# On Some Inequalities of Uncertainty Principles Type in Quantum Calculus

Authors: Ahmed Fitouhi; Néji Bettaibi; Rym H. Bettaieb; Wafa Binous
Journal: International Journal of Mathematics and Mathematical Sciences
 (2008)

---

## Abstract

The aim of this paper is to generalize theq-Heisenberg uncertainty principles
studied by Bettaibi et al. (2007),  to state local uncertainty principles for the q-Fourier-cosine, the q-Fourier-sine,
and the q-Bessel-Fourier transforms, then to provide an inequality of Heisenberg-Weyl-type for the q-Bessel-Fourier transform.


---

## Body

## 1. Introduction

The uncertainty principle is a metatheorem in harmonic
analysis that asserts, with   the use of some inequalities, that a function and
its Fourier transform cannot be sharply localized. We refer to the survey
article by Folland and Sitaram [1] and the book of Havin and Jöricke [2] for various classical
uncertainty principles of different nature which may be found in the literature.In [3], the authors gave q-analogues of the Heisenberg uncertainty
principle for the q-Fourier-cosine and the q-Fourier-sine transforms. One of the aims of
this paper is to provide a generalization of their work next to state local
uncertainty principles for various q-Fourier transforms.This paper is organized as follows. In Section2, we
present some preliminaries results and notations that will be useful in the
sequel. In Section 3, we prove a density theorem and a q-analogue of the Hausdorff-Young inequality.
Then, we state a generalization of the q-Heisenberg uncertainty principle for the q-Fourier-cosine and the q-Fourier-sine transforms. In Section 4, we
state local uncertainty principles for the q-Fourier-cosine, q-Fourier-sine, and q-Bessel-Fourier transforms. Then, we give a
Heisenberg-Weyl-type inequality for some q-Bessel-Fourier transform.


## 2. Notations and Preliminaries

Throughout
this paper, we assumeq
∈
]
0
,
1
[.
We recall some usual notions and notations used in the q-theory (see [4, 5]). We refer to the book by
Gasper and Rahman [4] for the definitions, notations, and properties of the q-shifted factorials and the q-hypergeometric functions.We writeℝ
q
=
{
±
q
n
:
n
∈
ℤ
}, ℝ
q
,
+
=
{
q
n
:
n
∈
ℤ
},
and(2.1)
[
x
]
q
=
1
−
q
x
1
−
q
,

x
∈
ℂ
,


[
n
]
q
!
=
(
q
;
q
)
n
(
1
−
q
)
n
,

n
∈
ℕ
.The q-derivative of a function f is given by(2.2)
(
D
q
f
)
(
x
)
=
f
(
x
)
−
f
(
q
x
)
(
1
−
q
)
x

if
x
⁢
≠
0,
(
D
q
f
)
(
0
)
=
lim
k
→
+
∞
(
D
q
f
)
(
q
k
),
provided that the limit exists.Theq-Jackson integrals from 0 to a and from 0 to ∞,
of a function f,
are (see [6])(2.3)
∫

0
a
f
(
x
)
d
q
x
=
(
1
−
q
)
a
∑
n
=
0
∞
f
(
a
q
n
)
q
n
,


∫

0
∞
f
(
x
)
d
q
x
=
(
1
−
q
)
∑
n
=
−
∞
∞
f
(
q
n
)
q
n
,provided that the sums converge
absolutely.Theq-Jackson integral in a generic interval [
a
,
b
] is given by (see [6])(2.4)
∫
a
b
f
(
x
)
d
q
x
=
∫
0
b
f
(
x
)
d
q
x
−
∫
0
a
f
(
x
)
d
q
x
.The q -integration by parts rule is given, for
suitable functions f and g
,  by(2.5)
∫
a
b
g
(
x
)
D
q
f
(
x
)
d
q
x
=
f
(
b
)
g
(
b
)
−
f
(
a
)
g
(
a
)
−
∫
a
b
f
(
q
x
)
D
q
g
(
x
)
d
q
x
.Jackson (see
[6]) defined a q-analogue of the Gamma function
by(2.6)
Γ
q
(
x
)
=
(
q
;
q
)
∞
(
q
x
;
q
)
∞
(
1
−
q
)
1
−
x
,

x
⁢
≠
0,
−
1,
−
2
,
….The third Jackson (2.7)q-Bessel function (see [,]) is(2.7)
J
ν
(
z
;
q
2
)
=
z
ν
(
1
−
q
2
)
ν
Γ
q
2
(
ν
+
1
)
⁢
1
φ
1
(
0
;
q
2
ν
+
2
;
q
2
,
q
2
z
2
)
, and the q-trigonometric functions (q-cosine and q-sine) are defined by (see [9])(2.8)
cos
⁡
(
x
;
q
2
)
=
Γ
q
2
(
1
/
2
)
q
(
1
+
q
−
1
)
1
/
2
x
1
/
2
J
−
1
/
2
(
1
−
q
q
x
;
q
2
)
=
∑
n
=
0
∞
(
−
1
)
n
q
n
(
n
−
1
)
x
2
n
[
2
n
]
q
!
,
sin
⁡
(
x
;
q
2
)
=
Γ
q
2
(
1
/
2
)
(
1
+
q
−
1
)
1
/
2
x
1
/
2
J
1
/
2
(
1
−
q
q
x
;
q
2
)
=
∑
n
=
0
∞
(
−
1
)
n
q
n
(
n
−
1
)
x
2
n
+
1
[
2
n
+
1
]
q
!
.They verify (2.9)
D
q
cos
⁡
(
x
;
q
2
)
=
−
1
q
sin
⁡
(
q
x
;
q
2
)
,


D
q
sin
⁡
(
x
;
q
2
)
=
cos
⁡
(
x
;
q
2
)
.We need the following spaces and
norms.(i)
𝒮
*
q
(
ℝ
q
) is the space of even functions f on ℝ
q satisfying (2.10)
∀
n
,
m
∈
N
,

P
n
,
m
,
q
(
f
)
=
sup
⁡
x
∈
R
q
;
0
≤
k
≤
n
|
(
1
+
x
2
)
m
D
q
k
f
(
x
)
|
<
+
∞
.
(ii)
L
q
n
(
ℝ
q
,
+
,
x
2
ν
+
1
d
q
x
), n
≥
1, ν
≥
−
1
/
2,
is the set of all functions defined on ℝ
q
,
+ such that(2.11)
∥
f
∥
n
,
ν
,
q
=
{
∫
0
∞
|
f
(
x
)
|
n
x
2
ν
+
1
d
q
x
}
1
/
n
<
∞
.
(iii)
L
q
n
(
ℝ
q
,
+
)
=
L
q
n
(
ℝ
q
,
+
,
d
q
x
), n
≥
1,
and ∥
⋅
∥
n
,
q
=
∥
⋅
∥
n
,
−
1
/
2
,
q
.
(iv)
L
q
∞
(
ℝ
q
,
+
) is the set of all bounded functions on ℝ
q
,
+.  We write ∥
f
∥
∞
,
q
=
sup
x
∈
ℝ
q
,
+
|
f
(
x
)
|
.


## 3. Generalization of the Heisenberg Uncertainty Principle

Theq-Fourier-cosine and the q-Fourier-sine transforms are defined as (see
[8, 9])(3.1)
ℱ
q
(
f
)
(
x
)
=
c
q
∫
0
∞
f
(
t
)
cos
⁡
(
x
t
;
q
2
)
d
q
t
,


ℱ
q
(
f
)
(
x
)
=
c
q
∫
0
∞
f
(
t
)
sin
(
x
t
;
q
2
)
d
q
t
,where(3.2)
c
q
=
(
1
+
q
−
1
)
1
/
2
Γ
q
2
(
1
/
2
)
.Letting q
↑
1 subject to the condition (
Log
(
1
−
q
)
/
Log
(
q
)
)
∈
ℤ gives, at least formally, the classical
Fourier transforms (see [3, 10]). In the remainder of the present paper, we assume
that this condition holds.It was shown in [8, 9] that we have
the following result.Proposition 3.1.
(1) Forf
∈
L
q
1
(
ℝ
q
,
+
),
one has ℱ
q
(
f
)
∈
L
q
∞
(
ℝ
q
,
+
) and (3.3)
∥
ℱ
q
(
f
)
∥
∞
,
q
≤
(
1
+
q
−
1
)
1
/
2
Γ
q
2
(
1
/
2
)
(
q
;
q
)
∞
2
∥
f
∥
1,
q
.
(2)ℱ
q is an isomorphism of L
q
2
(
ℝ
q
,
+
) (resp., S
*
,
q
(
ℝ
q
)) onto itself. Moreover, one
has ℱ
q
−
1
=
ℱ
q and the following Plancherel formula:(3.4)
∥
ℱ
q
(
f
)
∥
2,
q
=
∥
f
∥
2,
q
,

f
∈
L
q
2
(
R
q
,
+
)
.Similarly, it was shown in [3, 8] that the q-Fourier-sine transform verifies the following
properties.Proposition 3.2.
(1) Forf
∈
L
q
1
(
ℝ
q
,
+
),
one has ℱ
q
(
f
)
∈
L
q
∞
(
ℝ
q
,
+
) and (3.5)
∥
ℱ
q
(
f
)
∥
∞
,
q
≤
(
1
+
q
−
1
)
1
/
2
Γ
q
2
(
1
/
2
)
(
q
;
q
)
∞
2
∥
f
∥
1,
q
.
(2)ℱ
q is an isomorphism of L
q
2
(
ℝ
q
,
+
) onto itself; its inverse is given by ℱ
q
−
1
=
(
1
/
q
2
)
q
ℱ. One has the following Plancherel formula:(3.6)
∥
q
ℱ
(
f
)
∥
2,
q
=
q
∥
f
∥
2,
q
,

