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CUBO A Mathematical Journal
Vol.17, No¯ 02, (01–14). June 2015

Reproducing inversion formulas for the Dunkl-Wigner
transforms

Fethi Soltani
1

Department of Mathematics, Faculty of Science,

Jazan University,

P.O.Box 277, Jazan 45142, Saudi Arabia,

fethisoltani10@yahoo.com

ABSTRACT

We define and study the Fourier-Wigner transform associated with the Dunkl operators,

and we prove for this transform a reproducing inversion formulas and a Plancherel

formula. Next, we introduce and study the extremal functions associated to the Dunkl-

Wigner transform.

RESUMEN

Definimos y estudiamos la transformada de Fourier-Wigner asociada a los operadores

de Dunkl, y probamos una fórmula de inversion y una formula de Plancherel para esta

transformada. Luego introducimos y estudiamos las funciones extramales asociadas a

la transformada de Dunkl-Wigner.

Keywords and Phrases: Dunkl transform; Dunkl-Wigner transform; inversion formulas; ex-

tremal functions.

2010 AMS Mathematics Subject Classification: 42B10; 44A20; 46F12.

1Author partially supported by the DGRST research project LR11ES11 and CMCU program 10G/1503



2 Fethi Soltani CUBO
17, 2 (2015)

1 Introduction

In this paper, we consider Rd with the Euclidean inner product 〈., .〉 and norm |y| :=
√

〈y, y〉. For
α ∈ Rd\{0}, let σα be the reflection in the hyperplane Hα ⊂ Rd orthogonal to α:

σαy := y −
2〈α, y〉
|α|2

α.

A finite set Re ⊂ Rd\{0} is called a root system, if Re ∩R.α = {−α, α} and σα Re = Re for
all α ∈ Re. We assume that it is normalized by |α|2 = 2 for all α ∈ Re. For a root system Re,
the reflections σα, α ∈ Re, generate a finite group G. The Coxeter group G is a subgroup of the
orthogonal group O(d). All reflections in G, correspond to suitable pairs of roots. For a given

β ∈ Rd\
⋃

α∈Re Hα, we fix the positive subsystem Re+ := {α ∈ Re : 〈α, β〉 > 0}. Then for each
α ∈ Re either α ∈ Re+ or −α ∈ Re+.

Let k : Re → C be a multiplicity function on Re (a function which is constant on the orbits

under the action of G). As an abbreviation, we introduce the index γ = γk :=
∑

α∈Re+ k(α).

Throughout this paper, we will assume that k(α) ≥ 0 for all α ∈ Re. Moreover, let wk
denote the weight function wk(y) :=

∏
α∈Re+ |〈α, y〉|

2k(α), for all y ∈ Rd, which is G-invariant
and homogeneous of degree 2γ.

Let ck be the Mehta-type constant given by ck := (
∫
Rd

e−|y|
2/2wk(y)dy)

−1. We denote by

µk the measure on R
d given by dµk(y) := ckwk(y)dy; and by L

p(µk), 1 ≤ p ≤ ∞, the space of
measurable functions f on Rd, such that

‖f‖Lp(µk) :=
(

∫

Rd

|f(y)|pdµk(y)
)1/p

< ∞, 1 ≤ p < ∞,

‖f‖L∞(µk) := ess sup
y∈Rd

|f(y)| < ∞,

and by L
p
rad(µk) the subspace of L

p(µk) consisting of radial functions.

For f ∈ L1(µk) the Dunkl transform of f is defined (see [3]) by

Fk(f)(x) :=
∫

Rd

Ek(−ix, y)f(y)dµk(y), x ∈ Rd,

where Ek(−ix, y) denotes the Dunkl kernel. (For more details see the next section.)

The Dunkl translation operators τx, x ∈ Rd, [18] are defined on L2(µk) by

Fk(τxf)(y) = Ek(ix, y)Fk(f)(y), y ∈ Rd.

Let g ∈ L2rad(µk). The Dunkl-Wigner transform Vg is the mapping defined for f ∈ L2(µk) by

Vg(f)(x, y) :=

∫

Rd

f(t)τxgk,y(−t)dµk(t),



CUBO
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Reproducing inversion formulas for the Dunkl-Wigner transforms 3

where

gk,y(z) := Fk
(
√

τy|Fk(g)|2
)

(z).

