BEURLING-HÖRMANDER UNCERTAINTY PRINCIPLE FOR THE SPHERICAL MEAN OPERATOR

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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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BEURLING-HÖRMANDER UNCERTAINTY PRINCIPLE FOR THE SPHERICAL MEAN

OPERATOR

N. MSEHLI L.T. RACHDI

Département de mathématiques et d’informatique Département de Mathématiques Institut national des sciences appliquées Faculté des Sciences de Tunis

et de technologie de Tunis El Manar II,

1080 Tunis, Tunisia 2092 Tunis, Tunisia

EMail:n.msehli@yahoo.fr EMail:lakhdartannech.rachdi@fst.rnu.tn

Received: 15 November, 2008

Accepted: 01 May, 2009

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.

Abstract: We establish the Beurling-Hörmander theorem for the Fourier transform con- nected 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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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 functionf and its Fourier transformfbcannot 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 iff is an integrable function onRwith respect to the Lebesgue measure and if

Z Z

R²

|f(x)||fˆ(y)|e^|xy|dxdy <+∞, thenf = 0almost everywhere.

A strong multidimensional version of this theorem has been established by Bonami, Demange and Jaming [4] (see also [19]) who have showed that iff is a square integrable function onRⁿwith respect to the Lebesgue measure, then

Z

Rⁿ

Z

Rⁿ

|f(x)||fˆ(y)|

(1 +|x|+|y|)^de^|hx/yi|dxdy <+∞, d≥0;

if and only iff 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−n₂ .

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In particular ford≤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 be- cause 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 onR×Rⁿ, even with respect to the first variable) by

R(f)(r, x) = Z

Sⁿ

f rη, x+rξ

dσ_n(η, ξ), whereSⁿis the unit sphere

(η, ξ)∈R×Rⁿ; η²+|ξ|² = 1 inR×Rⁿandσ_nis the surface measure onSⁿnormalized to have total measure one.

The dual operator^tRofR is defined by

tR(g)(r, x) = Γ ⁿ⁺¹₂ πⁿ⁺¹²

Z

Rⁿ

gp

r²+|x−y|², y

dy,

wheredyis the Lebesgue measure onRⁿ.

The spherical mean operator R and its dual^tR play an important role and have many applications, for example; in the image processing of so-called synthetic aper- ture radar (SAR) data [17, 18], or in the linearized inverse scattering problem in

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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 operatorR the Fourier transformF defined by

F(f)(µ, λ) = Z ∞

0

Z

Rⁿ

f(r, x)ϕ_µ,λ(r, x)dν_n(r, x), where

• ϕ_µ,λ(r, x) = R cos(µ.)e^−ihλ/·i (r, x)

• dν_nis the measure defined on[0,+∞[×Rⁿby dν_n(r, x) = 1

2ⁿ⁻¹² Γ(ⁿ⁺¹₂ )rⁿdr⊗ dx (2π)ⁿ².

They have constructed the harmonic analysis related to the transformF (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 transformF, 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 transformF.

In the third section, we establish the main result of this paper, that is, from the Beurling Hörmander theorem:

• Letfbe a measurable function onR×Rⁿ; even with respect to the first variable and such thatf ∈L²(dν_n). If

Z Z

Γ⁺

Z ∞ 0

Z

Rⁿ

|f(r, x)||F(f)(µ, λ)|e|(r,x)||θ(µ,λ)|

(1 +|(r, x)|+|θ(µ, λ)|)^ddν_n(r, x)d˜γ_n(µ, λ)<+∞; d≥0,

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then

i. Ford≤n+ 1; f = 0;

ii. Ford > n+ 1;there exists a positive constantaand a polynomialP onR×Rⁿ even with respect to the first variable, such that

f(r, x) =P(r, x)e^−a(r²^+|x|²⁾ withdegree(P)< ^d−(n+1)₂ ;

where

• Γ₊is the set given by

Γ₊= [0,+∞[×Rⁿ∪

(iµ, λ); (µ, λ)∈R×Rⁿ; 0≤µ≤ |λ|

• θis the bijective function defined onΓ₊by θ(µ, λ) =p

µ²+|λ|², λ

• d˜γ_nis the measure defined onΓ₊by Z Z

Γ⁺

g(µ, λ)d˜γn(µ, λ) = r2

π 1 (2π)ⁿ²

×

"

Z ∞ 0

Z

Rⁿ

g(µ, λ) µdµdλ pµ²+λ² +

Z

Rⁿ

Z |λ|

0

g(iµ, λ) µdµdλ pλ²−µ²

# .

The last section of this paper is devoted to the Gelfand-Shilov and Cowling Price theorems for the transformF.

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• Letp, q be two conjugate exponents; p, q ∈]1,+∞[. Letη, 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 thatf ∈L²(dν_n).

