BEURLING-HÖRMANDER UNCERTAINTY
PRINCIPLE FOR THE SPHERICAL MEAN
OPERATOR
Uncertainty Principle for
the Spherical Mean Operator
N. Msehli and L.T. Rachdi
N. MSEHLI
L.T. RACHDI
Département de mathématiques et d’informatique
Institut national des sciences appliquées
et de technologie de Tunis
1080 Tunis, Tunisia
EMail: n.msehli@yahoo.fr
Département de Mathématiques
Faculté des Sciences de Tunis
El Manar II,
2092 Tunis, Tunisia
EMail: lakhdartannech.rachdi@fst.rnu.tn
vol. 10, iss. 2, art. 38, 2009
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Received:
15 November, 2008
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Accepted:
01 May, 2009
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Communicated by:
J.M. Rassias
2000 AMS Sub. Class.:
42B10, 43A32.
Key words:
Uncertainty principle, Beurling-Hörmander theorem, Gelfand-Shilov theorem,
Cowling-Price theorem, Fourier transform, Spherical mean operator.
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Abstract:
We establish the Beurling-Hörmander theorem for the Fourier transform connected with the spherical mean operator. Applying this result, we prove the
Gelfand-Shilov and Cowling-Price type theorems for this transform.
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Contents
1
Introduction
3
2
The Spherical Mean Operator
9
3
The Beurling-Hörmander Theorem for the Spherical Mean Operator
17
4
Applications of the Beurling-Hörmander Theorem
33
Uncertainty Principle for
the Spherical Mean Operator
N. Msehli and L.T. Rachdi
vol. 10, iss. 2, art. 38, 2009
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1.
Introduction
Uncertainty principles play an important role in harmonic analysis and have been
studied by many authors, from many points of view [13, 19]. These principles
state that a function f and its Fourier transform fb cannot be simultaneously sharply
localized. Many aspects of such principles have been studied, for example the
Heisenberg-Pauli-Weyl inequality [16] has been established for various Fourier transforms [26, 31, 32] and several generalized forms of this inequality are given in
[28, 29, 30]. See also the theorems of Hardy, Morgan, Beurling and Gelfand-Shilov
[7, 15, 23, 25, 26]. The most recent Beurling-Hörmander theorem has been proved
by Hörmander [20] using an idea of Beurling [3]. This theorem states that if f is an
integrable function on R with respect to the Lebesgue measure and if
ZZ
|f (x)||fˆ(y)|e|xy| dxdy < +∞,
R2
then f = 0 almost everywhere.
A strong multidimensional version of this theorem has been established by Bonami,
Demange and Jaming [4] (see also [19]) who have showed that if f is a square integrable function on Rn with respect to the Lebesgue measure, then
Z Z
|f (x)||fˆ(y)| |hx/yi|
e
dxdy < +∞,
d ≥ 0;
d
Rn Rn (1 + |x| + |y|)
if and only if f can be written as
f (x) = P (x)e−hAx/xi ,
where A is a real positive definite symmetric matrix and P is a polynomial with
degree(P ) < d−n
.
2
Uncertainty Principle for
the Spherical Mean Operator
N. Msehli and L.T. Rachdi
vol. 10, iss. 2, art. 38, 2009
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In particular for d ≤ n; f is identically zero.
The Beurling-Hörmander uncertainty principle has been studied by many authors
for various Fourier transforms. In particular, Trimèche [33] has shown this uncertainty principle for the Dunkl transform, Kamoun and Trimèche [21] have proved an
analogue of the Beurling-Hörmander theorem for some singular partial differential
operators, Bouattour and Trimèche [5] have shown this theorem for the hypergroup
of Chébli-Trimèche. We cite also Yakubovich [37], who has established the same
result for the Kontorovich-Lebedev transform.
Many authors are interested in the Beurling-Hörmander uncertainty principle because this principle implies other well known quantitative uncertainty principles such
as those of Gelfand-Shilov [14], Cowling Price [7], Morgan [2, 23], and the one of
Hardy [15].
On the other hand, the spherical mean operator is defined on C∗ (R × Rn ) (the
space of continuous functions on R × Rn , even with respect to the first variable) by
Z
R(f )(r, x) =
f rη, x + rξ dσn (η, ξ),
Sn
where S n is the unit sphere (η, ξ) ∈ R × Rn ; η 2 + |ξ|2 = 1 in R × Rn and σn is
the surface measure on S n normalized to have total measure one.
The dual operator t R of R is defined by
Z
p
Γ n+1
t
2
2
2
g
R(g)(r, x) =
r + |x − y| , y dy,
n+1
π 2
Rn
where dy is the Lebesgue measure on Rn .
The spherical mean operator R and its dual t R play an important role and have
many applications, for example; in the image processing of so-called synthetic aperture radar (SAR) data [17, 18], or in the linearized inverse scattering problem in
Uncertainty Principle for
the Spherical Mean Operator
N. Msehli and L.T. Rachdi
vol. 10, iss. 2, art. 38, 2009
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acoustics [11]. These operators have been studied by many authors from many points
of view [1, 8, 11, 24, 27].
In [24] (see also [8, 27]); the second author with others, associated to the spherical
mean operator R the Fourier transform F defined by
Z ∞Z
F (f )(µ, λ) =
f (r, x)ϕµ,λ (r, x)dνn (r, x),
Rn
0
Uncertainty Principle for
the Spherical Mean Operator
where
N. Msehli and L.T. Rachdi
• ϕµ,λ (r, x) = R cos(µ.)e−ihλ/·i (r, x)
vol. 10, iss. 2, art. 38, 2009
n
• dνn is the measure defined on [0, +∞[×R by
dνn (r, x) =
1
2
n−1
2
Γ( n+1
)
2
rn dr ⊗
dx
n .
(2π) 2
They have constructed the harmonic analysis related to the transform F (inversion formula, Plancherel formula, Paley-Wiener theorem, Plancherel theorem).
Our purpose in the present work is to study the Beurling-Hörmander uncertainty
principle for the Fourier transform F , from which we derive the Gelfand-Shilov and
Cowling -Price type theorems for this transform.
More precisely, we collect some basic harmonic analysis results for the Fourier
transform F .
In the third section, we establish the main result of this paper, that is, from the
Beurling Hörmander theorem:
• Let f be a measurable function on R×Rn ; even with respect to the first variable
and such that f ∈ L2 (dνn ). If
Z Z Z ∞Z
|F (f )(µ, λ)|e|(r,x)||θ(µ,λ)|
|f (r, x)|
dνn (r, x)dγ̃n (µ, λ) < +∞; d ≥ 0,
(1 + |(r, x)| + |θ(µ, λ)|)d
Γ+ 0 Rn
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then
i. For d ≤ n + 1; f = 0;
ii. For d > n + 1; there exists a positive constant a and a polynomial P on R × Rn
even with respect to the first variable, such that
f (r, x) = P (r, x)e−a(r
with degree(P ) <
2 +|x|2 )
Uncertainty Principle for
the Spherical Mean Operator
d−(n+1)
;
2
N. Msehli and L.T. Rachdi
vol. 10, iss. 2, art. 38, 2009
where
• Γ+ is the set given by
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n
n
Γ+ = [0, +∞[×R ∪ (iµ, λ); (µ, λ) ∈ R × R ; 0 ≤ µ ≤ |λ|
• θ is the bijective function defined on Γ+ by
p
θ(µ, λ) =
µ2 + |λ|2 , λ
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• dγ̃n is the measure defined on Γ+ by
r
ZZ
2 1
g(µ, λ)dγ̃n (µ, λ) =
n
π (2π) 2
Γ+
"Z Z
#
Z Z |λ|
∞
µdµdλ
µdµdλ
×
g(µ, λ) p
+
g(iµ, λ) p
.
µ2 + λ2
λ2 − µ2
0
Rn
Rn 0
The last section of this paper is devoted to the Gelfand-Shilov and Cowling Price
theorems for the transform F .
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• Let p, q be two conjugate exponents; p, q ∈]1, +∞[. Let η, ξ be two positive
real numbers such that ξη ≥ 1. Let f be a measurable function on R × Rn ;
even with respect to the first variable such that f ∈ L2 (dνn ).