We study some of its properties, and we prove reproducing inversion formulas for this transform.

Next, Building on the ideas of Matsuura et al. [6], Saitoh [11, 13] and Yamada et al. [20], and

using the theory of reproducing kernels [10], we give best approximation of the mapping Vg on the

Sobolev-Dunkl spaces Hs(µk). More precisely, for all λ > 0, h ∈ L2(µk ⊗ µk), the infimum

inf
f∈Hs(µk)

{
λ‖f‖2Hs(µk) + ‖h − Vg(f)‖

2
L2(µk⊗µk)

}
,

is attained at one function f∗λ,h, called the extremal function, and given by

f∗λ,h(y) =

∫

Rd

∫

Rd

Ek(iy, z)
√

τt|Fk(g)|2(z)Fk(h(., t))(z)
λ(1 + |z|2)s + ‖g‖2

L2
rad

(µk)

dµk(t)dµk(z).

In the Dunkl setting, the extremal functions are studied in several directions [14, 15, 16, 17].

In the classical case, the Fourier-Wigner transforms are studied by Weyl [21] and Wong [22].

In the Bessel-Kingman hypergroups, these operators are studied by Dachraoui [1].

This paper is organized as follows. In Section 2, we recall some properties of harmonic analysis

for the Dunkl operators. Next, we define the Fourier-Wigner transform Vg in the Dunkl setting,

and we have established for it a reproducing inversion formulas. In Section 3, we introduce and

study the extremal functions associated to the Dunkl-Wigner transform Vg.

2 The Dunkl-Wigner transform

The Dunkl operators Dj; j = 1, ..., d, on Rd associated with the finite reflection group G and
multiplicity function k are given, for a function f of class C1 on Rd, by

Djf(y) :=
∂

∂yj
f(y) +

∑

α∈Re+
k(α)αj

f(y) − f(σαy)

〈α, y〉
.

For y ∈ Rd, the initial problem Dju(., y)(x) = yju(x, y), j = 1, ..., d, with u(0, y) = 1 admits
a unique analytic solution on Rd, which will be denoted by Ek(x, y) and called Dunkl kernel [2, 4].

This kernel has a unique analytic extension to Cd × Cd (see [7]). The Dunkl kernel has the
Laplace-type representation [8]

Ek(x, y) =

∫

Rd

e〈y,z〉dΓx(z), x ∈ Rd, y ∈ Cd, (2.1)

where 〈y, z〉 :=
∑d

i=1 yizi and Γx is a probability measure on R
d, such that

supp(Γx) ⊂ {z ∈ Rd : |z| ≤ |x|}. In our case,

|Ek(ix, y)| ≤ 1, x, y ∈ Rd. (2.2)



4 Fethi Soltani CUBO
17, 2 (2015)

The Dunkl kernel gives rise to an integral transform, which is called Dunkl transform on Rd,

and was introduced by Dunkl in [3], where already many basic properties were established. Dunkl’s

results were completed and extended later by De Jeu [4]. The Dunkl transform of a function f in

L1(µk), is defined by

Fk(f)(x) :=
∫

Rd

Ek(−ix, y)f(y)dµk(y), x ∈ Rd.

We notice that F0 agrees with the Fourier transform F that is given by

F(f)(x) := (2π)−d/2
∫

Rd

e−i〈x,y〉f(y)dy, x ∈ Rd.

Some of the properties of Dunkl transform Fk are collected bellow (see [3, 4]).
Theorem 2.1. (i) L1 − L∞-boundedness. For all f ∈ L1(µk), Fk(f) ∈ L∞(µk), and

‖Fk(f)‖L∞(µk) ≤ ‖f‖L1(µk).

(ii) Inversion theorem. Let f ∈ L1(µk), such that Fk(f) ∈ L1(µk). Then

f(x) = F(Fk(f))(−x), a.e. x ∈ Rd.

(iii) Plancherel theorem. The Dunkl transform Fk extends uniquely to an isometric isomor-
phism of L2(µk) onto itself. In particular, we have

‖f‖L2(µk) = ‖Fk(f)‖L2(µk).

