Volume 10 (2009), Issue 2, Article 38, 22 pp.
BEURLING-HÖRMANDER UNCERTAINTY PRINCIPLE FOR THE SPHERICAL
MEAN OPERATOR
N. MSEHLI AND L.T. RACHDI
D ÉPARTEMENT DE MATHÉMATIQUES ET D ’ INFORMATIQUE
I NSTITUT NATIONAL DES SCIENCES APPLIQUÉES ET DE TECHNOLOGIE DE T UNIS
1080 T UNIS , T UNISIA
n.msehli@yahoo.fr
D ÉPARTEMENT DE M ATHÉMATIQUES
FACULTÉ DES S CIENCES DE T UNIS
2092 E L M ANAR II
T UNIS , T UNISIA
lakhdartannech.rachdi@fst.rnu.tn
Received 15 November, 2008; accepted 01 May, 2009
Communicated by J.M. Rassias
A BSTRACT. 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 CowlingPrice type theorems for this transform.
Key words and phrases: Uncertainty principle, Beurling-Hörmander theorem, Gelfand-Shilov theorem, Cowling-Price theorem, Fourier transform, Spherical mean operator.
2000 Mathematics Subject Classification. 42B10, 43A32.
1. I NTRODUCTION
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.
312-08
2
N. M SEHLI AND L.T. R ACHDI
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
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 BeurlingHörmander 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 GelfandShilov [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 + |x − y|2 , y dy,
R(g)(r, x) =
g
r
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 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
where
• ϕµ,λ (r, x) = R cos(µ.)e−ihλ/·i (r, x)
• dνn is the measure defined on [0, +∞[×Rn by
dνn (r, x) =
1
2
n−1
2
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 38, 22 pp.
Γ( n+1
)
2
rn dr ⊗
dx
n .
(2π) 2
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U NCERTAINTY P RINCIPLE FOR THE S PHERICAL M EAN O PERATOR
3
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
ZZ 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
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
2 +|x|2 )
;
with degree(P ) < d−(n+1)
2
where
• Γ+ is the set given by
Γ+ = [0, +∞[×Rn ∪ (iµ, λ); (µ, λ) ∈ R × Rn ; 0 ≤ µ ≤ |λ|
• θ is the bijective function defined on Γ+ by
p
µ2 + |λ|2 , λ
θ(µ, λ) =
• 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
∞
×
0
µdµdλ
g(µ, λ) p
+
µ2 + λ 2
Rn
Z
Rn
Z
0
|λ|
#
µdµdλ
g(iµ, λ) p
.
λ 2 − µ2
The last section of this paper is devoted to the Gelfand-Shilov and Cowling Price theorems
for the transform F .
• 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
ξq |(r,x)|q
ZZ
|F (f )(µ, λ)|e q
dγ̃n (µ, λ) < +∞; d ≥ 0,
(1 + |θ(µ, λ)|)d
Γ+
then
; f = 0.
i. For d ≤ n+1
2
n+1
ii. For d > 2 ; we have
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 38, 22 pp.
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4
N. M SEHLI AND L.T. R ACHDI
– 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
< +∞
(1 + |(·, ·)|)w1 p,νn
and
η|θ(·,·)|2
e
F
(f
)
(1 + |θ(·, ·)|)w2
< +∞,
q,γ̃n
then
i. For ξη > 14 ; f = 0.
ii. For ξη = 14 ; 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
2 +|x|2 )
.
2. T HE S PHERICAL M EAN O PERATOR
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
p
2
2
r µ + λ e−ihλ/xi ,
where
• λ2 = λ21 + · · · + λ2n ; λ = (λ1 , . . . , λn ) ∈ Cn ;
• hλ/xi = λ1 x1 + · · · + λn xn ; x = (x1 , . . . , xn ) ∈ Rn ;
• j n−1 is the modified Bessel function given by
2
(s)
n−1
n + 1 J n−1
2
(2.2)
j n−1 (s) = 2 2 Γ
n−1
2
2
s 2
∞
s 2k
n+1 X
(−1)k
=Γ
;
2
k!Γ(k + n+1
) 2
2
k=0
and J n−1 is the usual Bessel function of first kind and order
2
n−1
2
[9, 10, 22, 36].
