Besov-Type Spaces on Rd and Integrability for the Dunkl Transform
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...
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irk-123456789-1492532019-02-20T01:26:56Z Besov-Type Spaces on Rd and Integrability for the Dunkl Transform Abdelkefi, C. Sassi, F. Sifi, M. Anker, Jean-Philippe 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. 2009 Article Besov-Type Spaces on Rd and Integrability for the Dunkl Transform / C. Abdelkefi, Jean-Philippe Anker, F. Sassi, M. Sifi // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 23 назв. — англ. 1815-0659 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. http://dspace.nbuv.gov.ua/handle/123456789/149253 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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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. |
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Abdelkefi, C. Sassi, F. Sifi, M. Anker, Jean-Philippe |
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Abdelkefi, C. Sassi, F. Sifi, M. Anker, Jean-Philippe Besov-Type Spaces on Rd and Integrability for the Dunkl Transform Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Abdelkefi, C. Sassi, F. Sifi, M. Anker, Jean-Philippe |
| author_sort |
Abdelkefi, C. |
| title |
Besov-Type Spaces on Rd and Integrability for the Dunkl Transform |
| title_short |
Besov-Type Spaces on Rd and Integrability for the Dunkl Transform |
| title_full |
Besov-Type Spaces on Rd and Integrability for the Dunkl Transform |
| title_fullStr |
Besov-Type Spaces on Rd and Integrability for the Dunkl Transform |
| title_full_unstemmed |
Besov-Type Spaces on Rd and Integrability for the Dunkl Transform |
| title_sort |
besov-type spaces on rd and integrability for the dunkl transform |
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Інститут математики НАН України |
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2009 |
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http://dspace.nbuv.gov.ua/handle/123456789/149253 |
| citation_txt |
Besov-Type Spaces on Rd and Integrability for the Dunkl Transform / C. Abdelkefi, Jean-Philippe Anker, F. Sassi, M. Sifi // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 23 назв. — англ. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
| work_keys_str_mv |
AT abdelkefic besovtypespacesonrdandintegrabilityforthedunkltransform AT sassif besovtypespacesonrdandintegrabilityforthedunkltransform AT sifim besovtypespacesonrdandintegrabilityforthedunkltransform AT ankerjeanphilippe besovtypespacesonrdandintegrabilityforthedunkltransform |
| first_indexed |
2025-07-12T21:42:01Z |
| last_indexed |
2025-07-12T21:42:01Z |
| _version_ |
1837478998194520064 |
| fulltext |
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 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 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
denoted by BDβ,k
p,q in this paper, is the subspace of functions f ∈ Lp
k(R
d) satisfying
‖f‖BDβ,k
p,q
=
(∑
j∈Z
(2jβ‖ϕj ∗k f‖p,k)q
) 1
q
< +∞ if q < +∞
?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
mailto:chokri.abdelkefi@ipeit.rnu.tn
mailto:feriel.sassi@ipeit.rnu.tn
mailto:Jean-Philippe.Anker@univ-orleans.fr
mailto:mohamed.sifi@fst.rnu.tn
http://dx.doi.org/10.3842/SIGMA.2009.019
http://www.emis.de/journals/SIGMA/Dunkl_operators.html
2 C. Abdelkefi, J.-Ph. Anker, F. Sassi and M. Sifi
and
‖f‖BDβ,k
p,∞
= sup
j∈Z
2jβ‖ϕj ∗k f‖p,k < +∞ if q = +∞,
where (ϕj)j∈Z is a sequence of functions in S(Rd)rad such that
(i) suppFk(ϕj) ⊂ Aj =
{
x ∈ Rd ; 2j−1 ≤ ‖x‖ ≤ 2j+1
}
for j ∈ Z;
(ii) sup
j∈Z
‖ϕj‖1,k < +∞;
(iii)
∑
j∈Z
Fk(ϕj)(x) = 1, for x ∈ Rd\{0}.
S(Rd)rad being the subspace of functions in the Schwartz space S(Rd) which are radial.
Put A =
{
φ ∈ S(Rd)rad : suppFk(φ) ⊂ {x ∈ Rd; 1 ≤ ‖x‖ ≤ 2}
}
. Given φ ∈ A, we denote
by Cφ,β,k
p,q the subspace of functions f ∈ Lp
k(R
d) satisfying(∫ +∞
0
(
‖f ∗k φt‖p,k
tβ
)q dt
t
) 1
q
< +∞ if q < +∞
and
sup
t∈(0,+∞)
‖f ∗k φt‖p,k
tβ
< +∞ if q = +∞,
where φt(x) = 1
t2(γ+ d
2 )
φ(x
t ), for all t ∈ (0,+∞) and x ∈ Rd.
