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volume 6, issue 3, article 84, 2005.

Received 11 October, 2004;

accepted 22 July, 2005.

Communicated by:S.S. Dragomir

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Journal of Inequalities in Pure and Applied Mathematics

LITTLEWOOD-PALEY g-FUNCTION IN THE DUNKL ANALYSIS ON R^d

FETHI SOLTANI

Department of Mathematics Faculty of Sciences of Tunis

University of EL-Manar Tunis 2092 Tunis, Tunisia.

EMail:Fethi.Soltani@fst.rnu.tn

c

2000Victoria University ISSN (electronic): 1443-5756 183-04

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Littlewood-Paleyg-Function in the Dunkl Analysis onR^d

Fethi Soltani

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J. Ineq. Pure and Appl. Math. 6(3) Art. 84, 2005

http://jipam.vu.edu.au

Abstract

We proveL^p-inequality for the Littlewood-Paleyg-function in the Dunkl case on R^d.

2000 Mathematics Subject Classification:42B15, 42B25.

Key words: Dunkl operators, Generalized Poisson integral,g-function.

The author is very grateful to the referee for many comments on this paper.

1. Introduction

In the Euclidean case, the Littlewood-Paleyg-function is given by

g(f)(x) :=

"

Z ∞ 0

∂

∂tu(x, t)

2

+|∇xu(x, t)|²

! t dt

#¹₂

, x∈R^d,

whereuis the Poisson integral off and∇is the usual gradient. TheL^p-norm of this operator is comparable with theL^p-norm off forp ∈]1,∞[(see [19]).

Next, this operator plays an important role in questions related to multipliers, Sobolev spaces and Hardy spaces (see [19]).

Over the past twenty years considerable effort has been made to extend the Littlewood-Paley g-function on generalized hypergroups [20, 1, 2], and com- plete Riemannian manifolds [4].

In this paper we consider the differential-difference operatorsTj;j = 1, . . . , d, onR^dintroduced by Dunkl in [5] and aptly called Dunkl operators in the litera- ture. These operators extend the usual partial derivatives by additional reflection terms and give generalizations of many multi-variable analytic structures like the exponential function, the Fourier transform, the convolution product and the Poisson integral (see [12,23,16] and [13]).

During the last years, these operators have gained considerable interest in various fields of mathematics and in certain parts of quantum mechanics; one expects that the results in this paper will be useful when discussing the boundedness property of the Littlewood-Paleyg-function in the Dunkl analysis onR^d. Moreover they are naturally connected with certain Schrödinger operators for Calogero-Sutherland-type quantum many body systems [3,9].

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The main purpose of this paper is to give theL^p-inequality for the Littlewood- Paley g-function in the Dunkl case onR^dby using continuity properties of the Dunkl transformF_k, the Dunkl translation operators of radial functions and the generalized convolution product∗_k. We will adapt to this case techniques Stein used in [18,19].

The paper is organized as follows. In Section 2 we recall some basic harmonic analysis results related to the Dunkl operators on R^d. In particular, we list some basic properties of the Dunkl transform F_k and the generalized convolution product∗_k(see [8,23,15]).

In Section3we study the Littlewood-Paleyg-function:

g(f)(x) :=

"

Z ∞ 0

∂

∂tuk(x, t)

2

+|∇xuk(x, t)|²

! t dt

#¹₂

, x∈R^d,

whereu_k(·, t)is the generalized Poisson integral off. We prove thatgisL^p-boundedness forp∈]1,2].

Throughout the paper c denotes a positive constant whose value may vary from line to line.

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2. The Dunkl Analysis on R

^d

We considerR^dwith the Euclidean inner producth·,·iand normkxk=p hx, xi.

Forα∈R^d\{0}, letσ_αbe the reflection in the hyperplaneH_α ⊂R^dorthog- onal toα:

σ_αx:=x−

2hα, xi kαk²

α.

