We provide characterization of these spaces by the Dunkl convolution

CHARACTERIZATION OF BESOV SPACES FOR THE DUNKL OPERATOR ON THE REAL LINE

CHOKRI ABDELKEFI AND MOHAMED SIFI DEPARTMENT OFMATHEMATICS

PREPARATORYINSTITUTE OFENGINEERSTUDIES OFTUNIS

1089 MONFLEURYTUNIS, TUNISIA

chokri.abdelkefi@ipeit.rnu.tn DEPARTMENT OFMATHEMATICS

FACULTY OFSCIENCES OFTUNIS

1060 TUNIS, TUNISIA

mohamed.sifi@fst.rnu.tn

Received 07 February, 2007; accepted 28 August, 2007 Communicated by S.S. Dragomir

ABSTRACT. In this paper, we define subspaces ofL^pby differences using the Dunkl translation operators that we call Besov-Dunkl spaces. We provide characterization of these spaces by the Dunkl convolution.

Key words and phrases: Dunkl operators, Dunkl transform, Dunkl translation operators, Dunkl convolution, Besov-Dunkl spaces.

2000 Mathematics Subject Classification. 46E30, 44A15, 44A35.

1. INTRODUCTION

On the real line, Dunkl operators are differential-difference operators introduced in 1989, by C. Dunkl in [5] and are denoted byΛ_α,whereαis a real parameter>−¹₂. These operators, are associated with the reflection groupZ2 onR. The Dunkl kernelE_α is used to define the Dunkl transformF_α which was introduced by C. Dunkl in [6]. Rösler in [13] showed that the Dunkl kernel satisfies a product formula. This allows us to define the Dunkl translationτ_x,x∈R. As a result, we have the Dunkl convolution∗_α.

There are many ways to define Besov spaces (see [4, 12, 16]). This paper deals with Besov- Dunkl spaces (see [1, 2, 8]). Letβ > 0, 1 ≤ p < +∞ and1 ≤ q ≤ +∞, the Besov-Dunkl space denoted byBD_β,α^p,q is the subspace of functionsf ∈L^p(µα)satisfying

Z +∞

0

kτ_x(f) +τ−x(f)−2fk_p,α x^β

q

dx

x <+∞ if q <+∞

The authors thank the referee for his remarks and suggestions.

048-07

and

sup

x∈(0,+∞)

kτx(f) +τ−x(f)−2fkp,α

x^β <+∞ if q = +∞,

whereµ_αis a weighted Lebesgue measure onR(see next section).

Put

A=

φ∈ S∗(R) : Z +∞

0

φ(x)dµα(x) = 0

,

whereS_∗(R)is the space of even Schwartz functions onR. Givenφ ∈ A, we shall denote by C_φ,β,α^p,q the subspace of functionsf ∈L^p(µ_α)satisfying

Z +∞

0

kf∗αφtkp,α

t^β

q

dt

t <+∞ if q <+∞

and

sup

t∈(0,+∞)

kf ∗_αφ_tk_p,α

t^β <+∞ if q= +∞, whereφ_t(x) = _t2(α+1)¹ φ(^x_t), for allt∈(0,+∞)andx∈R.

Our objective will be to prove thatBD^p,q_β,α ⊂ C_φ,β,α^p,q and when1< p < +∞,0 < β <1then BD_β,α^p,q =C_φ,β,α^p,q .

Observe that the Besov-Dunkl spaces are independent of the specific selection ofφinAand for 1 < p < +∞, 0 < β < 1, we have BD_β,α^p,q ⊂ BDe ^p,q_β,α, where BDe _β,α^p,q is the subspace of functionsf ∈L^p(µ_α)satisfying

Z +∞

0

kτ_x(f)−fk_p,α x^β

q

dx

x <+∞ if q <+∞

and

sup

x∈(0,+∞)

kτ_x(f)−fk_p,α

x^β <+∞ if q = +∞, (see Remark 3.7 in Section 3, below).

Analogous results have been obtained for the weighted Besov spaces (see [3]).

The contents of this paper are as follows. In Section 2, we collect some basic definitions and results about harmonic analysis associated with Dunkl operators .In Section 3, we prove the results about inclusion and coincidence between the spacesBD_β,α^p,q andC_φ,β,α^p,q .

