Besov-Type Spaces on
R
dand 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 estab-lish 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 operatorsTi, 1≤i≤d, onRd, associated with a positive
root systemR+and a non negative multiplicity functionk, 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 transformFk. K. Trim`eche 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 functionτx(f) is unknown
in general. However, such formula is known whenf is a radial function and theLp-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 denoted by BDβ,kp,q in this paper, is the subspace of functionsf ∈Lpk(Rd) satisfying
kfkBDβ,k p,q =
X
j∈Z
(2jβkϕj∗kfkp,k)q 1
q
<+∞ if q <+∞
⋆This paper is a contribution to the Special Issue on Dunkl Operators and Related Topics. The full collection
and
kfkBDβ,k
p,∞ = sup j∈Z
2jβkϕj∗kfkp,k<+∞ if q= +∞,
where (ϕj)j∈Z is a sequence of functions inS(Rd)rad such that
(i) suppFk(ϕj)⊂Aj =
x∈Rd; 2j−1 ≤ kxk ≤2j+1 forj∈Z; (ii) sup
j∈Z
kϕjk1,k<+∞;
(iii) X
j∈Z
Fk(ϕj)(x) = 1, forx∈Rd\{0}.
S(Rd)rad being the subspace of functions in the Schwartz spaceS(Rd) which are radial.
Put A=φ∈ S(Rd)rad : suppFk(φ)⊂ {x∈Rd; 1≤ kxk ≤2} . Given φ ∈ A, we denote by Cp,qφ,β,k the subspace of functions f ∈Lpk(Rd) satisfying
Z +∞
0
kf ∗kφtkp,k tβ
q dt
t 1
q
<+∞ if q <+∞
and
sup
t∈(0,+∞)
kf ∗kφtkp,k
tβ <+∞ if q = +∞,
where φt(x) = 1
t2(γ+d2 )
φ(xt), for allt∈(0,+∞) andx∈Rd.
In this paper we show forβ >0 the inclusion of the Schwartz space inBDβ,kp,q for 1≤p, q≤+∞
and the density when 1 ≤p, q < +∞. We prove an interpolation formula for the Besov–Dunkl spaces by the real method. We compare these spaces with Cp,qφ,β,k which extend to the Dunkl
operators on Rd some results obtained in [4, 5, 21]. Finally we establish further results of integrability of Fk(f) when f is in a suitable Besov–Dunkl space BDp,qβ,k 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 M´oricz in [13] for a classical Fourier transform on R and for q = 1,+∞ by Betancor and Rodr´ıguez-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 onR and in radial case on Rd. The argument used in [1,2] to establish such integrability is theLp-boundedness of the Dunkl translation operators, making it difficult to extend the results onRd. 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 Section2 we collect some basic definitions and results about harmonic analysis associated with Dunkl operators. In Section 3 we show the inclusion and the density of the Schwartz space in BDβ,kp,q, we prove an interpolation formula
for the Besov–Dunkl spaces by the real method and we compare these spaces with Cp,qφ,β,k. In
Section4we establish our results concerning integrability of the Dunkl transform of function in the Besov–Dunkl spaces.
Along this paper we denote by h·,·i the usual Euclidean inner product in Rd as well as its extension toCd×Cd, we write forx∈Rd,kxk=phx, xiand we represent byca suitable positive constant which is not necessarily the same in each occurrence. Furthermore we denote by
• E(Rd) the space of infinitely differentiable functions onRd;
• S(Rd) the Schwartz space of functions in E(Rd) which are rapidly decreasing as well as their derivatives;
2
Preliminaries
LetW be a finite reflection group onRd, associated with a root systemR and R+ the positive subsystem ofR (see [8,10,11,12,18,19]). We denote byka nonnegative multiplicity function defined on R with the property that kis W-invariant. We associate withk the index
γ =γ(R) = X
ξ∈R+
k(ξ)≥0,
and the weight function wk defined by
wk(x) = Y
ξ∈R+
|hξ, xi|2k(ξ), x∈Rd.
Further we introduce the Mehta-type constantck by
ck= Z
Rd
e−kxk 2
2 wk(x)dx −1
.
For every 1 ≤ p ≤ +∞ we denote by Lpk(Rd) the space Lp(Rd, wk(x)dx), Lp
k(Rd)rad the
subspace of those f ∈Lpk(Rd) that are radial and we usek · kp,k as a shorthand fork · k
Lpk(Rd).
