. (6.9)

v) For ally ∈Rd_{, the function(x, t)→p}k

t(x, y) satisfies

Hkpkt(x, y) = 0, on Rd×(0,∞). (6.10)

Remark 6.1. To give the new results of this paper, we consider a second multiplicity function l : R → [0,∞[. We consider also the cherednik opera-tors T_jl, j = 1,2, ..., d, the Opdam-cherednik’s kernel Gl_λ, the transmutation operatorsVland its dualtVl, the hypergeometric Fourier transformHl, the

hy-pergeometric translation operatorTl

x and its dualtTxl, the fundamental solution E_tl of the operator H_l and of the heat kernelpl_t(x.y).

7. The Cherednik-Trim`eche’s transmutation operator U_kl and its dual tU_kl

Definition 7.1. The Cherednik-Trim`eche’s transmutation operator Uklis

defined onE(Rd_{) by}

∀x∈Rd_{, Ukl(f)(x) =Vk◦V}−1

l (f)(x). (7.1)

By using the properties of the transmutation operatorsVk and Vl given in

Section 3, we obtain the following properties of the operator Ukl.

Theorem 7.1. i) For k=l, we have

ii) The operatorU_kl is the unique topological isomorphism fromE(Rd_)onto

itself satisfying the condition

Ukl(f)(0) =f(0). (7.3)

iii) The inverse operatorU_kl−1 is given forf inE(Rd_{) by} ∀x∈Rd_{, U}−1

kl (f)(x) =Vl◦Vk−1(f)(x) =Ulk(f)(x). (7.4) iv) The operatorU_klsatisfies for all f inE(Rd_{) the following transmutation}

relations

∀x∈Rd, T_jk(U_kl(f))(x) =U_kl(T_jl(f))(x), j= 1,2, ..., d. (7.5)

v) We have

∀λ∈Cd,∀x∈Rd, U_kl(Gl_λ)(x) =Gk_λ(x). (7.6)

vi) We have

Ukl(1) = 1. (7.7)

Definition 7.2. The dual of the Cherednik-Trim`eche’s transmutation operator Ukl is the operator tUkl defined onD(Rd) (resp. S2(Rd)) by

∀y∈Rd,tU_kl(g)(y) =tV_l−1◦tV_k(g)(y). (7.8) The properties of the operator tV_k and tV_l, given in Section 3, imply the following properties of the operator tU_kl.

Theorem 7.2. i) For k=l, we have t_U

kl =Id. (7.9)

ii) The operatortU_klis a topological isomorphism fromD(Rd_{)(resp. S2(}Rd₎₎

onto itself. iii) The inverse operator tU_kl−1 is given for g in D(Rd_{) (resp.} S2(Rd)) by

∀y∈Rd_, t_U−1

kl (g)(y) =tVk−1◦tVl(g)(y) =tUlk(g)(y). (7.10) iv) The operatort_U

klsatisfies for allginD(Rd)(resp. S2(Rd))the following

transmutation relation

∀y∈Rd_, t_Ukl((Tk

Proposition 7.1. The operators U_kl and tU_kl are related for f inE(Rd₎

and g inD(Rd_{) (resp. S2(}Rd_{) by the following duality relation} Z

Rd

U_kl(f)(x)g(x)Ak(x)dx=

Z

Rd

t_U

kl(g)(y)f(y)Al(y)dy. (7.12)

Corollary 7.1. The operatorstU_klsatisfys for allginD(Rd_{)(resp. S2(}Rd₎

the following expression:

∀y∈Rd, tU_kl(g)(y) = (Hl)−1◦ Hk(g)(y). (7.13)

Proof. From the relations (7.12),(7.6), we have

∀λ∈Rd_,Hk_{(g)(λ) =H}l₍t_Ukl(g))(λ). We deduce (7.13) from this relation and Theorem 4.2.