f
∈
L
q
2
(
R
q
,
+
)
.Let us now
state the following useful density result.Proposition 3.3.
For alln
≥
1, S
*
,
q
(
ℝ
q
) is dense in L
q
n
(
ℝ
q
,
+
).Proof.
Letn
≥
1 and f
∈
L
q
n
(
ℝ
q
,
+
).
For p
∈
ℕ,
put f
p
=
f
⋅
χ
[
q
p
,
q
−
p
],
where χ
[
q
p
,
q
−
p
] is the characteristic function of [
q
p
,
q
−
p
].
It is clear that for allp
∈
ℕ, f
p
∈
S
*
,
q
(
ℝ
q
) and |
f
−
f
p
|
n
≤
|
f
|
n.
So, the Lebesgue theorem implies that (
f
p
)
p converges to f in L
q
n
(
ℝ
q
,
+
).Remark 3.4.
Using the density ofS
*
,
q
(
ℝ
q
) in L
q
n
(
ℝ
q
,
+
) (n
≥
1), one can see that the q-Fourier-cosine (resp., q-Fourier-sine) transform has a unique
continuous extension on L
q
n
(
ℝ
q
,
+
),
that will also be denoted as ℱ
q (resp., ℱ
q). We have the following q-analogue of the Hausdorff-Young inequality.Theorem 3.5.
Letn
∈
]
1
,
2
] (resp., n
=
1) and m
=
n
/
(
n
−
1
) (resp., m
=
∞) be the dual exponent of n.
For all f in L
q
n
(
ℝ
q
,
+
),
the functions ℱ
q
(
f
) and ℱ
q
(
f
) belong to L
q
m
(
ℝ
q
,
+
),
and one has(3.7)
∥
ℱ
q
(
f
)
∥
m
,
q
≤
C
1
∥
f
∥
n
,
q
,


∥
ℱ
q
(
f
)
∥
m
,
q
≤
C
2
∥
f
∥
n
,
q
,where(3.8)
C
1
=
(
(
1
+
q
−
1
)
1
/
2
Γ
q
2
(
1
/
2
)
(
q
;
q
)
∞
2
)
1
−
2
(
(
n
−
1
)
/
n
)
,


C
2
=
(
(
1
+
q
−
1
)
1
/
2
Γ
q
2
(
1
/
2
)
(
q
;
q
)
∞
2
)
1
−
2
(
(
n
−
1
)
/
n
)
q
2
(
(
n
−
1
)
/
n
)
.
Proof.
The result is a
direct consequence of [11, Theorem 1.3.4, page 35], and Propositions 3.1 and 3.2, by
taking S
*
,
q
(
ℝ
q
) as a set of simple functions.The following lemma gives relations between the two
Fourierq-trigonometric transforms.Lemma 3.6.
(1) Forf
∈
L
q
2
(
ℝ
q
,
+
) such that D
q
f
∈
L
q
2
(
ℝ
q
,
+
),
one has(3.9)
ℱ
q
(
D
q
f
)
(
λ
)
=
−
λ
q
ℱ
q
(
f
)
(
λ
q
)
,

λ
∈
R
q
,
+
.
(2) Additionally, iflim
n
→
+
∞
f
(
q
n
)
=
0,
then(3.10)
ℱ
q
(
D
q
f
)
(
λ
)
=
λ
q
2
ℱ
q
(
f
)
(
λ
)
,

λ
∈
R
q
,
+
.Proof.
The same steps as in the
proof of  [3, Lemma 2]; the q-integration by parts rule and the fact that(3.11)
∫
0
∞
f
(
t
)
d
q
t
=
lim
⁡
n
→
+
∞
∫
q
n
q
−
n
f
(
t
)
d
q
tgive the result.In [3], the authors proved the following q-analogues of the Heisenberg uncertainty
principle.Theorem 3.7.
Letf be in L
q
2
(
ℝ
q
,
+
) such that D
q
f is in L
q
2
(
ℝ
q
,
+
).
Then,(3.12)
∥
t
f
∥
2,
q
∥
λ
ℱ
q
(
f
)
∥
2,
q
≥
q
q
3
/
2
+
1
∥
f
∥
2,
q
2
.In addition, if lim
n
→
+
∞
f
(
q
n
)
=
0,
one has(3.13)
∥
t
f
∥
2,
q
∥
λ
ℱ
q
(
f
)
∥
2,
q
≥
q
q
−
3
/
2
+
1
∥
f
∥
2,
q
2
.Now, we are in a position to generalize Theorem3.7. One obvious way to
generalize it is to replace the L
q
2 norms by L
q
n norms. This is the purpose of the following
result.Theorem 3.8.
For1
≤
n
≤
2 and f
∈
L
q
2
(
ℝ
q
,
+
),
one has(3.14)
∥
f
∥
2,
q
2
≤
C
1
′
∥
x
f
∥
n
,
q
∥
λ
ℱ
q
(
f
)
∥
n
,
q
,
(3.15)
∥
f
∥
2,
q
2
≤
C
2
′
∥
x
f
∥
n
,
q
∥
λ
ℱ
q
(
f
)
∥
n
,
q
,where(3.16)
C
1
′
=
q
−
1
+
1
/
n
(
1
+
q
−
(
n
+
1
)
/
n
)
C
2
,


C
2
′
=
q
−
1
(
1
+
q
−
(
n
+
1
)
/
n
)
C
1
,with C
1 and C
2 being given by (3.8).Proof.
The casen
=
2 has been dealt with in Theorem 3.7. Now, assume 1
≤
n
<
2 and let m be the dual exponent of n.
Let f
∈
S
*
,
q
(
ℝ
q
) such that lim
t
→
0
f
(
t
)
=
0.
From the relation(3.17)
D
q
(
f
f
¯
)
(
t
)
=
D
q
f
(
t
)
f
¯
(
t
)
+
f
(
q
t
)
D
q
f
¯
(
t
)
,the q-integration by parts rule, and the Hölder
inequality, we have, since t
|
f
(
t
)
|
2 tends to 0 as t tends to ∞ in ℝ
q
,
+,(3.18)
1
q
∫
0
∞
|
f
(
t
)
|
2
d
q
t
=
|
∫
0
∞
t
D
q
(
f
f
¯
)
(
t
)
d
q
t
|
≤
∫
0
∞
|
t
D
q
f
(
t
)
f
¯
(
t
)
|
d
q
t
+
∫
0
∞
|
t
f
(
q
t
)
D
q
f
¯
(
t
)
|
d
q
t
≤
(
∫
0
∞
|
t
f
¯
(
t
)
|
n
d
q
t
)
1
/
n
(
∫
0
∞
|
D
q
f
(
t
)
|
m
d
q
t
)
1
/
m

+
(
∫
0
∞
|
t
f
(
q
t
)
|
n
d
q
t
)
1
/
n
(
∫
0
∞
|
D
q
f
¯
(
t
)
|
m
d
q
t
)
1
/
m
.However, the change of variable u
=
q
t gives(3.19)
(
∫
0
∞
|
t
f
(
q
t
)
|
n
d
q
t
)
1
/
n
=
q
−
(
n
+
1
)
/
n
(
∫
0
∞
|
t
f
(
t
)
|
n
d
q
t
)
1
/
n
.So,(3.20)
1
q
∫
0
∞
|
f
(
t
)
|
2
d
q
t
≤
(
1
+
q
−
(
n
+
1
)
/
n
)
∥
t
f
∥
n
,
q
∥
D
q
(
f
)
∥
m
,
q
.On the other hand, we have D
q
(
f
)
=
ℱ
q
[
ℱ
q
(
D
q
(
f
)
)
]
=
q
−
2
ℱ
q
[
ℱ
q
(
D
q
(
f
)
)
] since D
q
(
f
) is in L
q
2
(
ℝ
q
,
+
).
Then, by using Lemma 3.6 and the q-analogue of the Hausdorff-Young inequality,
we obtain(3.21)
∥
D
q
(
f
)
∥
m
,
q
≤
C
1
∥
ℱ
q
(
D
q
(
f
)
)
∥
n
,
q
=
C
1
q
2
∥
λ
ℱ
q
(
f
)
∥
n
,
q
,
∥
D
q
(
f
)
∥
m
,
q
≤
q
−
2
C
2
∥
q
ℱ
(
D
q
(
f
)
)
∥
n
,
q
=
q
−
2
C
2
∥
λ
q
ℱ
q
(
f
)
(
λ
q
)
∥
n
,
q
=
q
−
2
+
1
/
n
C
2
∥
λ
ℱ
q
(
f
)
∥
n
,
q
.
 Thus,(3.22)
∥
f
∥
2,
q
2
≤
q
−
1
(
1
+
q
−
(
n
+
1
)
/
n
)
C
1
∥
t
f
∥
n
,
q
∥
λ
ℱ
q
(
f
)
∥
n
,
q
,
∥
f
∥
2,
q
2
≤
q
−
1
+
1
/
n
(
1
+
q
−
(
n
+
1
)
/
n
)
C
2
∥
t
f
∥
n
,
q
∥
λ
ℱ
q
(
f
)
∥
n
,
q
.Now, let f
∈
L
q
2
(
ℝ
q
,
+
);
it is easy to see that for all p
∈
ℕ, f
p
=
f
χ
[
q
p
,
q
−
p
]
∈
S
*
,
q
(
ℝ
q
), lim
t
→
0
f
p
(
t
)
=
0,
and (
f
p
)
p converges to f in L
q
2
(
ℝ
q
,
+
).
Moreover, if the right-hand side of (3.14) (resp., (3.15)) is finite, then the
functions t
f and λ
ℱ
q
(
f
) (resp., λ
ℱ
q
(
f
)) are in L
q
n
(
ℝ
q
,
+
)
, and they are limits in L
q
n
(
ℝ
q
,
+
) (as p tends to ∞) of t
f
p and λ
ℱ
q
(
f
p
) (resp., λ
ℱ
q
(
f
p
)), respectively. Finally, the substitution of f
p in (3.22) and a passage to the limit
when p tends to ∞ complete the proof.