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 allx∈Rd. The functionF is integrable with respect to the measure r2γ+d−1dr on [0,+∞) and we have
Z
Sd−1
wk(x)dσ(x) =
c−k1
2γ+d2−1Γ(γ+d 2)
,
where Sd−1 is the unit sphere onRd with the normalized surface measuredσ and
Z
Rd
f(x)wk(x)dx= Z +∞
0
Z
Sd−1
wk(ry)dσ(y)
F(r)rd−1dr
= c
−1
k
2γ+d2−1Γ(γ+d 2)
Z +∞
0
F(r)r2γ+d−1dr. (1)
Introduced by C.F. Dunkl in [9] the Dunkl operators Tj, 1≤ j ≤d, on Rd associated with
the reflection groupW and the multiplicity functionkare the first-order differential- difference operators given by
Tjf(x) = ∂f
∂xj
(x) + X
α∈R+
k(α)αj f(x)−f(σα(x))
hα, xi , f ∈ E(R
d), x∈Rd,
where αj =hα, eji, (e1, e2, . . . , ed) being the canonical basis of Rd.
The Dunkl kernelEk on Rd×Rdhas been introduced by C.F. Dunkl in [10]. For y∈Rd the
functionx7→Ek(x, y) can be viewed as the solution onRd of the following initial problem
Tju(x, y) =yju(x, y), 1≤j≤d, u(0, y) = 1.
This kernel has a unique holomorphic extension to Cd×Cd. M. R¨osler has proved in [17] the following integral representation for the Dunkl kernel
Ek(x, z) = Z
Rd
where µkx is a probability measure on Rd with support in the closed ball B(0,kxk) of center 0 and radiuskxk.
We have for all λ ∈ C and z, z′ ∈ Cd Ek(z, z′) = Ek(z′, z), Ek(λz, z′) = Ek(z, λz′) and for
x, y∈Rd |Ek(x, iy)| ≤1 (see [10,17,18,19,22]).
The Dunkl transformFk which was introduced by C.F. Dunkl in [11] (see also [8]) is defined
forf ∈ D(Rd) by
Fk(f)(x) =ck Z
Rd
f(y)Ek(−ix, y)wk(y)dy, x∈Rd.
According to [8,11,18] we have the following results:
i) The Dunkl transform of a functionf ∈L1k(Rd) has the following basic property
kFk(f)k∞,k ≤ kfk1,k. (2)
ii) The Schwartz spaceS(Rd) is invariant under the Dunkl transformFk.
iii) When both f and Fk(f) are inL1k(Rd), we have the inversion formula
f(x) =
Z
Rd
Fk(f)(y)Ek(ix, y)wk(y)dy, x∈Rd.
iv) (Plancherel’s theorem) The Dunkl transform onS(Rd) extends uniquely to an isometric isomorphism on L2k(Rd).
By (2), Plancherel’s theorem and the Marcinkiewicz interpolation theorem (see [20]) we get forf ∈Lpk(Rd) with 1≤p≤2 andp′ such that 1
p +p1′ = 1,
kFk(f)kp′,k ≤ckfkp,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. Trim`eche has introduced in [23] the Dunkl translation operators τx, x ∈ Rd, on E(Rd). For f ∈ S(Rd) and x, y∈Rdwe have
Fk(τx(f))(y) =Ek(ix, y)Fk(f)(y)
and
τx(f)(y) =ck Z
Rd
Fk(f)(ξ)Ek(ix, ξ)Ek(iy, ξ)wk(ξ)dξ. (4)
Notice that for all x, y∈Rd τx(f)(y) =τy(f)(x), and for fixed x∈Rd
τx is a continuous linear mapping fromE(Rd) intoE(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
The Dunkl convolution product∗kof two functionsf andginL2k(Rd) (see [19,23]) is given by
(f ∗kg)(x) = Z
Rd
τx(f)(−y)g(y)wk(y)dy, x∈Rd.
The Dunkl convolution product is commutative and forf, g∈ D(Rd) we have
Fk(f ∗k g) =Fk(f)Fk(g). (6)
It was shown in [19, Theorem 4.1] that wheng is a bounded function inL1k(Rd)rad, then
(f∗kg)(x) = Z
Rd
f(y)τx(g)(−y)wk(y)dy, x∈Rd,
initially defined on the intersection of L1k(Rd) andL2k(Rd) extends to all Lpk(Rd), 1≤p ≤+∞
as a bounded operator. In particular,
kf ∗kgkp,k≤ kfkp,kkgk1,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), for x∈Rd. (8)
3
Interpolation and characterization
for the Besov–Dunkl spaces
In this section we establish the inclusion and the density of S(Rd) in BDp,qβ,k and we prove an interpolation formula for the Besov–Dunkl spaces by the real method. Finally we compare the spacesBDp,qβ,k withCp,qφ,β,k. Before, we start with some useful results.