8. The Cherednik-Trim`eche’s translation operator and its dual Definition 8.1. For x, y∈Rd<sub>, the Cherednik-Trim`eche’s translation</sub> op-eratorTkl

x is defined for all f inE(Rd) by

T_xkl(f)(y) = (Vl)x((Vk)y[V_k−1(f)(x+y)]. (8.1)

Definition 8.2. For all x, y∈ Rd_{, the dual of the Cherednik-Trim`eche’s} translation operator t_Tkl

x is defined for all g inD(Rd) (resp. S2(Rd)) by

t_Tkl

x (g)(z) = (Vl)x(tV_k−1)z[tVk(g)(z−x)]. (8.2)

From the properties of the operator Vl, Vk and tVk given in Section 3, we

obtain the following properties of the operators Tkl

x andtTxkl.

Proposition 8.1. i) For allf inE(Rd_{), the functionT}kl

x (f)(y) is of class C∞onRd_{with respect to the variablesxand y. ii) For allginD(}Rd_)(resp. S2(Rd)) the function tTxkl(g)(z) is of class Rd with respect to the variables x, and belongs to D(Rd_{) (resp. S2(}Rd_{)) with respect to the variable z. More}

precisely if the support of g is contained in the ball of center 0 and radius a >0, andx∈Rd_{, we have have}

iii) For allf inE(Rd_{) and} x∈Rd_{, we have}

T_xkl(f)(0) =Vl◦V_k−1(f)(x). (8.4) iv) For all g inD(Rd_{) (resp. S2(}Rd_{))and z∈}Rd_{, we have}

t_Tkl

0 (g)(z) =g(z). (8.5)

v) For allx, y∈Rd_{and λ∈}Cd_{, we have}

T_xkl(Gk_λ)(y) =Gl_λ(x)Gk_λ(y). (8.6)

Proposition 8.2. The operators Tkl

x and tTxkl are related for all f in

E(Rd_{)and g inD(}Rd_{) (resp. S2(}Rd_{)) by the following duality relation:} Z

Rd

T_xkl(f)(y)g(y)Ak(y)dy =

Z

Rd

f(z)tT_xkl(g)(z)Ak(z)dz. (8.7)

Corollary 8.1. For allx∈Rd_{, the dual}t_Tkl

x of the Cherednik-Trim`eche’s translation operator, satisfies for all g in D(Rd_{) (resp. S2(}Rd_{)) the following}

relations:

i) ∀λ∈Rd_{, H}k₍t_Tkl

x (g))(λ) =Glλ(x)Hk(g)(λ), (8.8)

ii) ∀z∈Rd_, t_Tkl

x (g)(z) =

Z

Rd

Gl_λ(x)Gk_λ(−z)Hk(g)(λ)Ck(λ)dλ. (8.9)

Proof. i) By applying thr relation (8.7) withf(z) =Gk_λ(z), z∈Rd_,λ∈Rd_, we obtain

Z

Rd

Gk_λ(z)tT_xkl(g)(z)Ak(z)dz=

Z

Rd

T_xkl(Gk_λ)(y)g(y)Ak(y)dy.

We deduce relation (8.8) from this relation and relations (8.6),(4.1).

9. The Cherednik-Trim`eche’s convolution product

Definition 9.1. The Cherednik-Trim`eche’s convolution product of the functions f, gin D(Rd_{) (resp. S2(}Rd_{)) is the functionf ∗klg defined by}

∀y∈Rd_{, f ∗klg(y) =} Z

Rd

t_Tkl

x (f)(y)g(x)Ak(x)dx. (9.1)

Proposition 9.1. i) For all functions f, g in D(Rd_{) (resp. S2(}Rd_{)) the}

functionf ∗klg belongs toD(Rd) (resp. S2(Rd)).

ii) For allf, g inD(Rd_{) (resp. S2(}Rd_{)), we have the following relations:} ∀λ∈Rd_{, H}k_{(f∗klg)(λ) =H}l_{(g∗klf)(λ), (9.2)} ∀λ∈Rd_{, H}k_{(f ∗klg)(λ) =H}k_(f)(λ)Hl_{(g)(λ). (9.3)}

Proposition 9.2. For all functionsf, ginD(Rd_{) (resp. S2(}Rd_{)), we have} ∀y ∈Rd, tV_k(f∗klg)(y) =tVk(f)∗tVl(g)(y), (9.4) where∗ is the classical convolution product of functions onRd_.