If
ξp |(r,x)|p
Z ∞Z
|f (r, x)|e p
dνn (r, x) < +∞
d
0
Rn (1 + |(r, x)|)
and
ZZ
Γ+
ξq |(r,x)|q
|F (f )(µ, λ)|e q
(1 + |θ(µ, λ)|)d
dγ̃n (µ, λ) < +∞;
d ≥ 0,
Uncertainty Principle for
the Spherical Mean Operator
N. Msehli and L.T. Rachdi
vol. 10, iss. 2, art. 38, 2009
then
i. For d ≤
ii. For d >
n+1
;
2
n+1
;
2
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f = 0.
Contents
we have
– f = 0 for ξη > 1
– f = 0 for ξη = 1 and p 6= 2
2
2
– f (r, x) = P (r, x)e−a(r +|x| ) for ξη = 1 and p = q = 2, where a > 0
and P is a polynomial on R×Rn even with respect to the first variable,
with degree (P ) < d − n+1
.
2
• Let η, ξ, w1 and w2 be non negative real numbers such that ηξ ≥ 14 . Let p, q be
two exponents, p, q ∈ [1, +∞] and let f be a measurable function on R × Rn ,
even with respect to the first variable such that f ∈ L2 (dνn ).
If
ξ|(·,·)|2
e
< +∞
w
1
(1 + |(·, ·)|) p,νn
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and
η|θ(·,·)|2
e
F
(f
)
(1 + |θ(·, ·)|)w2
< +∞,
q,γ̃n
then
i. For ξη > 14 ; f = 0.
1
;
4
n
ii. For ξη = there exists a positive constant a and a polynomial P on R × R ,
even with respect to the first variable such that
Uncertainty Principle for
the Spherical Mean Operator
N. Msehli and L.T. Rachdi
vol. 10, iss. 2, art. 38, 2009
−a(r 2 +|x|2 )
f (r, x) = P (r, x)e
.
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2.
The Spherical Mean Operator
For all (µ, λ) ∈ C × Cn ; if we denote by ϕµ,λ the function defined by
ϕµ,λ (r, x) = R cos(µ.)e−ihλ/·i (r, x),
then we have
ϕµ,λ (r, x) = j n−1
(2.1)
2
Uncertainty Principle for
the Spherical Mean Operator
p
r µ2 + λ2 e−ihλ/xi ,
N. Msehli and L.T. Rachdi
vol. 10, iss. 2, art. 38, 2009
where
• λ2 = λ21 + · · · + λ2n ; λ = (λ1 , . . . , λn ) ∈ Cn ;
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• hλ/xi = λ1 x1 + · · · + λn xn ; x = (x1 , . . . , xn ) ∈ Rn ;
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• j n−1 is the modified Bessel function given by
2
j n−1 (s) = 2
(2.2)
=Γ
J n−1 (s)
n+1
2
X
∞
n+1
n−1
2
2
Γ
2
k=0
2
n−1
2
s
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k
s 2k
(−1)
k!Γ(k + n+1
) 2
2
and J n−1 is the usual Bessel function of first kind and order
2
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;
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n−1
2
[9, 10, 22, 36].
Also, the modified Bessel function j n−1 has the following integral representation,
2
for all z ∈ C:
Z
2Γ( n+1
) 1
n
2
j n−1 (z) = √
(1 − t2 ) 2 −1 cos(zt)dt.
n
2
π Γ( 2 ) 0
Close
Thus, for all z ∈ C; we have
n−1
j 2 (z) ≤ e| Im z| .
(2.3)
Using the relation (2.1) and the properties of the function j n−1 , we deduce that the
2
function ϕµ,λ satisfies the following properties [24, 27]:
•
sup
(2.4)
(r,x)∈R×Rn
Uncertainty Principle for
the Spherical Mean Operator
ϕµ,λ (r, x) = 1
N. Msehli and L.T. Rachdi
vol. 10, iss. 2, art. 38, 2009
if and only if (µ, λ) belongs to the set Γ defined by
(2.5)
Γ = R × Rn ∪ (iµ, λ); (µ, λ) ∈ R × Rn ; |µ| ≤ |λ| .
• For all (µ, λ) ∈ C × Cn ; the function ϕµ,λ is a unique solution of the system
 ∂u
(r, x) = −i λj u(r, x); 1 ≤ j ≤ n


 ∂xj
Lu(r, x) = −µ2 u(r, x)



u(0, 0) = 1; ∂u
(0, x1 , . . . , xn ) = 0; ∀ (x1 , . . . , xn ) ∈ Rn
∂r
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where
n
X
n ∂
∂2
+
−
L =
∂r2
r ∂r j=1
∂
∂xj
2
.
In the following, we denote by
• dmn+1 the measure defined on [0, +∞[×Rn ; by
r
2 1
dmn+1 (r, x) =
n dr ⊗ dx,
π (2π) 2
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where dx is the Lebesgue measure on Rn .
• Lp (dmn+1 ); p ∈ [1, +∞], the space of measurable functions f on [0, +∞[×Rn
satisfying
 R R
p1
∞
p

|f
(r,
x)|
dm
(r,
x)
< +∞, if 1 ≤ p < +∞;

n+1
n
 0 R
kf kp,mn+1 =



|f (r, x)| < +∞,
ess sup
if p = +∞.
(r,x)∈[0,+∞[×Rn
Uncertainty Principle for
the Spherical Mean Operator
N. Msehli and L.T. Rachdi
vol. 10, iss. 2, art. 38, 2009
• dνn the measure defined on [0, +∞[×Rn by
dνn (r, x) =
rn dr
2
n−1
2
Γ( n+1
)
2
⊗
dx
n .
(2π) 2
• Lp (dνn ), p ∈ [1, +∞], the space of measurable functions f on [0, +∞[×Rn
such that kf kp,νn < +∞.
• Γ+ the subset of Γ, given by
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Γ+ = [0, +∞[×Rn ∪ (iµ, λ); (µ, λ) ∈ R × Rn ; 0 ≤ µ ≤ |λ| .
• BΓ+ the σ−algebra defined on Γ+ by
(2.6)
BΓ+ = θ−1 (B); B ∈ Bor [0, +∞[×Rn ,
where θ is the bijective function defined on Γ+ by
p
θ(µ, λ) =
µ2 + |λ|2 , λ .
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• dγn the measure defined on BΓ+ by
∀A ∈ BΓ+ ; γn (A) = νn θ(A) .
• Lp (dγn ), p ∈ [1, +∞], the space of measurable functions g on Γ+ such that
kgkp,γn < +∞.
• dγ̃n the measure defined on BΓ+ by
n
2
dγ̃n (µ, λ) =
2 Γ
√
n+1
2
π
Uncertainty Principle for
the Spherical Mean Operator
dγn (µ, λ)
n .
(µ2 + |λ|2 ) 2
• Lp (dγ̃n ), p ∈ [1, +∞], the space of measurable functions g on Γ+ such that
kgkp,γ̃n < +∞.
• S∗ (R × Rn ) the Schwarz space formed by the infinitely differentiable functions
on R × Rn , rapidly decreasing together with all their derivatives, and even with
respect to the first variable.
Proposition 2.1.
i. For all non negative measurable functions g on Γ+ (respectively integrable on
Γ+ with respect to the measure dγn ), we have
ZZ
g(µ, λ)dγn (µ, λ)
Γ+
Z ∞ Z
n−1
1
= n−1 n+1
g(µ, λ)(µ2 + |λ|2 ) 2 µdµdλ
n
2 2 Γ( 2 )(2π) 2
0
Rn
!
Z Z |λ|
2
2 n−1
+
g(iµ, λ)(|λ| − µ ) 2 µdµdλ) .
Rn
0
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vol. 10, iss. 2, art. 38, 2009
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ii. For all non negative measurable functions f on [0, +∞[×Rn (respectively integrable on [0, +∞[×Rn with respect to the measure dmn+1 ), the function f ◦ θ
is measurable on Γ+ (respectively integrable on Γ+ with respect to the measure
dγn ) and we have
Z ∞Z
ZZ
f ◦ θ(µ, λ)dγn (µ, λ) =
f (r, x)dνn (r, x).