(iv) Parseval theorem. For f, g ∈ L2(µk), we have

〈f, g〉L2(µk) = 〈Fk(f), Fk(g)〉L2(µk).

The Dunkl transform Fk allows us to define a generalized translation operators on L2(µk) by
setting

Fk(τxf)(y) = Ek(ix, y)Fk(f)(y), y ∈ Rd. (2.3)

It is the definition of Thangavelu and Xu given in [18]. It plays the role of the ordinary translation

τxf = f(x + .) in R
d, since the Euclidean Fourier transform satisfies F(τxf)(y) = eixyF(f)(y).

Note that from (2.2) and Theorem 2.1 (iii), the definition (2.3) makes sense, and

‖τxf‖L2(µk) ≤ ‖f‖L2(µk), f ∈ L
2
(µk). (2.4)



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Reproducing inversion formulas for the Dunkl-Wigner transforms 5

Rösler [9] introduced the Dunkl translation operators for radial functions. If f are radial

functions, f(x) = F(|x|), then

τxf(y) =

∫

Rd

F
(

√

|x|2 + |y|2 + 2〈y, z〉
)

dΓx(z); x, y ∈ Rd,

where Γx is the representing measure given by (2.1).

This formula allows us to establish the following results [18, 19].

Proposition 2.2. (i) For all p ∈ [1, 2] and for all x ∈ Rd, the Dunkl translation τx : Lprad(µk) →
Lp(µk) is a bounded operator, and for f ∈ Lprad(µk), we have

‖τxf‖Lp(µk) ≤ ‖f‖Lprad(µk).

(ii) Let f ∈ L1rad(µk). Then, for all x ∈ Rd, we have
∫

Rd

τxf(y)dµk(y) =

∫

Rd

f(y)dµk(y).

The Dunkl convolution product ∗k of two functions f and g in L2(µk) is defined by

f ∗k g(x) :=
∫

Rd

τxf(−y)g(y)dµk(y), x ∈ Rd. (2.5)

We notice that ∗k generalizes the convolution ∗ that is given by

f ∗ g(x) := (2π)−d/2
∫

Rd

f(x − y)g(y)dy, x ∈ Rd.

The Proposition 2.2 allows us to establish the following properties for the Dunkl convolution

on Rd (see [18]).

Proposition 2.3. (i) Assume that p ∈ [1, 2] and q, r ∈ [1, ∞] such that 1/p + 1/q = 1 + 1/r .
Then the map (f, g) → f ∗k g extends to a continuous map from Lprad(µk) × Lq(µk) to Lr(µk), and

‖f ∗k g‖Lr(µk) ≤ ‖f‖Lprad(µk)‖g‖Lq(µk).

(ii) For all f ∈ L1rad(µk) and g ∈ L2(µk), we have

Fk(f ∗k g) = Fk(f) Fk(g).

(iii) Let f ∈ L2rad(µk) and g ∈ L2(µk). Then f∗k g belongs to L2(µk) if and only if Fk(f)Fk(g)
belongs to L2(µk), and

Fk(f ∗k g) = Fk(f)Fk(g), in the L2(µk) − case.



6 Fethi Soltani CUBO
17, 2 (2015)

(iv) Let f ∈ L2rad(µk) and g ∈ L2(µk). Then
∫

Rd

|f ∗ g(x)|2dµk(x) =
∫

Rd

|Fk(f)(z)|2|Fk(g)(z)|2dµk(z),

where both sides are finite or infinite.

Let g ∈ L2rad(µk) and y ∈ Rd. The modulation of g by y is the function gk,y defined by

gk,y(z) := Fk
(
√

τy|Fk(g)|2
)

(z), z ∈ Rd.

Thus,

‖gk,y‖L2(µk) = ‖g‖L2rad(µk). (2.6)

Let g ∈ L2rad(µk). The Fourier-Wigner transform associated to the Dunkl operators, is the
mapping Vg defined for f ∈ L2(µk) by

Vg(f)(x, y) :=

∫

Rd

f(t)τxgk,y(−t)dµk(t), x, y ∈ Rd. (2.7)

Proposition 2.4. Let (f, g) ∈ L2(µk) × L2rad(µk).

