Symmetry, Integrability and Geometry: Methods and Applications
SIGMA 5 (2009), 019, 15 pages
Besov-Type Spaces on Rd and Integrability
for the Dunkl Transform?
Chokri ABDELKEFI † , Jean-Philippe ANKER ‡ , Feriel SASSI
†
and Mohamed SIFI
§
†
Department of Mathematics, Preparatory Institute of Engineer Studies of Tunis,
1089 Monf leury Tunis, Tunisia
E-mail: chokri.abdelkefi@ipeit.rnu.tn, feriel.sassi@ipeit.rnu.tn
‡
Department of Mathematics, University of Orleans & CNRS, Federation Denis Poisson
(FR 2964), Laboratoire MAPMO (UMR 6628), B.P. 6759, 45067 Orleans cedex 2, France
E-mail: Jean-Philippe.Anker@univ-orleans.fr
§
Department of Mathematics, Faculty of Sciences of Tunis, 1060 Tunis, Tunisia
E-mail: mohamed.sifi@fst.rnu.tn
Received August 28, 2008, in final form February 05, 2009; Published online February 16, 2009
doi:10.3842/SIGMA.2009.019
Abstract. In this paper, we show the inclusion and the density of the Schwartz space
in Besov–Dunkl spaces and we prove an interpolation formula for these spaces by the real
method. We give another characterization for these spaces by convolution. Finally, we establish further results concerning integrability of the Dunkl transform of function in a suitable
Besov–Dunkl space.
Key words: Dunkl operators; Dunkl transform; Dunkl translations; Dunkl convolution;
Besov–Dunkl spaces
2000 Mathematics Subject Classification: 42B10; 46E30; 44A35
1
Introduction
We consider the differential-difference operators Ti , 1 ≤ i ≤ d, on Rd , associated with a positive
root system R+ and a non negative multiplicity function k, introduced by C.F. Dunkl in [9] and
called Dunkl operators (see next section). These operators can be regarded as a generalization of
partial derivatives and lead to generalizations of various analytic structure, like the exponential
function, the Fourier transform, the translation operators and the convolution (see [8, 10, 11,
16, 17, 18, 19, 22]). The Dunkl kernel Ek has been introduced by C.F. Dunkl in [10]. This
kernel is used to define the Dunkl transform Fk . K. Trimèche has introduced in [23] the Dunkl
translation operators τx , x ∈ Rd , on the space of infinitely differentiable functions on Rd . At
the moment an explicit formula for the Dunkl translation operator of function τx (f ) is unknown
in general. However, such formula is known when f is a radial function and the Lp -boundedness
of τx for radial functions is established. As a result, we have the Dunkl convolution ∗k .
There are many ways to define the Besov spaces (see [6, 15, 21]) and the Besov spaces for
the Dunkl operators (see [1, 2, 3, 4, 14]). Let β > 0, 1 ≤ p, q ≤ +∞, the Besov–Dunkl space
β,k
denoted by BDp,q
in this paper, is the subspace of functions f ∈ Lpk (Rd ) satisfying
kf kBDβ,k =
X
p,q
jβ
q
(2 kϕj ∗k f kp,k )
1
q
< +∞
if q < +∞
j∈Z
?
This paper is a contribution to the Special Issue on Dunkl Operators and Related Topics. The full collection
is available at http://www.emis.de/journals/SIGMA/Dunkl operators.html
2
C. Abdelkefi, J.-Ph. Anker, F. Sassi and M. Sifi
and
kf kBDβ,k = sup 2jβ kϕj ∗k f kp,k < +∞
p,∞
if q = +∞,
j∈Z
where (ϕj )j∈Z is a sequence of functions in S(Rd )rad such that
(i) supp Fk (ϕj ) ⊂ Aj = x ∈ Rd ; 2j−1 ≤ kxk ≤ 2j+1 for j ∈ Z;
(ii) sup kϕj k1,k < +∞;
j∈Z
(iii)
X
Fk (ϕj )(x) = 1, for x ∈ Rd \{0}.
j∈Z
d
S(Rd )rad being
of functions in the Schwartz space S(R
the subspace
) which are radial.
d rad
d
Put A = φ ∈ S(R ) : supp Fk (φ) ⊂ {x ∈ R ; 1 ≤ kxk ≤ 2} . Given φ ∈ A, we denote
φ,β,k
by Cp,q
the subspace of functions f ∈ Lpk (Rd ) satisfying
Z +∞ 1
kf ∗k φt kp,k q dt q
< +∞
if q < +∞
t
tβ
0
and
kf ∗k φt kp,k
< +∞
tβ
t∈(0,+∞)
sup
where φt (x) =
1
d
t2(γ+ 2 )
if q = +∞,
φ( xt ), for all t ∈ (0, +∞) and x ∈ Rd .
β,k
In this paper we show for β > 0 the inclusion of the Schwartz space in BDp,q
for 1 ≤ p, q ≤ +∞
and the density when 1 ≤ p, q < +∞. We prove an interpolation formula for the Besov–Dunkl
φ,β,k
which extend to the Dunkl
spaces by the real method. We compare these spaces with Cp,q
d
operators on R some results obtained in [4, 5, 21]. Finally we establish further results of
β,k
integrability of Fk (f ) when f is in a suitable Besov–Dunkl space BDp,q
for 1 ≤ p ≤ 2 and
1 ≤ q ≤ +∞. Using the characterization of the Besov spaces by differences analogous results
of integrability have been obtained in the case q = 1 by Giang and Móricz in [13] for a classical
Fourier transform on R and for q = 1, +∞ by Betancor and Rodrı́guez-Mesa in [7] for the
Hankel transform on (0, +∞) in Lipschitz–Hankel spaces. Later Abdelkefi and Sifi in [1, 2]
have established similar results of integrability for the Dunkl transform on R and in radial case
on Rd . The argument used in [1, 2] to establish such integrability is the Lp -boundedness of the
Dunkl translation operators, making it difficult to extend the results on Rd . We take a different
approach based on the the characterization of the Besov spaces by convolution to establish our
results on higher dimension.
