# Uncertainty Principles for the Dunkl-Wigner Transforms

Authors: Fethi Soltani
Journal: Journal of Operators
 (2016)

---

## Abstract

We prove a version of Heisenberg-type uncertainty principle for the Dunkl-Wigner transform of magnitudes
>
0; and we deduce a local uncertainty principle for this transform.


---

## Body

## 1. Introduction

In this paper, we considerR
d with the Euclidean inner product 〈
·
,
·
〉 and norm |
y
|
≔
〈
y
,
y
〉. For α
∈
R
d
∖
{
0
}, let σ
α be the reflection in the hyperplane H
α
⊂
R
d orthogonal to α:(1)
σ
α
y
≔
y
-
2
α
,
y
α
2
α
.A finite setR
⊂
R
d
∖
{
0
} is called a root system, if R
∩
R
.
α
=
{
-
α
,
α
} and σ
α
R
=
R for all α
∈
R. We assume that it is normalized by |
α
|
2
=
2 for all α
∈
R. For a root system R, the reflections σ
α, α
∈
R, 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
∈
R
d
∖
⋃
α
∈
R
H
α, we fix the positive subsystem R
+
≔
{
α
∈
R
:
α
,
β
>
0
}. Then for each α
∈
R either α
∈
R
+ or -
α
∈
R
+.Letk
:
R
→
C be a multiplicity function on R (a function which is constant on the orbits under the action of G). As an abbreviation, we introduce the index γ
=
γ
k
≔
∑
α
∈
R
+
k
(
α
).Throughout this paper, we will assume thatk
(
α
)
≥
0 for all α
∈
R. Moreover, let w
k denote the weight function w
k
(
y
)
≔
∏
α
∈
R
+
α
,
y
2
k
(
α
), for all y
∈
R
d, which is G-invariant and homogeneous of degree 2
γ.Letc
k be the Mehta-type constant given by c
k
≔
(
∫
R
d
e
-
|
y
|
2
/
2
w
k
(
y
)
d
y
)
-
1. We denote by μ
k the measure on R
d given by d
μ
k
(
y
)
≔
c
k
w
k
(
y
)
d
y, by L
p
(
μ
k
), 1
≤
p
≤
∞, the space of measurable functions f on R
d, such that (2)
f
L
p
μ
k
≔
∫
R
d
f
y
p
d
μ
k
y
1
/
p
<
∞
,
1
≤
p
<
∞
,
f
L
∞
μ
k
≔
ess
sup
y
∈
R
d
⁡
f
y
<
∞
,and by L
r
a
d
p
(
μ
k
) the subspace of L
p
(
μ
k
) consisting of radial functions.Forf
∈
L
1
(
μ
k
) the Dunkl transform of f is defined (see [1]) by (3)
F
k
f
x
≔
∫
R
d
E
k
-
i
x
,
y
f
y
d
μ
k
y
,
x
∈
R
d
,where E
k
(
-
i
x
,
y
) denotes the Dunkl kernel. (For more details see the next section.)Many uncertainty principles have already been proved for the Dunkl transformF
k, namely, by Rösler [2] and Shimeno [3] who established the Heisenberg-type uncertainty inequality for this transform, by showing that for f
∈
L
2
(
μ
k
), (4)
f
L
2
μ
k
2
≤
2
2
γ
+
d
x
f
L
2
μ
k
y
F
k
f
L
2
μ
k
.Recently, the author [4–7] proved general forms of the Heisenberg-type inequality for the Dunkl transform F
k.The Dunkl translation operatorsτ
x, x
∈
R
d, [8] are defined on L
2
(
μ
k
) by (5)
F
k
τ
x
f
y
=
E
k
i
x
,
y
F
k
f
y
,
y
∈
R
d
.Letg
∈
L
rad
2
(
μ
k
). The Dunkl-Wigner transform V
g is the mapping defined for f
∈
L
2
(
μ
k
) by (6)
V
g
f
x
,
y
≔
∫
R
d
f
t
τ
x
g
k
,
y
-
t
¯
d
μ
k
t
,where (7)
g
k
,
y
z
≔
F
k
τ
y
F
k
g
2
z
.This transform is studied in [9, 10] where the author established some applications (Plancherel formula, inversion formula, Calderón’s reproducing formula, extremal function, etc.).In this paper we use formula (4); we prove uncertainty principle intervening F
k and V
g of magnitudes a
,
b
≥
1; that is, for every f
∈
L
2
(
μ
k
),(8)
x
a
V
g
f
L
2
μ
k
⊗
μ
k
b
z
b
F
k
f
L
2
μ
k
a
≥
γ
+
d
2
a
b
f
L
2
μ
k
a
+
b
g
L
r
a
d
2
μ
k
b
.Next, we prove a Heisenberg-type uncertainty principle for the Dunkl-Wigner transform V
g of magnitude s
>
0; that is, there exists a constant c
(
k
,
s
)
>
0 such that, for f
∈
L
2
(
μ
k
), (9)
x
s
V
g
f
L
2
μ
k
⊗
μ
k
y
s
V
g
f
L
2
μ
k
⊗
μ
k
≥
c
k
,
s
f
L
2
μ
k
2
g
L
r
a
d
2
μ
k
2
.Finally, we prove a local uncertainty principle for the Dunkl-Wigner transform V
g; that is, there exists a constant b
(
k
,
s
)
>
0 such that, for f
∈
L
2
(
μ
k
) and for measurable subset E of R
d
×
R
d such that 0
<
μ
k
⊗
μ
k
(
E
)
<
∞, (10)
χ
E
V
g
f
L
2
μ
k
⊗
μ
k
≤
b
k
,
s
μ
k
⊗
μ
k
E
1
/
2
x
,
y
s
V
g
f
L
2
μ
k
⊗
μ
k
,where χ
E is the indicator function of the set E.In the classical case, the Fourier-Wigner transforms are studied by Weyl [11] and Wong [12]. In the Bessel-Kingman hypergroups, these operators are studied by Dachraoui [13].This paper is organized as follows. In Section2, we recall some properties of the Dunkl-Wigner transform V
g. In Section 3, we prove a Heisenberg-type uncertainty principle for the Dunkl-Wigner transform V
g of magnitude s
>
0; and we deduce a local uncertainty principle for this transform.