Proof. We deduce relation (9.4) from relations (9.1) and (8.2).

10. The Cherednik-Trim`eche’s heat kernel

Definition 10.1. For allx, y∈Rd_{andt∈(0,∞), we define the} Cherednik-Trim`eche’s heat kernelpkl_t (x, y) by

pkl_t (x, y) =tT_xkl(E_tk)(y), (10.1)

whereE_tk(x) is the fundamental solution of the operatorH_kgiven by the relation (6.4).

Proposition 10.1. i) For all x∈Rd_,t∈(0,∞) and λ∈Rd_{, we have} Hk(pkl_t (x, .))(λ) =e−t(kλk2+kρkk2)Gl_λ(x). (10.2)

ii) For allx, y∈Rd _{andt∈(0,∞), we have}

pkl_t (x, y) = Z

Rd

Proof. i) From the relations (10.1),(8.7) we have

∀(x, t)∈Rd_×(0,∞), λ∈Rd,Hk(p_tkl(x, .))(λ) =Gl_λ(x)Hk(E_tk)(λ).

We deduce relation (10.2) from relation (6.5).

ii) The relations (10.1), (10.2), (8.9) (8.10) imply relation (10.3).

Proposition 10.2. Forx∈Rd_{and t∈(0,∞), we have} ∀y∈Rd_, t_Ukl(pkl

t (x, .))(y) =e−t(kρ

k

k2

−kρl

k2

)_pl

t(x, y). (10.4)

Proof. From the relations (7.13),(10.2), we have for ally ∈Rd_,

t_U

kl(pklt (x, .))(y) = (Hl)−1[e−t(kλk

2

+kρk

k2

)_Gl

λ(x)](y)

= e−t(kρkk2−kρlk2)(Hl)−1[e−t(kλk2+kρlk2)Gl_λ(x)](y).

By using the relation (4.2) we obtain for all y∈Rd_,

t_U

kl(pklt (x, .))(y) =e−t(kρ

k

k2_−kρl

k2₎Z Rd

e−t(kλk2+kρlk2)Gl_λ(x)Gl_λ(−y)Cl(λ)dλ.

Thus the relation (6.1) implies

∀y∈Rd_, t_Ukl(pkl

t (x, .))(y) =e−t(kρ

k

k2_−kρl

k2₎

pl_t(x, y).

In coming papers we plan to study:

1. The Harmonic Analysis associated to the Cherednik-Trim`eche’s trans-mutation operators on Rd _{in theW-invariant case.}

2. Applications of the Harmonic Analysis associated to the Cherednik-Trim`eche’s operators onRd_.

11. Acknowledgments

References

[1] I. Cherednik, A unification of Knizhnik-Zamolod-dchnikov equations and Dunkl operators via affine Hecke algebras,Invent. Math. Res.,106(1991), 411-432.

[2] G.J. Heckman, E.M. Opdam, Root systems and hypergeometric functions I, Compositio Math.,64(1987), 329-352.

[3] E.M. Opdam, Harmonic analysis for certain representations of graded Hecke algebras,Acta Math., 175(1995), 75-121.

[4] B. Schapira, Contribution to the hypergeometric function theory of Heck-man and Opdam; sharp estimates, Schwartz spaces, heat kernel, Geom. Funct. Anal.,18 (2008), 222-250.

[5] K. Trim`eche, The trigonometric Dunkl intertwining operator and its dual associated with the Cherednik operators and the Heckman Opdam theory,

Adv. Pure Appl. Math.,1 (2010), 293-323.

[6] K. Trim`eche, Harmonic analysis associated with the Cherednik operators and the Heckman-Opdam theory,Adv. Pure Appl. Math.,2(2011), 23-46. [7] K. Trim`eche, The positivity of the hypergeometric translation operators associated to the Cherednik operators and the Heckman-Opdam theory attached to the root system of type BC2, International Journal of