CHARACTERIZATION OF BESOV SPACES FOR THE
DUNKL OPERATOR ON THE REAL LINE
Characterization of
Besov-Dunkl spaces
CHOKRI ABDELKEFI
MOHAMED SIFI
Department of Mathematics
Preparatory Institute of Engineer Studies of Tunis
1089 Monfleury Tunis, Tunisia
EMail: chokri.abdelkefi@ipeit.rnu.tn
Department of Mathematics
Faculty of Sciences of Tunis
1060 Tunis, Tunisia
EMail: mohamed.sifi@fst.rnu.tn
Chokri Abdelkefi and Mohamed Sifi
vol. 8, iss. 3, art. 73, 2007
Title Page
Contents
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 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.
Acknowledgements:
The authors thank the referee for his remarks and suggestions.
JJ
II
J
I
Page 1 of 21
Go Back
Full Screen
Close
Contents
1
Introduction
3
2
Preliminaries
5
3
Characterization of Besov-Dunkl Spaces
9
Characterization of
Besov-Dunkl spaces
Chokri Abdelkefi and Mohamed Sifi
vol. 8, iss. 3, art. 73, 2007
Title Page
Contents
JJ
II
J
I
Page 2 of 21
Go Back
Full Screen
Close
1.
Introduction
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 Besov-Dunkl spaces (see [1, 2, 8]). Let β > 0, 1 ≤ p < +∞ and 1 ≤ q ≤ +∞,
p,q
the Besov-Dunkl 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 < +∞
xβ
x
0
and
kτx (f ) + τ−x (f ) − 2f kp,α
sup
< +∞
xβ
x∈(0,+∞)
if q = +∞,
where µα is a weighted Lebesgue measure on R (see next section).
Put
Z +∞
A = φ ∈ S∗ (R) :
φ(x)dµα (x) = 0 ,
0
where S∗ (R) is the space of even Schwartz functions on R. Given φ ∈ A, we shall
p,q
denote by 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
Characterization of
Besov-Dunkl spaces
Chokri Abdelkefi and Mohamed Sifi
vol. 8, iss. 3, art. 73, 2007
Title Page
Contents
JJ
II
J
I
Page 3 of 21
Go Back
Full Screen
Close
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 < +∞,
p,q
p,q
0 < β < 1 then BDβ,α = Cφ,β,α .
Observe that the Besov-Dunkl spaces are independent of the specific selection of
p,q
e p,q , where BD
e p,q is
φ in A and for 1 < p < +∞, 0 < β < 1, we have BDβ,α
⊂ BD
β,α
β,α
the subspace of functions f ∈ Lp (µα ) satisfying
q
Z +∞ kτx (f ) − f kp,α
dx
< +∞
if q < +∞
β
x
x
0
and
Characterization of
Besov-Dunkl spaces
Chokri Abdelkefi and Mohamed Sifi
vol. 8, iss. 3, art. 73, 2007
Title Page
Contents
kτx (f ) − f kp,α
< +∞
xβ
x∈(0,+∞)
sup
if
q = +∞,
(see Remark 2 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 spaces
p,q
p,q
BDβ,α
and Cφ,β,α
.
In what follows, c represents a suitable positive constant which is not necessarily
the same in each occurence.
JJ
II
J
I
Page 4 of 21
Go Back
Full Screen
Close
2.
Preliminaries
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]).
Let µα be the weighted Lebesgue measure on R given by
2α+1
dµα (x) =
|x|
2α+1 Γ(α
+ 1)
dx.
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 )
Characterization of
Besov-Dunkl spaces
Chokri Abdelkefi and Mohamed Sifi
vol. 8, iss. 3, art. 73, 2007
Title Page
Contents
JJ
II
J
I
Page 5 of 21
Go Back
Full Screen
Close
where
bx,y,z =


x2 +y 2 −z 2
2xy
 0
and
∆α (x, y, z) =
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
where
Sx,y
h
i
= ||x| − |y|| , |x| + |y| .