Γ+
0
Rn
iii. For all non negative measurable functions f on [0, +∞[×Rn (respectively integrable on [0, +∞[×Rn with respect to the measure dmn+1 ), we have
ZZ
Z ∞Z
(2.7)
f ◦ θ(µ, λ)dγ̃n (µ, λ) =
f (r, x)dmn+1 (r, x),
Γ+
0
Rn
Uncertainty Principle for
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N. Msehli and L.T. Rachdi
vol. 10, iss. 2, art. 38, 2009
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where θ is the function given by the relation (2.6).
In the sequel, we shall define the Fourier transform associated with the spherical
mean operator and give some properties.
Definition 2.2. The Fourier transform F associated with the spherical mean operator is defined on L1 (dνn ) by
Z ∞Z
∀(µ, λ) ∈ Γ; F (f )(µ, λ) =
f (r, x)ϕµ,λ (r, x)dνn (r, x),
0
Rn
where ϕµ,λ is the function given by the relation (2.1) and Γ is the set defined by (2.5).
Remark 1. For all (µ, λ) ∈ Γ, we have
(2.8)
F (f )(µ, λ) = F˜ (f ) ◦ θ(µ, λ),
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where
F˜ (f )(µ, λ) =
(2.9)
Z
∞
0
Z
Rn
f (r, x)j n−1 (rµ)e−ihλ/xi dνn (r, x)
2
and j n−1 is the modified Bessel function given by the relation (2.2).
2
Moreover, by the relation (2.4), the Fourier transform F is a bounded linear
operator from L1 (dνn ) into L∞ (dγn ) and for all f ∈ L1 (dνn ):
N. Msehli and L.T. Rachdi
kF (f )k∞,γn ≤ kf k1,νn .
(2.10)
vol. 10, iss. 2, art. 38, 2009
Theorem 2.3 (Inversion formula). Let f ∈ L (dνn ) such that F (f ) ∈ L (dγn ),
then for almost every (r, x) ∈ [0, +∞[×Rn , we have
ZZ
(2.11)
f (r, x) =
F (f )(µ, λ)ϕµ,λ (r, x)dγn (µ, λ)
Γ+
Z ∞Z
=
F˜ (f )(µ, λ)j n−1 (rµ)eihλ/xi dνn (µ, λ).
1
0
Rn
Uncertainty Principle for
the Spherical Mean Operator
1
2
Lemma 2.4. Let R n−1 be the mapping defined for all non negative measurable func2
tions g on [0, +∞[×Rn by
Z r
2Γ n+1
n
2 1−n
(2.12)
R n−1 (g)(r, x) = √
r
(r2 − t2 ) 2 −1 g(t, x)dt
n
2
πΓ 2
0
n+1 Z 1
2Γ
n
(1 − t2 ) 2 −1 g(tr, x)dt,
= √ 2n πΓ 2 0
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then for all non negative measurable functions f, g on [0, +∞[×Rn , we have
Z ∞Z
R n−1 (g)(r, x)f (r, x)dνn (r, x)
(2.13)
2
0
Rn
Z ∞Z
=
g(t, x)W n−1 (f )(t, x)dmn+1 (t, x)
0
2
Rn
where W n−1 is the classical Weyl transform defined for all non negative measurable
2
functions g on [0, +∞[×Rn by
Z ∞
n
1
(2.14)
W n−1 (f )(t, x) = n n
(r2 − t2 ) 2 −1 f (r, x)2rdr.
2
2 2 Γ( 2 ) t
Proposition 2.5. For all f ∈ L1 (dνn ), the function W n−1 (f ) given by the relation
2
(2.14) is defined almost every where, belongs to the space L1 (dmn+1 ) and we have
≤ kf k1,νn .
(2.15)
W n−1 (f )
2
1,mn+1
Moreover,
(2.16)
Uncertainty Principle for
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vol. 10, iss. 2, art. 38, 2009
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F˜ (f )(µ, λ) = Λn+1 ◦ W n−1 (f )(µ, λ),
2
where Λn+1 is the usual Fourier cosine transform defined on L1 (dmn+1 ) by
Z ∞Z
Λn+1 (g)(µ, λ) =
g(r, x) cos(rµ)e−ihλ,xi dmn+1 (r, x).
0
Rn
and F˜ is the Fourier-Bessel transform defined by the relation (2.9).
Remark 2. It is well known [34, 35] that the Fourier transforms F˜ and Λn+1 are
topological isomorphisms from S∗ (R × Rn ) onto itself. Then, by the relation (2.16),
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we deduce that the classical Weyl transform W n−1 is also a topological isomorphism
2
from S∗ (R × Rn ) onto itself, and the inverse isomorphism is given by [24]
[ n2 ]+1 !
n
∂
−1
(2.17)
W n−1
(f )(r, x) = (−1)[ 2 ]+1 F[ n2 ]− n2 +1
f (r, x),
2
∂t2
where Fa ; a > 0 is the mapping defined on S∗ (R × Rn ) by
Z ∞
1
(2.18)
Fa (f )(r, x) = a
(t2 − r2 )a−1 f (t, x)2tdt
2 Γ(a) r
and
∂
∂r2
is the singular partial differential operator defined by
∂
1 ∂f (r, x)
f (r, x) =
.
2
∂r
r ∂r
Uncertainty Principle for
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vol. 10, iss. 2, art. 38, 2009
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3.
The Beurling-Hörmander Theorem for the Spherical Mean
Operator
This section contains the main result of this paper, that is the Beurling-Hörmander
theorems for the Fourier transform F associated with the spherical mean operator.
We firstly recall the following result that has been established by Bonami, Demange and Jaming [4].
Theorem 3.1. Let f be a measurable function on R × Rn , even with respect to the
first variable such that f ∈ L2 (dmn+1 ) and let d be a real number, d ≥ 0. If
Z ∞Z Z ∞Z
|f (r, x)||Λn+1 (f )(s, y)| |(r,x)||(s,y)|
e
dmn+1 (r, x)dmn+1 (s, y) < +∞,
d
0
Rn 0
Rn(1 + |(r, x)| + |(s, y)|)
then there exist a positive constant a and a polynomial P on R×Rn even with respect
to the first variable, such that
−a(r2 +|x|2 )
f (r, x) = P (r, x)e
,
with degree(P ) < d−(n+1)
.
2
In particular, f = 0 for d ≤ (n + 1).
Lemma 3.2. Let f ∈ L2 (dνn ) and let d be a real number, d ≥ 0. If
Z Z Z ∞Z
|f (r, x)||F (f )(µ, λ)|e|(r,x)||θ(µ,λ)|
dνn (r, x)dγ̃n (µ, λ) < +∞,
d
Γ+ 0
Rn
1 + |(r, x)| + |θ(µ, λ)|
then the function f belongs to the space L1 (dνn ).
Uncertainty Principle for
the Spherical Mean Operator
N. Msehli and L.T. Rachdi
vol. 10, iss. 2, art. 38, 2009
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Proof. Let f ∈ L2 (dνn ), f 6= 0. From the relations (2.7) and (2.8), we obtain
Z Z Z ∞Z
|f (r, x)||F (f )(µ, λ)| |(r,x)||θ(µ,λ)|
dνn (r, x)dγ̃n (µ, λ)
d e
Γ+ 0
Rn 1 + |(r, x)| + |θ(µ, λ)|
Z ∞Z Z ∞Z
|f (r, x)||F˜ (f )(µ, λ)| |(r,x)||(µ,λ)|
=
e
dνn (r, x)dmn+1 (µ, λ) < +∞.
d
0
Rn 0
Rn (1 + |(r, x)| + |(µ, λ)|)
Then for almost every (µ, λ) ∈ [0, +∞[×Rn ,
Z ∞Z
|f (r, x)|e|(r,x)||(µ,λ)|
˜
dνn (r, x) < +∞.
F (f )(µ, λ)
d
0
Rn (1 + |(r, x)| + |(µ, λ)|)
In particular, there exists (µ0 , λ0 ) ∈ [0, +∞[×Rn \ {(0, 0)} such that
Z ∞Z
|f (r, x)|e|(r,x)||(µ0 ,λ0 )|
˜
dνn (r, x) < +∞.
