Volume 8 (2007), Issue 3, Article 73, 11 pp.
CHARACTERIZATION OF BESOV SPACES FOR THE DUNKL OPERATOR ON
THE REAL LINE
CHOKRI ABDELKEFI AND MOHAMED SIFI
D EPARTMENT OF M ATHEMATICS
P REPARATORY I NSTITUTE OF E NGINEER S TUDIES OF T UNIS
1089 M ONFLEURY T UNIS , T UNISIA
chokri.abdelkefi@ipeit.rnu.tn
D EPARTMENT OF M ATHEMATICS
FACULTY OF S CIENCES OF T UNIS
1060 T UNIS , T UNISIA
mohamed.sifi@fst.rnu.tn
Received 07 February, 2007; accepted 28 August, 2007
Communicated by S.S. Dragomir
A BSTRACT. In this paper, we define subspaces of Lp by 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. I NTRODUCTION
On the real line, Dunkl operators are differential-difference operators introduced in 1989, by
C. Dunkl in [5] and are denoted by Λα , where α is a real parameter > − 12 . These operators, are
associated with the reflection group Z2 on R. The Dunkl kernel Eα is used to define the Dunkl
transform Fα 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 translation τx , 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 BesovDunkl spaces (see [1, 2, 8]). Let β > 0, 1 ≤ p < +∞ and 1 ≤ q ≤ +∞, the Besov-Dunkl
p,q
space denoted by BDβ,α
is the subspace of functions f ∈ Lp (µα ) satisfying
q
Z +∞ kτx (f ) + τ−x (f ) − 2f kp,α
dx
< +∞
if q < +∞
β
x
x
0
The authors thank the referee for his remarks and suggestions.
048-07
2
C HOKRI A BDELKEFI AND M OHAMED S IFI
and
kτx (f ) + τ−x (f ) − 2f kp,α
< +∞
if q = +∞,
xβ
x∈(0,+∞)
where µα is a weighted Lebesgue measure on R (see next section).
Put
Z +∞
A = φ ∈ S∗ (R) :
φ(x)dµα (x) = 0 ,
sup
0
where S∗ (R) is the space of even Schwartz functions on R. Given φ ∈ A, we shall denote by
p,q
Cφ,β,α
the subspace of functions f ∈ Lp (µα ) satisfying
q
Z +∞ kf ∗α φt kp,α
dt
< +∞
if q < +∞
β
t
t
0
and
kf ∗α φt kp,α
< +∞
tβ
t∈(0,+∞)
sup
if
q = +∞,
1
where φt (x) = t2(α+1)
φ( xt ), for all t ∈ (0, +∞) and x ∈ R.
p,q
p,q
Our objective will be to prove that BDβ,α
⊂ Cφ,β,α
and when 1 < p < +∞, 0 < β < 1 then
p,q
p,q
BDβ,α
= Cφ,β,α
.
Observe that the Besov-Dunkl spaces are independent of the specific selection of φ in A and
p,q
e p,q , where BD
e p,q is the subspace of
for 1 < p < +∞, 0 < β < 1, we have BDβ,α
⊂ BD
β,α
β,α
functions f ∈ Lp (µα ) satisfying
q
Z +∞ kτx (f ) − f kp,α
dx
< +∞
if q < +∞
xβ
x
0
and
kτx (f ) − f kp,α
< +∞
if q = +∞,
xβ
x∈(0,+∞)
(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
p,q
p,q
results about inclusion and coincidence between the spaces BDβ,α
and Cφ,β,α
.
In what follows, c represents a suitable positive constant which is not necessarily the same in
each occurence.
sup
2. P RELIMINARIES
On the real line, we consider the first-order differential-difference operator defined by
df
2α + 1 f (x) − f (−x)
1
Λα (f )(x) =
(x) +
, f ∈ E(R), α > − ,
dx
x
2
2
which is called the Dunkl operator. For λ ∈ C, the Dunkl kernel Eα (λ .) on R was introduced
by C. Dunkl in [5] and is given by
λx
Eα (λx) = jα (iλx) +
jα+1 (iλx), x ∈ R,
2(α + 1)
where jα is the normalized Bessel function of the first kind of order α (see [17]). The Dunkl
kernel Eα (λ .) is the unique solution on R of the initial problem for the Dunkl operator (see
[5]).
