Volume 9 (2008), Issue 3, Article 88, 23 pp.
HEISENBERG-PAULI-WEYL UNCERTAINTY PRINCIPLE FOR THE
RIEMANN-LIOUVILLE OPERATOR
S. OMRI AND L.T. RACHDI
D EPARTMENT OF M ATHEMATICS
FACULTY OF S CIENCES OF T UNIS
2092 M ANAR 2 T UNIS
T UNISIA .
lakhdartannech.rachdi@fst.rnu.tn
Received 29 March, 2008; accepted 08 August, 2008
Communicated by J.M. Rassias
A BSTRACT. The Heisenberg-Pauli-Weyl inequality is established for the Fourier transform connected with the Riemann-Liouville operator.Also, a generalization of this inequality is proved.
Lastly, a local uncertainty principle is studied.
Key words and phrases: Heisenberg-Pauli-Weyl Inequality, Riemann-Liouville operator, Fourier transform, local uncertainty
principle.
2000 Mathematics Subject Classification. 42B10, 33C45.
1. I NTRODUCTION
Uncertainty principles play an important role in harmonic analysis and have been studied by
many authors and from many points of view [8].These principles state that a function f and
its Fourier transform fb cannot be simultaneously sharply localized. The theorems of Hardy,
Morgan, Beurling, ... are established for several Fourier transforms in [4], [9], [13] and [14]. In
this context, a remarkable Heisenberg uncertainty principle [10] states, according to Weyl [25]
who assigned the result to Pauli, that for all square integrable functions f on Rn with respect to
the Lebesgue measure, we have
2
Z
Z
Z
1
2
2
2 b
2
2
ξj |f (ξ)| dξ ≥
|f (x)| dx , j ∈ {1, . . . , n}.
xj |f (x)| dx
4
Rn
Rn
Rn
This inequality is called the Heisenberg-Pauli-Weyl inequality for the classical Fourier transform.
Recently, many works have been devoted to establishing the Heisenberg-Pauli-Weyl inequality for various Fourier transforms, Rösler [21] and Shimeno [22] have proved this inequality for
the Dunkl transform, in [20] Rösler and Voit have established an analogue of the HeisenbergPauli-Weyl inequality for the generalized Hankel transform. In the same context, Battle [3] has
proved this inequality for wavelet states, and Wolf [26], has studied this uncertainty principle
100-08
2
S. O MRI
AND
L.T. R ACHDI
for Gelfand pairs. We cite also De Bruijn [5] who has established the same result for the classical Fourier transform by using Hermite Polynomials, and Rassias [17, 18, 19] who gave several
generalized forms for the Heisenberg-Pauli-Weyl inequality.
In [2], the second author with others considered the singular partial differential operators
defined by

∂

∆1 = ∂x ,

∆ =
2
∂2
∂r2
2α+1 ∂
r ∂r
+
−
∂2
;
∂x2
(r, x) ∈]0, +∞[×R; α ≥ 0.
and they associated to ∆1 and ∆2 the following integral transform, called the Riemann-Liouville
operator, defined on C∗ (R2 ) (the space of continuous functions on R2 , even with respect to the
first variable) by
 α R1 R1
√
1
2 , x + rt (1 − t2 )α− 2 (1 − s2 )α−1 dtds; if α > 0,
f
rs
1
−
t

 π −1 −1
Rα (f )(r, x) =
√
R1

 π1 −1
f r 1 − t2 , x + rt √ dt 2 ;
if α = 0.
(1−t )
In addition, a convolution product and a Fourier transform Fα connected with the mapping Rα
have been studied and many harmonic analysis results have been established for the Fourier
transform Fα (Inversion formula, Plancherel formula, Paley-Winer and Plancherel theorems,
...).
Our purpose in this work is to study the Heisenberg-Pauli-Weyl uncertainty principle for the
Fourier transform Fα connected with Rα .More precisely, using Laguerre and Hermite polynomials we establish firstly the Heisenberg-Pauli-Weyl inequality for the Fourier transform Fα ,
that is
• For all f ∈ L2 (dνα ), we have
Z
0
+∞
Z
12
(r + x )|f (r, x)| dνα (r, x)
2
2
2
R
Z Z
×
12
(µ + 2λ )|Fα (f )(µ, λ)| dγα (µ, λ)
2
2
2
Γ+
2α + 3
≥
2
Z
+∞
0
Z
|f (r, x)| dνα (r, x) ,
2
R
with equality if and only if
f (r, x) = Ce
−r
2 +x2
2t2
0
; C ∈ C, t0 > 0,
where
• dνα (r, x) is the measure defined on R+ × R by
dνα (r, x) =
r2α+1
√ dr ⊗ dx.
2α Γ(α + 1) 2π
• dγα (µ, λ) is the measure defined on the set
Γ+ = R+ × R ∪ (it, x); (t, x) ∈ R+ × R; t ≤ |x| ,
J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 88, 23 pp.
http://jipam.vu.edu.au/
H EISENBERG -PAULI -W EYL U NCERTAINTY P RINCIPLE
3
by
Z Z
1
√
g(µ, λ)dγα (µ, λ) =
α
2 Γ(α + 1) 2π
Γ+
+∞
Z
Z
0
g(µ, λ)(µ2 + λ2 )α µdµdλ
R
Z Z
!
|λ|
g(iµ, λ)(λ2 − µ2 )α µdµdλ .
+
0
R
Next, we give a generalization of the Heisenberg-Pauli-Weyl inequality, that is
• For all f ∈ L2 (dνα ), a, b ∈ R; a, b ≥ 1 and η ∈ R such that ηa = (1 − η)b, we have
+∞
Z
0
Z
η2
(r + x ) |f (r, x)| dνα (r, x)
2
2 a
2
R
Z Z
×
1−η
2
(µ + 2λ ) |Fα (f )(µ, λ)| dγα (µ, λ)
2
2 b
2
Γ+
≥
2α + 3
2
aη Z
+∞
Z
0
12
|f (r, x)| dνα (r, x) ,
2
R
with equality if and only if
a = b = 1 and
f (r, x) = Ce
−r
2 +x2
2t2
0
C ∈ C;
;
t0 > 0.
In the last section of this paper, building on the ideas of Faris [7], and Price [15, 16], we
develop a family of inequalities in their sharpest forms, which constitute the principle of local
uncertainty.
Namely, we have established the following main results
• For all real numbers ξ; 0 < ξ < 2α+3
, there exists a positive constant Kα,ξ such that for
2
2
all f ∈ L (dνα ), and for all measurable subsets E ⊂ Γ+ ; 0 < γα (E) < +∞, we have
Z Z
2
|Fα (f )(µ, λ)| dγα (µ, λ) < Kα,ξ
2ξ
γα (E) 2α+3
+∞
Z
E
0
Z
(r2 + x2 )ξ |f (r, x)|2 dνα (r, x).
R
, there exists a positive constant Mα,ξ such that for all
• For all real number ξ; ξ > 2α+3
2
f ∈ L2 (dνα ), and for all measurable subsets E ⊂ Γ+ ; 0 < γα (E) < +∞, we have
Z Z
|Fα (f )(µ, λ)|2 dγα (µ, λ) < Mα,ξ γα (E)
Z
E
+∞
Z
0
Z
+∞
R
Z
×
0
2ξ−2α−3
2ξ
|f (r, x)|2 dνα (r, x)
2α+3
2ξ
(r + x ) |f (r, x)| dνα (r, x)
,
2
2 ξ
2
R
where Mα,ξ is the best (the smallest) constant satisfying this inequality.
J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 88, 23 pp.
http://jipam.vu.edu.au/
4
S. O MRI
AND
L.T. R ACHDI
2. T HE F OURIER T RANSFORM A SSOCIATED WITH
O PERATOR
THE
R IEMANN -L IOUVILLE
It is well known [2] that for all (µ, λ) ∈ C2 , the system


∆1 u(r, x) = −iλu(r, x),




∆2 u(r, x) = −µ2 u(r, x),




u(0, 0) = 1, ∂u (0, x) = 0, ∀x ∈ R,
∂r
admits a unique solution ϕµ,λ , given by
(2.1)
∀(r, x) ∈ R2 ;
where
jα (x) = 2α Γ(α + 1)
p
ϕµ,λ (r, x) = jα r µ2 + λ2 e−iλx ,
+∞
x 2n
X
(−1)n
Jα (x)
=
Γ(α
+
1)
,
xα
n!Γ(α
+
n
+
1)
2
n=0
and Jα is the Bessel function of the first kind and index α [6, 11, 12, 24].