If

Z ∞ 0

Z

Rⁿ

|f(r, x)|e^ξp|(r,x)|

p p

(1 +|(r, x)|)^d dν_n(r, x)<+∞

and

Z Z

Γ⁺

|F(f)(µ, λ)|e^ξq^|(r,x)|

q q

(1 +|θ(µ, λ)|)^d d˜γ_n(µ, λ)<+∞; d≥0, then

i. Ford≤ ⁿ⁺¹₂ ; f = 0.

ii. Ford > ⁿ⁺¹₂ ; we have – f = 0forξη >1

– f = 0forξη = 1andp6= 2

– f(r, x) = P(r, x)e^−a(r²^+|x|²⁾ forξη = 1andp =q = 2, wherea > 0 andP is a polynomial onR×Rⁿeven with respect to the first variable, withdegree (P)< d−ⁿ⁺¹₂ .

• Letη, ξ, w₁ andw₂be non negative real numbers such thatηξ ≥ ¹₄. Letp, q be two exponents,p, q ∈[1,+∞]and letf be a measurable function onR×Rⁿ, even with respect to the first variable such thatf ∈L²(dν_n).

If

e^ξ|(·,·)|² (1 +|(·,·)|)^w¹

p,νn

<+∞

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and

e^{η|θ(·,·)|}²

(1 +|θ(·,·)|)^w²F(f) q,˜γn

<+∞, then

i. Forξη > ¹₄; f = 0.

ii. Forξη = ¹₄; there exists a positive constantaand a polynomialP onR×Rⁿ, even with respect to the first variable such that

f(r, x) = P(r, x)e^−a(r²^+|x|²⁾.

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2. The Spherical Mean Operator

For all(µ, λ)∈C×Cⁿ; if we denote byϕ_µ,λthe function defined by ϕ_µ,λ(r, x) =R cos(µ.)e^−ihλ/·i

(r, x), then we have

(2.1) ϕ_µ,λ(r, x) =jⁿ⁻¹

2

rp

µ²+λ²

e^−ihλ/xi,

where

• λ² =λ²₁+· · ·+λ²_n; λ= (λ1, . . . , λn)∈Cⁿ;

• hλ/xi=λ₁x₁+· · ·+λ_nx_n; x= (x₁, . . . , x_n)∈Rⁿ;

• jⁿ⁻¹

2 is the modified Bessel function given by jⁿ⁻¹

2 (s) = 2ⁿ⁻¹² Γ

n+ 1 2

Jⁿ⁻¹

2 (s) sⁿ⁻¹² (2.2)

= Γ

n+ 1 2

^∞ X

k=0

(−1)^k k!Γ(k+ ⁿ⁺¹₂ )

s 2

2k

; andJⁿ⁻¹

2 is the usual Bessel function of first kind and order ⁿ⁻¹₂ [9,10,22,36].

Also, the modified Bessel functionjⁿ⁻¹

2 has the following integral representation, for allz ∈C:

jⁿ⁻¹

2 (z) = 2Γ(ⁿ⁺¹₂ )

√πΓ(ⁿ₂) Z 1

0

(1−t²)ⁿ²⁻¹cos(zt)dt.

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Thus, for allz ∈C; we have (2.3)

jⁿ⁻¹

2 (z)

≤e^|Im^z|.

Using the relation (2.1) and the properties of the function jⁿ⁻¹

2 , we deduce that the functionϕ_µ,λsatisfies the following properties [24,27]:

•

(2.4) sup

(r,x)∈R×Rⁿ

ϕ_µ,λ(r, x) = 1 if and only if(µ, λ)belongs to the setΓdefined by (2.5) Γ =R×Rⁿ ∪

(iµ, λ); (µ, λ)∈R×Rⁿ; |µ| ≤ |λ| .

• For all(µ, λ)∈C×Cⁿ; the functionϕµ,λis a unique solution of the system











∂u

∂xj(r, x) = −i λ_j u(r, x); 1≤j ≤n Lu(r, x) = −µ²u(r, x)

u(0,0) = 1; ^∂u_∂r (0, x₁, . . . , x_n) = 0; ∀ (x₁, . . . , x_n)∈Rⁿ where

L = ∂²

∂r² +n r

∂

∂r −

n

X

j=1

∂

∂x_j 2

.

In the following, we denote by

• dm_n+1 the measure defined on[0,+∞[×Rⁿ; by dm_n+1(r, x) =

r2 π

1

(2π)ⁿ² dr⊗dx,

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wheredxis the Lebesgue measure onRⁿ.