A finite setR ⊂R^d\{0}is called a root system, ifR∩R, α={−α, α}and σαR = Rfor all α ∈ R. We assume that it is normalized bykαk² = 2for all α ∈R.

For a root systemR, the reflectionsσ_α,α ∈ Rgenerate a finite group G⊂ O(d), the reflection group associated withR. All reflections inG, correspond to suitable pairs of roots. For a givenβ ∈H :=R^dS

α∈RH_α, we fix the positive subsystem:

R₊:={α∈R /hα, βi>0}.

Then for eachα ∈Reitherα∈R₊or−α∈R₊.

Let k : R → C be a multiplicity function on R (i.e. a function which is constant on the orbits under the action of G). For brevity, we introduce the index:

γ =γ(k) := X

α∈R+

k(α).

Moreover, letw_kdenote the weight function:

w_k(x) := Y

α∈R₊

|hα, xi|^2k(α), x∈R^d,

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which isG-invariant and homogeneous of degree2γ.

We introduce the Mehta-type constantc_k, by (2.1) ck :=

Z

R^d

e^−kxk²dµk(x) −1

, where dµk(x) := wk(x)dx.

The Dunkl operatorsT_j; j = 1, . . . , d, on R^d associated with the finite reflection groupGand multiplicity functionkare given for a functionf of class C¹ onR^d, by

T_jf(x) := ∂

∂x_jf(x) + X

α∈R+

k(α)α_jf(x)−f(σ_αx) hα, xi .

The generalized Laplacian∆_kassociated withGandk, is defined by∆_k :=

Pd

j=1T_j². It is given explicitly by

(2.2) ∆_kf(x) :=L_kf(x)−2 X

α∈R₊

k(α)f(x)−f(σ_αx) hα, xi² , with the singular elliptic operator:

(2.3) Lkf(x) := ∆f(x) + 2 X

α∈R+

k(α)h∇f(x), αi hα, xi , where∆denotes the usual Laplacian.

The operatorL_k can also be written in divergence form:

(2.4) L_kf(x) = 1

w_k(x)

d

X

i=1

∂

∂x_i

w_k(x) ∂

∂x_i

.

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This is a canonical multi-variable generalization of the Sturm-Liouville operator for the classical spherical Bessel function [1,2,20].

Fory∈R^d, the initial value problemT_ju(x,·)(y) = x_ju(x, y);j = 1, . . . , d, withu(0, y) = 1admits a unique analytic solution onR^d, which will be denoted byE_k(x, y)and called a Dunkl kernel [6,14,16,23].

This kernel has the Bochner-type representation (see [12]):

(2.5) E_k(x, z) = Z

R^d

e^hy,zidΓ_x(y); x∈R^d, z ∈C^d, wherehy, zi := Pd

i=1yizi andΓx is a probability measure onR^d with support in the closed ballB_d(o,kxk)of centeroand radiuskxk.

Example 2.1 (see [23, p. 21]). IfG=Z2, the Dunkl kernel is given by Eγ(x, z) = Γ γ+¹₂

√πΓ(γ) ·sgn(x)

|x|^2γ Z |x|

−|x|

e^yz(x²−y²)^γ−1(x+y)dy.

Notation. We denote byD(R^d)the space ofC^∞−functions onR^dwith compact support.

The Dunkl kernel gives an integral transform, called the Dunkl transform on R^d, which was studied by de Jeu in [8]. The Dunkl transform of a functionfin D(R^d)is given by

F_k(f)(x) :=

Z

R^d

E_k(−ix, y)f(y)dµ_k(y), x∈R^d.

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Note thatF₀ agrees with the Fourier transformF onR^d: F(f)(x) :=

Z

R^d

e^−ihx,yif(y)dy, x∈R^d.