In this paper we show for β > 0 the inclusion of the Schwartz space in BDβ,k
p,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 Cφ,β,k
p,q 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 BDβ,k
p,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 Rodŕıguez-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
inclusion and the density of the Schwartz space in BDβ,k
p,q , we prove an interpolation formula
for the Besov–Dunkl spaces by the real method and we compare these spaces with Cφ,β,k
p,q . In
Section 4 we establish our results concerning integrability of the Dunkl transform of function in
the Besov–Dunkl spaces.
Along this paper we denote by 〈·, ·〉 the usual Euclidean inner product in Rd as well as its
extension to Cd×Cd, we write for x ∈ Rd, ‖x‖ =
√
〈x, x〉 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 3
2 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
γ = γ(R) =
∑
ξ∈R+
k(ξ) ≥ 0,
and the weight function wk defined by
wk(x) =
∏
ξ∈R+
|〈ξ, x〉|2k(ξ), x ∈ Rd.
Further we introduce the Mehta-type constant ck by
ck =
(∫
Rd
e−
‖x‖2
2 wk(x)dx
)−1
.
For every 1 ≤ p ≤ +∞ we denote by Lp
k(R
d) the space Lp(Rd, wk(x)dx), L
p
k(R
d)rad the
subspace of those f ∈ Lp
k(R
d) that are radial and we use ‖ · ‖p,k as a shorthand for ‖ · ‖Lp
k(Rd).
By using the homogeneity of wk it is shown in [18] that for f ∈ L1
k(Rd)rad there exists
a function F on [0,+∞) such that f(x) = F (‖x‖), for all x ∈ Rd. The function F is integrable
with respect to the measure r2γ+d−1dr on [0,+∞) and we have∫
Sd−1
wk(x)dσ(x) =
c−1
k
2γ+ d
2
−1Γ(γ + d
2)
,
where Sd−1 is the unit sphere on Rd with the normalized surface measure dσ and∫
Rd
f(x)wk(x)dx =
∫ +∞
0
(∫
Sd−1
wk(ry)dσ(y)
)
F (r)rd−1dr
=
c−1
k
2γ+ d
2
−1Γ(γ + d
2)
∫ +∞
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 group W and the multiplicity function k are the first-order differential- difference
operators given by
Tjf(x) =
∂f
∂xj
(x) +
∑
α∈R+
k(α)αj
f(x)− f(σα(x))
〈α, x〉
, f ∈ E(Rd), x ∈ Rd,
where αj = 〈α, ej〉, (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
Tju(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
Ek(x, z) =
∫
Rd
e〈y,z〉dµk
x(y), x ∈ Rd, z ∈ Cd,
4 C. Abdelkefi, J.-Ph. Anker, F. Sassi and M. Sifi
where µk
x is a probability measure on Rd with support in the closed ball B(0, ‖x‖) of center 0
and radius ‖x‖.
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 transform Fk which was introduced by C.F. Dunkl in [11] (see also [8]) is defined
for f ∈ D(Rd) by
Fk(f)(x) = ck
∫
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 function f ∈ L1
k(Rd) has the following basic property
‖Fk(f)‖∞,k ≤ ‖f‖1,k. (2)
ii) The Schwartz space S(Rd) is invariant under the Dunkl transform Fk .
iii) When both f and Fk(f) are in L1
k(Rd), we have the inversion formula
f(x) =
∫
Rd
Fk(f)(y)Ek(ix, y)wk(y)dy, x ∈ Rd.
iv) (Plancherel’s theorem) The Dunkl transform on S(Rd) extends uniquely to an isometric
isomorphism on L2
k(Rd).
By (2), Plancherel’s theorem and the Marcinkiewicz interpolation theorem (see [20]) we get
for f ∈ Lp
k(R
d) with 1 ≤ p ≤ 2 and p′ such that 1
p + 1
p′ = 1,
‖Fk(f)‖p′,k ≤ c‖f‖p,k. (3)
The Dunkl transform of a function in L1
k(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
τx(f)(y) = ck
∫
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 from E(Rd) into E(Rd). (5)
As an operator on L2
k(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 Lp
k(R
d)rad, 1 ≤ p ≤ 2 and for f ∈ Lp
k(R
d)rad
we have
‖τx(f)‖p,k ≤ ‖f‖p,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 L2
k(Rd) (see [19, 23]) is given by
(f ∗k g)(x) =
∫
Rd
τx(f)(−y)g(y)wk(y)dy, x ∈ 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 L1
k(Rd)rad, then
(f ∗k g)(x) =
∫
Rd
f(y)τx(g)(−y)wk(y)dy, x ∈ Rd,
initially defined on the intersection of L1
k(Rd) and L2
k(Rd) extends to all Lp
k(R
d), 1 ≤ p ≤ +∞
as a bounded operator. In particular,
‖f ∗k g‖p,k ≤ ‖f‖p,k‖g‖1,k. (7)
The Dunkl Laplacian ∆k is defined by ∆k :=
d∑
i=1
T 2
i . From [16] we have for each λ > 0
λI −∆k maps S(Rd) onto itself and
Fk((λI −∆k)f)(x) =
(
λ+ ‖x‖2
)
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 BDβ,k
p,q and we prove an
interpolation formula for the Besov–Dunkl spaces by the real method. Finally we compare the
spaces BDβ,k
p,q with Cφ,β,k
p,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) suppFk(ϕj) ⊂ Aj = {x ∈ Rd; 2j−1 ≤ ‖x‖ ≤ 2j+1} for j ∈ Z;
(ii) sup
j∈Z
‖ϕj‖1,k < +∞;
(iii)
∑
j∈Z
Fk(ϕj)(x) = 1, for x ∈ Rd\{0}.