## 2. The Dunkl-Wigner Transform

The Dunkl operatorsD
j, j
=
1
,
…
,
d, on R
d associated with the finite reflection group G and multiplicity function k are given, for a function f of class C
1 on R
d, by (11)
D
j
f
y
≔
∂
∂
y
j
f
y
+
∑
α
∈
R
+
k
α
α
j
f
y
-
f
σ
α
y
α
,
y
.Fory
∈
R
d, the initial value problem D
j
u
(
·
,
y
)
(
x
)
=
y
j
u
(
x
,
y
), j
=
1
,
…
,
d, with u
(
0
,
y
)
=
1 admits a unique analytic solution on R
d, which will be denoted by E
k
(
x
,
y
) and called Dunkl kernel [14, 15]. This kernel has a unique analytic extension to C
d
×
C
d (see [16]). The Dunkl kernel has the Laplace-type representation [17] (12)
E
k
x
,
y
=
∫
R
d
e
y
,
z
d
Γ
x
z
,
x
∈
R
d
,
y
∈
C
d
,where y
,
z
≔
∑
i
=
1
d
y
i
z
i and Γ
x is a probability measure on R
d, such that supp
⁡
(
Γ
x
)
⊂
{
z
∈
R
d
:
z
≤
x
}. In our case, (13)
E
k
±
i
x
,
y
≤
1
,
x
,
y
∈
R
d
.The Dunkl kernel gives rise to an integral transform, which is called Dunkl transform onR
d, and was introduced by Dunkl in [1], where already many basic properties were established. Dunkl’s results were completed and extended later by de Jeu [15]. The Dunkl transform of a function f in L
1
(
μ
k
) is defined by (14)
F
k
f
x
≔
∫
R
d
E
k
-
i
x
,
y
f
y
d
μ
k
y
,
x
∈
R
d
.We notice that F
0 agrees with the Fourier transform F that is given by (15)
F
f
x
≔
2
π
-
d
/
2
∫
R
d
e
-
i
x
,
y
f
y
d
y
,
x
∈
R
d
.The Dunkl transform of a functionf
∈
L
r
a
d
1
(
μ
k
) which is radial is again radial and could be computed via the associated Fourier-Bessel transform F
γ
+
d
/
2
-
1
B (see [18], Proposition 4); that is, (16)
F
k
f
x
=
F
γ
+
d
/
2
-
1
B
F
x
,where f
(
x
)
=
F
(
x
) and (17)
F
γ
+
d
/
2
-
1
B
F
x
≔
∫
0
∞
F
r
j
γ
+
d
/
2
-
1
x
r
2
γ
+
d
/
2
-
1
Γ
γ
+
d
/
2
r
2
γ
+
d
-
1
d
r
.Here j
γ is the spherical Bessel function (see [19]).Some of the properties of Dunkl transformF
k are collected below (see [1, 15]).Theorem 1.
(i)L
1
-
L
∞-Boundedness. For all f
∈
L
1
(
μ
k
), F
k
(
f
)
∈
L
∞
(
μ
k
), and (18)
F
k
f
L
∞
μ
k
≤
f
L
1
μ
k
.(ii) Inversion Theorem. Let f
∈
L
1
(
μ
k
), such that F
k
(
f
)
∈
L
1
(
μ
k
). Then (19)
f
x
=
F
k
F
k
f
-
x
,
a
.
e
.
x
∈
R
d