Also, the modified Bessel function j n−1 has the following integral representation, for all
2
z ∈ C:
Z
) 1
2Γ( n+1
n
2
(1 − t2 ) 2 −1 cos(zt)dt.
j n−1 (z) = √
n
2
π Γ( 2 ) 0
Thus, for all z ∈ C; we have
(2.3)
j n−1 (z) ≤ e| Im z| .
2
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 38, 22 pp.
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U NCERTAINTY P RINCIPLE FOR THE S PHERICAL M EAN O PERATOR
5
Using the relation (2.1) and the properties of the function j n−1 , we deduce that the function ϕµ,λ
2
satisfies the following properties [24, 27]:
•
(2.4)
sup ϕµ,λ (r, x) = 1
(r,x)∈R×Rn
(2.5)
if and only if (µ, λ) belongs to the set Γ defined by
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
where
n
X
∂2
n ∂
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
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

< +∞, if 1 ≤ p < +∞;

n |f (r, x)| dmn+1 (r, x)
0
R

kf kp,mn+1 =

ess sup
|f (r, x)| < +∞,
if p = +∞.


n
(r,x)∈[0,+∞[×R
• dνn the measure defined on [0, +∞[×Rn by
dx
rn dr
dνn (r, x) = n−1 n+1 ⊗
n .
2 2 Γ( 2 ) (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
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 .
• dγn the measure defined on BΓ+ by
∀A ∈ BΓ+ ; γn (A) = νn θ(A) .
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 38, 22 pp.
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6
N. M SEHLI AND L.T. R ACHDI
• 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 2 Γ n+1
dγn (µ, λ)
√ 2
dγ̃n (µ, λ) =
n .
π
(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|
n−1
+
g(iµ, λ)(|λ|2 − µ2 ) 2 µdµdλ) .
Rn
0
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
ZZ
Z ∞Z
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+
Rn
0
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 2.1. 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),
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 38, 22 pp.
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U NCERTAINTY P RINCIPLE FOR THE S PHERICAL M EAN O PERATOR
7
where
F˜ (f )(µ, λ) =
(2.9)
Z
∞
Z
f (r, x)j n−1 (rµ)e−ihλ/xi dνn (r, x)
2
Rn
0
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 ):
kF (f )k∞,γn ≤ kf k1,νn .
(2.10)
Theorem 2.2 (Inversion formula). Let f ∈ L1 (dνn ) such that F (f ) ∈ L1 (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).
0
2
Rn
Lemma 2.3. Let R n−1 be the mapping defined for all non negative measurable functions g on
2
[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
= √ 2n (1 − t2 ) 2 −1 g(tr, x)dt,
πΓ 2 0
then for all non negative measurable functions f, g on [0, +∞[×Rn , we have
Z ∞Z
(2.13)
R n−1 (g)(r, x)f (r, x)dνn (r, x)
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 functions g
2
on [0, +∞[×Rn by
Z ∞
n
1
(r2 − t2 ) 2 −1 f (r, x)2rdr.
(2.14)
W n−1 (f )(t, x) = n n
2
2 2 Γ( 2 ) t
Proposition 2.4. For all f ∈ L1 (dνn ), the function W n−1 (f ) given by the relation (2.14) is
2
defined almost every where, belongs to the space L1 (dmn+1 ) and we have
(2.15)
≤ kf k1,νn .
W n−1 (f )
2
1,mn+1
Moreover,
(2.16)
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).
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 38, 22 pp.
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8
N. M SEHLI AND L.T. R ACHDI
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), we deduce that the
classical Weyl transform W n−1 is also a topological isomorphism from S∗ (R × Rn ) onto itself,
2
and the inverse isomorphism is given by [24]
[ n2 ]+1 !
n
∂
−1
f (r, x),
(f )(r, x) = (−1)[ 2 ]+1 F[ n2 ]− n2 +1
(2.17)
W n−1
2
∂t2
where Fa ; a > 0 is the mapping defined on S∗ (R × Rn ) by
Z ∞
1
(t2 − r2 )a−1 f (t, x)2tdt
(2.18)
Fa (f )(r, x) = a
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
3. T HE B EURLING -H ÖRMANDER T HEOREM
FOR THE
S PHERICAL M EAN O PERATOR
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
2
2
f (r, x) = P (r, x)e−a(r +|x| ) ,
.
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
ZZ 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 ).
Proof. Let f ∈ L2 (dνn ), f 6= 0. From the relations (2.7) and (2.8), we obtain
ZZ 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)|)
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 38, 22 pp.