We shall denote byΦthe set of all sequences of functions (ϕj)j∈Z in S(Rd)rad satisfying
(i) suppFk(ϕj)⊂Aj ={x∈Rd; 2j−1 ≤ kxk ≤2j+1}forj∈Z;
(ii) sup
j∈Z
kϕjk1,k<+∞;
(iii) P
j∈Z
Fk(ϕj)(x) = 1, for x∈Rd\{0}.
Proposition 1. Let β >0 and 1≤p, q≤+∞, thenBDp,qβ,k is independent of the choice of the
sequence in Φ.
Proof . Fix (φj)j∈Z, (ψj)j∈Z in Φ and f ∈ BDp,qβ,k for q < +∞. Using the properties (i) for
(φj)j∈Z and (i) and (iii) for (ψj)j∈Z, we have forj∈Zφj =φj∗k(ψj−1+ψj+ψj+1). Then by
the property (ii) for (φj)j∈Z, (7) and H¨older’s inequality for j∈Zwe obtain
kf ∗kφjkqp,k≤c3q−1 j+1
X
s=j−1
kψs∗kfkqp,k.
Thus summing over j with weights 2jβq we get
X
j∈Z
2jβkφj ∗kfkp,k q
≤cX j∈Z
2jβkψj∗kfkp,k q
.
Hence by symmetry we get the result of our proposition. When q = +∞ we make the usual
Remark 1. Letβ >0, 1≤p, q≤+∞, we denote by ¨BDp,qβ,kthe subspace of functionsf ∈Lpk(Rd)
satisfying
X
j∈N
(2jβkϕj ∗kfkp,k)q 1
q
<+∞ if q <+∞
and
sup
j∈N
2jβkϕj∗kfkp,k<+∞ if q= +∞,
where (ϕj)j∈N is a sequence of functions inS(Rd)rad such that
i) suppFk(ϕ0) ⊂ {x∈Rd;kxk ≤2} and suppFk(ϕj) ⊂Aj ={x∈Rd; 2j−1 ≤ kxk ≤2j+1}
forj∈N\{0};
ii) sup
j∈N
kϕjk1,k<+∞;
iii) P
j∈N
Fk(ϕj)(x) = 1, forx∈Rd\{0}.
As the Besov–Dunkl spaces these spaces are also independent of the choice of the sequence (ϕj)j∈Nsatisfying the previous properties.
Proposition 2. For β >0 and1≤p, q≤+∞ we have ¨
BDβ,kp,q =BDp,qβ,k.
Proof . Since both spaces ¨BDp,qβ,k and BDp,qβ,k 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 functionφ∈ S(Rd)rad such that
• suppFk(φ)⊂ {x∈Rd; 12 ≤ kxk ≤2}; • Fk(φ)(x)>0 for 12 <kxk<2;
• P j∈Z
Fk(φ2−j)(x) = 1, x∈Rd\{0}.
If we consider the sequences (ψj)j∈Z and (ϕj)j∈N in S(Rd)rad defined respectively for BDp,qβ,k
and ¨BDβ,kp,q by ψj = φ2−j ∀j ∈Z and ϕ0 = P
j∈Z−
φ2−j, ϕj = φ2−j ∀j ∈ N∗, we can assert that
¨
BDp,qβ,k=BDp,qβ,k.
Remark 2. By Proposition 2and [21, Proposition 2] we have the following embeddings.
1. Let 1≤q1≤q2 ≤+∞and β >0. Then
BDβ,kp,q1 ⊂ BDp,qβ,k2 if 1≤p≤+∞.
2. Let 1≤q1, q2 ≤+∞,β >0 and ε >0. Then
BDβp,q+1ε,k ⊂ BDβ,kp,q2 if 1≤p≤+∞.
Proposition 3. For β >0 and1≤p, q≤+∞ we have
S(Rd)⊂ BDβ,kp,q.