Proposition 1. Let β > 0 and 1 ≤ p, q ≤ +∞, then BDβ,k
p,q is independent of the choice of the
sequence in Φ.
Proof. Fix (φj)j∈Z, (ψj)j∈Z in Φ and f ∈ BDβ,k
p,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
‖f ∗k φj‖q
p,k ≤ c 3q−1
j+1∑
s=j−1
‖ψs ∗k f‖q
p,k.
Thus summing over j with weights 2jβq we get∑
j∈Z
(
2jβ‖φj ∗k f‖p,k
)q ≤ c
∑
j∈Z
(
2jβ‖ψj ∗k f‖p,k
)q
.
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
Remark 1. Let β > 0, 1 ≤ p, q ≤ +∞, we denote by B̈Dβ,k
p,q the subspace of functions f ∈ Lp
k(R
d)
satisfying(∑
j∈N
(2jβ‖ϕj ∗k f‖p,k)q
) 1
q
< +∞ if q < +∞
and
sup
j∈N
2jβ‖ϕj ∗k f‖p,k < +∞ if q = +∞,
where (ϕj)j∈N is a sequence of functions in S(Rd)rad such that
i) suppFk(ϕ0) ⊂ {x ∈ Rd; ‖x‖ ≤ 2} and suppFk(ϕj) ⊂ Aj = {x ∈ Rd; 2j−1 ≤ ‖x‖ ≤ 2j+1}
for j ∈ N\{0};
ii) sup
j∈N
‖ϕj‖1,k < +∞;
iii)
∑
j∈N
Fk(ϕj)(x) = 1, for x ∈ Rd\{0}.
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
B̈Dβ,k
p,q = BDβ,k
p,q .
Proof. Since both spaces B̈Dβ,k
p,q and BDβ,k
p,q are in Lp
k(R
d) 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
• suppFk(φ) ⊂ {x ∈ Rd; 1
2 ≤ ‖x‖ ≤ 2};
• Fk(φ)(x) > 0 for 1
2 < ‖x‖ < 2;
•
∑
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 BDβ,k
p,q
and B̈Dβ,k
p,q by ψj = φ2−j ∀ j ∈ Z and ϕ0 =
∑
j∈Z−
φ2−j , ϕj = φ2−j ∀ j ∈ N∗, we can assert that
B̈Dβ,k
p,q = BDβ,k
p,q . �
Remark 2. By Proposition 2 and [21, Proposition 2] we have the following embeddings.
1. Let 1 ≤ q1 ≤ q2 ≤ +∞ and β > 0. Then
BDβ,k
p,q1
⊂ BDβ,k
p,q2
if 1 ≤ p ≤ +∞.
2. Let 1 ≤ q1, q2 ≤ +∞, β > 0 and ε > 0. Then
BDβ+ε,k
p,q1
⊂ BDβ,k
p,q2
if 1 ≤ p ≤ +∞.
Proposition 3. For β > 0 and 1 ≤ p, q ≤ +∞ we have
S(Rd) ⊂ BDβ,k
p,q .
If 1 ≤ p, q < +∞, then S(Rd) is dense in BDβ,k
p,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
fact that BDβ,k
p,∞ ⊂ BDβ′,k
p,q for β > 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 number L such that
sup
j∈N
2jβ‖ϕj ∗k f‖p,k ≤ sup
j∈N
2jβ‖(1 + ‖x‖2)L(ϕj ∗k f)‖∞,k.
Since ϕj ∈ S(Rd)rad, then for x ∈ Rd F−1
k (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
j∈N∗
2jβ‖ϕj ∗k f‖p,k ≤ sup
j∈N∗
2jβ‖Fk[(I −∆k)L(Fk(ϕj)F−1
k (f))]‖∞,k
≤ sup
j∈N∗
2jβ‖(I −∆k)L(Fk(ϕj)F−1
k (f))‖1,k
≤ sup
j∈N∗
cj 2jβ sup
x∈Aj
|(I −∆k)L(Fk(ϕj)F−1
k (f))(x)|,
where cj =
∫
Aj
wk(x)dx. Hence there exists a sufficiently large natural number M such that
cj2
jβ
(1+22(j−1))M ≤ 1, ∀ j ∈ N∗ and we get
sup
j∈N∗
2jβ‖ϕj ∗k f‖p,k ≤ sup
j∈N∗
sup
x∈Aj
|
(
1 + ‖x‖2
)M (I −∆k)L(Fk(ϕj)F−1
k (f))(x)|.