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U NCERTAINTY P RINCIPLE FOR THE S PHERICAL M EAN O PERATOR
9
In particular, there exists (µ0 , λ0 ) ∈ [0, +∞[×Rn \ {(0, 0)} such that
Z ∞Z
|f (r, x)|e|(r,x)||(µ0 ,λ0 )|
˜
F (f )(µ0 , λ0 ) 6= 0 and
dνn (r, x) < +∞.
d
0
Rn (1 + |(r, x)| + |(µ0 , λ0 )|)
Let h be the function defined on [0, +∞[ by
es|(µ0 ,λ0 )|
,
(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 )|.
h(s) =
Consequently,
Z ∞Z
|f (r, x)|dνn (r, x) ≤
0
Rn
1
h(s0 )
Z
∞
Z
Rn
0
|f (r, x)|e|(r,x)||(µ0 ,λ0 )|
dνn (r, x) < +∞.
(1 + |(r, x)| + |(µ0 , λ0 )|)d
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
Z ∞Z
|f (r, x)|e|(r,x)||(µ0 ,λ0 )|
dνn (r, x)
d
0
Rn (1 + |(r, x)| + |(µ0 , λ0 )|)
Z ∞Z
e|(r,x)|(|(µ0 ,λ0 )|−(n+1)a)
=
|f (r, x)|e(n+1)a|(r,x)|
dνn (r, x) < +∞.
(1 + |(r, x)| + |(µ0 , λ0 )|)d
0
Rn
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 )|,
if
d
|(µ0 ,λ0 )|−(n+1)a
≤ 1 + |(µ0 , λ0 )|.
Consequently,
Z ∞Z
(3.1)
0
|f (r, x)|e(n+1)a|(r,x)| dνn (r, x)
Rn
Z ∞Z
1
|f (r, x)|e|(r,x)||(µ0 ,λ0 )|
≤
dνn (r, x) < +∞.
d
g(s0 ) 0
Rn (1 + |(µ0 , λ0 )| + |(r, x)|)
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 38, 22 pp.
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10
N. M SEHLI
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L.T. R ACHDI
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
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
(3.2)
j n−1 (rµ)e−ihλ,xi ≤ 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. 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 ×
n
R , |(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.
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 )|
|f (r, x)|
dνn (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
eσt
t 7−→
(1 + t + σ)d
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
ZZ 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
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 38, 22 pp.
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U NCERTAINTY P RINCIPLE FOR THE S PHERICAL M EAN O PERATOR
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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) < +∞.
Rn
0
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.
2
Rn
0
In particular, the function f is bounded and
kf k∞,νn ≤ F˜ (f )
(3.4)
.
1,νn
By virtue of the relation (2.14), we get
Z ∞
n
1
(r2 − t2 ) 2 −1 |f (r, x)|2rdr
(f )(r, x) ≤ n n
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
Using Minkowski’s inequality for integrals [12], we get:
Z ∞ Z 12
2
(3.5)
W n−1 (f )(r, x) dmn+1 (r, x)
0
2
Rn
1
≤ n n
2 2 Γ( 2 )
"Z
1
≤ n n
2 2 Γ( 2 )
Z
∞
Rn
0
∞
Z
Z
Z
∞
rn (y 2 − 1)
n
−1
2
1
= n −1 n
2 2 Γ( 2 )
∞
Z
Γ( 14 )
= n 2n+1
2 2 Γ( 4 )
2n
∞
2
2
(y − 1)
n
−1
2
y
−n+ 12
n−2
12
|f (ry, x)| dmn+1 (r, x) 2ydy
2
Z
0
∞
Z
∞
Z
dy
0
1
Z
dmn+1 (r, x)
Rn
0
Z
|f (ry, x)|2ydy
1
r (y − 1)
1
# 12
2
12
s |f (s, x)| dmn+1 (s, x)
2n
2
Rn
12
s2n |f (s, x)|2 dmn+1 (s, x) .
Rn
Using the relations (3.1), (3.4) and (3.5), we deduce that
Z ∞ Z 21
2
(f )
=
(f ) (r, x)dmn+1 (r, x)
W n−1
W n−1
2
2
2,mn+1
0
Rn
Z ∞Z
≤ Kn
|f (s, x)|e(n+1)a|(s,x)| dνn (s, x) < +∞,
0
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 38, 22 pp.