.(iii) Plancherel Theorem. The Dunkl transform F
k extends uniquely to an isometric isomorphism of L
2
(
μ
k
) onto itself. In particular, one has (20)
f
L
2
μ
k
=
F
k
f
L
2
μ
k
.(iv) Parseval Theorem. For f
,
g
∈
L
2
(
μ
k
), one has (21)
f
,
g
L
2
μ
k
=
F
k
f
,
F
k
g
L
2
μ
k
.The Dunkl transformF
k allows us to define a generalized translation operators on L
2
(
μ
k
) by setting(22)
F
k
τ
x
f
y
=
E
k
i
x
,
y
F
k
f
y
,
y
∈
R
d
.It is the definition of Thangavelu and Xu given in [8]. It plays the role of the ordinary translation τ
x
f
=
f
(
x
+
·
) in R
d, since the Euclidean Fourier transform satisfies F
(
τ
x
f
)
(
y
)
=
e
i
〈
x
,
y
〉
F
(
f
)
(
y
). Note that, from (13) and Theorem 1(iii), relation (22) makes sense, and τ
x
f
L
2
(
μ
k
)
≤
f
L
2
(
μ
k
), for all f
∈
L
2
(
μ
k
).Rösler [20] introduced the Dunkl translation operators for radial functions. If f are radial functions, f
(
x
)
=
F
(
x
), then (23)
τ
x
f
y
=
∫
R
d
F
x
2
+
y
2
+
2
y
,
z
d
Γ
x
z
;
x
,
y
∈
R
d
,where Γ
x is the representing measure given by (12).This formula allows us to establish the following results [8, 21].Proposition 2.
(i) For allp
∈
[
1,2
] and for all x
∈
R
d, the Dunkl translation τ
x
:
L
r
a
d
p
(
μ
k
)
→
L
p
(
μ
k
) is a bounded operator, and for f
∈
L
r
a
d
p
(
μ
k
), one has (24)
τ
x
f
L
p
μ
k
≤
f
L
r
a
d
p
μ
k
.(ii) Let f
∈
L
r
a
d
1
(
μ
k
). Then, for all x
∈
R
d, one has (25)
∫
R
d
τ
x
f
y
d
μ
k
y
=
∫
R
d
f
y
d
μ
k
y
.The Dunkl convolution product∗
k of two functions f and g in L
2
(
μ
k
) is defined by (26)
f
∗
k
g
x
≔
∫
R
d
τ
x
f
-
y
g
y
d
μ
k
y
,
x
∈
R
d
.We notice that ∗
k generalizes the convolution ∗ that is given by (27)
f
∗
g
x
≔
2
π
-
d
/
2
∫
R
d
f
x
-
y
g
y
d
y
,
x
∈
R
d
.Proposition2 allows us to establish the following properties for the Dunkl convolution on R
d (see [8]).Proposition 3.
(i) Assume thatp
∈
[
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 L
r
a
d
p
(
μ
k
)
×
L
q
(
μ
k
) to L
r
(
μ
k
), and (28)
f
∗
k
g
L
r
μ
k
≤
f
L
r
a
d
p
μ
k
g
L
q
μ
k
.(ii) For all f
∈
L
r
a
d
1
(
μ
k
) and g
∈
L
2
(
μ
k
), one has (29)
F
k
f
∗
k
g
=
F
k
f
F
k
g
.(iii) Let f
∈
L
r
a
d
2
(
μ
k
) and g
∈
L
2
(
μ
k
). Then f
∗
k
g belongs to L