(i) Vg(f)(x, y) = gk,y ∗k f(x).

(ii) Vg(f)(x, y) =

∫

Rd

Ek(ix, z)Fk(f)(z)
√

τy|Fk(g)|2(z)dµk(z).

(iii) The function Vg(f) belongs to L
∞(µk ⊗ µk), and

‖Vg(f)‖L∞(µk⊗µk) ≤ ‖f‖L2(µk)‖g‖L2rad(µk).

Proof. (i) follows from (2.5), (2.7) and the fact that τxgk,y(−t) = τxgk,y(−t).

(ii) By Theorem 2.1 (iv) and (2.3) we have

Vg(f)(x, y) =

∫

Rd

Ek(ix, z)Fk(f)(z)Fk(gk,y)(−z)dµk(z).

We obtain the result from the fact that

Fk(gk,y)(−z) = Fk(gk,y)(z) =
√

τy|Fk(g)|2(z).

(iii) follows from (2.7), by using Hölder’s inequality, (2.4) and (2.6). ✷

Theorem 2.5. Let g ∈ L2rad(µk).

(i) Plancherel formula: For every f ∈ L2(µk), we have

‖Vg(f)‖L2(µk⊗µk) = ‖g‖L2rad(µk)‖f‖L2(µk).



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Reproducing inversion formulas for the Dunkl-Wigner transforms 7

(ii) Parseval formula: For every f, h ∈ L2(µk), we have

〈Vg(f), Vg(h)〉L2(µk⊗µk) = ‖g‖
2
L2
rad

(µk)
〈f, h〉L2(µk).

(iii) Inversion formula: For all f ∈ L1 ∩ L2(µk) such that Fk(f) ∈ L1(µk), we have

f(z) =
1

‖g‖2
L2
rad

(µk)

∫

Rd

∫

Rd

Vg(f)(x, y)τzgk,y(−x)dµk(x)dµk(y).

Proof. (i) From Theorem 2.1 (iii), Proposition 2.2 (ii), Proposition 2.3 (iv) and Proposition 2.4

(i), we obtain

∫

Rd

∫

Rd

|Vg(f)(x, y)|
2dµk(x)dµk(y) =

∫

Rd

∫

Rd

|gk,y ∗k f(x)|2dµk(x)dµk(y)

=

∫

Rd

∫

Rd

|Fk(gk,y)(z)|2|Fk(f)(z)|2dµk(z)dµk(y)

=

∫

Rd

∫

Rd

τy|Fk(g)|2(z)|Fk(f)(z)|2dµk(z)dµk(y)

= ‖g‖2
L2
rad

(µk)

∫

Rd

|Fk(f)(z)|2dµk(z).

(ii) follows from (i) by polarization.

(iii) From Theorem 2.1 (iv), Proposition 2.3 (ii) and (iii), we have

∫

Rd

∫

Rd

Vg(f)(x, y)τzgk,y(−x)dµk(x)dµk(y)

=

∫

Rd

∫

Rd

τy|Fk(g)|2(t)Fk(f)(t)Ek(iz, t)dµk(t)dµk(y).

Then, by Fubini’s theorem, Theorem 2.1 (ii) and Proposition 2.2 (ii) we deduce that

∫

Rd

∫

Rd

Vg(f)(x, y)τzgk,y(−x)dµk(x)dµk(y) = ‖g‖2L2
rad

(µk)

∫

Rd

Fk(f)(t)Ek(iz, t)dµk(t)

= ‖g‖2
L2
rad

(µk)
f(z).

✷

In the following we establish reproducing inversion formula of Calderón’s type for the Dunkl-

Wigner transform on Rd.