## 4. Local Uncertainty Principles

In the literature, the first classical local
inequalities were obtained by  Faris (see [12]) in 1978, and they were
generalized by  Price (see [13, 14]) in 1983 and 1987. In this section, we will
generalize Price's results by giving their q-analogues.


### 4.1. Local Uncertainty Principles for theq-Fourier Trigonometric Transforms

Theorem 4.1.
If0
<
a
<
1
/
2,
there is a constant K
=
K
(
a
,
q
) such that for all bounded subset E of ℝ
q
,
+ and all f
∈
L
q
2
(
ℝ
q
,
+
),
one has(4.1)
∫
E
|
ℱ
q
(
f
)
(
λ
)
|
2
d
q
λ
≤
K
|
E
|
2
a
∥
x
a
f
∥
2,
q
2
.Here, |
E
|
=
∫
0
∞
χ
E
(
x
)
d
q
x and K
=
(
(
c
˜
q
/
[
1
−
2
a
]
q
)
(
(
1
−
2
a
)
/
2
a
)
)
4
a
(
1
/
(
1
−
2
a
)
2
),
where c
˜
q
=
(
1
+
q
−
1
)
1
/
2
/
Γ
q
2
(
1
/
2
)
(
q
;
q
)
∞
2
.
Proof.
Forr
>
0,
let χ
r
=
χ
[
0
,
r
] be the characteristic function of [
0
,
r
] and χ
˜
r
=
1
−
χ
r.
Then, forr
>
0,
we have, since f
⋅
χ
r
∈
L
q
1
(
ℝ
q
,
+
),(4.2)
(
∫
E
|
ℱ
q
(
f
)
(
λ
)
|
2
d
q
λ
)
1
/
2
=
∥
ℱ
q
(
f
)
χ
E
∥
2,
q
≤
∥
ℱ
q
(
f
⋅
χ
r
)
χ
E
∥
2,
q
+
∥
ℱ
q
(
f
⋅
χ
˜
r
)
χ
E
∥
2,
q
≤
|
E
|
1
/
2
∥
ℱ
q
(
f
⋅
χ
r
)
∥
∞
,
q
+
∥
ℱ
q
(
f
⋅
χ
˜
r
)
∥
2,
q
,and by the use of the Hölder
inequality, we obtain(4.3)
∥
ℱ
q
(
f
⋅
χ
r
)
∥
∞
,
q
≤
c
˜
q
∥
f
⋅
χ
r
∥
1,
q


=
c
˜
q
∥
x
−
a
χ
r
⋅
x
a
f
∥
1,
q
≤
c
˜
q
∥
x
−
a
χ
r
∥
2,
q
∥
x
a
f
∥
2,
q
≤
c
˜
q
[
1
−
2
a
]
q
r
1
/
2
−
a
∥
x
a
f
∥
2,
q
.On the other hand, since f
∈
L
q
2
(
ℝ
q
,
+), we have f
⋅
χ
˜
r
∈
L
q
2
(
ℝ
q
,
+
),
and by the Plancherel formula, we get(4.4)
∥
ℱ
q
(
f
⋅
χ
˜
r
)
∥
2,
q
=
∥
f
⋅
χ
˜
r
∥
2,
q
=
∥
x
−
a
χ
˜
r
.
x
a
f
∥
2,
q
≤
∥
x
−
a
χ
˜
r
∥
∞
,
q
∥
x
a
f
∥
2,
q
≤
r
−
a
∥
x
a
f
∥
2,
q
.So,(4.5)
(
∫
E
|
ℱ
q
(
f
)
(
λ
)
|
2
d
q
λ
)
1
/
2
≤
(
c
˜
q
[
1
−
2
a
]
q
|
E
|
1
/
2
r
1
/
2
−
a
+
r
−
a
)
∥
x
a
f
∥
2,
q
.The desired result is obtained
by minimizing the right-hand side of the previous inequality over r
>
0.Corollary 4.2.
For0
<
a
<
1
/
2 and b
>
0,
there is a constant K
a
,
b such that for all f
∈
L
q
2
(
ℝ
q
,
+
),
one has (4.6)
∥
f
∥
2,
q
(
a
+
b
)
≤
K
a
,
b
∥
x
a
f
∥
2,
q
b
∥
λ
b
ℱ
q
(
f
)
∥
2,
q
a
.Proof.
Forr
>
0,
put E
r
=
[
0
,
r
[
∩
ℝ
q
,
+ and E
˜
r
=
[
r
,
+
∞
[
∩
ℝ
q
,
+.
It is easy to see that E
r is a bounded subset of ℝ
q
,
+ and |
E
r
|
≤
r
.
Then, from the Plancherel formula and Theorem4.1, we
have(4.7)
∥
f
∥
2,
q
2
=
∥
ℱ
q
(
f
)
∥
2,
q
2
=
∫
E
r
|
ℱ
q
(
f
)
|
2
(
λ
)
d
q
λ
+
∫
E
˜
r
|
ℱ
q
(
f
)
|
2
(
λ
)
d
q
λ
≤
K
r
2
a
∥
x
a
f
∥
2,
q
2
+
r
−
2
b
∥
λ
b
ℱ
q
(
f
)
∥
2,
q
2
.Choosing r
>
0 so as to minimize the right-hand side of the
inequality, we obtain ∥
f
∥
2,
q
2
≤
(
K
a
,
b
∥
x
a
f
∥
2,
q
b
∥
λ
b
ℱ
q
(
f
)
∥
2,
q
a
)
2
/
(
a
+
b
),
with K
a
,
b
=
(
(
a
/
b
)
b
/
(
a
+
b
)
+
(
b
/
a
)
a
/
(
a
+
b
)
)
(
a
+
b
)
/
2
K
b
/
2,
and K is the constant given in Theorem 4.1.In the same way, one can prove the following local
uncertainty principle for theq-Fourier-sine transform.Theorem 4.3.
If0
<
a
<
1
/
2,
there is a constant K
′
=
K
′
(
a
,
q
) such that for all bounded subset E of ℝ
q
,
+ and all f
∈
L
q
2
(
ℝ
q
,
+
),
one has(4.8)
∫
E
|
q
ℱ
(
f
)
(
λ
)
|
2
d
q
λ
≤
K
′
|
E
|
2
a
∥
x
a
f
∥
2,
q
2
,where K
′
=
(
(
c
˜
q
/
[
1
−
2
a
]
q
)
(
(
1
−
2
a
)
/
2
q
a
)
)
4
a
[
1
+
2
q
a
/
(
1
−
2
a
)
]
2.Corollary 4.4.
For0
<
a
<
1
/
2 and b
>
0,
there is a constant K
a
,
b
′ such that for all f
∈
L
q
2
(
ℝ
q
,
+
),
one has(4.9)
∥
f
∥
2,
q
(
a
+
b
)
≤
K
a
,
b
′
∥
x
a
f
∥
2,
q
b
∥
λ
b
F
q
(
f
)
∥
2,
q
a
,with K
a
,
b
′
=
(
(
a
/
b
)
b
/
(
a
+
b
)
+
(
b
/
a
)
a
/
(
a
+
b
)
)
(
a
+
b
)
/
2
(
K
′
)
b
/
2
q
−
(
a
+
b
)
.Proof.
The same steps of Corollary4.2 give the
result.Theorem 4.5.
Ifa
>
1
/
2,
there is a constant K
1
=
K
1
(
a
,
q
) such that for all bounded subset E of ℝ
q
,
+ and f
∈
L
q
2
(
ℝ
q
,
+
),
one has
(4.10)
∫
E
|
ℱ
q
(
f
)
(
λ
)
|
2
d
q
λ
≤
K
1
|
E
|
∥
f
∥
2,
q
(
2
−
1
/
a
)
∥
x
a
f
∥
2,
q
1
/
a
,
(4.11)
∫
E
|
ℱ
q
(
f
)
(
λ
)
|
2
d
q
λ
≤
K
1
|
E
|
∥
f
∥
2,
q
(
2
−
1
/
a
)
∥
x
a
f
∥
2,
q
1
/
a
.The proof of this result needs the following lemmas.Lemma 4.6.
Supposea
>
1
/
2,
then for all f
∈
L
q
2
(
ℝ
q
,
+
)
, such that x
a
f
∈
L
q
2
(
ℝ
q
,
+
),(4.12)
∥
f
∥
1,
q
2
≤
K
2
[
∥
f
∥
2,
q
2
+
∥
x
a
f
∥
2,
q
2
]
,where K
2
=
K
2
(
a
,
q
)
=
(
1
−
q
)
(
(
q
2
a
,
q
2
a
,
−
q
,
−
q
2
a
−
1
;
q
2
a
)
∞
/
(
q
,
q
2
a
−
1
,
−
q
2
a
,
−
1
;
q
2
a
)
∞
).Proof.
From
[15, Example 1], and the Hölder inequality, we have(4.13)
∥
f
∥
1,
q
2
=
[
∫
0
+
∞
(
1
+
x
2
a
)
1
/
2
|
f
(
x
)
|
(
1
+
x
2
a
)
−
1
/
2
d
q
x
]
2
≤
K
2
[
∥
f
∥
2,
q
2
+
∥
x
a
f
∥
2,
q
2
]
,where K
2
=
∫
0
+
∞
(
1
+
x
2
a
)
−
1
d
q
x
=
(
1
−
q
)
(
(
q
2
a
,
q
2
a
,
−
q
,
−
q
2
a
−
1
;
q
2
a
)
∞
/
(
q
,
q
2
a
−
1
,
−
q
2
a
,
−
1
;
q
2
a
)
∞
).Lemma 4.7.
Supposea
>
1
/
2,
then for all f
∈
L
q
2
(
ℝ
q
,
+
),
such that x
a
f
∈
L
q
2
(
ℝ
q
,
+
),
one has(4.14)
∥
f
∥
1,
q
≤
K
3
∥
f
∥
2,
q
(
1
−
1
/
2
a
)
∥
x
a
f
∥
2,
q
1
/
2
a
,where K
3
=
K
3
(
a
,
q
)
=
[
2
a
K
2
(
2
a
q
−
q
)
1
/
2
a
−
1
]
1
/
2
.Proof.
Fors
∈
ℝ
q
,
+
, define the function f
s by f
s
(
x
)
=
f
(
s
x
)
,
x
∈
ℝ
q
,
+.
We have∥
f
s
∥
1,
q
=
s
−
1
∥
f
∥
1,
q, ∥
x
a
f
s
∥
2,
q
2
=
s
−
2
a
−
1
∥
x
a
f
∥
2,
q
2
.
Replacement off by f
s in Lemma 4.6 gives(4.15)
∥
f
∥
1,
q
2
≤
K
2
[
s
∥
f
∥
2,
q
2
+
s
−
2
a
+
1
∥
x
a
f
∥
2,
q
2
]
.Now, for all r
>
0
, put α
(
r
)
=
Log
(
r
)
/
Log
(
q
)
−
E
(
Log
(
r
)
/
Log
(
q
)
)
. We have s
=
(
r
/
q
α
(
r
)
)
∈
ℝ
q
,
+ and r
≤
s
<
r
/
q
. Then, for all r
>
0
,
(4.16)
∥
f
∥
1,
q
2
≤
K
2
[
r
q
∥
f
∥
2,
q
2
+
r
−
2
a
+
1
∥
x
a
f
∥
2,
q
2
]
.The right-hand side of this
inequality is minimized by choosing(4.17)
r
=
(
2
a
−
1
)
1
/
2
a
q
1
/
2
a
∥
f
∥
2,
q
−
1
/
a
∥
x
a
f
∥
2,
q
1
/
a
.When this is done, we obtain the
result.Proof of Theorem4.5.
Since the proofs of the two statements are  similar, it is sufficient to
prove (4.11).
LetE be a bounded subset of ℝ
q
,
+.
When the right-hand side of the inequality (4.11) is finite, Lemma 4.6 implies that f
∈
L
q
1
(
ℝ
q
,
+
);
so ℱ
q
(
f
) is defined and bounded on ℝ
q
,
+.
Using Proposition 3.1, Lemma 4.7, and the fact that(4.18)
∫
E
|
ℱ
q
(
f
)
(
λ
)
|
2
d
q
λ
≤
|
E
|
∥
ℱ
q
(
f
)
∥
∞
,
q
2
,we obtain the result with K
1
=
(
(
1
+
q
−
1
)
/
Γ
q
2
2
(
1
/
2
)
(
q
;
q
)
∞
4
)
K
3
2.Remark 4.8.
By the same technique as in the proof of Corollary4.2,
we can show that Theorem 4.5 leads to inequalities (4.6) and (4.9) with some
different constants.