Proof . In order to prove the inclusion, we may restrict ourself toq = +∞. This follows from the fact that BDβ,kp,∞⊂ BDβ
′,k
p,q forβ > β′ >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 numberL such that
the properties (i) and (iii) for (ϕj)j∈Z we can assert that fN =
N P
j=−N
ϕj∗kfN+1 which gives
fNθn= N P
j=−N
ϕj∗kfN+1θn. Using the properties (i), (ii), (iii) for (ϕj)j∈Z and (7) we obtain
kfN −fNθnkq BDp,qβ,k
≤c NX+1
j=−N−1
2jβqkfN+1−fN+1θnkqp,k.
The dominated convergence theorem implies that
kfN+1−fN+1θnkp,k →0 as n→+∞.
Hence we deduce that
kfN −fNθnkBDβ,k
p,q →0 as n→+∞, (10)
Combining (9) and (10) we conclude thatS(Rd) is dense inBDp,qβ,k. This completes the proof of
Proposition3.
For 0< θ <1,β0, β1 >0, 1≤p, q0, q1 ≤+∞and 1≤q ≤+∞, the real interpolation Besov– Dunkl space denoted by (BDβ0,k
p,q0,BD β1,k
p,q1)θ,q is the subspace of functions f ∈ BD β0,k p,q0 +BD
β1,k p,q1 satisfying
Z +∞
0
t−θKp,k(t, f;β0, q0;β1, q1)
q dt t
1
q
<+∞ if q <+∞,
and
sup
t∈(0,+∞)
t−θKp,k(t, f;β0, q0;β1, q1)<+∞ if q= +∞,
with Kp,k is the PeetreK-functional given by
Kp,k(t, f;β0, q0;β1, q1) = inf
n
kf0kBDβ0,k
p,q0 +tkf1kBDp,qβ1,k1
o ,
where the infinimum is taken over all representations of f of the form
f =f0+f1, f0 ∈ BDβp,q0,k0, f1 ∈ BD β1,k p,q1.
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
BDβ0,k p,q0,BD
β1,k p,q1
θ,q =BD β,k p,q.
Proof . We start with the proof of the inclusion (BDβ0,k
p,∞,BDβp,1∞,k)θ,q ⊂ BDβ,kp,q.We may assume
that β0 > β1. Let q < +∞, for f = f0 +f1 with f0 ∈ BDβp,0∞,k and f1 ∈ BDβp,1∞,k we get by Proposition2
+∞ X
l=0
2qlβkϕl∗kfkqp,k≤c
+∞ X
l=0
2−θql(β0−β1)2lβ0kϕ
l∗kf0kp,k+ 2l(β0−β1)2lβ1kϕl∗kf1kp,k q
≤c
+∞ X
l=0
2−θql(β0−β1)kf
0kBDβ0,k p,∞ + 2
l(β0−β1)kf
1kBDβ1,k p,∞
Then we deduce that
which proves the result. When q= +∞, we make the usual modification. For 1≤s≤q0, q1 Remark 2gives
Then in order to complete the proof of the theorem we have to show only that
+c
Here when q= +∞ we make the usual modification. Our theorem is proved.
Theorem 2. Let β >0 and 1≤p, q≤+∞. Then for allφ∈ A, we have
Using H¨older’s inequality forj ∈Zwe get
kf ∗kφ2−jukqp,k≤ kφkq1,k3q−1
Here when q= +∞, we make the usual modification. This completes the proof.
Theorem 3. Letβ >0 and 1≤p, q≤+∞, then forφ∈ Asuch that P
j∈Z
Fk(φ2−ju)(x) = 1, for
all 1≤u≤2 and x∈Rd we have
Proof .P By Theorem 2 we have only to show that Cp,qφ,β,k ⊂ BDβ,kp,q. Let φ ∈ A such that
Integrating with respect to u over (1,2) we obtain
2jβkϕj∗kfkp,k
When q= +∞ we make the usual modification. Our result is proved.
Remark 3. We observe that the spacesCp,qφ,β,k are independent of the specific selection ofφ∈ A
satisfying the assumption of Theorem 3.
Remark 4. In the cased= 1, W =Z2,α >−12 and
4
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 functionf onRd, 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
Rd
kxk2(γ+d2)(p−2)|Fk(f)(x)|pwk(x)dx≤ckfkp
p,k. (11)
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
L(f)(x) =kxk2(γ+d2)Fk(f)(x), x∈Rd.