Rn
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12
N. M SEHLI
AND
L.T. R ACHDI
where
Γ
Kn = n
22 Γ
1
4
2n+1
4
r
π
Γ
2
n+1
2
2
n−1
2
max(sn e−(n+1)as )kf k∞,νn
s≥0
21
.
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
Rn
0
∞
Z
Z
Rn
0
˜
(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
Proof. From the hypothesis, the relations (2.7), (2.8) and Fubini’s theorem, we have
ZZ Z ∞ Z
e|(r,x)||θ(µ,λ)|
(3.6)
|f (r, x)||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)
d
0
Rn
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 (3.7)
(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
0
Z ∞RZ 0 R
Z ∞Z
˜
|(r,x)||(µ,λ)|
≤
W n−1 (|f |)(r, x)e
dmn+1 (r, x) dmn+1 (µ, λ)
F (f )(µ, λ)
2
0
Rn
0
Rn
Z ∞Z Z ∞Z
˜
|(·,·)||(µ,λ)|
)(r, x)dνn (r, x) dmn+1(µ, λ).
≤
|f (r, x)|R n−1 (e
F (f )(µ, λ)
Rn
0
0
2
Rn
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
0
∞
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
Rn 0
Z ∞ ZR Z ∞Z
˜
≤
|f (r, x)|e|(r,x)||(µ,λ)| dνn (r, x)dmn+1 (µ, λ) < +∞.
F (f )(µ, λ)
0
Rn
0
Rn
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 38, 22 pp.
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U NCERTAINTY P RINCIPLE FOR THE S PHERICAL M EAN O PERATOR
13
ii. For d > 0, let Bd = {(r, x) ∈ [0, +∞[×Rn ; |(r, x)| ≤ d}. We have
Z ∞Z 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)
d
0
Rn
0
Rn (1 + |(r, x)| + |(µ, λ)|)
!
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)
d
Bdc
0
Rn (1 + |(r, x)| + |(µ, λ)|)
!
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
n
Bd
0
R
From the relation (2.13), we deduce that
!
ZZ Z ∞ Z W n−1 (|f |)(r, x)e|(r,x)||(µ,λ)|
˜
2
(3.9)
dmn+1 (r, x) dmn+1 (µ, λ)
F (f )(µ, λ)
c
(1
+ |(r, x)| + |(µ, λ)|)d
Bd
0
Rn
ZZ Z ∞Z
˜
=
|f (r, x)|
F (f )(µ, λ)
Bdc
× R n−1
2
0
|(·,·)||(µ,λ)|
Rn
e
(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|(·,·)||(µ,λ)|
e|(r,x)||(µ,λ)|
(3.10)
R n−1
(r,
x)
≤
.
2
(1 + |(·, ·)| + |(µ, λ)|)d
(1 + |(r, x)| + |(µ, λ)|)d
Combining the relations (3.6), (3.9) and (3.10), we get
!
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)
d
Bdc
0
Rn (1 + |(r, x)| + |(µ, λ)|)
ZZ Z ∞ Z
|f (r, x)|e|(r,x)||(µ,λ)|
˜
dνn (r, x) dmn+1 (µ, λ)
≤
F (f )(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)|)
< +∞.
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)
d
Bd
Bd (1 + |(r, x)| + |(µ, λ)|)
Z Z Z Z ˜
d2
(f )(r, x) dmn+1 (r, x)
≤e
F (f )(µ, λ) dmn+1 (µ, λ)
W n−1
2
Bd
Bd
d2
.
≤ e mn+1 (Bd )kF (f )k∞,γn W n−1 (f )
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)
d
Bd
Bd (1 + |(r, x)| + |(µ, λ)|)
2
≤ ed mn+1 (Bd )kf k21,νn < +∞.
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 38, 22 pp.
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14
N. M SEHLI
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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
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)
F (f )(µ, λ)
(1
+ |(r, x)| + |(µ, λ)|)d
n
Bd
0
R
ZZ 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
ZZ
However, by ii) of Remark 3 and the relation (2.10), we deduce that for all (µ, λ) ∈ Bd :
e|(·,·)||(µ,λ)|
ed|(r,x)|
c
R n−1
1
(·,
·)
(r,
x)
≤
1B c (r, x).