The kernel Wα (see [13]) is even and we have
Characterization of
Besov-Dunkl spaces
Chokri Abdelkefi and Mohamed Sifi
vol. 8, iss. 3, art. 73, 2007
Title Page
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.
Contents
JJ
II
J
I
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
Page 6 of 21
Go Back
Full Screen
Close
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
kτx (f )kp,α ≤ 4kf kp,α .
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)) dνα (θ),
f (Ψ) + f (−Ψ) +
Ψ
0
(2.3)
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(x)
sgn(z)
(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]).
Characterization of
Besov-Dunkl spaces
Chokri Abdelkefi and Mohamed Sifi
vol. 8, iss. 3, art. 73, 2007
Title Page
Contents
JJ
II
J
I
Page 7 of 21
Go Back
Full Screen
Close
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)
Characterization of
Besov-Dunkl spaces
Fα (f ∗α g) = Fα (f )Fα (g)
Chokri Abdelkefi and Mohamed Sifi
vol. 8, iss. 3, art. 73, 2007
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.
Title Page
Contents
JJ
II
J
I
Page 8 of 21
Go Back
Full Screen
Close
3.
Characterization of Besov-Dunkl Spaces
Let β > 0, 1 ≤ p < +∞ and 1 ≤ q ≤ +∞. We say that a measurable function f on
p,q
R is in 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
< +∞
xβ
x∈(0,+∞)
Characterization of
Besov-Dunkl spaces
Chokri Abdelkefi and Mohamed Sifi
if q = +∞.
Remark 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 3.1. Let 0 < β < 1, 1 ≤ p < +∞, 1 ≤ q ≤ +∞ and f ∈ Lp (µα ). If
p,q
Λα (f ) ∈ Lp (µα ) 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
vol. 8, iss. 3, art. 73, 2007
Title Page
Contents
JJ
II
J
I
Page 9 of 21
Go Back
Full Screen
Close
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
Z +∞ kτx (f ) + τ−x (f ) − 2f kp,α
dx
xβ
x
0
q
q
Z A
Z +∞ xkΛα (f )kp,α
dx
dx
kf kp,α
≤c
+c
< +∞.
β
β
x
x
x
x
0
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 3.1, we
can assert that
p,q
1. S(R), Cc1 (R) ⊂ BDβ,α
.
Characterization of
Besov-Dunkl spaces
Chokri Abdelkefi and Mohamed Sifi
vol. 8, iss. 3, art. 73, 2007
Title Page
2. The functions g, hn defined on R, by g(x) = e−|x| and hn (x) =
p,q
are in BDβ,α
.
xn
cosh x
,n∈N
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 3.2. 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
Contents
JJ
II
J
I
Page 10 of 21
Go Back
Full Screen
(3.1) kφt ∗α f kp,α
Z
≤c
r +∞
x 2(α+1)
t
dx
min
,
kτx (f ) + τ−x (f ) − 2f kp,α .
t
x
x
0
Proof. Let t > 0, we have
Z +∞
Z
φt (x)dµα (x) =
0
0
+∞
φ(x)dµα (x) = 0
Close
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 +∞
=2
φt (x) [τx (f )(y) + τ−x (f )(y) − 2f (y)] dµα (x).
Characterization of
Besov-Dunkl spaces
Chokri Abdelkefi and Mohamed Sifi
vol. 8, iss. 3, art. 73, 2007
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) x
dx
x ≤c
(3.2)
φ
kτx (f ) + τ−x (f ) − 2f kp,α
t
t
x
Z0 +∞ 2(α+1)
x
dx
(3.3)
≤c
kτx (f ) + τ−x (f ) − 2f kp,α .
t
x
0
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
t
dx
(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).
Title Page
Contents
JJ
II
J
I
Page 11 of 21
Go Back
Full Screen
Close
Lemma 3.3. 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
dt
fε,δ (y) =
(φt ∗α φt ∗α f )(y) ,
t
ε
Characterization of
Besov-Dunkl spaces
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
(3.6)
k(τx (fε,δ ) + τ−x (fε,δ ) − 2fε,δ kp,α
Z δ
dt
≤
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
Chokri Abdelkefi and Mohamed Sifi
vol. 8, iss. 3, art. 73, 2007
Title Page
Contents
JJ
II
J
I
Page 12 of 21
Go Back
Full Screen
Close
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
ZR ZR y z
t−2(α+1) dµα (y).