F (f )(µ0 , λ0 ) 6= 0 and
d
0
Rn (1 + |(r, x)| + |(µ0 , λ0 )|)
Let h be the function defined on [0, +∞[ by
Uncertainty Principle for
the Spherical Mean Operator
N. Msehli and L.T. Rachdi
vol. 10, iss. 2, art. 38, 2009
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es|(µ0 ,λ0 )|
h(s) =
,
(1 + s + |(µ0 , λ0 )|)d
then the function h has an absolute minimum attained at:
(
d
− 1 − |(µ0 , λ0 )|; if |(µ0d,λ0 )| > 1 + |(µ0 , λ0 )|;
|(µ0 ,λ0 )|
s0 =
0;
if |(µ0d,λ0 )| ≤ 1 + |(µ0 , λ0 )|.
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Consequently,
Z ∞Z
|f (r, x)|dνn (r, x)
0
Rn
Z ∞Z
|f (r, x)|e|(r,x)||(µ0 ,λ0 )|
1
≤
dνn (r, x) < +∞.
d
h(s0 ) 0
Rn (1 + |(r, x)| + |(µ0 , λ0 )|)
Uncertainty Principle for
the Spherical Mean Operator
N. Msehli and L.T. Rachdi
Lemma 3.3. Let f ∈ L2 (dνn ) and let d be a real number, d ≥ 0. If
ZZ Z ∞ Z
|f (r, x)||F (f )(µ, λ)|e|(r,x)||θ(µ,λ)|
dνn (r, x)dγ̃n (µ, λ) < +∞,
(1 + |(r, x)| + |θ(µ, λ)|)d
Γ+ 0
Rn
then there exists a > 0 such that the function F˜ (f ) is analytic on the set
(µ, λ) ∈ C × Cn ; | Im µ| < a, | Im λj | < a; ∀j ∈ {1, . . . , n} .
Proof. From the proof of Lemma 3.2, there exists (µ0 , λ0 ) ∈ [0, +∞[×Rn \ {(0, 0)}
such that
Z ∞Z
|f (r, x)|e|(r,x)||(µ0 ,λ0 )|
dνn (r, x) < +∞.
d
0
Rn (1 + |(r, x)| + |(µ0 , λ0 )|)
Let a be a real number such that 0 < (n + 1)a < |(µ0 , λ0 )|. Then we have
vol. 10, iss. 2, art. 38, 2009
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Z
0
∞
|(r,x)||(µ0 ,λ0 )|
|f (r, x)|e
dνn (r, x)
d
Rn (1 + |(r, x)| + |(µ0 , λ0 )|)
Z ∞Z
e|(r,x)|(|(µ0 ,λ0 )|−(n+1)a)
dνn (r, x) < +∞.
=
|f (r, x)|e(n+1)a|(r,x)|
(1 + |(r, x)| + |(µ0 , λ0 )|)d
0
Rn
Z
Let g be the function defined on [0, +∞[ by
g(s) =
es(|(µ0 ,λ0 )|−(n+1)a)
,
(1 + s + |(µ0 , λ0 )|)d
then g admits a minimum attained at

d
 |(µ0 ,λ0 )|−(n+1)a
− 1 − |(µ0 , λ0 )|;
s0 =
 0;
if
d
|(µ0 ,λ0 )|−(n+1)a
> 1 + |(µ0 , λ0 )|,
Uncertainty Principle for
the Spherical Mean Operator
if
d
|(µ0 ,λ0 )|−(n+1)a
≤ 1 + |(µ0 , λ0 )|.
N. Msehli and L.T. Rachdi
vol. 10, iss. 2, art. 38, 2009
Consequently,
Z ∞Z
(3.1)
|f (r, x)|e(n+1)a|(r,x)| dνn (r, x)
n
0
R
Z ∞Z
|f (r, x)|e|(r,x)||(µ0 ,λ0 )|
1
≤
dνn (r, x) < +∞.
d
g(s0 ) 0
Rn (1 + |(µ0 , λ0 )| + |(r, x)|)
On the other hand, from the relation (2.2), we deduce that for all (r, x) ∈ [0, +∞[×Rn ;
the function
(µ, λ) 7−→ j n−1 (rµ)e−ihλ/xi
2
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n
is analytic on C × C [6], even with respect to the first variable and by the relation
(2.3), we deduce that ∀ (r, x) ∈ [0, +∞[×Rn , ∀(µ, λ) ∈ C × Cn ,
Pn
−ihλ,xi (3.2)
j n−1 (rµ)e
≤ er| Im µ|+ j=1 | Im λj ||xj |
2
≤ e|(r,x)|[| Im µ|+
Pn
j=1
| Im λj |]
.
Then the result follows from the relations (2.9), (3.1), (3.2) and by the analyticity
theorem.
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Corollary 3.4. Let f ∈ L2 (dνn ), f 6= 0 and let d be a real number, d ≥ 0. If
ZZ Z ∞ Z
|f (r, x)||F (f )(µ, λ)| |(r,x)||θ(µ,λ)|
e
dνn (r, x)dγ̃n (µ, λ) < +∞,
d
Γ+ 0
Rn (1 + |(r, x)| + |θ(µ, λ)|)
then for all real numbers a, a > 0, we have mn+1 (Aa ) > 0, where
n
o
(3.3)
Aa = (µ, λ) ∈ R × Rn ; F˜ (f )(µ, λ) 6= 0 and |(µ, λ)| > a .
Proof. Let f be a function satisfying the hypothesis. From Lemma 3.2, the function
f belongs to L1 (dνn ) and consequently the function F˜ (f ) is continuous on R × Rn ,
even with respect to the first variable. Then for all a > 0, the set Aa given by the
relation (3.3) is an open subset of R × Rn .
So, if mn+1 (Aa ) = 0, then this subset is empty. This means that for every (µ, λ) ∈
R × Rn , |(µ, λ)| > a, we have F˜ (f )(µ, λ) = 0.
From Lemma 3.2, and by analytic continuation, we deduce that F˜ (f ) = 0, and
by the inversion formula (2.11), it follows that f = 0.
Remark 3.
Uncertainty Principle for
the Spherical Mean Operator
N. Msehli and L.T. Rachdi
vol. 10, iss. 2, art. 38, 2009
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i. Let f be a function satisfying the hypothesis of Corollary 3.4, then for all real
numbers a, a > 0, there exists (µ0 , λ0 ) ∈ [0, +∞[×Rn such that |(µ0 , λ0 )| > a
and
Z ∞Z
e|(r,x)||(µ0 ,λ0 )|
dνn (r, x) < +∞.
|f (r, x)|
(1 + |(r, x)| + |(µ0 , λ0 )|)d
0
Rn
ii. Let d and σ be non negative real numbers, σ + σ 2 ≥ d. Then the function
t 7−→
eσt
(1 + t + σ)d
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is not decreasing on [0, +∞[.
Lemma 3.5. Let f be a measurable function on R × Rn even with respect to the first
variable, and f ∈ L2 (dνn ). Let d be real number, d ≥ 0. If
Z Z Z ∞Z
e|(r,x)||θ(µ,λ)|
|F (f )(µ, λ)||f (r, x)|
dνn (r, x)dγ̃n (µ, λ) < +∞,
(1 + |(r, x)| + |θ(µ, λ)|)d
Γ+ 0
Rn
then the function W n−1 (f ) defined by the relation (2.14) belongs to the space L2 (dmn+1 ).
2
Proof. From the hypothesis and the relations (2.7) and (2.8), we have
ZZ Z ∞ Z
|f (r, x)|e|(r,x)||θ(µ,λ)|
|F (f )(µ, λ)|
dνn (r, x)dγ̃n (µ, λ)
(1 + |(r, x)| + |θ(µ, λ)|)d
Γ+ 0
Rn
Z ∞Z Z ∞Z |f (r, x)|e|(r,x)||(µ,λ)|
˜
=
dνn (r, x)dmn+1 (µ, λ)
F (f )(µ, λ)
(1 + |(r, x)| + |(µ, λ)|)d
0
Rn 0
Rn
< +∞.
In the same manner as the proof of the inequality (3.1) in Lemma 3.2, there exists
b ∈ R, b > 0 such that
Z ∞Z
|F˜ (f )(µ, λ)|eb|(µ,λ)| dmn+1 (µ, λ) < +∞.