For every f ∈L2(Rd) we have from Plancherel’s theorem
Z
Rd
|L(f)(x)|2 wk(x) kxk4(γ+d2)
dx !1/2
=kFk(f)k2,k=c−k1kfk2,k, (12)
Moreover, according to (1) and (2) we get forλ∈]0,+∞) and f ∈L1k(Rd)
Z
{x∈Rd:L(f)(x)|>λ}
wk(x) kxk4(γ+d2)
dx≤ Z
kxk>(kfkλ 1,k)
1 2(γ+d2 )
wk(x) kxk4(γ+d2)
dx
≤c Z +∞
( λ
kfk1,k)
1 2(γ+d2 )
r2γ+d−1 r4(γ+d2)
dr≤ckfk1,k
λ . (13)
Hence by (12) and (13) L is an operator of strong-type (2,2) and weak-type (1,1) between the spaces (Rd, wk(x)dx) and (Rd, wk(x)
kxk4(γ+d2 ) dx).
Using Marcinkiewicz interpolation’s theorem we can assert thatL is an operator of strong-type (p, p) for 1< p≤2, between the spaces under consideration. We conclude that
Z
Rd
|L(f)(x)|p wk(x) kxk4(γ+d2)
dx=
Z
Rd
kxk2(γ+d2)(p−2)|Fk(f)(x)|pwk(x)dx≤ckfkp p,k,
thus we obtain the result.
Now in order to prove the following two theorems we denote byAethe subset of functions φ
inA such that
∃c >0; |Fk(φ)(x)| ≥ckxk2 if 1≤ kxk ≤2. (14)
Let β >0 and 1≤p, q≤+∞. From Theorem 2 we have obviously for allφ∈Ae,
BDβ,kp,q ⊂ Cp,qφ,β,k. (15)
For 1≤p≤2 we takep′ such that 1p+p1′ = 1. We recall thatFk(f)∈Lp ′
k(Rd) for allf ∈L p k(Rd).
Theorem 4. Let 1< p≤2. Iff ∈ BD 2(γ+d2 )
p ,k
p,1 , then
Proof . Let f ∈ BD
Integrating with respect to t overR+, applying Fubini’s theorem and using (15), it yields
Z
This complete the proof of the theorem.
≤ Z
Gt
|Fk(f)(x)|p ′
kxk2p′wk(x)dx
s/p′Z
Gt
kxk−p ′s
p′−sw
k(x)dx 1−ps′
≤ctβ−2 Z 2
t1/s
1
t1/s
r2γ+d−1− p ′s
p′−sdr
!1−ps′
≤ct−1+β−2(γ+d2)( 1
s−
1
p′).
Integrating with respect to t over (0,1) and applying Fubini’s theorem, it yields
Z
kxk≥1
|Fk(f)(x)|swk(x)dx≤c Z 1
0
t−1+β−2(γ+d2)(1s−p1′)dt <+∞.
Since Lpk′(B(0,1), wk(x)dx)⊂Lsk(B(0,1), wk(x)dx) we deduce that Fk(f) is inLsk(Rd). ii) Assume now β > 2(γ+
d
2)
p . For p6= 1 by proceeding in the same manner as in the proof of i) withs= 1, we obtain the desired result.
Forp= 1, using (3) and (6), we have fort∈(0,+∞)
kFk(f ∗kφt)k∞,k =kFk(f)Fk(φt)k∞,k≤ckf∗kφtk1,k.
Then from (14) and (15) we obtain
t2khtFk(f)k∞,k ≤ckf ∗kφtk1,k ≤ctβ, (18)
whereht(x) =χt(x)kxk2 withχtis the characteristic function of the set{x∈Rd:1t ≤ kxk ≤ 2t}.
By H¨older’s inequality, (1) and (18) we have
Z
1
t≤kxk≤
2
t
|Fk(f)(x)|kxkwk(x)dx≤ khtFk(f)k∞,k Z
Rd
|χt(x)|kxk−1wk(x)dx
≤ctβ−2 Z 2
t
1
t
r2γ+d−2dr≤ctβ−2(γ+d2)−1.
Integrating with respect to t over (0,1) and applying Fubini’s theorem we obtain
Z
kxk≥1
|Fk(f)(x)|wk(x)dx≤c Z 1
0
tβ−2(γ+d2)−1dt <+∞.
Since L∞k (B(0,1), wk(x)dx) ⊂ L1k(B(0,1), wk(x)dx) we deduce that Fk(f) is in L1k(Rd). Our
theorem is proved.
Remark 5.