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 show the
β,k
inclusion and the density of the Schwartz space in BDp,q
, we prove an interpolation formula
φ,β,k
for the Besov–Dunkl spaces by the real method and we compare these spaces with Cp,q
. In
Section 4 we establish our results concerning integrability of the Dunkl transform of function in
the Besov–Dunkl spaces.
d
Along this paper we denote by h·, ·i the usual
p Euclidean inner product in R as well as its
d
d
d
extension to C ×C , we write for x ∈ R , kxk = hx, xi and we represent by c a suitable positive
constant which is not necessarily the same in each occurrence. Furthermore we denote by
• E(Rd ) the space of infinitely differentiable functions on Rd ;
• S(Rd ) the Schwartz space of functions in E(Rd ) which are rapidly decreasing as well as
their derivatives;
• D(Rd ) the subspace of E(Rd ) of compactly supported functions.
Besov-Type Spaces on Rd and Integrability for the Dunkl Transform
2
3
Preliminaries
Let W be a finite reflection group on Rd , associated with a root system R and R+ the positive
subsystem of R (see [8, 10, 11, 12, 18, 19]). We denote by k a nonnegative multiplicity function
defined on R with the property that k is W -invariant. We associate with k the index
X
k(ξ) ≥ 0,
γ = γ(R) =
ξ∈R+
and the weight function wk defined by
Y
|hξ, xi|2k(ξ) ,
x ∈ Rd .
wk (x) =
ξ∈R+
Further we introduce the Mehta-type constant ck by
Z
−1
kxk2
− 2
ck =
e
wk (x)dx
.
Rd
For every 1 ≤ p ≤ +∞ we denote by Lpk (Rd ) the space Lp (Rd , wk (x)dx), Lpk (Rd )rad the
subspace of those f ∈ Lpk (Rd ) that are radial and we use k · kp,k as a shorthand for k · kLp (Rd ) .
k
By using the homogeneity of wk it is shown in [18] that for f ∈ L1k (Rd )rad there exists
a function F on [0, +∞) such that f (x) = F (kxk), for all x ∈ Rd . The function F is integrable
with respect to the measure r2γ+d−1 dr on [0, +∞) and we have
Z
c−1
k
wk (x)dσ(x) =
,
d
−1
γ+
d−1
S
2 2 Γ(γ + d2 )
where S d−1 is the unit sphere on Rd with the normalized surface measure dσ and
Z
Z +∞ Z
f (x)wk (x)dx =
wk (ry)dσ(y) F (r)rd−1 dr
Rd
S d−1
−1
ck
d
2γ+ 2 −1 Γ(γ + d2 )
0
=
Z
+∞
F (r)r2γ+d−1 dr.
(1)
0
Introduced by C.F. Dunkl in [9] the Dunkl operators Tj , 1 ≤ j ≤ d, on Rd associated with
the reflection group W and the multiplicity function k are the first-order differential- difference
operators given by
Tj f (x) =
X
∂f
f (x) − f (σα (x))
(x) +
k(α)αj
,
∂xj
hα, xi
f ∈ E(Rd ),
x ∈ Rd ,
α∈R+
where αj = hα, ej i, (e1 , e2 , . . . , ed ) being the canonical basis of Rd .
The Dunkl kernel Ek on Rd × Rd has been introduced by C.F. Dunkl in [10]. For y ∈ Rd the
function x 7→ Ek (x, y) can be viewed as the solution on Rd of the following initial problem
Tj u(x, y) = yj u(x, y),
1 ≤ j ≤ d,
u(0, y) = 1.
This kernel has a unique holomorphic extension to Cd × Cd . M. Rösler has proved in [17] the
following integral representation for the Dunkl kernel
Z
Ek (x, z) =
ehy,zi dµkx (y),
x ∈ Rd ,
z ∈ Cd ,
Rd
4
C. Abdelkefi, J.-Ph. Anker, F. Sassi and M. Sifi
where µkx is a probability measure on Rd with support in the closed ball B(0, kxk) of center 0
and radius kxk.
We have for all λ ∈ C and z, z 0 ∈ Cd Ek (z, z 0 ) = Ek (z 0 , z), Ek (λz, z 0 ) = Ek (z, λz 0 ) and for
x, y ∈ Rd |Ek (x, iy)| ≤ 1 (see [10, 17, 18, 19, 22]).
The Dunkl transform Fk which was introduced by C.F. Dunkl in [11] (see also [8]) is defined
for f ∈ D(Rd ) by
Z
Fk (f )(x) = ck
f (y)Ek (−ix, y)wk (y)dy,
x ∈ Rd .
Rd
According to [8, 11, 18] we have the following results:
i) The Dunkl transform of a function f ∈ L1k (Rd ) has the following basic property
kFk (f )k∞,k ≤ kf k1,k .
(2)
ii) The Schwartz space S(Rd ) is invariant under the Dunkl transform Fk .
iii) When both f and Fk (f ) are in L1k (Rd ), we have the inversion formula
Z
f (x) =
Fk (f )(y)Ek (ix, y)wk (y)dy,
x ∈ Rd .
Rd
iv) (Plancherel’s theorem) The Dunkl transform on S(Rd ) extends uniquely to an isometric
isomorphism on L2k (Rd ).
By (2), Plancherel’s theorem and the Marcinkiewicz interpolation theorem (see [20]) we get
for f ∈ Lpk (Rd ) with 1 ≤ p ≤ 2 and p0 such that p1 + p10 = 1,
kFk (f )kp0 ,k ≤ ckf kp,k .
(3)
The Dunkl transform of a function in L1k (Rd )rad is also radial and could be expressed via the
Hankel transform (see [18, Proposition 2.4]).