In what follows,crepresents a suitable positive constant which is not necessarily the same in each occurence.

2. PRELIMINARIES

On the real line, we consider the first-order differential-difference operator defined by Λ_α(f)(x) = df

dx(x) + 2α+ 1 x

f(x)−f(−x) 2

, f ∈ E(R), α >−1 2,

which is called the Dunkl operator. Forλ ∈ C, the Dunkl kernelE_α(λ .)onRwas introduced by C. Dunkl in [5] and is given by

E_α(λx) =j_α(iλx) + λx

2(α+ 1)j_α+1(iλx), x∈R,

wherej_α is the normalized Bessel function of the first kind of orderα (see [17]). The Dunkl kernelE_α(λ .) is the unique solution on R of the initial problem for the Dunkl operator (see [5]).

Letµ_αbe the weighted Lebesgue measure onRgiven by dµ_α(x) = |x|^2α+1

2^α+1Γ(α+ 1)dx.

For every 1 ≤ p ≤ +∞, we denote byL^p(µ_α) the spaceL^p(R, dµ_α)and we use k · k_p,α as a shorthand fork · k_L^p_(µα).

The Dunkl transformF_αwhich was introduced by C. Dunkl in [6], is defined forf ∈L¹(µ_α) by

F_α(f)(x) = Z

R

E_α(−ixy)f(y)dµ_α(y), x∈R. For allx, y, z∈R, consider

(2.1) Wα(x, y, z) = (Γ(α+ 1)²) 2^α−1√

πΓ(α+ ¹₂)(1−bx,y,z+bz,x,y+bz,y,x)∆α(x, y, z), where

b_x,y,z =







x²+y²−z²

2xy ifx, y ∈R\{0}, z ∈R

0 otherwise

and

∆_α(x, y, z) =







([(|x|+|y|)²−z²][z²−(|x|−|y|)²])^α−12

|xyz|^2α if|z| ∈S_x,y

0 otherwise

where

S_x,y =h

||x| − |y||, |x|+|y|i . The kernelW_α(see [13]) is even and we have

W_α(x, , y, z) =W_α(y, x, z) =W_α(−x, z, y) = W_α(−z, y,−x)

and Z

R

|W_α(x, y, z)|dµ_α(z)≤4.

In the sequel we consider the signed measureγ_x,y, onR, given by

(2.2) dγ_x,y(z) =











W_α(x, y, z)dµ_α(z) ifx, y ∈R\{0}

dδ_x(z) ify= 0

dδ_y(z) ifx= 0.

Forx, y ∈Randf a continuous function onR, the Dunkl translation operatorτ_xis given by τx(f)(y) =

Z

R

f(z)dγx,y(z).

According to [9], forx ∈R, τxis a continuous linear operator fromE(R)into itself and for all f ∈ E(R), we have

τ_x(f)(y) =τ_y(f)(x), τ₀(f)(x) =f(x), forx, y ∈R, whereE(R)denotes the space ofC^∞-functions onR.

According to [14, 15], the operator τ_x can be extended toL^p(µ_α), 1 ≤ p ≤ +∞ and for f ∈L^p(µ_α)we have

(2.3) kτ_x(f)k_p,α ≤4kfk_p,α.

Using the change of variablez = Ψ(x, y, θ) =p

x²+y²−2xycosθ, we have also (2.4) τx(f)(y) = cα

Z π 0

f(Ψ) +f(−Ψ) + x+y

Ψ (f(Ψ)−f(−Ψ))

dνα(θ), wheredν_α(θ) = (1−cosθ) sin^2αθdθandc_α = ₂^√Γ(α+1)_πΓ(α+1

2).

From the generalized Taylor formula with integral remainder (see [11, Theorem 2, p. 349]), we have forf ∈ E(R)andx, y ∈R

(τx(f)−f)(y) = Z |x|

−|x|

sgn(x)

2|x|^2α+1 + sgn(z) 2|z|^2α+1

τz(Λαf)(y)dµα(z).

The Dunkl convolutionf ∗_αg, of two continuous functionsf andgonRwith compact support, is defined by

(f ∗_α g)(x) = Z

R

τ_x(f)(−y)g(y)dµ_α(y), x∈R. The convolution∗α is associative and commutative (see [13]).