• L^p(dm_n+1); p∈[1,+∞],the space of measurable functionsf on[0,+∞[×Rⁿ satisfying

kfk_p,m_n+1 =









 R∞

0

R

Rⁿ|f(r, x)|^pdm_n+1(r, x)¹_p

<+∞, if1≤p < +∞;

ess sup

(r,x)∈[0,+∞[×Rⁿ

|f(r, x)|<+∞, ifp= +∞.

• dνnthe measure defined on[0,+∞[×Rⁿby dν_n(r, x) = rⁿdr

2ⁿ⁻¹² Γ(ⁿ⁺¹₂ )⊗ dx (2π)ⁿ².

• L^p(dνn), p ∈ [1,+∞], the space of measurable functionsf on[0,+∞[×Rⁿ such thatkfkp,νn <+∞.

• Γ₊the subset ofΓ, given by Γ₊ = [0,+∞[×Rⁿ∪

(iµ, λ); (µ, λ)∈R×Rⁿ; 0 ≤µ≤ |λ| .

• BΓ+ theσ−algebra defined onΓ₊by

(2.6) BΓ+ =

θ⁻¹(B); B ∈Bor [0,+∞[×Rⁿ , whereθ is the bijective function defined onΓ₊by

θ(µ, λ) =p

µ²+|λ|², λ .

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• dγ_nthe measure defined onBΓ+ by

∀A∈BΓ+;γ_n(A) = ν_n θ(A) .

• L^p(dγn), p ∈ [1,+∞], the space of measurable functions g on Γ+ such that kgk_p,γ_n <+∞.

• d˜γnthe measure defined onBΓ+ by

d˜γ_n(µ, λ) = 2ⁿ²Γ ⁿ⁺¹₂

√π

dγ_n(µ, λ) (µ²+|λ|²)ⁿ² .

• L^p(d˜γ_n), p ∈ [1,+∞], the space of measurable functions g on Γ₊ such that kgk_p,˜_γ_n <+∞.

• S∗(R×Rⁿ)the Schwarz space formed by the infinitely differentiable functions onR×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 functionsg onΓ₊(respectively integrable on Γ₊with respect to the measuredγ_n), we have

Z Z

Γ+

g(µ, λ)dγ_n(µ, λ)

= 1

2ⁿ⁻¹² Γ(ⁿ⁺¹₂ )(2π)ⁿ²

Z ∞ 0

Z

Rⁿ

g(µ, λ)(µ²+|λ|²)ⁿ⁻¹² µdµdλ +

Z

Rⁿ

Z |λ|

0

g(iµ, λ)(|λ|²−µ²)ⁿ⁻¹² µdµdλ)

! .

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ii. For all non negative measurable functions f on[0,+∞[×Rⁿ (respectively in- tegrable on[0,+∞[×Rⁿwith respect to the measuredm_n+1), the functionf◦θ is measurable onΓ₊(respectively integrable onΓ₊with respect to the measure dγ_n) and we have

Z Z

Γ+

f◦θ(µ, λ)dγ_n(µ, λ) = Z ∞

0

Z

Rⁿ

f(r, x)dν_n(r, x).

iii. For all non negative measurable functions f on[0,+∞[×Rⁿ (respectively in- tegrable on[0,+∞[×Rⁿwith respect to the measuredm_n+1), we have

(2.7)

Z Z

Γ+

f ◦θ(µ, λ)d˜γ_n(µ, λ) = Z ∞

0

Z

Rⁿ

f(r, x)dm_n+1(r, x), whereθis 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 transformF associated with the spherical mean oper- ator is defined onL¹(dν_n)by

∀(µ, λ)∈Γ; F(f)(µ, λ) = Z ∞

0

Z

Rⁿ

f(r, x)ϕ_µ,λ(r, x)dν_n(r, x),

whereϕµ,λis the function given by the relation (2.1) andΓis 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

(2.9) F˜(f)(µ, λ) = Z ∞

0

Z

Rⁿ

f(r, x)jⁿ⁻¹

2 (rµ)e^−ihλ/xidν_n(r, x) andjⁿ⁻¹

2 is the modified Bessel function given by the relation (2.2).

Moreover, by the relation (2.4), the Fourier transform F is a bounded linear operator fromL¹(dν_n)intoL^∞(dγ_n)and for allf ∈L¹(dν_n):

(2.10) kF(f)k∞,γn ≤ kfk_1,ν_n.

Theorem 2.3 (Inversion formula). Let f ∈ L¹(dνn)such that F(f) ∈ L¹(dγn), then for almost every(r, x)∈[0,+∞[×Rⁿ, we have

f(r, x) = Z Z

Γ⁺

F(f)(µ, λ)ϕ_µ,λ(r, x)dγ_n(µ, λ) (2.11)

= Z ∞

0

Z

Rⁿ

F˜(f)(µ, λ)jⁿ⁻¹

2 (rµ)e^ihλ/xidν_n(µ, λ).