0
Rn
Consequently, the function F˜ (f ) belongs to the space L1 (dνn ) and by the inversion
formula for F˜ , we deduce that
Z ∞Z
f (r, x) =
F˜ (f )(µ, λ)j n−1 (rµ)eihλ/xi dνn (µ, λ). a.e.
0
Rn
2
Uncertainty Principle for
the Spherical Mean Operator
N. Msehli and L.T. Rachdi
vol. 10, iss. 2, art. 38, 2009
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In particular, the function f is bounded and
˜ (3.4)
kf k∞,νn ≤ F (f )
.
1,νn
By virtue of the relation (2.14), we get
Z ∞
n
1
(f )(r, x) ≤ n n
(r2 − t2 ) 2 −1 |f (r, x)|2rdr
W n−1
2
2
2 Γ( 2 ) t
Z ∞
n
rn
= n n
(y 2 − 1) 2 −1 |f (ry, x)|2ydy.
2 2 Γ( 2 ) 1
Uncertainty Principle for
the Spherical Mean Operator
N. Msehli and L.T. Rachdi
vol. 10, iss. 2, art. 38, 2009
Using Minkowski’s inequality for integrals [12], we get:
Z ∞ Z 21
2
(3.5)
W n−1 (f )(r, x) dmn+1 (r, x)
0
Contents
2
Rn
1
≤ n n
2 2 Γ( 2 )
Title Page
" Z Z Z
∞
∞
n
2
r (y − 1)
Rn
0
n
−1
2
2
|f (ry, x)|2ydy
# 12
dmn+1 (r, x)
1
12
Z ∞Z ∞Z
1
r2n (y 2 − 1)n−2 |f (ry, x)|2 dmn+1 (r, x) 2ydy
≤ n n
2 2 Γ( 2 ) 1
0
Rn
Z ∞
n
1
1
2
−1
−n+
2 dy
(y − 1) 2 y
= n −1 n
2 2 Γ( 2 ) 1
Z ∞ Z
12
2n
2
×
s |f (s, x)| dmn+1 (s, x)
Rn
0
Γ( 14 )
= n 2n+1
2 2 Γ( 4 )
Z
0
∞
Z
Rn
12
s |f (s, x)| dmn+1 (s, x) .
2n
2
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Using the relations (3.1), (3.4) and (3.5), we deduce that
2
∞
Z
W n−1 (f )
Z
=
2,mn+1
0
∞
Z
Rn
12
2
(f ) (r, x)dmn+1 (r, x)
W n−1
2
Z
≤ Kn
0
|f (s, x)|e(n+1)a|(s,x)| dνn (s, x) < +∞,
Rn
Uncertainty Principle for
the Spherical Mean Operator
where
Kn =
1
4
2n+1
4
Γ
2 Γ
n
2
r
π
Γ
2
n+1
2
2
n−1
2
max(sn e−(n+1)as )kf k∞,νn
s≥0
N. Msehli and L.T. Rachdi
21
.
vol. 10, iss. 2, art. 38, 2009
Title Page
Theorem 3.6. Let f ∈ L2 (dνn ); f 6= 0 and let d be a real number; d ≥ 0.
If
ZZ Z ∞ Z
|f (r, x)||F (f )(µ, λ)|e|(r,x)||θ(µ,λ)|
dνn (r, x)dγ̃n (µ, λ) < +∞;
(1 + |(r, x)| + |θ(µ, λ)|)d
Γ+ 0
Rn
then
Z ∞Z
0
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Z
Rn
0
∞
Z
Rn
(f )(r, x) F˜ (f )(µ, λ)
W n−1
2
e|(r,x)||(µ,λ)|
dmn+1 (r, x)dmn+1 (µ, λ) < +∞
×
(1 + |(r, x)| + |(µ, λ)|)d
where W n−1 is the Weyl transform defined by the relation (2.14).
2
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Proof. From the hypothesis, the relations (2.7), (2.8) and Fubini’s theorem, we have
ZZ Z ∞ Z
(3.6)
|f (r, x)||F (f )(µ, λ)|
Γ+
Rn
0
e|(r,x)||θ(µ,λ)|
dνn (r, x)dγ̃n (µ, λ)
(1 + |(r, x)| + |θ(µ, λ)|)d
Z ∞Z ˜
=
F
(f
)(µ,
λ)
0
Rn
Z ∞Z
|f (r, x)|e|(r,x)||(µ,λ)|
×
dνn (r, x)dmn+1 (µ, λ)
d
0
Rn (1 + |(r, x)| + |(µ, λ)|)
< +∞.
×
i. If d = 0, then by the relation (2.13) and Fubini’s theorem, we get
Z ∞Z Z ∞Z ˜
n−1
(3.7)
W
(f
)(r,
x)
F
(f
)(µ,
λ)
2
0
Rn 0
Rn
|(r,x)||(µ,λ)|
×e
dmn+1 (r, x)dmn+1 (µ, λ)
Z ∞Z ˜
≤
F (f )(µ, λ)
0
Rn
Z ∞Z
|(r,x)||(µ,λ)|
×
W n−1 (|f |)(r, x)e
dmn+1 (r, x) dmn+1 (µ, λ)
2
0
Rn
Z ∞Z ˜
≤
F (f )(µ, λ)
0
Rn
Z ∞Z
|(·,·)||(µ,λ)|
×
|f (r, x)|R n−1 (e
)(r, x)dνn (r, x) dmn+1(µ, λ).
0
Rn
2
Uncertainty Principle for
the Spherical Mean Operator
N. Msehli and L.T. Rachdi
vol. 10, iss. 2, art. 38, 2009
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However, by (2.12), we deduce that for all (r, x) ∈ [0, +∞[×Rn ,
(3.8)
R n−1 e|(·,·)||(µ,λ)| (r, x) ≤ e|(r,x)||(µ,λ)| .
2
Combining the relations (3.6), (3.7) and (3.8), we deduce that
Z ∞Z Z ∞Z (f )(r, x) F˜ (f )(µ, λ) e|(r,x)||(µ,λ)| dmn+1 (r, x)dmn+1 (µ, λ)
W n−1
2
n
n
R
Z0 ∞ ZR 0
Z ∞Z
˜
≤
|f (r, x)|e|(r,x)||(µ,λ)| dνn (r, x)dmn+1 (µ, λ) < +∞.
F (f )(µ, λ)
Rn
0
Rn
0
ii. For d > 0, let Bd = {(r, x) ∈ [0, +∞[×Rn ; |(r, x)| ≤ d}. We have
∞
Z
Z
Rn
0
ZZ
≤
Bdc
ZZ
+
Bd
Z
˜
F (f )(µ, λ)
˜
F (f )(µ, λ)
˜
F (f )(µ, λ)
∞
Z
Z
×
dmn+1 (r, x)dmn+1 (µ, λ)
d
0
Rn (1 + |(r, x)| + |(µ, λ)|)
!
Z ∞ Z W n−1 (|f |)(r, x)e|(r,x)||(µ,λ)|
2
dmn+1 (r, x) dmn+1 (µ, λ)
(1
+ |(r, x)| + |(µ, λ)|)d
n
0
R
!
Z ∞ Z W n−1 (|f |)(r, x)e|(r,x)||(µ,λ)|
2
dmn+1 (r, x) dmn+1 (µ, λ).
(1
+ |(r, x)| + |(µ, λ)|)d
n
0
R
W n−1 (|f |)(r, x)e|(r,x)||(µ,λ)|
2
0
Rn
vol. 10, iss. 2, art. 38, 2009
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2
(1 + |(r, x)| + |(µ, λ)|)d
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Bdc
∞
N. Msehli and L.T. Rachdi
|W n−1 (f )(r, x)|e|(r,x)||(µ,λ)|
From the relation (2.13), we deduce that
ZZ ˜
(3.9)
F (f )(µ, λ)
Z
Uncertainty Principle for
the Spherical Mean Operator
!
dmn+1 (r, x) dmn+1 (µ, λ)
ZZ
=
Bdc
Z
˜
F (f )(µ, λ)
0
× R n−1
2
∞
Z
|f (r, x)|
Rn
e|(·,·)||(µ,λ)|
(1 + |(·, ·)| + |(µ, λ)|)d
(r, x)dνn (r, x)dmn+1 (µ, λ).