K. Trimèche has introduced in [23] the Dunkl translation operators τx , x ∈ Rd , on E(Rd ).
For f ∈ S(Rd ) and x, y ∈ Rd we have
Fk (τx (f ))(y) = Ek (ix, y)Fk (f )(y)
and
Z
Fk (f )(ξ)Ek (ix, ξ)Ek (iy, ξ)wk (ξ)dξ.
τx (f )(y) = ck
(4)
Rd
Notice that for all x, y ∈ Rd τx (f )(y) = τy (f )(x), and for fixed x ∈ Rd
τx is a continuous linear mapping from E(Rd ) into E(Rd ).
(5)
As an operator on L2k (Rd ), τx is bounded. A priori it is not at all clear whether the translation
operator can be defined for Lp -functions with p different from 2. However, according to [19,
Theorem 3.7] the operator τx can be extended to Lpk (Rd )rad , 1 ≤ p ≤ 2 and for f ∈ Lpk (Rd )rad
we have
kτx (f )kp,k ≤ kf kp,k .
Besov-Type Spaces on Rd and Integrability for the Dunkl Transform
5
The Dunkl convolution product ∗k of two functions f and g in L2k (Rd ) (see [19, 23]) is given by
Z
(f ∗k g)(x) =
τx (f )(−y)g(y)wk (y)dy,
x ∈ Rd .
Rd
The Dunkl convolution product is commutative and for f, g ∈ D(Rd ) we have
Fk (f ∗k g) = Fk (f )Fk (g).
(6)
It was shown in [19, Theorem 4.1] that when g is a bounded function in L1k (Rd )rad , then
Z
f (y)τx (g)(−y)wk (y)dy,
x ∈ Rd ,
(f ∗k g)(x) =
Rd
initially defined on the intersection of L1k (Rd ) and L2k (Rd ) extends to all Lpk (Rd ), 1 ≤ p ≤ +∞
as a bounded operator. In particular,
kf ∗k gkp,k ≤ kf kp,k kgk1,k .
(7)
The Dunkl Laplacian ∆k is defined by ∆k :=
d
P
i=1
Ti2 . From [16] we have for each λ > 0
λI − ∆k maps S(Rd ) onto itself and
Fk ((λI − ∆k )f )(x) = λ + kxk2 Fk (f )(x),
3
for x ∈ Rd .
(8)
Interpolation and characterization
for the Besov–Dunkl spaces
β,k
In this section we establish the inclusion and the density of S(Rd ) in BDp,q
and we prove an
interpolation formula for the Besov–Dunkl spaces by the real method. Finally we compare the
β,k
φ,β,k
spaces BDp,q
with Cp,q
. Before, we start with some useful results.
We shall denote by Φ the set of all sequences of functions (ϕj )j∈Z in S(Rd )rad satisfying
(i) supp Fk (ϕj ) ⊂ Aj = {x ∈ Rd ; 2j−1 ≤ kxk ≤ 2j+1 } for j ∈ Z;
(ii) sup kϕj k1,k < +∞;
j∈Z
(iii)
P
Fk (ϕj )(x) = 1, for x ∈ Rd \{0}.
j∈Z
β,k
Proposition 1. Let β > 0 and 1 ≤ p, q ≤ +∞, then BDp,q
is independent of the choice of the
sequence in Φ.
β,k
Proof . Fix (φj )j∈Z , (ψj )j∈Z in Φ and f ∈ BDp,q
for q < +∞. Using the properties (i) for
(φj )j∈Z and (i) and (iii) for (ψj )j∈Z , we have for j ∈ Z φj = φj ∗k (ψj−1 + ψj + ψj+1 ). Then by
the property (ii) for (φj )j∈Z , (7) and Hölder’s inequality for j ∈ Z we obtain
kf ∗k φj kqp,k ≤ c 3q−1
j+1
X
kψs ∗k f kqp,k .
s=j−1
Thus summing over j with weights 2jβq we get
X
X
q
q
2jβ kφj ∗k f kp,k ≤ c
2jβ kψj ∗k f kp,k .
j∈Z
j∈Z
Hence by symmetry we get the result of our proposition. When q = +∞ we make the usual
modification.
6
C. Abdelkefi, J.-Ph. Anker, F. Sassi and M. Sifi
β,k
Remark 1. Let β > 0, 1 ≤ p, q ≤ +∞, we denote by B̈Dp,q
the subspace of functions f ∈ Lpk (Rd )
satisfying
X
jβ
q
1
q
(2 kϕj ∗k f kp,k )
< +∞
if q < +∞
j∈N
and
sup 2jβ kϕj ∗k f kp,k < +∞
if q = +∞,
j∈N
where (ϕj )j∈N is a sequence of functions in S(Rd )rad such that
i) supp Fk (ϕ0 ) ⊂ {x ∈ Rd ; kxk ≤ 2} and supp Fk (ϕj ) ⊂ Aj = {x ∈ Rd ; 2j−1 ≤ kxk ≤ 2j+1 }
for j ∈ N\{0};
ii) sup kϕj k1,k < +∞;
j∈N
iii)
P
Fk (ϕj )(x) = 1, for x ∈ Rd \{0}.
j∈N
As the Besov–Dunkl spaces these spaces are also independent of the choice of the sequence
(ϕj )j∈N satisfying the previous properties.
Proposition 2. For β > 0 and 1 ≤ p, q ≤ +∞ we have
β,k
β,k
B̈Dp,q
= BDp,q
.