Theorem 2.6. Let ∆ =
∏d

j=1[aj, bj], −∞ < aj < bj < ∞; and let g ∈ L2rad(µk) such that
Fk(g) ∈ L∞(µk). Then, for f ∈ L2(µk), the function f∆ given by

f∆(z) =
1

‖g‖L2
rad

(µk)

∫

∆

∫

Rd

Vg(f)(x, y)τzgk,y(−x)dµk(x)dµk(y),



8 Fethi Soltani CUBO
17, 2 (2015)

belongs to L2(µk) and satisfies

lim
aj→−∞
bj→+∞

‖f∆ − f‖L2(µk) = 0. (2.8)

Proof. From Theorem 2.1 (iii), Proposition 2.3 (iv) and Proposition 2.4 (i), we have

f∆(z) =
1

‖g‖2
L2
rad

(µk)

∫

∆

∫

Rd

τy|Fk(g)|2(t)Fk(f)(t)Ek(iz, t)dµk(t)dµk(y).

By Fubini’s theorem we get

f∆(z) =

∫

Rd

K∆(t)Fk(f)(t)Ek(iz, t)dµk(t). (2.9)

where

K∆(t) =
1

‖g‖2
L2
rad

(µk)

∫

∆

τy|Fk(g)|2(t)dµk(y).

It is easily to see that ‖K∆‖L∞(µk) ≤ 1. On the other hand, by Hölder’s inequality, we deduce that

|K∆(t)|
2 ≤

µk(∆)

‖g‖4
L2
rad

(µk)

∫

∆

|τy|Fk(g)|2(t)|2dµk(y).

Hence, by (2.4) we find

‖K∆‖2L2(µk) ≤
(µk(∆))

2

‖g‖4
L2
rad

(µk)

∫

Rd

|Fk(g)(t)|4dµk(t) ≤
(µk(∆))

2‖Fk(g)‖2L∞(µk)
‖g‖2

L2
rad

(µk)

.

Thus K∆ ∈ L∞ ∩ L2(µk). Therefore and by (2.9) we obtain

Fk(f∆)(t) = K∆(t)Fk(f)(t).

From this relation and Theorem 2.1 (iii), it follows that f∆ ∈ L2(µk) and

‖f∆ − f‖2L2(µk) =
∫

Rd

|Fk(f)(t)|2(1 − K∆(t))2dµk(t).

But by Proposition 2.2 (ii) we have

lim
aj→−∞
bj→+∞

K∆(t) = 1, for all t ∈ Rd,

and

|Fk(f)(t)|2(1 − K∆(t))2 ≤ |Fk(f)(t)|2, for all t ∈ Rd.

So, the relation (2.8) follows from the dominated convergence theorem. ✷



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Reproducing inversion formulas for the Dunkl-Wigner transforms 9

3 Extremal functions for the mapping Vg

Let s ≥ 0. We define the Sobolev-Dunkl space of order s, that will be denoted Hs(µk), as the set
of all f ∈ L2(µk) such that (1 + |z|2)s/2Fk(f) ∈ L2(µk). The space Hs(µk) provided with the inner
product

〈f, g〉Hs(µk) =
∫

Rd

(1 + |z|2)sFk(f)(z)Fk(g)(z)dµk(z),

and the norm

‖f‖Hs(µk) =
[∫

Rd

(1 + |z|2)s|Fk(f)(z)|2dµk(z)
]1/2

.

The space Hs(µk) satisfies the following properties.

(a) H0(µk) = L
2(µk).

(b) For all s > 0, the space Hs(µk) is continuously contained in L
2(µk) and ‖f‖L2(µk) ≤

‖f‖Hs(µk).

(c) For all s, t > 0, such that t > s, the space Ht(µk) is continuously contained in H
s(µk) and

‖f‖Hs(µk) ≤ ‖f‖Ht(µk).

(d) The space Hs(µk), s ≥ 0 provided with the inner product 〈., .〉Hs(µk) is a Hilbert space.
Remark 3.1. For s > γ + d/2, the function y → (1 + |z|2)−s/2 belongs to L2(µk). Hence for all

f ∈ Hs(µk), we have ‖Fk(f)‖L2(µk) ≤ ‖f‖Hs(µk), and by Hölder’s inequality

‖Fk(f)‖L1(µk) ≤
[∫

Rd

dµk(z)

(1 + |z|2)s

]1/2

‖f‖Hs(µk) .

Then the function Fk(f) belongs to L1 ∩ L2(µk), and therefore

f(x) =

∫

Rd

Ek(ix, z)Fk(f)(z)dµk(z), a.e. x ∈ Rd.

