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

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Characterization of Besov-Dunkl spaces Chokri Abdelkefi and Mohamed Sifi

vol. 8, iss. 3, art. 73, 2007

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

CHOKRI ABDELKEFI MOHAMED SIFI

Department of Mathematics Department of Mathematics

Preparatory Institute of Engineer Studies of Tunis Faculty of Sciences of Tunis

1089 Monfleury Tunis, Tunisia 1060 Tunis, Tunisia

EMail:chokri.abdelkefi@ipeit.rnu.tn EMail:mohamed.sifi@fst.rnu.tn

Received: 07 February, 2007

Accepted: 28 August, 2007

Communicated by: S.S. Dragomir 2000 AMS Sub. Class.: 46E30, 44A15, 44A35.

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

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.

Acknowledgements: The authors thank the referee for his remarks and suggestions.

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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 groupZ2onR. The Dunkl kernel Eα is used to define the Dunkl transformFα 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(µ_α) satisfying

Z +∞

0

kτx(f) +τ−x(f)−2fkp,α

x^β

q

dx

x <+∞ if q <+∞

and

sup

x∈(0,+∞)

kτ_x(f) +τ−x(f)−2fk_p,α

x^β <+∞ if q = +∞,

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

Put

A=

φ∈ S_∗(R) : Z +∞

0

φ(x)dµ_α(x) = 0

,

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

Z +∞

0

kf∗_αφ_tk_p,α t^β

q

dt

t <+∞ if q <+∞

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and

sup

t∈(0,+∞)

kf ∗αφtkp,α

t^β <+∞ if q= +∞, whereφ_t(x) = _t_2(α+1)¹ φ(^x_t), for allt∈(0,+∞)andx∈R.

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

Observe that the Besov-Dunkl spaces are independent of the specific selection of φinAand for1< p < +∞,0< β <1, we haveBD_β,α^p,q ⊂BDe ^p,q_β,α,whereBDe ^p,q_β,α 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 Remark2in Section3, 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 Section3, we prove the results about inclusion and coincidence between the spaces BD^p,q_β,α andC_φ,β,α^p,q .

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

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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,

where j_α is the normalized Bessel function of the first kind of order α (see [17]).

The Dunkl kernelE_α(λ .)is the unique solution onRof the initial problem for the Dunkl operator (see [5]).

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

2^α+1Γ(α+ 1)dx.

For every1 ≤ p ≤ +∞, we denote by L^p(µ_α)the space L^p(R, dµ_α) and we use k · k_p,αas a shorthand fork · k_L^p_(µ_α₎.

The Dunkl transformF_αwhich was introduced by C. Dunkl in [6], is defined for f ∈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) W_α(x, y, z) = (Γ(α+ 1)²) 2^α−1√

πΓ(α+¹₂)(1−b_x,y,z+b_z,x,y+b_z,y,x)∆_α(x, y, z),

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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 operator τ_x is given by

τ_x(f)(y) = Z

R

f(z)dγ_x,y(z).

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According to [9], forx∈R, τ_x is a continuous linear operator fromE(R)into itself and for allf ∈ 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 operatorτ_x can be extended toL^p(µ_α),1≤ p ≤+∞

and forf ∈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 convolution f ∗_α g , of two continuous functions f and g on R with 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]).

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

(2.5) kf ∗α gkr,α ≤4kfkp,αkgkq,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(µ_α),g ∈L^p(µ_α)and1≤p < ∞, we get (2.7) τ_t(f ∗_α g) = τ_t(f) ∗_αg =f ∗_ατ_t(g), t ∈R.

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3. Characterization of Besov-Dunkl Spaces

Letβ >0,1≤p <+∞and1≤q ≤+∞. We say that a measurable functionf on Ris 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,+∞)

kτ_x(f) +τ−x(f)−2fk_p,α

x^β <+∞ if q = +∞.

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

Lemma 3.1. Let0 < β < 1, 1 ≤ p < +∞, 1 ≤ q ≤ +∞and f ∈ L^p(µα). If Λ_α(f)∈L^p(µ_α)thenf ∈ BD^p,q_β,α.

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

kτ_x(f) +τ−x(f)−2fk_p,α ≤ kτ_x(f)−fk_p,α+kτ−x(f)−fk_p,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.