The Dunkl transform of a functionf ∈ D(R^d)which is radial is again radial, and could be computed via the associated Fourier-Bessel transform F_γ+d/2−1^B [11, p. 586] that is:

F_k(f)(x) = 2^γ+d/2c⁻¹_k F_γ+d/2−1^B (F)(kxk), wheref(x) = F(kxk), and

F_γ+d/2−1^B (F)(kxk) :=

Z ∞ 0

F(r) j_γ+d/2−1(kxkr)

2^γ+d/2−1Γ γ+^d₂r^2γ+d−1dr.

Herej_γis the spherical Bessel function [24].

Notations. We denote byL^p_k(R^d),p∈[1,∞], the space of measurable functions f onR^d, such that

kfk_L^p

k :=

Z

R^d

|f(x)|^pdµ_k(x) ¹_p

<∞, p∈[1,∞[, kfk_L^∞

k := esssup

x∈R^d

|f(x)|<∞, whereµkis the measure given by (2.1).

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Theorem 2.1 (see [7]).

i) Plancherel theorem: the normalized Dunkl transform2^−γ−d/2c_kF_k is an isometric automorphism onL²_k(R^d). In particular,

kfk_L²

k = 2^−γ−d/2c_kkF_k(f)k_L²

k.

ii) Inversion formula: let f be a function in L¹_k(R^d), such that F_k(f) ∈ L¹_k(R^d). Then

F_k⁻¹(f)(x) = 2^−2γ−dc²_kFk(f)(−x), a.e.x∈R^d.

In [6], Dunkl defines the intertwining operatorV_k on P := C[R^d] (theC- algebra of polynomial functions onR^d), by

V_k(p)(x) :=

Z

R^d

p(y)dΓ_x(y), x∈R^d, whereΓ_xis the representing measure onR^dgiven by (2.5).

Next, Rösler proved the positivity properties of this operator (see [12]).

Notation. We denote by E(R^d)and byE⁰(R^d)the spaces ofC^∞−functions on R^dand of distributions onR^dwith compact support respectively.

In [22, Theorem 6.3], Trimèche has proved the following results:

Proposition 2.2.

i) The operatorVkcan be extended to a topological automorphism onE(R^d).

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ii) For all x ∈ R^d, there exists a unique distribution η_k,x in E⁰(R^d) with supp(ηk,x)⊂ {y∈R^d /kyk ≤ kxk}, such that

(V_k)⁻¹(f)(x) = hη_k,x, fi, f ∈ E(R^d).

Next in [23], the author defines:

• The Dunkl translation operatorsτ_x,x∈R^d, onE(R^d), by τ_xf(y) := (V_k)_x⊗(V_k)_y[(V_k)⁻¹(f)(x+y)], y∈R^d. These operators satisfy forx, y andz ∈R^dthe following properties:

(2.6) τ0f =f, τxf(y) = τyf(x), E_k(x, z)E_k(y, z) =τ_x(E_k(·, z))(x), and

(2.7) F_k(τ_xf)(y) = E_k(ix, y)F_k(f)(y), f ∈ D(R^d).

Thus by (2.7), the Dunkl translation operators can be extended onL²_k(R^d), and forx∈R^dwe have

kτ_xfk_L²

k ≤ kfk_L²

k, f ∈L²_k(R^d).

• The generalized convolution product∗_kof two functionsfandginL²_k(R^d), by

f ∗_kg(x) :=

Z

R^d

τ_xf(−y)g(y)dµ_k(y), x∈R^d.

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Note that∗₀ agrees with the standard convolution∗onR^d: f∗g(x) :=

Z

R^d

f(x−y)g(y)dy, x∈R^d. The generalized convolution∗_k satisfies the following properties:

i) Letf, g ∈ D(R^d). Then

F_k(f ∗_kg) =F_k(f)F_k(g).

ii) Letf, g ∈L²_k(R^d). Thenf∗_kgbelongs toL²_k(R^d)if and only ifF_k(f)F_k(g) belongs toL²_k(R^d)and we have

F_k(f ∗_kg) = F_k(f)F_k(g), in theL²_k−case.