Since (I −∆k)L is linear and continuous from S(Rd) into itself, we deduce that
sup
j∈N∗
2jβ‖ϕj ∗k f‖p,k ≤ c sup
j∈N∗
sup
x∈Rd
|Fk(ϕj)(x)| sup
x∈Rd
|(1 + ‖x‖2)MF−1
k (f)(x)|,
which gives by the property ii) of (ϕj)j∈N
sup
j∈N∗
2jβ‖ϕj ∗k f‖p,k ≤ c sup
j∈N∗
‖ϕj‖1,k sup
x∈Rd
|(1 + ‖x‖2)MF−1
k (f)(x)| < +∞.
By Proposition 2 we conclude that S(Rd) ⊂ BDβ,k
p,q .
Let us now prove the density of S(Rd) in BDβ,k
p,q for p, q < +∞. Assume f ∈ BDβ,k
p,q and
(ϕj)j∈Z ∈ Φ, then we put for N ∈ N\{0}, fN =
N∑
s=−N
ϕs ∗k f. It’s clear that fN ∈ BDβ,k
p,q . We
have
∑
j∈Z
2jβq‖ϕj ∗k (fN − f)‖q
p,k =
∑
j∈Z
2jβq
∥∥∥∥∥
( N∑
s=−N
ϕs
)
∗k ϕj ∗k f − ϕj ∗k f
∥∥∥∥∥
q
p,k
.
Using the properties (i) and (iii) for (ϕj)j∈Z we get
‖f − fN‖q
BDβ,k
p,q
≤ c
∑
|j|≥N
2jβq‖ϕj ∗k f‖q
p,k.
Since f ∈ BDβ,k
p,q , then we deduce that
lim
N→+∞
‖f − fN‖BDβ,k
p,q
= 0. (9)
Next we take a function θ ∈ D(Rd) such that θ(0) = 1. For n ∈ N\{0} we put θn(x) = θ(n−1x),
x ∈ Rd. From (5) we have for N ∈ N\{0} fN ∈ E(Rd), then fNθn ∈ 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∑
j=−N
ϕj ∗k fN+1 which gives
fNθn =
N∑
j=−N
ϕj ∗k fN+1θn. Using the properties (i), (ii), (iii) for (ϕj)j∈Z and (7) we obtain
‖fN − fNθn‖q
BDβ,k
p,q
≤ c
N+1∑
j=−N−1
2jβq‖fN+1 − fN+1θn‖q
p,k.
The dominated convergence theorem implies that
‖fN+1 − fN+1θn‖p,k → 0 as n→ +∞.
Hence we deduce that
‖fN − fNθn‖BDβ,k
p,q
→ 0 as n→ +∞, (10)
Combining (9) and (10) we conclude that S(Rd) is dense in BDβ,k
p,q . This completes the proof of
Proposition 3. �
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 ∈ BDβ0,k
p,q0 + BDβ1,k
p,q1
satisfying(∫ +∞
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 Peetre K-functional given by
Kp,k(t, f ;β0, q0;β1, q1) = inf
{
‖f0‖BDβ0,k
p,q0
+ t‖f1‖BDβ1,k
p,q1
}
,
where the infinimum is taken over all representations of f of the form
f = f0 + f1, f0 ∈ BDβ0,k
p,q0
, f1 ∈ BDβ1,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
,BDβ1,k
p,q1
)
θ,q
= BDβ,k
p,q .
Proof. We start with the proof of the inclusion (BDβ0,k
p,∞ ,BDβ1,k
p,∞ )θ,q ⊂ BDβ,k
p,q . We may assume
that β0 > β1. Let q < +∞, for f = f0 + f1 with f0 ∈ BDβ0,k
p,∞ and f1 ∈ BDβ1,k
p,∞ we get by
Proposition 2
+∞∑
l=0
2qlβ‖ϕl ∗k f‖q
p,k ≤ c
+∞∑
l=0
2−θql(β0−β1)
(
2lβ0‖ϕl ∗k f0‖p,k + 2l(β0−β1)2lβ1‖ϕl ∗k f1‖p,k
)q
≤ c
+∞∑
l=0
2−θql(β0−β1)
(
‖f0‖BDβ0,k
p,∞
+ 2l(β0−β1)‖f1‖BDβ1,k
p,∞
)q
.