Lemma 2.4. LetRⁿ⁻¹

2 be the mapping defined for all non negative measurable func- tionsg on[0,+∞[×Rⁿby

Rⁿ⁻¹

2 (g)(r, x) = 2Γ ⁿ⁺¹₂

√πΓ ⁿ₂r¹⁻ⁿ Z r

0

(r²−t²)ⁿ²⁻¹g(t, x)dt (2.12)

= 2Γ ⁿ⁺¹₂

√πΓ ⁿ₂ Z 1

0

(1−t²)ⁿ²⁻¹g(tr, x)dt,

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then for all non negative measurable functionsf, g on[0,+∞[×Rⁿ, we have Z ∞

0

Z

Rⁿ

Rⁿ⁻¹

2 (g)(r, x)f(r, x)dν_n(r, x) (2.13)

= Z ∞

0

Z

Rⁿ

g(t, x)Wⁿ⁻¹

2 (f)(t, x)dm_n+1(t, x) whereWⁿ⁻¹

2 is the classical Weyl transform defined for all non negative measurable functionsg on[0,+∞[×Rⁿby

(2.14) Wⁿ⁻¹

2 (f)(t, x) = 1 2ⁿ²Γ(ⁿ₂)

Z ∞ t

(r²−t²)ⁿ²⁻¹f(r, x)2rdr.

Proposition 2.5. For all f ∈ L¹(dν_n), the function Wⁿ⁻¹

2 (f)given by the relation (2.14) is defined almost every where, belongs to the spaceL¹(dmn+1)and we have (2.15)

Wⁿ⁻¹

2 (f) 1,mn+1

≤ kfk_1,ν_n.

Moreover,

(2.16) F˜(f)(µ, λ) = Λ_n+1◦Wⁿ⁻¹

2 (f)(µ, λ),

whereΛ_n+1 is the usual Fourier cosine transform defined onL¹(dm_n+1)by Λn+1(g)(µ, λ) =

Z ∞ 0

Z

Rⁿ

g(r, x) cos(rµ)e^−ihλ,xidmn+1(r, x).

andF˜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 Λ_n+1 are topological isomorphisms fromS∗(R×Rⁿ)onto itself. Then, by the relation (2.16),

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we deduce that the classical Weyl transformWⁿ⁻¹

2 is also a topological isomorphism fromS∗(R×Rⁿ)onto itself, and the inverse isomorphism is given by [24]

(2.17) Wⁿ⁻¹⁻¹

2

(f)(r, x) = (−1)^[ⁿ²^]+1F_[ⁿ

2]−ⁿ

2+1

∂

∂t² [ⁿ₂]+1

f

! (r, x), whereF_a; a >0is the mapping defined onS∗(R×Rⁿ)by

(2.18) F_a(f)(r, x) = 1 2^aΓ(a)

Z ∞ r

(t²−r²)^a−1f(t, x)2tdt and _∂r^∂2 is the singular partial differential operator defined by

∂

∂r²

f(r, x) = 1 r

∂f(r, x)

∂r .

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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 transformF associated with the spherical mean operator.

We firstly recall the following result that has been established by Bonami, De- mange and Jaming [4].

Theorem 3.1. Letf be a measurable function onR×Rⁿ, even with respect to the first variable such thatf ∈L²(dm_n+1)and letdbe a real number,d≥0. If

Z ∞ 0

Z

Rⁿ

Z ∞ 0

Z

Rⁿ

|f(r, x)||Λ_n+1(f)(s, y)|

(1 +|(r, x)|+|(s, y)|)^de|(r,x)||(s,y)|

dm_n+1(r, x)dm_n+1(s, y)<+∞, then there exist a positive constantaand a polynomialP onR×Rⁿeven with respect to the first variable, such that

f(r, x) = P(r, x)e^−a(r²^+|x|²⁾, withdegree(P)< ^d−(n+1)₂ .

In particular,f = 0ford≤(n+ 1).

Lemma 3.2. Letf ∈L²(dν_n)and letdbe a real number,d ≥0. If Z Z

Γ⁺

Z ∞ 0

Z

Rⁿ

|f(r, x)||F(f)(µ, λ)|e|(r,x)||θ(µ,λ)|

1 +|(r, x)|+|θ(µ, λ)|d dνn(r, x)d˜γn(µ, λ)<+∞, then the functionf belongs to the spaceL¹(dν_n).