The modified Bessel function jα has the following integral representation [1, 11], for all
µ ∈ C, and r ∈ R we have

R
 √2Γ(α+1)1 1 (1 − t2 )α− 12 cos(rµt)dt, if α > − 1 ;
2
πΓ(α+ 2 ) 0
jα (rµ) =
 cos(rµ),
if α = − 12 .
In particular, for all r, s ∈ R, we have
(2.2)
Z 1
1
2Γ(α + 1)
|jα (rs)| ≤ √
(1 − t2 )α− 2 | cos(rst)|dt
1
πΓ α + 2 0
Z 1
1
2Γ(α + 1)
(1 − t2 )α− 2 dt = 1.
≤√
1
πΓ(α + 2 ) 0
From the properties of the Bessel function, we deduce that the eigenfunction ϕµ,λ satisfies
the following properties
•
(2.3)
sup |ϕµ,λ (r, x)| = 1,
(r,x)∈R2
if and only if (µ, λ) belongs to the set
Γ = R2 ∪ {(it, x); (t, x) ∈ R2 ; |t| ≤ |x|}.
• The eigenfunction ϕµ,λ has the following Mehler integral representation
 R1 R1
√
1
 απ −1 −1 cos(µrs 1 − t2 )e−iλ(x+rt) (1 − t2 )α− 2 (1 − s2 )α−1 dtds; if α > 0,
ϕµ,λ (r, x) =
 1 R 1 cos(rµ√1 − t2 )e−iλ(x+rt) √ dt ;
if α = 0.
π −1
1−t2
In [2], using this integral representation, the authors have defined the Riemann-Liouville
integral transform associated with ∆1 , ∆2 by
√
 α R1 R1
1
 π −1 −1 f rs 1 − t2 , x + rt (1 − t2 )α− 2 (1 − s2 )α−1 dtds; if α > 0,
Rα (f )(r, x) =
√
R
 1 1 f r 1 − t2 , x + rt √ dt ;
if α = 0.
π −1
2
(1−t )
J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 88, 23 pp.
http://jipam.vu.edu.au/
H EISENBERG -PAULI -W EYL U NCERTAINTY P RINCIPLE
5
where f is a continuous function on R2 , even with respect to the first variable.
The transform Rα generalizes the "mean operator" defined by
Z 2π
1
R0 (f )(r, x) =
f (r sin θ, x + r cos θ)dθ.
2π 0
In the following we denote by
• dνα the measure defined on R+ × R, by
dνα (r, x) =
r2α+1
√ dr ⊗ dx.
2α Γ(α + 1) 2π
• Lp (dνα ) the space of measurable functions f on R+ × R such that
Z +∞ Z
p1
p
kf kp,να =
|f (r, x)| dνα (r, x)
< ∞, if p ∈ [1, +∞[,
0
R
kf k∞,να = ess sup |f (r, x)| < ∞,
if p = +∞.
(r,x)∈R+ ×R
• h / iνα the inner product defined on L2 (dνα ) by
Z +∞ Z
hf /giνα =
f (r, x)g(r, x)dνα (r, x).
0
R
• Γ+ = R+ × R ∪ (it, x); (t, x) ∈ R+ × R; t ≤ |x| .
• BΓ+ the σ-algebra defined on Γ+ by
BΓ+ = {θ−1 (B), B ∈ B(R+ × R)},
(2.4)
where θ is the bijective function defined on the set Γ+ by
p
θ(µ, λ) =
µ2 + λ2 , λ .
• dγα the measure defined on BΓ+ by
∀ A ∈ BΓ+ ; γα (A) = να (θ(A))
(2.5)
p
• L (dγα ) the space of measurable functions f on Γ+ , such that
Z Z
p1
kf kp,γα =
|f (µ, λ)|p dγα (µ, λ) < ∞, if p ∈ [1, +∞[,
Γ+
kf k∞,γα = ess sup |f (µ, λ)| < ∞,
if p = +∞.
(µ,λ)∈Γ+
• h / iγα the inner product defined on L2 (dγα ) by
Z Z
hf /giγα =
f (µ, λ)g(µ, λ)dγα (µ, λ).
Γ+
Then, we have the following properties.
Proposition 2.1.
i) For all non negative measurable functions g on Γ+ , we have
Z +∞ Z
Z Z
1
√
(2.6)
g(µ, λ)dγα (µ, λ) =
g(µ, λ)(µ2 + λ2 )α µdµdλ
α
2 Γ(α + 1) 2π
Γ+
0
R
!
Z Z |λ|
+
g(iµ, λ)(λ2 − µ2 )α µdµdλ .
R
J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 88, 23 pp.
0
http://jipam.vu.edu.au/
6
S. O MRI
AND
L.T. R ACHDI
In particular
µ2 + λ2
dγα (µ, λ).
2(α + 1)
ii) For all measurable functions f on R+ × R, the function f ◦ θ is measurable on Γ+ .
Furthermore, if f is non negative or an integrable function on R+ × R with respect to
the measure dνα , then we have
Z Z
Z +∞ Z
(2.8)
(f ◦ θ)(µ, λ)dγα (µ, λ) =
f (r, x)dνα (r, x).
dγα+1 (µ, λ) =
(2.7)
Γ+
0
R
In the following, we shall define the Fourier transform Fα associated with the operator Rα
and we give some properties that we use in the sequel.
Definition 2.1. The Fourier transform Fα associated with the Riemann-liouville operator Rα
is defined on L1 (dνα ) by
Z +∞ Z
(2.9)
∀(µ, λ) ∈ Γ; Fα (f )(µ, λ) =
f (r, x)ϕµ,λ (r, x)dνα (r, x).
0
R
By the relation (2.3), we deduce that the Fourier transform Fα is a bounded linear operator
from L1 (dνα ) into L∞ (dγα ), and that for all f ∈ L1 (dνα ), we have
kFα (f )k∞,γα ≤ kf k1,να .
(2.10)
Theorem 2.2 (Inversion formula). Let f ∈ L1 (dνα ) such that Fα (f ) ∈ L1 (dγα ), then for
almost every (r, x) ∈ R+ × R, we have
Z Z
f (r, x) =
Fα (f )(µ, λ)ϕµ,λ (r, x)dγα (µ, λ).
Γ+
Theorem 2.3 (Plancherel). The Fourier transform Fα can be extended to an isometric isomorphism from L2 (dνα ) onto L2 (dγα ).
In particular, for all f, g ∈ L2 (dνα ), we have the following Parseval’s equality
Z Z
Z +∞ Z
(2.11)
Fα (f )(µ, λ)Fα (g)(µ, λ)dγα (µ, λ) =
f (r, x)g(r, x)dνα (r, x).
Γ+
0
R
3. H ILBERT BASIS OF THE S PACES L2 (dνα ), AND L2 (dγα )
In this section, using Laguerre and Hermite polynomials, we construct a Hilbert basis of the
spaces L2 (dνα ) and L2 (dγα ), and establish some intermediate results that we need in the next
section.
It is well known [11, 23] that for every α ≥ 0, the Laguerre polynomials Lαm are defined by
the following Rodriguez formula
1 r −α dm m+α −r Lαm (r) =
er
r
e ;
m ∈ N.
m!
drm
Also, the Hermite polynomials are defined by the Rodriguez formula
n
n x2 d
−x2
Hn (x) = (−1) e
e
;
n ∈ N.
dxn
Moreover, the families
(s
)
)
(s
m!
1
α
√ Hn
L
and
Γ(α + m + 1) m
2n n! π
m∈N
J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 88, 23 pp.
n∈N
http://jipam.vu.edu.au/
H EISENBERG -PAULI -W EYL U NCERTAINTY P RINCIPLE
7
2
are respectively a Hilbert basis of the Hilbert spaces L2 (R+ , e−r rα dr) and L2 (R, e−x dx).