### 4.2. Local Uncertainty Principles for theq-Bessel-Fourier Transform

Theq-Bessel-Fourier transform is defined (see
[16]) for f
∈
L
q
1
(
ℝ
q
,
+
,
x
2
ν
+
1
d
q
x
) by(4.19)
ℱ
ν
,
q
(
f
)
(
λ
)
=
c
ν
,
q
∫
0
∞
f
(
x
)
j
ν
(
λ
x
;
q
2
)
x
2
ν
+
1
d
q
x
,where(4.20)
j
ν
(
z
;
q
2
)
=
(
1
−
q
2
)
ν
Γ
q
2
(
ν
+
1
)
(
(
1
−
q
)
q
−
1
z
)
−
ν
J
ν
(
(
1
−
q
)
q
−
1
z
;
q
2
)is the normalized third Jackson q-Bessel function, and(4.21)
c
ν
,
q
=
(
1
+
q
−
1
)
−
ν
Γ
q
2
(
ν
+
1
)
.It was shown in [10] that for ν
≥
−
1
/
2,
we have the following result.Theorem 4.9.
(1) Forf
∈
L
q
1
(
ℝ
q
,
+
,
x
2
ν
+
1
d
q
x
),
one has ℱ
ν
,
q
(
f
)
∈
L
q
∞
(
ℝ
q
,
+
) and(4.22)
∥
ℱ
ν
,
q
(
f
)
∥
∞
,
q
≤
c
ν
,
q
(
q
;
q
2
)
∞
2
∥
f
∥
1,
ν
,
q
.
(2)ℱ
ν
,
q is an isomorphism of L
q
2
(
ℝ
q
,
+
,
x
2
ν
+
1
d
q
x
) onto itself, ℱ
ν
,
q
−
1
=
q
4
ν
+
2
ℱ
ν
,
q,
and one has the following Plancherel formula:(4.23)
∀
f
∈
L
q
2
(
R
q
,
+
,
x
2
ν
+
1
d
q
x
)
,