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Proof. Letf ∈L²(dν_n), f 6= 0. From the relations (2.7) and (2.8), we obtain Z Z

Γ⁺

Z ∞ 0

Z

Rⁿ

|f(r, x)||F(f)(µ, λ)|

1 +|(r, x)|+|θ(µ, λ)|de|(r,x)||θ(µ,λ)|

dν_n(r, x)d˜γ_n(µ, λ)

= Z ∞

0

Z

Rⁿ

Z ∞ 0

Z

Rⁿ

|f(r, x)||F˜(f)(µ, λ)|

(1 +|(r, x)|+|(µ, λ)|)^de|(r,x)||(µ,λ)|

dν_n(r, x)dm_n+1(µ, λ)<+∞.

Then for almost every(µ, λ)∈[0,+∞[×Rⁿ,

F˜(f)(µ, λ)

Z ∞ 0

Z

Rⁿ

|f(r, x)|e|(r,x)||(µ,λ)|

(1 +|(r, x)|+|(µ, λ)|)^ddνn(r, x)<+∞.

In particular, there exists(µ₀, λ₀)∈[0,+∞[×Rⁿ\ {(0,0)}such that F˜(f)(µ₀, λ₀)6= 0and

Z ∞ 0

Z

Rⁿ

|f(r, x)|e^|(r,x)||(µ⁰^,λ⁰^)|

(1 +|(r, x)|+|(µ₀, λ₀)|)^ddν_n(r, x) < +∞.

Lethbe the function defined on[0,+∞[by h(s) = e^s|(µ⁰^,λ⁰^)|

(1 +s+|(µ₀, λ₀)|)^d, then the functionhhas an absolute minimum attained at:

s₀ =

( _d

|(µ₀,λ0)|−1− |(µ0, λ0)|; if _|(µ^d

0,λ0)| >1 +|(µ0, λ0)|;

0; if _|(µ^d

0,λ0)| ≤ 1 +|(µ₀, λ₀)|.

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Consequently, Z ∞

0

Z

Rⁿ

|f(r, x)|dν_n(r, x)

≤ 1 h(s₀)

Z ∞ 0

Z

Rⁿ

|f(r, x)|e^|(r,x)||(µ⁰^,λ⁰^)|

(1 +|(r, x)|+|(µ₀, λ₀)|)^ddν_n(r, x)<+∞.

Lemma 3.3. Letf ∈L²(dν_n)and letdbe a real number,d ≥0. If Z Z

Γ+

Z ∞ 0

Z

Rⁿ

|f(r, x)||F(f)(µ, λ)|e|(r,x)||θ(µ,λ)|

(1 +|(r, x)|+|θ(µ, λ)|)^d dνn(r, x)d˜γn(µ, λ)<+∞, then there existsa >0such that the functionF˜(f)is analytic on the set

(µ, λ)∈C×Cⁿ; |Imµ|< a, |Imλj|< a; ∀j ∈ {1, . . . , n} .

Proof. From the proof of Lemma3.2, there exists(µ₀, λ₀)∈[0,+∞[×Rⁿ\ {(0,0)}

such that

Z ∞ 0

Z

Rⁿ

|f(r, x)|e^|(r,x)||(µ⁰^,λ⁰^)|

(1 +|(r, x)|+|(µ₀, λ₀)|)^ddν_n(r, x)<+∞.

Letabe a real number such that0<(n+ 1)a <|(µ₀, λ₀)|. Then we have Z ∞

0

Z

Rⁿ

|f(r, x)|e^|(r,x)||(µ⁰^,λ⁰^)|

(1 +|(r, x)|+|(µ₀, λ₀)|)^ddν_n(r, x)

= Z ∞

0

Z

Rⁿ

|f(r, x)|e(n+1)a|(r,x)| e|(r,x)|(|(µ0,λ0)|−(n+1)a)

(1 +|(r, x)|+|(µ₀, λ₀)|)^ddν_n(r, x)<+∞.

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Letg be the function defined on[0,+∞[by

g(s) = e^s(|(µ⁰^,λ⁰^)|−(n+1)a) (1 +s+|(µ₀, λ₀)|)^d, theng admits a minimum attained at

s0 =







d

|(µ₀,λ0)|−(n+1)a −1− |(µ₀, λ₀)|; if _|(µ ^d

0,λ0)|−(n+1)a >1 +|(µ₀, λ₀)|,

0; if _|(µ ^d

0,λ0)|−(n+1)a ≤ 1 +|(µ₀, λ₀)|.

Consequently, (3.1)

Z ∞ 0

Z

Rⁿ

|f(r, x)|e(n+1)a|(r,x)|

dν_n(r, x)

≤ 1 g(s₀)

Z ∞ 0

Z

Rⁿ

|f(r, x)|e^|(r,x)||(µ⁰^,λ⁰^)|

(1 +|(µ₀, λ₀)|+|(r, x)|)^ddνn(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−→jⁿ⁻¹

2 (rµ)e^−ihλ/xi

is analytic onC×Cⁿ[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,

jⁿ⁻¹

2 (rµ)e^−ihλ,xi

≤e^r|Im^µ|+^Pⁿ^j=1^|^Im^λ^j^||x^j^| (3.2)

≤e^|(r,x)|[^|Im^µ|+^Pⁿj=1|Imλj|].