J. Inequal. Pure and Appl. Math., 8(3) (2007), Art. 73, 11 pp.
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C HARACTERIZATION OF B ESOV-D UNKL SPACES
3
Let µα be the weighted Lebesgue measure on R given by
dµα (x) =
|x|2α+1
dx.
2α+1 Γ(α + 1)
For every 1 ≤ p ≤ +∞, we denote by Lp (µα ) the space Lp (R, dµα ) and we use k · kp,α as a
shorthand for k · kLp (µα ) .
The Dunkl transform Fα which was introduced by C. Dunkl in [6], is defined for f ∈ L1 (µα )
by
Z
Fα (f )(x) =
Eα (−ixy)f (y)dµα (y),
x ∈ R.
R
For all x, y, z ∈ R, consider
(2.1)
Wα (x, y, z) =
(Γ(α + 1)2 )
√
(1 − bx,y,z + bz,x,y + bz,y,x )∆α (x, y, z),
2α−1 πΓ(α + 12 )
where
bx,y,z =


x2 +y 2 −z 2
2xy
 0
and
∆α (x, y, z) =
where
if x, y ∈ R\{0}, z ∈ R
otherwise
1


([(|x|+|y|)2 −z 2 ][z 2 −(|x|−|y|)2 ])α− 2
|xyz|2α
if |z| ∈ Sx,y

0
otherwise
h
i
Sx,y = ||x| − |y|| , |x| + |y| .
The kernel Wα (see [13]) is even and we have
Wα (x, , y, z) = Wα (y, x, z) = Wα (−x, z, y) = Wα (−z, y, −x)
and
Z
|Wα (x, y, z)|dµα (z) ≤ 4.
R
In the sequel we consider the signed measure γx,y , on R, given by

Wα (x, y, z)dµα (z) if x, y ∈ R\{0}



dδx (z)
if y = 0
(2.2)
dγx,y (z) =



dδy (z)
if x = 0.
For x, y ∈ R and f a continuous function on R, the Dunkl translation operator τx is given by
Z
τx (f )(y) =
f (z)dγx,y (z).
R
According to [9], for x ∈ R, τx is a continuous linear operator from E(R) into itself and for all
f ∈ E(R), we have
τx (f )(y) = τy (f )(x),
τ0 (f )(x) = f (x) ,
for x, y ∈ R,
where E(R) denotes the space of C ∞ -functions on R.
According to [14, 15], the operator τx can be extended to Lp (µα ), 1 ≤ p ≤ +∞ and for
f ∈ Lp (µα ) we have
(2.3)
kτx (f )kp,α ≤ 4kf kp,α .
J. Inequal. Pure and Appl. Math., 8(3) (2007), Art. 73, 11 pp.
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4
C HOKRI A BDELKEFI AND M OHAMED S IFI
p
Using the change of variable z = Ψ(x, y, θ) = x2 + y 2 − 2xy cos θ, we have also
Z π
x+y
(2.4)
τx (f )(y) = cα
f (Ψ) + f (−Ψ) +
(f (Ψ) − f (−Ψ)) dνα (θ),
Ψ
0
where dνα (θ) = (1 − cos θ) sin2α θdθ and cα = 2√Γ(α+1)
.
πΓ(α+ 12 )
From the generalized Taylor formula with integral remainder (see [11, Theorem 2, p. 349]),
we have for f ∈ E(R) and x, y ∈ R
Z |x| sgn(z)
sgn(x)
(τx (f ) − f )(y) =
+
τz (Λα f )(y) dµα (z).
2|x|2α+1 2|z|2α+1
−|x|
The Dunkl convolution f ∗α g , of two continuous functions f and g on R with compact support,
is defined by
Z
(f ∗α g)(x) =
τx (f )(−y)g(y)dµα (y),
x ∈ R.
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 p1 + 1q = 1 + 1r (the Young condition). Then
the map (f, g) → f ∗α g defined on Cc (R) × Cc (R), extends to a continuous map from
Lp (µα ) × Lq (µα ) to Lr (µα ) and we have
(2.5)
kf ∗α gkr,α ≤ 4kf kp,α kgkq,α .
ii) For all f ∈ L1 (µα ) and g ∈ L2 (µα ), we have
(2.6)
Fα (f ∗α g) = Fα (f )Fα (g)
and for f ∈ L1 (µα ), g ∈ Lp (µα ) and 1 ≤ p < ∞, we get
(2.7)
τt (f ∗α g) = τt (f ) ∗α g = f ∗α τt (g),
t ∈ R.