∥
ℱ
ν
,
q
∥
2,
ν
,
q
=
q
2
ν
+
1
∥
f
∥
2,
ν
,
q
.
The following
result states a local uncertainty principle for theq-Bessel-Fourier transform.Theorem 4.10.
Forν
≥
−
1
/
2 and 0
<
a
<
ν
+
1,
there is a constant K
a
,
ν
=
K
(
a
,
ν
,
q
) such that for all f
∈
L
q
2
(
ℝ
q
,
+
,
x
2
ν
+
1
d
q
x
) and all bounded subset E of ℝ
q
,
+,
one has(4.24)
∫
E
|
ℱ
ν
,
q
(
f
)
(
λ
)
|
2
λ
2
ν
+
1
d
q
λ
≤
K
a
,
ν
|
E
|
ν
a
/
(
ν
+
1
)
∥
x
a
f
∥
2,
ν
,
q
2
.Here, |
E
|
ν
=
∫
0
∞
χ
E
(
x
)
x
2
ν
+
1
d
q
x, c
˜
ν
,
q
=
c
ν
,
q
/
(
q
;
q
2
)
∞
2,
and
(4.25)
K
a
,
ν
=
(
c
˜
ν
,
q
[
2
ν
+
2
−
2
a
]
q
)
2
a
/
(
ν
+
1
)
[
(
a
q
2
ν
+
1
ν
+
1
−
a
)
1
−
a
/
(
ν
+
1
)
+
q
2
ν
+
1
(
a
q
2
ν
+
1
ν
+
1
−
a
)
−
a
/
(
ν
+
1
)
]
2.Proof
Let (4.25)ν
≥
−
1
/
2, (4.25)
0
<
a
<
ν
+
1, (4.25)
f
∈
L
q
2
(
R
q
,
+
,
x
2
ν
+
1
d
q
x
),
and let (4.25)
E be a bounded subset of (4.25)
R
q
,
+.
For (4.25)r
>
0,
we have, since (4.25)
f
.
χ
r
∈
L
q
1
(
R
q
,
+
,
x
2
ν
+
1
d
q
x
),(4.25)
(
∫
E
|
ℱ
ν
,
q
(
f
)
(
λ
)
|
2
λ
2
ν
+
1
d
q
λ
)
1
/
2
=
∥
ℱ
ν
,
q
(
f
)
χ
E
∥
2,
ν
,
q
≤
∥
ℱ
ν
,
q
(
f
⋅
χ
r
)
χ
E
∥
2,
ν
,
q
+
∥
ℱ
ν
,
q
(
f
⋅
χ
˜
r
)
χ
E
∥
2,
ν
,
q
≤
|
E
|
ν
1
/
2
∥
ℱ
ν
,
q
(
f
⋅
χ
r
)
∥
∞
,
q
+
∥
ℱ
ν
,
q
(
f
⋅
χ
˜
r
)
∥
2,
ν
,
q
.Proof.
Letν
≥
−
1
/
2, 0
<
a
<
ν
+
1, f
∈
L
q
2
(
ℝ
q
,
+
,
x
2
ν
+
1
d
q
x
),
and let E be a bounded subset of ℝ
q
,
+.
Forr
>
0,
we have, since f
.
χ
r
∈
L
q
1
(
ℝ
q
,
+
,
x
2
ν
+
1
d
q
x
),(4.26)
∥
ℱ
ν
,
q
(
f
⋅
χ
r
)
∥
∞
,
q
≤
c
˜
ν
,
q
∥
f
⋅
χ
r
∥
1,
q
=
c
˜
q
∥
x
−
a
χ
r
.
x
a
f
∥
1,
ν
,
q
≤
c
˜
ν
,
q
∥
x
−
a
χ
r
∥
2,
ν
,
q
∥
x
a
f
∥
2,
ν
,
q
.However, by the use of the
Hölder inequality, we obtain(4.27)
∥
x
−
a
χ
r
∥
2,
ν
,
q
2
=
∫
0
∞
x
−
2
a
χ
r
(
x
)
x
2
ν
+
1
d
q
x
=
∫
0
q
k
x
2
ν
+
1
−
2
a
d
q
x
=
q
2
k
(
ν
+
1
−
a
)
[
2
ν
+
2
−
2
a
]
q
≤
r
2
(
ν
+
1
−
a
)
[
2
ν
+
2
−
2
a
]
q
.Now, if k is the integer such that q
k
≤
r
<
q
k
−
1,
we get, since a
<
ν
+
1,(4.28)
∥
ℱ
ν
,
q
(
f
⋅
χ
r
)
∥
∞
,
q
≤
c
˜
ν
,
q
[
2
ν
+
2
−
2
a
]
q
r
(
ν
+
1
−
a
)
∥
x
a
f
∥
2,
ν
,
q
.Then,(4.29)
∥
ℱ
ν
,
q
(
f
⋅
χ
˜
r
)
∥
2,
ν
,
q
=
q
2
ν
+
1
∥
f
⋅
χ
˜
r
∥
2,
ν
,
q
=
q
2
ν
+
1
∥
x
−
a
χ
˜
r
⋅
x
a
f
∥
2,
ν
,
q
≤
q
2
ν
+
1
∥
x
−
a
χ
˜
r
∥
∞
,
q
∥
x
a
f
∥
2,
q
≤
q
2
ν
+
1
r
−
a
∥
x
a
f
∥
2,
ν
,
q
.On the other hand, since f
∈
L
q
2
(
ℝ
q
,
+
,
x
2
ν
+
1
d
q
x
),
we have f
.
χ
˜
r
∈
L
q
2
(
ℝ
q
,
+
,
x
2
ν
+
1
d
q
x
)
, and by the Plancherel formula (4.23), we
obtain(4.30)
(
∫
E
|
ℱ
ν
,
q
(
f
)
(
λ
)
|
2
λ
2
ν
+
1
d
q
λ
)
1
/
2
≤
(
c
˜
ν
,
q
[
2
ν
+
2
−
2
a
]
q
|
E
|
ν
1
/
2
r
(
ν
+
1
−
a
)
+
q
2
ν
+
1
r
−
a
)
∥
x
a
f
∥
2,
ν
,
q
.So,(4.31)
∫
E
|
ℱ
ν
,
q
(
f
)
(
λ
)
|
2
λ
2
ν
+
1
d
q
λ
≤
K
a
,
ν
′
|
E
|
∥
f
∥
2,
ν
,
q
2
(
1
−
(
ν
+
1
)
/
a
)
∥
x
a
f
∥
2,
ν
,
q
2
(
(
ν
+
1
)
/
a
)
.By minimization of the
right-hand side of the previous inequality over r
>
0 and by easy computation, we obtain the desired
result.Theorem 4.11.
Forν
≥
−
1
/
2 and a
>
ν
+
1,
there exists a constant K
a
,
ν
′ such that for all bounded subset E of ℝ
q
,
+ and all f in L
q
2
(
ℝ
q
,
+
,
x
2
ν
+
1
d
q
x
),
one has(4.32)
K
a
,
ν
′
=
(
q
2
a
,
q
2
a
,
−
q
2
ν
+
2
,
−
q
2
(
a
−
ν
−
1
)
;
q
2
a
)
∞
(
q
2
ν
+
2
,
q
2
(
a
−
ν
−
1
)
,
−
q
2
a
,
−
1
;
q
2
a
)
∞
c
ν
,
q
′
,
c
ν
,
q
′
=
(
1
−
q
)
(
c
ν
,
q
(
q
;
q
2
)
∞
2
)
2
(
a
ν
+
1
−
1
)
(
ν
+
1
)
/
a
(
a
a
−
ν
−
1
)
q
−
2
(
ν
+
1
)
(
(
a
−
ν
−
1
)
/
a
)
.Proof.
Sincea
>
ν
+
1,
the same steps as in the proof of Theorem 4.5 and the relation (4.22) give the
result with(4.33)
∥
f
∥
2,
ν
,
q
(
a
+
b
)
≤
K
a
,
b
,
ν
∥
x
a
f
∥
2,
ν
,
q
b
∥
λ
b
ℱ
ν
,
q
(
f
)
∥
2,
ν
,
q
a
,Corollary 4.12.
Forν
≥
−
1
/
2 and a
,
b
>
0,
there is a constant K
a
,
b
,
ν
=
K
(
a
,
b
,
ν
,
q
) such that for all f
∈
L
q
2
(
ℝ
q
,
+
,
x
2
ν
+
1
d
q
x
),
one has(4.34)
K
a
,
b
,
ν


=


{
[
(
b
a
)
a
/
(
a
+
b
)
+
(
a
b
)
b
/
(
a
+
b
)
]
(
a
+
b
)
/
2
(
K
a
,
ν
)
b
/
2
q
−
(
2
ν
+
1
)
(
a
+
b
)
(
[
2
ν
+
2
]
q
)
a
b
/
2
(
ν
+
1
)




















if
a
<
ν
+
1,

(

K
a
,
ν
′
[
2
ν
+
2
]
q

)
a
b
/
(
2
ν
+
2
)

(

q

−
(
4
ν
+
2
)

[

(
b
ν
+
1
)
(
ν
+
1
)
/
(
ν
+
b
+
1
)


+

(
b
ν
+
1
)

−
b
/
(
ν
+
b
+
1
)

]
)
a
(
ν
+
b
+
1
)
/
2
(
ν
+
1
)























if
a
>
ν
+
1,with (4.35)
q
4
ν
+
2
∥
f
∥
2,
ν
,
q
2
=
∥
ℱ
ν
,
q
(
f
)
∥
2,
ν
,
q
2
=
∫
E
r
|
ℱ
ν
,
q
(
f
)
|
2
(
λ
)
λ
2
ν
+
1
d
q
λ
+
∫
E
˜
r
|
ℱ
ν
,
q
(
f
)
|
2
(
λ
)
λ
2
ν
+
1
d
q
λ
≤
{
K
a
,
ν
|
E
r
|
ν
a
/
(
ν
+
1
)
∥
x
a
f
∥
2,
ν
,
q
2
+
r
−
2
b
∥
λ
b
ℱ
ν
,
q
(
f
)
∥
2,
ν
,
q
2
if
a
<
ν
+
1,
K
a
,
ν
′
|
E
r
|
∥
f
∥
2,
ν
,
q
2
(
a
−
ν
−
1
)
/
a
∥
x
a
f
∥
2,
ν
,
q
2
(
ν
+
1
)
/
a
+
r
−
2
b
∥
λ
b
ℱ
ν
,
q
(
f
)
∥
2,
ν
,
q
2
if
a
>
ν
+
1,
≤
{
K
a
,
ν
[
2
ν
+
2
]
q
a
/
(
ν
+
1
)
r
2
a
∥
x
a
f
∥
2,
ν
,
q
2
+
r
−
2
b
∥
λ
b
ℱ
ν
,
q
(
f
)
∥
2,
ν
,
q
2
if
a
<
ν
+
1,
K
a
,
ν
′
r
2
ν
+
2
[
2
ν
+
2
]
q
∥
f
∥
2,
ν
,
q
2
(
a
−
ν
−
1
)
/
a
∥
x
a
f
∥
2,
ν
,
q
2
(
ν
+
1
)
/
a
+
r
−
2
b
∥
λ
b
ℱ
ν
,
q
(
f
)
∥
2,
ν
,
q
2
if
a
>
ν
+
1. where K
a
,
ν (resp., K
a
,
ν
′) is the constant given in Theorem 4.10 (resp.,
Theorem 4.11).Proof.
Forr
>
0,
we put E
r
=
[
0
,
r
[
∩
ℝ
q
,
+ and E
˜
r
=
[
r
,
+
∞
[
∩
ℝ
q
,
+.
We haveE
r is a bounded subset of ℝ
q
,
+ and |
E
r
|
ν
≤
r
2
ν
+
2
/
[
2
ν
+
2
]
q
. Then, the Plancherel formula (4.23) and Theorems
4.10 and 4.11 lead to (4.36)
∥
f
∥
2,
ν
,
q
2
≤
K
1,1,
ν
∥
x
f
∥
2,
ν
,
q
∥
λ
ℱ
ν
,
q
(
f
)
∥
2,
ν
,
q
.The desired result follows by
minimizing the right expressions over r
>
0.Remark that whena
=
b
=
1,
we obtain a Heisenberg-Weyl-type inequality for the q-Bessel-Fourier transform.Corollary 4.13.
Forν
≥
−
1
/
2,
ν
≠
0,
one has for all f
∈
L
q
2
(
ℝ
q
,
+
,
x
2
ν
+
1
d
q
x
),(4.37)
∥
f
∥
2,
ν
,
q
2
≤
K
1,1,
ν
∥
x
f
∥
2,
ν
,
q
∥
λ
ℱ
ν
,
q
(
f
)
∥
2,
ν
,
q
.