However, from the relation (2.12) and ii) of Remark 3, we deduce that for all (µ, λ) ∈
Bdc , we have
e|(r,x)||(µ,λ)|
e|(·,·)||(µ,λ)|
n−1
(r, x) ≤
.
(3.10)
R
2
(1 + |(·, ·)| + |(µ, λ)|)d
(1 + |(r, x)| + |(µ, λ)|)d
Combining the relations (3.6), (3.9) and (3.10), we get
ZZ
Bdc
Z
˜
F (f )(µ, λ)
∞
0
Z
W n−1 (|f |)(r, x)e
|(r,x)||(µ,λ)|
2
Rn
(1 + |(r, x)| +
|(µ, λ)|)d
Z
˜
F (f )(µ, λ)
Contents
dmn+1 (r, x) dmn+1 (µ, λ)
∞Z
Bd
vol. 10, iss. 2, art. 38, 2009
Title Page
We have
ZZ Z Z |W n−1 (f )(r, x)|e|(r,x)||(µ,λ)|
˜
2
dmn+1 (r, x)dmn+1 (µ, λ)
F (f )(µ, λ)
(1
+
|(r, x)| + |(µ, λ)|)d
Bd
Bd
Z Z Z Z ˜
d2
≤e
(f )(r, x) dmn+1 (r, x)
F (f )(µ, λ) dmn+1 (µ, λ)
W n−1
2
Bd
N. Msehli and L.T. Rachdi
!
|f (r, x)|e|(r,x)||(µ,λ)|
≤
dνn (r, x) dmn+1 (µ, λ)
d
Bdc
0
Rn (1 + |(r, x)| + |(µ, λ)|)
Z ∞Z Z ∞ Z
|f (r, x)|e|(r,x)||(µ,λ)|
˜
≤
dνn (r, x) dmn+1 (µ, λ)
F (f )(µ, λ)
d
0
Rn
0
Rn (1 + |(r, x)| + |(µ, λ)|)
< +∞.
ZZ
Uncertainty Principle for
the Spherical Mean Operator
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≤ e mn+1 (Bd )kF (f )k∞,γn W n−1 (f )
d2
2
.
1,mn+1
By the relations (2.10) and (2.15), we deduce that
ZZ Z Z W n−1 (f )(r, x) e|(r,x)||(µ,λ)|
˜
2
dmn+1 (r, x)dmn+1 (µ, λ)
F (f )(µ, λ)
(1
+ |(r, x)| + |(µ, λ)|)d
Bd
Bd
d2
≤e
mn+1 (Bd )kf k21,νn
< +∞.
By the relation (2.13), we get


|(r,x)||(µ,λ)|
ZZ Z
Z
n−1 (f )(r, x) e
W
˜
2
dmn+1 (r, x) dmn+1 (µ, λ)
F (f )(µ, λ) 
d
c
(1
+
|(r,
x)|
+
|(µ,
λ)|)
Bd
Bd
ZZ ˜
≤
F (f )(µ, λ)
Bd
Z
∞
Z
×
W n−1 (|f |)(r, x)e|(r,x)||(µ,λ)|
2
1Bdc (r, x)dmn+1 (r, x)dmn+1 (r, x)
(1 + |(r, x)| + |(µ, λ)|)d
Z ∞ Z
˜
|f (r, x)|
F (f )(µ, λ)
Bd
0
Rn
e|(·,·)||(µ,λ)|
× R n−1
1B c (·, ·) (r, x)dνn (r, x) dmn+1 (µ, λ).
2
(1 + |(·, ·)| + |(µ, λ)|)d d
0
Rn
Uncertainty Principle for
the Spherical Mean Operator
N. Msehli and L.T. Rachdi
vol. 10, iss. 2, art. 38, 2009
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Contents
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Page 28 of 44
ZZ
=
However, by ii) of Remark 3 and the relation (2.10), we deduce that for all (µ, λ) ∈
Bd :
ed|(r,x)|
e|(·,·)||(µ,λ)|
c (·, ·)
R n−1
1
(r,
x)
≤
1B c (r, x).
B
2
(1 + |(·, ·)| + |(µ, λ)|)d d
(1 + |(r, x)| + d)d d
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Thus,
ZZ
(3.11)
˜
F (f )(µ, λ)
Bd


Z Z W n−1 (f )(r, x) e|(r,x)||(µ,λ)|
2
dmn+1 (r, x) dmn+1 (µ, λ)
×
c
(1
+ |(r, x)| + |(µ, λ)|)d
Bd
ZZ
ed|(r,x)|
≤ kf k1,νn mn+1 (Bd )
|f (r, x)|
dνn (r, x).
(1 + |(r, x)| + d)d
Bdc
On the other hand, from i) of Remark 3, there exists (µ0 , λ0 ) ∈ [0, +∞[×Rn ,
|(µ0 , λ0 )| > d such that
Z ∞Z
e|(r,x)||(µ0 ,λ0 )| |f (r, x)|
dνn (r, x) < +∞.
d
0
Rn (1 + |(r, x)| + |(µ0 , λ0 )|)
Again, by ii) of Remark 3, we have
ZZ
ed|(r,x)|
(3.12)
|f (r, x)|
dνn (r, x)
(1 + |(r, x)| + d)d
Bdc
ZZ
e|(r,x)||(µ0 ,λ0 )|
dνn (r, x) < +∞.
≤
|f (r, x)|
(1 + |(r, x)| + |(µ0 , λ0 )|)d
Bdc
The relations (3.11) and (3.12) imply that


ZZ ZZ W n−1 (f )(r, x) e|(r,x)||(µ,λ)|
2
F˜ (f )(µ, λ)
dmn+1 (r, x)dmn+1 (µ, λ) < +∞,
d
c
(1
+
|(r,
x)|
+
|(µ,
λ)|)
Bd
Bd
and the proof of Theorem 3.1 is complete.
Uncertainty Principle for
the Spherical Mean Operator
N. Msehli and L.T. Rachdi
vol. 10, iss. 2, art. 38, 2009
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Theorem 3.7 (Beurling Hörmander for R). Let f be a measurable function on
R × Rn , even with respect to the first variable and such that f ∈ L2 (dνn ).
Let d be a real number, d ≥ 0. If
ZZ Z ∞ Z
|F (f )(µ, λ)|e|(r,x)||θ(µ,λ)|
|f (r, x)|
dνn (r, x)dγ̃n (µ, λ) < +∞,
(1 + |(r, x)| + |θ(µ, λ)|)d
Γ+ 0
Rn
Uncertainty Principle for
the Spherical Mean Operator
then
N. Msehli and L.T. Rachdi
• For d ≤ n + 1, f = 0.
vol. 10, iss. 2, art. 38, 2009
• For d > n + 1, there exist a positive constant a and a polynomial P on R × R
even with respect to the first variable, such that
n
Title Page
−a(r 2 +|x|2 )
f (r, x) = P (r, x)e
with degree(P ) <
d−(n+1)
.
2
Proof. Let f be a function satisfying the hypothesis. Then, from Theorem 3.1, we
have
Z ∞Z Z ∞Z ˜
(3.13)
W n−1 (f )(r, x) F (f )(µ, λ)
0
Rn
0
Rn
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2
e|(r,x)||(µ,λ)|
×
dmn+1 (r, x)dmn+1 (µ, λ) < +∞.
(1 + |(r, x)| + |(µ, λ)|)d
On the other hand, from Proposition 2.1, Lemma 3.2 and Lemma 3.3, we deduce that
the function W n−1 (f ) belongs to the space L1 (dmn+1 ) ∩ L2 (dmn+1 ) and by (2.16),
2
we have
˜
n−1
F (f ) = Λn+1 W (f ) .
2
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Substituting into (3.13), we get
Z ∞Z Z ∞Z n−1
n−1
W (f )(r, x) Λn+1 W (f ) (µ, λ)
Rn
0
0
2
Rn
×
2
e|(r,x)||(µ,λ)|
dmn+1 (r, x)dmn+1 (µ, λ) < +∞.