3. C HARACTERIZATION OF B ESOV-D UNKL S PACES
Let β > 0, 1 ≤ p < +∞ and 1 ≤ q ≤ +∞. We say that a measurable function f on R is in
p,q
the Besov-Dunkl space BDβ,α
if f ∈ Lp (µα ) with
q
Z +∞ kτx (f ) + τ−x (f ) − 2f kp,α
dx
< +∞
if q < +∞
β
x
x
0
and
kτx (f ) + τ−x (f ) − 2f kp,α
sup
< +∞
if q = +∞.
xβ
x∈(0,+∞)
Remark 3.1. Note that for f ∈ Lp (µα ) the function R → R+ , x 7→ kτx (f ) + τ−x (f ) − 2f kp,α
is measurable (see [10, Lemma 1, (ii)]).
Lemma 3.2. Let 0 < β < 1, 1 ≤ p < +∞, 1 ≤ q ≤ +∞ and f ∈ Lp (µα ). If Λα (f ) ∈ Lp (µα )
p,q
then f ∈ BDβ,α
.
Proof. Using the generalized Taylor formula, Minkowski’s inequality for integrals and (2.3),
we can write
kτx (f ) + τ−x (f ) − 2f kp,α ≤ kτx (f ) − f kp,α + kτ−x (f ) − f kp,α
Z x
1
1
≤ c kΛα (f )kp,α
+
dµα (z) ,
2|x|2α+1 2|z|2α+1
−x
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C HARACTERIZATION OF B ESOV-D UNKL SPACES
5
hence we obtain for x > 0,
kτx (f ) + τ−x (f ) − 2f kp,α ≤ c x kΛα (f )kp,α .
Then it follows that for A > 0
q
q
Z +∞ Z A
dx
dx
kτx (f ) + τ−x (f ) − 2f kp,α
xkΛα (f )kp,α
≤c
β
β
x
x
x
x
0
0
q
Z +∞ kf kp,α
dx
+c
< +∞.
β
x
x
A
Here when q = +∞, we make the usual modification. This completes the proof.
Example 3.1. Let 0 < β < 1, 1 ≤ p < +∞ and 1 ≤ q ≤ +∞. By Lemma 3.2, we can assert
that
p,q
(1) S(R), Cc1 (R) ⊂ BDβ,α
.
xn
(2) The functions g, hn defined on R, by g(x) = e−|x| and hn (x) = cosh
, n ∈ N are in
x
p,q
BDβ,α .
p,q
p,q
Now, in order to establish that for all φ ∈ A, BDβ,α
⊂ Cφ,β,α
and for 1 < p < +∞,
p,q
p,q
0 < β < 1, BDβ,α = Cφ,β,α , we need to show some useful lemmas.
Lemma 3.3. Let φ ∈ A, 1 ≤ p < +∞ and r > 0, then there exists a constant c > 0 such that
for all f ∈ Lp (µα ) and t > 0, we have
r Z +∞
t
dx
x 2(α+1)
,
kτx (f ) + τ−x (f ) − 2f kp,α .
(3.1)
kφt ∗α f kp,α ≤ c
min
t
x
x
0
Proof. Let t > 0, we have
Z
+∞
Z
φt (x)dµα (x) =
0
+∞
φ(x)dµα (x) = 0
0
and
Z
(φt ∗α f )(y) =
φt (x)τy (f )(−x)dµα (x)
ZR
=
φt (x)τy (f )(x)dµα (x),
R
then we can write for y ∈ R
Z
2(φt ∗α f )(y) =
φt (x) [τy (f )(x) + τy (f )(−x) − 2f (y)] dµα (x)
R
Z
+∞
φt (x) [τx (f )(y) + τ−x (f )(y) − 2f (y)] dµα (x).