## 4.1. Local Uncertainty Principles for theq-Fourier Trigonometric Transforms

Theorem 4.1.
If0
<
a
<
1
/
2,
there is a constant K
=
K
(
a
,
q
) such that for all bounded subset E of ℝ
q
,
+ and all f
∈
L
q
2
(
ℝ
q
,
+
),
one has(4.1)
∫
E
|
ℱ
q
(
f
)
(
λ
)
|
2
d
q
λ
≤
K
|
E
|
2
a
∥
x
a
f
∥
2,
q
2
.Here, |
E
|
=
∫
0
∞
χ
E
(
x
)
d
q
x and K
=
(
(
c
˜
q
/
[
1
−
2
a
]
q
)
(
(
1
−
2
a
)
/
2
a
)
)
4
a
(
1
/
(
1
−
2
a
)
2
),
where c
˜
q
=
(
1
+
q
−
1
)
1
/
2
/
Γ
q
2
(
1
/
2
)
(
q
;
q
)
∞
2
.
Proof.
Forr
>
0,
let χ
r
=
χ
[
0
,
r
] be the characteristic function of [
0
,
r
] and χ
˜
r
=
1
−
χ
r.
Then, forr
>
0,
we have, since f
⋅
χ
r
∈
L
q
1
(
ℝ
q
,
+
),(4.2)
(
∫
E
|
ℱ
q
(
f
)
(
λ
)
|
2
d
q
λ
)
1
/
2
=
∥
ℱ
q
(
f
)
χ
E
∥
2,
q
≤
∥
ℱ
q
(
f
⋅
χ
r
)
χ
E
∥
2,
q
+
∥
ℱ
q
(
f
⋅
χ
˜
r
)
χ
E
∥
2,
q
≤
|
E
|
1
/
2
∥
ℱ
q
(
f
⋅
χ
r
)
∥
∞
,
q
+
∥
ℱ
q
(
f
⋅
χ
˜
r
)
∥
2,
q
,and by the use of the Hölder
inequality, we obtain(4.3)
∥
ℱ
q
(
f
⋅
χ
r
)
∥
∞
,
q
≤
c
˜
q
∥
f
⋅
χ
r
∥
1,
q


=
c
˜
q
∥
x
−
a
χ
r
⋅
x
a
f
∥
1,
q
≤
c
˜
q
∥
x
−
a
χ
r
∥
2,
q
∥
x
a
f
∥
2,
q
≤
c
˜
q
[
1
−
2
a
]
q
r
1
/
2
−
a
∥
x
a
f
∥
2,
q
.On the other hand, since f
∈
L
q
2
(
ℝ
q
,
+), we have f
⋅
χ
˜
r
∈
L
q
2
(
ℝ
q
,
+
),
and by the Plancherel formula, we get(4.4)
∥
ℱ
q
(
f
⋅
χ
˜
r
)
∥
2,
q
=
∥
f
⋅
χ
˜
r
∥
2,
q
=
∥
x
−
a
χ
˜
r
.
x
a
f
∥
2,
q
≤
∥
x
−
a
χ
˜
r
∥
∞
,
q
∥
x
a
f
∥
2,
q
≤
r
−
a
∥
x
a
f
∥
2,
q
.So,(4.5)
(
∫
E
|
ℱ
q
(
f
)
(
λ
)
|
2
d
q
λ
)
1
/
2
≤
(
c
˜
q
[
1
−
2
a
]
q
|
E
|
1
/
2
r
1
/
2
−
a
+
r
−
a
)
∥
x
a
f
∥
2,
q
.The desired result is obtained
by minimizing the right-hand side of the previous inequality over r
>
0.Corollary 4.2.
For0
<
a
<
1
/
2 and b
>
0,
there is a constant K
a
,
b such that for all f
∈
L
q
2
(
ℝ
q
,
+
),
one has (4.6)
∥
f
∥
2,
q
(
a
+
b
)
≤
K
a
,
b
∥
x
a
f
∥
2,
q
b
∥
λ
b
ℱ
q
(
f
)
∥
2,
q
a
.Proof.
Forr
>
0,
put E
r
=
[
0
,
r
[
∩
ℝ
q
,
+ and E
˜
r
=
[
r
,
+
∞
[
∩
ℝ
q
,
+.
It is easy to see that E
r is a bounded subset of ℝ
q
,
+ and |
E
r
|
≤
r
.
Then, from the Plancherel formula and Theorem4.1, we
have(4.7)
∥
f
∥
2,
q
2
=
∥
ℱ
q
(
f
)
∥
2,
q
2
=
∫
E
r
|
ℱ
q
(
f
)
|
2
(
λ
)
d
q
λ
+
∫
E
˜
r
|
ℱ
q
(
f
)
|
2
(
λ
)
d
q
λ
≤
K
r
2
a
∥
x
a
f
∥
2,
q
2
+
r
−
2
b
∥
λ
b
ℱ
q
(
f
)
∥
2,
q
2
.Choosing r
>
0 so as to minimize the right-hand side of the
inequality, we obtain ∥
f
∥
2,
q
2
≤
(
K
a
,
b
∥
x
a
f
∥
2,
q
b
∥
λ
b
ℱ
q
(
f
)
∥
2,
q
a
)
2
/
(
a
+
b
),
with K
a
,
b
=
(
(
a
/
b
)
b
/
(
a
+
b
)
+
(
b
/
a
)
a
/
(
a
+
b
)
)
(
a
+
b
)
/
2
K
b
/
2,
and K is the constant given in Theorem 4.1.In the same way, one can prove the following local
uncertainty principle for theq-Fourier-sine transform.Theorem 4.3.
If0
<
a
<
1
/
2,
there is a constant K
′
=
K
′
(
a
,
q
) such that for all bounded subset E of ℝ
q
,
+ and all f
∈
L
q
2
(
ℝ
q
,
+
),
one has(4.8)
∫
E
|
q
ℱ
(
f
)
(
λ
)
|
2
d
q
λ
≤
K
′
|
E
|
2
a
∥
x
a
f
∥
2,
q
2
,where K
′
=
(
(
c
˜
q
/
[
1
−
2
a
]
q
)
(
(
1
−
2
a
)
/
2
q
a
)
)
4
a
[
1
+
2
q
a
/
(
1
−
2
a
)
]
2.Corollary 4.4.
For0
<
a
<
1
/
2 and b
>
0,
there is a constant K
a
,
b
′ such that for all f
∈
L
q
2
(
ℝ
q
,
+
),
one has(4.9)
∥
f
∥
2,
q
(
a
+
b
)
≤
K
a
,
b
′
∥
x
a
f
∥
2,
q
b
∥
λ
b
F
q
(
f
)
∥
2,
q
a
,with K
a
,
b
′
=
(
(
a
/
b
)
b
/
(
a
+
b
)
+
(
b
/
a
)
a
/
(
a
+
b
)
)
(
a
+
b
)
/
2
(
K
′
)
b
/
2
q
−
(
a
+
b
)
.Proof.
The same steps of Corollary4.2 give the
result.Theorem 4.5.
Ifa
>
1
/
2,
there is a constant K
1
=
K
1
(
a
,
q
) such that for all bounded subset E of ℝ
q
,
+ and f
∈
L
q
2
(
ℝ
q
,
+
),
one has
(4.10)
∫
E
|
ℱ
q
(
f
)
(
λ
)
|
2
d
q
λ
≤
K
1
|
E
|
∥
f
∥
2,
q
(
2
−
1
/
a
)
∥
x
a
f
∥
2,
q
1
/
a
,
(4.11)
∫
E
|
ℱ
q
(
f
)
(
λ
)
|
2
d
q
λ
≤
K
1
|
E
|
∥
f
∥
2,
q
(
2
−
1
/
a
)
∥
x
a
f
∥
2,
q
1
/
a
.The proof of this result needs the following lemmas.Lemma 4.6.
Supposea
>
1
/
2,
then for all f
∈
L
q
2
(
ℝ
q
,
+
)
, such that x
a
f
∈
L
q
2
(
ℝ
q
,
+
),(4.12)
∥
f
∥
1,
q
2
≤
K
2
[
∥
f
∥
2,
q
2
+
∥
x
a
f
∥
2,
q
2
]
,where K
2
=
K
2
(
a
,
q
)
=
(
1
−
q
)
(
(
q
2
a
,
q
2
a
,
−
q
,
−
q
2
a
−
1
;
q
2
a
)
∞
/
(
q
,
q
2
a
−
1
,
−
q
2
a
,
−
1
;
q
2
a
)
∞
).Proof.
From
[15, Example 1], and the Hölder inequality, we have(4.13)
∥
f
∥
1,
q
2
=
[
∫
0
+
∞
(
1
+
x
2
a
)
1
/
2
|
f
(
x
)
|
(
1
+
x
2
a
)
−
1
/
2
d
q
x
]
2
≤
K
2
[
∥
f
∥
2,
q
2
+
∥
x
a
f
∥
2,
q
2
]
,where K
2
=
∫
0
+
∞
(
1
+
x
2
a
)
−
1
d
q
x
=
(
1
−
q
)
(
(
q
2
a
,
q
2
a
,
−
q
,
−
q
2
a
−
1
;
q
2
a
)
∞
/
(
q
,
q
2
a
−
1
,
−
q
2
a
,
−
1
;
q
2
a
)
∞
).Lemma 4.7.
Supposea
>
1
/
2,
then for all f
∈
L
q
2
(
ℝ
q
,
+
),
such that x
a
f
∈
L
q
2
(
ℝ
q
,
+
),
one has(4.14)
∥
f
∥
1,
q
≤
K
3
∥
f
∥
2,
q
(
1
−
1
/
2
a
)
∥
x
a
f
∥
2,
q
1
/
2
a
,where K
3
=
K
3
(
a
,
q
)
=
[
2
a
K
2
(
2
a
q
−
q
)
1
/
2
a
−
1
]
1
/
2
.Proof.
Fors
∈
ℝ
q
,
+
, define the function f
s by f
s
(
x
)
=
f
(
s
x
)
,
x
∈
ℝ
q
,
+.
We have∥
f
s
∥
1,
q
=
s
−
1
∥
f
∥
1,
q, ∥
x
a
f
s
∥
2,
q
2
=
s
−
2
a
−
1
∥
x
a
f
∥
2,
q
2
.
Replacement off by f
s in Lemma 4.6 gives(4.15)
∥
f
∥
1,
q
2
≤
K
2
[
s
∥
f
∥
2,
q
2
+
s
−
2
a
+
1
∥
x
a
f
∥
2,
q
2
]
.Now, for all r
>
0
, put α
(
r
)
=
Log
(
r
)
/
Log
(
q
)
−
E
(
Log
(
r
)
/
Log
(
q
)
)
. We have s
=
(
r
/
q
α
(
r
)
)
∈
ℝ
q
,
+ and r
≤
s
<
r
/
q
. Then, for all r
>
0
,
(4.16)
∥
f
∥
1,
q
2
≤
K
2
[
r
q
∥
f
∥
2,
q
2
+
r
−
2
a
+
1
∥
x
a
f
∥
2,
q
2
]
.The right-hand side of this
inequality is minimized by choosing(4.17)
r
=
(
2
a
−
1
)
1
/
2
a
q
1
/
2
a
∥
f
∥
2,
q
−
1
/
a
∥
x
a
f
∥
2,
q
1
/
a
.When this is done, we obtain the
result.Proof of Theorem4.5.
Since the proofs of the two statements are  similar, it is sufficient to
prove (4.11).
LetE be a bounded subset of ℝ
q
,
+.
When the right-hand side of the inequality (4.11) is finite, Lemma 4.6 implies that f
∈
L
q
1
(
ℝ
q
,
+
);
so ℱ
q
(
f
) is defined and bounded on ℝ
q
,
+.
Using Proposition 3.1, Lemma 4.7, and the fact that(4.18)
∫
E
|
ℱ
q
(
f
)
(
λ
)
|
2
d
q
λ
≤
|
E
|
∥
ℱ
q
(
f
)
∥
∞
,
q
2
,we obtain the result with K
1
=
(
(
1
+
q
−
1
)
/
Γ
q
2
2
(
1
/
2
)
(
q
;
q
)
∞
4
)
K
3
2.Remark 4.8.
By the same technique as in the proof of Corollary4.2,
we can show that Theorem 4.5 leads to inequalities (4.6) and (4.9) with some
different constants.