β,k
β,k
Proof . Since both spaces B̈Dp,q
and BDp,q
are in Lpk (Rd ) and are independent of the specific
selection of sequence of functions, then according to [5, Lemma 6.1.7, Theorem 6.3.2] we can
take a function φ ∈ S(Rd )rad such that
• supp Fk (φ) ⊂ {x ∈ Rd ;
1
2
≤ kxk ≤ 2};
1
2
• Fk (φ)(x) > 0 for < kxk < 2;
P
•
Fk (φ2−j )(x) = 1, x ∈ Rd \{0}.
j∈Z
β,k
If we consider the sequences (ψj )j∈Z and (ϕj )j∈N in S(Rd )rad defined respectively for BDp,q
P
β,k
and B̈Dp,q
by ψj = φ2−j ∀ j ∈ Z and ϕ0 =
φ2−j , ϕj = φ2−j ∀ j ∈ N∗ , we can assert that
j∈Z−
β,k
β,k
B̈Dp,q
= BDp,q
.
Remark 2. By Proposition 2 and [21, Proposition 2] we have the following embeddings.
1. Let 1 ≤ q1 ≤ q2 ≤ +∞ and β > 0. Then
β,k
β,k
BDp,q
⊂ BDp,q
1
2
if 1 ≤ p ≤ +∞.
2. Let 1 ≤ q1 , q2 ≤ +∞, β > 0 and ε > 0. Then
β+ε,k
β,k
BDp,q
⊂ BDp,q
1
2
if 1 ≤ p ≤ +∞.
Proposition 3. For β > 0 and 1 ≤ p, q ≤ +∞ we have
β,k
S(Rd ) ⊂ BDp,q
.
β,k
If 1 ≤ p, q < +∞, then S(Rd ) is dense in BDp,q
.
Besov-Type Spaces on Rd and Integrability for the Dunkl Transform
7
Proof . In order to prove the inclusion, we may restrict ourself to q = +∞. This follows from the
β,k
β 0 ,k
fact that BDp,∞
⊂ BDp,q
for β > β 0 > 0 and 1 ≤ p, q ≤ +∞ (see Remark 2, 2). Let f ∈ S(Rd )
and (ϕj )j∈N a sequence of functions in S(Rd )rad satisfying the properties of Remark 1, there
exists a sufficiently large natural number L such that
sup 2jβ kϕj ∗k f kp,k ≤ sup 2jβ k(1 + kxk2 )L (ϕj ∗k f )k∞,k .
j∈N
j∈N
Since ϕj ∈ S(Rd )rad , then for x ∈ Rd Fk−1 (ϕj )(x) = Fk (ϕj )(−x) = Fk (ϕj )(x), so using (8) and
the property i) of (ϕj )j∈N (see Remark 1) we obtain
sup 2jβ kϕj ∗k f kp,k ≤ sup 2jβ kFk [(I − ∆k )L (Fk (ϕj )Fk−1 (f ))]k∞,k
j∈N∗
j∈N∗
≤ sup 2jβ k(I − ∆k )L (Fk (ϕj )Fk−1 (f ))k1,k
j∈N∗
≤ sup cj 2jβ sup |(I − ∆k )L (Fk (ϕj )Fk−1 (f ))(x)|,
j∈N∗
x∈Aj
where cj =
R
cj 2jβ
(1+22(j−1) )M
≤ 1, ∀ j ∈ N∗ and we get
Aj
wk (x)dx. Hence there exists a sufficiently large natural number M such that
sup 2jβ kϕj ∗k f kp,k ≤ sup sup | 1 + kxk2
j∈N∗
j∈N∗ x∈Aj
M
(I − ∆k )L (Fk (ϕj )Fk−1 (f ))(x)|.
Since (I − ∆k )L is linear and continuous from S(Rd ) into itself, we deduce that
sup 2jβ kϕj ∗k f kp,k ≤ c sup sup |Fk (ϕj )(x)| sup |(1 + kxk2 )M Fk−1 (f )(x)|,
j∈N∗
j∈N∗ x∈Rd
x∈Rd
which gives by the property ii) of (ϕj )j∈N
sup 2jβ kϕj ∗k f kp,k ≤ c sup kϕj k1,k sup |(1 + kxk2 )M Fk−1 (f )(x)| < +∞.
j∈N∗
j∈N∗
x∈Rd
β,k
By Proposition 2 we conclude that S(Rd ) ⊂ BDp,q
.
β,k
β,k
d
and
Let us now prove the density of S(R ) in BDp,q for p, q < +∞. Assume f ∈ BDp,q
N
P
β,k
(ϕj )j∈Z ∈ Φ, then we put for N ∈ N\{0}, fN =
ϕs ∗k f . It’s clear that fN ∈ BDp,q
. We
s=−N
have
X
2jβq kϕj ∗k (fN − f )kqp,k =
j∈Z
X
j∈Z
q
N
X
2jβq ϕs ∗k ϕj ∗k f − ϕj ∗k f .
s=−N
p,k
Using the properties (i) and (iii) for (ϕj )j∈Z we get
X
kf − fN kq β,k ≤ c
2jβq kϕj ∗k f kqp,k .
BDp,q
|j|≥N
β,k
Since f ∈ BDp,q
, then we deduce that
lim kf − fN kBDβ,k = 0.
N →+∞
p,q
(9)
Next we take a function θ ∈ D(Rd ) such that θ(0) = 1. For n ∈ N\{0} we put θn (x) = θ(n−1 x),
x ∈ Rd . From (5) we have for N ∈ N\{0} fN ∈ E(Rd ), then fN θn ∈ S(Rd ). Again from
8
C. Abdelkefi, J.-Ph. Anker, F. Sassi and M. Sifi
the properties (i) and (iii) for (ϕj )j∈Z we can assert that fN =
N
P
ϕj ∗k fN +1 which gives
j=−N
N
P
fN θn =
ϕj ∗k fN +1 θn . Using the properties (i), (ii), (iii) for (ϕj )j∈Z and (7) we obtain
j=−N
kfN −
fN θn kq β,k
BDp,q
N
+1
X
≤c
2jβq kfN +1 − fN +1 θn kqp,k .
j=−N −1
The dominated convergence theorem implies that
kfN +1 − fN +1 θn kp,k → 0
as
n → +∞.