1. For β >0, 1≤p≤2 et 1≤q ≤+∞, using Remark 2, the results of Theorem5 are true forBDp,qβ,k.
2. From Remark 2we get BDp,β,k∞⊂ BD 2(γ+d2 )
p ,k
p,1 forβ > 2(γ+d
2)
p . Using Theorem4 we recover
the result of Theorem 5,ii) with 1< p≤2.
3. Letβ >2(γ+d2), by Theorem 5,ii) we can assert that
i) BDβ,k1,∞ is an example of space where we can apply the inversion formula;
ii) BDβ,k1,∞ is contained in L1k(Rd)∩L∞
k (Rd) and hence is a subspace of L2k(Rd). By (4)
we obtain forf ∈ BDβ,k1,∞
τy(f)(x) =ck Z
Rd
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.
References
[1] Abdelkefi C., Sifi M., On the uniform convergence of partial Dunkl integrals in Besov–Dunkl spaces,Fract. Calc. Appl. Anal.9(2006), 43–56.
[2] Abdelkefi C., Sifi M., Further results of integrability for the Dunkl transform, Commun. Math. Anal. 2 (2007), 29–36.
[3] Abdelkefi C., Dunkl transform on Besov spaces and Herz spaces,Commun. Math. Anal.2(2007), 35–41. [4] Abdelkefi C., Sifi M., Characterization of Besov spaces for the Dunkl operator on the real line,JIPAM. J.
Inequal. Pure Appl. Math.8(2007), no. 3, Article 73, 11 pages.
[5] Bergh J., L¨ofstr¨om J., Interpolation spaces. An introduction, Grundlehren der Mathematischen Wis-senschaften, no. 223, Springer-Verlag, Berlin – New York, 1976.
[6] Besov O.V., On a family of function spaces in connection with embeddings and extentions, Trudy. Mat. Inst. Steklov.60(1961), 42–81 (in Russian).
[7] Betancor J.J., Rodr´ıguez-Mesa L., Lipschitz–Hankel spaces and partial Hankel integrals,Integral Transform. Spec. Funct.7(1998), 1–12.
[8] de Jeu M.F.E., The Dunkl transform,Invent. Math.113(1993), 147–162.
[9] Dunkl C.F., Differential-difference operators associated to reflection groups,Trans. Amer. Math. Soc.311 (1989), 167–183.
[10] Dunkl C.F., Integral kernels with reflection group invariance,Canad. J. Math.43, (1991), 1213–1227. [11] Dunkl C.F., Hankel transforms associated to finite reflection groups, in Proc. of Special Session on
Hyper-geometric Functions on Domains of Positivity, Jack Polynomials and Applications (Tampa, 1991),Contemp. Math.138(1992), 123–138.
[12] Dunkl C.F., Xu Y., Orthogonal polynomials of several variables, Encyclopedia of Mathematics and its Applications, Vol. 81, Cambridge University Press, Cambridge, 2001.
[13] Giang D.V., M´orciz F., On the uniform and the absolute convergence of Dirichlet integrals of functions in Besov-spaces,Acta Sci. Math. (Szeged)59(1994), 257–265.
[14] Kamoun L., Besov-type spaces for the Dunkl operator on the real line,J. Comput. Appl. Math.199(2007), 56–67.
[15] Pelczynski A., Wojciechowski M., Molecular decompositions and embbeding theorems for vector-valued Sobolev spaces with gradient norm,Studia Math.107(1993), 61–100.
[16] R¨osler M., Generalized Hermite polynomials and the heat equation for Dunkl operators,Comm. Math. Phys.
192(1998), 519–542,q-alg/9703006.
[17] R¨osler M., Positivity of Dunkl’s intertwining operator,Duke Math. J.98(1999), 445–463,q-alg/9710029. [18] R¨osler M., Voit M., Markov processes with Dunkl operators,Adv. in Appl. Math.21(1998), 575–643. [19] Thangavelyu S., Xu Y., Convolution operator and maximal function for Dunkl transform,J. Anal. Math.
97(2005), 25–55.
[20] Titchmarsh E.C., Introduction to the theory of Fourier integrals, Clarendon Press, Oxford, 1937.
[21] Triebel H., Theory of function spaces,Monographs in Mathematics, Vol. 78, Birkh¨auser, Verlag, Basel, 1983.
[22] Trim`eche K., The Dunkl intertwining operator on spaces of functions and distributions and integral repre-sentation of its dual,Integral Transforms Spec. Funct.12(2001), 349–374.