=2
0
Using Minkowski’s inequality for integrals, we obtain
Z +∞
kφt ∗α f kp,α ≤
|φt (x)| kτx (f ) + τ−x (f ) − 2f kp,α dµα (x)
0
Z +∞ 2(α+1) dx
x
x (3.2)
≤c
φ
kτx (f ) + τ−x (f ) − 2f kp,α
t
t
x
0
Z +∞ 2(α+1)
x
dx
(3.3)
≤c
kτx (f ) + τ−x (f ) − 2f kp,α .
t
x
0
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6
C HOKRI A BDELKEFI AND M OHAMED S IFI
On the other hand, since the function φ belongs to S∗ (R), then for r > 0 there exists a constant
c such that
x 2(α+1)+r x φ
≤ c.
t
t
By (3.2), we obtain
Z +∞ r
dx
t
(3.4)
kφt ∗α f kp,α ≤ c
kτx (f ) + τ−x (f ) − 2f kp,α .
x
x
0
From (3.3) and (3.4), we deduce (3.1).
Lemma 3.4. Let φ ∈ A and 1 < p < +∞, then there exists a constant c > 0 such that for all
f ∈ Lp (µα ) and x > 0, we have
Z +∞
n xo
dt
(3.5)
kτx (f ) + τ−x (f ) − 2f kp,α ≤ c
min 1,
kφt ∗α f kp,α .
t
t
0
Proof. Put for 0 < ε < δ < +∞
Z
δ
(φt ∗α φt ∗α f )(y)
fε,δ (y) =
ε
dt
,
t
y ∈ R.
By interchanging the orders of integration and (2.7), we obtain
Z δ
dt
τx (fε,δ )(y) =
τx (φt ∗α φt ∗α f )(y)
t
ε
Z δ
dt
=
(τx (φt ) ∗α φt ∗α f )(y) , y ∈ R, x ∈ (0, +∞),
t
ε
so we can write for x ∈ (0, +∞) and y ∈ R,
Z δ
dt
(τx (fε,δ ) + τ−x (fε,δ ) − 2fε,δ )(y) =
[(τx (φt ) + τ−x (φt ) − 2φt ) ∗α φt ∗α f ](y) .
t
ε
Using Minkowski’s inequality for integrals and (2.5), we get
Z δ
dt
(3.6) k(τx (fε,δ ) + τ−x (fε,δ ) − 2fε,δ kp,α ≤
k(τx (φt ) + τ−x (φt ) − 2φt ) ∗α φt ∗α f kp,α
t
ε
Z δ
dt
≤c
kτx (φt ) + τ−x (φt ) − 2φt k1,α kφt ∗α f kp,α .
t
ε
For x, t ∈ (0, +∞), we have
kτx (φt ) + τ−x (φt ) − 2φt k1,α
Z Z
dµα (y)
=
φ
(z)(dγ
(z)
+
dγ
(z))
−
2φ
(y)
t
x,y
−x,y
t
R
R
Z Z y z
t−2(α+1) dµα (y).
=
φ
(dγx,y (z) + dγ−x,y (z)) − 2φ
t
t R
R
By (2.1) and the change of variable z 0 =
z
t
, we have
Wα (x, y, z 0 t)t2(α+1) = Wα
x y , , z0 ,
t t
then from (2.2), we get
dγx,y (z) = dγ xt , yt (z 0 ) and dγ−x,y (z) = dγ −x , y (z 0 ) ,
t
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t
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C HARACTERIZATION OF B ESOV-D UNKL SPACES
7
hence
(3.7)
kτx (φt ) + τ−x (φt ) − 2φt k1,α
Z Z
y 0
0
0
t−2(α+1) dµα (y)
=
φ(z ) dγ xt , yt (z ) + dγ −x , y (z ) − 2φ
t t
t
ZR h R y i
y
x
+ τ −x (φ)
t−2(α+1) − 2φt (y) dµα (y)
=
τ t (φ)
t
t
t
R
= τ xt (φ) + τ −x (φ) − 2φ t
t 1,α
x
−x
= τ t (φ) + τ (φ) − 2φ .
t
1,α
Since φ ∈ S∗ (R), then using (2.4) and [7, Theorem 2.1] (see also [11, Theorem 2, p. 349]), we
can assert that
x
x
x
(φ) − 2φ ≤ c kφ0 k1,α ≤ c .
τ t (φ) + τ −x
t
t
t
1,α
On the other hand, by (2.3) we have
x
−x
τ
(φ)
+
τ
(φ)
−
2φ
t
≤ 10kφk1,α ≤ c,
t
1,α
then we get,
x
τ t (φ) + τ −x (φ) − 2φ
(3.8)
t
1,α
n xo
.