2
(
μ
k
) if and only if F
k
(
f
)
F
k
(
g
) belongs to L
2
(
μ
k
), and (30)
F
k
f
∗
k
g
=
F
k
f
F
k
g
,
in
the
L
2
μ
k
—
case
.(iv) Let f
∈
L
r
a
d
2
(
μ
k
) and g
∈
L
2
(
μ
k
). Then (31)
∫
R
d
f
∗
g
x
2
d
μ
k
x
=
∫
R
d
F
k
f
z
2
F
k
g
z
2
d
μ
k
z
,where both sides are finite or infinite.Letg
∈
L
r
a
d
2
(
μ
k
) and y
∈
R
d. The modulation of g by y is the function g
k
,
y defined by (32)
g
k
,
y
z
≔
F
k
τ
y
F
k
g
2
z
,
z
∈
R
d
.Thus, (33)
g
k
,
y
L
2
μ
k
=
g
L
r
a
d
2
μ
k
.Letg
∈
L
r
a
d
2
(
μ
k
). The Fourier-Wigner transform associated with the Dunkl operators is the mapping V
g defined for f
∈
L
2
(
μ
k
) by (34)
V
g
f
x
,
y
≔
∫
R
d
f
t
τ
x
g
k
,
y
-
t
¯
d
μ
k
t
,
x
,
y
∈
R
d
.In the following we recall some properties of the Dunkl-Wigner transform (Plancherel formula, inversion formula, reproducing inversion formula of Calderón’s type, etc.).Proposition 4 (see [10]).
Let(
f
,
g
)
∈
L
2
(
μ
k
)
×
L
r
a
d
2
(
μ
k
). Then(i)
V
g
(
f
)
(
x
,
y
)
=
g
k
,
y
¯
∗
k
f
(
x
).
(ii)
V
g
f
x
,
y
=
∫
R
d
E
k
i
x
,
z
F
k
f
z
τ
y
|
F
k
(
g
)
|
2
(
z
)
d
μ
k
(
z
).
(iii)
The functionV
g
(
f
) belongs to L
∞
(
μ
k
⊗
μ
k
), and (35)
V
g
f
L
∞
μ
k
⊗
μ
k
≤
f
L
2
μ
k
g
L
r
a
d
2
μ
k
.Theorem 5 (see [10]).
Letg
∈
L
r
a
d
2
(
μ
k
) be a nonzero function. Then one has the following.(i)
Plancherel formula: for everyf
∈
L
2
(
μ
k
), one has (36)
V
g
f
L
2
μ
k
⊗
μ
k
=
f
L
2
μ
k
g
L
r
a
d
2
μ
k
.
(ii)
Parseval formula: for everyf
,
h
∈
L
2
(
μ
k
), one has (37)
V
g
f
,
V
g
h
L
2
μ
k
⊗
μ
k
=
g
L
r
a
d
2
μ
k
2
f
,
h
L
2
μ
k
.
(iii)
Inversion formula: for allf
∈
L
1
∩
L
2
(
μ
k
) such that F
k
(
f
)
∈
L
1
(
μ
k
), one has (38)
f
z
=
1
g
L
r
a
d
2
μ
k
2
∬
R
d
V
g
f
x
,
y
τ
z
g
k
,
y
-
x
¯
d
μ
k
x
d
μ
k
y
.Theorem 6 (Calderón’s reproducing inversion formula; see [10]).
LetΔ
=
∏
j
=
1
d
‍
[
a
j
,
b
j
], -
∞
<
a
j
<
b
j
<
∞, and let g
∈
L
r
a
d
2
(
μ
k
) be a nonzero function, such that F
k
(
g
)
∈
L
∞
(
μ
k
). Then, for f
∈
L
2
(
μ
k
), the function f
Δ given by (39)
f
Δ
z
=
1
g
L
r
a
d
2
μ
k
∫
Δ
∫
R
d
V
g
f
x
,
y
τ
z
g
k
,
y
-
x
¯
d
μ
k
x
d
μ
k
ybelongs to L
2
(
μ
k
) and satisfies (40)
lim
a
j
→
-
∞
b
j
→
+
∞
⁡
f
Δ
-
f
L
2
μ
k
=
0
.