We have the following results (see [14]).

i) Assume that p, q, r ∈ [1,+∞[satisfying ¹_p + ¹_q = 1 + ¹_r (the Young condition). Then the map(f, g)→f ∗_α gdefined onC_c(R)×C_c(R), extends to a continuous map from L^p(µ_α)×L^q(µ_α)toL^r(µ_α)and we have

(2.5) kf ∗_α gk_r,α ≤4kfk_p,αkgk_q,α. ii) For allf ∈L¹(µ_α)andg ∈L²(µ_α), we have

(2.6) F_α(f ∗_αg) = F_α(f)F_α(g)

and forf ∈L¹(µα),g ∈L^p(µα)and1≤p < ∞, we get

(2.7) τ_t(f ∗_α g) =τ_t(f) ∗_αg =f ∗_ατ_t(g), t∈R. 3. CHARACTERIZATION OFBESOV-DUNKLSPACES

Letβ >0,1≤p < +∞and1 ≤q ≤+∞. We say that a measurable functionf onRis in the Besov-Dunkl spaceBD^p,q_β,αiff ∈L^p(µ_α)with

Z +∞

0

kτ_x(f) +τ−x(f)−2fk_p,α x^β

q

dx

x <+∞ if q <+∞

and

sup

x∈(0,+∞)

kτx(f) +τ−x(f)−2fkp,α

x^β <+∞ if q = +∞.

Remark 3.1. Note that forf ∈L^p(µ_α)the functionR→R⁺,x7→ kτ_x(f) +τ_−x(f)−2fk_p,α is measurable (see [10, Lemma 1, (ii)]).

Lemma 3.2. Let0< β < 1,1≤p <+∞,1≤q≤+∞andf ∈L^p(µ_α). IfΛ_α(f)∈L^p(µ_α) thenf ∈ BD_β,α^p,q.

Proof. Using the generalized Taylor formula, Minkowski’s inequality for integrals and (2.3), we can write

kτx(f) +τ−x(f)−2fkp,α ≤ kτx(f)−fkp,α+kτ−x(f)−fkp,α

≤ckΛ_α(f)k_p,α Z x

−x

1

2|x|^2α+1 + 1 2|z|^2α+1

dµ_α(z),

hence we obtain forx >0,

kτ_x(f) +τ−x(f)−2fk_p,α ≤c xkΛ_α(f)k_p,α. Then it follows that forA >0

Z +∞

0

kτx(f) +τ−x(f)−2fkp,α

x^β

q

dx x ≤c

Z A 0

xkΛα(f)kp,α

x^β

q

dx x +c

Z +∞

A

kfk_p,α x^β

q

dx

x <+∞.

Here whenq= +∞, we make the usual modification. This completes the proof.

Example 3.1. Let0< β < 1,1≤ p <+∞and1 ≤q ≤ +∞. By Lemma 3.2, we can assert that

(1) S(R), C_c¹(R)⊂ BD^p,q_β,α.

(2) The functionsg, h_n defined on R, by g(x) = e^−|x| andh_n(x) = _cosh^xn_x, n ∈ N are in BD_β,α^p,q.

Now, in order to establish that for all φ ∈ A, BD^p,q_β,α ⊂ C_φ,β,α^p,q and for 1 < p < +∞, 0< β <1,BD_β,α^p,q =C_φ,β,α^p,q , we need to show some useful lemmas.

Lemma 3.3. Letφ ∈ A, 1≤p < +∞andr >0, then there exists a constantc >0such that for allf ∈L^p(µ_α)andt >0, we have

(3.1) kφ_t∗_αfk_p,α ≤c Z +∞

0

min x

t

2(α+1)

, t

x r

kτ_x(f) +τ−x(f)−2fk_p,αdx x . Proof. Lett >0, we have

Z +∞

0

φ_t(x)dµ_α(x) = Z +∞

0

φ(x)dµ_α(x) = 0 and

(φ_t∗_αf)(y) = Z

R

φ_t(x)τ_y(f)(−x)dµ_α(x)

= Z

R

φt(x)τy(f)(x)dµα(x), then we can write fory∈R

2(φ_t∗_αf)(y) = Z

R

φ_t(x) [τ_y(f)(x) +τ_y(f)(−x)−2f(y)]dµ_α(x)

= 2 Z +∞

0

φ_t(x) [τ_x(f)(y) +τ−x(f)(y)−2f(y)]dµ_α(x).