Besov-Type Spaces on Rd and Integrability for the Dunkl Transform 9
Then we deduce that
+∞∑
l=0
2qlβ‖ϕl ∗k f‖q
p,k ≤ c
+∞∑
l=0
2−θql(β0−β1)
(
Kp,k(2l(β0−β1), f ;β0,∞;β1,∞)
)q
≤ c
∫ +∞
0
(
t−θKp,k(t, f ;β0,∞;β1,∞)
)q dt
t
< +∞,
which proves the result. When q = +∞, we make the usual modification.
For 1 ≤ s ≤ q0, q1 Remark 2 gives
(BDβ0,k
p,s ,BDβ1,k
p,s )θ,q ⊂ (BDβ0,k
p,q0
,BDβ1,k
p,q1
)θ,q ⊂ (BDβ0,k
p,∞ ,BDβ1,k
p,∞ )θ,q ⊂ BDβ,k
p,q .
Then in order to complete the proof of the theorem we have to show only that
BDβ,k
p,q ⊂ (BDβ0,k
p,s ,BDβ1,k
p,s )θ,q for 1 ≤ s ≤ q.
Suppose that β0 > β1 again. Let q < +∞, we have∫ +∞
0
(
t−θKp,k(t, f ;β0, s;β1, s)
)q dt
t
=
∫ 1
0
+
∫ +∞
1
= I1 + I2.
Since β > β1, by Remark 2 we get
Kp,k(t, f ;β0, s;β1, s) ≤ ct‖f‖BDβ1,k
p,s
≤ ct‖f‖BDβ,k
p,q
,
hence we deduce
I1 ≤ c‖f‖q
BDβ,k
p,q
.
To estimate I2 take f0 =
l∑
j=0
ϕj ∗k f and f1 =
+∞∑
j=l+1
ϕj ∗k f . Using the properties of the sequence
(ϕj)j∈N we obtain
‖f0‖s
BDβ0,k
p,s
≤ c
l+1∑
j=0
2jβ0s‖ϕj ∗k f‖s
p,k and ‖f1‖s
BDβ1,k
p,s
≤ c
+∞∑
j=l
2jβ1s‖ϕj ∗k f‖s
p,k.
Hence we can write
I2 ≤ c
+∞∑
l=0
2−θql(β0−β1)
(
Kp,k(2l(β0−β1), f ;β0, s;β1, s)
)q
≤ c
+∞∑
l=0
2−θql(β0−β1)
( l+1∑
j=0
2jβ0s‖ϕj ∗k f‖s
p,k
)1/s
+ 2l(β0−β1)
( +∞∑
j=l
2jβ1s‖ϕj ∗k f‖s
p,k
)1/s
q
≤ c
+∞∑
l=0
2qlβ
l+1∑
j=0
2(j−l)β0s‖ϕj ∗k f‖s
p,k +
+∞∑
j=l
2(j−l)β1s‖ϕj ∗k f‖s
p,k
q/s
.
For s = q it is easy to see that I2 ≤ c‖f‖q
BDβ,k
p,q
.
For s < q we take u > s such that s
q + s
u = 1 and β1 < α1 < β < α0 < β0, then by Hölder’s
inequality we have
I2 ≤ c
+∞∑
l=0
2ql(β−β0)
( l+1∑
j=0
2(β0−α0)ju
)q/u( l+1∑
j=0
2α0jq‖ϕj ∗k f‖q
p,k
)
10 C. Abdelkefi, J.-Ph. Anker, F. Sassi and M. Sifi
+ c
+∞∑
l=0
2ql(β−β1)
( +∞∑
j=l
2(β1−α1)ju
)q/u( +∞∑
j=l
2α1jq‖ϕj ∗k f‖q
p,k
)
≤ c
+∞∑
l=0
2ql(β−α0)
l+1∑
j=0
2α0jq‖ϕj ∗k f‖q
p,k + c
+∞∑
l=0
2ql(β−α1)
+∞∑
j=l
2α1jq‖ϕj ∗k f‖q
p,k
≤ c
+∞∑
j=0
2α0jq‖ϕj ∗k f‖q
p,k
+∞∑
l=j−1
2ql(β−α0) + c
+∞∑
j=0
2α1jq‖ϕj ∗k f‖q
p,k
j∑
l=0
2ql(β−α1)
≤ c‖f‖q
BDβ,k
p,q
.
Finally we deduce∫ +∞
0
(
t−θKp,k(t, f ;β0, s;β1, s)
)q dt
t
≤ c‖f‖q
BDβ,k
p,q
.
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
BDβ,k
p,q ⊂ Cφ,β,k
p,q .
Proof. For φ ∈ A and 1 ≤ u ≤ 2, we get suppFk(φ2−ju) ⊂ Aj , ∀ j ∈ Z. Then we can write
Fk(φ2−ju) = Fk(φ2−ju)(Fk(ϕj−1) + Fk(ϕj) + Fk(ϕj+1)), which gives φ2−ju = φ2−ju ∗k (ϕj−1 +
ϕj + ϕj+1), ∀ j ∈ Z.