=
(dγx,y (z) + dγ−x,y (z)) − 2φ
φ
t
t R
R
By (2.1) and the change of variable z 0 =
z
t
Wα (x, y, z 0 t)t2(α+1)
Characterization of
Besov-Dunkl spaces
Chokri Abdelkefi and Mohamed Sifi
, we have
x y = Wα
, , z0 ,
t t
vol. 8, iss. 3, art. 73, 2007
Title Page
then from (2.2), we get
Contents
0
0
dγx,y (z) = dγ xt , yt (z ) and dγ−x,y (z) = dγ −x , y (z ) ,
JJ
II
hence
J
I
(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 i
y
y
x
−2(α+1)
−x
=
+ τ (φ)
t
− 2φt (y) dµα (y)
τ t (φ)
t
t
t
R
x
= τ t (φ) + τ −x (φ) − 2φ t
t
1,α
x
= τ t (φ) + τ −x (φ) − 2φ .
Page 13 of 21
t
t
1,α
t
Go Back
Full Screen
Close
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
−x
τ t (φ) + τ t (φ) − 2φ ≤ c kφ0 k1,α ≤ c .
t
t
1,α
On the other hand, by (2.3) we have
x
τ t (φ) + τ −x (φ) − 2φ
t
1,α
≤ 10kφk1,α ≤ c,
Chokri Abdelkefi and Mohamed Sifi
then we get,
vol. 8, iss. 3, art. 73, 2007
x
τ t (φ) + τ −x (φ) − 2φ
(3.8)
Characterization of
Besov-Dunkl spaces
t
1,α
n xo
≤ c min 1,
.
t
Title Page
From (3.6), (3.7) and (3.8), we obtain
Contents
n xo
δ
dt
≤c
min 1,
kφt ∗α f kp,α .
t
t
ε
Z
(3.9)
kτx (fε,δ ) + τ−x (fε,δ ) − 2fε,δ kp,α
Using (2.6), observe that
Z
(φ ∗α φ)(x)|x|2α+1 dx = 2α+1 Γ(α + 1)Fα (φ ∗α φ)(0)
JJ
II
J
I
Page 14 of 21
Go Back
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
Full Screen
Close
Then, by the Calderón reproducing formula related to the Dunkl operators (see [10,
Theorem 3]), we have
lim
ε→0, δ→+∞
in Lp (µα ).
fε,δ = c f,
From (2.3) and (3.9), we deduce (3.5).
Lemma 3.4. Let 0 ≤ ε, r < +∞ and r > β > 0, then there exists constants c1 ,
c2 > 0 such that we have
r Z +∞ β
y ε z
dy
y
(3.10)
min
,
≤ c1 , z ∈ (0, +∞)
z
z
y
y
0
Characterization of
Besov-Dunkl spaces
Chokri Abdelkefi and Mohamed Sifi
vol. 8, iss. 3, art. 73, 2007
and
Z
(3.11)
+∞
y β
z
0
r y ε z
dz
min
,
≤ c2 ,
z
y
z
Proof. We can write
r Z +∞ β
y
y ε z
dy
min
,
z
z
y
y
0
Z z
Z
−(β+ε)
β+ε−1
r−β
=z
y
dy + z
0
and
Z
0
Title Page
y ∈ (0, +∞).
Contents
JJ
II
J
I
Page 15 of 21
+∞
y β−r−1 dy ≤ c1 ,
z ∈ (0, +∞)
Go Back
z
Full Screen
+∞
r y ε z
dz
min
,
z
z
y
z
Z y
Z
β−r
−β+r−1
β+ε
=y
z
dz + y
Close
y β
0
which proves the results.
y
+∞
z −β−ε−1 dz ≤ c2 ,
y ∈ (0, +∞),
Theorem 3.5.