Using Minkowski’s inequality for integrals, we obtain kφ_t∗_αfk_p,α ≤

Z +∞

0

|φ_t(x)| kτ_x(f) +τ−x(f)−2fk_p,αdµ_α(x)

≤c Z +∞

0

x t

2(α+1) φx

t

kτ_x(f) +τ−x(f)−2fk_p,αdx (3.2) x

≤c Z +∞

0

x t

2(α+1)

kτx(f) +τ−x(f)−2fkp,α

dx x . (3.3)

On the other hand, since the functionφbelongs toS∗(R), then forr >0there exists a constant csuch that

x t

2(α+1)+r φx

t

≤c.

By (3.2), we obtain

(3.4) kφ_t∗_αfk_p,α ≤c Z +∞

0

t x

r

kτ_x(f) +τ−x(f)−2fk_p,αdx x .

From (3.3) and (3.4), we deduce (3.1).

Lemma 3.4. Letφ ∈ Aand1 < p < +∞, then there exists a constantc > 0such that for all f ∈L^p(µ_α)andx >0, we have

(3.5) kτ_x(f) +τ_−x(f)−2fk_p,α ≤c Z +∞

0

minn 1,x

t

okφ_t∗_αfk_p,αdt t . Proof. Put for0< ε < δ <+∞

f_ε,δ(y) = Z δ

ε

(φ_t∗_αφ_t∗_αf)(y)dt

t , y∈R. By interchanging the orders of integration and (2.7), we obtain

τx(fε,δ)(y) = Z δ

ε

τx(φt∗αφt∗αf)(y)dt t

= Z δ

ε

(τ_x(φ_t)∗_αφ_t∗_αf)(y)dt

t , y ∈R, x∈(0,+∞), so we can write forx∈(0,+∞)andy∈R,

(τ_x(f_ε,δ) +τ−x(f_ε,δ)−2f_ε,δ)(y) = Z δ

ε

[(τ_x(φ_t) +τ−x(φ_t)−2φ_t)∗_αφ_t∗_αf](y)dt t . Using Minkowski’s inequality for integrals and (2.5), we get

k(τ_x(f_ε,δ) +τ−x(f_ε,δ)−2f_ε,δk_p,α ≤ Z δ

ε

k(τ_x(φ_t) +τ−x(φ_t)−2φ_t)∗_αφ_t∗_αfk_p,αdt (3.6) t

≤c Z δ

ε

kτ_x(φ_t) +τ−x(φ_t)−2φ_tk_1,αkφ_t∗_αfk_p,αdt t . Forx, t∈(0,+∞), we have

kτx(φt) +τ−x(φt)−2φtk1,α

= Z

R

Z

R

φ_t(z)(dγ_x,y(z) +dγ−x,y(z))

−2φ_t(y)

dµ_α(y)

= Z

R

Z

R

φz t

(dγ_x,y(z) +dγ−x,y(z))

−2φy t

t^−2(α+1)dµ_α(y).

By (2.1) and the change of variablez⁰ = ^z_t , we have W_α(x, y, z⁰t)t^2(α+1)=W_αx

t,y t, z⁰

, then from (2.2), we get

dγ_x,y(z) =dγ^x

t,^y_t(z⁰) and dγ_−x,y(z) = dγ^−x

t ,^y_t(z⁰),

hence

kτ_x(φ_t) +τ_−x(φ_t)−2φ_tk_1,α (3.7)

= Z

R

Z

R

φ(z⁰) dγ^x

t,^y_t(z⁰) +dγ^−x

t ,^y_t(z⁰)

−2φy t

t^−2(α+1)dµ_α(y)

= Z

R

h

τ^x

t(φ)y t

+τ^−x

t (φ)y t

i

t^−2(α+1)−2φ_t(y)

dµ_α(y)

=

τ^x

t(φ) +τ^−x

t (φ)−2φ

t

1,α

= τ^x

t(φ) +τ^−x

t (φ)−2φ 1,α.