Therefore the families
)
(s
)
(s
x2
1
2α+1 Γ(α + 1)m! − r2 α 2
√ e− 2 Hn
e 2 Lm (r )
and
n
Γ(α + m + 1)
2 n! π
n∈N
m∈N
2α+1
are respectively a Hilbert basis of the Hilbert spaces L2 R+ , 2αrΓ(α+1) dr and L2 R, √dx2π ,
n
o
α
hence the family em,n
defined by
(m,n)∈N2
eαm,n (r, x) =
2α+1 Γ(α + 1)m!
n− 12
2
! 12
n!Γ(m + α + 1)
e−
r 2 +x2
2
Lαm (r2 )Hn (x),
is a Hilbert basis of the space L2 (dνα ).
n
o
α
Using the relation (2.8), we deduce that the family ξm,n
(m,n)∈N2
α+1
α
(µ, λ) = (eαm,n ◦ θ)(µ, λ) =
ξm,n
! 12
2
Γ(α + 1)m!
n− 12
n!Γ(m + α + 1)
2
e−
, defined by
µ2 +2λ2
2
Lαm (µ2 + λ2 )Hn (λ),
is a Hilbert basis of the space L2 (dγα ), where θ is the function defined by the relation (2.4).
In the following, we agree that the Laguerre and Hermite polynomials with negative index
are zero.
Proposition 3.1. For all (m, n) ∈ N2 , (r, x) ∈ R+ × R and (µ, λ) ∈ Γ+ , we have
(3.1)
xeαm,n (r, x)
(3.2)
α
λξm,n
(µ, λ)
(3.3) r2 eα+1
m,n (r, x) =
(3.4)
r
n+1 α
e
(r, x) +
2 m,n+1
r
n α
e
(r, x).
2 m,n−1
r
n+1 α
ξ
(µ, λ) +
2 m,n+1
r
n α
ξ
(µ, λ).
2 m,n−1
=
=
p
p
2(α + 1)(α + m + 1)eαm,n (r, x) − 2(α + 1)(m + 1)eαm+1,n (r, x).
α+1
(µ, λ) =
(µ2 + λ2 )ξm,n
p
α
2(α + 1)(α + m + 1)ξm,n
(µ, λ)
p
α
− 2(α + 1)(m + 1)ξm+1,n
(µ, λ).
Proof. We know [11] that the Hermite polynomials satisfy the following recurrence formula
Hn+1 (x) − 2xHn (x) + 2nHn−1 (x) = 0;
J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 88, 23 pp.
n ∈ N,
http://jipam.vu.edu.au/
8
S. O MRI
AND
L.T. R ACHDI
Therefore, for all (r, x) ∈ R+ × R, we have
xeαm,n (r, x) =
2α+1 Γ(α + 1)m!
! 12
n− 12
e−
r 2 +x2
2
Lαm (r2 )xHn (x)
2
n!Γ(m + α + 1)
! 12
r
2
2
2α+1 Γ(α + 1)m!
n+1
− r +x
2
=
e
Lαm (r2 )Hn+1 (x)
1
n+
2
2 2 (n + 1)!Γ(m + α + 1)
! 12
r
α+1
|(r,x)|2
2 Γ(α + 1)m!
n
− 2
e
Lαm (r2 )Hn−1 (x)
+
3
n−
2 2 2 (n − 1)!Γ(m + α + 1)
r
r
n+1 α
n α
=
em,n+1 (r, x) +
e
(r, x)
2
2 m,n−1
α
and it is obvious that the same relation holds for the elements ξm,n
.
On the other hand
α+2
r2 eα+1
m,n (r, x) =
2
Γ(α + 2)m!
n− 12
n!Γ(m + α + 2)
2
! 12
e−
r 2 +x2
2
2
r2 Lα+1
m (r )Hn (x).
However, the Laguerre polynomials satisfy the following recurrence formulas
(m + 1)Lαm+1 (r) + (r − α − 2m − 1)Lαm (r) + (m + α)Lαm−1 (r) = 0;
m ∈ N,
and
α+1
α
Lα+1
m (r) − Lm−1 (r) = Lm (r);
m ∈ N.
Hence, we deduce that
r2 eα+1
m,n (r, x)
=
2α+2 Γ(α + 2)m!
n− 12
2
n!Γ(m + α + 2)
! 12
e−
r 2 +x2
2
Hn (x)
2
α+1
2
α+1
2
× (α + 2m + 2)Lα+1
m (r ) − (m + 1)Lm+1 (r ) − (α + m + 1)Lm−1 (r )
! 12
2
2
2α+2 Γ(α + 2)m!
− r +x
2
(α
+
m
+
1)e
Lαm (r2 )Hn (x)
=
n− 12
2
n!Γ(m + α + 2)
! 12
2
2
2α+2 Γ(α + 2)m!
− r +x
2
−
(m
+
1)e
Lαm+1 (r2 )Hn (x)
1
n− 2
2
n!Γ(m + α + 2)
p
p
= 2(α + 1)(α + m + 1)eαm,n (r, x) − 2(α + 1)(m + 1)eαm+1,n (r, x).
Proposition 3.2. For all (m, n) ∈ N2 , and (µ, λ) ∈ Γ+ , we have
(3.5)
α
Fα (eαm,n )(µ, λ) = (−i)2m+n ξm,n
(µ, λ).
J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 88, 23 pp.
http://jipam.vu.edu.au/
H EISENBERG -PAULI -W EYL U NCERTAINTY P RINCIPLE
9
Proof. It is clear that for all (m, n) ∈ N2 , the function eαm,n belongs to the space L1 (dνα ) ∩
L2 (dνα ), hence by using Fubini’s theorem, we get
Fα (eαm,n )(µ, λ) =
2α+1 Γ(α + 1)m!
! 12
1
2n− 2 n!Γ(m + α + 1)
Z +∞
p
r2
×
e− 2 Lαm (r2 )jα r µ2 + λ2
r2α+1
dr
2α Γ(α + 1)
0
Z
2
dx
− x2 −iλx
×
e
Hn (x) √
,
2π
R
and then the required result follows from the following equalities [11]:
Z +∞
y
r
α
α
√
∀m ∈ N;
e− 2 Lαm (r)Jα ( ry)r 2 dr = (−1)m 2e− 2 y 2 Lαm (y),
0
and
√
y2
x2
eixy e− 2 Hn (x)dx = in 2πe− 2 Hn (y),
Z
∀n ∈ N;
R
where Jα denotes the Bessel function of the first kind and index α defined for all x > 0 by
+∞
x 2n+α
X
(−1)n
Jα (x) =
.
n!Γ(α
+
n
+
1)
2
n=0
Proposition 3.3. Let f ∈ L2 (dνα ) ∩ L2 (dνα+1 ) such that Fα (f ) ∈ L2 (dγα+1 ), then for all
(m, n) ∈ N2 , we have
s
s
α
+
m
+
1
m+1
(3.6)
hf /eα+1
hf /eαm,n iνα −
hf /eαm+1,n iνα ,
m,n iνα+1 =
2(α + 1)
2(α + 1)
and
s
α+1
(3.7) hFα (f )/ξm,n
iγα+1 =
α+m+1
(−i)2m+n hf /eαm,n iνα
2(α + 1)
s
m+1
+
(−i)2m+n hf /eαm+1,n iνα .