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 =
conjugate of q when 1 < q < +∞.
Characterization of
Besov-Dunkl spaces
q
q−1
the
• We start with the proof of the inclusion (3.12). Suppose that 1 ≤ p < +∞,
p,q
1 ≤ q ≤ +∞, φ ∈ A, r > β and f ∈ BDβ,α
.
Case when q = 1. By (3.1) and Fubini’s theorem, we have
Z +∞
kf ∗α φt kp,α dt
tβ
t
0
r Z +∞ Z +∞
x 2(α+1)
t
dx
≤c
min
,
ωpα (f )(x)t−β−1 dt
t
x
x
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 x
Z +∞
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
.
Chokri Abdelkefi and Mohamed Sifi
vol. 8, iss. 3, art. 73, 2007
Title Page
Contents
JJ
II
J
I
Page 16 of 21
Go Back
Full Screen
Close
Case when q = +∞. By (3.1), we have
Z t Z +∞ r
x 2(α+1) α
dx
t
dx
α
kφt ∗α f kp,α ≤ c
ωp (f )(x) +
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
Characterization of
Besov-Dunkl spaces
ωpα (f )(x)
≤ ct
sup
,
xβ
x∈(0,+∞)
β
Chokri Abdelkefi and Mohamed Sifi
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
x 2(α+1)
t
kφt ∗α f kp,α
x
≤
c
min
,
.
tβ
t
t
x
xβ
x
0
Put
t r
β
2(α+1)
K(x, t) =
min
,
.
t
t
x
Using Hölder’s inequality and (3.10), we can write
Z +∞
α
1
1 ωp (f )(x)
dx
kφt ∗α f kp,α
0
q
q
(K(x, t))
≤c
(K(x, t))
β
β
x
x
t
0
Z +∞
α
q 1q
ωp (f )(x) dx
.
≤c
K(x, t)
xβ
x
0
x
x
Then by Fubini’s theorem and (3.11), we have
q
Z +∞
Z +∞ α
Z +∞ ωp (f )(x) q
dt
dt dx
kφt ∗α f kp,α
≤c
K(x, t)
tβ
t
xβ
t
x
0
0
0
vol. 8, iss. 3, art. 73, 2007
Title Page
Contents
JJ
II
J
I
Page 17 of 21
Go Back
Full Screen
Close
Z
≤c
0
+∞
ωpα (f )(x)
xβ
q
dx
< +∞,
x
which proves the result.
p,q
• Let us now prove the equality (3.13). Assume f ∈ Cφ,β,α
, φ ∈ A and 0 < β <
p,q
1. For 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 +∞ Z +∞
n xo
+∞ ω α (f )(x)
dx
dt
p
≤
c
min
1,
kφt ∗α f kp,α x−β−1 dx
β
x
x
t
t
0
Z +∞
Z0 +∞ 0
n xo
dt
−β−1
≤c
kφt ∗α f kp,α
min 1,
x
dx
t
t
0
0Z 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
Z
then we obtain the result.
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
t dt + x
t dt
≤ c sup
tβ
t∈(0,+∞)
0
x
≤ cxβ
kφt ∗α f kp,α
,
tβ
t∈(0,+∞)
sup
Characterization of
Besov-Dunkl spaces
Chokri Abdelkefi and Mohamed Sifi
vol. 8, iss. 3, art. 73, 2007
Title Page
Contents
JJ
II
J
I
Page 18 of 21
Go Back
Full Screen
Close
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
β
x
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
q
q
≤c
(R(x, t))
(R(x, t))
β
β
x
t
t
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 +∞ kφt ∗α f kp,α
dt
≤c
< +∞,
β
t
t
0
thus the result is established.