B
2
(1 + |(·, ·)| + |(µ, λ)|)d d
(1 + |(r, x)| + d)d d
Thus,
ZZ
(3.11)
Bd

ZZ
˜
F (f )(µ, λ) 
(f )(r, x) e|(r,x)||(µ,λ)|
W n−1
2

dmn+1 (r, x) dmn+1 (µ, λ)
(1 + |(r, x)| + |(µ, λ)|)d
ZZ
ed|(r,x)|
≤ kf k1,νn mn+1 (Bd )
|f (r, x)|
dνn (r, x).
(1 + |(r, x)| + d)d
Bdc
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 )|
≤
|f (r, x)|
dνn (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) < +∞,
(1 + |(r, x)| + |(µ, λ)|)d
Bd
Bdc
and the proof of Theorem 3.1 is complete.
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
then
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 38, 22 pp.
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U NCERTAINTY P RINCIPLE FOR THE S PHERICAL M EAN O PERATOR
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• For d ≤ n + 1, f = 0.
• For d > n + 1, 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
with degree(P ) <
2 +|x|2 )
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)
Rn
0
Rn
0
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), we have
2
F˜ (f ) = Λn+1 W n−1 (f ) .
2
Substituting into (3.13), we get
Z ∞Z Z ∞Z W n−1 (f )(r, x) Λn+1 W n−1 (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
2
• If d ≤ n + 1, W (f ) = 0 and by Remark 2, we have f = 0.
• 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−1
2
2
2
W n−1 (f )(r, x) = Q(r, x)e−a(r +|x| )
2
X
2
2
=
ak,α r2k xα e−a(r +|x| ) ;
xα = xα1 1 · · · xαnn .
2k+|α|≤m
In particular, the function W n−1 (f ) lies in S∗ (R × Rn ) and by Remark 2, the function f
2
belongs to S∗ (R × Rn ) and we have
2
2 −1
Q(r, x)e−a(r +|x| ) .
f = W n−1
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
n
∂
2
2
Q(t, y)e−a(t +|y| ) (r, x)
=(−1)[ 2 ]+1 F[ n2 ]− n2 +1
∂t2
"
#
[ n2 ]+1
X
n
∂
2
2
ak,α F[ n2 ]− n2 +1
(t2k y α e−a(t +|y| ) ) (r, x).
=(−1)[ 2 ]+1
∂t2
2
"
2k+|α|≤m
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 38, 22 pp.
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16
N. M SEHLI
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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)
2
2 Fb t2k y α e−a(t +|y| ) (r, x) =
k
X
1
2b Γ(b)
Ckj
j=0
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
f (r, x) = P (r, x)e−a(r
2 +|x|2 )
,
where P is a polynomial, even with respect to the first variable and degree(P ) = degree(Q).
4. A PPLICATIONS OF THE B EURLING -H ÖRMANDER T HEOREM
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 ,
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)
ϕ(t) ∼ Cm tm
(t −→ +∞),
where
Z
Cm =
am (w)dσn (w) > 0.
Sn
Now, by (4.1), there exists A > 0 such that
∀ t ≥ A;
p(t) ≥
Cm m
t .
2
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 38, 22 pp.
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U NCERTAINTY P RINCIPLE FOR THE S PHERICAL M EAN O PERATOR
17
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
ξq |(r,x)|q
ZZ
|F (f )(µ, λ)|e q
dγ̃n (µ, λ) < +∞, d ≥ 0,
(1 + |θ(µ, λ)|)d
Γ+
then
i. For d ≤ n+1
, f = 0.
2
n+1
ii. For d > 2 , 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)
ZZ Z ∞ Z
|f (r, x)||F (f )(µ, λ)|
eηξ|(r,x)||θ(µ,λ)| dνn (r, x)dγ̃n (µ, λ)
≤
d
d
Γ+ 0
Rn (1 + |(r, x)|) (1 + |θ(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)|)
< +∞.
Then from the Beurling-Hörmander theorem, we deduce that
i. For d ≤ n+1
, f = 0.
2
ii. For d > n+1
, there exist a positive constant a and a polynomial P on R × Rn , even with
2
respect to the first variable such that
f (r, x) = P (r, x)e−a(r
(4.3)
with degree(P ) <
(4.4)
2d−(n+1)
,
2
2 +|x|2 )
and using standard calculus, we obtain
1
2
2
F˜ (f )(µ, λ) = Q(µ, λ)e− 4a (µ +|λ| ) ,
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) < +∞.