Let λ > 0. We denote by 〈., .〉λ,Hs(µk) the inner product defined on the space Hs(µk) by

〈f, h〉λ,Hs(µk) := λ〈f, h〉Hs(µk) + 〈Vg(f), Vg(h)〉L2(µk⊗µk) ,

and the norm ‖f‖λ,Hs(µk) :=
√

〈f, f〉λ,Hs(µk) .

In the next we suppose that g ∈ L2rad(µk). By Theorem 2.5 (ii), the inner product 〈., .〉λ,Hs(µk)
can be written

〈f, h〉λ,Hs(µk) = λ〈f, h〉Hs(µk) + ‖g‖
2
L2
rad

(µk)
〈f, h〉L2(µk) . (3.1)

Theorem 3.2. Let λ > 0 and s > γ+d/2 and let g ∈ L2rad(µk). The space (Hs(µk), 〈., .〉λ,Hs(µk))
has the reproducing kernel

Ks(x, y) =

∫

Rd

Ek(ix, z)Ek(−iy, z)

λ(1 + |z|2)s + ‖g‖2
L2
rad

(µk)

dµk(z), (3.2)



10 Fethi Soltani CUBO
17, 2 (2015)

that is

(i) For all y ∈ Rd, the function x → Ks(x, y) belongs to Hs(µk).

(ii) The reproducing property: for all f ∈ Hs(µk) and y ∈ Rd,

〈f, Ks(., y)〉λ,Hs(µk) = f(y).

Proof. (i) Let y ∈ Rd. From (2.2), the function Φy : z → Ek(−iy,z)λ(1+|z|2)s+‖g‖2
L2
rad

(µk)

belongs to

L1 ∩ L2(µk). Then, the function Ks is well defined and by Theorem 2.1 (ii), we have

Ks(x, y) = F−1k (Φy)(x), x ∈ R
d.

From Theorem 2.1 (iii), it follows that Ks(., y) belongs to L
2(µk), and we have

Fk(Ks(., y))(z) =
Ek(−iy, z)

λ(1 + |z|2)s + ‖g‖2
L2
rad

(µk)

, z ∈ Rd. (3.3)

Then by (2.2), we obtain

|Fk(Ks(., y))(z)| ≤
1

λ(1 + |z|2)s
,

and

‖Ks(., y)‖2Hs(µk) ≤
1

λ2

∫

Rd

dµk(z)

(1 + |z|2)s
< ∞.

This proves that for all y ∈ Rd the function Ks(., y) belongs to Hs(µk).

(ii) Let f ∈ Hs(µk) and y ∈ Rd. From (3.1) and (3.3), we have

〈f, Ks(., y)〉λ,Hs(µk) =
∫

Rd

Ek(iy, z)Fk(f)(z)dµk(z),

and from Remark 3.1, we obtain the reproducing property:

〈f, Ks(., y)〉λ,Hs(µk) = f(y).

This completes the proof of the theorem. ✷

The main result of this subsection can then be stated as follows.

Theorem 3.3. Let s > γ + d/2 and g ∈ L2rad(µk). For any h ∈ L2(µk ⊗ µk) and for any λ > 0,
there exists a unique function f∗λ,g, where the infimum

inf
f∈Hs(µk)

{
λ‖f‖2Hs(µk) + ‖h − Vg(f)‖

2
L2(µk⊗µk)

}
(3.4)

is attained. Moreover, the extremal function f∗λ,h is given by

f∗λ,h(y) =

∫

Rd

∫

Rd

h(x, t)Qs(x, y, t)dµk(t)dµk(x),



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Reproducing inversion formulas for the Dunkl-Wigner transforms 11

where

Qs(x, y, t) =

∫

Rd

Ek(−ix, z)Ek(iy, z)
√

τt|Fk(g)|2(z)
λ(1 + |z|2)s + ‖g‖2

L2
rad

(µk)

dµk(z).

Proof. The existence and unicity of the extremal function f∗λ,h satisfying (3.4) is given by Kimel-

dorf and Wahba [5], Matsuura et al. [6] and Saitoh [12]. Especially, f∗λ,h is given by the reproducing

kernel of Hs(µk) with ‖.‖λ,Hs(µk) norm as

f∗λ,h(y) = 〈h, Vg(Ks(., y))〉L2(µk⊗µk), (3.5)

where Ks is the kernel given by (3.2).