Let f ∈ BDβ,k
p,q for 1 ≤ q < +∞, we can assert that∫ +∞
0
(
‖f ∗k φt‖p,k
tβ
)q dt
t
≤
∑
j∈Z
∫ 2
1
(
‖f ∗k φ2−ju‖p,k
(2−ju)β
)q du
u
.
Using Hölder’s inequality for j ∈ Z we get
‖f ∗k φ2−ju‖
q
p,k ≤ ‖φ‖q
1,k3
q−1
j+1∑
s=j−1
‖ϕs ∗k f‖q
p,k,
hence we obtain∫ +∞
0
(
‖f ∗k φt‖p,k
tβ
)q dt
t
≤ c‖φ‖q
1,k
∑
s∈Z
∫ 2
1
(
‖ϕs ∗k f‖p,k
(2−su)β
)q du
u
≤ c
∑
s∈Z
(2sβ‖ϕs ∗k f‖p,k)q < +∞.
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
∑
j∈Z
Fk(φ2−ju)(x) = 1, for
all 1 ≤ u ≤ 2 and x ∈ Rd we have
Cφ,β,k
p,q = BDβ,k
p,q .
Besov-Type Spaces on Rd and Integrability for the Dunkl Transform 11
Proof. By Theorem 2 we have only to show that Cφ,β,k
p,q ⊂ BDβ,k
p,q . Let φ ∈ A such that∑
j∈Z
Fk(φ2−ju)(x) = 1, for x ∈ Rd and 1 ≤ u ≤ 2. Then we can assert that
Fk(ϕj) = Fk(ϕj)(Fk(φ2−j−1u) + Fk(φ2−ju) + Fk(φ2−j+1u)),
this implies that ϕj = ϕj ∗k (φ2−j−1u + φ2−ju + φ2−j+1u), ∀ j ∈ Z.
Let f ∈ Cφ,β,k
p,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
‖ϕj ∗k f‖q
p,k ≤ ‖ϕj‖q
1,k3
q−1
j+1∑
s=j−1
‖f ∗k φ2−su‖
q
p,k ≤ c
j+1∑
s=j−1
‖f ∗k φ2−su‖
q
p,k.
Integrating with respect to u over (1, 2) we obtain
(
2jβ‖ϕj ∗k f‖p,k
)q ≤ c
j+1∑
s=j−1
∫ 2
1
(
‖f ∗k φ2−su‖p,k
(2−su)β
)q du
u
.
Hence∑
j∈Z
(
2jβ‖ϕj ∗k f‖p,k
)q ≤ c
∫ +∞
0
(
‖f ∗k φt‖p,k
tβ
)q dt
t
< +∞.
When q = +∞ we make the usual modification. Our result is proved. �
Remark 3. We observe that the spaces Cφ,β,k
p,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 > −1
2 and
T1(f)(x) =
df
dx
(x) +
2α+ 1
x
[
f(x)− f(−x)
2
]
, f ∈ E(R),
we can characterize the Besov–Dunkl spaces by differences using the Dunkl translation operators.
Observe that{
φ ∈ A :
∑
j∈Z
Fk(φ2−ju)(x) = 1, ∀ 1 ≤ u ≤ 2, ∀x ∈ R
}
⊂ H,
where H =
{
φ ∈ S∗(R) :
∫ +∞
0 φ(x)dµα(x) = 0
}
with dµα(x) = |x|2α+1
2α+1Γ(α+1)
dx and S∗(R) the
space of even Schwartz functions on R. Then we can assert from Theorem 3 and [4, Theo-
rem 3.6] that for 1 < p < +∞, 1 ≤ q ≤ +∞ and 0 < β < 1 we have
BDβ,k
p,q = BDp,q
α,β ⊂ B̃Dp,q
α,β,
where BDp,q
α,β is the subspace of functions f ∈ Lp(µα) satisfying(∫ +∞
0
(
wp,α(f)(x)
xβ
)q dx
x
) 1
q
< +∞ if q < +∞
and
sup
x∈(0,+∞)
wp,α(f)(x)
xβ
< +∞ if q = +∞,
with wp,α(f)(x) = ‖τx(f) + τ−x(f) − 2f‖p,α. For the space B̃Dp,q
α,β we replace wp,α(f)(x) by
w̃p,α(f)(x) = ‖τx(f)− f‖p,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
f ∈ BDp,q
α,β ⇐⇒ f ∈ B̃Dp,q
α,β.
12 C. Abdelkefi, J.-Ph. Anker, F. Sassi and M. Sifi
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
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 ∈ Lp
k(R
d) for some 1 < p ≤ 2, then∫
Rd
‖x‖2(γ+ d
2
)(p−2)|Fk(f)(x)|pwk(x)dx ≤ c‖f‖p
p,k. (11)
Proof. To see (11) we will make use of the Marcinkiewicz interpolation theorem (see [20]). For
f ∈ Lp
k(R
d) with 1 ≤ p ≤ 2 consider the operator
L(f)(x) = ‖x‖2(γ+ d
2
)Fk(f)(x), x ∈ Rd.