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 38, 22 pp.
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18
N. M SEHLI
L.T. R ACHDI
AND
So,
∞
Z
Z
(4.5)
Rn
0
ϕ(t)
ψ(ρ) ξηtρ −at2 − ρ2 2n n
4a t
e e
ρ dtdρ < +∞,
(1 + t)d (1 + ρ)d
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
ϕ(t) ≥ Ctm
∀t ≥ A,
and
ψ(ρ) ≥ Cρm .
∀ρ ≥ A,
(4.6)
(4.7)
Then the inequality (4.5) leads to
Z ∞Z ∞
2
eξηtρ
−at2 − ρ4a
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
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
eεA
eερt
≥
(1 + t)d (1 + ρ)d
(1 + A)2d
and by (4.7), it follows that
Z ∞Z
∞
(4.8)
A
ρ2
2
ecρt−at e− 4a dtdρ < +∞.
A
ρ2
R∞
ecρt− 4a dρ, then the function F can be written as
Z ∞
Z t
2
2
2 s2
ac2 t2
− ρ4a
−A
cAs−ac
e
ds .
F (t) = e
e dρ + 2aγe 4a
Let F (t) =
A
0
A
In particular,
ac2 t2
∞
Z
ρ2
e− 4a dρ.
F (t) ≥ e
A
Thus,
Z
∞
Z
∞
e
A
Z
2
cρt−at2 − ρ4a
∞
dtdρ ≥
A
a(c2 −1)t2
e
A
Z
∞
dt
ρ2
e− 4a dρ = +∞
A
because c > 1. This contradicts the relation (4.8) and shows that f = 0.
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 38, 22 pp.
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U NCERTAINTY P RINCIPLE FOR THE S PHERICAL M EAN O PERATOR
19
• 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
η q ρq
q
ρ2
∞
Z
ψ(ρ)e− 4a e
(1 + ρ)d
(4.9)
0
ρn dρ < +∞.
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
∞
A
ρ2
η q ρq
e− 4a e q
dρ < +∞,
(1 + ρ)d
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 ≥ 41 . 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)
w
1
(1 + |(·, ·)|) p,νn
and
(4.11)
2
eη|θ(·,·)|
F
(f
)
w
2
(1 + |θ(·, ·)|)
< +∞,
q,γ̃n
then
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.
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
(1 + |(·, ·)|)d1 /p 0
(1 + |(·, ·)|)w1 p ,νn
p,νn
10
Z
Z
eξ|(·,·)|2 f ∞
p
dνn (r, x)
=
< +∞.
d1
(1 + |(·, ·)|)w1 0
Rn (1 + |(r, x)|)
p,νn
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 38, 22 pp.
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20
N. M SEHLI
AND
L.T. R ACHDI
and
ZZ
Γ+
2
|F (f )(µ, λ)|eη|θ(µ,λ)|
dγ̃n (µ, λ)
(1 + |θ(µ, λ)|)w2 +d2 /q0
2
eη|θ(·,·)|
≤
F (f )
w
2
(1 + |θ(·, ·)|)
q,γ̃n
1
0
(1 + |θ(·, ·)|)d2 /q .
q 0 ,γ̃n
By the relation (2.7), we obtain
ZZ
2
|F (f )(µ, λ)|eη|θ(µ,λ)|
(4.13)
dγ̃n (µ, λ)
w2 +d2 /q 0
Γ+ (1 + |θ(µ, λ)|)
Z ∞ Z
1
η|θ(·,·)|2
dmn+1 (µ, λ) q0
e
≤
F (f )
< +∞.
d2
(1 + |θ(·, ·)|)w2
0
Rn (1 + |(µ, λ)|)
q,γ̃n
Let d > max w1 + dp10 , w2 + dq20 , n+1
, then from the relations (4.12) and (4.13), we have
2
Z
0
∞
Z
Rn
2
|f (r, x)|eξ|(r,x)|
dνn (r, x) < +∞
(1 + |(r, x)|)d
and
2
|F (f )(µ, λ)|eη|θ(µ,λ)|
dγ̃n (µ, λ) < +∞.
(1 + |θ(µ, λ)|)d
Γ+
Then the desired result follows from Theorem 4.2.
ZZ
Remark 4. The Hardy theorem is a special case of Theorem 4.2, when p = q = +∞.
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