But by Proposition 2.4 (ii) and (3.3), we have

Vg(Ks(., y))(x, t) =

∫

Rd

Ek(ix, z)Fk(Ks(., y))(z)
√

τt|Fk(g)|2(z)dµk(z)

=

∫

Rd

Ek(ix, z)Ek(−iy, z)
√

τt|Fk(g)|2(z)
λ(1 + |z|2)s + ‖g‖2

L2
rad

(µk)

dµk(z).

This clearly yields the result. ✷

Theorem 3.4. Let s > γ + d/2 and g ∈ L2rad(µk). For any h ∈ L2(µk ⊗ µk) and for any λ > 0,
we have

(i) |f∗λ,h(y)| ≤
‖h‖

L2(µk⊗µk)

2
√
λ

[∫

Rd

dµk(z)

(1 + |z|2)s

]1/2

.

(ii) ‖f∗λ,h‖2L2(µk) ≤
1

4λ

∫

Rd

∫

Rd

|h(x, t)|2e(|x|
2
+|t|2)/2dµk(t)dµk(x).

Proof. (i) From (3.5) and Theorem 2.5 (i), we have

|f∗λ,h(y)| ≤ ‖h‖L2(µk⊗µk)‖Vg(Ks(., y))‖L2(µk⊗µk)
≤ ‖h‖L2(µk⊗µk)‖g‖L2rad(µk)‖Ks(., y)‖L2(µk).

Then, by Theorem 2.1 (iii) and (3.3), we deduce that

|f∗λ,g(y)| ≤ ‖h‖L2(µk⊗µk)‖g‖L2rad(µk)‖Fk(Ks(., y))‖L2(µk)

≤ ‖h‖L2(µk⊗µk)‖g‖L2rad(µk)
[

∫

Rd

dµk(z)

[λ(1 + |z|2)s + ‖g‖2
L2
rad

(µk)
]2

]1/2

.

Using the fact that
[

λ(1 + |z|2)s + ‖g‖2
L2
rad

(µk)

]2

≥ 4λ(1 + |z|2)s‖g‖2
L2
rad

(µk)
, we obtain the result.

(ii) We write

f∗λ,h(y) =

∫

Rd

∫

Rd

e−(|x|
2
+|t|2)/4e(|x|

2
+|t|2)/4h(x, t)Qs(x, y, t)dµk(t)dµk(x).

Applying Hölder’s inequality, we obtain

|f∗λ,h(y)|
2 ≤

∫

Rd

∫

Rd

|h(x, t)|2e(|x|
2
+|t|2)/2

|Qs(x, y, t)|
2dµk(t)dµk(x).



12 Fethi Soltani CUBO
17, 2 (2015)

Thus and from Fubini-Tonnelli’s theorem, we get

‖f∗λ,h‖2L2(µk) ≤
∫

Rd

∫

Rd

|h(x, t)|2e(|x|
2
+|t|2)/2‖Qs(x, ., t)‖2L2(µk)dµk(t)dµk(x).

The function z →
Ek(−ix,z)

√
τt|Fk(g)|2(z)

λ(1+|z|2)s+‖g‖2
L2
rad

(µk)

belongs to L1 ∩ L2(µk), then by Theorem 2.1 (ii), we get

Qs(x, y, t) = F−1k
(

Ek(−ix, z)
√

τt|Fk(g)|2(z)
λ(1 + |z|2)s + ‖g‖2

L2
rad

(µk)

)

(y).

Thus, by Theorem 2.1 (iii) we deduce that

‖Qs(x, ., t)‖2L2(µk) =
∫

Rd

|Fk(Qs(x, ., t))(z)|2dµk(z)

≤
∫

Rd

τt|Fk(g)|2(z)dµk(z)
[λ(1 + |z|2)s + ‖g‖2

L2
rad

(µk)
]2
.

Then

‖Q(x, ., t)‖2L2(µk) ≤
1

4λ‖g‖2
L2
rad

(µk)

∫

Rd

τt|Fk(g)|2(z)dµk(z) ≤
1

4λ
.