Proof. The assertion i) is shown in [23, Theorem 7.2]. We can prove ii) in the same manner demonstrated in [21, p. 101–103].

Theorem 2.4. Let p, q, r ∈ [1,∞]satisfy the Young’s condition: 1/p+ 1/q = 1 + 1/r. Assume thatf ∈ L^p_k(R^d)andg ∈ L^q_k(R^d). Ifkτ_xfk_L^q

k ≤ ckfk_L^q

k for allx∈R^d, then

kf ∗_kgk_L^r

k ≤ckfk_L^p

kkgk_L^q

k.

Proof. The assumption thatτ_xis a bounded operator onL^p_k(R^d)ensures that the usual proof of Young’s inequality (see [25, p. 37]) works.

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i) Iff(x) =F(kxk)inE(R^d), then we have τ_xf(y) =

Z

Ax,y

F p

kxk²+kyk²+ 2hy, ξi

dΓ_x(ξ); x, y ∈R^d, where

A_x,y =

ξ∈R^d/ min

g∈Gkx+gyk ≤ kξk ≤max

g∈G kx+gyk

,

andΓ_xthe representing measure given by(2.5).

ii) For allx∈R^dand forf ∈L^p_k(R^d), radial,p∈[1,∞], kτ_xfk_L^p

k ≤ kfk_L^p

k.

iii) Let p, q, r ∈ [1,∞]satisfy the Young’s condition: 1/p+ 1/q = 1 + 1/r.

Assume thatf ∈L^p_k(R^d), radial, andg ∈L^q_k(R^d), then kf ∗_kgk_L^r

k ≤ kfk_L^p

kkgk_L^q

k.

Proof. The assertion i) is shown by Rösler in [13, Theorem 5.1].

ii) Sincef is a radial function, the explicit formula ofτ_xf shows that

|τ_xf(y)| ≤τ_x(|f|)(y).

Hence, it follows readily from(2.6)that kτ_xfk_L¹

k ≤ kfk_L¹

k.

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By duality the same inequality holds forp=∞.

Thus by interpolation we obtain the result forp∈]1,∞[.

iii) follows directly from Theorem2.4.

Notation. For allx, y, z ∈R, we put

Wγ(x, y, z) := [1−σx,y,z+σz,x,y+σz,y,x]Bγ(|x|,|y|,|z|), where

σ_x,y,z :=







x²+y²−z²

2xy , ifx, y ∈R\{0}

0, otherwise

andBγ is the Bessel kernel given by Bγ(|x|,|y|,|z|)

:=







d_γ[((|x|+|y|)²−z²) (z²−(|x| − |y|)²)]^γ−1

|xyz|^2γ−1 , if|z| ∈A_x,y

0, otherwise,

d_γ = 2^−2γ+1Γ γ+¹₂

√πΓ(γ) , A_x,y =h

|x| − |y|

,|x|+|y|i . Remark 1 (see [10]). The signed kernelWγis even and satisfies:

W_γ(x, y, z) =W_γ(y, x, z) =W_γ(−x, z, y), Wγ(x, y, z) = Wγ(−z, y,−x) =Wγ(−x,−y,−z),

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and Z

R

|W_γ(x, y, z)|dz ≤4.

We consider the signed measuresν_x,y(see [10]) defined by

dν_x,y(z) :=











W_γ(x, y, z)|z|^2γdz, ifx, y ∈R\{0}

dδ_x(z), ify= 0 dδy(z), ifx= 0.

The measuresν_x,yhave the following properties:

supp(ν_x,y) = A_x,y∪(−A_x,y), kν_x,yk:=

Z

R

d|ν_x,y| ≤4.