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. Letf ∈L²(dν_n), f 6= 0and letdbe a real number,d ≥0. If Z Z

Γ+

Z ∞ 0

Z

Rⁿ

|f(r, x)||F(f)(µ, λ)|

(1 +|(r, x)|+|θ(µ, λ)|)^de|(r,x)||θ(µ,λ)|dν_n(r, x)d˜γ_n(µ, λ)<+∞, then for all real numbersa, a >0, we havem_n+1(A_a)>0, where

(3.3) A_a=n

(µ, λ)∈R×Rⁿ; F˜(f)(µ, λ)6= 0and|(µ, λ)|> ao .

Proof. Letf be a function satisfying the hypothesis. From Lemma3.2, the function f belongs toL¹(dν_n)and consequently the functionF˜(f)is continuous onR×Rⁿ, even with respect to the first variable. Then for all a > 0, the setA_a given by the relation (3.3) is an open subset ofR×Rⁿ.

So, ifm_n+1(A_a) = 0, then this subset is empty. This means that for every(µ, λ)∈ R×Rⁿ, |(µ, λ)|> a, we haveF˜(f)(µ, λ) = 0.

From Lemma3.2, and by analytic continuation, we deduce thatF˜(f) = 0,and by the inversion formula (2.11), it follows thatf = 0.

Remark 3.

i. Letf be a function satisfying the hypothesis of Corollary3.4, then for all real numbersa, a >0, there exists(µ₀, λ₀)∈[0,+∞[×Rⁿsuch that|(µ₀, λ₀)|> a and

Z ∞ 0

Z

Rⁿ

|f(r, x)| e^|(r,x)||(µ⁰^,λ⁰^)|

(1 +|(r, x)|+|(µ₀, λ₀)|)^ddν_n(r, x)<+∞.

ii. Letdandσbe non negative real numbers,σ+σ² ≥d.Then the function t7−→ e^σt

(1 +t+σ)^d

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is not decreasing on[0,+∞[.

Lemma 3.5. Letf be a measurable function onR×Rⁿeven with respect to the first variable, andf ∈L²(dν_n). Letdbe real number,d≥0. If

Z Z

Γ⁺

Z ∞ 0

Z

Rⁿ

|F(f)(µ, λ)||f(r, x)| e|(r,x)||θ(µ,λ)|

(1 +|(r, x)|+|θ(µ, λ)|)^ddν_n(r, x)d˜γ_n(µ, λ)<+∞, then the functionWⁿ⁻¹

2 (f)defined by the relation (2.14) belongs to the spaceL²(dm_n+1).

Proof. From the hypothesis and the relations (2.7) and (2.8), we have Z Z

Γ⁺

Z ∞ 0

Z

Rⁿ

|F(f)(µ, λ)| |f(r, x)|e|(r,x)||θ(µ,λ)|

(1 +|(r, x)|+|θ(µ, λ)|)^ddν_n(r, x)d˜γ_n(µ, λ)

= Z ∞

0

Z

Rⁿ

Z ∞ 0

Z

Rⁿ

F˜(f)(µ, λ)

|f(r, x)|e|(r,x)||(µ,λ)|

(1 +|(r, x)|+|(µ, λ)|)^ddν_n(r, x)dm_n+1(µ, λ)

<+∞.

In the same manner as the proof of the inequality (3.1) in Lemma3.2, there exists b∈R, b > 0such that

Z ∞ 0

Z

Rⁿ

|F˜(f)(µ, λ)|e^b|(µ,λ)|dm_n+1(µ, λ)<+∞.

Consequently, the functionF˜(f)belongs to the spaceL¹(dν_n)and by the inversion formula forF˜, we deduce that

f(r, x) = Z ∞

0

Z

Rⁿ

F˜(f)(µ, λ)jⁿ⁻¹

2 (rµ)e^ihλ/xidν_n(µ, λ).a.e.

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In particular, the functionf is bounded and

(3.4) kfk∞,ν_n ≤

F˜(f) 1,νn

.

By virtue of the relation (2.14), we get

Wⁿ⁻¹

2 (f)(r, x)

≤ 1 2ⁿ²Γ(ⁿ₂)

Z ∞ t

(r² −t²)ⁿ²⁻¹|f(r, x)|2rdr

= rⁿ 2ⁿ²Γ(ⁿ₂)

Z ∞ 1

(y²−1)ⁿ²⁻¹|f(ry, x)|2ydy.