Hence we deduce that
kfN − fN θn kBDβ,k → 0
as
p,q
n → +∞,
(10)
β,k
. This completes the proof of
Combining (9) and (10) we conclude that S(Rd ) is dense in BDp,q
Proposition 3.
For 0 < θ < 1, β0 , β1 > 0, 1 ≤ p, q0 , q1 ≤ +∞ and 1 ≤ q ≤ +∞, the real interpolation Besov–
β0 ,k
β1 ,k
β0 ,k
β1 ,k
Dunkl space denoted by (BDp,q
0 , BDp,q1 )θ,q is the subspace of functions f ∈ BDp,q0 + BDp,q1
satisfying
+∞ Z
t
−θ
0
q dt 1q
Kp,k (t, f ; β0 , q0 ; β1 , q1 )
< +∞
t
if q < +∞,
and
sup
t−θ Kp,k (t, f ; β0 , q0 ; β1 , q1 ) < +∞
if q = +∞,
t∈(0,+∞)
with Kp,k is the Peetre K-functional given by
o
n
Kp,k (t, f ; β0 , q0 ; β1 , q1 ) = inf kf0 kBDβ0 ,k + tkf1 kBDβ1 ,k ,
p,q0
p,q1
where the infinimum is taken over all representations of f of the form
f = f0 + f1 ,
β0 ,k
f0 ∈ BDp,q
,
0
β1 ,k
f1 ∈ BDp,q
.
1
Theorem 1. Let 0 < θ < 1 and 1 ≤ p, q, q0 , q1 ≤ +∞. For β0 , β1 > 0, β0 6= β1 and β =
(1 − θ)β0 + θβ1 we have
β0 ,k
β1 ,k
β,k
BDp,q
, BDp,q
= BDp,q
.
0
1 θ,q
β0 ,k
β1 ,k
β,k
Proof . We start with the proof of the inclusion (BDp,∞
, BDp,∞
)θ,q ⊂ BDp,q
. We may assume
β0 ,k
β1 ,k
that β0 > β1 . Let q < +∞, for f = f0 + f1 with f0 ∈ BDp,∞ and f1 ∈ BDp,∞
we get by
Proposition 2
+∞
X
2qlβ kϕl ∗k f kqp,k ≤ c
l=0
+∞
X
q
2−θql(β0 −β1 ) 2lβ0 kϕl ∗k f0 kp,k + 2l(β0 −β1 ) 2lβ1 kϕl ∗k f1 kp,k
l=0
≤c
+∞
X
q
2−θql(β0 −β1 ) kf0 kBDβ0 ,k + 2l(β0 −β1 ) kf1 kBDβ1 ,k .
p,∞
l=0
p,∞
Besov-Type Spaces on Rd and Integrability for the Dunkl Transform
9
Then we deduce that
+∞
X
2qlβ kϕl ∗k f kqp,k ≤ c
l=0
+∞
X
q
2−θql(β0 −β1 ) Kp,k (2l(β0 −β1 ) , f ; β0 , ∞; β1 , ∞)
l=0
+∞ Z
≤c
0
q dt
t−θ Kp,k (t, f ; β0 , ∞; β1 , ∞)
< +∞,
t
which proves the result. When q = +∞, we make the usual modification.
For 1 ≤ s ≤ q0 , q1 Remark 2 gives
β0 ,k
β1 ,k
β0 ,k
β1 ,k
β0 ,k
β1 ,k
β,k
(BDp,s
, BDp,s
)θ,q ⊂ (BDp,q
, BDp,q
) ⊂ (BDp,∞
, BDp,∞
)θ,q ⊂ BDp,q
.
0
1 θ,q
Then in order to complete the proof of the theorem we have to show only that
β,k
β0 ,k
β1 ,k
BDp,q
⊂ (BDp,s
, BDp,s
)θ,q
for 1 ≤ s ≤ q.
Suppose that β0 > β1 again. Let q < +∞, we have
Z +∞ q dt Z 1 Z +∞
−θ
=
+
= I1 + I2 .
t Kp,k (t, f ; β0 , s; β1 , s)
t
0
1
0
Since β > β1 , by Remark 2 we get
Kp,k (t, f ; β0 , s; β1 , s) ≤ ctkf kBDβ1 ,k ≤ ctkf kBDβ,k ,
p,q
p,s
hence we deduce
I1 ≤ ckf kq
β,k
BDp,q
.
To estimate I2 take f0 =
l
P
ϕj ∗k f and f1 =
j=0
+∞
P
ϕj ∗k f . Using the properties of the sequence
j=l+1
(ϕj )j∈N we obtain
kf0 ks β0 ,k
BDp,s
≤c
l+1
X
jβ0 s
2
kϕj ∗k f ksp,k
kf1 ks β1 ,k
BDp,s
and
j=0
≤c
+∞
X
2jβ1 s kϕj ∗k f ksp,k .
j=l
Hence we can write
I2 ≤ c
+∞
X
q
2−θql(β0 −β1 ) Kp,k (2l(β0 −β1 ) , f ; β0 , s; β1 , s)
l=0
≤c
+∞
X


X
1/s
X
1/s q
l+1
+∞

2−θql(β0 −β1 ) 
2jβ0 s kϕj ∗k f ksp,k
+ 2l(β0 −β1 )
2jβ1 s kϕj ∗k f ksp,k
j=0
l=0
j=l

q/s
+∞
l+1
+∞
X
X
X
≤c
2qlβ 
2(j−l)β0 s kϕj ∗k f ksp,k +
2(j−l)β1 s kϕj ∗k f ksp,k  .
l=0
j=0
j=l
For s = q it is easy to see that I2 ≤ ckf kq
For s < q we take u > s such that
inequality we have
I2 ≤ c
+∞
X
l=0
2ql(β−β0 )
X
l+1
j=0
s
q
2(β0 −α0 )ju
+
s
u
β,k
BDp,q
.