≤ c min 1,
t
From (3.6), (3.7) and (3.8), we obtain
Z
(3.9)
kτx (fε,δ ) + τ−x (fε,δ ) − 2fε,δ kp,α ≤ c
ε
δ
n xo
dt
min 1,
kφt ∗α f kp,α .
t
t
Using (2.6), observe that
Z
(φ ∗α φ)(x)|x|2α+1 dx = 2α+1 Γ(α + 1)Fα (φ ∗α φ)(0)
R
= 2α+1 Γ(α + 1)(Fα (φ)(0))2
Z
2
α+1
= 2 Γ(α + 1)
φ(z)dµα (z) = 0,
R
and since φ ∗α φ is in the Schwarz space S(R), we have
Z
| log |x|| |φ ∗α φ(x)| |x|2α+1 dx < +∞.
R
Then, by the Calderón reproducing formula related to the Dunkl operators (see [10, Theorem
3]), we have
lim
fε,δ = c f, in Lp (µα ).
ε→0, δ→+∞
From (2.3) and (3.9), we deduce (3.5).
Lemma 3.5. Let 0 ≤ ε, r < +∞ and r > β > 0, then there exists constants c1 , c2 > 0 such
that we have
r Z +∞ β
y
y ε z
dy
(3.10)
min
,
≤ c1 , z ∈ (0, +∞)
z
z
y
y
0
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8
C HOKRI A BDELKEFI AND M OHAMED S IFI
and
Z
+∞
(3.11)
y β
0
z
r y ε z
dz
min
,
≤ c2 ,
z
y
z
y ∈ (0, +∞).
Proof. We can write
Z
+∞
y β
z
0
r y ε z
dy
min
,
z
y
y
Z z
Z
−(β+ε)
β+ε−1
r−β
=z
y
dy + z
0
+∞
y β−r−1 dy ≤ c1 ,
z ∈ (0, +∞)
z −β−ε−1 dz ≤ c2 ,
y ∈ (0, +∞),
z
and
Z
+∞
y β
z
0
r y ε z
dz
min
,
z
y
z
Z y
Z
β−r
−β+r−1
β+ε
=y
z
dz + y
0
+∞
y
which proves the results.
Theorem 3.6.
(1) Let 1 ≤ p < +∞, 1 ≤ q ≤ +∞ and β > 0, then we have for all φ ∈ A
(3.12)
p,q
p,q
BDβ,α
⊂ Cφ,β,α
.
(2) Let 1 < p < +∞, 1 ≤ q ≤ +∞ and 0 < β < 1, then we have for all φ ∈ A
(3.13)
p,q
p,q
BDβ,α
= Cφ,β,α
.
Proof. Put ωpα (f )(x) = kτx (f ) + τ−x (f ) − 2f kp,α for f ∈ Lp (µα ) and q 0 =
of q when 1 < q < +∞.
q
q−1
the conjugate
• We start with the proof of the inclusion (3.12). Suppose that 1 ≤ p < +∞, 1 ≤ q ≤
p,q
+∞, φ ∈ A, r > β and f ∈ BDβ,α
.
Case when q = 1. By (3.1) and Fubini’s theorem, we have
r Z +∞
Z +∞ Z +∞
kf ∗α φt kp,α dt
x 2(α+1)
t
dx
≤c
min
,
ωpα (f )(x)t−β−1 dt
β
t
t
t
x
x
0
0
0
Z +∞
r Z +∞
x 2(α+1)
t
dx
α
−β−1
≤c
ωp (f )(x)
min
,
t
dt
t
x
x
0
0
Z +∞
Z x
Z +∞
dx
α
−r
r−β−1
2(α+1)
−β−2α−3
≤c
ωp (f )(x) x
t
dt + x
t
dt
x
0
0
x
Z +∞ α
ωp (f )(x) dx
≤c
< +∞,
xβ
x
0
p,1
hence f ∈ Cφ,β,α
.
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C HARACTERIZATION OF B ESOV-D UNKL SPACES
9
Case when q = +∞. By (3.1), we have
Z t Z +∞ r
dx
dx
x 2(α+1) α
t
α
ωp (f )(x) +
kφt ∗α f kp,α ≤ c
ωp (f )(x)
t
x
x
x
0
t
Z t
Z +∞
α
ωp (f )(x) −2(α+1)
2α+1+β
r
−β−r−1
≤ c sup
t
x
dx + t
x
dx
xβ
x∈(0,+∞)
0
t
≤ ctβ
ωpα (f )(x)
,
xβ
x∈(0,+∞)
sup
p,+∞
then we deduce that f ∈ Cφ,β,α
.