From this inequality we deduce the result. ✷

Theorem 3.5. Let s > γ + d/2 and g ∈ L2rad(µk). For any h ∈ L2(µk ⊗ µk) and for any λ > 0,
we have

(i) f∗λ,h(y) =

∫

Rd

∫

Rd

Ek(iy, z)
√

τt|Fk(g)|2(z)Fk(h(., t))(z)
λ(1 + |z|2)s + ‖g‖2

L2
rad

(µk)

dµk(t)dµk(z).

(ii) Fk(f∗λ,h)(z) =

∫

Rd

√

τt|Fk(g)|2(z)Fk(h(., t))(z)dµk(t)

λ(1 + |z|2)s + ‖g‖2
L2
rad

(µk)

.

(iii) ‖f∗λ,h‖Hs(µk) ≤
1

2
√
λ
‖h‖L2(µk⊗µk).

Proof. (i) From Theorem 3.3 and Fubini’s theorem, we have

f∗λ,h(y) =

∫

Rd

∫

Rd

Ek(iy, z)
√

τt|Fk(g)|2(z)
λ(1 + |z|2)s + ‖g‖2

L2
rad

(µk)

[∫

Rd

h(x, t)Ek(−ix, z)dµk(x)

]

dµk(t)dµk(z)

=

∫

Rd

∫

Rd

Ek(iy, z)
√

τt|Fk(g)|2(z)Fk(h(., t))(z)
λ(1 + |z|2)s + ‖g‖2

L2
rad

(µk)

dµk(t)dµk(z).

(ii) The function z →

∫

Rd

√

τt|Fk(g)|2(z)Fk(h(., t))(z)dµk(t)
λ(1+|z|2)s+‖g‖2

L2
rad

(µk)

belongs to L1 ∩ L2(µk). Then



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Reproducing inversion formulas for the Dunkl-Wigner transforms 13

by Theorem 2.1 (ii) and (iii), it follows that f∗λ,h belongs to L
2(µk), and

Fk(f∗λ,h)(z) =

∫

Rd

√

τt|Fk(g)|2(z)Fk(h(., t))(z)dµk(t)

λ(1 + |z|2)s + ‖g‖2
L2
rad

(µk)

.

(iii) From (ii), Hölder’s inequality and (2.6) we have

|Fk(f∗λ,h)(z)|2 ≤
‖g‖2

L2
rad

(µk)

[λ(1 + |z|2)s + ‖g‖2
L2
rad

(µk)
]2

∫

Rd

|Fk(h(., t))(z)|2dµk(t).

Thus,

‖f∗λ,h‖2Hs(µk) ≤
∫

Rd

(1 + |z|2)s‖g‖2
L2
rad

(µk)

[λ(1 + |z|2)s + ‖g‖2
L2
rad

(µk)
]2

[∫

Rd

|Fk(h(., t))(z)|2dµk(t)
]

dµk(z)

≤ 1
4λ

∫

Rd

[∫

Rd

|Fk(h(., t))(z)|2dµk(t)
]

dµk(z) =
1

4λ
‖h‖2L2(µk⊗µk),

which ends the proof. ✷

Theorem 3.6. Let s > γ + d/2 and g ∈ L2rad(µk). For any h ∈ L2(µk ⊗ µk) and for any λ > 0,
we have

Vg(f
∗
λ,h)(x, y) =

∫

Rd

∫

Rd

Ek(ix, z)
√

τt|Fk(g)|2(z)τy|Fk(g)|2(z)Fk(h(., t))(z)
λ(1 + |z|2)s + ‖g‖2

L2
rad

(µk)

dµk(t)dµk(z).

Proof. From Proposition 2.4 (ii), we have

Vg(f
∗
λ,h)(x, y) =

∫

Rd

Ek(ix, z)Fk(f∗λ,h)(z)
√

τy|Fk(g)|2(z)dµk(z).

Then by Theorem 3.5 (ii), we obtain the result. ✷

Received: January 2015. Accepted: April 2015.

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	Introduction
	The Dunkl-Wigner transform
	Extremal functions for the mapping Vg