Using Minkowski’s inequality for integrals [12], we get:

Z ∞ 0

Z

Rⁿ

Wⁿ⁻¹

2 (f)(r, x)

2

dm_n+1(r, x) ¹₂ (3.5)

≤ 1

2ⁿ²Γ(ⁿ₂)

"

Z ∞ 0

Z

Rⁿ

Z ∞ 1

rⁿ(y² −1)ⁿ²⁻¹|f(ry, x)|2ydy 2

dm_n+1(r, x)

#¹₂

≤ 1

2ⁿ²Γ(ⁿ₂) Z ∞

1

Z ∞ 0

Z

Rⁿ

r²ⁿ(y²−1)ⁿ⁻²|f(ry, x)|²dm_n+1(r, x) ¹₂

2ydy

= 1

2ⁿ²⁻¹Γ(ⁿ₂) Z ∞

1

(y²−1)ⁿ²⁻¹y⁻ⁿ⁺¹²dy

× Z ∞

0

Z

Rⁿ

s²ⁿ|f(s, x)|²dm_n+1(s, x) ¹₂

= Γ(¹₄) 2ⁿ²Γ(²ⁿ⁺¹₄ )

Z ∞ 0

Z

Rⁿ

s²ⁿ|f(s, x)|²dm_n+1(s, x) ¹₂

.

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Using the relations (3.1), (3.4) and (3.5), we deduce that

Wⁿ⁻¹

2 (f) 2,mn+1

= Z ∞

0

Z

Rⁿ

Wⁿ⁻¹

2 (f)

2

(r, x)dmn+1(r, x) ¹₂

≤K_n Z ∞

0

Z

Rⁿ

|f(s, x)|e(n+1)a|(s,x)|dν_n(s, x)<+∞, where

Kn= Γ ¹₄ 2ⁿ²Γ ²ⁿ⁺¹₄

rπ 2Γ

n+ 1 2

2ⁿ⁻¹² max

s≥0 (sⁿe^−(n+1)as)kfk∞,νn

¹2

.

Theorem 3.6. Letf ∈L²(dνn); f 6= 0and letdbe a real number;d≥0.

If Z Z

Γ⁺

Z ∞ 0

Z

Rⁿ

|f(r, x)||F(f)(µ, λ)|e|(r,x)||θ(µ,λ)|

(1 +|(r, x)|+|θ(µ, λ)|)^d dν_n(r, x)d˜γ_n(µ, λ)<+∞;

then Z ∞

0

Z

Rⁿ

Z ∞ 0

Z

Rⁿ

Wⁿ⁻¹

2 (f)(r, x)

F˜(f)(µ, λ)

× e|(r,x)||(µ,λ)|

(1 +|(r, x)|+|(µ, λ)|)^ddm_n+1(r, x)dm_n+1(µ, λ)<+∞

whereWⁿ⁻¹

2 is the Weyl transform defined by the relation (2.14).

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Proof. From the hypothesis, the relations (2.7), (2.8) and Fubini’s theorem, we have Z Z

Γ⁺

Z ∞ 0

Z

Rⁿ

|f(r, x)||F(f)(µ, λ)|

(3.6)

× e|(r,x)||θ(µ,λ)|

(1 +|(r, x)|+|θ(µ, λ)|)^ddν_n(r, x)d˜γ_n(µ, λ)

= Z ∞

0

Z

Rⁿ

F˜(f)(µ, λ)

× Z ∞

0

Z

Rⁿ

|f(r, x)|e|(r,x)||(µ,λ)|

(1 +|(r, x)|+|(µ, λ)|)^ddν_n(r, x)dm_n+1(µ, λ)

<+∞.

i. Ifd= 0, then by the relation (2.13) and Fubini’s theorem, we get Z ∞

0

Z

Rⁿ

Z ∞ 0

Z

Rⁿ

Wⁿ⁻¹

2 (f)(r, x)

F˜(f)(µ, λ) (3.7)

×e|(r,x)||(µ,λ)|

dmn+1(r, x)dmn+1(µ, λ)

≤ Z ∞

0

Z

Rⁿ

F˜(f)(µ, λ)

× Z ∞

0

Z

Rⁿ

Wⁿ⁻¹

2 (|f|)(r, x)e|(r,x)||(µ,λ)|

dm_n+1(r, x)

dm_n+1(µ, λ)

≤ Z ∞

0

Z

Rⁿ

F˜(f)(µ, λ)

× Z ∞

0

Z

Rⁿ

|f(r, x)|Rⁿ⁻¹

2 (e|(·,·)||(µ,λ)|

)(r, x)dν_n(r, x)

dm_n+1(µ, λ).