(1 + |(r, x)| + |(µ, λ)|)d
Applying Theorem 3.1 when f is replaced by W n−1 (f ), we deduce that
Uncertainty Principle for
the Spherical Mean Operator
2
• If d ≤ n + 1, W n−1 (f ) = 0 and by Remark 2, we have f = 0.
2
• If d > n + 1, there exist a > 0 and a polynomial Q on R × Rn , even with
respect to the first variable such that
N. Msehli and L.T. Rachdi
vol. 10, iss. 2, art. 38, 2009
Title Page
−a(r2 +|x|2 )
W n−1 (f )(r, x) = Q(r, x)e
2
X
2
2
ak,α r2k xα e−a(r +|x| ) ;
=
Contents
xα = xα1 1 · · · xαnn .
2k+|α|≤m
In particular, the function W n−1 (f ) lies in S∗ (R × R ) and by Remark 2, the
2
function f belongs to S∗ (R × Rn ) and we have
2
2 −1
f = W n−1
Q(r, x)e−a(r +|x| ) .
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n
2
Now, using the relation (2.17), we obtain
(3.14)
2 +|y|2 )
−1
f (r, x) =W n−1
(Q(t, y)e−a(t
)(r, x)
#
"
[ n2 ]+1
∂
2 +|y|2 )
[n
]+1
−a(t
=(−1) 2 F[ n2 ]− n2 +1
Q(t, y)e
(r, x)
∂t2
2
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[n
]+1
2
= (−1)
X
"
ak,α F[ n2 ]− n2 +1
2k+|α|≤m
∂
∂t2
[ n2 ]+1
#
−a(t2 +|y|2 )
(t2k y α e
) (r, x).
However, for all l ∈ N,
l
∂
2
2
(3.15)
(t2k y α e−a(t +|y| ) )
2
∂t


min(l,k)
X j 2j k!
2
2
=
Cl
(−2a)k−j t2(k−j)  y α e−a(t +|y| )
(k − j)!
j=0
and for all b > 0,
(3.16)
Uncertainty Principle for
the Spherical Mean Operator
N. Msehli and L.T. Rachdi
vol. 10, iss. 2, art. 38, 2009
Title Page
2
2 Fb t2k y α e−a(t +|y| ) (r, x)
=
1
2b Γ(b)
Contents
k
X
j=0
Ckj
Γ(b + k − j) 2j
r
aµ+k−j r2j
!
xα e−a(r
2 +|x|2 )
,
where the transform Fb is defined by the relation (2.18).
Combining the relations (3.14), (3.15) and (3.16), we deduce that
−a(r2 +|x|2 )
f (r, x) = P (r, x)e
,
where P is a polynomial, even with respect to the first variable and degree(P ) =
degree(Q).
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4.
Applications of the Beurling-Hörmander Theorem
This section is devoted to giving some applications of the Beurling-Hörmander theorem for the spherical mean operator. More precisely, we prove a Gelfand-Shilov
theorem for the Fourier transform F and establish a Cowling Price type theorem for
this transform.
Lemma 4.1. Let P be a polynomial on R × Rn ; P 6= 0 with degree(P ) = m. Then
there exist two positive constants A and C such that
Z
∀ t ≥ A, ϕ(t) =
|P (tw)|dσn (w) ≥ Ctm ,
Uncertainty Principle for
the Spherical Mean Operator
N. Msehli and L.T. Rachdi
vol. 10, iss. 2, art. 38, 2009
Sn
where dσn is the surface measure on the unit sphere S n of R × Rn .
Proof. Let P be a polynomial on R × Rn , P 6= 0 and degree (P ) = m. Then we
have
Z X
m
k
ϕ(t) =
ak (w)t dσn (w),
Sn k=0
where ak , 0 ≤ k ≤ m are continuous functions on S n and am 6= 0.
Then the function ϕ is continuous on [0, +∞[ and by the dominated convergence
theorem, we have
(4.1)
where
ϕ(t) ∼ Cm tm (t −→ +∞),
Z
am (w)dσn (w) > 0.
Cm =
Sn
Now, by (4.1), there exists A > 0 such that
∀ t ≥ A;
p(t) ≥
Cm m
t .
2
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Theorem 4.2 (Gelfand-Shilov). Let p, q be two conjugate exponents, p, q ∈]1, +∞[.
Let η, ξ be two positive real numbers such that ξη ≥ 1.
Let f be a measurable function on R × Rn , even with respect to the first variable
such that f ∈ L2 (dνn ).
If
ξp |(r,x)|p
Z ∞Z
|f (r, x)|e p
dνn (r, x) < +∞
d
0
Rn (1 + |(r, x)|)
and
N. Msehli and L.T. Rachdi
ξq |(r,x)|q
q
ZZ
Γ+
|F (f )(µ, λ)|e
(1 + |θ(µ, λ)|)d
Uncertainty Principle for
the Spherical Mean Operator
vol. 10, iss. 2, art. 38, 2009
dγ̃n (µ, λ) < +∞,
d ≥ 0,
then
Title Page
i. For d ≤
n+1
,
2
ii. For d >
n+1
,
2
Contents
f = 0.
we have:
• f = 0 for ξη > 1;
• f = 0 for ξη = 1 and p 6= 2;
2
2
• f (r, x) = P (r, x)e−a(r +|x| ) for ξη = 1 and p = q = 2, where a > 0 and
P is a polynomial on R × Rn even with respect to the first variable, with
degree (P ) < d − n+1
.
2
Proof. Let f be a function satisfying the hypothesis. Since ξη ≥ 1, by a convexity
argument we have
ZZ Z ∞ Z
|f (r, x)||F (f )(µ, λ)| |(r,x)||θ(µ,λ)|
(4.2)
dνn (r, x)dγ̃n (µ, λ)
2d e
Γ+ 0
Rn 1 + |(r, x)| + |θ(µ, λ)
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∞
|f (r, x)||F (f )(µ, λ)|
d
d
Γ+ 0
Rn (1 + |(r, x)|) (1 + |θ(µ, λ)|)
× eηξ|(r,x)||θ(µ,λ)| dνn (r, x)dγ̃n (µ, λ)
η q |θ(µ,λ)|q
ξp |(r,x)|p
ZZ
Z ∞Z
q
|F (f )(µ, λ)|e
|f (r, x)|e p
≤
dγ̃n (µ, λ)
dνn (r, x)
d
(1 + |θ(µ, λ)|)d
Γ+
0 Rn (1 + |(r, x)|)
< +∞.
ZZ
Z
Z
≤
Then from the Beurling-Hörmander theorem, we deduce that
i. For d ≤ n+1
, f = 0.
2
n+1
ii. For d > 2 , there exist a positive constant a and a polynomial P on R × Rn ,
even with respect to the first variable such that
Uncertainty Principle for
the Spherical Mean Operator
N. Msehli and L.T. Rachdi
vol. 10, iss. 2, art. 38, 2009
Title Page
−a(r2 +|x|2 )
f (r, x) = P (r, x)e
(4.3)
2d−(n+1)
,
2
with degree(P ) <
and using standard calculus, we obtain
1
2
2
F˜ (f )(µ, λ) = Q(µ, λ)e− 4a (µ +|λ| ) ,
(4.4)
where Q is a polynomial on R × Rn , even with respect to the first variable, with
degree (Q) = degree(P ).
On the other hand, from the relations (2.7), (2.8), (4.2), (4.3) and (4.4), we get
Z ∞Z Z ∞Z
|P (r, x)||Q(µ, λ)|eξη|(r,x)||(µ,λ)|
d
d
0
Rn 0
Rn (1 + |(r, x)|) (1 + |(µ, λ)|)
−(µ2 +|λ|2 )
2
2
4a
×e
e−a(r +|x| ) dνn (r, x)dmn+1 (µ, λ) < +∞.
So,
Z
∞
Z
(4.5)
0
Rn
ϕ(t)
ψ(ρ) ξηtρ −at2 − ρ2 2n n
4a t
e e
ρ dtdρ < +∞,
(1 + t)d (1 + ρ)d
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where
Z
|P (tw)||w1 |n dσn (w)
ϕ(t) =
Sn
and
Z
|Q(ρw)|dσn (w).