Remark 2. By proceeding in the same manner as in Lemma 3.3 and (2) of Theorem
p,q
e p,q ,
3.5, we can assert that for 1 < p < +∞ and 0 < β < 1, we have Cφ,β,α
⊂ BD
β,α
p,q
e p,q .
hence from (3.13) we conclude that BDβ,α
⊂ BD
β,α
Characterization of
Besov-Dunkl spaces
Chokri Abdelkefi and Mohamed Sifi
vol. 8, iss. 3, art. 73, 2007
Title Page
Contents
JJ
II
J
I
Page 19 of 21
Go Back
Full Screen
Close
References
[1] C. ABDELKEFI AND M. SIFI, On the uniform convergence of partial Dunkl
integrals in Besov-Dunkl spaces, Fractional Calculus and Applied Analysis,
9(1) (2006), 43–56.
[2] C. ABDELKEFI AND M. SIFI, Further results of integrability for the Dunkl
transform, Communications in Mathematical Analysis, 2(1) (2007), 29–36.
[3] J.L. ANSORENA AND O. BLASCO, Characterization of weighted Besov
spaces, Math. Nachr., 171 (1995), 5–17.
[4] O.V. BESOV, On a family of function spaces in connection with embeddings
and extentions, Trudy Mat. Inst. Steklov, 60 (1961), 42–81.
[5] C.F. DUNKL, Differential-difference operators associated to reflection groups,
Trans.Amer. Math. Soc., 311(1) (1989), 167–183.
[6] C.F. DUNKL, Hankel transforms associated to finite reflection groups in Proc.
of Special Session on Hypergeometric Functions on Domains of Positivity, Jack
Polynomials and Applications. Proceedings, Tampa 1991, Contemp. Math., 138
(1992), 123–138.
[7] J. GOSSELIN AND K. STEMPAK, A weak type estimate for Fourier-Bessel
multipliers, Proc. Amer. Math. Soc., 106 (1989), 655–662.
[8] L. KAMOUN, Besov-type spaces for the Dunkl operator on the real line, J.
Comp. and Appl. Math., 199(1) (2007), 56–67.
[9] M.A. MOUROU, Transmutation operators associated with a Dunkl-type
differential-difference operator on the real line and certain of their applications,
Integral transforms Spec. Funct., 12(1) (2001), 77–88.
Characterization of
Besov-Dunkl spaces
Chokri Abdelkefi and Mohamed Sifi
vol. 8, iss. 3, art. 73, 2007
Title Page
Contents
JJ
II
J
I
Page 20 of 21
Go Back
Full Screen
Close
[10] M.A. MOUROU AND K. TRIMÈCHE, Calderon’s reproducing formula related
to the Dunkl operator on the real line, Monatshefte für Mathematik, 136 (2002),
47–65.
[11] M.A. MOUROU, Taylor series associated with a differential-difference operator on the real line, J. Comp. and Appl. Math., 153 (2003), 343–354.
[12] J. PEETRE, New Thoughts on Besov Spaces, Duke Univ. Math. Series,
Durham, NC, 1976.
[13] M. RÖSLER, Bessel-Type signed hypergroup on R, in Probability measure on
groups and related structures, Proc. Conf Oberwolfach, (1994), H. Heyer and
A. Mukherjea (Eds) World scientific Publ, 1995, pp 292–304.
Characterization of
Besov-Dunkl spaces
Chokri Abdelkefi and Mohamed Sifi
vol. 8, iss. 3, art. 73, 2007
Title Page
p
[14] F. SOLTANI, L -Fourier multipliers for the Dunkl operator on the real line, J.
Funct. Analysis, 209 (2004), 16–35.
[15] S. THANGAVELYU AND Y. XU, Convolution operator and maximal function
for Dunkl transform, J. Anal. Math., 97 (2005), 25–56.
[16] A. TORCHINSKY, Real-variable Methods in Harmonic Analysis, Academic
Press, 1986.
[17] H. TRIEBEL, Theory of Function Spaces, Monographs in Math., Vol. 78,
Birkhäuser Verlag, Basel, 1983.
[18] G.N. WATSON, A Treatise on the Theory of Bessel Functions, Camb. Univ.
Press, Cambridge 1966.
Contents
JJ
II
J
I
Page 21 of 21
Go Back
Full Screen
Close