## 3. Uncertainty Principles for the MappingV
g

In this section we establish Heisenberg-type uncertainty principle for the Dunkl-Wigner transformV
g. We begin by the following theorem.Theorem 7.
Letg
∈
L
r
a
d
2
(
μ
k
) be a nonzero function. Then, for f
∈
L
2
(
μ
k
), one has (41)
x
V
g
f
L
2
μ
k
⊗
μ
k
z
F
k
f
L
2
μ
k
≥
γ
+
d
2
f
L
2
μ
k
2
g
L
r
a
d
2
μ
k
.Proof.
Letf
∈
L
2
(
μ
k
). Assume that x
V
g
(
f
)
L
2
(
μ
k
⊗
μ
k
)
+
z
F
k
(
f
)
L
2
(
μ
k
)
<
∞. Inequality (4) leads to (42)
∫
R
d
V
g
f
x
,
y
2
d
μ
k
x
≤
2
2
γ
+
d
∫
R
d
x
2
V
g
f
x
,
y
2
d
μ
k
x
1
/
2
∫
R
d
z
2
F
k
V
g
f
·
,
y
z
2
d
μ
k
z
1
/
2
.Integrating with respect to d
μ
k
(
y
) and using the Schwarz inequality, we get (43)
V
g
f
L
2
μ
k
⊗
μ
k
2
≤
2
2
γ
+
d
∬
R
d
x
2
V
g
f
x
,
y
2
d
μ
k
x
d
μ
k
y
1
/
2
∬
R
d
z
2
F
k
V
g
f
·
,
y
z
2
d
μ
k
z
d
μ
k
y
1
/
2
.But by Proposition 4(ii), Fubini-Tonelli’s theorem, (16), Proposition 2(ii), and Theorem 1(iii), we have (44)
∬
R
d
z
2
F
k
V
g
f
·
,
y
z
2
d
μ
k
z
d
μ
k
y
=
∬
R
d
z
2
τ
y
F
k
g
2
z
F
k
f
z
2
d
μ
k
z
d
μ
k
y
=
g
L
r
a
d
2
μ
k
2
∫
R
d
z
2
F
k
f
z
2
d
μ
k
z
.This yields the result and completes the proof of the theorem.Theorem 8.
Letg
∈
L
r
a
d
2
(
μ
k
) be a nonzero function and s
≥
1. Then, for f
∈
L
2
(
μ
k
), one has(45)
x
s
V
g
f
L
2
μ
k
⊗
μ
k
z
s
F
k
f
L
2
μ
k
≥
γ
+
d
2
s
f
L
2
μ
k
2
g
L
r
a
d
2
μ
k
.Proof.
Lets
≥
1 and let f
∈
L
2
(
μ
k
), f
≠
0, such that x
s
V
g
(
f
)
L
2
(
μ
k
⊗
μ
k
)
+
z
s
F
k
(
f
)
L
2
(
μ
k
)
<
∞. Then, for s
>
1, we have (46)
x
s
V
g
f
L
2
μ
k
⊗
μ
k
2
/
s
V
g
f
L
2
μ
k
⊗
μ
k
2
/
s
′
=
x
2
V
g
f
2
/
s
L
s
μ
k
⊗
μ
k
V
g
f
2
/
s
′
L
s
′
μ
k
⊗
μ
k
,where s
′ is defined as usual by 1
/
s
+
1
/
s
′
=
1. By Hölder’s inequality we get (47)
x
s
V
g
f
L
2
μ
k
⊗
μ
k
1
/
s
V
g
f
L
2
μ
k
⊗
μ
k
1
/
s
′
≥
x
V
g
f
L
2
μ
k
⊗
μ
k
.Thus, for all s
≥
1, we have (48)
x
s
V
g
f
L
2
μ
k
⊗
μ
k
1
/
s
≥
x
V
g
f
L
2
μ
k
⊗
μ
k
V
g
f
L
2
μ
k
⊗
μ
k
1
-
1
/
s
,with equality if s
=
1. In the same manner and using Theorem 1(iii), we have, for s
≥
1, (49)
z
s
F
k
f
L
2
μ
k
1
/
s
≥
z
F
k
f
L
2
μ
k
F
k
f
L
2
μ
k