= 1 and β1 < α1 < β < α0 < β0 , then by Hölder’s
q/u X
l+1
j=0
2α0 jq kϕj ∗k f kqp,k
10
C. Abdelkefi, J.-Ph. Anker, F. Sassi and M. Sifi
+c
+∞
X
2
ql(β−β1 )
X
+∞
l=0
≤c
+∞
X
≤c
ql(β−α0 )
2
l+1
X
α0 jq
2
α1 jq
2
kϕj ∗k f kqp,k
j=l
α0 jq
2
kϕj ∗k f kqp,k
+c
j=0
+∞
X
2
ql(β−α1 )
l=0
+∞
X
kϕj ∗k f kqp,k
j=0
≤
2
q/u X
+∞
j=l
l=0
+∞
X
(β1 −α1 )ju
2
ql(β−α0 )
+c
+∞
X
+∞
X
2α1 jq kϕj ∗k f kqp,k
j=l
2
α1 jq
kϕj ∗k f kqp,k
j=0
l=j−1
j
X
2ql(β−α1 )
l=0
ckf kq β,k .
BDp,q
Finally we deduce
Z
0
+∞ q dt
≤ ckf kq β,k .
t−θ Kp,k (t, f ; β0 , s; β1 , s)
BDp,q
t
Here when q = +∞ we make the usual modification. Our theorem is proved.
Theorem 2. Let β > 0 and 1 ≤ p, q ≤ +∞. Then for all φ ∈ A, we have
β,k
φ,β,k
BDp,q
⊂ Cp,q
.
Proof . For φ ∈ A and 1 ≤ u ≤ 2, we get supp Fk (φ2−j u ) ⊂ Aj , ∀ j ∈ Z. Then we can write
Fk (φ2−j u ) = Fk (φ2−j u )(Fk (ϕj−1 ) + Fk (ϕj ) + Fk (ϕj+1 )), which gives φ2−j u = φ2−j u ∗k (ϕj−1 +
ϕj + ϕj+1 ), ∀ j ∈ Z.
β,k
for 1 ≤ q < +∞, we can assert that
Let f ∈ BDp,q
Z
0
+∞ kf ∗k φt kp,k
tβ
q
dt X
≤
t
2
Z
j∈Z
1
kf ∗k φ2−j u kp,k
(2−j u)β
q
du
.
u
Using Hölder’s inequality for j ∈ Z we get
j+1
X
kf ∗k φ2−j u kqp,k ≤ kφkq1,k 3q−1
kϕs ∗k f kqp,k ,
s=j−1
hence we obtain
Z +∞ X Z 2 kϕs ∗k f kp,k q du
kf ∗k φt kp,k q dt
q
≤
ckφk
1,k
t
u
tβ
(2−s u)β
0
1
s∈Z
≤c
X
(2sβ kϕs ∗k f kp,k )q < +∞.
s∈Z
Here when q = +∞, we make the usual modification. This completes the proof.
Theorem 3. Let β > 0 and 1 ≤ p, q ≤ +∞, then for φ ∈ A such that
P
j∈Z
all 1 ≤ u ≤ 2 and x ∈ Rd we have
φ,β,k
β,k
.
Cp,q
= BDp,q
Fk (φ2−j u )(x) = 1, for
Besov-Type Spaces on Rd and Integrability for the Dunkl Transform
11
φ,β,k
β,k
Proof
. By Theorem 2 we have only to show that Cp,q
⊂ BDp,q
. Let φ ∈ A such that
P
Fk (φ2−j u )(x) = 1, for x ∈ Rd and 1 ≤ u ≤ 2. Then we can assert that
j∈Z
Fk (ϕj ) = Fk (ϕj )(Fk (φ2−j−1 u ) + Fk (φ2−j u ) + Fk (φ2−j+1 u )),
this implies that ϕj = ϕj ∗k (φ2−j−1 u + φ2−j u + φ2−j+1 u ), ∀ j ∈ Z.
φ,β,k
Let f ∈ Cp,q
for 1 ≤ q < +∞, using Hölder’s inequality for j ∈ Z and the property ii) of
the sequence of functions (ϕj )j∈Z we get
j+1
X
kϕj ∗k f kqp,k ≤ kϕj kq1,k 3q−1
kf ∗k φ2−s u kqp,k ≤ c
s=j−1
j+1
X
kf ∗k φ2−s u kqp,k .
s=j−1
Integrating with respect to u over (1, 2) we obtain
j+1 Z 2 X
q
kf ∗k φ2−s u kp,k q du
jβ
2 kϕj ∗k f kp,k ≤ c
.
u
(2−s u)β
1
s=j−1
Hence
X
jβ
2 kϕj ∗k f kp,k
q
Z
≤c
0
j∈Z
+∞ kf ∗k φt kp,k
tβ
q
dt
< +∞.
t
When q = +∞ we make the usual modification. Our result is proved.
φ,β,k
Remark 3. We observe that the spaces Cp,q
are independent of the specific selection of φ ∈ A
satisfying the assumption of Theorem 3.
Remark 4. In the case d = 1, W = Z2 , α > − 21 and
df
2α + 1 f (x) − f (−x)
T1 (f )(x) =
(x) +
,
dx
x
2
f ∈ E(R),
we can characterize the Besov–Dunkl spaces by differences using the Dunkl translation operators.