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However, by (2.12), we deduce that for all(r, x)∈[0,+∞[×Rⁿ,

(3.8) Rⁿ⁻¹

2 e|(·,·)||(µ,λ)|

(r, x)≤e|(r,x)||(µ,λ)|

. Combining the relations (3.6), (3.7) and (3.8), we deduce that

Z ∞ 0

Z

Rⁿ

Z ∞ 0

Z

Rⁿ

Wⁿ⁻¹

2 (f)(r, x)

F˜(f)(µ, λ)

e|(r,x)||(µ,λ)|

dm_n+1(r, x)dm_n+1(µ, λ)

≤ Z ∞

0

Z

Rⁿ

F˜(f)(µ, λ)

Z ∞ 0

Z

Rⁿ

|f(r, x)|e|(r,x)||(µ,λ)|

dν_n(r, x)dm_n+1(µ, λ)<+∞.

ii. Ford >0, letB_d={(r, x)∈[0,+∞[×Rⁿ; |(r, x)| ≤d}. We have Z ∞

0

Z

Rⁿ

F˜(f)(µ, λ)

Z ∞ 0

Z

Rⁿ

|Wⁿ⁻¹

2 (f)(r, x)|e|(r,x)||(µ,λ)|

(1 +|(r, x)|+|(µ, λ)|)^d dm_n+1(r, x)dm_n+1(µ, λ)

≤ Z Z

B_d^c

F˜(f)(µ, λ)

Z ∞ 0

Z

Rⁿ

Wⁿ⁻¹

2 (|f|)(r, x)e|(r,x)||(µ,λ)|

(1 +|(r, x)|+|(µ, λ)|)^d dm_n+1(r, x)

!

dm_n+1(µ, λ) +

Z Z

Bd

F˜(f)(µ, λ)

Z ∞ 0

Z

Rⁿ

Wⁿ⁻¹

2 (|f|)(r, x)e|(r,x)||(µ,λ)|

(1 +|(r, x)|+|(µ, λ)|)^d dm_n+1(r, x)

!

dm_n+1(µ, λ).

From the relation (2.13), we deduce that (3.9)

Z Z

B^c_d

F˜(f)(µ, λ)

× Z ∞

0

Z

Rⁿ

Wⁿ⁻¹

2 (|f|)(r, x)e|(r,x)||(µ,λ)|

(1 +|(r, x)|+|(µ, λ)|)^d dm_n+1(r, x)

!

dm_n+1(µ, λ)

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= Z Z

B_d^c

F˜(f)(µ, λ)

Z ∞ 0

Z

Rⁿ

|f(r, x)|

×Rⁿ⁻¹

2

e|(·,·)||(µ,λ)|

(1 +|(·,·)|+|(µ, λ)|)^d

(r, x)dν_n(r, x)dm_n+1(µ, λ).

However, from the relation (2.12) and ii) of Remark3, we deduce that for all(µ, λ)∈ B_d^c, we have

(3.10) Rⁿ⁻¹

2

e|(·,·)||(µ,λ)|

(1 +|(·,·)|+|(µ, λ)|)^d

(r, x) ≤ e|(r,x)||(µ,λ)|

(1 +|(r, x)|+|(µ, λ)|)^d. Combining the relations (3.6), (3.9) and (3.10), we get

Z Z

B_d^c

F˜(f)(µ, λ)

Z ∞ 0

Z

Rⁿ

Wⁿ⁻¹

2 (|f|)(r, x)e|(r,x)||(µ,λ)|

(1 +|(r, x)|+|(µ, λ)|)^d dmn+1(r, x)

!

dmn+1(µ, λ)

≤ Z Z

B_d^c

F˜(f)(µ, λ)

Z ∞ 0

Z

Rⁿ

|f(r, x)|e|(r,x)||(µ,λ)|

(1 +|(r, x)|+|(µ, λ)|)^ddνn(r, x)

dmn+1(µ, λ)

≤ Z ∞

0

Z

Rⁿ

F˜(f)(µ, λ)

Z ∞ 0

Z

Rⁿ

|f(r, x)|e|(r,x)||(µ,λ)|

(1 +|(r, x)|+|(µ, λ)|)^ddνn(r, x)

dmn+1(µ, λ)

<+∞.

We have Z Z

Bd

F˜(f)(µ, λ)

Z Z

Bd

|Wⁿ⁻¹

2 (f)(r, x)|e|(r,x)||(µ,λ)|

(1 +|(r, x)|+|(µ, λ)|)^d dmn+1(r, x)dmn+1(µ, λ)

≤e^d² Z Z

Bd

F˜(f)(µ, λ)

dm_n+1(µ, λ)

Z Z

Bd

Wⁿ⁻¹

2 (f)(r, x)

dm_n+1(r, x)