2(α + 1)
Proof. We have
hf /eα+1
m,n iνα+1
Z
+∞ Z
f (r, x)eα+1
m,n (r, x)dνα+1 (r, x)
0
R
Z +∞ Z
1
=
f (r, x)r2 eα+1
m,n (r, x)dνα (r, x)
2(α + 1) 0
R
1
=
hf /r2 eα+1
m,n iνα ,
2(α + 1)
=
hence by using the relation (3.3), we deduce that
s
s
α
+
m
+
1
m+1
hf /eα+1
hf /eαm,n iνα −
hf /eαm+1,n iνα .
m,n iνα+1 =
2(α + 1)
2(α + 1)
J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 88, 23 pp.
http://jipam.vu.edu.au/
10
S. O MRI AND L.T. R ACHDI
In the same manner, and by virtue of the relation (2.7), we have
Z Z
α+1
α+1
hFα (f )/ξm,n iγα+1 =
Fα (f )(µ, λ)ξm,n
(µ, λ)dγα+1 (µ, λ)
Γ+
Z Z
1
α+1
=
Fα (f )(µ, λ)(µ2 + λ2 )ξm,n
(µ, λ)dγα (µ, λ),
2(α + 1)
Γ+
using the relations (3.4) and (3.5), we deduce that
Z Z
√
1
α+1
α
hFα (f )/ξm,n iγα+1 = p
Fα (f )(µ, λ) α + m + 1ξm,n
(µ, λ)
2(α + 1)
Γ+
√
α
− m + 1ξm+1,n (µ, λ) dγα (µ, λ)
s
α+m+1
hFα (f )/(i)2m+n Fα (eαm,n )iγα
=
2(α + 1)
s
m+1
−
hFα (f )/(i)2m+2+n Fα (eαm+1,n )iγα ,
2(α + 1)
hence, according to the Parseval’s equality (2.11), we obtain
s
α+m+1
α+1
hFα (f )/ξm,n
iγα+1 =
(−i)2m+n hf /eαm,n iνα
2(α + 1)
s
m+1
+
(−i)2m+n hf /eαm+1,n iνα .
2(α + 1)
4. H EISENBERG -PAULI -W EYL I NEQUALITY FOR THE F OURIER T RANSFORM Fα
In this section, we will prove the main result of this work, that is the Heisenberg-Pauli-Weyl
inequality for the Fourier transform Fα connected with the Riemann-Liouville operator Rα .
Next we give a generalization of this result, for this we need the following important lemma.
Lemma 4.1. Let f ∈ L2 (dνα ), such that
k|(r, x)|f k2,να < +∞
and
1
k(µ2 + 2λ2 ) 2 Fα (f )k2,γα < +∞,
then
(4.1)
1
k|(r, x)|f k22,να + k(µ2 + 2λ2 ) 2 Fα (f )k22,γα =
+∞
X
2α + 4m + 2n + 3 |am,n |2 ,
m,n=0
where am,n = hf /eαm,n iνα ;
(m, n) ∈ N2 .
Proof. Let f ∈ L2 (dνα ), such that
∀(r, x) ∈ R+ × R;
f (r, x) =
+∞
X
am,n eαm,n (r, x),
m,n=0
and assume that
k|(r, x)|f k2,να < +∞
1
and k(µ2 + 2λ2 ) 2 Fα (f )k2,γα < +∞,
J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 88, 23 pp.
http://jipam.vu.edu.au/
H EISENBERG -PAULI -W EYL U NCERTAINTY P RINCIPLE
11
then the functions (r, x) 7−→ rf (r, x) and (r, x) 7−→ xf (r, x) belong to the space L2 (dνα ), in
particular f ∈ L2 (dνα ) ∩ L2 (dνα+1 ). In the same manner, the functions
1
(µ, λ) 7−→ (µ2 + λ2 ) 2 Fα (f )(µ, λ),
(µ, λ) 7−→ λFα (f )(µ, λ)
and
belong to the space L2 (dγα ). In particular, by the relation (2.7), we deduce that Fα (f ) ∈
L2 (dγα ) ∩ L2 (dγα+1 ), and we have
Z +∞ Z
2
krf k2,να =
r2 |f (r, x)|2 dνα (r, x)
0
R
= 2(α + 1)kf k22,να+1
+∞
X
2
hf /eα+1
= 2(α + 1)
m,n iνα+1 ,
m,n=0
hence, according to the relation (3.6), we obtain
+∞
X
√
√
α + m + 1am,n − m + 1am+1,n 2 .
=
krf k22,να
(4.2)
m,n=0
Similarly, we have
kxf k22,να
Z
+∞ Z
x2 |f (r, x)|2 dνα (r, x)
=
=
0
+∞
X
R
+∞
X
hxf /eαm,n iνα 2 =
hf /xeαm,n iνα 2 ,
m,n=0
m,n=0
and by the relation (3.1), we get
kxf k22,να
(4.3)
2
r
+∞ r
X
n
n+1
=
am,n+1 +
am,n−1 .
2
2
m,n=0
By the same arguments, and using the relations (3.2), (3.7) and the Parseval’s equality (2.11),
we obtain
(4.4)
2
2
k(µ + λ )
1
2
Fα (f )k22,γα
+∞
X
√
√
α + m + 1am,n + m + 1am+1,n 2 ,
=
m,n=0
and
(4.5)
kλFα (f )k22,γα
2
r
+∞ r
X
n
n+1
am,n+1 −
am,n−1 .
=
2
2
m,n=0
Combining now the relations (4.2), (4.3), (4.4) and (4.5), we deduce that
1
k|(r, x)|f k22,να + k(µ2 + 2λ2 ) 2 Fα (f )k22,γα
1
= krf k22,να + k(µ2 + λ2 ) 2 Fα (f )k22,γα + kxf k22,να + kλFα (f )k22,γα
J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 88, 23 pp.
http://jipam.vu.edu.au/
12
S. O MRI AND L.T. R ACHDI
=2
+∞ X
(α + m + 1)|am,n |2 + (m + 1)|am+1,n |2
m,n=0
+∞ X
n+1
n
2
2
+2
|am,n+1 | + |am,n−1 |
2
2
m,n=0
+∞
+∞
X
X
n+1
n
2
=2
(α + m + 1)|am,n | + 2
m|am,n | + 2
|am,n | + 2
|am,n |2
2
2
m,n=0
m,n=0
m,n=0
m,n=0
+∞
X
+∞
X
=
2
+∞
X
2
2α + 4m + 2n + 3 |am,n |2 .
m,n=0
Remark 1. From the relation (4.1), we deduce that for all f ∈ L2 (dνα ), we have
(4.6)
1
k|(r, x)|f k22,να + k(µ2 + 2λ2 ) 2 Fα (f )k22,γα ≥ (2α + 3)kf k22,να ,
with equality if and only if
∀(r, x) ∈ R+ × R;
f (r, x) = Ce−
r 2 +x2
2
;
C ∈ C.
Lemma 4.2. Let f ∈ L2 (dνα ) such that,
k|(r, x)|f k2,να < +∞
1
k(µ2 + 2λ2 ) 2 Fα (f )k2,γα < +∞,
and
then
1) For all t > 0,
1
1
k|(r, x)|f k22,να + t2 k(µ2 + 2λ2 ) 2 Fα (f )k22,γα ≥ (2α + 3)kf k22,να .
2
t
2) The following assertions are equivalent
1
2α + 3
i) k|(r, x)|f k2,να k(µ2 + 2λ2 ) 2 Fα (f )k2,γα =
kf k22,να .
2
ii) There exists t0 > 0, such that
1
k|(r, x)|ft0 k22,να + k(µ2 + 2λ2 ) 2 Fα (ft0 )k22,γα = (2α + 3)kft0 k22,να ,
where ft0 (r, x) = f (t0 r, t0 x).
Proof. 1) Let f ∈ L2 (dνα ) satisfy the hypothesis.For all t > 0 we put ft (r, x) = f (tr, tx), and
then by a simple change of variables, we get
1
(4.7)
kft k22,να = 2α+3 kf k22,να ,
t
and
1
(4.8)
k|(r, x)|ft k22,να = 2α+5 k|(r, x)|f k22,να .
t
For all (µ, λ) ∈ Γ,
1
µ λ
(4.9)
Fα (ft )(µ, λ) = 2α+3 Fα (f )
,
,
t
t t
and by using the relation (2.6), we deduce that
1
1
1
(4.10)
k(µ2 + 2λ2 ) 2 Fα (ft )k22,γα = 2α+1 k(µ2 + 2λ2 ) 2 Fα (f )k22,γα .
t
J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 88, 23 pp.
http://jipam.vu.edu.au/
H EISENBERG -PAULI -W EYL U NCERTAINTY P RINCIPLE
13
Then, the desired result follows by replacing f by ft in the relation (4.6).