Observe that
X
φ∈A:
Fk (φ2−j u )(x) = 1, ∀ 1 ≤ u ≤ 2, ∀ x ∈ R ⊂ H,
j∈Z
n
o
R +∞
|x|2α+1
where H = φ ∈ S∗ (R) : 0 φ(x)dµα (x) = 0 with dµα (x) = 2α+1
dx and S∗ (R) the
Γ(α+1)
space of even Schwartz functions on R. Then we can assert from Theorem 3 and [4, Theorem 3.6] that for 1 < p < +∞, 1 ≤ q ≤ +∞ and 0 < β < 1 we have
e p,q ,
BDβ,k = BDp,q ⊂ BD
p,q
p,q
BDα,β
α,β
α,β
is the subspace of functions f ∈ Lp (µα ) satisfying
1
+∞ wp,α (f )(x) q dx q
< +∞
if q < +∞
x
xβ
0
where
Z
and
wp,α (f )(x)
< +∞
xβ
x∈(0,+∞)
sup
if q = +∞,
e p,q we replace wp,α (f )(x) by
with wp,α (f )(x) = kτx (f ) + τ−x (f ) − 2f kp,α . For the space BD
α,β
w
ep,α (f )(x) = kτx (f ) − f kp,α .
Note that when f is an even function in Lp (µα ) we have τx (f )(y) = τ−x (f )(−y) for x, y ∈ R,
then we get
e p,q .
f ∈ BDp,q ⇐⇒ f ∈ BD
α,β
α,β
12
4
C. Abdelkefi, J.-Ph. Anker, F. Sassi and M. Sifi
Integrability of the Dunkl transform of function
in Besov–Dunkl space
In this section, we establish further results concerning integrability of the Dunkl transform of
function f on Rd , when f is in a suitable Besov–Dunkl space.
In the following lemma we prove the Hardy–Littlewood inequality for the Dunkl transform.
Lemma 1. If f ∈ Lpk (Rd ) for some 1 < p ≤ 2, then
Z
d
kxk2(γ+ 2 )(p−2) |Fk (f )(x)|p wk (x)dx ≤ ckf kpp,k .
(11)
Rd
Proof . To see (11) we will make use of the Marcinkiewicz interpolation theorem (see [20]). For
f ∈ Lpk (Rd ) with 1 ≤ p ≤ 2 consider the operator
d
L(f )(x) = kxk2(γ+ 2 ) Fk (f )(x),
x ∈ Rd .
For every f ∈ L2 (Rd ) we have from Plancherel’s theorem
!1/2
Z
w
(x)
k
|L(f )(x)|2
= kFk (f )k2,k = c−1
d dx
k kf k2,k ,
Rd
kxk4(γ+ 2 )
(12)
Moreover, according to (1) and (2) we get for λ ∈]0, +∞) and f ∈ L1k (Rd )
Z
Z
wk (x)
wk (x)
dx
1
d dx ≤
d
4(γ+ d2 )
{x∈Rd :L(f )(x)|>λ} kxk4(γ+ 2 )
kxk>( kf kλ ) 2(γ+ 2 ) kxk
1,k
Z +∞
kf k1,k
r2γ+d−1
1
≤c
.
d dr ≤ c
4(γ+
)
2(γ+ d
)
λ
λ
2
2
( kf k )
r
1,k
(13)
Hence by (12) and (13) L is an operator of strong-type (2, 2) and weak-type (1, 1) between the
k (x)
spaces (Rd , wk (x)dx) and (Rd , w4(γ+
d ) dx).
kxk
2
Using Marcinkiewicz interpolation’s theorem we can assert that L is an operator of strongtype (p, p) for 1 < p ≤ 2, between the spaces under consideration. We conclude that
Z
Z
d
p wk (x)
|L(f )(x)|
kxk2(γ+ 2 )(p−2) |Fk (f )(x)|p wk (x)dx ≤ ckf kpp,k ,
d dx =
Rd
Rd
kxk4(γ+ 2 )
thus we obtain the result.
Now in order to prove the following two theorems we denote by Ae the subset of functions φ
in A such that
∃ c > 0;
|Fk (φ)(x)| ≥ ckxk2
if 1 ≤ kxk ≤ 2.
(14)
e
Let β > 0 and 1 ≤ p, q ≤ +∞. From Theorem 2 we have obviously for all φ ∈ A,
β,k
φ,β,k
BDp,q
⊂ Cp,q
.
(15)
0
For 1 ≤ p ≤ 2 we take p0 such that p1 + p10 = 1. We recall that Fk (f ) ∈ Lpk (Rd ) for all f ∈ Lpk (Rd ).
2(γ+ d
2 ) ,k
p
Theorem 4. Let 1 < p ≤ 2. If f ∈ BDp,1
Fk (f ) ∈ L1k (Rd ).
, then
Besov-Type Spaces on Rd and Integrability for the Dunkl Transform
13
2(γ+ d
2 ) ,k
p
Proof . Let f ∈ BDp,1
with 1 < p ≤ 2. For φ ∈ Ae we can write from (6) and for
t ∈ (0, +∞), Fk (f ∗k φt )(x) = Fk (f )(x)Fk (φt )(x), a.e. x ∈ Rd . From Lemma 1 we obtain
Z
d
|Fk (f )(x)|p |Fk (φt )(x)|p kxk2(γ+ 2 )(p−2) wk (x)dx ≤ ckf ∗k φt kpp,k .
Rd
By (14) we get |Fk (φt )(x)| ≥ cktxk2 if 1 ≤ ktxk ≤ 2, then we can assert that
t
!1/p
Z
2
p
1
≤kxk≤ 2t
t
2(γ+ d2 )(p−2)+2p
|Fk (f )(x)| kxk
≤ ckf ∗k φt kp,k .
wk (x)dx
(16)
Then by Hölder’s inequality, (1) and (16) we have
Z
1
≤kxk≤ 2t
t
kf ∗k φt kp,k
kxk|Fk (f )(x)|wk (x)dx ≤ c
t2
≤c
2
t
Z
r
1
( 1−p
)[ 2(γ+ d2 )(p−2)+p ]
! 10
p
r2γ+d−1 dr
1
t
kf ∗k φt kp,k 1
.