1
-
1
/
s
,with equality if s
=
1. By (48) and (49), for all s
≥
1, we have (50)
x
s
V
g
f
L
2
μ
k
⊗
μ
k
1
/
s
z
s
F
k
f
L
2
μ
k
1
/
s
≥
x
V
g
f
L
2
μ
k
⊗
μ
k
z
F
k
f
L
2
μ
k
f
L
2
μ
k
2
g
L
r
a
d
2
μ
k
1
-
1
/
s
,with equality if s
=
1. Applying Theorem 7, we obtain (51)
x
s
V
g
f
L
2
μ
k
⊗
μ
k
z
s
F
k
f
L
2
μ
k
≥
γ
+
d
2
s
f
L
2
μ
k
2
g
L
r
a
d
2
μ
k
,which completes the proof of the theorem.From (48) and (49) we deduce the following remark.Remark 9.
Letg
∈
L
r
a
d
2
(
μ
k
) be a nonzero function and a
,
b
≥
1. Then, for f
∈
L
2
(
μ
k
), we have (52)
x
a
V
g
f
L
2
μ
k
⊗
μ
k
b
z
b
F
k
f
L
2
μ
k
a
≥
γ
+
d
2
a
b
f
L
2
μ
k
a
+
b
g
L
r
a
d
2
μ
k
b
.Forλ
>
0, we define the dilation of f
∈
L
2
(
μ
k
) by (53)
f
λ
x
≔
1
λ
γ
+
d
/
2
f
x
λ
,
x
∈
R
d
.Then (54)
f
λ
L
2
μ
k
=
f
L
2
μ
k
,
(55)
F
k
f
λ
z
=
λ
γ
+
d
/
2
F
k
f
λ
z
,
τ
x
f
λ
y
=
1
λ
γ
+
d
/
2
τ
x
/
λ
f
y
λ
.Let us now turn to establishing Heisenberg-type uncertainty principle for the Dunkl-Wigner transformV
g of magnitude s
>
0. Thus, we consider the following lemma.Lemma 10.
Letλ
>
0 and let g
∈
L
r
a
d
2
(
μ
k
) be a nonzero function. Then, for f
∈
L
2
(
μ
k
), one has (56)
V
g
λ
f
λ
x
,
y
=
V
g
f
x
λ
,
λ
y
,
x
,
y
∈
R
d
.Proof.
From Proposition4(ii), we have (57)
V
g
λ
f
λ
x
,
y
=
∫
R
d
E
k
i
x
,
z
F
k
f
λ
z
τ
y
F
k
g
λ
2
z
d
μ
k
z
.But by (55) we have (58)
τ
y
F
k
g
λ
2
z
=
λ
2
γ
+
d
τ
λ
y
F
k
g
2
λ
z
.Thus, (59)
V
g
λ
f
λ
x
,
y
=
λ
2
γ
+
d
∫
R
d
E
k
i
x
,
z
F
k
f
λ
z
τ
λ
y
F
k
g
2
λ
z
d
μ
k
z
=
∫
R
d
E
k
i
x
,
z
λ
F
k
f
z
τ
λ
y
F
k
g
2
z
d
μ
k
z
=
V
g
f
x
λ
,
λ
y
,which gives the result.Theorem 11 (Heisenberg-type uncertainty principle forV
g).
Lets
>
0. Then there exists a constant c
(
k
,
s
)
>
0 such that, for all f
∈
L
2
(
μ
k
) and g
∈
L
r
a
d
2
(
μ
k
), one has (60)
x
s
V
g
f
L
2
μ
k
⊗
μ
k
y
s
V
g
f
L
2
μ
k
⊗
μ
k
≥
c
k
,
s
f
L
2
μ
k
2
g
L
r
a
d
2
μ
k
2
.Proof.
Lets
,
r
0
>
0 and B
r
0
≔
{
(
x
,
y
)
∈
R
d
×
R
d
:
x
,
y
<
r
0
}, where (
x
,
y
)
=
(
x
2
+
y
2
)
1
/
2. Fix r
0 such that μ
k
⊗
μ
k
(
B
r
0
)
<
1. We write (61)
V
g
f
L
2
μ
k
⊗
μ
k
2
=
χ
B
r
0
V
g
f
L