Proposition 2.6 (see [10,15]). Ifd= 1andG=Z², then

i) For allx, y ∈Rand forf a continuous function onR, we have τ_xf(y) =

Z

Ax,y

f(ξ)dν_x,y(ξ) + Z

(−Ax,y)

f(ξ)dν_x,y(ξ).

ii) For allx∈Rand forf ∈L^p_γ(R),p∈[1,∞], kτxfk_L^p_γ ≤4kfk_L^p_γ.

iii) Assume that p, q, r ∈ [1,∞] satisfy the Young’s condition: 1/p+ 1/q = 1 + 1/r. Then the map(f, g)→f ∗_γgextends to a continuous map from L^p_γ(R)×L^q_γ(R)toL^r_γ(R)and we have

kf ∗_γgk_L^r_γ ≤4kfk_L^p_γkgk_L^q_γ.

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3. The Littlewood-Paley g-Function

By analogy with the case of Euclidean space [19, p. 61] we define, fort > 0, the functionsW_tandP_tonR^d, by

W_t(x) := 2^−2γ−dc²_k Z

R^d

e^−tkξk²E_k(ix, ξ)dµ_k(ξ), x∈R^d, and

P_t(x) := 2^−2γ−dc²_k Z

R^d

e^−tkξkE_k(ix, ξ)dµ_k(ξ), x∈R^d.

The functionW_t, may be called the generalized heat kernel and the functionP_t, the generalized Poisson kernel respectively.

From [23, p. 37] we have W_t(x) = c_k

(4t)^γ+d/2e^−kxk²^/4t, x∈R^d. Writing

(3.1) Pt(x) = 1

√π Z ∞

0

e^−s

√sW_t²_/4s(x)ds, x∈R^d, we obtain

(3.2) P_t(x) = a_kt

(t²+kxk²)^γ+(d+1)/2, a_k := c_kΓ γ+^d+1₂

√π .

However, fort >0and for allf ∈L^p_k(R^d),p∈[1,∞], we put:

uk(x, t) := Pt∗kf(x), x∈R^d.

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The functionu_kis called the generalized Poisson integral off, which was studied by Rösler in [11,13].

Let us consider the Littlewood-Paley g-function (in the Dunkl case). This auxiliary operator is defined initially forf ∈ D(R^d), by

g(f)(x) :=

"

Z ∞ 0

∂

∂tu_k(x, t)

2

+|∇_xu_k(x, t)|²

! t dt

#¹₂

, x∈R^d,

whereukis the generalized Poisson integral.

The main result of the paper is:

Theorem 3.1. For p ∈]1,2], there exists a constant A_p > 0 such that, for f ∈L^p_k(R^d),

kg(f)k_L^p

k ≤A_pkfk_L^p.

For the proof of this theorem we need the following lemmas:

Lemma 3.2. Letf ∈ D(R^d)be a positive function.

i) u_k(x, t)≥0and

∂^Nuk

∂t^N (x, t)

≤ _t2γ+d+N^c ; k∈Nandx∈R^d. ii) Forkxklarge we have

u_k(x, t)≤ c

(t²+kxk²)^γ+d/2 and

∂u_k

∂x_i(x, t)

≤ c

(t²+kxk²)^γ+(d+1)/2. Proof. i) If the generalized Poisson kernelP_tis a positive radial function, then from Proposition2.5i) we obtainuk(x, t)≥0.

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On the other hand from Proposition2.5iii) we have

∂^Nu_k

∂t^N (x, t)

≤ kfk_L¹

k

∂^NP_t

∂t^N L^∞_k

.

Then we obtain the result from the fact that

∂^NP_t

∂t^N L^∞_k

≤ c

t^2γ+d+N. ii) From Proposition2.5i) we can write

τ_xP_t(−y) =a_k Z

R^d

t dΓ_x(ξ)

[t²+kxk²+kyk²−2hy, ξi]^γ+(d+1)/2; x, y ∈R, wherea_kis the constant given by (3.2).