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Then it follows that forA >0 Z +∞

0

kτ_x(f) +τ−x(f)−2fk_p,α x^β

q

dx x

≤c Z A

0

xkΛ_α(f)k_p,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 Lemma3.1, we can assert that

1. S(R), C_c¹(R)⊂ BD_β,α^p,q.

2. The functions g, h_ndefined onR, byg(x) = e^−|x|andh_n(x) = _cosh^xⁿ_x, n ∈ N are inBD_β,α^p,q.

Now, in order to establish that for all φ ∈ A, BD_β,α^p,q ⊂ C_φ,β,α^p,q and for 1 < p <

+∞,0< β <1,BD^p,q_β,α =C_φ,β,α^p,q , we need to show some useful lemmas.

Lemma 3.2. Letφ∈ A,1≤p <+∞andr >0, then there exists a constantc >0 such 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

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and

(φ_t∗_αf)(y) = Z

R

φ_t(x)τ_y(f)(−x)dµ_α(x)

= Z

R

φ_t(x)τ_y(f)(x)dµ_α(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

φt(x) [τx(f)(y) +τ−x(f)(y)−2f(y)]dµα(x).

Using Minkowski’s inequality for integrals, we obtain kφt∗αfkp,α ≤

Z +∞

0

|φt(x)| kτx(f) +τ−x(f)−2fkp,αdµα(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)

kτ_x(f) +τ−x(f)−2fk_p,αdx x . (3.3)

On the other hand, since the functionφbelongs toS∗(R), then forr >0there exists a constantcsuch 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).

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Lemma 3.3. Letφ ∈ Aand1 < p <+∞, then there exists a constantc > 0such that for allf ∈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 τ_x(f_ε,δ)(y) =

Z δ ε

τ_x(φ_t∗_αφ_t∗_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 (3.6)

≤ Z δ

ε

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

≤c Z δ

ε

kτ_x(φ_t) +τ_−x(φ_t)−2φ_tk_1,αkφ_t∗_αfk_p,αdt t .

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Forx, t∈(0,+∞), we have kτ_x(φ_t) +τ−x(φ_t)−2φ_tk_1,α

= Z

R

Z

R

φt(z)(dγx,y(z) +dγ−x,y(z))

−2φt(y)

dµα(y)

= Z

R

Z

R

φ z

t

(dγx,y(z) +dγ−x,y(z))

−2φ y

t

t^−2(α+1)dµα(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)−2φt(y)

dµα(y)

=

τ^x

t(φ) +τ^−x

t (φ)−2φ

t

1,α

= τ^x

t(φ) +τ^−x

t (φ)−2φ 1,α

.

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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

tkφ⁰k1,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)dµα(z) 2

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

Z

R

|log|x|| |φ∗_αφ(x)| |x|^2α+1dx <+∞.

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

ε→0, δ→+∞lim f_ε,δ =c f, inL^p(µ_α).

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

Lemma 3.4. Let0 ≤ ε, r < +∞ andr > β > 0, then there exists constants c₁, 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.

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Theorem 3.5.

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_β,α^p,q =C_φ,β,α^p,q .

Proof. Putω_p^α(f)(x) =kτ_x(f) +τ−x(f)−2fk_p,αforf ∈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∗αφtkp,α

t^β

dt t

≤c Z +∞

0

Z +∞

0

min x

t

2(α+1)

, t

x r

ω^α_p(f)(x)t^−β−1dtdx x

≤c Z +∞

0

ω_p^α(f)(x)

Z +∞

0

min x

t

2(α+1)

, t

x r

t^−β−1dt dx

x

≤c Z +∞

0

ω_p^α(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 .

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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^α_p(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))¹^qω_p^α(f)(x) x^β

dx x

≤c

Z +∞

0

K(x, t)

ω_p^α(f)(x) x^β

^q dx

x ¹_q

. 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

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≤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 ∈ Aand0< β <

1. For1< 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

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

tkφ_t∗_αfk_p,αdt t

≤c sup

t∈(0,+∞)

kφt∗αfkp,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 ,

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so, we deduce thatf ∈ BD_β,α^p,+∞.

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))¹^qkφ_t∗_αfk_p,α t^β

dt t

≤c

Z +∞

0

R(x, t)

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

qZ +∞

0

R(x, t)dx x

dt t

≤c Z +∞

0

q

dt

t <+∞, thus the result is established.

Remark 2. By proceeding in the same manner as in Lemma3.3and (2) of Theorem 3.5, we can assert that for1< p < +∞and0 < β <1, we haveC_φ,β,α^p,q ⊂ BDe _β,α^p,q , hence from (3.13) we conclude thatBD_β,α^p,q ⊂BDe ^p,q_β,α.

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References

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