2(γ+ d
t
2)
t p
Integrating with respect to t over R+ , applying Fubini’s theorem and using (15), it yields
Z +∞
Z
kf ∗k φt kp,k dt
< +∞ .
|Fk (f )(x)| wk (x)dx ≤ c
2(γ+ d
t
2)
Rd
0
p
t
This complete the proof of the theorem.
β,k
Theorem 5. Let β > 0 and 1 ≤ p ≤ 2. If f ∈ BDp,∞
, then
i) for p 6= 1 and 0 < β ≤
2(γ+ d2 )
p
Fk (f ) ∈ Lsk (Rd )
ii) for β >
2(γ+ d2 )
p
we have
2(γ + d2 )p
provided that
βp + 2(γ + d2 )(p − 1)
< s ≤ p0 ;
we have Fk (f ) ∈ L1k (Rd ).
β,k
e
Proof . Let f ∈ BDp,∞
with 1 ≤ p ≤ 2 and φ ∈ A.
i) Suppose that p 6= 1 and 0 < β ≤
2(γ+ d2 )
.
p
Using (3) and (6) we have for t ∈ (0, +∞)
kFk (f ∗k φt )kp0 ,k = kFk (f )Fk (φt )kp0 ,k ≤ ckf ∗k φt kp,k .
Then from (14) and (15) we obtain
t
!1/p0
Z
2
p0
1
≤kxk≤ 2t
t
Let s ∈
|Fk (f )(x)| kxk
2(γ+ d2 )p
, p0 .
βp+2(γ+ d2 )(p−1)
2p0
(17)
0
Since Fk (f ) ∈ Lpk (Rd ), we have only to show the case s 6= p0 . For
t ≥ 1 put Gt the set of x in Rd such that
we have
Z
|Fk (f )(x)|s kxks wk (x)dx
Gt
≤ ckf ∗k φt kp,k ≤ ctβ .
wk (x)dx
1
t1/s
≤ kxk ≤
2
.
t1/s
By Hölder’s inequality, (1) and (17)
14
C. Abdelkefi, J.-Ph. Anker, F. Sassi and M. Sifi
Z
≤
p0
2p0
|Fk (f )(x)| kxk
s/p0 Z
wk (x)dx
1− s0
p
wk (x)dx
Gt
Gt
≤ ctβ−2
kxk
−p0 s
p0 −s
Z
2
t1/s
!1− s0
0
r
s
2γ+d−1− pp0 −s
p
−1+β−2(γ+ d2 )( 1s − p10 )
≤ ct
dr
.
1
t1/s
Integrating with respect to t over (0, 1) and applying Fubini’s theorem, it yields
Z 1
Z
−1+β−2(γ+ d2 )( 1s − p10 )
dt < +∞.
t
|Fk (f )(x)|s wk (x)dx ≤ c
kxk≥1
0
0
Since Lpk (B(0, 1), wk (x)dx) ⊂ Lsk (B(0, 1), wk (x)dx) we deduce that Fk (f ) is in Lsk (Rd ).
2(γ+ d )
2
ii) Assume now β >
. For p 6= 1 by proceeding in the same manner as in the proof of
p
i) with s = 1, we obtain the desired result.
For p = 1, using (3) and (6), we have for t ∈ (0, +∞)
kFk (f ∗k φt )k∞,k = kFk (f )Fk (φt )k∞,k ≤ ckf ∗k φt k1,k .
Then from (14) and (15) we obtain
t2 kht Fk (f )k∞,k ≤ ckf ∗k φt k1,k ≤ ctβ ,
(18)
where ht (x) = χt (x)kxk2 with χt is the characteristic function of the set {x ∈ Rd :
By Hölder’s inequality, (1) and (18) we have
Z
Z
|Fk (f )(x)|kxkwk (x)dx ≤ kht Fk (f )k∞,k
|χt (x)|kxk−1 wk (x)dx
1
≤kxk≤ 2t
t
1
t
≤ kxk ≤ 2t }.
Rd
≤ ctβ−2
2
t
Z
d
r2γ+d−2 dr ≤ ctβ−2(γ+ 2 )−1 .
1
t
Integrating with respect to t over (0, 1) and applying Fubini’s theorem we obtain
Z
Z 1
d
|Fk (f )(x)|wk (x)dx ≤ c
tβ−2(γ+ 2 )−1 dt < +∞.
kxk≥1
0
d
1
1
Since L∞
k (B(0, 1), wk (x)dx) ⊂ Lk (B(0, 1), wk (x)dx) we deduce that Fk (f ) is in Lk (R ). Our
theorem is proved.
Remark 5.
1. For β > 0, 1 ≤ p ≤ 2 et 1 ≤ q ≤ +∞, using Remark 2, the results of Theorem 5 are true
β,k
for BDp,q
.
β,k
BDp,∞
2(γ+ d
2 ) ,k
p
2. From Remark 2 we get
⊂ BDp,1
for β >
the result of Theorem 5, ii) with 1 < p ≤ 2.
2(γ+ d2 )
.
p
Using Theorem 4 we recover
3. Let β > 2(γ + d2 ), by Theorem 5, ii) we can assert that
β,k
i) BD1,∞
is an example of space where we can apply the inversion formula;
β,k
d
2
d
ii) BD1,∞
is contained in L1k (Rd ) ∩ L∞
k (R ) and hence is a subspace of Lk (R ). By (4)
β,k
we obtain for f ∈ BD1,∞
Z
τy (f )(x) = ck
Fk (f )(ξ)Ek (ix, ξ)Ek (−iy, ξ)wk (ξ)dξ,
Rd
x, y ∈ Rd .
Besov-Type Spaces on Rd and Integrability for the Dunkl Transform
15
Acknowledgements
The authors thank the referees for their remarks and suggestions. Work supported by the
DGRST research project 04/UR/15-02 and the program CMCU 07G 1501.
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