Sincef ∈ D(R^d), there existsa >0, such that supp(f)⊂B_d(o, a). Then u_k(x, t) = a_k

Z

B_d(o,a)

Z

Ax,y

t f(y)dΓ_x(ξ)dµ_k(y)

[t²+kxk²+kyk²−2hy, ξi]^γ+(d+1)/2. It is easily verified forkxklarge andy∈B_d(o, a)that

1

[t²+kxk²+kyk²−2hy, ξi]^γ+(d+1)/2 ≤ c

(t² +kxk²)^γ+(d+1)/2. Therefore and using the fact thatt ≤(t²+kxk²)^1/2, we obtain

u_k(x, t)≤ c t

(t²+kxk²)^γ+(d+1)/2 ≤ c

(t²+kxk²)^γ+d/2.

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Thus the first inequality is proven.

From (2.6) we can write u_k(x, t) =a_k

Z

Bd(o,a)

Z

Ax,y

t f(−y)dΓ_y(ξ)dµ_k(y)

[t²+kxk²+kyk²+ 2hx, ξi]^γ+(d+1)/2. By derivation under the integral sign we obtain

∂u_k

∂x_i(x, t) =a_k Z

Bd(o,a)

Z

Ax,y

−t(2x_i+ξ_i)f(−y)dΓ_y(ξ)dµ_k(y) [t² +kxk²+kyk²+ 2hx, ξi]^γ+(d+3)/2. But forkxklarge andy∈B_d(o, a)we have

t|2x_i+ξ_i|

[t²+kxk²+kyk²+ 2hx, ξi]^γ+(d+3)/2 ≤ t(2|x_i|+|ξ_i|) (t²+kxk²)^γ+(d+3)/2.

Using the fact that t(2|x_i|+|ξ_i|)≤(1 +|ξ_i|)(t²+kxk²) when kxk large, we obtain

∂uk

∂x_i(x, t)

≤ c

(t²+kxk²)^γ+(d+1)/2, which proves the second inequality.

Lemma 3.3. Letf ∈ D(R^d)be a positive function andp∈]1,∞[.

i) lim

N→∞

R

B_d(o,N)

RN 0

∂²u^p_k

∂t² (x, t)tdtdµ_k(x) =R

R^df^p(x)dµ_k(x).

ii) lim

N→∞

RN 0

R

Bd(o,N)L_ku^p_k(·, t)(x)dµ_k(x)tdt= 0,

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whereL_kis the singular elliptic operator given by(2.4).

Proof. i) Integrating by parts, we obtain Z

Bd(o,N)

Z N 0

∂²u^p_k

∂t² (x, t)tdtdµ_k(x)

= Z

Bd(o,N)

f^p(x)dµk(x)− Z

Bd(o,N)

u^p_k(x, N)dµk(x) +pN

Z

Bd(o,N)

u^p−1_k (x, N)∂u_k

∂t (x, N)dµ_k(x).

From Lemma3.2i), we easily get Z

Bd(o,N)

u^p_k(x, N)dµ_k(x)≤c N−(p−1)(2γ+d)

, and

N Z

Bd(o,N)

u^p−1_k (x, N)∂u_k

∂t (x, N)dµk(x)≤c N−(p−1)(2γ+d)

, which gives i).

ii) We have Z N

0

Z

Bd(o,N)

L_ku^p_k(·, t)(x)dµ_k(x)tdt=

d

X

i=1

I_i,N,

where I_i,N =

Z N 0

Z

Bd(o,N)

∂

∂x_i

w_k(x)∂u^p_k

∂x_i(x, t)

dxtdt, i= 1, . . . , d.

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Let us studyI_1,N: I_1,N =p

Z N 0

Z

Bd−1(o,N)

w_k(x^(N⁾)

u^p−1_k (x^(N), t)∂u_k

∂x₁(x^(N), t)

− u^p−1_k (−x^(N), t)∂u_k

∂x₁(−x^(N), t)

dx₂. . . dx_dtdt,

wherex^(N) = q

N²−Pd

i=2x²_i , x₂, . . . , x_d

.