For every f ∈ L2(Rd) we have from Plancherel’s theorem(∫
Rd
|L(f)(x)|2 wk(x)
‖x‖4(γ+ d
2
)
dx
)1/2
= ‖Fk(f)‖2,k = c−1
k ‖f‖2,k, (12)
Moreover, according to (1) and (2) we get for λ ∈]0,+∞) and f ∈ L1
k(Rd)∫
{x∈Rd:L(f)(x)|>λ}
wk(x)
‖x‖4(γ+ d
2
)
dx ≤
∫
‖x‖>( λ
‖f‖1,k
)
1
2(γ+ d
2 )
wk(x)
‖x‖4(γ+ d
2
)
dx
≤ c
∫ +∞
( λ
‖f‖1,k
)
1
2(γ+ d
2 )
r2γ+d−1
r4(γ+ d
2
)
dr ≤ c
‖f‖1,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)
‖x‖4(γ+ d
2 )
dx).
Using Marcinkiewicz interpolation’s theorem we can assert that L is an operator of strong-
type (p, p) for 1 < p ≤ 2, between the spaces under consideration. We conclude that∫
Rd
|L(f)(x)|p wk(x)
‖x‖4(γ+ d
2 )
dx =
∫
Rd
‖x‖2(γ+ d
2
)(p−2)|Fk(f)(x)|pwk(x)dx ≤ c‖f‖p
p,k,
thus we obtain the result. �
Now in order to prove the following two theorems we denote by à the subset of functions φ
in A such that
∃ c > 0; |Fk(φ)(x)| ≥ c‖x‖2 if 1 ≤ ‖x‖ ≤ 2. (14)
Let β > 0 and 1 ≤ p, q ≤ +∞. From Theorem 2 we have obviously for all φ ∈ Ã,
BDβ,k
p,q ⊂ Cφ,β,k
p,q . (15)
For 1 ≤ p ≤ 2 we take p′ such that 1
p + 1
p′ = 1. We recall that Fk(f) ∈ Lp′
k (Rd) for all f ∈ Lp
k(R
d).
Theorem 4. Let 1 < p ≤ 2. If f ∈ BD
2(γ+ d
2 )
p
,k
p,1 , then
Fk(f) ∈ L1
k(Rd).
Besov-Type Spaces on Rd and Integrability for the Dunkl Transform 13
Proof. Let f ∈ BD
2(γ+ d
2 )
p
,k
p,1 with 1 < p ≤ 2. For φ ∈ Ã 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∫
Rd
|Fk(f)(x)|p|Fk(φt)(x)|p‖x‖2(γ+ d
2
)(p−2)wk(x)dx ≤ c‖f ∗k φt‖p
p,k.
By (14) we get |Fk(φt)(x)| ≥ c‖tx‖2 if 1 ≤ ‖tx‖ ≤ 2, then we can assert that
t2
(∫
1
t
≤‖x‖≤ 2
t
|Fk(f)(x)|p‖x‖2(γ+ d
2
)(p−2)+2pwk(x)dx
)1/p
≤ c‖f ∗k φt‖p,k. (16)
Then by Hölder’s inequality, (1) and (16) we have
∫
1
t
≤‖x‖≤ 2
t
‖x‖|Fk(f)(x)|wk(x)dx ≤ c
‖f ∗k φt‖p,k
t2
(∫ 2
t
1
t
r
( 1
1−p
)[ 2(γ+ d
2
)(p−2)+p ]
r2γ+d−1dr
) 1
p′
≤ c
‖f ∗k φt‖p,k
t
2(γ+ d
2 )
p
1
t
.
Integrating with respect to t over R+, applying Fubini’s theorem and using (15), it yields∫
Rd
|Fk(f)(x)|wk(x)dx ≤ c
∫ +∞
0
‖f ∗k φt‖p,k
t
2(γ+ d
2 )
p
dt
t
< +∞ .
This complete the proof of the theorem. �
Theorem 5. Let β > 0 and 1 ≤ p ≤ 2. If f ∈ BDβ,k
p,∞, then
i) for p 6= 1 and 0 < β ≤ 2(γ+ d
2
)
p we have
Fk(f) ∈ Ls
k(Rd) provided that
2(γ + d
2)p
βp+ 2(γ + d
2)(p− 1)
< s ≤ p′;
ii) for β > 2(γ+ d
2
)
p we have Fk(f) ∈ L1
k(Rd).