## 4.2. Local Uncertainty Principles for theq-Bessel-Fourier Transform

Theq-Bessel-Fourier transform is defined (see
[16]) for f
∈
L
q
1
(
ℝ
q
,
+
,
x
2
ν
+
1
d
q
x
) by(4.19)
ℱ
ν
,
q
(
f
)
(
λ
)
=
c
ν
,
q
∫
0
∞
f
(
x
)
j
ν
(
λ
x
;
q
2
)
x
2
ν
+
1
d
q
x
,where(4.20)
j
ν
(
z
;
q
2
)
=
(
1
−
q
2
)
ν
Γ
q
2
(
ν
+
1
)
(
(
1
−
q
)
q
−
1
z
)
−
ν
J
ν
(
(
1
−
q
)
q
−
1
z
;
q
2
)is the normalized third Jackson q-Bessel function, and(4.21)
c
ν
,
q
=
(
1
+
q
−
1
)
−
ν
Γ
q
2
(
ν
+
1
)
.It was shown in [10] that for ν
≥
−
1
/
2,
we have the following result.Theorem 4.9.
(1) Forf
∈
L
q
1
(
ℝ
q
,
+
,
x
2
ν
+
1
d
q
x
),
one has ℱ
ν
,
q
(
f
)
∈
L
q
∞
(
ℝ
q
,
+
) and(4.22)
∥
ℱ
ν
,
q
(
f
)
∥
∞
,
q
≤
c
ν
,
q
(
q
;
q
2
)
∞
2
∥
f
∥
1,
ν
,
q
.
(2)ℱ
ν
,
q is an isomorphism of L
q
2
(
ℝ
q
,
+
,
x
2
ν
+
1
d
q
x
) onto itself, ℱ
ν
,
q
−
1
=
q
4
ν
+
2
ℱ
ν
,
q,
and one has the following Plancherel formula:(4.23)
∀
f
∈
L
q
2
(
R
q
,
+
,
x
2
ν
+
1
d
q
x
)
,