Then, by using Lemma3.2ii) and the fact thatwk(x^(N))≤2^γN^2γ we obtain forN large,

I_1,N ≤c N^2γ Z N

0

Z

Bd−1(o,N)

dx₂. . . dx_dtdt (t²+N²)(γ+d/2)p+1/2

≤c N−p(2γ+d)+2γ−1Z N 0

Z

Bd−1(o,N)

dx₂. . . dx_dtdt

≤c N−(p−1)(2γ+d)−(d−1)/2

.

The same result holds forIi,N,i= 2, . . . , d, which proves ii).

Lemma 3.4. Letf ∈ D(R^d)be a positive function. Define the maximal function M_k(f), by

(3.3) M_k(f)(x) := sup

t>0

(u_k(x, t)), x∈R^d.

Then forp∈]1,∞[, there exists a constantC_p >0such that, forf ∈L^p_k(R^d), kM_k(f)k_L^p

k ≤C_pkfk_L^p

k,

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moreover the operatorM_kis of weak type(1,1).

Proof. From (3.1) it follows that u_k(x, t) = t

8√ π

Z ∞ 0

W_s∗_kf(x)e^−t²^/4ss^−3/2ds,

which implies, as in [18, p. 49] that Mk(f)(x)≤csup

y>0

1 y

Z y 0

Qsf(x)ds

, x∈R^d,

where Q_sf(x) = W_s ∗_k f(x), which is a semigroup of operators on L^p_k(R^d).

Hence using the Hopf-Dunford-Schwartz ergodic theorem as in [18, p. 48], we get the boundedness ofMkonL^p_k(R^d)forp∈]1,∞]and weak type(1,1).

Proof of Theorem3.1. Letf ∈ D(R^d)be a positive function. From Lemma3.2 i) the generalized Poisson integralu_koff is positive.

First step: Estimate of the quantity

_∂t^∂u_k(x, t)

²+|∇_xu_k(x, t)|². LetH_k be the operator:

H_k:=L_k+ ∂²

∂t²,

whereL_kis the singular elliptic operator given by (2.3).

Using the fact that

∆_ku_k(·, t)(x) + ∂²

∂t²u_k(x, t) = 0,

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we obtain forp∈]1,∞[,

H_ku^p_k(x, t) =p(p−1)u^p−2_k (x, t)

"

∂

∂tu_k(x, t)

2

+|∇_xu_k(x, t)|²

#

+p X

α∈R₊

k(α)U_α(x, t) hα, xi² , where

U_α(x, t) := 2u^p−1_k (x, t) [u_k(x, t)−u_k(σ_αx, t)], α∈R₊. LetA, B ≥0, then the inequality

2A^p−1(A−B)≥(A^p−1+B^p−1)(A−B) is equivalent to

(A^p−1−B^p−1)(A−B)≥0, which holds ifA ≥BorA < B. Thus we deduce that

U_α(x, t)≥

u^p−1_k (x, t) +u^p−1_k (σ_αx, t)

[u_k(x, t)−u_k(σ_αx, t)], and therefore we get

(3.4)

∂

∂tu_k(x, t)

2

+|∇_xu_k(x, t)|²

≤ 1

p(p−1)u^2−p_k (x, t) [v_k(x, t) +H_ku^p_k(x, t)],

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where

v_k(x, t) =p X

α∈R+

k(α) hα, xi²

u^p−1_k (σ_αx, t) +u^p−1_k (x, t)

[u_k(σ_αx, t)−u_k(x, t)]. Second step: The inequalitykg(f)k_L^p

k ≤Apkfk_L^p

k, forp∈]1,2[.

From (3.4), we have [g(f)(x)]² ≤ 1

p(p−1) Z ∞

0

u^2−p_k (x, t) [v_k(x, t) +H_ku^p_k(x, t)]tdt

≤ 1

p(p−1)I_k(f)(x) [M_k(f)(x)]^2−p, x∈R^d, where

Ik(f)(x) :=

Z ∞ 0

[vk(x, t) +Hku^p_k(x, t)]tdt, andM_k(f)the maximal function given by (3.3).