Case when 1 < q < +∞. By (3.1) again, we have for t > 0
r α
Z +∞ β
ωp (f )(x) dx
kφt ∗α f kp,α
x
x 2(α+1)
t
≤
c
min
,
.
tβ
t
t
x
xβ
x
0
Put
K(x, t) =
x
β
t
x 2(α+1) t r
min
,
.
t
x
Using Hölder’s inequality and (3.10), we can write
Z +∞
α
1
1 ωp (f )(x)
kφt ∗α f kp,α
dx
0
≤c
(K(x, t)) q (K(x, t)) q
β
β
t
x
x
0
1
Z +∞
α
ωp (f )(x) q dx q
≤c
K(x, t)
.
xβ
x
0
Then by Fubini’s theorem and (3.11), we have
q
Z +∞
Z +∞ Z +∞ α
ωp (f )(x) q
kφt ∗α f kp,α
dt
dt dx
≤c
K(x, t)
tβ
t
xβ
t
x
0
0
0
q
Z +∞ α
ωp (f )(x) dx
≤c
< +∞,
xβ
x
0
which proves the result.
p,q
• Let us now prove the equality (3.13). Assume f ∈ Cφ,β,α
, φ ∈ A and 0 < β < 1. For
p,q
1 < p < +∞ and 1 ≤ q ≤ +∞, we have to show only that f ∈ BDβ,α
.
Case when q = 1. By (3.5) and Fubini’s theorem, we have
Z +∞ α
Z +∞ Z +∞
n xo
ωp (f )(x) dx
dt
≤
c
min
1,
kφt ∗α f kp,α x−β−1 dx
β
x
x
t
t
0
0
0
Z +∞
Z +∞
n xo
dt
−β−1
≤c
kφt ∗α f kp,α
min 1,
x
dx
t
t
0
0
Z t
Z +∞
Z +∞
1
dt
≤c
kφt ∗α f kp,α
x−β dx +
x−β−1 dx
t 0
t
0
t
Z +∞
kφt ∗α f kp,α dt
≤c
< +∞,
tβ
t
0
then we obtain the result.
J. Inequal. Pure and Appl. Math., 8(3) (2007), Art. 73, 11 pp.
http://jipam.vu.edu.au/
10
C HOKRI A BDELKEFI AND M OHAMED S IFI
Case when q = +∞. By (3.5), we get
Z x
Z +∞
dt
x
dt
α
ωp (f )(x) ≤ c
kφt ∗α f kp,α +
kφt ∗α f kp,α
t
t
t
0
x
Z x
Z +∞
kφt ∗α f kp,α
β−1
β−2
≤ c sup
t dt + x
t dt
tβ
t∈(0,+∞)
0
x
≤ cxβ
kφt ∗α f kp,α
,
tβ
t∈(0,+∞)
sup
p,+∞
so, we deduce that f ∈ BDβ,α
.
Case when 1 < q < +∞. By (3.5) again, we have for x > 0
Z +∞ β
n x o kφ ∗ f k dt
ωpα (f )(x)
t
t α
p,α
≤
c
min
1,
.
β
xβ
x
t
t
t
0
Put
n xo
t β
R(x, t) =
min 1,
.
x
t
Using Hölder’s inequality and (3.10), we can write
Z +∞
1
ωpα (f )(x)
1 kφt ∗α f kp,α
dt
0
≤c
(R(x, t)) q (R(x, t)) q
β
β
t
t
x
0
Z +∞
q 1q
kφt ∗α f kp,α
dt
≤c
R(x, t)
,
β
t
t
0
then by Fubini’s theorem and (3.11), we have
q Z +∞
Z +∞ α
Z +∞ ωp (f )(x) q dx
kφt ∗α f kp,α
dx dt
≤c
R(x, t)
xβ
x
tβ
x
t
0
0
0
q
Z +∞ dt
kφt ∗α f kp,α
≤c
< +∞,
β
t
t
0
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
p,q
e p,q , hence from (3.13)
can assert that for 1 < p < +∞ and 0 < β < 1, we have Cφ,β,α
⊂ BD
β,α
p,q
e p,q .
we conclude that BDβ,α
⊂ BD
β,α
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