2) Let f ∈ L2 (dνα ); f 6= 0.
• Assume that
1
k|(r, x)|f k2,να k(µ2 + 2λ2 ) 2 Fα (f )k2,γα =
2α + 3
kf k22,να .
2
1
By Theorem 2.2, we have k(µ2 + 2λ2 ) 2 Fα (f )k2,γα 6= 0, then for
s
k|(r, x)|f k2,να
t0 =
,
1
k(µ2 + 2λ2 ) 2 Fα (f )k2,γα
we have
1
1
k|(r, x)|f k22,να + t20 k(µ2 + 2λ2 ) 2 Fα (f )k22,γα = (2α + 3)kf k22,να ,
2
t0
and this is equivalent to
1
k|(r, x)|ft0 k22,να + k(µ2 + 2λ2 ) 2 Fα (ft0 )k22,γα = (2α + 3)kft0 k22,να .
• Conversely, suppose that there exists t1 > 0, such that
1
k|(r, x)|ft1 k22,να + k(µ2 + 2λ2 ) 2 Fα (ft1 )k22,γα = (2α + 3)kft1 k22,να .
This is equivalent to
1
1
(4.11)
k|(r, x)|f k22,να + t21 k(µ2 + 2λ2 ) 2 Fα (f )k22,γα = (2α + 3)kf k22,να .
2
t1
However, let h be the function defined on ]0, +∞[, by
1
1
h(t) = 2 k|(r, x)|f k22,να + t2 k(µ2 + 2λ2 ) 2 Fα (f )k22,γα ,
t
then, the minimum of the function h is attained at the point
s
k|(r, x)|f k2,να
t0 =
1
k(µ2 + 2λ2 ) 2 Fα (f )k2,γα
and
1
h(t0 ) = 2k|(r, x)|f k2,να k(µ2 + 2λ2 ) 2 Fα (f )k2,γα .
Thus by 1) of this lemma, we have
1
h(t1 ) ≥ h(t0 ) = 2k|(r, x)|f k2,να k(µ2 + 2λ2 ) 2 Fα (f )k2,γα ≥ (2α + 3)kf k22,να .
According to the relation (4.11), we deduce that
1
h(t1 ) = h(t0 ) = 2k|(r, x)|f k2,να k(µ2 + 2λ2 ) 2 Fα (f )k2,γα = (2α + 3)kf k22,να .
Theorem 4.3 (Heisenbeg-Pauli-Weyl inequality). For all f ∈ L2 (dνα ), we have
(4.12)
(2α + 3)
kf k22,να
2
1
k|(r, x)|f k2,να k(µ2 + 2λ2 ) 2 Fα (f )k2,γα ≥
with equality if and only if
∀(r, x) ∈ R+ × R;
f (r, x) = Ce
J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 88, 23 pp.
−r
2 +x2
2t2
0
;
t0 > 0, C ∈ C.
http://jipam.vu.edu.au/
14
S. O MRI AND L.T. R ACHDI
1
Proof. It is obvious that if f = 0, or if k|(r, x)|f k2,να = +∞, or k(µ2 + 2λ2 ) 2 Fα (f )k2,γα =
+∞, then the inequality (4.12) holds.
1
Let us suppose that k|(r, x)|f k2,να + k(µ2 + 2λ2 ) 2 Fα (f )k2,γα < +∞, and f 6= 0.
By 1) of Lemma 4.2, we have for all t > 0
1
2
2
2
2 12
k|(r,
x)|f
k
+
t
k(µ
+
2λ
) Fα (f )k22,γα ≥ (2α + 3)kf k22,να ,
2,ν
α
2
t
and the result follows if we pick
s
k|(r, x)|f k2,να
t=
.
1
2
k(µ + 2λ2 ) 2 Fα (f )k2,γα
By 2) of Lemma 4.2, we have
1
k|(r, x)|f k2,να k(µ2 + 2λ2 ) 2 Fα (f )k2,γα =
2α + 3
kf k22,να ,
2
if and only if there exists t0 , such that
1
k|(r, x)|ft0 k22,να + k(µ2 + 2λ2 ) 2 Fα (ft0 )k22,γα = (2α + 3)kft0 k22,να ,
and according to Remark 1, this is equivalent to
ft0 (r, x) = Ce−
r 2 +x2
2
which means that
f (r, x) = Ce
−r
2 +x2
2t2
0
;
C ∈ C,
;
C ∈ C.
The following result gives a generalization of the Heisenberg-Pauli-Weyl inequality.
Theorem 4.4. Let a, b ≥ 1 and η ∈ R such that ηa = (1 − η)b, then for all f ∈ L2 (dνα ) we
have
aη
2α + 3
η
1−η
a
2
2 2b
k|(r, x)| f k2,να k(µ + 2λ ) Fα (f )k2,γα ≥
kf k2,να
2
with equality if and only if a = b = 1 and
∀(r, x) ∈ R+ × R;
f (r, x) = Ce
−r
2 +x2
2t2
0
;
t0 > 0, C ∈ C.
Proof. Let f ∈ L2 (dνα ), f 6= 0, such that
b
k|(r, x)|a f k2,να + k(µ2 + 2λ2 ) 2 Fα (f )k2,γα < +∞.
Then for all a > 1, we have
1
1
2
1
2
1
a
a0
2
a0 k 20
k|(r, x)|a f k2,ν
kf k2,ν
= k|(r, x)|2 |f | a ka,ν
α k|f |
a ,να ,
α
α
where a0 is defined as usual by a0 =
a
.
a−1
By Hölder’s inequality we get
1
1
a
a0
k|(r, x)|a f k2,ν
kf k2,ν
> k|(r, x)|f k2,να .
α
α
The strict inequality here is justified by the fact that if f 6= 0, then the functions |(r, x)|2a |f |2
and |f |2 cannot be proportional.Thus for all a ≥ 1, we have
1
(4.13)
a
k|(r, x)|a f k2,ν
≥
α
k|(r, x)|f k2,να
1
.
a0
kf k2,ν
α
with equality if and only if a = 1.
J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 88, 23 pp.
http://jipam.vu.edu.au/
H EISENBERG -PAULI -W EYL U NCERTAINTY P RINCIPLE
15
In the same manner and using Plancherel’s Theorem 2.3, we have for all b ≥ 1
1
1
b
b
2
k(µ + 2λ ) Fα (f )k2,γα ≥
2
(4.14)
2
k(µ2 + 2λ2 ) 2 Fα (f )k2,γα
1
b0
kFα (f )k2,γ
α
1
≥
k(µ2 + 2λ2 ) 2 Fα (f )k2,γα
1
.
b0
kf k2,ν
α
with equality if and only if b = 1.
b
Let η = a+b
, then by the relations (4.13), (4.14) and for all a, b ≥ 1, we have

ηa
2
2 21
b
2λ ) Fα (f )k2,γα 
 k|(r, x)|f k2,να k(µ +
k|(r, x)|a f kη2,να k(µ2 + 2λ2 ) 2 Fα (f )k1−η
,
1
2,γα ≥
+ b10
a0
kf k2,ν
α
with equality if and only if a = b = 1.
Applying Theorem 4.3, we obtain
a
k|(r, x)|
f kη2,να k(µ2
2
+ 2λ )
b
2
Fα (f )k1−η
2,γα
≥
2α + 3
2
ηa
kf k2,να ,
with equality if and only if a = b = 1 and
∀(r, x) ∈ R+ × R;
f (r, x) = Ce
−r
2 +x2
2t2
0
;
t0 > 0, C ∈ C.
Remark 2. In the particular case when a = b = 2, the previous result gives us the HeisenbergPauli-Weyl inequality for the fourth moment of Heisenberg
2
2α + 3
2
2
2
k|(r, x)| f k2,να k(µ + 2λ )Fα (f )k2,γα >
kf k22,να .