∥
ℱ
ν
,
q
∥
2,
ν
,
q
=
q
2
ν
+
1
∥
f
∥
2,
ν
,
q
.
The following
result states a local uncertainty principle for theq-Bessel-Fourier transform.Theorem 4.10.
Forν
≥
−
1
/
2 and 0
<
a
<
ν
+
1,
there is a constant K
a
,
ν
=
K
(
a
,
ν
,
q
) such that for all f
∈
L
q
2
(
ℝ
q
,
+
,
x
2
ν
+
1
d
q
x
) and all bounded subset E of ℝ
q
,
+,
one has(4.24)
∫
E
|
ℱ
ν
,
q
(
f
)
(
λ
)
|
2
λ
2
ν
+
1
d
q
λ
≤
K
a
,
ν
|
E
|
ν
a
/
(
ν
+
1
)
∥
x
a
f
∥
2,
ν
,
q
2
.Here, |
E
|
ν
=
∫
0
∞
χ
E
(
x
)
x
2
ν
+
1
d
q
x, c
˜
ν
,
q
=
c
ν
,
q
/
(
q
;
q
2
)
∞
2,
and
(4.25)
K
a
,
ν
=
(
c
˜
ν
,
q
[
2
ν
+
2
−
2
a
]
q
)
2
a
/
(
ν
+
1
)
[
(
a
q
2
ν
+
1
ν
+
1
−
a
)
1
−
a
/
(
ν
+
1
)
+
q
2
ν
+
1
(
a
q
2
ν
+
1
ν
+
1
−
a
)
−
a
/
(
ν
+
1
)
]
2.Proof
Let (4.25)ν
≥
−
1
/
2, (4.25)
0
<
a
<
ν
+
1, (4.25)
f
∈
L
q
2
(
R
q
,
+
,
x
2
ν
+
1
d
q
x
),
and let (4.25)
E be a bounded subset of (4.25)
R
q
,
+.
For (4.25)r
>
0,
we have, since (4.25)
f
.
χ
r
∈
L
q
1
(
R
q
,
+
,
x
2
ν
+
1
d
q
x
),(4.25)
(
∫
E
|
ℱ
ν
,
q
(
f
)
(
λ
)
|
2
λ
2
ν
+
1
d
q
λ
)
1
/
2
=
∥
ℱ
ν
,
q
(
f
)
χ
E
∥
2,
ν
,
q
≤
∥
ℱ
ν
,
q
(
f
⋅
χ
r
)
χ
E
∥
2,
ν
,
q
+
∥
ℱ
ν
,
q
(
f
⋅
χ
˜
r
)
χ
E
∥
2,
ν
,
q
≤
|
E
|
ν
1
/
2
∥
ℱ
ν
,
q
(
f
⋅
χ
r
)
∥
∞
,
q
+
∥
ℱ
ν
,
q
(
f
⋅
χ
˜
r
)
∥
2,
ν
,
q
.Proof.
Letν
≥
−
1
/
2, 0
<
a
<
ν
+
1, f
∈
L
q
2
(
ℝ
q
,
+
,
x
2
ν
+
1
d
q
x
),
and let E be a bounded subset of ℝ
q
,
+.
Forr
>
0,
we have, since f
.
χ
r
∈
L
q
1
(
ℝ
q
,
+
,
x
2
ν
+
1
d
q
x
),(4.26)
∥
ℱ
ν
,
q
(
f
⋅
χ
r
)
∥
∞
,
q
≤
c
˜
ν
,
q
∥
f
⋅
χ
r
∥
1,
q
=
c
˜
q
∥
x
−
a
χ
r
.
x
a
f
∥
1,
ν
,
q
≤
c
˜
ν
,
q
∥
x
−
a
χ
r
∥
2,
ν
,
q
∥
x
a
f
∥
2,
ν
,
q
.However, by the use of the
Hölder inequality, we obtain(4.27)
∥
x
−
a
χ
r
∥
2,
ν
,
q
2
=
∫
0
∞
x
−
2
a
χ
r
(
x
)
x
2
ν
+
1
d
q
x
=
∫
0
q
k
x
2
ν
+
1
−
2
a
d
q
x
=
q
2
k
(
ν
+
1
−
a
)
[
2
ν
+
2
−
2
a
]
q
≤
r
2
(
ν
+
1
−
a
)
[
2
ν
+
2
−
2
a
]
q
.Now, if k is the integer such that q
k
≤
r
<
q
k
−
1,
we get, since a
<
ν
+
1,(4.28)
∥
ℱ
ν
,
q
(
f
⋅
χ
r
)
∥
∞
,
q
≤
c
˜
ν
,
q
[
2
ν
+
2
−
2
a
]
q
r
(
ν
+
1
−
a
)
∥
x
a
f
∥
2,
ν
,
q
.Then,(4.29)
∥
ℱ
ν
,
q
(
f
⋅
χ
˜
r
)
∥
2,
ν
,
q
=
q
2
ν
+
1
∥
f
⋅
χ
˜
r
∥
2,
ν
,
q
=
q
2
ν
+
1
∥
x
−
a
χ
˜
r
⋅
x
a
f
∥
2,
ν
,
q
≤
q
2
ν
+
1
∥
x
−
a
χ
˜
r
∥
∞
,
q
∥
x
a
f
∥
2,
q
≤
q
2
ν
+
1
r
−
a
∥
x
a
f
∥
2,
ν
,
q
.On the other hand, since f
∈
L
q
2
(
ℝ
q
,
+
,
x
2
ν
+
1
d
q
x
),
we have f
.
χ
˜
r
∈
L
q
2
(
ℝ
q
,
+
,
x
2
ν
+
1
d
q
x
)
, and by the Plancherel formula (4.23), we
obtain(4.30)
(
∫
E
|
ℱ
ν
,
q
(
f
)
(
λ
)
|
2
λ
2
ν
+
1
d
q
λ
)
1
/
2
≤
(
c
˜
ν
,
q
[
2
ν
+
2
−
2
a
]
q
|
E
|
ν
1
/
2
r
(
ν
+
1
−
a
)
+
q
2
ν
+
1
r
−
a
)
∥
x
a
f
∥
2,
ν
,
q
.So,(4.31)
∫
E
|
ℱ
ν
,
q
(
f
)
(
λ
)
|
2
λ
2
ν
+
1
d
q
λ
≤
K
a
,
ν
′
|
E
|
∥
f
∥
2,
ν
,
q
2
(
1
−
(
ν
+
1
)
/
a
)
∥
x
a
f
∥
2,
ν
,
q
2
(
(
ν
+
1
)
/
a
)
.By minimization of the
right-hand side of the previous inequality over r
>
0 and by easy computation, we obtain the desired
result.Theorem 4.11.
Forν
≥
−
1
/
2 and a
>
ν
+
1,
there exists a constant K
a
,
ν
′ such that for all bounded subset E of ℝ
q
,
+ and all f in L
q
2
(
ℝ
q
,
+
,
x
2
ν
+
1
d
q
x
),
one has(4.32)
K
a
,
ν
′
=
(
q
2
a
,
q
2
a
,
−
q
2
ν
+
2
,
−
q
2
(
a
−
ν
−
1
)
;
q
2
a
)
∞
(
q
2
ν
+
2
,
q
2
(
a
−
ν
−
1
)
,
−
q
2
a
,
−
1
;
q
2
a
)
∞
c
ν
,
q
′
,
c
ν
,
q
′
=
(
1
−
q
)
(
c
ν
,
q
(
q
;
q
2
)
∞
2
)
2
(
a
ν
+
1
−
1
)
(
ν
+
1
)
/
a
(
a
a
−
ν
−
1
)
q
−
2
(
ν
+
1
)
(
(
a
−
ν
−
1
)
/
a
)
.Proof.
Sincea
>
ν
+
1,
the same steps as in the proof of Theorem 4.5 and the relation (4.22) give the
result with(4.33)
∥
f
∥
2,
ν
,
q
(
a
+
b
)
≤
K
a
,
b
,
ν
∥
x
a
f
∥
2,
ν
,
q
b
∥
λ
b
ℱ
ν
,
q
(
f
)
∥
2,
ν
,
q
a
,Corollary 4.12.
Forν
≥
−
1
/
2 and a
,
b
>
0,
there is a constant K
a
,
b
,
ν
=
K
(
a
,
b
,
ν
,
q
) such that for all f
∈
L
q
2
(
ℝ
q
,
+
,
x
2
ν
+
1
d
q
x
),
one has(4.34)
K
a
,
b
,
ν


=


{
[
(
b
a
)
a
/
(
a
+
b
)
+
(
a
b
)
b
/
(
a
+
b
)
]
(
a
+
b
)
/
2
(
K
a
,
ν
)
b
/
2
q
−
(
2
ν
+
1
)
(
a
+
b
)
(
[
2
ν
+
2
]
q
)
a
b
/
2
(
ν
+
1
)




















if
a
<
ν
+
1,

(

K
a
,
ν
′
[
2
ν
+
2
]
q

)
a
b
/
(
2
ν
+
2
)

(

q

−
(
4
ν
+
2
)

[

(
b
ν
+
1
)
(
ν
+
1
)
/
(
ν
+
b
+
1
)


+

(
b
ν
+
1
)

−
b
/
(
ν
+
b
+
1
)

]
)
a
(
ν
+
b
+
1
)
/
2
(
ν
+
1
)























if
a
>
ν
+
1,with (4.35)
q
4
ν
+
2
∥
f
∥
2,
ν
,
q
2
=
∥
ℱ
ν
,
q
(
f
)
∥
2,
ν
,
q
2
=
∫
E
r
|
ℱ
ν
,
q
(
f
)
|
2
(
λ
)
λ
2
ν
+
1
d
q
λ
+
∫
E
˜
r
|
ℱ
ν
,
q
(
f
)
|
2
(
λ
)
λ
2
ν
+
1
d
q
λ
≤
{
K
a
,
ν
|
E
r
|
ν
a
/
(
ν
+
1
)
∥
x
a
f
∥
2,
ν
,
q
2
+
r
−
2
b
∥
λ
b
ℱ
ν
,
q
(
f
)
∥
2,
ν
,
q
2
if
a
<
ν
+
1,
K
a
,
ν
′
|
E
r
|
∥
f
∥
2,
ν
,
q
2
(
a
−
ν
−
1
)
/
a
∥
x
a
f
∥
2,
ν
,
q
2
(
ν
+
1
)
/
a
+
r
−
2
b
∥
λ
b
ℱ
ν
,
q
(
f
)
∥
2,
ν
,
q
2
if
a
>
ν
+
1,
≤
{
K
a
,
ν
[
2
ν
+
2
]
q
a
/
(
ν
+
1
)
r
2
a
∥
x
a
f
∥
2,
ν
,
q
2
+
r
−
2
b
∥
λ
b
ℱ
ν
,
q
(
f
)
∥
2,
ν
,
q
2
if
a
<
ν
+
1,
K
a
,
ν
′
r
2
ν
+
2
[
2
ν
+
2
]
q
∥
f
∥
2,
ν
,
q
2
(
a
−
ν
−
1
)
/
a
∥
x
a
f
∥
2,
ν
,
q
2
(
ν
+
1
)
/
a
+
r
−
2
b
∥
λ
b
ℱ
ν
,
q
(
f
)
∥
2,
ν
,
q
2
if
a
>
ν
+
1. where K
a
,
ν (resp., K
a
,
ν
′) is the constant given in Theorem 4.10 (resp.,
Theorem 4.11).Proof.
Forr
>
0,
we put E
r
=
[
0
,
r
[
∩
ℝ
q
,
+ and E
˜
r
=
[
r
,
+
∞
[
∩
ℝ
q
,
+.
We haveE
r is a bounded subset of ℝ
q
,
+ and |
E
r
|
ν
≤
r
2
ν
+
2
/
[
2
ν
+
2
]
q
. Then, the Plancherel formula (4.23) and Theorems
4.10 and 4.11 lead to (4.36)
∥
f
∥
2,
ν
,
q
2
≤
K
1,1,
ν
∥
x
f
∥
2,
ν
,
q
∥
λ
ℱ
ν
,
q
(
f
)
∥
2,
ν
,
q
.The desired result follows by
minimizing the right expressions over r
>
0.Remark that whena
=
b
=
1,
we obtain a Heisenberg-Weyl-type inequality for the q-Bessel-Fourier transform.Corollary 4.13.
Forν
≥
−
1
/
2,
ν
≠
0,
one has for all f
∈
L
q
2
(
ℝ
q
,
+
,
x
2
ν
+
1
d
q
x
),(4.37)
∥
f
∥
2,
ν
,
q
2
≤
K
1,1,
ν
∥
x
f
∥
2,
ν
,
q
∥
λ
ℱ
ν
,
q
(
f
)
∥
2,
ν
,
q
.



---

*Source: 465909-2008-05-28.xml*