Thus it is proven that kg(f)k^p_L^p

k ≤

1 p(p−1)

^p₂ Z

R^d

[Ik(f)(x)]^p/2[Mk(f)(x)]^(2−p)p/2dµk(x).

By applying Hölder’s inequality, we obtain (3.5) kg(f)k^p_Lp

k

≤

1 p(p−1)

^p₂

kI_k(f)k^p/2_L1 k

kM_k(f)k^(2−p)p/2_Lp k

.

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Since v_k(x, t) + H_ku^p_k(x, t) ≥ 0, we can apply Fubini-Tonnelli’s Theorem to obtain

kI_k(f)k_L¹

k = lim

N→∞

Z N 0

Z

Bd(o,N)

[v_k(x, t) +H_ku^p_k(x, t)]dµ_k(x)tdt.

Puttingy = σ_αxand using the fact thatσ²_α = id;hσ_αy, αi =−hy, αi, then as in the argument of [16, p. 390] we obtain

Z

B_d(o,N)

v_k(x, t)dµ_k(x) = − Z

B_d(o,N)

v_k(y, t)dµ_k(y).

Thus Z

Bd(o,N)

v_k(x, t)dµ_k(x) = 0.

Hence from Lemma3.3, we deduce that (3.6) kI_α(f)k_L¹

k = lim

N→∞

Z

Bd(o,N)

Z N 0

H_ku^p_k(x, t)tdtdµ_k(x) = kfk^p_L^p

k

.

On the other hand from Lemma3.4we have

(3.7) kM_k(f)k_L^p

k ≤C_pkfk_L^p

k. Finally, from (3.5), (3.6) and (3.7), we obtain

kg(f)k_L^p

k ≤A_pkfk_L^p

k, A_p =

1 p(p−1)

¹₂

C_p^(2−p)/2.

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Since the operator g is sub-linear, we obtain the inequality for f ∈ D(R^d).

And by an easy limiting argument one shows that the result is also true for any f ∈L^p_k(R^d),p∈]1,2[.

For the casep= 2, using (3.4) and (3.6) we get kg(f)k²_L2

k ≤ 1 2

Z

R^d

Z ∞ 0

vk(x, t) +Hku²_k(x, t)

tdtdµk(x) = 1 2kfk²_L2

k, which completes the proof of the theorem.

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References

[1] A. ACHOUR AND K. TRIMÈCHE, La g-fonction de Littlewood-Paley associée à un opérateur différentiel singulier sur(0,∞), Ann. Inst. Fourier, Grenoble, 33 (1983), 203–226.

[2] H. ANNABI ANDA. FITOUHI, Lag-fonction de Littlewood-Paley asso- ciée à une classe d’opérateurs différentiels sur]0,∞[contenant l’opérateur de Bessel, C.R. Acad. Sc. Paris, 303 (1986), 411–413.

[3] T.H. BAKER AND P.J. FORRESTER, The Calogero-Sutherland model and generalized classical polynomials, Comm. Math. Phys., 188 (1997), 175–216.

[4] T. COULHON, X.T. DUONGANDX.D. LI, Littlewood-Paley-Stein functions on manifolds1≤p≤2, Studia Math., 154 (2003), 37–57.

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[8] M.F.E. de JEU, The Dunkl transform, Inv. Math., 113 (1993), 147–162.

[9] L. LAPOINTE AND L. VINET, Exact operator solution of the Calogero- Sutherland model, Comm. Math. Phys., 178 (1996), 425–452.

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[10] M. RÖSLER, Bessel-type signed hypergroups on R, In : H.Heyer, A.Mukherjea (Eds.), Proc XI, Probability measures on groups and re- lated structures, Oberwolfach, 1994, World Scientific, Singapore, (1995), p. 292–304.

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[19] E.M. STEIN, Singular integrals and differentiability properties of func- tions, Princeton Univ. Press, Princeton, New Jersey, 1971.

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