2
5. T HE L OCAL U NCERTAINTY P RINCIPLE
Theorem 5.1. Let ξ be a real number such that 0 < ξ < 2α+3
, then for all f ∈ L2 (dνα ), f 6= 0,
2
and for all measurable subsets E ⊂ Γ+ ; 0 < γα (E) < +∞, we have
ZZ
2ξ
(5.1)
|Fα (f )(µ, λ)|2 dγα (µ, λ) < Kα,ξ γα (E) 2α+3 k|(r, x)|ξ f k22,να ,
E
where
Kα,ξ =
2ξ
! 2α+3
2α + 3 − 2ξ
5
ξ 2 2α+ 2 Γ α +
3
2
2α + 3
2α + 3 − 2ξ
2
.
Proof. For all s > 0, we put
Bs = (r, x) ∈ R+ × R ; r2 + x2 < s2 .
Let f ∈ L2 (dνα ). By Minkowski’s inequality, we have
Z Z
12
2
(5.2)
|Fα (f )(µ, λ)| dγα (µ, λ)
E
= kFα (f )1E k2,γα
≤ kFα (f 1Bs )1E k2,γα + kFα (f 1Bsc )1E k2,γα
1
≤ γα (E) 2 kFα (f 1Bs )k∞,γα + kFα (f 1Bsc )k2,γα .
J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 88, 23 pp.
http://jipam.vu.edu.au/
16
S. O MRI AND L.T. R ACHDI
Applying the relation (2.10), we deduce that for every s > 0, we have
Z Z
21
1
2
(5.3)
|Fα (f )(µ, λ)| dγα (µ, λ)
≤ γα (E) 2 kf 1Bs k1,να + kFα (f 1Bsc )k2,γα .
E
On the other hand, by Hölder’s inequality we have
(5.4)
kf 1Bs k1,να ≤ k|(r, x)|ξ f k2,να k|(r, x)|−ξ 1Bs k2,να
s
ξ
= k|(r, x)| f (r, x)k2,να (5.5)
α+ 12
2
Γ α+
2α+3−2ξ
2
3
2
12 .
(2α + 3 − 2ξ)
By Plancherel’s theorem 2.3, we have also
kFα (f 1Bsc )k2,γα = kf 1Bsc k2,να
(5.6)
≤ k|(r, x)|ξ f k2,να k|(r, x)|−ξ 1Bsc k∞,να
= s−ξ k|(r, x)|ξ f k2,να .
Combining the relations (5.3), (5.5) and (5.6), we deduce that for all s > 0 we have
Z Z
21
(5.7)
|Fα (f )(µ, λ)|2 dγα (µ, λ)
≤ gα,ξ (s)k|(r, x)|ξ f (r, x)k2,να ,
E
where gα,ξ is the function defined on ]0, +∞[ by
gα,ξ (s) = s−ξ +
α+ 12
2
γα (E)
Γ α + 32 (2α + 3 − 2ξ)
! 12
s
2α+3−2ξ
2
.
Thus, the inequality (5.7) holds for
s0 =
! 1
5
ξ 2 2α+ 2 Γ( α + 32 ) 2α+3
,
γα (E)(2α + 3 − 2ξ)
however
1
ξ
2
gα,ξ (s0 ) = γα (E) 2α+3 Kα,ξ
,
where
Kα,ξ =
2α + 3 − 2ξ
2ξ
! 2α+3
2α + 3
2α + 3 − 2ξ
2
.
5
ξ 2 2α+ 2 Γ(α + 32 )
Let us prove that the equality in (5.1) cannot hold. Indeed, suppose that
Z Z
2ξ
|Fα (f )(µ, λ)|2 dγα (µ, λ) = Kα,ξ γα (E) 2α+3 k|(r, x)|ξ f k22,να .
E
Then
Z Z
|Fα (f )(µ, λ)|2 dγα (µ, λ) = gα,ξ (s0 )2 k|(r, x)|ξ f k22,να ,
E
and therefore by the relations (5.2), (5.3) and (5.4), we get
1
(5.8)
kFα (f 1Bs0 )1E k2,γα = γα (E) 2 kFα (f 1Bs0 )k∞,γα ,
(5.9)
kf 1Bs0 k1,να = kFα (f 1Bs0 )k∞,γα ,
J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 88, 23 pp.
http://jipam.vu.edu.au/
H EISENBERG -PAULI -W EYL U NCERTAINTY P RINCIPLE
17
and
kf 1Bs0 k1,να = k|(r, x)ξ f k2,να k|(r, x)−ξ 1Bs0 k2,να .
(5.10)
However, if f satisfies the equality (5.10), then there exists C > 0, such that
|(r, x)|2ξ f (r, x)2 = C|(r, x)|−2ξ 1Bs0 ,
hence
∀(r, x) ∈ R+ × R;
(5.11)
f (r, x) = CeiΦ(r,x) |(r, x)|−2ξ 1Bs0 ,
where Φ is a real measurable function on R+ × R.
But if f satisfies the relation (5.9), then there exists (µ0 , λ0 ) ∈ Γ+ , such that
kf k1,να = kFα (f )k∞,γα = Fα (f )(µ0 , λ0 ).
So, there exists θ0 ∈ R satisfying
Fα (f )(µ0 , λ0 ) = eiθ0 kf k1,να ,
and therefore
q
Z +∞ Z
−2ξ
iθ0
iΦ(r,x)−iλ0 x−iθ0
|(r, x)| 1Bs0 (r, x) e
Ce
jα r µ20 + λ20 − 1 dνα (r, x) = 0.
0
R
This implies that for almost every (r, x) ∈ R+ × R,
q
iΦ(r,x)−iλ0 x−iθ0
2
2
e
jα r µ0 + λ0 = 1.
Hence, we deduce that for all r ∈ R+ ,
q
jα r µ20 + λ20 = 1.
Using the relation (2.2), it follows that µ20 + λ20 = 0, or µ0 = i|λ0 |, and then
eiΦ(r,x) = eiθ0 +iλ0 x .
Replacing in (5.11), we get
f (r, x) = C 0iλ0 x |(r, x)|−2ξ 1Bs0 (r, x).
Now, the relation (5.8) means that for almost every (µ, λ) ∈ E, we have
Fα (f )(µ, λ) = kFα (f )k∞,γα = Fα (f )(µ0 , λ0 ),
which implies that for almost every (µ, λ) ∈ E, we have
Fα (f )(µ, λ) = eiψ(µ,λ) Fα (f )(µ0 , λ0 ),
where ψ is a real measurable function on E, and therefore
Z +∞ Z
p
0iψ(µ,λ)
C
|(r, x)|−2ξ 1Bs0 (r, x) e−iλx+iλ0 x−iψ(µ,λ) jα r µ2 + λ2 − 1 dνα (r, x) = 0.
0
R
Consequently for all (r, x) ∈ R+ × R,
p
e−iλx+iλ0 x−iψ(µ,λ) jα r µ2 + λ2 = 1,
which implies that λ = λ0 and µ = µ0 .
However, since γα (E) > 0, this contradicts the fact that for almost every (µ, λ) ∈ E,
Fα (f )(µ, λ) = Fα (f )(µ0 , λ0 ),
and shows that the inequality in (5.1) is strictly satisfied.
J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 88, 23 pp.
http://jipam.vu.edu.au/
18
S. O MRI AND L.T. R ACHDI
Lemma 5.2. Let ξ be a real number such that ξ >
R+ × R we have
2−
2α+3
,
2
(2α+3)
ξ
kf k21,να ≤ Mα,ξ kf k2,να
(5.12)
then for all measurable function f on
(2α+3)
k|(r, x)|ξ f k2,ναξ ,
where
Mα,ξ =
π
α+ 12
2
Γ α+
3
2
(2ξ − 2α − 3)
2ξ−2α+3
2ξ
(2α + 3)
2α+3
2ξ
.
2α+3
sin π 2ξ
with equality in (5.12) if and only if there exist a, b > 0 such that
|f (r, x)| = (a + b|(r, x)|2ξ )−1 .
Proof. We suppose naturally that f 6= 0. It is obvious that the inequality (5.12) holds if
kf k2,να = +∞ or k|(r, x)|ξ f k2,να = +∞.