2
μ
k
⊗
μ
k
2
+
χ
R
d
∖
B
r
0
V
g
f
L
2
μ
k
⊗
μ
k
2
≤
χ
B
r
0
V
g
f
L
2
μ
k
⊗
μ
k
2
+
r
0
-
2
s
x
,
y
s
V
g
f
L
2
μ
k
⊗
μ
k
2
.But from Hölder’s inequality and Proposition 4(iii) we have (62)
χ
B
r
0
V
g
f
L
2
μ
k
⊗
μ
k
2
≤
μ
k
⊗
μ
k
B
r
0
V
g
f
L
∞
μ
k
⊗
μ
k
2
≤
μ
k
⊗
μ
k
B
r
0
f
L
2
μ
k
2
g
L
r
a
d
2
μ
k
2
.Therefore, by Theorem 5(i),(63)
r
0
2
s
1
-
μ
k
⊗
μ
k
B
r
0
f
L
2
μ
k
2
g
L
r
a
d
2
μ
k
2
≤
x
,
y
s
V
g
f
L
2
μ
k
⊗
μ
k
2
.Using the fact that |
(
x
,
y
)
|
s
=
(
|
x
|
2
+
|
y
|
2
)
s
/
2
≤
2
s
/
2
(
|
x
|
s
+
|
y
|
s
) we deduce that(64)
2
c
k
,
s
f
L
2
μ
k
2
g
L
r
a
d
2
μ
k
2
≤
x
s
V
g
f
L
2
μ
k
⊗
μ
k
2
+
y
s
V
g
f
L
2
μ
k
⊗
μ
k
2
,where (65)
c
k
,
s
=
r
0
2
s
2
s
+
1
1
-
μ
k
⊗
μ
k
B
r
0
.Replacing f and g by f
λ and g
λ, respectively, in the previous inequality, we obtain by Lemma 10 and by a suitable change of variables (66)
2
c
k
,
s
f
L
2
μ
k
2
g
L
r
a
d
2
μ
k
2
≤
λ
2
s
x
s
V
g
f
L
2
μ
k
⊗
μ
k
2
+
λ
-
2
s
y
s
V
g
f
L
2
μ
k
⊗
μ
k
2
.By setting λ
=
(
y
s
V
g
(
f
)
L
2
(
μ
k
⊗
μ
k
)
/
x
s
V
g
(
f
)
L
2
(
μ
k
⊗
μ
k
)
)
1
/
2
s in the right-hand side of the previous inequality we obtain the desired result.We will now prove a local uncertainty principle for the Dunkl-Wigner transformV
g, which extends the result of Faris [22].Theorem 12 (local uncertainty principle forV
g).
Lets
>
0. Then there exists a constant b
(
k
,
s
)
>
0 such that, for all f
∈
L
2
(
μ
k
) and g
∈
L
r
a
d
2
(
μ
k
) and for all measurable subset E of R
d
×
R
d such that 0
<
μ
k
⊗
μ
k
(
E
)
<
∞, one has(67)
χ
E
V
g
f
L
2
μ
k
⊗
μ
k
≤
b
k
,
s
μ
k
⊗
μ
k
E
1
/
2
x
,
y
s
V
g
f
L
2
μ
k
⊗
μ
k
.Proof.
Lets
>
0 and let E be a measurable subset of R
d
×
R
d such that 0
<
μ
k
⊗
μ
k
(
E
)
<
∞. From Hölder’s inequality and Proposition 4(iii) we have (68)
χ
E
V
g
f
L
2
μ
k
⊗
μ
k
≤
μ
k
⊗
μ
k
E
1
/
2
V
g
f
L
∞
μ
k
⊗
μ
k
≤
μ
k
⊗
μ
k
E
1
/
2
f
L
2
μ
k
g
L
r
a
d
2
μ
k
.From (63) there exists b
(
k
,
s
)
>
0 such that (69)
f
L
2
μ
k
g
L
r
a
d
2
μ
k
≤
b
k
,
s
x
,
y
s
V
g
f
L
2
μ
k
⊗
μ
k
.Therefore we obtain the desired result.



---

*Source: 7637346-2016-09-27.xml*