Sinceφ ∈ S∗(R), then using (2.4) and [7, Theorem 2.1] (see also [11, Theorem 2, p. 349]), we can assert that

τ^x

t(φ) +τ^−x

t (φ)−2φ

_1,α≤cx

tkφ⁰k_1,α ≤cx t. On the other hand, by (2.3) we have

τ^x

t(φ) +τ^−x

t (φ)−2φ 1,α

≤10kφk_1,α ≤c, then we get,

(3.8)

τ^x

t(φ) +τ^−x

t (φ)−2φ 1,α

≤cminn 1,x

t o

.

From (3.6), (3.7) and (3.8), we obtain

(3.9) kτ_x(f_ε,δ) +τ−x(f_ε,δ)−2f_ε,δk_p,α ≤c Z δ

ε

minn 1,x

t

okφ_t∗_αfk_p,αdt t . Using (2.6), observe that

Z

R

(φ∗_αφ)(x)|x|^2α+1dx= 2^α+1Γ(α+ 1)F_α(φ∗_αφ)(0)

= 2^α+1Γ(α+ 1)(F_α(φ)(0))²

= 2^α+1Γ(α+ 1) Z

R

φ(z)dµ_α(z) 2

= 0, and sinceφ∗_αφis in the Schwarz spaceS(R), we have

Z

R

|log|x|| |φ∗αφ(x)| |x|^2α+1dx <+∞.

Then, by the Calderón reproducing formula related to the Dunkl operators (see [10, Theorem 3]), we have

ε→0, δ→+∞lim fε,δ =c f, inL^p(µα).

From (2.3) and (3.9), we deduce (3.5).

Lemma 3.5. Let0 ≤ ε, r < +∞andr > β >0, then there exists constantsc₁, c₂ > 0such that we have

(3.10)

Z +∞

0

y z

β

min y

z ε

, z

y r

dy

y ≤c₁, z ∈(0,+∞)

and

(3.11)

Z +∞

0

y z

β

min y

z ε

, z

y r

dz

z ≤c2, y∈(0,+∞).

Proof. We can write

Z +∞

0

y z

β

min y

z ε

, z

y r

dy y

=z^−(β+ε) Z z

0

y^β+ε−1dy+z^r−β Z +∞

z

y^β−r−1dy≤c1, z ∈(0,+∞) and

Z +∞

0

y z

β

min y

z ε

, z

y r

dz z

=y^β−r Z y

0

z^−β+r−1dz+y^β+ε Z +∞

y

z^{−β−ε−1}dz ≤c₂, y ∈(0,+∞),

which proves the results.

Theorem 3.6.

(1) Let1≤p < +∞,1≤q≤+∞andβ >0, then we have for allφ∈ A

(3.12) BD_β,α^p,q ⊂ C_φ,β,α^p,q .

(2) Let1< p < +∞,1≤q≤+∞and0< β <1, then we have for allφ ∈ A

(3.13) BD^p,q_β,α =C_φ,β,α^p,q .

Proof. Putω_p^α(f)(x) = kτ_x(f) +τ−x(f)−2fk_p,α forf ∈ L^p(µ_α)andq⁰ = _q−1^q the conjugate ofqwhen1< q <+∞.

• We start with the proof of the inclusion (3.12). Suppose that 1 ≤ p < +∞, 1 ≤ q ≤ +∞,φ∈ A,r > β andf ∈ BD^p,q_β,α.

Case whenq= 1. By (3.1) and Fubini’s theorem, we have Z +∞

0

kf ∗_αφ_tk_p,α t^β

dt t ≤c

Z +∞

0

Z +∞

0

min x

t

2(α+1)

, t

x r

ω_p^α(f)(x)t^−β−1dtdx x

≤c Z +∞

0

ω^α_p(f)(x)

Z +∞

0

min x

t

2(α+1)

, t

x r

t^−β−1dt dx

x

≤c Z +∞

0

ω^α_p(f)(x)

x^−r Z x

0

t^r−β−1dt+x^2(α+1) Z +∞

x

t^{−β−2α−3}dt dx

x

≤c Z +∞

0

ω_p^α(f)(x) x^β

dx

x <+∞, hencef ∈ C_φ,β,α^p,1 .