Assume that kf k2,να + k|(r, x)|ξ f k2,να < +∞.From the hypothesis 2ξ > 2α + 3, we deduce
that for all a, b > 0, the function
(r, x) 7−→ (a + b|(r, x)|2ξ )−1
belongs to L1 (dνα ) ∩ L2 (dνα ) and by Hölder’s inequality, we have
2
1 2
2ξ − 12 kf k21,να ≤ (1 + |(r, x)|2ξ ) 2 f (5.13)
(1
+
|(r,
x)|
)
2,να
2,να
1 2
≤ kf k22,να + k|(r, x)|ξ f k22,να (1 + |(r, x)|2ξ )− 2 .
2,να
However, by standard calculus, we have
2
2ξ − 21 =
(1 + |(r, x)| ) 2,να
α+ 32
ξ2
π
.
Γ α + 32 sin π 2α+3
2ξ
Thus
(5.14)
kf k21,να ≤
α+ 32
ξ2
π
kf k22,να + k|(r, x)|ξ f k22,να ,
Γ α + 32 sin π 2α+3
2ξ
with equality in (5.14) if and only if we have equality in (5.13), that is there exists C > 0
satisfying
1
1
(1 + |(r, x)|2ξ ) 2 |f (r, x)| = C(1 + |(r, x)|2ξ )− 2 ,
or
(5.15)
|f (r, x)| = C(1 + |(r, x)|2ξ )−1 .
For t > 0, we put as above, ft (r, x) = f (tr, tx), then we have
(5.16)
kft k21,να =
1
t4α+6
kf k21,να ,
and
(5.17)
k|(r, x)|ξ ft k22,να =
1
t2ξ+2α+3
k|(r, x)|ξ f k22,να .
Replacing f by ft in the relation (5.14), we deduce that for all t > 0, we have
π
2
2α+3
2
2α+3−2ξ
ξ
2
t
kf k2,να + t
k|(r, x)| f k2,να .
kf k1,να ≤
3
ξ2α+ 2 Γ α + 32 sin π 2α+3
2ξ
J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 88, 23 pp.
http://jipam.vu.edu.au/
H EISENBERG -PAULI -W EYL U NCERTAINTY P RINCIPLE
19
In particular, for
t = t0 =
(2ξ − 2α − 3)k|(r, x)|ξ f k22,να
(2α + 3)kf k22,να
! 2ξ1
,
we get
2−
(2α+3)
ξ
kf k21,να ≤ Mα,ξ kf k2,να
(2α+3)
k|(r, x)|ξ f k2,ναξ ,
where
Mα,ξ =
π
α+ 12
2
Γ α+
3
2
(2ξ − 2α − 3)
2ξ−2α+3
2ξ
(2α + 3)
2α+3
2ξ
.
2α+3
sin π 2ξ
Now suppose that we have equality in the last inequality. Then we have equality in (5.14) for
ft0 and by means of (5.15), we obtain
|ft0 (r, x)| = C(1 + |(r, x)|2ξ )−1 ,
and then
|f (r, x)| = (a + b|(r, x)|2ξ )−1 .
Theorem 5.3. Let ξ be a real number such that ξ > 2α+3
. Then for all f ∈ L2 (dνα ), f 6= 0,
2
and for all measurable subsets E ⊂ Γ+ ; 0 < γα (E) < +∞, we have
Z Z
2α+3
2− 2α+3
(5.18)
|Fα (f )(µ, λ)|2 dγα (µ, λ) < Mα,ξ γα (E)kf k2,να ξ k|(r, x)|ξ f k2,νξα ,
E
where
Mα,ξ =
π
α+ 12
2
Γ α+
3
2
(2ξ − 2α − 3)
2ξ−2α+3
2ξ
(2α + 3)
2α+3
2ξ
.
2α+3
sin π 2ξ
Moreover, Mα,ξ is the best (the smallest) constant satisfying (5.18).
Proof.
• Suppose that the right-hand side of (5.18) is finite. Then, according to Lemma 5.2,
the function f belongs to L1 (dνα ) and we have
Z Z
|Fα (f )(µ, λ)|2 dγα (µ, λ) ≤ γα (E)kFα (f )k2∞,γα
E
≤ γα (E)kf k21,να
2−
(2α+3)
ξ
≤ γα (E)Mα,ξ kf k2,να
(2α+3)
k|(r, x)|ξ f k2,ναξ ,
where
Mα,ξ =
π
1
2α+ 2 Γ α +
3
2
(2ξ − 2α − 3)
2ξ−2α+3
2ξ
(2α + 3)
2α+3
2ξ
.
sin π 2α+3
2ξ
• Let us prove that the equality in (5.18) cannot hold.
Indeed, suppose that
Z Z
2α+3
2− 2α+3
|Fα (f )(µ, λ)|2 dγα (µ, λ) = Mα,ξ γα (E)kf k2,να ξ k|(r, x)|ξ f k2,νξα .
E
Then
Z Z
(5.19)
|Fα (f )(µ, λ)|2 dγα (µ, λ) = γα (E)kFα (f )k2∞,γα ,
Γ+
(5.20)
kFα (f )k∞,γα = kf k1,να ,
J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 88, 23 pp.
http://jipam.vu.edu.au/
20
S. O MRI AND L.T. R ACHDI
and
2− 2α+3
(2α+3)
kf k21,να = Mα,ξ kf k2,να ξ k|(r, x)|ξ f k2,ναξ .
(5.21)
Applying Lemma 5.2 and the relation (5.21), we deduce that
∀(r, x) ∈ R+ × R;
(5.22)
f (r, x) = ϕ(r, x)(a + b|(r, x)|2ξ )−1 ,
with |ϕ(r, x)| = 1; a, b > 0.
On the other hand, there exists (µ0 , λ0 ) ∈ Γ+ such that
kFα (f )k∞,γα = |Fα (f )(µ0 , λ0 )| = eiθ0 Fα (f )(µ0 , λ0 );
(5.23)
θ0 ∈ R.
Combining now the relations (5.20), (5.22) and (5.23), we get
q
Z +∞ Z iθ0
−iλ0 x
2
2
1 − e ϕ(r, x)jα r µ0 + λ0 e
(a + b|(r, x)|2ξ )−1 dνα (r, x) = 0.
0
R
This implies that for almost every (r, x) ∈ R+ × R,
q
−iλ0 x iθ0
e
e ϕ(r, x)jα r µ20 + λ20 = 1.
Since |ϕ(r, x)| = 1, we deduce that for all r ∈ R+ ,
q
jα r µ20 + λ20 = 1.
Using the relation (2.2), it follows that µ20 + λ20 = 0, or µ0 = i|λ0 |, and therefore
ϕ(r, x) = e−iθ0 eiλ0 x .
Replacing in (5.22), we get
f (r, x) = Ceiλ0 x (a + b|(r, x)|2ξ )−1 ;
|C| = 1.
Now, the relation (5.19) means that
Z Z
kFα (f )k2∞,γα − |Fα (f )(µ, λ)|2 dγα (µ, λ) = 0.
E
Hence, for almost every (µ, λ) ∈ E,
|Fα (f )(µ, λ)| = kFα (f )k∞,γα = eiθ0 Fα (f )(µ0 , λ0 ).
(5.24)
Let Ψ(µ, λ) ∈ R, such that
|Fα (f )(µ, λ)| = eiΨ(µ,λ) Fα (f )(µ, λ).
Then from (5.24), for almost every (µ, λ) ∈ E,
eiΨ(µ,λ) Fα (f )(µ, λ) = eiθ0 Fα (f )(µ0 , λ0 ),
and therefore
Z +∞ Z
0
h
p
i
(a + b|(r, x)|2ξ )−1 1 − ei(λ0 −λ)x eiΨ(µ,λ) jα r µ2 + λ2 dνα (r, x) = 0.
R
Consequently, for all (r, x) ∈ R+ × R,
i(λ0 −λ)x iΨ(µ,λ)
e
e
p
2
2
jα r µ + λ = 1,
which implies that λ = λ0 and µ = µ0 .
However, since γα (E) > 0, this contradicts the fact that for almost every (µ, λ) ∈ E,
eiΨ(µ,λ) Fα (f )(µ, λ) = eiθ0 Fα (f )(µ0 , λ0 ),
J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 88, 23 pp.
http://jipam.vu.edu.au/
H EISENBERG -PAULI -W EYL U NCERTAINTY P RINCIPLE
21
and shows that the inequality in (5.18) is strictly satisfied.