Proof. Let f ∈ BDβ,k
p,∞ with 1 ≤ p ≤ 2 and φ ∈ Ã.
i) Suppose that p 6= 1 and 0 < β ≤ 2(γ+ d
2
)
p . Using (3) and (6) we have for t ∈ (0,+∞)
‖Fk(f ∗k φt)‖p′,k = ‖Fk(f)Fk(φt)‖p′,k ≤ c‖f ∗k φt‖p,k.
Then from (14) and (15) we obtain
t2
(∫
1
t
≤‖x‖≤ 2
t
|Fk(f)(x)|p′‖x‖2p′wk(x)dx
)1/p′
≤ c‖f ∗k φt‖p,k ≤ ctβ. (17)
Let s ∈
] 2(γ+ d
2
)p
βp+2(γ+ d
2
)(p−1)
, p′
]
. Since Fk(f) ∈ Lp′
k (Rd), we have only to show the case s 6= p′. For
t ≥ 1 put Gt the set of x in Rd such that 1
t1/s ≤ ‖x‖ ≤ 2
t1/s . By Hölder’s inequality, (1) and (17)
we have∫
Gt
|Fk(f)(x)|s ‖x‖swk(x)dx
14 C. Abdelkefi, J.-Ph. Anker, F. Sassi and M. Sifi
≤
(∫
Gt
|Fk(f)(x)|p′‖x‖2p′wk(x)dx
)s/p′ (∫
Gt
‖x‖
−p′s
p′−swk(x)dx
)1− s
p′
≤ ctβ−2
(∫ 2
t1/s
1
t1/s
r
2γ+d−1− p′s
p′−sdr
)1− s
p′
≤ ct
−1+β−2(γ+ d
2
)( 1
s
− 1
p′ ).
Integrating with respect to t over (0, 1) and applying Fubini’s theorem, it yields∫
‖x‖≥1
|Fk(f)(x)|swk(x)dx ≤ c
∫ 1
0
t
−1+β−2(γ+ d
2
)( 1
s
− 1
p′ )dt < +∞.
Since Lp′
k (B(0, 1), wk(x)dx) ⊂ Ls
k(B(0, 1), wk(x)dx) we deduce that Fk(f) is in Ls
k(Rd).
ii) Assume now β >
2(γ+ d
2
)
p . For p 6= 1 by proceeding in the same manner as in the proof of
i) with s = 1, we obtain the desired result.
For p = 1, using (3) and (6), we have for t ∈ (0,+∞)
‖Fk(f ∗k φt)‖∞,k = ‖Fk(f)Fk(φt)‖∞,k ≤ c‖f ∗k φt‖1,k.
Then from (14) and (15) we obtain
t2‖htFk(f)‖∞,k ≤ c‖f ∗k φt‖1,k ≤ ctβ, (18)
where ht(x) = χt(x)‖x‖2 with χt is the characteristic function of the set {x ∈ Rd : 1
t ≤ ‖x‖ ≤ 2
t }.
By Hölder’s inequality, (1) and (18) we have∫
1
t
≤‖x‖≤ 2
t
|Fk(f)(x)|‖x‖wk(x)dx ≤ ‖htFk(f)‖∞,k
∫
Rd
|χt(x)|‖x‖−1wk(x)dx
≤ ctβ−2
∫ 2
t
1
t
r2γ+d−2 dr ≤ ctβ−2(γ+ d
2
)−1.
Integrating with respect to t over (0, 1) and applying Fubini’s theorem we obtain∫
‖x‖≥1
|Fk(f)(x)|wk(x)dx ≤ c
∫ 1
0
tβ−2(γ+ d
2
)−1dt < +∞.
Since L∞k (B(0, 1), wk(x)dx) ⊂ L1
k(B(0, 1), wk(x)dx) we deduce that Fk(f) is in L1
k(Rd). 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
for BDβ,k
p,q .
2. From Remark 2 we get BDβ,k
p,∞ ⊂ BD
2(γ+ d
2 )
p
,k
p,1 for β > 2(γ+ d
2
)
p . Using Theorem 4 we recover
the result of Theorem 5, ii) with 1 < p ≤ 2.
3. Let β > 2(γ + d
2), by Theorem 5, ii) we can assert that
i) BDβ,k
1,∞ is an example of space where we can apply the inversion formula;
ii) BDβ,k
1,∞ is contained in L1
k(Rd) ∩ L∞k (Rd) and hence is a subspace of L2
k(Rd). By (4)
we obtain for f ∈ BDβ,k
1,∞
τy(f)(x) = ck
∫
Rd
Fk(f)(ξ)Ek(ix, ξ)Ek(−iy, ξ)wk(ξ)dξ, 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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http://arxiv.org/abs/q-alg/9703006
http://arxiv.org/abs/q-alg/9710029
1 Introduction
2 Preliminaries
3 Interpolation and characterization for the Besov-Dunkl spaces
4 Integrability of the Dunkl transform of function in Besov-Dunkl space
References
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