ψ(ρ) =
Sn
• Suppose that ξη > 1. If f 6= 0, then each of the polynomials P and Q is not
identically zero. Let m = degree(P ) = degree(Q).
From Lemma 4.1, there exist two positive constants A and C such that
∀t ≥ A,
Uncertainty Principle for
the Spherical Mean Operator
N. Msehli and L.T. Rachdi
vol. 10, iss. 2, art. 38, 2009
ϕ(t) ≥ Ctm
Title Page
and
∀ρ ≥ A,
ψ(ρ) ≥ Cρm .
Then the inequality (4.5) leads to
Z ∞Z ∞
2
eξηtρ
−at2 − ρ4a
(4.6)
e
e
dtdρ < +∞.
d
d
A
A (1 + t) (1 + ρ)
Let ε > 0 such that c = ηξ − ε > 1. The relation (4.6) implies that
Z ∞Z ∞
2
eερt
cρt −at2 − ρ4a
(4.7)
e
e
e
dtdρ < +∞.
d
d
A
A (1 + t) (1 + ρ)
However, for all t ≥ A ≥
d
ε
and ρ ≥ A, we have
2
eερt
eεA
≥
(1 + t)d (1 + ρ)d
(1 + A)2d
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and by (4.7), it follows that
Z ∞Z
(4.8)
A
Let F (t) =
R∞
∞
ρ2
2
ecρt−at e− 4a dtdρ < +∞.
A
2
cρt− ρ4a
dρ, then the function F can be written as
Z ∞
Z t
2
2
2 s2
ac2 t2
−A
cAs−ac
− ρ4a
F (t) = e
e dρ + 2aγe 4a
e
ds .
A
e
A
0
Uncertainty Principle for
the Spherical Mean Operator
N. Msehli and L.T. Rachdi
In particular,
F (t) ≥ eac
2 t2
Z
∞
vol. 10, iss. 2, art. 38, 2009
ρ2
e− 4a dρ.
A
Thus,
Title Page
Z
∞
A
Z
∞
A
ecρt−at
2−
ρ2
4a
Z
∞
dtdρ ≥
A
ea(c
2 −1)t2
Z
∞
dt
2
− ρ4a
e
Contents
dρ = +∞
A
because c > 1. This contradicts the relation (4.8) and shows that f = 0.
• Suppose that ξη = 1 and p 6= 2.
In this case, we have p > 2 or q > 2.
Suppose that q > 2. Then from the second hypothesis and the relations (2.7),
(2.8) and (4.4), we get
q q
Z ∞
ρ2 η ρ
ψ(ρ)e− 4a e q n
(4.9)
ρ dρ < +∞.
(1 + ρ)d
0
If f 6= 0, then the polynomial Q is not identically zero, and by Lemma 4.1 and
the relation (4.9), it follows that there exists A > 0 such that
Z ∞ − ρ2 ηqqρq
e 4a e
dρ < +∞,
(1 + ρ)d
A
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which is impossible because q > 2.
The proof of Theorem 4.2 is thus complete.
Theorem 4.3 (Cowling-Price for spherical mean operator). Let η, ξ, w1 and w2
be non negative real numbers such that ηξ ≥ 14 . Let p, q be two exponents, p, q ∈
[1, +∞] and let f be a measurable function on R × Rn , even with respect to the first
variable such that f ∈ L2 (dνn ). If
eξ|(·,·)|2 f (4.10)
< +∞
(1 + |(·, ·)|)w1 Uncertainty Principle for
the Spherical Mean Operator
N. Msehli and L.T. Rachdi
vol. 10, iss. 2, art. 38, 2009
p,νn
and
(4.11)
Title Page
η|θ(·,·)|2
e
F
(f
)
(1 + |θ(·, ·)|)w2
Contents
< +∞,
q,γ̃n
then
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Page 38 of 44
i. For ξη > 14 , f = 0.
ii. For ξη = 14 , there exist a positive constant a and a polynomial P on R × Rn ,
even with respect to the first variable such that
f (r, x) = P (r, x)e−a(r
2 +|x|2 )
.
Proof. Let p0 and q 0 be the conjugate exponents of p respectively q.
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Let us pick d1 , d2 ∈ R such that d1 > 2n+1 and d2 > n+1. Then from Hölder’s
inequality and the relations (4.10) and (4.11), we deduce that
Z ∞Z
2
|f (r, x)|eξ|(r,x)|
(4.12)
dνn (r, x)
w1 +d1 /p0
0
Rn (1 + |(r, x)|)
eξ|(·,·)|2 f 1
≤
0
w
d
/p
(1 + |(·, ·)|) 1 (1 + |(·, ·)|) 1
p0 ,νn
p,νn
10
Z ∞Z
eξ|(·,·)|2 f p
dνn (r, x)
< +∞.
=
d1
(1 + |(·, ·)|)w1 0
Rn (1 + |(r, x)|)
Uncertainty Principle for
the Spherical Mean Operator
N. Msehli and L.T. Rachdi
vol. 10, iss. 2, art. 38, 2009
p,νn
Title Page
and
Contents
2
ZZ
Γ+
|F (f )(µ, λ)|eη|θ(µ,λ)|
dγ̃n (µ, λ)
(1 + |θ(µ, λ)|)w2 +d2 /q0
η|θ(·,·)|2
e
≤
F (f )
w
2
(1 + |θ(·, ·)|)
q,γ̃n
1
0
(1 + |θ(·, ·)|)d2 /q .
q 0 ,γ̃n
By the relation (2.7), we obtain
(4.13)
2
q,γ̃n
Let d > max w1 +
d1
,
p0
w2 +
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|F (f )(µ, λ)|eη|θ(µ,λ)|
dγ̃n (µ, λ)
w2 +d2 /q 0
Γ+ (1 + |θ(µ, λ)|)
Z ∞ Z
1
η|θ(·,·)|2
e
dmn+1 (µ, λ) q0
≤
F (f )
< +∞.
d2
(1 + |θ(·, ·)|)w2
0
Rn (1 + |(µ, λ)|)
ZZ
JJ
d2 n+1
, 2
q0
, then from the relations (4.12) and (4.13),
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we have
Z
∞
0
and
ZZ
Γ+
Z
Rn
2
|f (r, x)|eξ|(r,x)|
dνn (r, x) < +∞
(1 + |(r, x)|)d
2
|F (f )(µ, λ)|eη|θ(µ,λ)|
dγ̃n (µ, λ) < +∞.
(1 + |θ(µ, λ)|)d
Then the desired result follows from Theorem 4.2.
Remark 4. The Hardy theorem is a special case of Theorem 4.2, when p = q = +∞.
Uncertainty Principle for
the Spherical Mean Operator
N. Msehli and L.T. Rachdi
vol. 10, iss. 2, art. 38, 2009
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References
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Uncertainty Principle for
the Spherical Mean Operator
N. Msehli and L.T. Rachdi
vol. 10, iss. 2, art. 38, 2009
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Contents
[5] L. BOUATTOUR AND K. TRIMÈCHE, An analogue of the BeurlingHörmander’s theorem for the Chébli-Trimèche transform, Glob. J. Pure Appl.
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[6] B. CHABAT, Introduction à l’Analyse Complexe: Fonctions de Plusieurs Variables, Éditions Mir Moscou, 3rd ed., 1984.
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[8] M. DZIRI, M. JELASSI AND L.T. RACHDI, Spaces of DLp type and a convolution product associated with the spherical mean operator, Int. J. Math. Math.
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[10] A. ERDELY et al, Tables of Integral Transforms, Vol. 2, McGraw-Hill Book
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[11] J.A. FAWCETT, Inversion of N-dimensional spherical means, SIAM. J. Appl.
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Pure and Applied Mathematics (New York), John Wiley and Sons, New York,
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the Spherical Mean Operator
N. Msehli and L.T. Rachdi
[13] G.B. FOLLAND AND A. SITARAM, The uncertainty principle: a mathematical survey, J. Fourier Anal. Appl., 3 (1997), 207–238.
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[14] I.M. GELFAND AND G.E. SHILOV, Fourier transforms of rapidly increasing
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Uspekhi Mat. Nauk, 8 (1953), 3–54.
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[15] G.H. HARDY, A theorem concerning Fourier transform, J. London Math.
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