• Let us prove that the constant Mα,ξ is the best one satisfying (5.18).
Let A be a positive constant, such that for all f ∈ L2 (dνα ),
Z Z
2α+3
2− 2α+3
(5.25)
|Fα (f )(µ, λ)|2 dγα (µ, λ) ≤ Aγα (E)kf k2,να ξ k|(r, x)|ξ f k2,νξα .
E
We assume that k|(r, x)|ξ f k2,να < +∞. Then by Lemma 5.2, f belongs to L1 (dνα ).
Replacing f by ft (r, x) = f (tr, tx) in (5.25), and using the relations (4.7), (4.9) and (5.17), we
deduce that for all t > 0
2
Z Z 2α+3
2α+3
ξ
ξ
ξ
Fα (f ) µ , λ dγα (µ, λ) ≤ Akf k2−
k|(r,
x)|
f
k
.
2,ν
2,ν
α
α
t t
E
Using the dominate convergence theorem and the relations (2.1), (2.3), (2.9) and (2.10), we
deduce that
2
Z Z µ λ 2
lim
Fα (f ) t , t dγα (µ, λ) = |Fα (f )(0, 0)| γα (E).
t→+∞
E
Consequently, for all f ∈ L2 (dνα ), such that k|(r, x)|ξ f k2,να < +∞,
(5.26)
2− 2α+3
2α+3
|Fα (f )(0, 0)|2 ≤ Akf k2,να ξ k|(r, x)|ξ f k2,νξα .
Now, let f ∈ L2 (dνα ), such that k|(r, x)|ξ f k2,να < +∞, and let (µ, λ) ∈ Γ. Putting
p
g(r, x) = jα r µ2 + λ2 e−iλx f (r, x),
then g ∈ L2 (dνα ), and k|(r, x)|ξ gk2,να < +∞. Moreover, Fα (g)(0, 0) = Fα (f )(µ, λ), and by
(5.26), it follows that
2− 2α+3
2α+3
|Fα (f )(µ, λ)|2 ≤ Akf k2,να ξ k|(r, x)|ξ f k2,νξα .
Thus, for all f ∈ L2 (dνα ) such that k|(r, x)|ξ f k2,να < +∞, we have
(5.27)
2− 2α+3
2α+3
kFα (f )k2∞,γα ≤ Akf k2,να ξ k|(r, x)|ξ f k2,νξα .
Taking f0 (r, x) = (1 + |(r, x)|2ξ )−1 , we have

kFα (f0 )k2∞,γα = kf0 k21,να = 
kf0 k22,να =
α+ 32
ξ2
π
 ,
Γ α + 32 sin π 2α+3
2ξ
(2ξ − 2α − 3)π
,
3
2α+3
Γ α + 2 sin π 2ξ
α+ 52
ξ22
k|(r, x)|ξ f0 k22,να =
2
α+ 52
ξ22
(2α + 3)π
.
Γ α + 32 sin π 2α+3
2ξ
Replacing f by f0 in the relation (5.27), we obtain A ≥ Mα,ξ .
This completes the proof of Theorem 5.3.
J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 88, 23 pp.
http://jipam.vu.edu.au/
22
S. O MRI AND L.T. R ACHDI
R EFERENCES
[1] G. ANDREWS, R. ASKEY AND R. ROY, Special Functions. Cambridge University Press, 1999.
[2] C. BACCAR, N.B. HAMADI AND L.T. RACHDI, Inversion formulas for the Riemann-Liouville
transform and its dual associated with singular partial differential operators, International Journal
of Mathematics and Mathematical Sciences, 2006 (2006), Article ID 86238, 1-26.
[3] G. BATTLE, Heisenberg inequalities for wavelet states, Appl. and Comp. Harmonic Anal., 4(2)
(1997), 119–146.
[4] M.G. COWLING AND J.F. PRICE, Generalizations of Heisenberg’s inequality in Harmonic analysis (Cortona, 1982), Lecture Notes in Math., 992, Springer, Berlin (1983), 443–449. MR0729369
(86g:42002b).
[5] N.G. DE BRUIJN, Uncertainty principles in Fourier analysis, Inequalities (Proc. Sympos. WrightPatterson Air Force Base, Ohio, 1965), Academic Press, New York, (1967), 57–71.
[6] A. ERDELY, Asymptotic Expansions. Dover Publications, Inc.New-York, 1956.
[7] W.G. FARIS, Inequalities and uncertainty principles, J. Math. Phys., 19 (1978), 461–466.
[8] G.B. FOLLAND AND A. SITARAM, The uncertainty principle: a mathematical survey, J. Fourier
Anal. Appl., 3 (1997), 207–238.
[9] G.H. HARDY, A theorem concerning Fourier transform, J. London Math. Soc., 8 (1933), 227–231.
[10] W. HEISENBERG, Über den anschaulichen Inhalt der quantentheoretischen Kinematic und
Mechanik. Zeit. Physik, 43, 172 (1927); The Physical Principles of the Quantum Theory (Dover,
New York, 1949; The Univ. Chicago Press, 1930).
[11] N.N. LEBEDEV, Special Functions and their Applications, Dover Publications Inc., New-York,
1972.
[12] W. MAGNUS, F. OBERHETTINGER AND F.G. TRICOMI, Tables of Integral Transforms,
McGraw-Hill Book Compagny, Inc. New York/London/Toronto, 1954.
[13] G.W. MORGAN, A note on Fourier transforms, J. London. Math. Soc., 9 (1934), 178–192.
[14] S. OMRI AND L.T. RACHDI, An Lp − Lq version of Morgan’s theorem associated with RiemannLiouville transform, International Journal of Mathematical Analysis, 1(17) (2007), 805–824.
[15] J.F. PRICE, Inequalities and local uncertainty principles, J. Math. Phys., 24 (1983), 1711–1714.
[16] J.F. PRICE, Sharp local uncertainty inequalities, Studia Math., 85 (1987), 37–45.
[17] J.M. RASSIAS, On the Heisenberg-Pauli-Weyl inequality, J. Inequ. Pure and Appl. Math., 5(1)
(2004), Art. 4. [ONLINE: http://jipam.vu.edu.au/article.php?sid=356].
[18] J.M. RASSIAS, On the Heisenberg-Weyl inequality, J. Inequ. Pure and Appl. Math., 6(1) (2005),
Art. 11. [ONLINE: http://jipam.vu.edu.au/article.php?sid=480].
[19] J.M. RASSIAS, On the sharpened Heisenberg-Weyl inequality, J. Inequ. Pure and Appl. Math.,
7(3) (2006), Art. 80. [ONLINE: http://jipam.vu.edu.au/article.php?sid=727].
[20] M. RÖSLER and M. VOIT, An uncertainty principle for Hankel transforms, American Mathematical Society, 127(1) (1999), 183–194.
[21] M. RÖSLER, An uncertainty principle for the Dunkl transform, Bull. Austral. Math. Soc., 59
(1999), 353–360.
[22] N. SHIMENO, A note on the uncertainty principle for the Dunkl transform, J. Math. Sci. Univ.
Tokyo, 8 (2001), 33–42.
[23] G. SZEGÖ, Orthogonal polynomials, Amer. Math. Soc. Colloquium, Vol. 23, New York, 1939.
J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 88, 23 pp.
http://jipam.vu.edu.au/
H EISENBERG -PAULI -W EYL U NCERTAINTY P RINCIPLE
23
[24] G.N. WATSON, A Treatise on the Theory of Bessel Functions, 2nd ed, Cambridge Univ. Press, Inc.
Cambridge, 1959.
[25] H. WEYL, Gruppentheorie und Quantenmechanik. S. Hirzel, Leipzig, 1928; and Dover edition,
New York, 1950.
[26] J.A. WOLF, The Uncertainty principle for Gelfand pairs, Nova J. Alg. Geom., 1 (1992), 383–396.
J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 88, 23 pp.
http://jipam.vu.edu.au/