Case whenq= +∞. By (3.1), we have kφ_t∗_αfk_p,α ≤c

Z t 0

x t

2(α+1)

ω_p^α(f)(x)dx x +

Z +∞

t

t x

r

ω^α_p(f)(x)dx x

≤c sup

x∈(0,+∞)

ω_p^α(f)(x) x^β

t^−2(α+1) Z t

0

x^2α+1+βdx+t^r Z +∞

t

x^{−β−r−1}dx

≤ct^β sup

x∈(0,+∞)

ω^α_p(f)(x) x^β , then we deduce thatf ∈ C_φ,β,α^p,+∞.

Case when1< q <+∞. By (3.1) again, we have fort >0 kφ_t∗_αfk_p,α

t^β ≤c

Z +∞

0

x t

β

min x

t

2(α+1)

, t

x

rω_p^α(f)(x) x^β

dx x . Put

K(x, t) =x t

βmin x

t

2(α+1), t

x

r

.

Using Hölder’s inequality and (3.10), we can write kφ_t∗_αfk_p,α

t^β ≤c

Z +∞

0

(K(x, t))^q10

(K(x, t))^1qω_p^α(f)(x) x^β

dx x

≤c

Z +∞

0

K(x, t)

ω_p^α(f)(x) x^β

^q dx

x ¹q

.

Then by Fubini’s theorem and (3.11), we have Z +∞

0

kφ_t∗_αfk_p,α t^β

q

dt t ≤c

Z +∞

0

ω^α_p(f)(x) x^β

^qZ +∞

0

K(x, t)dt t

dx x

≤c Z +∞

0

ω^α_p(f)(x) x^β

q

dx

x <+∞, which proves the result.

• Let us now prove the equality (3.13). Assumef ∈ C_φ,β,α^p,q , φ ∈ A and0 < β < 1. For 1< p <+∞and1≤q ≤+∞, we have to show only thatf ∈ BD^p,q_β,α.

Case whenq= 1. By (3.5) and Fubini’s theorem, we have Z +∞

0

ω_p^α(f)(x) x^β

dx x ≤c

Z +∞

0

Z +∞

0

minn 1,x

t

okφ_t∗_αfk_p,αx^−β−1dt t dx

≤c Z +∞

0

kφ_t∗_αfk_p,α

Z +∞

0

minn 1,x

t o

x^−β−1dx dt

t

≤c Z +∞

0

kφ_t∗_αfk_p,α 1

t Z t

0

x^−βdx+ Z +∞

t

x^−β−1dx dt

t

≤c Z +∞

0

kφt∗αfkp,α

t^β

dt

t <+∞, then we obtain the result.

Case whenq= +∞. By (3.5), we get ω_p^α(f)(x)≤c

Z x 0

kφ_t∗_αfk_p,αdt t +

Z +∞

x

x

tkφ_t∗_αfk_p,αdt t

≤c sup

t∈(0,+∞)

kφ_t∗_αfk_p,α t^β

Z x 0

t^β−1dt+x Z +∞

x

t^β−2dt

≤cx^β sup

t∈(0,+∞)

kφ_t∗_αfk_p,α t^β , so, we deduce thatf ∈ BD^p,+∞_β,α .

Case when1< q <+∞. By (3.5) again, we have forx >0 ω_p^α(f)(x)

x^β ≤c Z +∞

0

t x

β

minn 1,x

t

okφ_t∗_αfk_p,α t^β

dt t . Put

R(x, t) = t

x

βminn 1,x

t o

. Using Hölder’s inequality and (3.10), we can write

ω_p^α(f)(x) x^β ≤c

Z +∞

0

(R(x, t))^q10

(R(x, t))^1qkφ_t∗_αfk_p,α t^β

dt t

≤c

Z +∞

0

R(x, t)

kφ_t∗_αfk_p,α t^β

q

dt t

¹_q , then by Fubini’s theorem and (3.11), we have

Z +∞

0

ω^α_p(f)(x) x^β

^q dx

x ≤c Z +∞

0

kφt∗αfkp,α

t^β

qZ +∞

0

R(x, t)dx x

dt t

≤c Z +∞

0

kφ_t∗_αfk_p,α t^β

q

dt

t <+∞,

thus the result is established.

Remark 3.7. By proceeding in the same manner as in Lemma 3.4 and (2) of Theorem 3.6, we can assert that for1< p < +∞and0< β < 1, we haveC_φ,β,α^p,q ⊂BDe ^p,q_β,α , hence from (3.13) we conclude thatBD_β,α^p,q ⊂BDe _β,α^p,q.

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